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--- abstract: 'We introduce a new method for category-level pose estimation which produces a distribution over predicted poses by integrating 3D shape estimates from a generative object model with segmentation information. Given an input depth-image of an object, our variable-time method uses a mixture density network architecture to produce a multi-modal distribution over 3DOF poses; this distribution is then combined with a prior probability encouraging silhouette agreement between the observed input and predicted object pose. Our approach significantly outperforms the current state-of-the-art in category-level 3DOF pose estimation—which outputs a point estimate and does not explicitly incorporate shape and segmentation information—as measured on the Pix3D and ShapeNet datasets.' author: - | Benjamin Burchfiel\ Department of Computer Science\ Duke University\ Durham, NC 27708\ `bcburch@cs.duke.edu`\ George Konidaris\ Department of Computer Science\ Brown University\ Providence, RI 02912\ `gdk@cs.brown.edu`\ bibliography: - 'main.bib' title: 'Probabilistic Category-Level Pose Estimation via Segmentation and Predicted-Shape Priors' --- Introduction ============ A critical task in many robotics and computer vision applications is determining an observed object’s 3D pose. While some approaches attempt to estimate only a subset of 3D pose parameters [@sun2018pix3d; @Su_2015_ICCV], general solutions must produce a full estimate of the object’s 3D orientation. Broadly speaking, full 3D pose estimation falls into two categories: instance-level and category-level. While the instance-level case assumes availability of a full 3D ground-truth model of the object, category-level pose estimation must deal with a novel object for which no ground-truth model is available. In the category-level case, it is common to define a canonical pose—a default pose that is well defined with respect to a particular class of objects. For instance, the canonical pose of a car might be with the front bumper directly facing the camera and all wheels on the ground plane. Category-level pose estimation approaches generally learn to transform an object into these canonical poses directly from training data (often implicitly). Recent work in category-level pose estimation includes both discrete pose estimators [@sun2018pix3d; @Su_2015_ICCV; @NIPS2012_4562], which treat pose estimation as a classification task by predicting a single pose bucket (generally with width around 15 degrees), and continuous methods which directly output pose parameters [@beo2017; @hbeo2018]. Notably, while some of these approaches estimate both object pose and 3D object shape from a single input depth [@hbeo2018] or RGB image [@sun2018pix3d], they fail to explicitly verify that their estimates of shape and pose are consistent with the observed depth or RGB input. We propose a novel pose estimation approach that leverages a generative model of 3D shape to verify that predicted pose matches the segmented input. Our method predicts a distribution over possible object poses from a single segmented depth-image, and constructs a pose consistency prior based on agreement between the predicted object silhouette and input image. We modify an existing generative object model, HBEOs [@hbeo2018], to produce a multimodal distribution over poses instead of a point-estimate, and develop an efficient 2D pose-consistency score by projecting shape-pose estimates back into the input image. We use this consistency score as a prior probability over poses and provide a sampling-based approximation to the *maximum a posteriori* estimate of 3D pose. We evaluate our approach empirically on several thousand 3D objects across three classes from the ShapeNet dataset [@shapenet2015] as well as roughly three thousand chairs from the Pix3D dataset [@sun2018pix3d]. Our results show a dramatic decrease in gross pose error ($\epsilon > 15^\circ$) compared to previous state-of-the-art and a significant performance contribution from both the density prediction and input-verification components. Furthermore, by dynamically varying the number of samples used to produce a pose-estimate, our approach becomes an any-time method; while it significantly outperforms existing methods with only a few samples, it produces increasingly high-quality pose estimates when given a longer budget. Background ========== Single-instance pose estimation has been widely studied and can be largely broken down into optimization-based and learning-based approaches. Optimization approaches formulate pose estimation as the alignment of an object’s known model to observed input; these methods, which include work such as Iterative Closest Points [@ICP] and Bingham-Distribution filters [@Glover-RSS-11], attempt to minimize error between object models and observations. More recently, learning-based approaches—which attempt to directly predict object pose—have begun to outperform optimization-based methods due to their robustness to local minima and ability to learn highly discriminative features in input space [@tulsiani2015viewpoints; @wohlhart2015learning; @deep_im; @pose_cnn]. There has been less work on category-level pose estimation—predicting the pose of novel objects [@sun2018pix3d; @Su_2015_ICCV; @NIPS2012_4562; @beo2017; @hbeo2018] given a canonical notion of pose for each object category. While RGB approaches such as @sun2018pix3d still tend to be limited to predicting pose in course bins (of roughly 15 degrees of width), a method called HBEOs [@hbeo2018] directly predicts an object’s continuous 3D orientation from a single input depth-image. The pose-estimation portion of HBEOs is still not reliable enough for robust robot interaction with objects, however, routinely producing pose estimates with errors of over fifteen degrees. Hybrid Bayesian Eigenobjects ---------------------------- HBEOs are a generative representation of 3D objects; they explicitly model a low-dimensional object-space and are capable of representing novel and partially observed objects from previously encountered categories. HBEOs take as input a segmented depth-image of a novel observed object and estimates the object’s 3D shape—in voxel form—along with its class and 3DOF orientation [@hbeo2018]. Similarly to other methods that complete 3D objects, HBEOs are trained from a set of aligned 3D meshes. Each mesh is voxelized and each training voxel object of size $m\times n\times o$ is reshaped to produce a size $d=m\cdot n\cdot o$ vector. A Bayesian variant of Principal Component Analysis [@pca], VBPCA [@vbpca], is used to learn a loose low-dimensional subspace that captures the shape variation in each class of training objects. We denote the learned basis-matrix and mean-vector for class $i$ as $\mathbf{W_i}$ and $\boldsymbol{\mu_i}$, respectively; from these class-specific subspaces, a single shared subspace is created by finding a basis spanning each subspace: $$\mathbf{W} = [\mathbf{W_1},...,\mathbf{W_m}, \boldsymbol{\mu_1},...,\boldsymbol{\mu_m}]. \label{eq:bigSubspace}$$ To ensure this basis is maximally compact, singular value decomposition is used to find a minimal orthonormal representation of $\mathbf{W}$. The projection of a voxel object in vector form, $\mathbf{o}$, into this subspace is $$\mathbf{o}'=\mathbf{W}^T\mathbf{o} \label{eq:project_complete}$$ and any point in this subspace can be mapped back into voxel-space via $$\mathbf{\hat{o}}= \mathbf{W}\mathbf{o}'. \label{eq:reconst}$$ HBEOs then train a CNN to directly estimate an object’s projection onto this space, $\mathbf{o}'$, given a single depth-view of that object. Because $\mathbf{W}$ has been learned during training, at inference time the HBEO prediction $\hat{\mathbf{o}}'$ is sufficient to estimate full 3D geometry using equation \[eq:reconst\]. HBEOs also simultaneously predict a probability distribution over object classes and a point-estimate of pose in the form of three axis-angle pose parameters. Mixture Density Networks ------------------------ Mixture Density Networks (MDNs) were first proposed by @mdns as a method for predicting distributions, rather than point-estimates, when using neural networks for regression. MDNs explicitly estimate a distribution that explains the training data; let $x$ be inputs to the model and $y$ be the desired regression target. A conventional regression network predicts $\hat{y}$ directly from $x$ while an MDN predicts a conditional density function $P(y | x)$. MDNs model $P(y | x)$ as a mixture of parameterized distributions; a common choice being the multivariate Gaussian mixture model (MV-GMM) with probability density function: $$p({\boldsymbol {y |\theta}})=\sum _{i=1}^{c}{\pi_{i}}{\mathcal {N}}(y | {\boldsymbol {{\mu _{i}},{\Sigma _{i}}}}), \label{eq:pose_likelihood}$$ where $\theta_i=\{\boldsymbol {\pi_i,{\mu _{i}},{\Sigma _{i}}}\}$ is the mixing coefficient, mean, and covariance of component $i$. The network predicts these mixture parameters by learning the function $\theta = f(x)$ while a conventional regression network learns $y=f(x)$. As a result, if $y\in\mathbb{R}^n$, a network that directly predicts $y$ would have size $n$ output while an MV-GMM-MDN would produce output of size $c(n^2+n+1)$, where $c$ is the number of mixture components. To reduce the output size of MDNs, it is common to assume a diagonal covariance for each component, in which case the output size becomes $c(2n+1)$. During training, each gradient update seeks to minimize the negative log-likelihood of observed data: $$loss(y , \theta) = -ln{\sum _{i=1}^{c}{\pi_{i}}{\mathcal {N}}(y | {\boldsymbol {{\mu _{i}},{\Sigma _{i}}}})}. \label{eq:mdn_loss}$$ MDN networks have some significant advantages over direct regression including non-unimodality and measurable uncertainty. While direct regression learns to predict the conditional mean of the data, MDNs can learn to model the shape of the underlying distribution. In contexts where the target is ill-posed (i.e. there are multiple valid mappings from $x$ to $y$), the mean over good solutions may not actually be a reasonable solution itself. [r]{}[0.4]{} ![image](Figures/density_example.pdf){width="40.00000%"} \[fig:multimode\_example\] Figure \[fig:multimode\_example\] illustrates this in a simple univariate scenario; the function learned by direct regression with a least-squares loss is unable to accurately represent the underlying data. MDNs also allow multiple solutions to be sampled and provide a measure of confidence for each sample via the predicted conditional PDF. This explicit likelihood estimate provides a natural avenue for incorporating prior information into estimates in a straightforward and principled way. Predicting Pose Posteriors ========================== We improve pose estimation in two primary ways: 1) predicting multi-modal pose distributions instead of point-estimates and 2) incorporating a prior probability dependent on agreement between predicted pose, shape, and the input depth-image. We construct a distribution-predicting network using the MDN formulation and show how approximate maximum likelihood and *maximum a posteriori* estimates may be obtained via sampling from its output. Using Predicted-Shape Priors for Predicting Pose Distributions -------------------------------------------------------------- In order to incorporate pose priors, we require a probabilistic estimate of pose distribution conditioned on input observation. To obtain this, we modify the HBEO architecture [@hbeo2018] to estimate a pose distribution by converting the final layer of the network to a multivariate Gaussian MDN layer. Let $z_i$ be the sum of input signals to output neuron $i$ and assume predicted components have diagonal covariance. Each covariance value is estimated using a translated exponential-linear unit (ELU) at the final layer: $$\sigma_{ii} = \left\{\begin{array}{lr} z_i, & \text{if} x > 0\\ \alpha~exp(z_i) + \epsilon, & \text{if } x \leq 0\\ \end{array}\right\}.$$ This translated ELU ensures that elements along the diagonal of the predicted covariance matrix will be strictly positive and that the resulting matrix will be positive semidefinite. Component mean parameters are estimated straightforwardly with a linear layer and component mixing coefficients, $\pi\in\Pi$, are estimated with a softmax layer, ensuring $\Pi$ forms a valid discrete probability distribution: $$\pi_i=\frac{e^{z_i}}{\Sigma_{j=1}^ce^{z_j}}.$$ We fixed the number of components to $c=5$ and used equation \[eq:mdn\_loss\] to calculate the pose portion of our loss function: $$loss_{pose}(y, \theta) = -ln{\sum _{i=1}^{5}{\pi_{i}}}\frac {\exp \left(-{\frac {1}{2}}({\mathbf {y} }-{\boldsymbol {\mu }})^{\mathrm {T} }{\boldsymbol {\Sigma }}^{-1}({\mathbf {y} }-{\boldsymbol {\mu }})\right)}{\sqrt {(2\pi )^{k}|{\boldsymbol {\Sigma }}|}}, \label{eq:pose_loss}$$ where $\theta$ are the distribution parameters predicted by the network for input image $x$ and $y$ is the true pose of the object depicted in $x$. We maintain the same structure used in @hbeo2018 for the lower layers of the network and for the class and shape output layers—softmax and linear layers, respectively. The full loss for the network is $$\mathcal{L}(y) = \lambda_{p}loss_{pose}(y) + \lambda_{s}loss_{shape}(y) + \lambda_{c}loss_{class}(y)$$ where $loss_{shape}(y)$ is a Euclidean loss over subspace projection coefficients, $loss_{class}(y)$ is the multiclass categorical cross-entropy loss over possible classes, and $\lambda_{p}$, $\lambda_{s}$, $\lambda_{c}$ are weighting coefficients over the pose, shape, and class terms.[^1] Beyond allowing multiple possible poses to be sampled, HBEO-MDNs are more robust to training data pose noise and object symmetry than HBEOs because they can explicitly model multiple pose modalities. Furthermore, MDNs naturally compensate for the representational discontinuity present in axis-angle formulations of pose. As an example, consider predicting only the z-axis rotation component of an object’s pose. If the true pose z-component is $\pi$, and target poses are are in the range of $(-\pi, \pi]$, then the HBEO network would receive a small loss for predicting $p_z=\pi-\epsilon$ and a large loss for predicting $p_z=\pi+\epsilon$, despite the fact that both predictions are close to the true pose. While other loss functions or pose representations may alleviate this particular issue, they do so at the expense of introducing problems such as double coverage, causing the network’s prediction target to no longer be well defined. By comparison, the HBEO-MDN approach suffers from none of these issues and can model object symmetry and discontinuities in prediction space explicitly by predicting multi-modal distributions over pose. Pose Priors from Shape and Segmentation --------------------------------------- While generative models that predict an object’s 3D shape exist, those that also estimate object pose do not explicitly verify that these predictions are consistent with observed depth input. While the shape estimate produced by such models is noisy, there is still valuable information to be obtained from such a verification. Let $D$ be a segmented depth-image input to such a model, $\mathbf{\hat{o}}$ be the predicted shape of the object present in $D$, and $\mathbf{\hat{R}(\mathbf{\hat{o}})}$ be the estimated 3DOF rotation transforming $\mathbf{\hat{o}}$ from canonical pose to the pose depicted in $D$. Assuming known depth camera intrinsic parameters, we can project the estimated shape and pose of the object back into a 2D depth-image via $\hat{D_R} = f(\mathbf{\hat{R}(\mathbf{\hat{o}})})$ where the projection function $f(x)$ simulates a depth camera. Intuitively, if the shape and pose of the observed object are correctly estimated, and the segmentation and camera intrinsics are accurate, then $\Delta_D = ||D -\hat{D_R}||=0$ while errors in these estimates will result in a discrepancy between the predicted and observed depth-images. As prior work has shown pose to be the least reliable part of the pipeline [@beo2017; @hbeo2018; @sun2018pix3d], we assume that the error in $\hat{R}$ will generally dominate error in the other portions of the process and thus employ $\Delta_D$ to refine $\hat{R}$. Let $T=SDF(D)$ be the 2D *signed distance field* [@TSDF] calculated from $D$, we define an image-space error score between segmented depth-images as $$e_R=||SDF(D) - SDF(\hat{D_R})||_{f} \label{eq:depth_error}$$ where $SDF(D)$ considers all non-masked depth values to be part of the object and $||\cdot||_{f}$ denotes the Frobenius norm. Figure \[fig:sdf\_example\] illustrates the masked depth input and SDF for an example object. The calculation of error in depth-image-domain has several advantages; because it operates in 2D image space, distance fields are both more efficient to calculate than in the 3D case and better defined because the observed input is partially occluded in 3D but fully observable from the single 2D perspective of the camera. Furthermore, by using the SDF instead of raw depth values, our error gains some robustness to sensor noise and minor errors in predicted shape. To transform this error into a pose prior, we take the quartic-normalized inverse of the score, producing the density function $$p_{prior}(\hat{R}) = \frac{1}{e_R^4+\epsilon}. \label{eq:pose_prior}$$ Sampling Pose Estimates ----------------------- It is possible to obtain approximate maximum likelihood (MLE) and *maximum a posteriori* (MAP) pose estimates by sampling from the pose distribution induced by the predicted $\theta$. Let $\mathbf{R}$ denote the set of $n$ pose estimates sampled the HBEO-MDN network and $R_i\in \mathbf{R}$ be a single sampled pose. From equation \[eq:pose\_likelihood\], the approximate MLE pose estimate is $$\hat{R}_{MLE} = argmax_{R_i\in \mathbf{R}}~~\sum _{i=1}^{c}{\pi_{i}}{\mathcal {N}}(R_i | {\boldsymbol {{\mu _{i}},{\Sigma _{i}}}}) \label{eq:MLE}$$ while incorporating equation \[eq:pose\_prior\] produces an approximate MAP pose estimate of $$\hat{R}_{MAP} = argmax_{R_i\in \mathbf{R}}~~\frac{1}{e_{R_{i}}^4+\epsilon}\sum _{i=1}^{c}{\pi_{i}}{\mathcal {N}}(R_i | {\boldsymbol {{\mu _{i}},{\Sigma _{i}}}}). \label{eq:MAP}$$ As $n\to\infty$, equations \[eq:MLE\] and \[eq:MAP\] approach the true MLE and MAP pose estimates, respectively. As a result, HBEO-MDN is a variable-time method for pose estimation, with prediction accuracy improving as computation time increases. Experimental Evaluation ======================= We evaluated our approach via ablation on three datasets consisting of cars, planes, and couches taken from ShapeNet [@shapenet2015] for a total of $6659$ training objects and $3098$ test objects. We also compared to two RGB-based approaches on the standard Pix3D chair dataset. Depth-based approaches were provided with segmented depth-image input and RGB-based approaches were given tight bounding boxes around objects; in the wild, these segmentations could be estimated using dense semantic segmentation such as MASK-RCNN [@he2017mask]. For the ablation experiments, HBEOs [@hbeo2018] and HBEO-MDNs were trained for each category of object with $2798$ couches, $2986$ planes, and $875$ cars used. During training, depth-images from random views[^2] were generated for each object for a total of $2.7M$ training images. Evaluation datasets were constructed for each class containing 1500 views from $50$ cars, $2300$ views from $947$ planes, and $2101$ views from $368$ couches. The HBEO and HBEO-MDN models used identical subspaces of size $d=300$ for each object class, predicted size $64^3$ voxel objects, and were trained for $25$ epochs (both models converged at similar rates).[^3] We examined two forms of HBEO-MDN, an ablation that used the MLE approximation from equation \[eq:MLE\] (HBEO-MDN Likelihood) and the full method which uses the posterior approximation defined in equation \[eq:MAP\] (HBEO-MDN Posterior). The performance of HBEO-MDN Likelihood illustrates the contribution of the MDN portion of our approach while HBEO-MDN Posterior shows the efficacy of explicitly verifying possible solutions against the observed depth-image. To ablate the impact of the generative portion of our model, we also evaluated two baselines, Random Sample + Oracle, which uniformly sampled poses from $SO(3)$ and was provided with an oracle to determine which of the samples poses was closest to ground truth, and Random Sample + SDF Error, which uniformly sampled poses from $SO(3)$ and used equation \[eq:depth\_error\] to select a single pose estimate from these samples. [.5]{} ![Mean pose error per number of samples across three datasets, Cars (top), Planes (middle), and Couches (bottom). HBEO results (green) are constant since HBEOs do not sample solutions. Error ranges indicate the 95 percent confidence estimate of the mean. []{data-label="fig:pose_graphs"}](Figures/car_results.pdf "fig:"){width="1\linewidth"} [.5]{} ![Mean pose error per number of samples across three datasets, Cars (top), Planes (middle), and Couches (bottom). HBEO results (green) are constant since HBEOs do not sample solutions. Error ranges indicate the 95 percent confidence estimate of the mean. []{data-label="fig:pose_graphs"}](Figures/car_results_zoom.pdf "fig:"){width="1\linewidth"} [.5]{} ![Mean pose error per number of samples across three datasets, Cars (top), Planes (middle), and Couches (bottom). HBEO results (green) are constant since HBEOs do not sample solutions. Error ranges indicate the 95 percent confidence estimate of the mean. []{data-label="fig:pose_graphs"}](Figures/plane_results.pdf "fig:"){width="1\linewidth"} [.5]{} ![Mean pose error per number of samples across three datasets, Cars (top), Planes (middle), and Couches (bottom). HBEO results (green) are constant since HBEOs do not sample solutions. Error ranges indicate the 95 percent confidence estimate of the mean. []{data-label="fig:pose_graphs"}](Figures/plane_results_zoom.pdf "fig:"){width="1\linewidth"} [.5]{} ![Mean pose error per number of samples across three datasets, Cars (top), Planes (middle), and Couches (bottom). HBEO results (green) are constant since HBEOs do not sample solutions. Error ranges indicate the 95 percent confidence estimate of the mean. []{data-label="fig:pose_graphs"}](Figures/couch_results.pdf "fig:"){width="1\linewidth"} [.5]{} ![Mean pose error per number of samples across three datasets, Cars (top), Planes (middle), and Couches (bottom). HBEO results (green) are constant since HBEOs do not sample solutions. Error ranges indicate the 95 percent confidence estimate of the mean. []{data-label="fig:pose_graphs"}](Figures/couch_results_zoom.pdf "fig:"){width="1\linewidth"} [llllll]{}\ (r)[3-5]{} Method & N & Car & Plane & Couch & Runtime\ HBEO-MDN Likelihood & $5$ & $6.92^\circ$ & $11.43^\circ$ & $10.84^\circ$ & $0.07$s\ HBEO-MDN Likelihood & $25$ & $6.10^\circ$ & $10.81^\circ$ & $10.77^\circ$ & $0.13$s\ HBEO-MDN Likelihood & $100$ & $6.26^\circ$ & $10.64^\circ$ & $10.80^\circ$ & $0.36$s\ HBEO-MDN Posterior & $5$ & $4.83^\circ$ & $9.77^\circ$ & $9.59^\circ$ & $0.12$s\ HBEO-MDN Posterior & $25$ & $3.70^\circ$ & $8.45^\circ$ & $8.12^\circ$ & $0.32$s\ HBEO-MDN Posterior & $100$ & $\mathbf{3.23^\circ}$ & $\mathbf{8.06^\circ}$ & $\mathbf{7.50^\circ}$ & $1.06$s\ HBEO [@hbeo2018] & $N/A$ & $9.45^\circ$ & $13.29^\circ$ & $18.84^\circ$ & $\mathbf{0.01}$s\ [lllll]{}\ (r)[3-5]{} Method & N & Car & Plane & Couch\ HBEO-MDN Likelihood & $5$ & $3.73~\%$ & $10.87~\%$ & $9.57~\%$\ HBEO-MDN Likelihood & $25$ & $2.67~\%$ & $9.78~\%$ & $9.09~\%$\ HBEO-MDN Likelihood & $100$ & $2.80~\%$ & $9.65~\%$ & $8.95~\%$\ HBEO-MDN Posterior & $5$ & $1.80~\%$ & $8.13~\%$ & $8.09~\%$\ HBEO-MDN Posterior & $25$ & $1.20~\%$ & $6.22~\%$ & $6.38~\%$\ HBEO-MDN Posterior & $100$ & $\mathbf{0.93~\%}$ & $\mathbf{5.78~\%}$ & $\mathbf{5.95~\%}$\ HBEO [@hbeo2018] & $N/A$ & $6.87~\%$ & $13.57~\%$ & $31.27~\%$\ \ Figure \[fig:pose\_graphs\] gives the performance of the approach across all three ShapeNet evaluation datasets as a function of the number of samples used. Table \[table:results\_table\] contains performance and inference-time at various sampling values while Table \[table:results\_table2\] contains the frequency of large pose errors of at least $15$ degrees. HBEO-MDN Posterior substantially outperformed other approaches, even with a limited sample size. HBEOs, while having generally inferior performance to both HBEO-MDN varieties, struggled most significantly on the couch dataset. We hypothesize that this is due to the symmetry present in couches when viewed from the side, where only small aspects of the image disambiguate left from right. Because the MDN approaches can predict multi-modal pose estimates, they can better model this symmetry. Interestingly, our baseline of Sample + SDF Error performed extremely poorly and only slightly better than selecting a pose estimate at random. It appears that because shape error is computed using predicted object shape, and not ground truth, it is too noisy of a signal to be directly useful —it can disambiguate between highly probable pose-candidates but is not suitable for disambiguating between arbitrary poses. Furthermore, while pose estimation with HBEO-MDNs is slower than with HBEOs, the most expensive HBEO-MDN Posterior method can still produce multiple predictions a second; if more time is available, a higher quality estimate can be obtained by increasing the number of evaluated samples. For a fixed time budget, the Posterior version of HBEO-MDNs outperformed the Likelihood variety, even though approximating the MAP will necessitate using fewer samples than the MLE. We also compared against two RGB-based category-level pose estimation methods, Pix3D [@sun2018pix3d] and Render For CNN [@Su_2015_ICCV] on the Pix3D dataset. All methods were trained on the ShapeNet chair category and evaluated on the $2894$ non-occluded chairs in Pix3D; Pix3D and Render for CNN used RGB images as their input while HBEOs and HBEO-MDNs were provided rendered depth-images from the same viewpoint. Because Pix3D and Render for CNN produce discretized poses, the output of HBEO and HBEO-MDN was discretized for comparison. Table \[table:results\_table\_pix3d\] contains these results for multiple levels of discretization (a larger number of possible views equates to requiring more pose-estimation accuracy and the entries in the table indicate the proportion of objects with correctly estimated pose-bin). Our full method, HBEO-MDN-Posterior[^4] achieved the best performance, along both azimuth and elevation, of all models when the number of bins was high, with HBEOs performing competitively as the bin size became larger. Interestingly, HBEOs slightly outperformed HBEO-MDN with very coarse bins (90$^\circ$) which we hypothesize is due to chair symmetry. While chairs are highly asymmetrical vertically, some variants of chairs lack arms and are thus fairly symmetrical rotationally. Because HBEOs learn to pick the average of good solutions, their predictions may be more likely to fall within 90 degrees of the true solution than HBEO-MDNs—which will tend to predict a mode of the distribution instead of the mean. This is primarily an artifact of using a sampling strategy to select a single pose instead of evaluating the entire MDN-predicted pose distribution. [lccccccc]{}\ (r)[2-5]{} (r)[6-8]{} Number of Bins & 4 & 8 & 12 & 24 & 4 & 6 & 12\ Render For CNN [@Su_2015_ICCV] & $0.71$ & $0.63$ & $0.56$ & $0.40$ & $0.57$ & $0.56$ & $0.37$\ Pix3D [@sun2018pix3d] & $0.76$ & $0.73$ & $0.61$ & $0.49$ & $0.87$ & $0.70$ & $0.61$\ HBEO [@hbeo2018] & $\mathbf{0.87}$ & $0.76$ & $0.69$ & $0.53$ & $0.96$ & $0.91$ & $0.71$\ HBEO-MDN-Likelihood (Ours) & $0.78$ & $0.73$ & $0.70$ & $0.59$ & $\mathbf{0.97}$ & $\mathbf{0.93}$ & $\mathbf{0.75}$\ HBEO-MDN-Posterior (Ours) & $0.80$ & $\mathbf{0.76}$ & $\mathbf{0.73}$ & $\mathbf{0.62}$ & $\mathbf{0.97}$ & $\mathbf{0.93}$ & $\mathbf{0.75}$\ Conclusion ========== We introduced a depth-based 3DOF category-level pose estimation algorithm, applicable to novel objects, that leverages probabilistic output and a 3D shape prediction to verify estimated pose against the observed depth-image. Our method is agnostic to the underlying generative model, provided it produces explicit 3DOF-pose and 3D-shape estimates, and we use it to extend the HBEO framework into HBEO-MDNs. Experimentally, HBEO-MDNs significantly improved on the existing state-of-the-art, providing a significant reduction in average-case pose error and incidence of catastrophic pose-estimation failure; an ablation analysis demonstrated that both the probabilistic output and estimated pose-prior contribute to this result. Our approach is fast enough for use in robotics applications, performing inference at multiple hertz, and is variable-time, allowing increased time budgets or computational power to result in improved pose-estimation. [^1]: In our experiments, we found that $\lambda_{p} = \lambda_{s} = \lambda_{c} = 1$ yielded good results. [^2]: Azimuth and elevation were sampled across the full range of possible angles while roll was sampled from 0-mean Gaussian distribution with 99-percent mass within the range $[-25^\circ$, $25^\circ]$. [^3]: Models were trained using the Adam optimizer with $\alpha=0.001$ and evaluated on an Nvidia 1080ti GPU. [^4]: HBEO-MDN variants utilized 100 samples.
--- abstract: 'We consider the problem in regression analysis of identifying subpopulations that exhibit different patterns of response, where each subpopulation requires a different underlying model. Unlike statistical cohorts, these subpopulations are not known *a priori*; thus, we refer to them as cadres. When the cadres and their associated models are interpretable, modeling leads to insights about the subpopulations and their associations with the regression target. We introduce a discriminative model that simultaneously learns cadre assignment and target-prediction rules. Sparsity-inducing priors are placed on the model parameters, under which independent feature selection is performed for both the cadre assignment and target-prediction processes. We learn models using adaptive step size stochastic gradient descent, and we assess cadre quality with bootstrapped sample analysis. We present simulated results showing that, when the true clustering rule does not depend on the entire set of features, our method significantly outperforms methods that learn subpopulation-discovery and target-prediction rules separately. In a materials-by-design case study, our model provides state-of-the-art prediction of polymer glass transition temperature. Importantly, the method identifies cadres of polymers that respond differently to structural perturbations, thus providing design insight for targeting or avoiding specific transition temperature ranges. It identifies chemically meaningful cadres, each with interpretable models. Further experimental results show that cadre methods have generalization that is competitive with linear and nonlinear regression models and can identify robust subpopulations.' author: - bibliography: - 'sdmBib.bib' title: 'Cadre Modeling: Simultaneously Discovering Subpopulations and Predictive Models' --- Introduction ============ In many real-world analytics tasks, observations are divided into different cohorts using simple criteria, and a different predictive model is created for each cohort. For example, in insurance, it is well known that young adults have different accident rates than do adults. Thus, one would make different predictive models of accident rates for different age subpopulations[@life_insurance]. Similarly, an epidemiologist may make different models for survey subjects of different ethnicities or socioeconomic statuses[@epidemiology1]. For most problems, however, appropriate subpopulations are not known *a priori*. Identifying appropriate subpopulations becomes a valuable aspect of the learning task. Consider the task in drug discovery of predicting the toxicity of a molecule. Different types of molecules have different mechanisms of toxicity, so understanding these classes of molecules and the different factors that influence their toxicity can help accelerate the drug discovery process[@drugs]. In these examples and others, only a few features are used in defining the subpopulations. Thus, a method for discovering new subpopulations must be able to perform feature selection. If the possible existence of these subpopulations is ignored, a more complicated model such as a kernel support vector machine (SVM)[@svms] may be required to attain a good fit. However, it is far more informative when this single complicated model can be replaced with a set of simpler ones[@ensemble][@jordanHME], each associated with a subpopulation of the dataset. The proposed learning problem is to jointly discover these subpopulations and estimate their prediction models. To maximize the insight the model provides, we seek to make interpretable models for both characterizing the subpopulations and predicting the target within each subpopulation. Our way of solving this class of problem is the supervised cadre model (SCM). An SCM divides the input space into probabilistic subpopulations that we call cadres. Cadres are similar to clusters[@lloyd], but they are able to perform feature selection and weighting. They are also selected so as to be easily assigned a predictive model. An SCM has two major components. The first is the cadre membership function, which assigns every point in input space a distribution characterizing that point’s probability of belonging to each cadre. The second component of an SCM is a set of target-prediction models. In this work, we focus on the case where the target-prediction models are linear. Crucially, the cadre membership function and target-prediction functions are learned simultaneously. Rather than being chosen to minimize an unsupervised quantity such as within-cluster-sum-of-squares, cadres are selected to maximize the effectiveness of the target-prediction process. Only a small subset of features are used for the the cadre-assignment and target-prediction processes – their functions are sparse in the input space. To achieve sparseness, we impose elastic net priors[@ElasticNet] on the supervised cadre model’s parameters. The cadre membership function and each cadre’s target-prediction function are allowed to use a different set of features. In this paper, we develop the mathematical formalism of the supervised cadre model and its learning problem, focusing on the case where the learning task is regression. We assess the quality of a cadre structure in terms of generalization, interpretability, and subpopulation robustness. Experimental benchmarks show that the SCM has generalization that is competitive with linear, piecewise-linear, and nonlinear regression methods. The cadre models identify stable subpopulations with sparse membership and prediction functions. To demonstrate interpretability, we provide a cheminformatics case study in which the SCM is applied to the the problem of predicting the glass transition temperature of polymers. The cadres found represent different subpopulations of molecules based on their connectivity, polarizability, and distribution of negatively charged surface area. These features correlate with specific molecular structure characteristics, which can provide design advice for creating new polymers with desired properties[@design]. The cadre method identifies small subsets of features that are used uniquely by each cadres and shared features that are predictive across all cadres. An implementation of the SCM is available on GitHub[^1]. Motivating Example ------------------ We consider a simple two-dimensional example to demonstrate the usefulness of a modeling procedure that can jointly discover subpopulations and predict the target. It is inspired by a glass temperature transition[@Katritzky] analysis found in Section \[cheminformatics\], but the data are simulated. In this problem, the cadres identify groups of molecules that obey different underlying physics. The SCM can solve this problem very well, but existing methods that combine clustering and regression can fail to learn a good prediction rule. We seek to the predict the glass transition temperature of a polymer using two characteristics: its branching connectivity and electrical polarizability. Polarizability can inform understanding of a polymer’s response to external fields and also relates to the level of induced electrical attraction between polymer strands; connectivity characterizes the level of backbone branching of functional groups in a polymer. For illustration purposes, imagine that, for molecules with low levels of branching and steric bulkiness, electrical polarizability is positively correlated with glass transition temperature. This means that polymer strands are able to associate with each other more strongly and remain associated at higher temperatures. For molecules with medium or high branching connectivity, the polarizability is negatively associated with glass transition temperatures, with different rates. The polymers can be grouped by their branching connectivity, as shown in Fig. \[toy\_true\] and \[toy\_true\_hist\]. In Fig. \[toy\_km\] and \[toy\_cadre\], a standard clustering method such as $K$-means[@lloyd] cannot discover the true subpopulations, but the SCM can. \ Related Work ============ Prior methods for combining a clustering task and a supervised learning task exist. For regression problems, there is the clusterwise linear regression model[@clusReg]. For classification problems, there is the clustered SVM[@clusSVM] and mixture of linear SVM[@mixSVM] models. The clusterwise linear regression and clustered SVM models learn hard partitions of the space, whereas we learn a soft partition. The idea of the mixture of linear SVMs is to approximate a nonlinear SVM with a large number of linear models. Because we want interpretable models, we are interested in problems in which a small number of carefully chosen linear models are sufficient for good prediction. In these prior models, there is no feature selection performed during the clustering and prediction processes. The input space is probabilistically partitioned, and each partition is assigned a simple supervised learner in the hierarchical mixture of experts (HME)[@jordanHME]. The SCM can be viewed as an HME with only one set of gates and experts and a special gating function. The standard HME gating function cannot discover subpopulations, unlike ours. Clustering has been combined with feature selection and weighting to yield the sparse $K$-means method[@sparse_cluster] and weighted fuzzy $c$-means method[@fuzzy], respectively. Both are intended for problems that lack a usable target feature, and sparse $K$-means focuses on those in the $p\gg n$ regime. We restrict our attention to supervised learning problems so that the subpopulations are defined relative to the behavior of a target feature. Recent papers [@Lipton][@DoshiVelez] have proposed different ways to quantify a model’s interpretability, including simulatability, decomposability, and amicability to post-hoc interpretation. Our model is simulatable because a human can easily replicate a model calculation. It is decomposable because all of its parameters have intuitive purposes. It also admits post-hoc interpretations because visualizations of feature distributions can be grouped by cadre for greater insight in a dataset’s structure. Additionally, the SCM is interpretable because its constituent functions are sparse[@lasso]. Model Development ================= Formalism --------- Let the input be $x \in \mathbb{R}^P$, and let the target be $y \in \mathcal{Y}$. We focus on regression problems where $\mathcal{Y} = \mathbb{R}$, but cadre models can be applied to classification problems where $\mathcal{Y} = \{1,\hdots,K\}$ if the loss function is changed. We make two assumptions. The first is that there exist subsets $X^1,\hdots,X^M$ of $\mathbb{R}^P$ such that the behavior of $y$ at points within each $X^m$ is more simply modeled than it is on the whole input space. We call these subsets cadres. The second is that some features affect only $x$’s cadre membership, and some features, given $x$’s cadre membership, affect only the predicted value of $y$. We define a prediction function $f:\mathbb{R}^P \to \mathbb{R}$ that may be decomposed as $$f(x) = \sum_m g_m(x) e_m(x)$$ where $g_m(x) = p(x \in X^m)$ and $e_m(x) = \mathbb{E}[y |x, x \in X^m]$. From these and $f$’s definition, we can show that $f(x) = \mathbb{E}[y | x]$. We parameterize each $g_m$ and $e_m$ as $$g_m(x) = \frac{e^{-\gamma ||x - c^m||_d^2}}{\sum_{m'}e^{-\gamma ||x - c^{m'}||^2_d}}\,\,\,\,\text{and}\,\,\,\,e_m(x) = \left(w^m\right)^T x + w_0^m.$$ Here: $||x||_d = \left(\sum_p |d_p| (x_p)^2\right)^{1/2}$ is a seminorm; $d$ is a feature-selection parameter used for cadre assignment; each $c^m$ is the center of the $m$th cadre, each pair $w^m, w^0_m$ characterizes the regression hyperplane for cadre $m$; and $\gamma > 0$ is a hyperparameter that controls the sharpness of the cadre-assignment process. Thus, the cadre membership of $x$ is a multinoulli random variable with probabilities $\{g_1(x),\hdots,g_M(x)\}$ characterized by weighted automatic relevance determination (ARD)[@Murphy] kernels $e^{-\gamma||x-c^m||_d^2}$. We can also interpret $g_m(x)$ as the softmax[@Murphy] of the set of weighted inverse-distances $\{\gamma||x-c^1||_d^{-2},\hdots,\gamma||x-c^M||_d^{-2}\}$. We let $m^*(x) = \arg\max_m g_m(x)$ be the cadre that observation $x$ is most likely to belong to. If we let $C$ and $W$ be matrices with columns $c^1,\hdots,c^M$ and $w^1,\hdots,w^M$, our model is fully specified by the the parameters $C \in \mathbb{R}^{P_C \times M}$, $d \in \mathbb{R}^{P_C}$, $W \in \mathbb{R}^{P_T \times M}$, $w^0 \in \mathbb{R}^M$, and the hyperparameter $\gamma>0$. We group a model’s parameters as $\Theta = \{C, d, W, w^0\}$. All the parameters have interpretations, ensuring model decomposability[@Lipton]. Each $c^m$ is the center of the $m$th cadre. The coefficient $d_p$ indicates how important the $p$th feature is for the cadre-assignment process. Each cadre has a regression hyperplane characterized by $w^m$ and $w^0_m$. We give a graphical representation of this model in Fig. \[gr\]. The hyperparameters $\lambda$ and $\alpha$ and the noise variance $\sigma^2$ are described in Section \[implementation\]. Learning a Cadre Model {#implementation} ---------------------- For regression problems, we use the standard $y | x \sim \mathcal{N}(f(x), \sigma^2)$. With a different loss model, an SCM can be specialized to other supervised learning tasks. For example, hinge-loss[@Murphy] could be used in classification. Models are learned with Bayesian point estimation. Let $X$ be the set of training observations, let $Y$ be the set of training target values, and let $N$ be number of training observations. We learn the supervised cadre model by solving the negative log-posterior minimization problem: $$\begin{aligned} \{\hat{\Theta}, \hat{\sigma}^2\} &=& \arg\min\{ -\log p(\Theta, \sigma^2 | X, Y)\}\\ &=& \arg\min\{ -\log(p(Y|\Theta,\sigma^2,X)p(\Theta|\sigma^2)p(\sigma^2))\}\end{aligned}$$ Expanding the term $-\log p(Y|\Theta,\sigma^2,X)$ will generate loss terms proportional to $\ell(x,y) = (f(x) - y)^2$. When $g_m(x)$ is close to 0 or 1, the gradient of this loss term is poorly conditioned. A tractable upper bound is $\ell(x,y) \leq \sum_m g_m(x)(e_m(x) - y)^2$. Its validity relies on the fact that $f(x)$ is a convex combination of $\{e_1(x),\hdots,e_M(x)\}$, and it is tight in the limiting case that $g_m(x) \in \{0,1\}$ for all $x$. The prior distribution for $\Theta$ is factored as $p(\Theta | \sigma^2) = p(d|\sigma^2) p(W|\sigma^2)$. We impose elastic net priors[@ElasticNet] on $W$ and $d$ with hyperparameters $\lambda = \{\lambda_d, \lambda_W\}$ and $\alpha = \{\alpha_d, \alpha_W\}$: $$p(d | \sigma^2) \propto \exp\left(-\frac{1}{2\sigma^2}R(d; \lambda_d, \alpha_d)\right)$$ $$p(W | \sigma^2) \propto \exp\left(-\frac{1}{2\sigma^2}R(W; \lambda_W, \alpha_W)\right),$$ where $R(\cdot; \lambda_i, \alpha_i) = \lambda_i(\alpha_i||\cdot||_1 + (1-\alpha_i)||\cdot||_2^2)$ is the elastic net regularization functional. Note that, for $W$, the norms are applied entrywise. The elastic net lets us find a compromise between the stability of an $\ell^2$ penalty and the sparseness of an $\ell^1$ penalty. Following the conventions of Bayesian linear regression[@Murphy], we impose an uninformative improper prior on $\sigma^2$: $p(\sigma^2) = 1 / \sigma^2$. The final augmented loss functional, minimized with respect to $\Theta=\{d,C,W,w^0\}$ and $\sigma^2$ is $$\begin{aligned} \mathcal{L}(\Theta, \sigma^2) &=& \frac{1}{2\sigma^2} \sum_{n=1}^N\sum_m g_m(x^n)(y_n - e_m(x^n))^2\\ &+& (1+N)\log \sigma^2\\ &+& \frac{\lambda_d}{2\sigma^2}(\alpha_d||d||_1 + (1-\alpha_d)||d||_2^2)\\ &+& \frac{\lambda_W}{2\sigma^2}(\alpha_W||W||_1 + (1-\alpha_W)||W||_2^2).\end{aligned}$$ Our augmented loss functional is nonsmooth and nonconvex, which makes it difficult to minimize with conventional optimization algorithms. We use the TensorFlow[@tensorflow] module for Python, which uses automatic differentiation for fast stochastic first-order optimization. Within TensorFlow, we use Adam[@Adam], an adaptive stepsize gradient decent method. The stochastic aspect of our optimization process makes our solver less susceptible to local minima: a point in parameter-space that is a bad local minimizer for one minibatch of the training data is not guaranteed to be a local minimizer for other minibatches of the training data[@bottou]. The learning problem is fully specified by the cadre-assignment sharpness hyperparameter $\gamma$, the elastic net mixing hyperparameters $\alpha$, the regularization strength hyperparameters $\lambda$, and the number of cadres $M$. Model Assessment {#assessment} ---------------- We evaluate the overall quality of a supervised cadre method in three ways: generalization, cadre quality, and interpretability. Generalization is how well the model works on new data. It can be measured by evaluating an accuracy metric such as mean squared error on a holdout set. A cadre is good if it accurately captures a subpopulation[@stability]. We develop two metrics of interpretability and examine a case study in depth. These metrics are applied to real datasets in Section \[empirical\]. ### Cadre Quality We use average best match[@Hopcroft] across bootstrapped samples as our metric of cadre quality. To properly gauge cadre goodness, the nonconvexity of the cadre model requires specially chosen parameter initializations. We begin by taking a bootstrap sample $\mathcal{D}_0$ of the data and learning a cadre model $\{\Theta_0, \sigma^2_0\}$ from this sample. We then apply Algorithm \[boot\] to learn $B$ more models. For each of the $B+1$ models, we find the cadre assignment of every observation in the original dataset. A cadre learned in bootstrap sample 0 is good if it also learned in many of the other bootstrap samples. So define $C_m^b$ to be the index set $$C^b_m = \{n \in \{1,\hdots,N\} : \arg\max g_{m'}(x^n) = m\}.$$ Then the match between two index sets $C^b_m$ and $C^{b'}_{m'}$ is $$\text{match}(C_m^b, C_{m'}^{b'}) = \min\left\{\frac{|C_m^b \cap C_{m'}^{b'}|}{|C_m^b|}, \frac{|C_m^b \cap C_{m'}^{b'}|}{|C_{m'}^{b'}|}\right\}.$$ Two sets $C^b_m$ and $C^{b'}_{m'}$ have a match score close to 1 when they share many elements in common and are approximately the same size. The match score decreases as either of those assumptions ceases to hold. Given a set of models from bootstrapped samples, we find the average best match[@Hopcroft] $ABM(m,M)$ for cadre $m$ in a model with $M$ total cadres. Average best match is given by $$ABM(m,M) = \frac{1}{B}\sum_{b>0}\max_{m'}\{ \text{match}(C^0_m, C^b_{m'})\}.$$ If a cadre has a high average best match, the model has identified a stable subpopulation. Given a set of bootstrap samples, the average best match assigns a number to every cadre in model 0. If we are comparing models with different total numbers of cadres, we can assign the entire model an average best match: $ABM(M) = \frac{1}{M}\sum_m ABM(m,M)$. ### Interpretability We separately measure the interpretability of an SCM’s cadre-assignment and target-prediction processes. For cadre-assignment, we consider how easily a domain expert can understand and validate the discovered subpopulations – i.e., post-hoc interpretability. This is easiest when the model is sparse[@Lipton]. Sparsity is summarized with the density rate $DR=||d||_0/P$, where $||d||_0$ is the number of nonzero entries in $d$ and $P$ is $d$’s dimension. For target-prediction, we introduce a statistic $\tau$ that measures how similar the set of learned linear models are to one another. When $\tau$ is small, then the linear models that each cadre has are very similar to each other. When $\tau$ is large, the linear models are very different. So $\tau$ provides high-level information about how each cadre’s regression weights vary from each other. We quantify this similarity by $$\tau = \frac{1}{P}\sum_{p=1}^P\text{StdDev}(\{w^1_p,\hdots,w^M_p\}).$$ So $\tau$ is the standard deviation of every model’s weight for the $p$th feature, averaged over all features. Empirical Evaluation {#empirical} ==================== In our empirical evaluation, first we present an in-depth case study on a cheminformatics problem. Then we present accuracy and cadre quality benchmarks on a variety of datasets. In all numerical results, we begin by centering and standardizing both the input and target data. The elastic net mixing parameter $\alpha$ was fixed at $\alpha_d = 0.95$ and $\alpha_W = 0.05$. All other hyperparameters, including those for competitor methods, were set by cross-validation over the training set. Cheminformatics Case Study {#cheminformatics} -------------------------- We demonstrate how the supervised cadre model may be applied to a regression problem. The task is to predict a polymer’s glass transition temperature (Tg), given various measurements of its characteristics[@Katritzky][@WuThesis]. Polymers transition from a brittle state to a viscous state if they are heated past their glass transition temperature. The glass transition temperature thus characterizes pure polymers, polymer blends, and copolymers. It can also be informative of a polymer’s miscibility. The relationship between the target and measurable features is known to be nonlinear. In addition, although a single glass transition temperature is measured, in truth, there are a range of temperatures over which a polymer undergoes its glass transition. Thus, predictions of transition temperature need not be perfectly accurate. In addition, previous work has suggested that polymers may possess a natural cadre structure[@WuThesis]. In our polymer dataset, there were 262 different polymers. We took an 80%-20% train-test split. The dataset had 119 features initially. Because of feature correlation, we followed [@WuThesis] and applied a correlation filter (threshold 0.7) to the training data. After this, 28 features remained. We compared the SCM’s predictive ability to that of other methods and determined the most natural number of cadres via cross-validation. These results are presented in Table \[table:chem\] and Fig. \[chem\_m\]. The models compared were the SCM, the $K$-means linear SVR, the $K$-means regression tree, the Gaussian Mixture linear SVR, the random forest, and the RBF SVR. In the $K$-means models, a $K$-means clustering was learned on the training set. Then, for each of the clusters, a separate supervised learning model was trained on the observations in each cluster. The Gaussian mixture linear SVR was defined similarly. A full covariance matrix was learned. The SCM outperforms the other models on this task. Model MSE $M^*$ ----------------------------- ---------- ------- Supervised cadres **0.14** 3 $K$-means ridge regression 0.26 1 $K$-means linear SVR 0.18 4 $K$-means regression tree 0.24 2 Gaussian Mixture linear SVR 0.17 3 Random forest 0.21 1 RBF SVR 0.22 1 : Generalization error and model complexity on Tg dataset. []{data-label="table:chem"} Now we discuss the subpopulations found by the SCM. Fig. \[tgYvsY\] shows the true Tg values versus their predicted values. Cadres 1 and 3 (respectively, the red circles and blue squares) contain polymers with low Tg. Cadre 2 (the green triangles) primarily contains polymers with high Tg. We provide the mean and standard deviation Tg of each cadre in Table \[table:chem2\]. Cadre \# Polymers Mean Tg (K) Std.Dev Tg ------- ------------- ------------- ------------ 1 121 300 68 2 62 456 106 3 79 330 72 : Cadre Characteristics. Std.Dev Tg is the standard deviation of the distribution of Tg scores in each cadre.[]{data-label="table:chem2"} We can also consider which features are important for determining cadre membership and how these features vary between cadres. Fig. \[memWei\] shows the distribution of cadre membership weights $d_p$. The optimal $d_p$ was very sparse: only 11 of the 28 features were used in the cadre-assignment process. So the cadre assignment process was simple. ![Distribution of cadre assignment weights $d_p$. Few features were used for cadre-assignment.[]{data-label="memWei"}](membershipWeights2.png){height="2.4in"} Looking at Fig. \[memWei\], the features important for determining cadre membership may be grouped into three categories: partial charge distribution, connectivity, and atomic polarizability. Four features related to partial charge surface distributions were important for cadre membership: poly\_PEOE\_VSA\_FNEG, poly\_PEOE\_VSA\_FPNEG, poly\_PEOE\_VSA\_maxNEG, and poly\_PEOE\_VSA+6. Three of them are used to distinguish between the two low-Tg cadres. Three features related to molecular bulk and backbone branching were important for cadre membership: polyb\_chi1v, B\_chi0, and polyb\_BB\_chi1v. The final important feature was polyb\_apol, a descriptor associated with induced electrostatic attraction between polymer chains. This examination of features shows how post-hoc interpretation helps to distinguish between the two low-Tg cadres and between the low-Tg cadres and the high-Tg cadres. In Fig. \[chemfeatures\], the feature distributions grouped by cadre reflect distinct subpopulations. \ \ \ Now we consider the regression models for each cadre. The weights $w^m_p$ are shown in Fig. \[predWeights\]. These weights have a $\tau$ score of 0.12: the cadres use distinct models but share some features. Some features, such as polyb\_BBrotN\_Weight (the weighted effect of rotatable bonds in the polymer repeat unit) and poly\_PEOE\_VSA+1 (related to surface charge), are used similarly in all three cadres. Some features are used differently between cadres. This includes sg\_weinerPath, a connectivity index, which is positively associated with Tg in the high-Tg cadre but negatively associated with Tg in the low-Tg cadres. Some features are not used by every cadre, such as poly\_PEOE\_VSA-3, which is positively associated with Tg in cadre 3, negatively associated with Tg in cadre2, and not used in cadre 1. These sparsity patterns help further characterize the differences between the two low-Tg cadres and the high-Tg cadre. Some features, such as polyb\_PEOE\_VSA+6, polyb\_PEOE\_VSA\_FPNEG, and B\_chi0, are very important for determining cadre membership but have small regression weights. We compare the polymers in cadres 1 and 3 to characterize their differences. Cadre 1 contains many elastic acrylates, but cadre 3 is primarily alkenes. The brittle, high-Tg carbonates are all placed into cadre 2. Within each cadre, the underlying patterns of physical factor controlling the Tg of each polymer are different and are represented by different but overlapping sets of descriptors. The descriptor patterns prominent in each cadre represent sets of latent variables that describe how the Tg of a polymer within that cadre would respond to specific structural changes. This ability to separate polymers with different sensitivities to structural alterations into cadres enables domain experts to design materials with desirable Tg properties using models with expanded and identifiable domains of applicability[@design]. ![Distribution of cadre regression weights $w^m_p$. The different cadres learned different linear models. []{data-label="predWeights"}](regressionHeatmap.PNG){width="3.3in"} Accuracy Benchmarks {#accuracy} ------------------- We want to empirically evaluate the SCM. Thus, we apply it to a variety of datasets taken from the UCI[@UCI] repository. The datasets were Airfoil Self Noise (Airfoil)[@AirfoilSelfNoise], AirQuality[@AirQuality], Boston Housing (Boston)[@Boston], Concrete Compressive Strength (Concrete)[@Concrete], Facebook Comment Volume (Facebook)[@Facebook], SkillCraft1 Master Table (GameSkill)[@GameSkill], and Parkinson’s Telemonitoring (Parkinson)[@Park]. We give the number of features and observations in Table \[table:sizes\]. Note that the Parkinson dataset contains two targets; we learn separate models for each. Our experimental design is as follows: We generated 20 different 75%-25% train-test splits for each dataset. For each split, we trained a model and recorded its mean squared error (MSE) when evaluated on the test set. The models we used were: a cadre model, a linear SVR (LSVR), an RBF SVR (KSVR), a $K$-means linear SVR (KM+LSVR), and a Gaussian Mixture Linear SVR (GMM+LSVR). By $K$-means linear SVR, we mean that we learned a $K$-means clustering on the training data and then trained a linear SVR on the observations in each cluster. The GMM+LSVR model is defined similarly. The Gaussian Mixture Model learned a full covariance matrix for each mixture component. Throughout, model selection was performed with cross-validation on the training set. For every dataset and every model, we calculated the mean and standard deviation of the test set MSE scores. These are shown in Table \[table:many\_errors\]. In the AirQuality and GameSkill datasets, the SCM had a significantly lower ($p<0.05$) mean test error than all other methods. In the Airfoil, Concrete, Parkinson1, and Parkinson2 datasets, the SCM had a significantly lower ($p<0.05$) mean test error than all methods save KSVR. In the Facebook dataset, the SCM had the lowest mean test error of all methods, but the differences were not significant. This shows that the SCM is competitive with powerful nonlinear regression methods and has performance that often exceeds naive piecewise-linear regression methods. \[table:many\_errors\] Cadre Goodness and Interpretability Evaluation ---------------------------------------------- We check that the SCM is capable of learning robust and interpretable cadres. This section’s results used the optimal number of cadres $M^*$ found in Section \[accuracy\] via cross-validation. For each of the datasets detailed in Section \[accuracy\], we generate 20 different bootstrap samples. For each split, we learn an SCM and evaluate our cadre goodness and interpretability metrics on that model; the results are in Table \[table:sizes\]. The average best matches (ABM) show that the SCM can learn stable cadre structures. The average $||d||_0$ (DR) shows that most models had quite sparse cadre-assignment mechanisms. The average variance $\tau$ measures how distinct the regression weights were: the Airfoil model had very distinct weights, and AirQuality, Facebook and GameSkill models had similar weights. The SCM is more interpretable than the competitors considered here. Unlike the SCM, Linear SVR, $K$-means, and gaussian mixtures are not sparse over the feature space. RBF SVR is nonlinear and nonparametric, so it is the least interpretable of any model considered here. Conclusion ========== We have developed a new framework for discovering informative subpopulations (cadres) while solving a predictive task. Here, we have focused on regression problems and linear prediction functions, but our notion of a cadre is applicable to learning problems in general. In an in-depth cheminformatics case study, we showed that the regression cadre model learns post-hoc interpretable groupings of polymers that each have accurate, informative models. The method identified cadres of polymers that respond differently to structural perturbations, thus providing design insight for targeting or avoiding specific transition temperature ranges. We assessed the quality of a cadre structure in terms of generalization, interpretability, and subpopulation robustness. On benchmark datasets, SCM performs well with respect to these metrics. For generalization, it outperforms linear and piecewise-linear models and is competitive with nonlinear ones. Due to its sparse linear aspects, it is more interpretable than non-sparse piecewise-linear and fully nonlinear models. In the future, cadre models can be customized to other subpopulation definitions, base functions, and learning tasks by changing the gating, emission and loss functions. We are currently applying cadre analysis to other classes of learning problems including classification and risk analysis. The key contribution is that SCM can discover informative subpopulations and thus enable a new type of precision analysis. This can augment expert knowledge in fields as diverse as drug discovery, fault detection, and health risk analysis. Acknowledgment {#acknowledgment .unnumbered} ============== Thanks to NSF Grant \#1331023 and Dr. Ke Wu for support of this project. This work is supported by IBM Research AI through the AI Horizons Network and by the Center for Biotechnology & Interdisciplinary Studies at Rensselaer. [^1]: https://github.com/newalexander/supervised-cadres
--- abstract: | We present broad band surface photometry of the radio galaxy 3C270 (NGC 4261). We find a distinct dust lane in the $V-R$ image of the galaxy, and determine its orientation and size. We use the major axis profile of the galaxy to estimate the optical depth of the dust lane, and discuss the significance of the lane to the shape of the galaxy. 0.4in [*Subject headings:*]{} Galaxies: active; Galaxies: radio; Galaxies: individual; Galaxies: interstellar medium; Galaxies: photometry; Galaxies: dust lane author: - | Ashish Mahabal, Ajit Kembhavi\ Inter-University Centre for Astronomy and Astrophysics\ Post Bag 4, Ganeshkhind, Pune 411 007, India,\ 0.1in K. P. Singh, P. N. Bhat\ Tata Institute of Fundamental Research\ Homi Bhabha Road Bombay 400 005, India - | T. P. Prabhu\ Indian Institute of Astrophysics\ Bangalore 560 034, India title: A Dust Lane in the Radio galaxy 3C270 --- =6.5truein =9truein =-1.5cm Introduction ============ 3C270 (NGC 4261) is a radio galaxy with a bright compact radio nucleus and prominent jets and lobes. While at first sight it appears to be a morphologically simple elliptical galaxy of type E2, closer examination reveals many interesting features. A dust lane roughly along the apparent major axis of the galaxy was reported by Mőllenhoff & Bender (1987), while others have failed to confirm its presence (e.g. Kormendy & Stauffer 1987; Peletier 1990). On a much smaller angular scale, recent observations with the Planetary Camera on the Hubble Space Telescope (Jaffe et al. 1993) have led to the detection of an absorbing disk of radius $\sim2\times10^{20}\rm\,cm$ which is made up of cool dust and gas. The plane of the disk is nearly normal to the radio jets, which are oriented close to the apparent minor axis of the galaxy. Long slit spectra (Davies & Birkinshaw 1986) have shown that the galaxy rotates round an apparent axis which lies only $6^\circ \pm 4^\circ$ from the apparent major axis, which suggests that the galaxy is prolate or close to prolate. In this paper we use $V$ and $R$ images of NGC 4261 to confirm the existence of the dust lane first noted by Mőllenhoff & Bender (1987). We demonstrate the existence of the lane in the combined $V-R$ colour image of the galaxy, determine its size and orientation, and estimate its optical depth from the colour excess relative to the region immediately outside the dust lane. We then independently estimate the optical depth of the dust lane by examining the deviation of the surface brightness profile of the galaxy from de Vaucouleurs’ law. We end with comments on the contribution of the dust lane to the observed boxiness of galaxy isophotes and the significance of the orientation of the lane to the radio structure and intrinsic shape of the galaxy. Observation and Data Reduction ============================== NGC 4261 was observed at the prime focus $(f/3.5)$ of the $2.3\rm\,m$ [*Vainu Bappu Telescope*]{} at Kavalur on the night of March 3, 1992 as part of a programme of broad band imaging of radio galaxies. The detector used was a GEC CCD with $576\times385$ pixels, each of $22\times22\,\mu\rm m^2$ in size, corresponding to $0.56\times0.56\rm\,arcsec^2$ at the prime focus. We obtained two $600\rm\,s$ exposure frames in $V$ and two $200\rm\,s$ frames in $R$, a number of twilight sky flats and bias frames spread throughout the night. We observed standard stars in the [*dipper asterism*]{} region of the open cluster M 67 for calibration; the procedure adopted has been described in detail by Anupama et al. (1994). We followed the standard procedure for bias subtraction and flat fielding, and performed sky subtraction for each frame using a mean sky value estimated from pixels in the four corners of the frame, all of which were unaffected by the galaxy. We combined the two frames in each filter into single $V$ and $R$ frames after performing translations and rotations so that the images were aligned to better than $0.1$ pixel. Since the night was not photometric, we did not apply any extinction corrections. Tasks from IRAF[^1] and STSDAS[^2] were used for all the data reduction and analysis. Boxiness is clearly discernible in the isophotes of NGC 4261, which nevertheless can be very well approximated by smooth ellipses. We fitted ellipses to the isophotes using routines in STSDAS, which are based on the method outlined by Jedrzejewski (1987). The semi-major axes of successive fitted ellipses were larger by a factor of $1.1$, and the fit extended from the centre to a semi-major axis length of $60''$, where the error in the surface brightness reaches $\sim 0.1\rm\,magnitude$. The semi-major axis brightness profile, ellipticity and position angle of the fitted ellipses as functions of semi-major axis length $a$ for the $V$ filter are shown in Figure 1. These parameters show similar behaviour in $R$. The rather large error-bars in case of ellipticity and position angle could be due to genuine departure of isophots from true ellipses. The deviation of an isophote from a perfect ellipse is estimated by expanding the difference in intensity between the isophote and the corresponding fitted ellipse as a Fourier series in the eccentric anomaly along the isophote. When boxiness is present, the coefficient $b4$ of the $\cos(4\phi)$ term is negative. We have shown in Figure 1 the ratio $b4/a$ as a function of $a$ for images in the $V$ filter. All the distributions in Figure 1 are in good agreement with the values obtained by Mőllenhoff & Bender (1987). Colour Image ============ The putative dust lane in NGC 4261 is too faint to be seen in a single direct image, and the best place to look for it is a 2-colour image of the galaxy. Since we do not have a $B$ frame with good signal-to-noise ratio, we investigate the $V-R$ image. The difference in the full width at half maximum (FWHM) in the point spread function (PSF) for the $V$ and $R$ images is $\sim 0''.06\rm$, compared to the FWHM of $\sim 2''.4\rm$. We therefore ignore the difference in further analysis. A narrow elongated structure is clearly seen in the $V-R$ image (Plate 1), oriented close to the apparent major axis of the galaxy. We show below that the feature is redder than its surroundings, and interpret it as a dust lane. In order to improve the appearance of the lane, we have also constructed a modified colour image adopting the procedure recommended by Sparks et al. (1985), which uses a smooth version of the higher wavelength image. This enhances the higher spatial frequency structures without affecting large scale colour features, and reduces noise in the 2-colour image as one of the components is smoothed. For this purpose we first fitted ellipses to the $R$ image following the procedure described in the previous section, then obtained a smooth image $R_{smooth}$ by interpolating between the isophotes using the ellipse parameters, and finally the modified colour image, $V-R_{smooth}$, which is shown in Plate 2. The elongated structure in the $V-R$ image is now more obvious than in Plate 1. The feature is also seen if the smooth image is obtained with the inclusion of higher harmonics which account for the boxiness of the isophotes. We wish to emphasize that all calculations presented in this paper are based on the original, unsmoothed $V$, $R$ and $V-R$ images. A cut across the dust lane in the $V-R$ image, averaged over ten rows around the galaxy centre, is shown in Figure 2. Even though the instrumental colour profile in the figure is noisy in the outer parts, it is clear that there is an overall shift to the blue as one moves towards the centre of the galaxy. A region obviously more red in colour than its immediate surroundings is seen straddling the centre. The red colour persists over the narrow feature, and we interpret it to be a thin disk of dust seen nearly edge on and appearing as a dust lane. From inspection of Plate 1, the extent of the dust lane is found to be $\sim21\times6\rm\,arcsec^2$. We have taken several $5$-pixel wide cuts across the dust lane, and assuming that the maximum in the $V-R$ profile in such a strip corresponds to the centre of the lane for that strip, obtained the coordinates of the centre along its length. A linear least square fit to these points then provides a straight line representation of the dust lane. This is shown in Figure 3, along with the direction of the major axis obtained from the ellipse fits, and the direction of the large scale radio jet as given by Birkinshaw & Davies (1985). The dust lane makes an angle of $9^{\circ}\pm1^{\circ}$ with the major axis, and an angle of $97^{\circ}\pm1^{\circ}$ with the radio jets. We have estimated the colour excess in the dust lane by comparing the colour in it with that of the surroundings. For this purpose we used a circular aperture with radius $1''.68$ ($3\rm\,pixels$) on the $V$ and $R$ frames separately to extract magnitudes along the lane and a region along the minor axis $10''$ from the centre, which is free of dust. The mean of the colour excess obtained in this manner is $$E(V-R) = (V-R)_{dust}-(V-R)_{galaxy} \simeq 0.05 \pm 0.01.$$ Assuming that the composition of the dust in the disk is similar to that of dust in our galaxy, and using standard relations from Savage & Mathius (1979), it follows that $E(B-V)=0.06$, $\tau_{V} = 0.17$ and $\tau_{R} = 0.08$, where $\tau$ is the optical depth. Because these numbers are small the dust lane is not immediately apparent in the individual direct images. However, the dust does affect the luminosity profile, and we explore that aspect in the next section. The optical depth obtained requires a correction since the dust disk obscures only those stars which are behind it, and not the ones that are between the disk and the observer. It can be shown (Brosch et al. 1990) that if the fraction of stars obscured is $f$, the relation between the observed extinction $A_{\lambda}^{obs}$ and the optical depth $\tau_{\lambda}$ is $$A_{\lambda}^{obs}=-2.5\log[1+f(e^{-\tau_{\lambda}}-1)].$$ For $\tau_{\lambda}\ll1$, the extinction is given by $A_{\lambda}^{obs}\simeq1.09f\tau_{\lambda}$, which may be compared to the usual relation $A_{\lambda}\simeq1.09\tau_{\lambda}$ obtained by assuming that all the stars are obscured. The value of $f$ depends on the geometry and orientation of the disk. When the disk is inclined to the line of sight, as in the present case, it is expected that the near side of the disk is darker than the far side [^3], and this could be used to fix the inclination when better data is available. Following Burstein & Heiles (1978), the column density of neutral hydrogen and the total neutral hydrogen content in the dust disk can be obtained from $$N(H) = 5.8\times10^{21} E(B-V){\rm atoms\ cm}^{-2}$$ and $$N^{H}_{tot}= D^2 \int (dust) N(H) d\Omega.$$ Using $E(B-V)=0.06$ and $D=14.7 Mpc$ we get $N^{H}_{tot} = 1.8 \times 10^{63}$, i.e. $M^{H}_{tot} = 1.7\times10^{6} M_{\odot}.$ $H$ here refers to the neutral hydrogen. Assuming that the dust to gas mass ratio is $\sim10^{-2}$ as in our galaxy, we get the dust mass $M_d = 1.5\times10^4 M_{ \odot }$, which is a lower limit since a faint disk can extend beyond the confines presently detected. The factor $f$ discussed above appears in the denominator of the integral defining $N^{H}_{tot}$. Therefore, since $f<1$, the actual neutral hydrogen content is larger than the estimate. Effect of Dust on the Luminosity Profile ======================================== It is known that de Vaucouleurs’ $r^{1/4}$ law describes very well the observed brightness distribution for elliptical galaxies within $0.1r_e \le a \le 1.5r_e$ (e.g. Burkert 1993), where $a$ and $r_e$ are the semi-major axis distance and effective radius respectively. The lower limit is mainly due to seeing effects. Following an iterative fitting procedure, Burkert finds that for NGC 4261, $r_e=34.5''$, so that de Vaucouleurs’ law should hold at least upto $\sim3''.5$. This procedure neglects the presence of the dust, which is expected to lead to departures from the law because of the absorption well above $0.1r_e$. The distribution of the surface brightness of NGC 4261 as a function of $r^{1/4}$, where $a$ is in arcsecond is shown in Fig. 1. It is clear from the linear part of the curve for $a > 11''$ that de Vaucouleurs’ law provides an excellent fit in this region. The departure from a straight line towards the centre is due to seeing, absorption and any real deprtures from the law at small radii. We have fit de Vaucouleurs’ law to the $V$ and $R$ profiles after excluding the inner $11''$. For the fit, a model galaxy was generated with assumed values of the effective radius $r_e$ and central surface brightness and using the observed distribution of ellipticity. The model was convolved with the point spread function (psf) obtained from stars in the frame, and the major axis profile was generated and compared with the observed profile. Parameters of the model were determined using the method of least squares. The effective radius obtained in this manner is $35''.8$. The best fit profile together with the observed surface brightness obtained from the ellipse fits described in Section 2 is shown in Figure 4, and the agreement between the two in the region used in the fit is seen to be excellent. If all points of the observed profile are used in the fit, the effective radius $r_e$ obtained is $42''.6$, which is in agreement with the value obtained by Peletier et al. (1990), but is an overestimate because of the neglect of absorption. The extrapolation of the best fit profile to the region of the dust disk lies above the observed points. If this is attributed to the absorption due to dust, which is certainly valid for $a \ge 3''.5$, the difference $\Delta V$ in magnitude between the fit and the observed surface brightness directly provides a measure of the extinction, with optical depth $\tau_V=\Delta V/1.086$. The optical depth obtained in this manner for the $V$ as well as the $R$ filter is shown in Figure 5. There are two points to be noted here : (1) The optical depth is independent of any assumptions regarding the properties of the dust, and therefore can be used to examine the validity of the assumptions made in Section 3 in estimating optical depth from the colour excess . (2) The observed surface brightness used in the determination of the extinction is obtained as a result of averaging over the best fit ellipses, which improves the signal-to-noise ratio and provides a smooth representation of the dependence of the optical depth on the semi-major axis length. For $8'' \le a \le 11''$ some points along the fitted ellipses lie outside the region covered by dust, and extinction obtained from the mean intensity as described above could be an underestimate. To check the magnitude of this effect, we have taken intensity cuts through the centre which extended only through the region covered by dust. These were then averaged and an intensity profile obtained. We found that this profile differed at most by 0.01 magnitude from the profile obtained using the fitted ellipses. We therefore use the latter over the entire dust region. The mean value of the optical depth over the dust region obtained using the above method separately for the two filters is $\tau_{v}=0.19\pm0.01$ and $\tau_{r}=0.10\pm0.01$, with the indicated error being primarily due to the uncertainty in deciding the extent of the dust obscured region. The optical depth here agrees within errors with the value in Section 3, which justifies the assumption made there that the dust in the disk is similar to the dust in our galaxy. It is possible to look for departures from de Vaucouleurs’ law in the image rather than in the radial surface brightness profiles. For this purpose, using the best fit de Vaucouleurs’ profile outside the putative dust region in the V filter, we have generated a 2-D model, extrapolated it to the region covered by the dust and convolved the whole with the point spread function. This provides a dust free representation $V_{df}$ of the galaxy. The residual image $V-V_{df}$ shows features similar to $V-R_{smooth}$, with some variations which arise because of the differences in model generation and smoothing used in the two cases. The similarity in the residual and colour images makes it reasonable to assume that de Vaucouleurs’ law modified by dust absorption provides a reasonable description of the inner region of NGC 4261. It is expected that the optical depth in $V$ is higher than that in $R$. However in Figure 5 it is seen that $\tau_{V}\simeq\tau_{R}$ for $a\ge4''$ and for $a<4''$ there is decrease in optical depth $\tau_{R}$ while $\tau_{V}$ continues to rise. Literally taken, this would mean increased red light towards the center. Some ellipticals are known to have red nuclei (e.g. Sparks et al. 1985) and red cores (e.g. Carter et al. 1983), but good spectroscopic data and observations in the $B$ band under better seeing conditions will be necessary to confirm these trends here, as well as the similarity of the dust in NGC 4261 and in our galaxy. Dust in the two galaxies could have significantly different properties because of different origin and environment. The active galactic nucleus in the radio galaxy could also have some effect on the dust (Begelman 1985; Shanbhag & Kembhavi 1988). The results at small radii would of course be affected by any colour dependent departures from de Vaucouleurs’ law. Dust and Isophote Boxiness ========================== The dust lane reduces the surface brightness at every point along its extent, which results in the isophotes appearing to be pulled inwards. When the dust lane is oriented along the major axis, this produces the appearance of boxiness, with the coefficient $b4$ of the $\cos(4\phi)$ term in the angular dependence of the differential intensity along the best fit ellipses becoming negative. To examine the magnitude of this effect, we have generated an elliptical galaxy with the central surface brightness and effective radius obtained for the best fit model in Section 3, using tasks from IRAF that allow the introduction of Poisson shot noise and characteristics of the CCD. After convolving the model with the observed psf, we have placed in it a dust lane which has the extent and optical depth observed in NGC 4261 in the $V$ band. We have then fitted ellipses to the isophotes, and obtained $b4/a$, which is shown in Figure 6. The magnitude of $b4/a$ is seen to be similar to the observed value where dust is present, and the circularizing effect of the seeing is not dominant. However the boxiness is colour dependent, unlike in the observed case, and reduces to insignificant values as soon as the dust is left behind. Nevertheless it is clear that observed isophotal boxiness could have a contribution from dust and could in fact be used as a diagnostic of major axis dust lanes which are too faint to be seen directly. In the same manner, dust lanes oriented in directions other than the apparent major axis will produce changes in isophotal shape which can be traced in the various Fourier coefficients. Discussion ========== The shape and orientation of a dust disk in an elliptical galaxy is determined by the allowed trajectories of the dust particles, which depend on the shape and the overall rotation of the galaxy. NGC 4261 shows rotation around the apparent major axis (Davies & Birkinshaw 1986) and therefore it cannot be an oblate spheroid (Binney 1985). Davies and Birkinshaw have argued from the kinematical data and modelling that the galaxy is prolate or triaxial and nearly prolate. In such a configuration, orbits in the equatorial plane (which is normal to the long axis) are stable, but these are not appropriate to describe the dust disk in NGC 4261, because the dust lane is observed to be aligned close to the major axis. However there is another class of orbits which is stable (see Kormendy & Djorgovoski (1989) and de Zeeuw & Franx (1991) for a review of the possibilities and references) and has different orientations at different radii which may be more appropriate to the present case. At small radii these orbits are polar, i.e. they lie in a plane containing the long and short axes; at larger radii the orbits are equatorial and skew at intermediate radii. The small scale absorption disk discovered using the Hubble Space Telescope (Jaffe et al. 1993) is inclined at an angle of $\sim64^{\circ}$ to the plane of the sky, while the inclination of the larger disk is $\sim75^{\circ}$. The two disks are to be viewed as the warped parts of a single disk, with the warping being due to the complex nature of the orbits, and the possible change in the shape of the galaxy from prolate to oblate towards the inner region, which will change the plane of stable orbits. The relative orientation of the different parts of the disk are also dependent on the extent to which the parts have settled in their final orbits, and projection effects which again depend on the shape of the galaxy and the direction of the line of sight. Higher signal-to-noise observations under better seeing conditions as well as spectroscopic data will be able to provide further information about the extent and shape of the dust disk which can be used in modelling the galaxy. The radio jet in NGC 4261 is oriented almost normal to the dust disk, which hints at a possible connection between the two. The direction of the angular momentum of the disk changes somewhat from region to region as the disk is warped. It would be reasonable for the direction of the base of the jet to be determined by the innermost parts of the disk, and precession of the disk would lead to a change in the direction of the jet. However the large scale jet may overall appear to be normal to the large scale disk. Based on the statistical analysis of radio jet directions (e.g. Palimaka et al. 1979; Kapahi & Saikia 1982), it has been argued that the jets are oriented along the apparent minor axis of ellipticals. When the galaxy is an oblate spheroid, the apparent minor axis always coincides with the projection of the short axis. The observation then means that jets are preferentially emitted close to the short axis. This coincidence could be related to the fact that in non-rotating or slowly rotating oblate spheroids the equatorial plane has stable orbits, so that the accretion disk is situated in this plane or close to it. When the galaxy is prolate, triaxial or rotating, the situation is more complex, and as mentioned above a variety of stable orbits is possible. If the jet direction is again taken to be determined by a disk, the relation between the jet direction and the apparent minor axis will depend on the the shape of the galaxy, the initial conditions of dust formation and the viewing direction. One should also expect to find a disk at least in those ellipticals which are powerful radio galaxies with well collimated jets. As in the case of NGC 4261, these disks may be too faint to be observed directly, but it would not be difficult to spot them using the techniques we have discussed above. If ellipticals are the results of mergers or cannibalism, one may well find dust disks in all of them. Conclusions =========== The main conclusions of the present paper are the following : 1. [The radio galaxy NGC 4261 contains a dust lane with dimensions $\sim21\times6\rm\,arcsec^2$, oriented close to the apparent major axis of the galaxy. The dust lane can be interpreted as the projection of a dust disk with inclination angle $\sim75^{\circ}$ to the plane of the sky. The observed colour excess is $E(V - R) = 0.05$. Assuming dust properties similar to those in our galaxy, the optical depth obtained from $A_{\lambda}\simeq 1.09\tau_{\lambda}$ is $\tau_{V} = 0.17$ and $\tau_{R} = 0.08$.]{} The actual optical depth could be a factor $\sim$2 higher than this value. 2. [The optical depth can be estimated directly from departures of the surface brightness profile from de Vaucouleurs’ law, and has values close to those obtained from the colour excess, confirming that the dust is similar to that in our galaxy.]{} 3. [Absorption due to a major axis dust lane can produce boxiness which is colour dependent.]{} Acknowledgements ================ The authors wish to thank the staff of the Vainu Bappu Observatory, Kavalur for help with the observations, and Professor D. Lynden-Bell, Dr. G. C. Anupama, Dr. S. K. Pandey for many enlightening discussions. Anupama, G. C., Kembhavi, A. K., Elvis, M., & Edelson, R. 1994, A&AS, 103, 315 Begelman, M. C. 1985, ApJ, 279, 492 Binney, J. 1985, MNRAS, 212, 767 Birkinshaw, M. & Davies, R. L. 1985, ApJ, 291, 32 Brosch, K., Almoznino, E., Grosbol, P., & Greenberg, J. M. 1990, A&A, 233, 341 Burkert, A. 1993, A&A, 278, 23 Burstein, D. & Heiles, C. 1978, ApJ, 225, 40 Capetti, A., Macchetto, F., Sparks, W. B., & Miley, G. 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T. 1963, ApJ, 137, 153 Mőllenhoff, C. & Bender, R. 1987, A&A, 174, 63 Mőllenhoff, C., Hummel, E., & Bender, R. 1992, A&A, 255, 35 Palimaka, J. J., Bridle, A. H., Fomalont, E. B., & Brandie, G. W. 1979, ApJ, 231, L7 Peletier, R., Davies, R. L., Illingworth, G. D., Davis, L. E., & Cawson, M. 1990, AJ, 100(4), 1091 Savage, B. D. & Mathis, J. S. 1979, ARA&A, 17, 73 Schweizer, F. 1979, ApJ, 233, 23 Shanbhag, S., & Kembhavi, A. K. 1988, ApJ, 334, 34 Sparks, W. B., Wall, J. V., Thorne, D. J., Jorden, P. R., van Breda, I. G., Rudd, P. J., & Jorgensen, H. E. 1985, MNRAS, 217, 87 [**Figure Captions**]{} 0.1in [**Figure 1 :** ]{} Semi-major axis profiles in the $V$ band of surface brightness, ellipticity, position angle and the [*boxiness coefficient*]{} $b4/a$. The surface brightness is shown as a function of $a^{1/4}$, where $a$ is the semi-major axis length, of the isophotes. 0.15in [**Figure 2 :** ]{} A cut across the dust lane, averaged over 10 rows around the galaxy centre. The instrumental colour $v-r$ is shown. 0.15in [**Figure 3 :** ]{} A schematic representation of NGC 4261. The major axis lies within the narrow confines of the dust lane and the radio jet is almost perpendicular to it. 0.15in [**Figure 4 :** ]{} Observed brightness profile along the semi-major axis, and a de Vaucouleurs’ law fit made using points beyond the small vertical bar shown at $a=11''$. The fit is extrapolated to the region inward of the bar. Deviation of the fit from the observed points in the inner region is indicative of the extinction due to dust. 0.15in [**Figure 5 :** ]{} Optical depth $\tau_{V}$ and $\tau_{R}$ as function of $a$. The value at a given $a$ is the mean over the best fit ellipse with that semi-major axis. 0.15in [**Figure 6 :** ]{} The [*boxiness coefficient*]{} for a simulated elliptical galaxy generated using the parameters obtained for NGC 4261. 0.15in [**Plate Captions**]{} 0.1in [**Plate 1 :** ]{} $V-R$ image of NGC 4261 in false colour. Each pixel is $0''.56.$ 0.15in [**Plate 2 :** ]{} $V-R_{smooth}$ image of NGC 4261 in false colour. Each pixel is $0''.56.$ [^1]: IRAF is distributed by the National Optical Astronomy Observatories, which is operated by the Association of Universities, Inc. (AURA) under cooperative agreement with the National Science Foundation. [^2]: The Space Telescope Science Data Analysis System STSDAS is distributed by the Space Telescope Science Institute. [^3]: We wish to thank an anonymous referee for pointing this out to us.
--- abstract: | In this paper we establish existence and nonexistence results concerning fully nontrivial minimal energy solutions of the nonlinear Schrödinger system $$\begin{aligned} \begin{gathered} -\Delta u + ~~\, u = |u|^{2q-2}u + b|u|^{q-2}u|v|^q \quad\text{in }{\mathbb{R}}^n, \\ -\Delta v + \omega^2 v = |v|^{2q-2}v + b|u|^q|v|^{q-2}v\quad\text{in }{\mathbb{R}}^n. \end{gathered} \end{aligned}$$ We consider the repulsive case $b<0$ and assume that the exponent $q$ satisfies $1<q<\frac{n}{n-2}$ in case $n\geq 3$ and $1<q<\infty$ in case $n=1$ or $n=2$. For space dimensions $n\geq 2$ and arbitrary $b<0$ we prove the existence of fully nontrivial nonnegative solutions which converge to a solution of some optimal partition problem as $b\to -\infty$. In case $n=1$ we prove that minimal energy solutions exist provided the coupling parameter $b$ has small absolute value whereas fully nontrivial solutions do not exist if $1<q\leq 2$ and $b$ has large absolute value. address: 'R. Mandel Department of Mathematics, Karlsruhe Institute of Technology (KIT)D-76128 Karlsruhe, Germany' author: - Rainer Mandel bibliography: - 'pub.bib' date: '15.03.2013' title: Minimal energy solutions for repulsive nonlinear Schrödinger systems --- Introduction ============ In this paper we are interested in fully nontrivial minimal energy solutions of the system $$\begin{aligned} \label{Mbrad Gl DGL} \begin{gathered} -\Delta u + ~~\, u = |u|^{2q-2}u + b|u|^{q-2}u|v|^q \quad\text{in }{\mathbb{R}}^n, \\ -\Delta v + \omega^2 v = |v|^{2q-2}v + b|u|^q|v|^{q-2}v\quad\text{in }{\mathbb{R}}^n \end{gathered} \end{aligned}$$ for parameter values $\omega\geq 1$ and $b\leq 0$. We henceforth assume that the exponent $q$ satisfies $1<q<\frac{n}{n-2}$ when $n\geq 3$ and $1<q<\infty$ when $n=1$ or $n=2$. For applications in physics the special case $q=2$ and $n\in\{1,2,3\}$ is of particular importance. For example, in photonic crystals the system is used to describe the approximate shape of so-called band gap solitons which are special nontrivial solitary wave solutions $E(x,t)=e^{-i\kappa t}\phi(x)$ of the time-dependent nonlinear Schrödinger equation (or Gross-Pitaevski equation) $$i\partial_t E = -\Delta E + V(x) E - |E|^2 E \qquad\text{in }[0,\infty)\times {\mathbb{R}}^n.$$ For a detailed exposition on that matter we refer to [@DoUe_Coupled_mode_equations]. During the last ten years many authors contributed to a better unterstanding of such nonlinear Schrödinger systems and various interesting results concerning the existence of nontrivial solutions have been proved using Ljusternik-Schnirelman theory [@WeWe_Radial_solutions_and], constrained minimization methods [@AmbCol_Bound_and_Ground],[@DeFLop_Solitary_waves_for],[@MaMoPe_Positive_solutions_for],[@Sir_Least_energy_solutions],[@WeWe_Nonradial_symmetric_bound] or bifurcation theory [@BaDaWa_A_Liouville_Theorem]. In the case of a positive coupling parameter $b$ many existence results for positive solutions of have been proved by investigations of appropriate constrained minimization problems. For instance Maia, Montefusco, Pellacci [@MaMoPe_Positive_solutions_for] proved the existence of nonnegative ground states of which, by definition, are solutions of minimal energy among all nontrivial solutions. Here, the energy corresponds to the Euler functional $I$ associated to which is given by $$I(u,v) = \frac{1}{2} \big( \|u\|^2 + \|v\|_\omega^2 \big) - \frac{1}{2q} \big(\|u\|_{2q}^{2q} + \|v\|_{2q}^{2q} +2b\|uv\|_q^q \big)$$ where $\|\cdot\|_{2q},\|\cdot\|_q$ denote Lebesgue norms and $\|\cdot\|,\|\cdot\|_\omega$ denote Sobolev space norms that we will define in . Moreover the authors gave sufficient conditions and necessary conditions for ground states to be positive in both components which basically require the coupling parameter $b$ to be positive and sufficiently large. In the special case $q=2$ additional sufficient conditions for the existence of positive ground states have been proved in [@AmbCol_Bound_and_Ground],[@DeFLop_Solitary_waves_for]. Furthermore, for $q=2$ and small positive values of $b$ Lin, Wei [@LiWe_Ground_states_of],[@LiWe_Erratum_ground_states] and Sirakov [@Sir_Least_energy_solutions] proved the existence of positive solutions which have minimal energy among all fully nontrivial solutions. From a technical point of view the approaches followed in [@MaMoPe_Positive_solutions_for] and [@Sir_Least_energy_solutions],[@LiWe_Ground_states_of],[@LiWe_Erratum_ground_states] differ in the following way. In [@MaMoPe_Positive_solutions_for] ground states are obtained by minimizing the Euler functional $I$ over the entire Nehari manifold $$\mathcal{N}_b = \Big\{ (u,v):\; u,v\in H^1({\mathbb{R}}^n),\;(u,v)\neq (0,0),\;\|u\|^2+\|v\|_\omega^2 = \|u\|_{2q}^{2q}+ \|v\|_{2q}^{2q}+2b\|uv\|_q^q\Big\}$$ whereas the positive solutions found in [@Sir_Least_energy_solutions],[@LiWe_Ground_states_of],[@LiWe_Erratum_ground_states] are minimizers of $I$ over the subset $\mathcal{M}_b$ of the Nehari manifold which is given by $$\begin{aligned} \mathcal{M}_b &= \Big\{ (u,v):\; u,v\in H^1({\mathbb{R}}^n),\; u,v\neq 0,\; \|u\|^2=\|u\|_{2q}^{2q}+b\|uv\|_q^q, \;\|v\|_\omega^2 = \|v\|_{2q}^{2q}+b\|uv\|_q^q\Big\}. \end{aligned}$$ When $b$ is negative, however, the analysis of these constrained minimization problems does not produce any fully nontrivial solutions. Indeed, for $b<0$ the minimizers of $I|_{\mathcal{N}_b}$ are given by the semitrivial solutions $(\pm u_0,0)$ or $(0,\pm u_0)$ (the latter being possible only for $\omega=1$) where $u_0$ is the unique positive function satisfying $-\Delta u_0 + u_0 = u_0^{2q-1}$ in ${\mathbb{R}}^n$, cf. [@MaMoPe_Positive_solutions_for],[@Kwo_Uniqueness_of_positive]. Furthermore it is known that $I|_{\mathcal{M}_b}$ does not admit minimizers in case $b<0$, cf. Theorem 1 in [@LiWe_Ground_states_of]. Therefore the case of negative coupling parameters $b<0$ has to be treated differently. In [@Sir_Least_energy_solutions] Sirakov considered the minimization problem $$\label{Mbrad Gl Defn kappabrad} \kappa_b^* := \inf_{\mathcal{M}_b^*} I\quad\text{where } \mathcal{M}_b^* = \big\{ (u,v)\in \mathcal{M}_b:\; u,v\text{ are radially symmetric}\big\}$$ and he proved the existence of a minimizer of $I|_{\mathcal{M}_b^*}$ for parameter values $q=2$ and $n\in\{2,3\}$, cf. Theorem 2 (i). Let us note that the indispensable condition $n\geq 2$ is missing in the statement of that theorem. The aim of this paper is to generalize Sirakov’s result to all space dimensions and to the full range of superlinear and subcritical exponents. In Theorem \[Mbrad Thm Grenzproblem Mbrad\] we first investigate the case $n\geq 2$. We show that minimizers $(u_b,v_b)$ of the functional $I|_{\mathcal{M}_b^*}$ exist and that, at least up to a subsequence, these minimizers of $I|_{\mathcal{M}_b^*}$ converge to a function $(\bar u,\bar v)$ with $\bar u\bar v=0$ and $$\label{Gl DGL bei -unendlich} -\Delta \bar u + \bar u = \bar u^{2q-1} \quad\text{in }\{\bar u\neq 0\},\qquad -\Delta \bar v + \omega^2 \bar v = \bar v^{2q-1} \quad\text{in }\{\bar v\neq 0\}$$ that solves the optimal partition problem $$\label{Mbrad Gl Defn kappa-inftyrad} \kappa_{-\infty}^* := \inf \Big\{ \frac{1}{2} \big( \|u\|^2 + \|v\|_\omega^2 \big) - \frac{1}{2q} \big(\|u\|_{2q}^{2q} + \|v\|_{2q}^{2q} \big) \; : \; (u,v)\in \mathcal{M}_{-\infty}^*\Big\}$$ where the set $\mathcal{M}_{-\infty}^*$ is defined by $$\label{Mbrad Gl Defn M-infty} \mathcal{M}_{-\infty}^* = \Big\{ (u,v):\; u,v\in H^1_r({\mathbb{R}}^n),\; u,v\neq 0,\; uv\equiv 0,\;\|u\|^2=\|u\|_{2q}^{2q}, \;\|v\|_\omega^2 = \|v\|_{2q}^{2q}\Big\}. $$ Here, $H^1_r({\mathbb{R}}^n)$ denotes the space of radially symmetric functions lying in $H^1({\mathbb{R}}^n)$. In particular, we find that the supports of $u_b,v_b$ separate as $b\to -\infty$. In general bounded domains $\Omega\subset{\mathbb{R}}^n$ these phenomena have been extensively studied in [@CoFe_Minimal_coexistence],[@CoFe_Global_minimizers_of],[@CoTeVe_A_variational_problem],[@TaTe_Sign-changin_solutions_of] and our Theorem \[Mbrad Thm Grenzproblem Mbrad\] can be considered as one kind of extension of their results. In case $n=1$, however, the situation turns out to be different. Since the embedding $H^1_r({\mathbb{R}}^n)\to L^{2q}({\mathbb{R}}^n)$ is not compact for $n=1$ the existence of minimizers of $I|_{\mathcal{M}_b^*}$ cannot be proved the same way as in the case $n\geq 2$. Therefore we approximate the original problem by the corresponding problem on intervals $B_R=(-R,R)$ for large $R>0$. In Theorem \[Mbrad Thm Kompaktheit n=1\] we show that for negative coupling parameters $b$ with small absolute value the corresponding minimizers converge to a minimizer of $I|_{\mathcal{M}_b^*}$ as $R\to\infty$. For negative $b$ with large absolute value, however, we prove in Theorem \[Mbrad Thm Nichtexistenz n=1 q leq 2\] that solutions of do not exist at least for exponents $1<q\leq 2$. Let us present the main results of this paper. The first one deals with the case $n\geq 2$. \[Mbrad Thm Grenzproblem Mbrad\] Let $n\geq 2,\omega\geq 1$. Then the following holds: - The value $\kappa_{-\infty}^*$ is attained at a nonnegative fully nontrivial solution of . - For $b\leq 0$ the value $\kappa_b^*$ is attained at a nonnegative fully nontrivial solution of . - As $b\to -\infty$ we have $\kappa_b^* \to \kappa_{-\infty}^*$ and every sequence of minimizers of ${I|_{\mathcal{M}_b^*}}$ has a subsequence $(u_b,v_b)$ such that $|b|^{1/q}u_bv_b\to 0$ in $L^q({\mathbb{R}}^n)$ and $(u_b,v_b)\to (\bar u,\bar v)$ where the latter function is a fully nontrivial solution of with $\bar u\bar v=0$. Since the proof of Theorem \[Mbrad Thm Grenzproblem Mbrad\] makes extensive use of the fact that $H^1_r({\mathbb{R}}^n)$ embeds compactly into $L^{2q}({\mathbb{R}}^n)$ when $n\geq 2$ one has to resort to different methods when the space dimension is one. In Theorem \[Mbrad Thm Nichtexistenz n=1 q leq 2\] we show that there is a threshold value $b^*(\omega,q)\in [-\infty,0)$ such that $\kappa_b^*$ is attained whenever $0\geq b>b^*(\omega,q)$ whereas $\kappa_b^*$ is not attained for $b<b^*(\omega,q)$. Moreover we find that $b^*(\omega,q)$ has the variational characterization $$\label{Gl Def b*(omega,q)} b^*(\omega,q) = \inf \max_{\alpha>0} \frac{(2+\omega^{\frac{q+1}{q-1}})^{1-q} \|u_0\|^{-2q}\|u_0\|_{2q}^{2q} (\|u\|^2+\alpha^2\|v\|_\omega^2)^q- \|u\|_{2q}^{2q}-\alpha^{2q}\|v\|_{2q}^{2q}}{2\alpha^q \|uv\|_q^q}$$ where the infimum is taken over all $u,v\in H^1_r({\mathbb{R}})$ with $uv\neq 0$. As above the function $u_0$ appearing in denotes the positive solution of $-\Delta u+u=u^{2q-1}$ in ${\mathbb{R}}^n$. Our first result dealing with the case $n=1$ reads as follows. \[Mbrad Thm Kompaktheit n=1\] Let $n=1,\omega\geq 1$. Then the following holds: - We have $\kappa_{-\infty}^* = (2+\omega^{\frac{q+1}{q-1}})I(u_0,0)$ and $\kappa_{-\infty}^*$ is not attained at any element of $\mathcal{M}_{-\infty}^*$. - If $b<b^*(\omega,q)$ then $\kappa_b^*=\kappa_{-\infty}^*$ and $\kappa_b^*$ is not attained at any element of $\mathcal{M}_b^*$. - If $0\geq b>b^*(\omega,q)$ then $\kappa_b^*<\kappa_{-\infty}^*$ and $\kappa_b^*$ is attained at a nonnegative fully nontrivial solution of . In view of part (iii) we may prove an explicit sufficient condition for the existence of a fully nontrivial solution of by estimating the value $b^*(\omega,q)$ from above. To this end we use $(u,v)=(u_0,u_0(\omega\cdot))$ as a test function in which leads to the following result. \[Mbrad Kor hinreichendes Krit Mbrad n=1\] Let $n=1,\omega\geq 1$. Then for all $b$ satisfying $$\begin{aligned} \label{Mbrad Gl hinreichendes Krit Mbrad n=1} 0\geq b > \max_{\alpha>0} \frac{(2+\omega^{\frac{q+1}{q-1}})^{1-q} (1+\alpha^2\omega)^q- 1-\alpha^{2q}\omega^{-1}}{2\alpha^q \omega^{-1/2}} \end{aligned}$$ the value $\kappa_b^*$ is attained at a nonnegative fully nontrivial solution of . In particular this is true in case $$\text{(i)}\quad q=2,\; b> -\frac{1}{\omega^{3/2}+\sqrt{2(1+\omega^3)}} \qquad\text{or}\qquad \text{(ii)}\quad \omega=1,\; b> (\frac{2}{3})^{q-1}-1.$$ In order to find necessary conditions for the existence of a minimizer one has to estimate the value $b^*(\omega,q)$ from below. For exponents $1<q\leq 2$ we may combine Theorem \[Mbrad Thm Kompaktheit n=1\] (iii) with the following nonexistence result to see that $b^*(\omega,q)$ must be larger than or equal to the right hand side in . \[Mbrad Thm Nichtexistenz n=1 q leq 2\] Let $n=1, 1<q\leq 2$ and assume $$\begin{aligned} \label{Gl Nichtexistenzkriterium n=1} b< \min_{z>0} \frac{(\omega^2-(q-1)\omega) z^{2q} - q z^2 - q\omega^3 z^{2q-2} - (\omega^2(q-1)-\omega)}{qz^{q+2}+(q-2)(\omega^2+\omega)z^q+q\omega^3 z^{q-2}}. \end{aligned}$$ Then the equation does not have any fully nontrivial solution. In particular this holds in case $q=2,\, b< -\frac{\omega^2+1}{2\omega}$ or $1<q\leq 2,\omega=1,\, b<-1$. - It is worth noticing that Theorem \[Mbrad Thm Nichtexistenz n=1 q leq 2\] not only applies to solutions of minimal energy but to all finite energy solutions. - It would be desirable to know whether a similar nonexistence result is true for exponents larger than 2. - From the strong minimum principle for nonnegative supersolutions of elliptic PDEs we know that the solutions $(u,v)$ of found in Theorem \[Mbrad Thm Grenzproblem Mbrad\] and Theorem \[Mbrad Thm Kompaktheit n=1\] satisfy $u>0$ and $v>0$ when $q\geq 2$. For $1<q<2$ we may apply the minimum principle to the function $u+v$ to conclude that $u+v$ is positive. It seems to be unclear, however, if both $u$ and $v$ are positive functions in that case. Finally let us illustrate our main results with two qualitative graphs of the map $b\mapsto \kappa_b^*$ in the cases $n\geq 2$ and $n=1,1<q\leq 2$. The monotonicity of this function is referred to at the end of the first step in the proof of Theorem \[Mbrad Thm Grenzproblem Mbrad\].     Notations and conventions ========================= In the following we always assume $n\in{\mathbb{N}}$ and $1<q<\frac{n}{n-2}$ whenever $n\geq 3$ and $1<q<\infty$ whenever $n=1$ or $n=2$ so that the Sobolev embedding $H^1_r({\mathbb{R}}^n)\to L^{2q}({\mathbb{R}}^n)$ exists and is compact in case $n\geq 2$. A function $(u,v)$ is called nontrivial if $u\neq 0$ or $v\neq 0$ and it is called fully nontrivial in case $u\neq 0$ and $v\neq 0$. The same way $(u,v)$ is nonnegative whenever $u\geq 0,v\geq 0$ and it is positive in case $u>0,v>0$. We always consider weak radially symmetric solutions of and where it is clear that all solutions of are twice continuously differentiable on ${\mathbb{R}}^n$ and smooth in the interior of each nodal domain. We use the symbols $\|\cdot\|_r = \|\cdot\|_{L^r({\mathbb{R}}^n)}$ to denote the standard Lebesgue norms for $1\leq r\leq \infty$ and we set $\|(u,v)\| := \sqrt{\|u\|^2+\|v\|_\omega^2}$ for $u,v\in H^1_r({\mathbb{R}}^n)$ where $$\begin{gathered} \label{Gl Defn norms} \|u\| := \Big( \int_{{\mathbb{R}}^n} |\nabla u|^2+u^2{\,dx}\Big)^{1/2},\quad \|v\|_\omega := \Big( \int_{{\mathbb{R}}^n} |\nabla v|^2+\omega^2 v^2{\,dx}\Big)^{1/2}. \end{gathered}$$ From the definition of $I$ we get $$\label{Gl für (u,v) in Nehari} I(u,v) = \frac{q-1}{2q} (\|u\|^2+\|v\|_\omega^2) \qquad\text{for all }(u,v)\in\mathcal{N}_b$$ and in particular for all elements of $\mathcal{M}_b$ or $\mathcal{M}_b^*$. The function $u_0\in H^1_r({\mathbb{R}}^n)$ is defined as above and for notational convenience we put $c_0:=I(u_0,0)$. We set $v_0:= \omega^{1/(q-1)}u_0(\omega\cdot)$ so that $v_0$ is the unique positive solution of $-\Delta v + \omega^2 v = v^{2q-1}$ in ${\mathbb{R}}^n$. A short calculation shows $$I(0,v_0)=\omega^{\frac{2q-n(q-1)}{q-1}} c_0.$$ We will use the facts that the functions $u_0,v_0$ are minimizers of the functionals $\frac{\|u\|_{~~}}{\|u\|_{2q}}, \frac{\|v\|_{\omega~}}{\|v\|_{2q}}$, respectively and that all minimizers of these functionals are translates of $u_0,v_0$. Moreover, we use that $(u_0,0)$ is a minimizer of the functional $I|_{\mathcal{N}_b}$ when $b<0$. Proof of Theorem \[Mbrad Thm Grenzproblem Mbrad\] ================================================= Throughout this section except for the first step we assume $n\geq 2$ according to the assumptions of Theorem \[Mbrad Thm Grenzproblem Mbrad\]. Its proof is given in four steps. First we prove variational characterizations for the values $\kappa_b^*,\kappa_{-\infty}^*$ which turn out to be more convenient than the original ones given by and . In the second step we use these characterizations to prove that minimizers of the functionals $I|_{\mathcal{M}_b^*}$ and $I|_{\mathcal{M}_{-\infty}^*}$ exist. In the third step we show that minimizers satisfy the corresponding Euler-Lagrange equation or so that the assertions (i) and (ii) of the theorem are proved. Finally we show part (iii) of the theorem. [*Step 1: A more convenient variational characterization for $\kappa_b^*,\kappa_{-\infty}^*$ ($n\geq 1$)*]{} For $s,t>0$ and $u,v\in H^1_r({\mathbb{R}}^n)$ with $u,v\neq 0$ one can check that $(su,tv)\in\mathcal{M}_b^*$ is equivalent to $(s,t)$ being a critical point of the function $\beta_{u,v}$ defined on ${\mathbb{R}}_{>0}\times{\mathbb{R}}_{>0}$ and given by $$\beta_{u,v}(\tilde s,\tilde t) := I(\tilde su,\tilde tv) = \frac{\tilde s^2}{2}\|u\|^2 +\frac{\tilde t}{2}\|v\|_\omega^2 - \frac{\tilde s^{2q}}{2q} \|u\|_{2q}^{2q} - \frac{\tilde t^{2q}}{2q} \|v\|_{2q}^{2q} - \frac{b\tilde s^q\tilde t^q}{q} \|uv\|_q^q.$$ A necessary and sufficient condition for such a critical point to exist is given by $$\label{Mbrad Gl Skalierungsbedingung} \|u\|_{2q}^q\|v\|_{2q}^q + b\|uv\|_q^q >0.$$ Indeed, in this case the functional $-\beta_{u,v}$ is coercive so that $\beta_{u,v}$ has a global maximum. Moreover one can show that the Hessian of the function $(\tilde s, \tilde t)\mapsto \beta_{u,v}(\tilde s^{1/2q},\tilde t^{1/2q})$ is positive definite on ${\mathbb{R}}_{>0}\times{\mathbb{R}}_{>0}$ so that the maximum is strict and no other critical point can exist. On the other hand a short calculation shows that is also a necessary condition for the existence of a critical point. From $$\begin{aligned} \max_{s,t>0} \beta_{u,v}(s,t) &= \max_{s,t>0} I(su,tv) \\ &= \max_{\alpha>0} \max_{s>0} I(su,s\alpha v) \\ &= \max_{\alpha>0} \max_{s>0} \frac{s^2}{2}(\|u\|^2 + \alpha^2 \|v\|_\omega^2) - \frac{s^{2q}}{2q} (\|u\|_{2q}^{2q} + \alpha^{2q} \|v\|_{2q}^{2q} +2b\alpha^q \|uv\|_q^q ).\\ &= \frac{q-1}{2q} \Big( \max_{\alpha>0}\frac{(\|u\|^2+\alpha^2\|v\|_\omega^2)^q }{\|u\|_{2q}^{2q}+\alpha^{2q}\|v\|_{2q}^{2q} + 2b\alpha^q \|uv\|_q^q} \Big)^{\frac{1}{q-1}} \end{aligned}$$ we obtain the following variational characterization for $\kappa_b^*$: $$\begin{aligned} \label{Mbrad Gl Charakerisierung kappabrad} \begin{aligned} &\kappa_b^* = \inf \Big\{ \frac{q-1}{2q} \hat J(u,v)^{\frac{1}{q-1}} : u,v\in H^1_r({\mathbb{R}}^n),\, (u,v) \text{ satisfies }\eqref{Mbrad Gl Skalierungsbedingung}\Big\} \\ &\text{where}\quad \hat J(u,v) = \max_{\alpha>0}\frac{(\|u\|^2+\alpha^2\|v\|_\omega^2)^q }{\|u\|_{2q}^{2q}+\alpha^{2q}\|v\|_{2q}^{2q} + 2b\alpha^q \|uv\|_q^q}. \end{aligned} \end{aligned}$$ Moreover if $(u,v)$ satisfies and minimizes $\hat J$ then $(su,tv)$ is a minimizer of $I|_{\mathcal{M}_b^*}$ provided $(s,t)$ is the unique maximizer of $\beta_{u,v}$. Similarly, one can show $$\begin{aligned} \label{Gl Grenzproblem bei -unendlich} \begin{aligned} &\kappa_{-\infty}^* = \inf \Big\{ \frac{q-1}{2q} \bar J(u,v)^{\frac{1}{q-1}} : u,v\in H^1_r({\mathbb{R}}^n), u,v\neq 0, uv= 0 \Big\} \\ &\text{where}\quad \bar J(u,v) = \max_{\alpha>0} \frac{(\|u\|^2+\alpha^2\|v\|_\omega^2)^q}{\|u\|_{2q}^{2q}+\alpha^{2q}\|v\|_{2q}^{2q}} = \Big(\big(\frac{\|u\|_{~~}}{\|u\|_{2q}} \big)^{\frac{2q}{q-1}} + \big(\frac{\|v\|_{\omega~}}{\|v\|_{2q}} \big)^{\frac{2q}{q-1}} \Big)^{q-1}. \end{aligned} \end{aligned}$$ Since the constraint $uv= 0$ is more restrictive than we obtain the inequality $$\label{Gl Ungleichung kappabrad kappa-unendlichrad} \kappa_b^* \leq \kappa_{-\infty}^* \qquad( b\leq 0).$$ Moreover, from it follows that the map $b\mapsto \kappa_b^*$ is nonincreasing. [*Step 2: Existence of nonnegative minimizers*]{} We prove that both $\kappa_b^*$ and $\kappa_{-\infty}^*$ are attained at nonnegative elements of $\mathcal{M}_b^*,\mathcal{M}_{-\infty}^*$, respectively. By the first step it suffices to show that the functionals $\hat J,\bar J$ defined in , admit fully nontrivial nonnegative minimizers. Since the reasonings for $\hat J$ and $\bar J$ are almost identical, we only give the proof for $\hat J$. Let $(u_j,v_j)$ be a minimizing sequence for $\hat J$ satisfying . Since $\hat J(u_j,v_j)=\hat J(s|u_j|,t|v_j|)$ for all $s,t>0$ we may assume ${u_j,v_j\geq 0}$ as well as $\|u_j\|_{2q}=\|v_j\|_{2q}=1$. Then $(u_j,v_j)$ is bounded and there is a subsequence $(u_j,v_j)$ that, due to the compactness of the embedding $H^1_r({\mathbb{R}}^n)\to L^{2q}({\mathbb{R}}^n)$, converges weakly, almost everywhere and in $L^{2q}({\mathbb{R}}^n)\times L^{2q}({\mathbb{R}}^n)$ to some nonnegative function $(u,v)$. This entails ${\|u\|_{2q}=\|v\|_{2q}=1}$ as well as $ u, v\geq 0$. Furthermore, $(u,v)$ satisfies because otherwise $\hat J(u_j,v_j)$ would tend to infinity as $j\to\infty$ contradicting its property of a minimizing sequence. Hence, for all $\alpha>0$ we have $$\begin{aligned} \frac{(\|u\|^2+\alpha^2\|v\|_\omega^2)^q }{\|u\|_{2q}^{2q}+\alpha^{2q}\|v\|_{2q}^{2q} + 2b\alpha^q \|uv\|_q^q} &\leq \liminf_{j\to\infty} \frac{(\|u_j\|^2+\alpha^2\|v_j\|_\omega^2)^q }{\|u_j\|_{2q}^{2q}+\alpha^{2q}\|v_j\|_{2q}^{2q} + 2b\alpha^q \|u_jv_j\|_q^q} \\ &\leq \liminf_{j\to\infty} \max_{\beta>0} \frac{(\|u_j\|^2+\beta^2\|v_j\|_\omega^2)^q }{\|u_j\|_{2q}^{2q}+\beta^{2q}\|v_j\|_{2q}^{2q} + 2b\beta^q \|u_jv_j\|_q^q}. \end{aligned}$$ Using we find $\hat J(u,v)\leq \liminf_{j\to\infty} \hat J(u_j,v_j)$ so that $(u,v)$ is a minimizer of $\hat J$. [*Step 3: The solution property*]{} We prove the following two statements: - In case $b\leq 0$ every minimizer of $I|_{\mathcal{M}_b^*}$ is a solution of . - Every minimizer of $I|_{\mathcal{M}_{-\infty}^*}$ is a solution of . Let us show assertion (i) first. For $u,v\in H^1_r({\mathbb{R}}^n)$ with $u,v\neq 0$ set $$H_1(u,v):= \|u\|^2- \|u\|_{2q}^{2q}-b\|uv\|_q^q,\quad\text{and}\quad H_2(u,v) := \|v\|_\omega^2-\|v\|_{2q}^{2q}-b\|uv\|_q^q.$$ so that $(u,v)\in \mathcal{M}_b^*$ if and only if $H_1(u,v)=H_2(u,v)=0$. Now, if $(u,v)\in \mathcal{M}_b^*$ is a minimizer of $I|_{\mathcal{M}_b^*}$ then $H_1(u,v)=H_2(u,v)=0$ implies $$\begin{aligned} H_1'(u,v)[u,0] &= 2\|u\|^2- 2q\|u\|_{2q}^{2q}- qb\|uv\|_q^q = (2-2q)\|u\|^2 + bq\|uv\|_q^q < 0, \\ H_2'(u,v)[0,v] &= 2\|v\|_\omega^2-2q\|v\|_{2q}^{2q}-q b\|uv\|_q^q = (2-2q)\|v\|_\omega^2 + bq\|uv\|_q^q < 0 \end{aligned}$$ so that Lagrange’s multiplier rule shows that there are $L_1,L_2\in{\mathbb{R}}$ such that $$\label{Gl ELequation} I'(u,v)+L_1 H_1'(u,v)+L_2 H_2'(u,v)=0.$$ It suffices to show $L_1=L_2=0$. Using $(u,0),(0,v)$ as test functions in we find ${\langleI'(u,v),(u,0)\rangle}=H_1(u,v)=0$ and ${\langleI'(u,v),(0,v)\rangle}=H_2(u,v)=0$ and thus $$\begin{aligned} 0 &= \Big((2-2q)\|u\|_{2q}^{2q}+(2-q)b\|uv\|_q^q\Big) L_1 - qb\|uv\|_q^q L_2, \\ 0 &= \Big((2-2q)\|v\|_{2q}^{2q}+(2-q)b\|uv\|_q^q\Big) L_2 - qb\|uv\|_q^q L_1. \end{aligned}$$ Assume $(L_1,L_2)\neq (0,0)$. Then $\|uv\|_q^q>0$ and the determinant of this system vanishes. We therefore get $$\begin{aligned} 0 &= \Big((2-2q)\|u\|_{2q}^{2q}+(2-q)b\|uv\|_q^q\Big)\cdot \Big((2-2q)\|v\|_{2q}^{2q}+(2-q)b\|uv\|_q^q\Big) - (qb\|uv\|_q^q)^2 \\ &= 4(1-q) \Big((b\|uv\|_q^q)^2 - \frac{q-2}{2} b\|uv\|_q^q(\|u\|_{2q}^{2q}+\|v\|_{2q}^{2q}) - (q-1)\|u\|_{2q}^{2q}\|v\|_{2q}^{2q} \Big). \end{aligned}$$ Solving for $b\|uv\|_q^q<0$ gives $$\begin{aligned} 4b\|uv\|_q^q &= (q-2)(\|u\|_{2q}^{2q}+\|v\|_{2q}^{2q})-\sqrt{(q-2)^2(\|u\|_{2q}^{2q}+\|v\|_{2q}^{2q})^2+ 16(q-1)\|u\|_{2q}^{2q}\|v\|_{2q}^{2q}}. \end{aligned}$$ Let now $A,B>0$ be given by $\|u\|_{2q}^{2q} = A\cdot |b|\|uv\|_q^q,\|v\|_{2q}^{2q}=B\cdot |b|\|uv\|_q^q$. Then $$\label{Mbrad Gl AB} -4 = (q-2)(A+B) - \sqrt{(q-2)^2(A+B)^2+16(q-1)AB}.$$ where $A,B$ are larger than 1 because of $$\|u\|_{2q}^{2q}-|b|\|uv\|_q^q = \|u\|^2 > 0,\qquad \|v\|_{2q}^{2q}-|b|\|uv\|_q^q = \|v\|_\omega^2 > 0.$$ Solving for $B$ we obtain $$B = \frac{2+(q-2)A}{2(q-1)A-(q-2)}$$ so that $A>1$ implies $B<1$, a contradiction. Hence, the assumption was false, i.e. ${I'(u,v)=0}$. Now consider (ii). Let $(\bar u,\bar v)\in\mathcal{M}_{-\infty}^*$ be a minimizer of the functional $I|_{\mathcal{M}_{-\infty}^*}$. Due to the one-dimensional Sobolev embedding we may choose $\bar u,\bar v$ to be a continuous function on ${\mathbb{R}}^n{\setminus}\{0\}$ so that the sets $\{\bar u\neq 0\},\{\bar v\neq 0\}$ are open. According to the first step we have $\bar J(\bar u,\bar v)\leq \bar J(\bar u+\varphi,\bar v+\psi)$ for all test functions $\varphi,\psi$ with $\operatorname{supp}(\varphi)~\subset~\operatorname{supp}(u)$ and $\operatorname{supp}(\psi)\subset \operatorname{supp}(v)$. In view of the second formula for $\bar J$ in we find that $(\bar u,\bar v)$ solves . [$\Box$]{} The above reasoning shows that all critical points and not only minimizers of $I|_{\mathcal{M}_b^*}$ or $I|_{\mathcal{M}_{-\infty}^*}$ satisfy the corresponding Euler-Lagrange equation. [*Step 4: Convergence to a fully nontrivial solution of as $b\to -\infty$*]{} Let $(b_j)$ be a sequence such that $b_j\to -\infty$ and let $(u_j,v_j)\in\mathcal{M}_{b_j}^*$ be a sequence of nonnegative fully nontrivial solutions of given by the second step, in particular ${I(u_j,v_j)=\kappa_{b_j}^*}$. Then $(u_j,v_j)$ is bounded and there is a subsequence $(u_j,v_j)$ that, due to the compactness of the embedding $H^1_r({\mathbb{R}}^n)\to L^{2q}({\mathbb{R}}^n)$, converges weakly, almost everywhere and in ${L^{2q}({\mathbb{R}}^n)\times L^{2q}({\mathbb{R}}^n)}$ to some nonnegative function $(\bar u,\bar v)$. From Sobolev’s inequality we get $$\begin{aligned} \|u_j\|^2 &= \|u_j\|_{2q}^{2q}+b_j\|u_jv_j\|_q^q \leq \|u_j\|_{2q}^{2q} \leq C \|u_j\|^{2q}, \\ \|v_j\|_\omega^2 &= \|v_j\|_{2q}^{2q}+b_j\|u_jv_j\|_q^q \leq \|v_j\|_{2q}^{2q} \leq C \|v_j\|_\omega^{2q} \end{aligned}$$ and thus $\|u_j\|_{2q},\|v_j\|_{2q}\geq c>0$ where $c,C$ are positive numbers which do not depend on $j$. It follows $\|\bar u\|_{2q},\|\bar v\|_{2q}\geq c$ and thus $\bar u,\bar v\neq 0$. In addition we find $$\label{Gl II Grenzproblem Mbrad} \|\bar u\|^2\leq \|\bar u\|_{2q}^{2q},\qquad \|\bar v\|_\omega^2\leq \|\bar v\|_{2q}^{2q}.$$ Since the sequence $(u_j,v_j)$ is bounded we get $\bar u\bar v\equiv 0$ from $$\|\bar u\bar v\|_q^q = \lim_{j\to\infty} \|u_jv_j\|_q^q = \lim_{j\to\infty} (\|u_j\|_{2q}^{2q}- \|u_j\|^2)|b_j|^{-1} \leq \liminf_{j\to\infty} C\cdot |b_j|^{-1} = 0.$$ Furthermore, from we obtain $\kappa_{b_j}^*\leq \kappa_{-\infty}^*$ so that implies $$\begin{aligned} \frac{q-1}{2q}(\| \bar u\|^2+\|\bar v\|_\omega^2) &\leq \frac{q-1}{2q} \liminf_{j\to\infty} (\|u_j\|^2+\|v_j\|_\omega^2) \\ &= \liminf_{j\to\infty} \kappa_{b_j}^* \\ &\leq \limsup_{j\to\infty} \kappa_{b_j}^* \\ &\leq \kappa_{-\infty}^* \\ &\leq \frac{q-1}{2q} \Big( \big(\frac{\|\bar u\|_{~~}}{\| \bar u\|_{2q}}\big)^{\frac{2q}{q-1}} + \big(\frac{\|\bar v\|_\omega~}{\|\bar v\|_{2q}}\big)^{\frac{2q}{q-1}} \Big) \\ &\leq \frac{q-1}{2q} (\|\bar u\|^2 + \|\bar v\|_\omega^2) \end{aligned}$$ where we used and in the last two inequalities. Hence, equality occurs in each line and thus $\|\bar u\|^2= \|\bar u\|_{2q}^{2q}$, $\|\bar v\|_\omega^2= \|\bar v\|_{2q}^{2q}$ as well as $\kappa_{b_j}^* \to \kappa_{-\infty}^*$, $(u_j,v_j)\to (\bar u,\bar v)$ as $b\to -\infty$. This entails $(\bar u,\bar v)\in\mathcal{M}_{-\infty}^*$ and $I(\bar u,\bar v)=\kappa_{-\infty}^*$ so that $(\bar u,\bar v)$ is a minimizer of $I|_{\mathcal{M}_{-\infty}^*}$ and thus a fully nontrivial nonnegative solution of . Finally, the assertion follows from $$\limsup_{j\to\infty} |b_j|\|u_jv_j\|_q^q = \limsup_{j\to\infty} \|u_j\|_{2q}^{2q}-\|u_j\|^2 = \|\bar u\|_{2q}^{2q}-\|\bar u\|^2 = 0.$$ Proof of Theorem \[Mbrad Thm Kompaktheit n=1\] and Corollary \[Mbrad Kor hinreichendes Krit Mbrad n=1\] ======================================================================================================= [*Proof of (i)*]{} First we show $\kappa_{-\infty}^*\geq (2+\omega^{\frac{q+1}{q-1}})c_0$ and that no element of $\mathcal{M}_{-\infty}^*$ attains this value. Let $(u,v)\in\mathcal{M}_{-\infty}^*$ and in particular $u(0)v(0)=0$. We first assume $u(0)=0$. Then the nontrivial functions $u_l:= u\cdot 1_{(-\infty,0)},u_r:= u\cdot 1_{(0,\infty)}$ lie in $H^1({\mathbb{R}})$, they have disjoint support and satisfy $u_r(r)=u_l(-r)$ due to $u\in H^1_r({\mathbb{R}})$. In particular from $\|u\|^2=\|u\|_{2q}^{2q}$ we infer $$\|u_l\|^2 = \|u_r\|^2 = \frac{1}{2}\|u\|^2 = \frac{1}{2}\|u\|_{2q}^{2q} = \|u_l\|_{2q}^{2q} = \|u_r\|_{2q}^{2q}.$$ This implies $(u_r,0),(u_l,0),(0,v)\in\mathcal{N}_b$ and using as well as $uv\equiv 0$ we obtain $$\begin{aligned} I(u,v) &= I(u_l,0) + I(u_r,0) + I(0,v) \\ &= \frac{q-1}{2q}\cdot \big( \|u_l\|^2 + \|u_r\|^2 + \|v\|_\omega^2 \big) \\ &= \frac{q-1}{2q}\cdot \Big( \big( \frac{\|u_l\|_{~~}}{\|u_l\|_{2q}}\big)^{\frac{2q}{q-1}} + \big( \frac{\|u_r\|_{~~}}{\|u_r\|_{2q}}\big)^{\frac{2q}{q-1}} + \big( \frac{\|v\|_{\omega~}}{\|v\|_{2q}}\big)^{\frac{2q}{q-1}} \Big). \intertext{Since the functions $u_0,v_0$ minimize the quotiens $\frac{\|u\|_{~~}}{\|u\|_{2q}},\frac{\|v\|_{\omega~}}{\|v\|_{2q}}$ we get} I(u,v) &\geq \frac{q-1}{2q}\cdot \Big( 2\cdot \big( \frac{\|u_0\|_{~~}}{\|u_0\|_{2q}}\big)^{\frac{2q}{q-1}} + \big( \frac{\|v_0\|_{\omega~}}{\|v_0\|_{2q}}\big)^{\frac{2q}{q-1}} \Big) \\ &= \frac{q-1}{2q}\cdot \big(2\|u_0\|^2+\|v_0\|_\omega^2 \big) \\ &= 2I(u_0,0) + I(0,v_0) \\ &= (2+\omega^{\frac{q+1}{q-1}})c_0. \end{aligned}$$ Analogously the assumption $v(0)=0$ leads to $$I(u,v) \geq (1+2\omega^{\frac{q+1}{q-1}})c_0 \geq (2+\omega^{\frac{q+1}{q-1}})c_0.$$ We therefore get $\kappa_{-\infty}^* \geq (2+\omega^{\frac{q+1}{q-1}})c_0$. Moreover we find that $\kappa_{-\infty}^*$ is not attained at any element of $\mathcal{M}_{-\infty}^*$ because in case $u(0)=0$ this would lead to the conclusion that $u_r,u_l$ are translates of $u_0$ which is impossible because of $\operatorname{supp}(u_r)\cap \operatorname{supp}(u_l)=\emptyset$. A similar reasoning shows that no element $(u,v)$ of $\mathcal{M}_{-\infty}^*$ with $v(0)=0$ can have energy $(2+\omega^{\frac{q+1}{q-1}})c_0$. Now let us prove the opposite inequality. To this end let $\chi_k:=\chi(k^{-1}\cdot)$ denote a suitable radially symmetric cut-off function with $\chi\equiv 1$ in $[-1,1]$ and $\chi\equiv 0$ outside of $(-2,2)$ then the sequence $$(u_k,v_k):= \Big( (u_0\chi_k)(2k+\cdot)+(u_0\chi_k)(-2k+\cdot),v_0\chi_k \Big)$$ lies in $\mathcal{M}_{-\infty}^*$ and $$\lim_{k\to\infty} I(u_k,v_k) = \lim_{k\to\infty} \big( 2 I(u_0\chi_k,0) + I(0,v_0\chi_k) \big) = 2 I(u_0,0) + I(0,v_0) = (2+\omega^{\frac{q+1}{q-1}})c_0$$ which proves $\kappa_{-\infty}^* \leq (2+\omega^{\frac{q+1}{q-1}})c_0$. Hence, we obtain $$\kappa_{-\infty}^* = (2+\omega^{\frac{q+1}{q-1}})c_0.$$ [*Proof of (ii)*]{} First we prove that $b<b^*(\omega,q)$ implies $\kappa_b^*=\kappa_{-\infty}^*$ and that $0\geq b>b^*(\omega,q)$ implies ${\kappa_b^*<\kappa_{-\infty}^*}$. From (i) and the variational characterization for $\kappa_b^*$ given by we get $\kappa_b^*<\kappa_{-\infty}^*$ if and only if there are functions $u,v\in H^1_r({\mathbb{R}}^n)$ with $u,v\neq 0$ and $\|u\|_{2q}^q\|v\|_{2q}^q>|b|\|uv\|_q^q$ that satisfy $$\max_{\alpha>0} \frac{(\|u\|^2+\alpha^2\|v\|_\omega^2)^q}{\|u\|_{2q}^{2q}+\alpha^{2q}\|v\|_{2q}^{2q}+2b\alpha^q\|uv\|_q^q} < \Big(\frac{2q}{q-1}\cdot (2+\omega^{\frac{q+1}{q-1}})c_0\Big)^{q-1} = (2+\omega^{\frac{q+1}{q-1}})^{q-1} \frac{\|u_0\|^{2q}}{\|u_0\|_{2q}^{2q}}.$$ In this case (i) and implies $uv\neq 0$ and thus $b>b^*(\omega,q)$ after some rearrangements of the above inequality. A short argument shows that $0\geq b>b^*(\omega,q)$ implies $\kappa_b^*<\kappa_{-\infty}^*$. Let $b<b^*(\omega,q)$ and assume that $\kappa_b^*=\kappa_{-\infty}^*$ is attained at some function $(u,v)$ satisfying the condition . Then $uv\neq 0$ since $\kappa_{-\infty}^*$ is not attained, see (i). Choose ${\varepsilon}>0$ such that $b+{\varepsilon}<b^*(\omega,q)$ so that $ \kappa_{b+{\varepsilon}}^* = \kappa_{-\infty}^*$. Then $(u,v)$ satisfies for $b+{\varepsilon}$ instead of $b$ and we get $$\begin{aligned} \kappa_b^* &= \frac{q-1}{2q} \Big( \max_{\alpha>0} \frac{(\|u\|^2 + \alpha^2\|v\|_\omega^2)^q}{\|u\|_{2q}^{2q} + \alpha^{2q}\|v\|_{2q}^{2q} + 2b\alpha^q\|uv\|_q^q} \Big)^{\frac{1}{q-1}} \\ &> \frac{q-1}{2q} \Big( \max_{\alpha>0} \frac{(\|u\|^2 + \alpha^2\|v\|_\omega^2)^q}{\|u\|_{2q}^{2q} + \alpha^{2q}\|v\|_{2q}^{2q} + 2(b+{\varepsilon})\alpha^q\|uv\|_q^q} \Big)^{\frac{1}{q-1}} \\ &\geq \kappa_{b+{\varepsilon}}^* \\ &= \kappa_{-\infty}^* \end{aligned}$$ which contradicts $\kappa_b^*=\kappa_{-\infty}^*$. Hence, $\kappa_b^*$ is not attained for $b<b^*(\omega,q)$. [*Proof of (iii)*]{} In order to prove (iii) we suppose $0\geq b>b^*(\omega,q)$. From the first statement in the proof of (ii) it follows that this implies $$\label{Gl Ungl kappab kappa-infty} \kappa_b^*<\kappa_{-\infty}^*=(2+\omega^{\frac{q+1}{q-1}})c_0.$$ For these values of $b$ let us investigate the behaviour of a special minimizing sequence for the functional $I|_{\mathcal{M}_b^*}$. We consider the corresponding problem on balls $B_R=(-R,R)$ where $R$ will be sent to infinity. We set $H^1_{0,r}(B_R) := \{ u\in H_0^1(B_R) : u \text{ is radially symmetric} \}$. All solutions $(u,v)\in H^1_{0,r}(B_R)\times H^1_{0,r}(B_R)$ of the boundary value problem $$\begin{aligned} \label{Mbrad Gl DGL uR vR} \begin{gathered} - u'' + ~~~ u = |u|^{2q-2}u + b|u|^{q-2}u|v|^q\quad\text{in }B_R, \\ - v'' + \omega^2 v = |v|^{2q-2}v + b|v|^{q-2}v|u|^q\quad\text{in }B_R, \\ u(-R)=u(R) = 0,\quad v(-R)=v(R)=0 \end{gathered} \end{aligned}$$ satisfy $$\begin{aligned} \int_{B_R} |u'|^2+ ~~~u^2{\,dx}&= \int_{B_R} |u|^{2q}+b|u|^q|v|^q{\,dx}, \label{Mbrad Gl MbstarR Bedingung 1} \\ \int_{B_R} |v'|^2+\omega^2 v^2{\,dx}&= \int_{B_R} |v|^{2q}+b|u|^q|v|^q{\,dx}. \label{Mbrad Gl MbstarR Bedingung 2} \end{aligned}$$ Following the approach of the last section we define $$\begin{aligned} \mathcal{M}_b^*(R) &:= \Big\{ (u,v)\in H^1_{0,r}(B_R)\times H^1_{0,r}(B_R) : u,v\neq 0,\, (u,v) \text{ satisfies } \eqref{Mbrad Gl MbstarR Bedingung 1},\eqref{Mbrad Gl MbstarR Bedingung 2} \Big\}. \end{aligned}$$ As before one can show that $\inf I|_{\mathcal{M}_b^*(R)}$ admits a variational characterization given by $$\begin{aligned} \label{Mbrad Gl Charakterisierung kappabradR} \kappa_b^*(R) := \inf_{\mathcal{M}_b^*(R)} I = \inf \Big\{ \frac{q-1}{2q} \hat J(u,v)^{\frac{1}{q-1}} : u,v\in H^1_{0,r}(B_R),\,(u,v) \text{ satisfies }\eqref{Mbrad Gl Skalierungsbedingung}\Big\}. \end{aligned}$$ Using the compactness of the embedding $H^1_{0,r}(B_R)\to L^{2q}(B_R)$ for all $R>0$ we obtain the following result: \[Mbrad Prop Mbrad-Minimierung n=1 Approx durch Kugeln\] Let $n=1$. For all $b\leq 0$ the value $\kappa_b^*(R)$ is attained at a fully nontrivial nonnegative solution $(u_R,v_R)\in H_{0,r}^1(B_R)\times H_{0,r}^1(B_R)$ of . Moreover, as $R\to\infty$ we have $\kappa_b^*(R)\to \kappa_b^*$. The existence of a fully nontrivial nonnegative minimizer of $I|_{\mathcal{M}_b^*(R)}$ can be shown as in the second step in the proof of Theorem \[Mbrad Thm Grenzproblem Mbrad\]. From the inclusion $\mathcal{M}_b^*(R)\subset \mathcal{M}_b^*$ it follows $$\label{Mbrad Gl Monotonie kappabradR} \kappa_b^*(R)\geq \kappa_b^*.$$ In order to show $\kappa_b^*(R)\to \kappa_b^*$ as $R\to\infty$ we choose a cut-off function $\chi$ with $\chi(x)=1$ for $|x|\leq \frac{1}{2}$ and $\chi(x)=0$ für $|x|\geq 1$, set $\chi_R(x):=\chi(R^{-1}x)$. Then for all $u,v\in H^1_r({\mathbb{R}})$ we have $u\chi_R,v\chi_R\in H^1_{0,r}(B_R)$. Moreover, if in addition $(u,v)$ satisfies so does $(u\chi_R,v\chi_R)$ for sufficiently large $R>0$ and we get from the inequality $$\limsup_{R\to\infty} \kappa_b^*(R) \leq \frac{q-1}{2q} \limsup_{R\to\infty} \hat J(u\chi_R,v\chi_R)^{\frac{1}{q-1}} = \frac{q-1}{2q} \hat J(u,v)^{\frac{1}{q-1}}.$$ Since this holds for all $u,v\in H^1_r({\mathbb{R}})$ satisfying we obtain from the estimate $$\label{Mbrad Gl Asymptotik kappabradR} \limsup_{R\to\infty} \kappa_b^*(R) \leq \kappa_b^*.$$ The inequalities and show $\kappa_b^*(R)\to \kappa_b^*$ as $R\to\infty$. Let now $(u_k,v_k):=(u_{R_k},v_{R_k})$ be the sequence of solutions on $(-R_k,R_k)$ given by Proposition \[Mbrad Prop Mbrad-Minimierung n=1 Approx durch Kugeln\] where $(R_k)$ is a fixed positive sequence going off to infinity as $k\to\infty$. Then $(u_k,v_k)$ lies in $\mathcal{M}_b^*(R_k)\subset\mathcal{M}^*_b$ and we have $I(u_k,v_k)=\kappa_b^*(R_k)\to \kappa_b^*$ as $k\to\infty$ by Proposition \[Mbrad Prop Mbrad-Minimierung n=1 Approx durch Kugeln\]. Since $(u_k,v_k)$ solves on $(-R_k,R_k)$ there is a real number $\alpha_k$ such that $$\begin{aligned} \label{Mbrad Gl Existenzres b kleiner 0 n gleich 1 I} -u_k'^2 - v_k'^2 + u_k^2 + \omega^2 v_k^2 - \frac{1}{q} (u_k^{2q}+v_k^{2q} + 2bu_k^qv_k^q) = \alpha_k \qquad \text{in } (-R_k,R_k) \end{aligned}$$ and $u_k(R_k)=v_k(R_k)=0$ implies $\alpha_k\leq 0$, see Proposition \[Mbrad Prop Identität n=1\] for the proof of a related result. The sequence $(u_k,v_k)$ is bounded in $H_r^1({\mathbb{R}})\times H_r^1({\mathbb{R}})$ and we may choose a subsequence again denoted by $(u_k,v_k)$ that converges weakly to some $(u,v)\in H^1_r({\mathbb{R}})\times H^1_r({\mathbb{R}})$. Then $R_k\to\infty$ implies that $(u,v)$ is a nonnegative solution of with $I(u,v)\leq \kappa_b^*$. It remains to show $u,v\neq 0$. We assume $u=0$. Since $u_k,v_k$ are radially symmetric we have $u_k'(0)=v_k'(0)=0$. From $b\leq 0$ and we get the inequality $$0\geq \alpha_k \geq u_k(0)^2 (q-u_k(0)^{2q-2}) + v_k(0)^2 (q\omega^2 -v_k(0)^{2q-2}).$$ From $u_k(0)\to u(0)=0$ it follows $v_k(0)\geq (q\omega^2)^{\frac{1}{2q-2}}$ for almost all $k$ and hence $v(0)>0$. It follows that $(u,v)=(0,v)$ is a solution of satisfying $v(0)>0$ as well as $v\geq 0$. Kwong’s uniqueness result [@Kwo_Uniqueness_of_positive] gives $v=v_0$ and we obtain $$\label{Mbrad Gl Konvergenz uk vk} (u_k,v_k) {\rightharpoonup}(0,v_0),\qquad (u_k,v_k) \to (0,v_0) \text{ in }C^2_{loc}({\mathbb{R}}).$$ Let now $x_k\in [0,R_k)$ be given by $$\max_{[-R_k,R_k]} u_k = u_k(x_k)=u_k(-x_k)>0.$$ From the differential equation and $b\leq 0$ we infer $$0 \leq - \frac{u_k''(x_k)}{u_k(x_k)} = \frac{u_k(x_k)^{2q-1}+bu_k(x_k)^{q-1}v_k(x_k)^q-u_k(x_k)}{u_k(x_k)} \leq u_k(x_k)^{2q-2} - 1$$ and thus $$\label{Mbrad Gl Hoehe Max uk} u_k(x_k)=u_k(-x_k)\geq 1.$$ From , we get $x_k\to +\infty$. Let now $(\tilde u_k^+,\tilde v_k^+),(\tilde u_k^-,\tilde v_k^-)$ be given by $$(\tilde u_k^+,\tilde v_k^+):=(u_k(\cdot + x_k),v_k(\cdot +x_k)),\qquad (\tilde u_k^-,\tilde v_k^-):=(u_k(\cdot -x_k),v_k(\cdot - x_k)).$$ These sequences are bounded in $H^1({\mathbb{R}})\times H^1({\mathbb{R}})$ and there are subsequences again denoted by $(\tilde u_k^+,\tilde v_k^+),(\tilde u_k^-,\tilde v_k^-)$ that converge weakly and locally uniformly to nonnegative functions $(\tilde u^+,\tilde v^+),(\tilde u^-,\tilde v^-)$, respectively. The inequality implies $\tilde u^+(0), \tilde u^-(0)>0$. Since the functions $(\tilde u^+,\tilde v^+),(\tilde u^-,\tilde v^-)$ are nontrivial solutions of on $(-\infty,a),(-a,\infty)$, respectively where $a:= \lim_{k\to\infty} (R_k-x_k)$ we obtain $(\tilde u^\pm,\tilde v^\pm)\in\mathcal{N}_b$ and gives $$\label{Mbrad Gl Energieabschaetzung tildeu tilde v} \frac{q-1}{2q}(\|\tilde u^\pm\|^2 + \|\tilde v^\pm\|_\omega^2) = I(\tilde u^\pm,\tilde v^\pm) \geq \min_{\mathcal{N}_b} I = I(u_0,0) = c_0.$$ Now let $\chi$ denote a cut-off-function satisfying $\chi(x)=1$ for $|x|\leq 1$ and $\chi(x)=0$ for $|x|\geq 2$, set $\chi_R(x):=\chi(R^{-1}x)$. Choosing $k_0(R)$ sufficiently large we obtain $x_k>2R$ for all $k\geq k_0(R)$. In particular for all $k\geq k_0(R)$ the sets $\operatorname{supp}(\chi_R)$, $\operatorname{supp}(\chi_R(\cdot-x_k))$, $\operatorname{supp}(\chi_R(\cdot+x_k))$ are pairwise disjoint and we get $$\begin{aligned} &\|(u_k,v_k)\|^2 \\ &\geq \|(u_k\chi_R,v_k\chi_R)\|^2 + \|(u_k\chi_R(\cdot-x_k),v_k\chi_R(\cdot-x_k))\|^2 + \|(u_k\chi_R(\cdot+x_k),v_k\chi_R(\cdot+x_k))\|^2 \\ &= \|(u_k\chi_R,v_k\chi_R)\|^2 + \|(\tilde u_k^+\chi_R,\tilde v_k^+\chi_R)\|^2 + \|(\tilde u_k^-\chi_R,\tilde v_k^-\chi_R)\|^2. \end{aligned}$$ From $(u_k,v_k){\rightharpoonup}(0,v_0), (\tilde u_k^+,\tilde v_k^+){\rightharpoonup}(\tilde u^+,\tilde v^+)$ and $(\tilde u_k^+,\tilde v_k^+){\rightharpoonup}(\tilde u^-,\tilde v^-)$ we infer $$\begin{aligned} \liminf_{k\to\infty} \|(u_k,v_k)\|^2 &\geq \|(0,v_0\chi_R)\|^2 + \|(\tilde u^+\chi_R,\tilde v^+\chi_R)\|^2 + \|(\tilde u^-\chi_R,\tilde v^-\chi_R)\|^2 \\ &= \|v_0\chi_R\|_\omega^2 + \|(\tilde u^+\chi_R,\tilde v^+\chi_R)\|^2 + \|(\tilde u^-\chi_R,\tilde v^-\chi_R)\|^2. \end{aligned}$$ Since this inequality holds for all $R>0$ we obtain $$\begin{aligned} \liminf_{k\to\infty} \|(u_k,v_k)\|^2 \geq \|v_0\|_\omega^2 + \|(\tilde u^+,\tilde v^+)\|^2 + \|(\tilde u^-,\tilde v^-)\|^2 \end{aligned}$$ and from the estimate and we get $$\begin{aligned} \kappa_b^* &= \lim_{k\to\infty} \kappa_b^*(R_k) \\ &= \frac{q-1}{2q} \lim_{k\to\infty} \|(u_k,v_k)\|^2 \\ &\geq \frac{q-1}{2q} \Big(\|v_0\|_\omega^2 + \|(\tilde u^+,\tilde v^+)\|^2 + \|(\tilde u^-,\tilde v^-)\|^2\Big) \\ &\geq (2+\omega^{\frac{q+1}{q-1}})c_0 \end{aligned}$$ which contradicts . Hence, $u\neq 0$. Analogously the assumption $v=0$ leads to the inequality $$\kappa_b^* \geq (1+2\omega^{\frac{q+1}{q-1}})c_0 \geq (2+\omega^{\frac{q+1}{q-1}})c_0,$$ which again gives a contradiction. It follows $u,v\neq 0$ and the proof is finished. [$\Box$]{} [*Proof of Corollary \[Mbrad Kor hinreichendes Krit Mbrad n=1\]*]{} Assume that $b$ is larger than the right hand side in . According to Theorem \[Mbrad Thm Kompaktheit n=1\] (iii) it suffices to show that this implies $b>b^*(\omega,q)$. To this end we estimate $b^*(\omega,q)$ from above using the test function $(u,v):=(u_0,u_0(\omega\cdot))$ in . We obtain $$\begin{aligned} b^*(\omega,q) &\leq \max_{\alpha>0} \frac{(2+\omega^{\frac{q+1}{q-1}})^{1-q} \|u_0\|^{-2q}\|u_0\|_{2q}^{2q} (\|u_0\|^2+\alpha^2\|u_0(\omega\cdot)\|_\omega^2)^q- \|u_0\|_{2q}^{2q}-\alpha^{2q}\|u_0(\omega\cdot)\|_{2q}^{2q}}{2\alpha^q \|u_0u_0(\omega\cdot)\|_q^q} \\ &=\max_{\alpha>0} \frac{(2+\omega^{\frac{q+1}{q-1}})^{1-q}( 1+\alpha^2\omega)^q -1 - \alpha^{2q}\omega^{-1}}{2\alpha^q}\cdot \frac{\|u_0\|_{2q}^{2q}}{\|u_0u_0(\omega\cdot)\|_q^q}. \end{aligned}$$ The numerator function is bounded from above by its negative maximum $2^{1-q}-1$ which is attained at $\alpha=2^{-1/2}\omega^{\frac{1}{q-1}}$. In particular, the right hand side is negative for all $\alpha>0$ so that the estimate ${\|u_0u_0(\omega\cdot)\|_q^q\leq \|u_0\|_{2q}^q\|u_0(\omega\cdot)\|_{2q}^q=\omega^{-1/2}\|u_0\|_{2q}^{2q}}$ leads to $$\begin{aligned} b^*(\omega,q) &\leq \max_{\alpha>0} \frac{(2+\omega^{\frac{q+1}{q-1}})^{1-q}( 1+\alpha^2\omega)^q -1 - \alpha^{2q}\omega^{-1}}{2\alpha^q\omega^{-1/2}} \end{aligned}$$ where the right hand side is smaller than $b$ by the assumption of Corollary \[Mbrad Kor hinreichendes Krit Mbrad n=1\]. As indicated above the result now follows from Theorem \[Mbrad Thm Kompaktheit n=1\] (iii). Finally, in the special cases $q=2$ or $1<q\leq 2,\omega=1$ we may determine the value of the right hand side in explicitly. In case $q=2$ the maximum is attained at $\alpha = \big(\frac{(1+\omega^3)\omega}{2}\big)^{1/4}$ and we get $$b^*(\omega,2)\leq -\frac{1}{\omega^{3/2}+\sqrt{2(1+\omega^3)}}.$$ In case $1<q\leq 2,\omega=1$ the maximum is attained at $\alpha=1$ and we obtain the value $$b^*(1,q)\leq (\frac{2}{3})^{q-1}-1.$$ Proof of Theorem \[Mbrad Thm Nichtexistenz n=1 q leq 2\] ======================================================== In the proof of Theorem \[Mbrad Thm Nichtexistenz n=1 q leq 2\] we will need the following elementary result. \[Mbrad Prop Identität n=1\] Let $n=1,\omega\geq 1$. Then every solution $(u,v)\in H^1({\mathbb{R}})\times H^1({\mathbb{R}})$ of satisfies $$\begin{aligned} \label{Mbrad Gl Identitaet n=1} - u'^2 - v'^2 + u^2 + \omega^2 v^2 - \frac{1}{q} (|u|^{2q}+|v|^{2q} + 2b|u|^q|v|^q) = 0 \qquad \text{in } {\mathbb{R}}. \end{aligned}$$ For a solution $(u,v)$ of the derivative of the left hand side in exists and equals zero. Hence there is some $\alpha\in{\mathbb{R}}$ such that $$- u'^2 - v'^2 + u^2 + \omega^2 v^2 - \frac{1}{q}(|u|^{2q}+|v|^{2q} + 2b|u|^q|v|^q) = \alpha \qquad \text{in } {\mathbb{R}}.$$ If $\alpha$ were not equal to zero then there would exist $\delta>0$ such that $u'^2+v'^2+u^2+\omega^2 v^2 \geq \delta$ in ${\mathbb{R}}$ which contradicts $u,v\in H^1({\mathbb{R}})$. [*Proof of Theorem \[Mbrad Thm Nichtexistenz n=1 q leq 2\]*]{} Let $b\in{\mathbb{R}}$ satisfy the inequality . We assume that there is a fully nontrivial solution $(u,v)\in H^1({\mathbb{R}})\times H^1({\mathbb{R}})$ of . Since the functions $(-u,v),(u,-v),(-u,-v)$ solve , too, we may assume that a maximal open interval $A\subset \{x\in{\mathbb{R}}: u(x)>0,v(x)>0\}$ is non-empty. We will prove later that the assumptions of the theorem imply that every critical point of $u^\omega v$ in $A$ is strict local minimizer. Once this is shown a contradiction can be achieved in the following way. In case $u^\omega v$ does not have any critical point in $A$ the function $u^\omega v$ is monotone on $A$ so that $A$ is unbounded and $(u^\omega v)(x)$ does not converge to 0 as $|x|\to\infty$. This contradicts $u,v\in H^1({\mathbb{R}})$. If, however, a critical point $x_0\in A$ exists, then $x_0$ is a strict local minimizer and therefore it must be the only critical point because any other critical point would have to be a strict local minimizer, too. It follows that $u^\omega v$ is increasing on $(x_0,\infty)\cap A$ and decreasing on $(-\infty,x_0)\cap A$ so that $(u^\omega v)(x)\geq (u^\omega v)(x_0)>0$ for all $x\in A$. Hence, $A={\mathbb{R}}$ from the maximality of $A$ and thus $(u^\omega v)(x)\geq (u^\omega v)(x_0)>0$ for all $x\in{\mathbb{R}}$ which contradicts $u,v\in H^1({\mathbb{R}})$. Now we show that every critical point of $u^\omega v$ in $A$ is a strict local minimizer. Clearly, for $x\in A$ such that $(u^\omega v)'(x)=0$ we have $$\label{Mbrad n=1 Gl 1} \omega u'(x)v(x) = -u(x)v'(x)$$ and a short calculation gives $$\label{Mbrad n=1 Gl 2} u'(x)v'(x) = - \omega u(x)v(x) \cdot \frac{u'(x)^2+v'(x)^2}{u(x)^2+\omega^2 v(x)^2}.$$ Using we obtain at the point $x$ $$\begin{aligned} (u^\omega v)'' &= \omega(\omega-1) u^{\omega-2}u'^2 v + \omega u'' u^{\omega-1} v + 2\omega u'u^{\omega-1}v' + u^\omega v'' \\ &= u^{\omega-1}\Big(\omega\frac{u'}{u}\cdot (\omega-1)u'v + \omega u'' v + 2\omega u'v' + uv''\Big) \\ &= u^{\omega-1}\Big( -(\omega-1)u'v' + \omega u'' v + + 2\omega u'v' + uv''\Big) \\ &= u^{\omega-1}\Big((\omega+1)u'v' + \omega u'' v + uv''\Big). \intertext{ From \eqref{Mbrad n=1 Gl 2} and the partial differential equation satisfied by $(u,v)$ we get} (u^\omega v)'' &= u^{\omega-1}\Big( -\omega(\omega+1)uv \cdot \frac{u'^2+v'^2}{u^2+\omega^2 v^2} \\ &\qquad\quad + \omega uv(1-u^{2q-2}-bu^{q-2}v^q) + uv(\omega^2 -v^{2q-2}-bv^{q-2}u^q) \Big). \intertext{Proposition \ref{Mbrad Prop Identität n=1} gives} (u^\omega v)'' &= u^{\omega}v\Big( -\omega(\omega+1)\cdot \Big( 1 - \frac{u^{2q}+v^{2q}+2bu^qv^q}{q(u^2+\omega^2 v^2)}\Big) \\ &\qquad\quad + \omega - \omega u^{2q-2}-b\omega u^{q-2}v^q + \omega^2 -v^{2q-2}-bv^{q-2}u^q \Big) \\ &= \frac{u^\omega v}{q(u^2+\omega^2 v^2)}\cdot \Big( \omega(\omega+1) (u^{2q}+v^{2q}+2bu^qv^q) - q(u^2+\omega^2 v^2)\cdot \\ &\qquad\qquad\qquad\qquad \big( \omega u^{2q-2} +b\omega u^{q-2}v^q + v^{2q-2} + bv^{q-2}u^q\big)\Big) \\ &= \frac{u^\omega v}{q(u^2+\omega^2 v^2)} \cdot \Big( -b q u^{q+2}v^{q-2} + (\omega^2-(q-1)\omega) u^{2q} - q u^2v^{2q-2} \\ &\qquad\qquad\qquad\qquad +b(2-q)(\omega^2+\omega) u^qv^q -q\omega^3 u^{2q-2}v^2 \\ &\qquad\qquad\qquad\qquad - (\omega^2(q-1)-\omega) v^{2q} - bq\omega^3 u^{q-2}v^{q+2} \Big) \\ &= \frac{u^\omega v^{2q+1}}{q(u^2+\omega^2 v^2)} \cdot \Big( -b q z^{q+2} + (\omega^2-(q-1)\omega) z^{2q} - q z^2 \\ &\qquad\qquad\qquad\qquad +b(2-q)(\omega^2+\omega) z^q -q\omega^3 z^{2q-2} \\ &\qquad\qquad\qquad\qquad - (\omega^2(q-1)-\omega) - bq\omega^3 z^{q-2} \Big) \end{aligned}$$ where $z:= \frac{u(x)}{v(x)}$. From assumption and $u(x),v(x),z>0$ we obtain $(u^\omega v)''(x)>0$ which proves the claim. We finish the proof of Theorem \[Mbrad Thm Nichtexistenz n=1 q leq 2\] considering the special cases $q=2$ and $1<q<2,\omega=1$. In case $q=2$ the minimum in is attained at $z=\sqrt{\omega}$ and we obtain that fully nontrivial solutions do not exist for parameter values $b < - \frac{\omega^2+1}{2\omega}$. In case $1<q<2,\omega=1$ we find that the following inequality holds for $b\leq -1$ and all $z>0$ $$\begin{aligned} (-b)\cdot \big(q z^{q+2} &-2(2-q)z^q + q z^{q-2}\big) + (2-q) z^{2q} - q z^2 - q z^{2q-2} - (q-2) \\ &\geq 1\cdot \big(q z^{q+2} -2(2-q)z^q + q z^{q-2}\big) + (2-q) z^{2q} - q z^2 - q z^{2q-2} - (q-2) \\ &= (2-q)(z^q-1)^2 + q(z^2-z^{q-2})(z^q-1) \\ &\geq 0 \end{aligned}$$ with equality if and only if $b=-1$ and $z=1$. Rearranging terms we see that the minimum in is 1 and it is attained at $z=1$. We obtain the nonexistence result for $b<-1$.[$\Box$]{} In the above reasoning we did not use the assumption $1<q\leq 2$ explicitly. Nevertheless we had to exclude the case $q>2$ case because the minimum in does not exist. Indeed, sending $z$ to $0$ and using $\omega\geq 1>\frac{1}{q-1}$ we find that the infimum is $-\infty$.
--- abstract: | The quasi-particle spectra in the metallic rutile and insulating monoclinic phases of VO$_2$ are shown to be dominated by local Coulomb interactions. In the rutile phase the small orbital polarization among V $3d$ $t_{2g}$ states leads to weak static but strong dynamical correlations. In the monoclinic phase the large $3d$ orbital polarization caused by the V–V Peierls distortion gives rise to strong static correlations which are shown to be the primary cause of the insulating behavior.\ \ PACS numbers: 71.20.Be, 71.27.+a. 79.60.Bm author: - 'A. Liebsch$^1$, H. Ishida$^2$, and G. Bihlmayer$^1$' title: 'Coulomb Correlations and Orbital Polarization in the Metal Insulator Transition of VO$_2$' --- Introduction ============ The metal insulator transition in VO$_2$ has been intensively studied for a long time. At 340 K the resistivity changes by several orders of magnitude.[@morin] The high-temperature metallic phase has a rutile structure, while the low-temperature insulating phase is monoclinic ($M_1$), with a zigzag-type pairing of V atoms along the $c$ axis. Both phases are non-magnetic. Although this transition is widely regarded as a Mott-Hubbard transition, [@pouget; @zilbersztejn; @mott; @rice] the role of the Peierls distortion of the crystal structure in the insulating phase has been the topic of intense debate. The discovery of a second insulating phase ($M_2$), [@pouget] in which only half of the V atoms form pairs while the other evenly spaced chains behave as magnetic insulators, suggested that both low temperature phases should be regarded as Mott-Hubbard and not as Peierls band insulators.[@zilbersztejn; @rice] Although the role of the Coulomb interaction in the metal insulator transition of VO$_2$ has been studied previously, [@huang; @korotin; @continenza; @laad] a consistent description of the rutile and monoclinic phases is not yet available. The aim of the present work is to elucidate the interplay of Coulomb correlations and orbital polarization in the quasi-particle spectra of VO$_2$. Analyzing recent photoemission data[@okazaki; @tjeng] we demonstrate that the metallic and insulating phases show evidence of strong local interactions which manifest themselves in distinctly different ways because of the different degree of orbital polarization in the rutile and monoclinic structures. The key aspect of the metallic phase is the small orbital polarization among V $3d$ $t_{2g}$ states, implying weak static but strong dynamical correlations. Thus, the spectra reveal band narrowing and reduced weight of the coherent peak near the Fermi level, and an incoherent satellite feature associated with the lower Hubbard band. The monoclinic phase, in contrast, is characterized by a pronounced $t_{2g}$ orbital polarization induced by the V–V Peierls distortion. The local Coulomb interaction therefore leads to strong static correlations and to the opening of an excitation gap. The influence of non-diagonal coupling among $t_{2g}$ orbitals on the size of the gap is investigated within the static limit and is found to be small. In the following sections we discuss results obtained within several theoretical approaches, such as the dynamical mean field theory (DMFT)[@DMFT], the local density approximation plus Hubbard $U$ (LDA+U)[@LDA+U] and $GW$[@GW] methods. We argue that none is presently capable of explaining all of the observed phenomena in a consistent manner. Instead, we focus on the merits and failures of these approaches with the aim of highlighting the different roles of Coulomb correlations and orbital polarization in the metallic and insulating phases of VO$_2$. Results and Discussion ====================== -3mm Fig. 1 shows photoemission spectra for VO$_2$ films (150 Å thick) grown on TiO$_2$(001) at binding energies corresponding to the V $3d$ bands (21.2 eV photon energy).[@okazaki] Emission from O $2p$ states extends from $-2$eV to about $-8$eV. As a result of compressive stress from the substrate [@muraoka] the transition temperature is lowered from 340 K to 290 K. The high temperature spectrum shows a Fermi edge characteristic of metallic behavior which is absent in the low temperature spectrum. While this trend is in agreement with previous photoemission measurements,[@PES; @shin] the data in Fig. 1 reveal two spectral features in the metallic phase: a peak close to $E_F$ and a second one near $-1$eV. In the insulating phase, the peak near $E_F$ disappears and the feature near $-1$eV becomes more intense. Recent VO$_2$ photoemission spectra taken at 520 eV photon energy [@tjeng] agree with the ones shown in Fig. 1, except for a considerably greater relative weight of the coherent peak near $E_F$ in the metallic phase. These spectral changes are consistent with the observation that satellite peaks in low photon energy spectra tend to be more pronounced as a result of a surface induced enhancement of Coulomb correlations. [@maiti; @sekiyama; @liebsch03] Before analyzing the photoemission data we discuss the single particle properties of VO$_2$ obtained using density functional theory. We have carried out full potential linearized augmented plane wave (LAPW) calculations for the rutile and monoclinic structures using the experimental lattice parameters and treating exchange correlation within the generalized gradient approximation (GGA).[@perdew] Due to the octahedral crystal field, the states near $E_F$ have V $3d$ $t_{2g}$ character. They are separated by a small gap from the empty $e_{g}$ states, and from the O $2p$ states by a gap of about 1.0 eV. The occupancy of the $t_{2g}$ manifold is $3d^1$. Our results qualitatively confirm previous LDA calculations. [@wentzcovitch; @eyert; @huang; @korotin] Fig. 2(a) shows a comparison of the V total $t_{2g}$ density of states for the rutile and monoclinic phases of VO$_2$. Although there are differences in detail, the overall width of these distributions and the shape of the occupied region are similar for both structures. Evidently, the GGA/LDA does not predict the insulating nature of the monoclinic phase. Moreover, on the basis of these densities one would not expect correlations to play very different roles in the two phases. However, if we plot the subband contributions to the $t_{2g}$ density, the two structures are very different, as shown in Figs. 2(b) and (c). Whereas in the rutile phase the $t_{2g}$ bands have similar occupation numbers, in the monoclinic phase the $d_{x^2-y^2}$ band is significantly more occupied than the $d_{xz,yz}$ bands. (We adopt the local coordinate system of Ref. , i.e., $x$ denotes the $c$ axis, while $y$ and $z$ point along the diagonals of the $a,b$ plane.) The origin of this orbital polarization is the V–V dimerization along the $c$ axis. The $d_{x^2-y^2}$ band splits into bonding and anti-bonding components, while the $d_{xz,yz}$ bands are pushed upwards due to shortening of V-O distances. Investigation of the energy bands shows that, in agreement with earlier work,[@wentzcovitch; @eyert] the top of the bonding $d_{x^2-y^2}$ bands is separated by a slight negative gap from the bottom of the $d_{xz,yz}$ bands. In the following subsection we show that different degree of orbital polarization in the rutile and monoclinic phases has a pronounced effect on the quasi-particle spectra of VO$_2$. -4mm Metallic Rutile Phase --------------------- Let us first discuss the rutile phase. Comparing the $t_{2g}$ density of states with the photoemission spectra it is plausible to associate the feature close to $E_F$ with emission from metallic V $3d$ states. The peak near $-1$eV, however, lies in the gap between V $3d$ and O $2p$ states and cannot be understood within the single particle picture. To describe the spectra in the rutile phase it is clearly necessary to account for dynamical Coulomb correlations. For the evaluation of the quasi-particle distributions we use the Dynamical Mean Field Theory combined with the multiband Quantum Monte Carlo (QMC) method. [@DMFT] Since hybridization among $t_{2g}$ states is weak the local self-energy is taken as diagonal in orbital space. The $t_{2g}$ density of states components shown in Fig. 2(b) then serve as input quantities accounting for the single particle properties of the rutile structure. The local Coulomb interaction defining the quantum impurity problem is characterized by intra- and inter-orbital matrix elements $U$ and $U'=U-2J$, where $J$ is the Hund’s rule exchange integral. According to constrained LDA calculations, $U\approx4.2$eV and $J\approx0.8$eV.[@korotin; @UJ] The calculations are performed at $T\approx500 $ K with up to $10^6$ sweeps. The quasi-particle distributions are obtained via maximum entropy reconstruction. [@jarrell] Fig. 3(a) shows that, in contrast to the single particle density of states, the calculated $t_{2g}$ quasi-particle spectra for the rutile phase of VO$_2$ exhibit two spectral features, a coherent peak near $E_F$ and a lower Hubbard band near $-1$eV, in agreement with experiment. The peak near $E_F$ accounts for the band narrowing and lifetime broadening of the metallic states whereas the Hubbard band is associated with incoherent emission. Since the $t_{2g}$ subbands have comparable single particle distributions their quasi-particle spectra reveal similar correlation features. Moreover, because of the weak orbital polarization in the rutile structure static correlations are negligible. Thus, in the metallic phase the spectral weight transfer between coherent and incoherent contributions to the spectrum is primarily the result of dynamical correlations. We point out that, in view of the approximate nature of the model underlying the DMFT, quantitative agreement with photoemission data cannot be expected. On the theoretical side, the consideration of purely on-site Coulomb interactions and the neglect of the momentum variation of the self-energy permit only a qualitative analysis of the spectra. In addition, there exists some uncertainty regarding the precise values of the Coulomb and exchange energies. Finally, the DMFT results depend on the temperature used in the QMC calculation. The comparison with results obtained for slightly different values of $U$, $J$ and $T$, however, gives us confidence that in the metallic phase the transfer of spectral weight from the coherent peak near $E_F$ to the satellite region near $-1$ eV is qualitatively reliable and consistent with analogous dynamical correlation effects in other $3d^1$ transition metal oxides, such as SrVO$_3$. [@sekiyama; @liebsch03; @pavarini] As we discuss below, the local DMFT treatment predicts the monoclinic phase to be also metallic. The metal insulator transition in VO$_2$ is therefore not achieved simply by lowering the temperature in the rutile phase. The lattice transformation from the rutile to monoclinic structure must be taken into account. Therefore, the DMFT results shown in Fig. 3(a) for the rutile structure at $T=500$ K can be regarded as representative of correlation induced behavior in the metallic domain. On the experimental side, as pointed out above, it is important to recall that photoemission data taken at low photon energies represent a superposition of bulk and surface contributions. Since correlation effects are observed to be more enhanced at surfaces, [@maiti; @sekiyama; @liebsch03] the relative intensity of the satellite peak near $-1$eV in the 21.2 eV spectra shown in Fig. 1 is considerably more pronounced than at high photon energies [@tjeng] at which primarily bulk-like valence states are detected. Dynamical effects in the metallic phase may also be evaluated within the $GW$ approach [@GW] which treats long range Coulomb interactions in the random phase approximation (RPA) and which has proven rather useful to describe excitation spectra of weakly correlated systems. [@GWreview] Because of the neglect of multiple electron-electron and hole-hole scattering processes, this scheme fails when local Coulomb interactions are important.[@gunnarsson] Presumably, therefore, for VO$_2$ the GW method does not reproduce the lower Hubbard band. The satellite in the rutile phase is, of course, also beyond the scope of the static LDA+U approach.[@comment] The qualitative agreement between the measured high-temperature spectra and the theoretical results shown in Fig. 2(a) suggests that the DMFT captures the key spectral weight rearrangement induced by dynamical correlations. -4mm Insulating Monoclinic Phase --------------------------- Turning now to the monoclinic phase we first calculated the $t_{2g}$ quasi-particle spectra within the DMFT. Because of the orbital polarization caused by the V–V Peierls distortion, correlation effects in the $d_{x^2-y^2}$ band are now stronger so that the intensity of the lower Hubbard band is enhanced compared to the rutile phase (not shown). While this trend agrees with the photoemission data, the coherent peak near $E_F$ persists. Thus, the single-site DMFT based on a diagonal self-energy does not reproduce the insulating nature of the monoclinic phase.[@heldVO2] This failure is in striking contrast to the results obtained within the LDA+U and $GW$ methods.[@huang; @continenza] Let us discuss first the LDA+U approach. In this scheme different orbital occupations give rise to orbital dependent potential terms which shift different $t_{2g}$ bands in opposite directions. Thus, orbital polarization leads to strong [*static*]{} correlation effects. Using the same Coulomb parameters as for the rutile phase we find that the local density of states calculated via the LAPW/LDA+U exhibits a gap of about 0.7 eV (see Fig. 3(b)), in agreement with optical data [@shin] and previous LDA+U results. [@huang] The bonding $d_{x^2-y^2}$ bands are now completely filled and the $d_{xz,yz}$ and anti-bonding $d_{x^2-y^2}$ bands are shifted upwards. Thus, the LDA orbital polarization in the monoclinic phase is further increased as a result of static correlations. In turn, this reduction of orbital degeneracy enhances the trend towards insulating behavior.[@koch] We emphasize that the LDA+U does not provide a complete description of correlation effects in the insulating phase. Essentially, it amounts to a fully self-consistent, non-local treatment of static screening within the dimerized structure. Genuine dynamical effects, such as the spectral weight transfer from the coherent to the incoherent peak as observed in the metallic phase are ignored. Neverteless, since static correlations appear to be the origin of the excitation gap in VO$_2$, it is instructive to inquire which features of the LDA+U contribute to the opening of the gap. Of particular interest is the role of the non-diagonal occupation matrix $n_{\alpha\beta}$ which is the key input quantity in the orbital dependent perturbation potential used in the LDA+U. Recent work by Pavarini [*et al.*]{} [@pavarini] on $3d^1$ perovskite materials exhibiting non-diagonal $t_{2g}$ orbital coupling caused by octahedral distortions suggests that this mechanism tends to suppress orbital fluctuations and to enhance insulating behavior. Fig. 3(c) compares the VO$_2$ LDA+U density of states with results of an approximate LDA+U treatment in which at each iteration only the diagonal elements $n_{\alpha\alpha}$ are retained. The size of the gap is seen to be only slightly reduced. In the case of VO$_2$, therefore, non-diagonal coupling among $t_{2g}$ orbitals is evidently not the main reason for the existence of the gap. To understand the gap formation obtained within the LDA+U, it is useful to formally express the $t_{2g}$ self-energy matrix as $\Sigma(\omega,k)=\Sigma^{\rm HF}(k)+\Delta\Sigma(\omega,k)$, where $\Sigma^{\rm HF}(k)= H^{\rm LDA+U}(k)-H^{\rm LDA}(k)$ is real and accounts for spectral changes associated with the LDA+U.[@HF-limit] $\Delta\Sigma(\omega,k)$ is complex and describes purely dynamical effects. The important point is that the LDA+U includes the full momentum variation of $\Sigma^{\rm HF}(k)$ within the true lattice geometry. Thus, in a site representation, static screening processes generate finite inter-site elements $\Sigma^{\rm HF}_{i\ne j}$ (each element is a matrix in orbital space) even if the bare LDA+U perturbation potential is site-diagonal. Thus, $\Sigma^{\rm HF}_{ij}(k)$ is more accurate than the static limit of the single-site DMFT which neglects the $k$ dependence and assumes the impurity environment to be isotropic, i.e., $\Sigma^{\rm DMFT}_{ij}(\omega)\sim\delta_{ij}$. The results shown in Fig. 3(b,c) suggest that the proper evaluation of $\Sigma^{\rm HF}(k)$ within the Brillouin Zone of the dimerized structure is the crucial ingredient to an adequate description of the insulating behavior in VO$_2$. The LDA+U amounts to a self-consistent treatment of $\Sigma^{\rm HF}(k)$ since the solution of the Schrödinger equation imposes no restrictions on how the wave functions adjust to the LDA+U potential. In order to go beyond static screening and include dynamical correlations in the monoclinic phase a cluster extension of the DMFT is most likely required. Such an extension is beyond the scope of the present work. A cluster DMFT would include the crucial inter-site elements $\Sigma^{\rm DMFT}_{i\ne j}(\omega)$ which arise naturally in a cluster representation of the lattice and of the impurity Green’s functions $G(\omega)$ and $G_0(\omega)$, even for a purely on-site Coulomb interaction. Preliminary results for VO$_2$ within a cluster DMFT[@sascha] show that dynamical screening processes beyond the static correlations included in the LDA+U cause a broadening of the LDA+U density distribution and a shift of the main spectral peak towards the Hubbard bands. Possibly, a multi-site extension of the DMFT might also identify the true origin of the metal insulator transition in VO$_2$, i.e., whether it is primarily caused by the lattice reconstruction or by Coulomb correlations, or whether these mechanisms mutually enhance each other. All one can say at present is that, given the orbital polarization induced by the lattice transition, Coulomb correlations have a pronounced effect on the quasi-particle spectra. As noted above, the $GW$ approach applied to the monoclinic phase of VO$_2$ also yields an excitation gap of the correct magnitude. [@continenza] These calculations utilize a model self-energy [@gygi] consisting of an approximate short-range contribution given by the local exchange correlation potential, and a correction due to incomplete screening of the Coulomb interaction. Essentially, in this simplified $GW$ scheme the self-energy correction consists of a “scissor” operator and additional, non-rigid shifts of energy eigenvalues.[@gygi] Presumably, the reason why this model self-energy yields a gap is that $\Sigma(q,\omega)$ is non-local and non-diagonal in site space, i.e., it includes the important static correlations in the dimerized structure in a similar fashion as the LDA+U. The approximate nature of the $GW$ method, in particular, the neglect of electron-electron and hole-hole ladder type interactions, affects mainly the remaining dynamical corrections caused by the strong local Coulomb energy. An adequate treatment of these corrections would require going beyond the RPA and should lead to a more realistic description of the position and width of the Hubbard bands. An interesting additional feature observed by Okazaki [*et al.*]{} [@okazaki] is the temperature dependence of the photoemission spectra below the metal insulator transition. Essentially, towards lower $T$ the excitation gap becomes more clearly defined and the main peak near $-1$ eV gets slightly sharper. These changes, however, are very small on the scale of the main discrepancy still existing between the LDA+U or $GW$ results and the experimental data. At present it is not clear whether cluster DMFT calculations in the monoclinic phase as a function of temperature will be able to explain the observed trend or whether an explicit treatment of electron-phonon coupling is required. Summary ======= The metal insulator transition in VO$_2$ appears to be remarkably complex and its origins are not yet fully understood. In the present work we focussed on the important role of two aspects, local Coulomb correlations and orbital polarization, in the low and high temperature photoemission spectra of VO$_2$. Whereas the metallic phase exhibits weak static and strong dynamical correlations, the monoclinic phase is dominated by static Coulomb correlations. Accordingly, the rutile spectra reveal a double-peak structure where the feature close to $E_F$ is identified with metallic V $3d$ states and the peak near $-1$eV with the lower Hubbard band. 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--- abstract: | We give a proof of perturbative renormalizability of SU(2) Yang–Mills theory in four-dimensional Euclidean space which is based on the Flow Equations of the renormalization group. The main motivation is to present a proof which does not make appear mathematically undefined objects (as for example dimensionally regularized generating functionals), which permits to parametrize the theory in terms of [*physical*]{} renormalization conditions, and which allows to control the singularities of the correlation functions of the theory in the infrared domain. Thus a large part of the proof is dedicated to bounds on massless correlation functions. ---------- ---------------- Preprint IPhT t17/055 RR 018-04-2017 ---------- ---------------- bibliography: - 'ym.bib' --- Renormalization of SU(2) Yang–Mills theory\ with Flow Equations Introduction ============ Renormalization theory based on the differential Flow Equations [@wilson; @wehoug; @pol] has allowed to build a unified approach to the analysis of renormalizability for a wide class of theories without recourse to Feynman graphs. This method was applied to show momentum bounds of massive $\phi^4$ theory [@kome], used to prove renormalization of spontaneously broken SU(2) Yang–Mills [@komu] and to establish uniform bounds on Schwinger functions of massless $\phi^4$ fields [@kogui]. Starting with the milestone works, [@YM54; @FP; @st2; @taylor; @LZJ; @tHV72; @brs; @tyutin; @st] (see [@Lai81] for more references), a variety of results on the renormalizability of nonabelian gauge theories flourished in the literature, in different contexts and with different level of mathematical rigor. Work on this problem in the context of Flow Equations (FE) includes [@wet; @bec; @bam; @ma], and, more recently, [@fhh]. The present work deals with perturbative renormalizability and shares certain aspects with some of these articles: from [@bec] we borrow the fruitful idea that the local operator describing the violation of Slavnov–Taylor identities (STI) for the one-particle irreducible (1PI) functions [@st; @KSZ75] is constrained by the nilpotency of the underlying ”Slavnov differential operator”; as in [@bam] (and in contrast with [@fhh]) we define the marginal correlators by physical boundary conditions at vanishing IR cutoff and nonexceptional momenta. Our main result is a proof of the renormalizability of Yang–Mills theory that complements the previous work in view of the following features: i) Rigorous control of the IR and UV behavior of the one-particle irreducible functions is established by means of uniform bounds in momentum space. In particular, from the bounds follow the existence of the IR limit of 1PI functions at nonexceptional momenta and the existence of the subsequent UV limit. ii) The vanishing of the STI violation in the UV limit is proven; iii) The relevant part of 1PI functions and STI violation at nonvanishing renormalization momenta, which gives initial conditions for the renormalization group FE, is thoroughly identified. Our proof refers explicitly to SU(2) Yang–Mills theory. However, it could be extended without important conceptual changes to other semi-simple compact Lie groups. We proceed as follows. In section \[sec\_327a\] we fix the notations, introduce the classical Yang–Mills action, generating functionals and regulators. We define the BRST transformations. Then we derive the FE of the renormalization group, for the connected amputated Schwinger functions and for the one-particle irreducible vertex functions. Finally we study the STI, the Antighost Equation (AGE) and their violation. Remark that in our context gauge invariance is broken through the presence of the cutoffs. Our proof of renormalizability of Yang-Mills theory is based on momentum bounds for the vertex bounds which permit to take the limits $\Lambda \to 0$ (IR-cutoff) and $\Lambda_0 \to \infty$ (UV-cutoff) for nonexceptional external momenta. These bounds are an extension of those in [@kogui] and are established inductively with the aid of the FE. They are expressed in terms of tree amplitudes and polynomials of logarithms. For our trees we only have to consider vertices of coordination numbers 1 and 3. In section \[sec\_21bnd\] we present the definitions of the tree structures to be considered, and the aforementioned bounds. These bounds permit to prove the existence of the vertex functions for nonexceptional external momenta on removal of the cutoffs. This statement also holds for vertex functions carrying an operator insertion related the the BRST transformation of the fields. These vertex functions are required to formulate the STI. Then we also have to consider vertex functions with operator insertions which permit to formulate the violation of the STI. Our bounds then permit to show that, for suitable renormalization conditions, these functions describing the violation of the STI vanish in the UV limit. Section \[sec\_327b\] is dedicated to the proof of those bounds. The rhs of the FE is a sum of products of vertex functions in lower loop order joined by free propagators. Our technique of proof then is based on the fact that applying our inductive bounds on these chains, which still have to be closed by a $\Lambda$-derived propagator and then integrated over the circulating loop momentum and over $\Lambda$, reproduce our inductive bounds at loop order $l$. The proof treats irrelevant terms first, then marginal and finally strictly relevant ones. Particular attention has to be paid to the renormalization conditions. Section \[sec\_327b\] ends with a proof of UV-convergence for $\Lambda_0 \to \infty$. In section \[sec\_sti\] we prove that the renormalization conditions required to prove the bounds of the previous sections can actually be imposed. These renormalization conditions are such that they leave us free to fix the physical coupling of the theory. At the same time they permit us to make vanish the relevant part of all functions describing the violation of the STI. This is required in the previous proof. In the Appendices \[sec\_gauss\], \[ex\_AAcc\], \[sec\_t\], \[sec\_b\] we present some facts on Gaussian measures, an example of chains of vertex functions, an analysis of linear independence of euclidean tensor structures, a large number of elementary bounds on integrals we encounter in the proofs. We also add a list with bounds on the propagators and their derivatives. In the two subsequent appendices \[sec\_gamma\], \[sec\_gamma2\] we analyse the generating functionals of the (inserted) vertex functions, as far as they have relevant content. Appendix \[sec\_rr\] is a supplement to section \[sec\_vsti\] and contains the generalization of the STI and AGE to the case of nonvanishing IR cutoff. In the last two appendices \[sec\_ren\], \[sec\_op\] we present the list of renormalization points and operator insertions to be considered. The formalism {#sec_327a} ============= Notations --------- $\mathbb{N}$ is the set of nonnegative integers. $|S|$ is the cardinality of a set $S$. Furthermore, $(a,b,c,...)$, $\{a,b,c,...\}$ denote a sequence and a set, respectively. Unless otherwise stated, sequence stands for finite sequence. For shortness we set $[a:b]:=\{i\in \mathbb{Z}: a\leqslant i \leqslant b\}$ and $[b]:=[1: b]$. Repeated indices are implicitly summed over, e.g. $A^a t_a:=\Sigma_a A^a t_a$. We choose the following basis of the Lie algebra $$\begin{aligned} t_a&:=\frac{1}{2}\sigma_a,&[t_a,t_b]&=i\epsilon_{abc}t_c,&a,b,c &\in \{1,2,3\},\end{aligned}$$ where $\sigma_a$ are the Pauli matrices and $\epsilon_{abc}$ is the Levi–Civita symbol, $\epsilon_{123}=1$. In this article we will deal with tensor fields on $\mathbb{R}^4$ in Cartesian coordinate systems with metric tensor $\delta_{\mu \nu}$. If $A,B$ are two Cartesian tensors of $\mathbb{R}^4$ of rank $r$ with components $A_{\mu_1...\mu_r}$ and $B_{\mu_1...\mu_r}$, respectively, then the scalar product $(A,B)$ is the contraction $$(A,B) := A^*_{\mu_1...\mu_r}B_{\mu_1...\mu_r}.\label{scalar_product}$$ Given a Cartesian tensor $T$, we use the norm $$|T|:=(T,T)^{\frac{1}{2}}.$$ For instance, for $p \in \mathbb{R}^4$, $|p|^2= \Sigma_\mu p^2_\mu$. Let $T$, $A$, and $B$ be Cartesian tensors such that $T_{\Vec{\mu} \Vec{\nu}}=A_{\Vec{\mu} \Vec{\rho}}\,B_{\Vec{\rho} \Vec{\nu}}$ where $\Vec{\mu}$, $\Vec{\rho}$, $\Vec{\nu}$ are multi-indices, for example $\Vec{\mu}:=(\mu_1,...,\mu_n)$. Then using the Cauchy–Schwarz inequality $$|T|^2=\sum_{\Vec{\mu},\Vec{\nu}} |\sum_{\Vec{\rho}}A_{\Vec{\mu}\Vec{\rho}}B_{\Vec{\rho}\Vec{\nu}}|^2\leqslant \sum_{\Vec{\mu},\Vec{\nu}}\sum_{\Vec{\rho},\Vec{\sigma}} |A_{\Vec{\mu}\Vec{\rho}}|^2|B_{\Vec{\sigma}\Vec{\nu}}|^2=|A|^2|B|^2\,.\label{eq_holder}$$ The integral over $\mathbb{R}^4$ of the product of two functions is denoted by $$\langle f_1,f_2 \rangle := \int d^4 x \, f_1(x)f_2(x) \,,$$ and the Fourier transform of a function is defined by $$f(p):=\int d^4x \, e^{-ipx} f(x).$$ The convolution of two functions $f$, $g$ is denoted as below $$(f * g)(x)= \int d^4 y \, f(y) g(x-y).$$ For functions $\phi_i(p_i)$ and $F(p_1,...,p_k)$ with $p_i \in \mathbb{R}^4$ the symbol $\langle F| \phi_1...\phi_{k};p\rangle$ denotes the following integral in momentum space & F| \_1...\_[k]{};p:= (2)\^4(\^k\_[j=1]{} p\_j+p) F(p\_1,...,p\_k) \_1(p\_1)...\_k(p\_[k]{}) \^k\_[i=1]{} .& We also use the shorthands $$\begin{aligned} \langle \phi_1...\phi_{k};p\rangle&:=\langle 1| \phi_1...\phi_{k};p\rangle,&\langle \phi_1...\phi_{k}\rangle&:=\langle 1| \phi_1...\phi_{k};0\rangle.\end{aligned}$$ A cumulative notation for the elementary fields and corresponding sources is $$\begin{aligned} \Phi&:=\begin{pmatrix}A^a_\mu,&B^a,&c^a,&\bar{c}^a\end{pmatrix},&K&:=\begin{pmatrix}j^a_\mu,&b^a,&\bar{\eta}^a,&\eta^a\end{pmatrix},\end{aligned}$$ where $c,\bar{c},\eta,\bar{\eta}$ are generators of an infinite-dimensional anticommuting algebra. Furthermore, we use the following shorthand $$K \cdot \Phi :=\langle j^a_\mu, A^a_\mu \rangle + \langle b^a, B^a \rangle + \langle \bar{\eta}^a, c^a \rangle + \langle \bar{c}^a, \eta^a \rangle.\label{eq_jA}$$ We will have to consider one-particle irreducible functions, also known as vertex functions, whose generating functional is denoted by $\Gamma$. These functions are translation-invariant in position space. Their *reduced* Fourier transforms $\Gamma^{\Vec{\phi}}$ are defined as follows $$\Gamma^{\Vec{\phi}}(p_1,\dots,p_{{\textsc{n}}-1}):= \int \Big(\prod^{{\textsc{n}}-1}_{i=1} d^4x_i\, e^{-i p_i x_i} \Big) \; \Gamma^{\Vec{\phi}}(0,x_1,\dots,x_{{\textsc{n}}-1})\,,$$ where $\Vec{\phi}:=(\phi_0,...,\phi_{{\textsc{n}}-1})$ is a sequence of field labels, $\phi_i \in \{A, B, c, \bar{c}\}$, and $$\Gamma^{\Vec{\phi}}(x_0,x_1,\dots,x_{{\textsc{n}}-1}):= \Big( \frac{\delta}{\delta \phi_{0}(x_{0})} \dots \frac{\delta}{\delta \phi_{{\textsc{n}}-1}(x_{{\textsc{n}}-1})}\Big) \Gamma \Big|_{\Vec{\phi}=0}\,.$$ Note that, with the exception of appendix \[sec\_gauss\], derivatives wrt Grassmannian variables are always intended to be left derivatives, see e.g. [@ber]. The complete Fourier transformed ${\textsc{n}}$-point vertex function then satisfies $$(2\pi)^{4({\textsc{n}}-1)} \Big(\frac{\delta }{\delta \phi_{0}(-p_{0})} \dots \frac{\delta }{\delta \phi_{{\textsc{n}}-1}(-p_{{\textsc{n}}-1})}\Big) \Gamma \Big |_{\Vec{\phi}=0}=\delta(\sum \limits^{{\textsc{n}}-1}_{i=0} p_i)\,\Gamma^{\Vec{\phi}}\,, \label{eq_f1}$$ where $\Gamma^{\Vec{\phi}}$ stands for $\Gamma^{\Vec{\phi}}(p_1,...,p_{{\textsc{n}}-1})$. The reduced and complete Fourier transforms with $n_\chi \geqslant 1$ composite operator insertions of sources $\Vec{\chi}=(\chi_0,...,\chi_{n_\chi-1})$ and $\Vec{\phi}:=(\phi_{n_\chi},...,\phi_{n-1})$ are correspondingly related by $$(2\pi)^{4(n-1)} \Big(\prod \limits^{n-1}_{i=n_\chi} \frac{\delta }{\delta \phi_i(-p_i)}\,\prod \limits^{n_\chi-1}_{i=0} \frac{\delta }{\delta \chi_i(-p_i)} \Big) \Gamma \Big|_{\substack{\Vec{\phi}=0\\\Vec{\chi}=0}}=\delta(\sum \limits^{n-1}_{i=0} p_i)\,\Gamma^{\Vec{\phi}}_{\Vec{\chi}}\,, \label{eq_f2}$$ where the order of the derivatives $\delta/\delta \phi_i$ is the same as before, the derivatives $\delta/\delta \chi_i$ are ordered with left-to-right increasing indices, and $\Gamma^{\Vec{\phi}}_{\Vec{\chi}}$ stands for $\Gamma^{\Vec{\phi}}_{\Vec{\chi}}(p_1,...,p_{n-1})$. Note that $n-1$ is the total number of arguments, e.g. for $\Gamma^{\Vec{\phi}}_{\Vec{\chi}}(p_1,...,p_{n-1})$ we have $n=n_\chi + {\textsc{n}}$. It will be useful to keep a bijective relation between momenta and field labels (including possible insertion labels), $p_i \leftrightarrow \phi_i$. Hence, we assume that $p_0$ is the negative subsum of all other momenta, $$\begin{aligned} \mathbb{P}_n&:=\{\Vec{p} \in \mathbb{R}^{4n}: \Vec{p}=(p_0,...,p_{n-1}),\; p_0=-\sum^{n-1}_{i=1} p_i\},& |\Vec{p}|^2&:=\sum \limits^{n-1}_{i=0} p^2_i,\label{eq_pn}\end{aligned}$$ and, referring to , , we use the notation $$\begin{aligned} \Gamma^{\Vec{\phi}}(\Vec{p})&=\Gamma^{\Vec{\phi}}(p_1,...,p_{{\textsc{n}}-1}),&\Vec{p}&=(p_0,...,p_{{\textsc{n}}-1}) \in \mathbb{P}_{{\textsc{n}}},\\ \Gamma^{\Vec{\phi}}_{\Vec{\chi}}(\Vec{p})&=\Gamma^{\Vec{\phi}}_{\Vec{\chi}}(p_1,...,p_{n-1}),&\Vec{p}&=(p_0,...,p_{n-1}) \in \mathbb{P}_{n}.\end{aligned}$$ A momentum configuration $\vec{p}\in\mathbb{P}_n$ is said **exceptional** iff there exists a nonempty $S\subset[1:n-1]$ such that $\sum_{i\in S}p_i=0$. If this is not the case $\vec{p}$ is said **nonexceptional**. We rely on the multi-index formalism for derivatives with respect to momenta. Taking in account that there are no derivatives wrt $p_0$, we set: $$\begin{aligned} \textbf{w} &:= (w_{0,1} , w_{0,2}, ... , w_{n-1,4} ),& w_{i,\mu } &\in \mathbb{N}, \quad w_{0,\mu}:=0,\\ w &:= (w_{0} , ... , w_{n-1} ),& w_i &:= \sum^{4}_{\mu=1 } w_{i,\mu },\\ \mathbb{W}_n&:=\{w\in \{0\} \times \mathbb{N}^{n-1}:\|w\|\leqslant w_{max} \},&\|w\|&:=\sum^{n-1}_{i=0} w_i,\label{eq_17w}\\ \partial ^{\,\textbf{w}} \Gamma& := \prod^{n-1}_{i=0} \prod_{\mu =1}^{4}\Bigl(\frac{\partial }{\partial p_{i,\mu }} \Bigr)^{w_{i,\mu }}\Gamma,&\partial^w\Gamma &:= \Big(\prod^{n-1}_{i=0} \partial^{w_i}_{p_i}\Big)\Gamma,\end{aligned}$$ where $\partial^{k}_{p}$ is the tensor with components $\partial_{p_{\mu_1}}...\partial_{p_{\mu_k}}$, and $w_{max}$ is an arbitrary integer $\geqslant4$ fixed throughout the paper. The following shorthands will be helpful: $$\begin{aligned} \dot{\Gamma}^{\Lambda\Lambda_0}&:=\partial_\Lambda \Gamma^{\Lambda\Lambda_0},&\partial_{\phi_i} \Gamma^{\Vec{\phi}}&:=\frac{\partial \Gamma^{\Vec{\phi}}}{\partial p_i},&\Gamma^{;w}&:=\partial^w \Gamma,\label{eq_11ns}\\ \tilde{\delta}_{\phi(p)}&:=(2\pi)^4\frac{\delta}{\delta \phi(-p)}\label{eq_409t},&\check{\phi}(p)&:=\phi(-p),&\log_+x&:=\log \max(1,x).\end{aligned}$$ The Classical Action -------------------- We start with a four-dimensional Euclidean classical field theory invariant under local SU(2) transformations and with a coupling $g \neq 0$. Given $G=e^{i g \alpha}$ with $\alpha=\alpha^a t_a$ an element of the Lie algebra, a gauge transformation reads: $$\begin{aligned} A_\mu & \mapsto \frac{i}{g}\, G\partial_\mu G^{-1} + G A_\mu G^{-1},& \phi & \mapsto G \phi G^{-1},\end{aligned}$$ and in infinitesimal form $$\begin{aligned} A_\mu &\mapsto A_\mu+\partial_\mu \alpha-ig[A_\mu,\alpha],& \phi & \mapsto \phi +ig[\alpha,\phi],\end{aligned}$$ where we also show the transformation for a field $\phi$ in the adjoint representation of the Lie algebra. The quantization of gauge theories using auxiliary ghost fields is presented, for example, by Slavnov and Faddeev in their book [@slfd]. Following [@fad; @thooft], with an appropriate linear gauge-fixing functional the semiclassical Lagrangian density takes the form $$\begin{aligned} \mathcal{L}^{tot}_{0}&=\frac{1}{4}F^a_{\mu \nu} F^a_{\mu \nu} + \frac{1}{2\xi}(\partial_\mu A^a_\mu)^2 - \partial_\mu \bar{c}^a (D_\mu c)^a,\\ D_\mu c &=\partial_\mu c -ig[A_\mu,c],\\ F_{\mu \nu} &= \partial_\mu A_\nu-\partial_\nu A_\mu - ig[A_\mu,A_\nu],\end{aligned}$$ where $\xi>0$ is the Feynman parameter and $\{F_{\mu \nu},A_\mu,c\}$ are elements of the su(2) algebra, e.g. $A_\mu=t^a A^a_\mu$. We will study the quantum theory in a framework which breaks local gauge invariance due to the presence of momentum space regulators. However, the Lagrangian respects global SU(2) symmetry, Euclidean isometries ISO(4) and has ghost number zero. We admit all counterterms compatible with these symmetries: $$\begin{aligned} \mathcal{L}_{ct}=&r^{0,\bar{c}c\bar{c}c} \bar{c}^bc^b \bar{c}^a c^a + r^{0,\bar{c}cAA}_1 \bar{c}^bc^b A^a_\mu A^a_\mu + r^{0,\bar{c}cAA}_2 \bar{c}^ac^b A^a_\mu A^b_\mu \nonumber\\ &+ r^{0,AAAA}_1 A^b_\mu A^b_\nu A^a_\mu A^a_\nu + r^{0,AAAA}_2 A^b_\nu A^b_\nu A^a_\mu A^a_\mu + 2\epsilon_{abc} r^{0,AAA} (\partial_\mu A^a_\nu) A^b_\mu A^c_\nu \nonumber\\ &-r^{0,A\bar{c}c}_1 \epsilon_{abd} (\partial_\mu \bar{c}^a) A^b_\mu c^d -r^{0,A\bar{c}c}_2 \epsilon_{abd} \bar{c}^a A^b_\mu \partial_\mu c^d + \Sigma^{0,\bar{c}c} \bar{c}^a \partial^2 c^a \nonumber \\ &- \frac{1}{2}\Sigma^{0,AA}_T A^a_{\mu}(\partial^2 \delta_{\mu \nu} - \partial_\mu \partial_\nu)A^a_{\nu} + \frac{1}{2\xi}\Sigma^{0,AA}_L(\partial_\mu A^a_\mu)^2\nonumber\\ &+\frac{1}{2}\delta m^2_{AA} A^a_\mu A^a_\mu - \delta m^2_{\bar{c} c} \bar{c}^a c^a. \label{eq_ct}\end{aligned}$$ There are eleven marginal counterterms and two strictly relevant counterterms. To denote the marginal counterterms we use the symbols $r^0$ and $\Sigma^0$. For the strictly relevant counterterms we use $\delta m^2_{AA}$ and $\delta m^2_{\bar{c} c}$. Generating functionals and Flow equations {#sec_gf} ----------------------------------------- In the next two sections we will introduce the essential structural tools required for our proof of renormalizability of nonabelian Yang–Mills theory. These tools are on the one hand the differential Flow Equations of the renormalization group, and on the other hand the (violated) Slavnov–Taylor identities. They are both obtained from the functional integral representation of the theory. Functional integrals are know to exist, beyond perturbation theory, if suitably regularized [@gljf]. To restrict their support to sufficiently regular functions in position space one has to introduce an ultraviolet (UV) cutoff. To avoid infrared problems one has to introduce a corresponding infrared cutoff, the most practical one being a finite volume cutoff, introduced by putting the system on a torus with periodic boundary conditions. For an analysis of the support properties of (Gaussian) functional integrals see [@ree]. We do not make explicit the finite volume cutoff in this paper, which would still blow up the manuscript to some degree. For some more details in this respect see [@mul]. On the other hand the UV cutoff is explicit. Once the finite volume cutoff has been introduced, the FEs (with UV regulator) can be shown to hold up to $\Lambda = 0$, $\Lambda$ being the flow parameter, even beyond perturbation theory (as far as the bosonic degrees of freedom are concerned). One can then construct perturbative solutions of the FEs, which are well-defined and regular in the open domain of nonexceptional external momenta, and which satisfy the Slavnov–Taylor identities, in the limits where the flow parameter is sent to zero and the UV cutoff is sent to infinity. If keeping the finite volume until the end, one then has to show that these solutions have uniform limits (in compact subsets of the open domain of nonexceptional momenta) once the volume is sent to infinity. Short-circuiting this reasoning, as we do here, is tantamount to postulating from the beginning that the FEs and the (violated) Slavnov-Taylor identities hold, for nonexceptional momenta, also in the limit $\Lambda \to 0$. By definition, the characteristic function of a finite measure $d \nu$ is its Fourier transform, see e.g. [@dlfm; @gljf], $$\begin{aligned} \chi(K)&:=\int d\nu(\Phi)\,e^{ \frac{i}{\hbar} K \cdot \Phi}\,. \label{def_cf}\end{aligned}$$ We consider a Gaussian measure $d\nu_{\Lambda \Lambda_0}$ with characteristic function $$\begin{aligned} \chi^{\Lambda\Lambda_0}(j,b,\bar{\eta},\eta)&=e^{\frac{1}{\hbar}\langle \bar{\eta}, S^{\Lambda\Lambda_0} \eta \rangle - \frac{1}{2\hbar}\langle j, C^{\Lambda\Lambda_0} j \rangle - \frac{1}{2 \hbar \xi} \langle b,b \rangle }, \label{def_gauss}\end{aligned}$$ and we denote as usual $$\begin{aligned} (C^{\Lambda \Lambda_0}j)_\mu(x)&=\int d^4 y\, C^{\Lambda \Lambda_0}_{\mu \nu}(x,y)j_\nu(y),\\ C^{\Lambda \Lambda_0}_{\mu \nu}(x,y)&=\int \frac{d^4 p}{(2 \pi)^4} \, e^{ip(x-y)}C^{\Lambda \Lambda_0}_{\mu \nu}(p),\end{aligned}$$ where in momentum space the regularized propagators are defined by the expressions $$\begin{aligned} C_{\mu \nu}(p)&:=\frac{1}{p^2}(\delta_{\mu \nu} + (\xi -1 )\frac{p_\mu p_\nu}{p^2}),&C^{\Lambda\Lambda_0}_{\mu \nu}(p)&:=C_{\mu \nu}(p)\,\sigma_{\Lambda\Lambda_0}(p^2)\,, \label{def_c}\\ S(p)&:=\frac{1}{p^2},&S^{\Lambda\Lambda_0}(p)&:=S(p)\,\sigma_{\Lambda\Lambda_0}(p^2)\,, \label{def_s}\\ \sigma_{\Lambda\Lambda_0}(s)&:=\sigma_{\Lambda_0}(s)-\sigma_{\Lambda}(s),&\sigma_{\lambda}(s)&:=\exp(-\frac{s^2}{\lambda^4})\,.\end{aligned}$$ For shortness we will also write $C^{-1}_{\Lambda \Lambda_0}$ instead of $(C^{\Lambda \Lambda_0})^{-1}$. The parameters $\Lambda$, $\Lambda_0$, such that $0 < \Lambda \leqslant \Lambda_0$, are respectively IR and UV cutoffs. Let $d\mu_{\Lambda\Lambda_0}$ be the measure defined by $$\begin{aligned} d \mu_{\Lambda\Lambda_0}(A,B, c,\bar{c}) := d \nu_{\Lambda\Lambda_0}(A,B-i\frac{1}{\xi}\partial A,c,\bar{c}). \label{eq_dm}\end{aligned}$$ For $\Phi=(A,B,c,\bar{c})$ and $K=(j,b , \bar{\eta}, \eta)$ and an infinitesimal variation $\delta \Phi=(\delta A,0,\delta c,\delta \bar{c})$, using the properties of Gaussian measures from appendix \[sec\_gauss\], we have $$\begin{aligned} d\mu_{\Lambda\Lambda_0}(\Phi+\delta \Phi)=&d\mu_{\Lambda\Lambda_0}(\Phi)\Big(1 + \frac{1}{\hbar}\langle \bar{c}, S^{-1}_{\Lambda\Lambda_0} \delta c \rangle + \frac{1}{\hbar}\langle \delta \bar{c}, S^{-1}_{\Lambda\Lambda_0} c \rangle \nonumber \\ &-\frac{1}{\hbar}\langle A, C^{-1}_{\Lambda\Lambda_0}\delta A\rangle -\frac{1}{\hbar} \langle i\partial(B -i \frac{1}{\xi}\partial A), \delta A \rangle \Big)\,. \label{eq_dm_delta}\end{aligned}$$ \[def\_z\] The free partition function $Z^{\Lambda\Lambda_0}_0$ is defined by $$\begin{aligned} Z^{\Lambda\Lambda_0}_0(K)&:=\int d \mu_{\Lambda\Lambda_0}(\Phi) e^{\frac{1}{\hbar} K \cdot \Phi}.\end{aligned}$$ It follows that $$\begin{aligned} Z^{\Lambda\Lambda_0}_0(j,b,\bar{\eta},\eta)&= \chi^{\Lambda\Lambda_0}(-i\,j-\frac{1}{\xi}\partial b,-i\,b,-i\,\bar{\eta},-i\,\eta),\label{eq_18z}\\ Z^{\Lambda \Lambda_0}_0(K)&=e^{\frac{1}{2\hbar}\langle K, \mathbf{C}^{\Lambda \Lambda_0} K \rangle},\end{aligned}$$ where $\mathbf{C}^{\Lambda \Lambda_0}$ is a 7x7 matrix, $$\mathbf{C}^{\Lambda \Lambda_0}:=\begin{pmatrix}C^{\Lambda \Lambda_0}_{\mu \nu},&S^{\Lambda \Lambda_0}p_\mu,&0,&0 \\-S^{\Lambda \Lambda_0}p_\nu,&\dfrac{1}{\xi}(1-\sigma_{\Lambda \Lambda_0}),&0,&0\\ 0,&0,&0,&-S^{\Lambda \Lambda_0}\\0,&0,&S^{\Lambda \Lambda_0},&0\end{pmatrix}.\label{eq_328s}$$ For $\Lambda<\Lambda_0$, $\mathbf{C}^{\Lambda \Lambda_0}$ is invertible: $$\mathbf{C}^{-1}_{\Lambda \Lambda_0}=\begin{pmatrix}C^{-1}_{\Lambda \Lambda_0; \mu \nu} - \dfrac{1}{\xi}p_\mu p_\nu,&-p_\mu,&0,&0\\p_\nu,&\xi,&0,&0\\0,&0,&0,&S^{-1}_{\Lambda \Lambda_0}\\0,&0,&-S^{-1}_{\Lambda \Lambda_0},&0\end{pmatrix} \label{eq_qform}.$$ We write the bosonic part of $\mathbf{C}^{-1}_{\Lambda \Lambda_0}$ as $\mathbf{P}^T \mathbf{Q}^{\Lambda \Lambda_0} \mathbf{P}$, where $\mathbf{P}$ is a diagonal matrix with $\mathbf{P}_{\mu \mu}=|p|$ for each index $\mu \in \{1,...4\}$ and $\mathbf{P}_{55}=1$. The eigenvalues $q_\alpha$ of $\mathbf{Q}^{\Lambda \Lambda_0}$ are: $$\begin{aligned} q_{1,2,3}&=\sigma^{-1}_{\Lambda \Lambda_0},&q_{4,5} &= \frac{\xi_{\Lambda \Lambda_0} \pm (\xi^2_{\Lambda \Lambda_0} -4\sigma^{-1}_{\Lambda \Lambda_0})^{\frac{1}{2}}}{2},&\xi_{\Lambda \Lambda_0}&:=\xi+\frac{1}{\xi}(\sigma^{-1}_{\Lambda \Lambda_0}-1).\end{aligned}$$ The fact that the real part of these eigenvalues is positive is known to be a prerequisite for the definition of a Gaussian measure for the bosonic part of the theory. A useful relation follows from , : $$\int d \mu_{\Lambda \Lambda_0}(\Phi) \, B \, e^{\frac{1}{\hbar}K \cdot \Phi }= \frac{1}{\xi} \int d \mu_{\Lambda \Lambda_0}(\Phi) \,(b + i\partial A ) \, e^{\frac{1}{\hbar}K \cdot \Phi }\,. \label{eq_17shift}$$ The partition function $Z^{\Lambda\Lambda_0}$ of $SU(2)$ Yang–Mills field theory is given by $$\begin{aligned} Z^{\Lambda\Lambda_0}(K)& := \int d \mu_{\Lambda\Lambda_0}(\Phi)\,e^{- \frac{1}{\hbar}L^{\Lambda_0\Lambda_0}}\,e^{\frac{1}{\hbar} K \cdot \Phi},& L^{\Lambda_0\Lambda_0}&:=\int d^4x \, \mathcal{L}^{\Lambda_0\Lambda_0} \label{eq_z}.\end{aligned}$$ The interaction Lagrangian density $\mathcal{L}^{\Lambda_0\Lambda_0}:=\mathcal{L}^{\Lambda_0\Lambda_0}_0 + \mathcal{L}^{\Lambda_0\Lambda_0}_{ct}$ is given by and $$\mathcal{L}^{\Lambda_0\Lambda_0}_0:=g\epsilon_{abc}(\partial_\mu A^a_\nu) A^b_\mu A^c_\nu + \frac{g^2}{4}\epsilon_{cab} \epsilon_{cds}A^a_\mu A^b_\nu A^d_\mu A^s_\nu - g\epsilon_{abc}(\partial_\mu\bar{c}^a) A^b_\mu c^c \,. \label{eq_l0}$$ Since we restrict to perturbation theory, all generating functionals are formal series in terms of $\hbar$ and of their source/field arguments. We also emphasize that $\mathcal{L}^{\Lambda_0\Lambda_0}_0$ does not depend on the $B$ field.\ From the expression for $\mathbf{C}^{-1}_{0 \infty}$ and equation one deduces the full tree level Lagrangian density $$\mathcal{L}^{tot}_{0}=\frac{1}{4}F_{\mu \nu}F_{\mu \nu} + \frac{\xi}{2}B^2 -iB\partial_\mu A_\mu - \partial_\mu \bar{c} D_\mu c\;.$$ The density $\mathcal{L}^{tot}_0$ is invariant under the infinitesimal BRST transformation [@brs], [@tyutin] $$\begin{aligned} \delta^{BRS} A&= \epsilon\,Dc,&\delta^{BRS} c&= \epsilon\,\frac{1}{2} ig \{c , c\} ,& \delta^{BRS} \bar{c}&= \epsilon\, i B ,& \delta^{BRS}B&=0,\end{aligned}$$ where $\epsilon$ is a Grassmann parameter, and $\{c,c\}^d=i\epsilon_{abd} c^a c^b $. Defining the classical operator $s$ by $\delta^{BRS} \Phi=\epsilon s\Phi$ one shows that $s$ is nilpotent. The generating functional of the Connected Schwinger (CS) functions is $$\begin{aligned} e^{\frac{1}{\hbar} W^{\Lambda\Lambda_0}(K)}:=Z^{\Lambda\Lambda_0}(K).\end{aligned}$$ The derivation of the FE is usually given considering the generating functional $L^{\Lambda\Lambda_0}$ of the Connected Amputated Schwinger (CAS) functions. The generating functional $L^{\Lambda\Lambda_0}$ of CAS functions is $$e^{-\frac{1}{\hbar}L^{\Lambda\Lambda_0}(\Phi)} := \int d \mu_{\Lambda\Lambda_0}(\Phi^\prime)\,e^{- \frac{1}{\hbar}L^{\Lambda_0\Lambda_0}(\Phi^\prime + \Phi)}.$$ From definition \[def\_z\] we have for any polynomial $P(\Phi)$ $$\frac{d}{d \Lambda} \int d \mu_{\Lambda \Lambda_0}(\Phi) P(\Phi)= \hbar \int d\mu_{\Lambda \Lambda_0}(\Phi) \langle \frac{\delta}{\delta \Phi},\hat{\mathbf{1}} \dot{\mathbf{C}}^{\Lambda \Lambda_0} \frac{\delta}{\delta \Phi} \rangle P(\Phi), \label{eq_poly}$$ where $$(\hat{\mathbf{1}})_{\phi \phi^\prime} = \left\{\begin{matrix} -\delta_{\phi \phi^\prime} & \mbox{if }\phi, \phi^\prime \in \{c,\bar{c}\},\\ \delta_{\phi \phi^\prime} & otherwise\,.\end{matrix}\right.$$ Using equation one obtains the FE, see e.g. [@gljf], [@kks], $$\dot{L}^{\Lambda \Lambda_0}(\Phi)= \frac{\hbar}{2} \langle \frac{\delta}{\delta \Phi}, \hat{\mathbf{1}} \dot{\mathbf{C}}^{\Lambda\Lambda_0} \frac{\delta}{\delta \Phi} \rangle L^{\Lambda \Lambda_0} -\frac{1}{2} \langle \frac{\delta L^{\Lambda \Lambda_0}}{\delta \Phi}, \hat{\mathbf{1}} \dot{\mathbf{C}}^{\Lambda\Lambda_0} \frac{\delta L^{\Lambda \Lambda_0}}{\delta \Phi} \rangle\;.\label{eq_fll}$$ From appendix \[sec\_gauss\] on Gaussian measures it follows that $$d \mu_{\Lambda \Lambda_0}(\Phi - \mathbf{C}^{\Lambda \Lambda_0}\delta \Phi)=d \mu_{\Lambda \Lambda_0}(\Phi) e^{-\frac{1}{2 \hbar} \langle \delta \Phi, \mathbf{C}^{\Lambda \Lambda_0}\delta \Phi\rangle}e^{\frac{1}{\hbar} \langle \delta \Phi, \Phi\rangle}.$$ This gives the relation between the generating functionals $W^{\Lambda \Lambda_0}$ and $L^{\Lambda \Lambda_0}$ $$W^{\Lambda \Lambda_0}(K)=\frac{1}{2} \langle K, \mathbf{C}^{\Lambda \Lambda_0} K \rangle - L^{\Lambda \Lambda_0}(\hat{\mathbf{1}}_{\bar{c}} \mathbf{C}^{\Lambda \Lambda_0}K),$$ where $$\begin{aligned} (\hat{\mathbf{1}}_{\bar{c}})_{\phi \phi^\prime} &= \left\{\begin{matrix} -1 & \mbox{if }\phi=\phi^\prime=\bar{c},\\ \delta_{\phi \phi^\prime} & otherwise,\end{matrix}\right.&(\hat{\mathbf{1}}_c)_{\phi \phi^\prime} &= \left\{\begin{matrix} -1 & \mbox{if }\phi=\phi^\prime=c,\\ \delta_{\phi \phi^\prime} & otherwise.\end{matrix}\right.\end{aligned}$$ \[def\_gamma\]For $0<\Lambda<\Lambda_0$, let $K^{\Lambda \Lambda_0}({\underline}{\Phi})$ be a solution of the system of equations $$\begin{aligned} {\underline}{A}-\frac{\delta W^{\Lambda \Lambda_0}}{\delta j}\Big|_{K^{\Lambda \Lambda_0}({\underline}{\Phi})}&=0,&{\underline}{B}-\frac{\delta W^{\Lambda \Lambda_0}}{\delta b}\Big|_{K^{\Lambda \Lambda_0}({\underline}{\Phi})}&=0,\label{627d}\\ {\underline}{c}-\frac{\delta W^{\Lambda \Lambda_0}}{\delta \bar{\eta}}\Big|_{K^{\Lambda \Lambda_0}({\underline}{\Phi})}&=0,&{\underline{\bar{c}}}+\frac{\delta W^{\Lambda \Lambda_0}}{\delta \eta}\Big|_{K^{\Lambda \Lambda_0}({\underline}{\Phi})}&=0.\label{627f}\end{aligned}$$ The effective action is $$\Gamma^{\Lambda \Lambda_0}({\underline}{\Phi}):=K^{\Lambda \Lambda_0}({\underline}{\Phi}) \cdot {\underline}{\Phi} - W^{\Lambda \Lambda_0}(K^{\Lambda \Lambda_0}({\underline}{\Phi}))\,.\label{eq_10gamma}$$ A solution of the system of equations , always exists as a formal series in $\hbar$ and in the fields. Same statement for the system . \[def\_gamme\] For $0<\Lambda\leqslant \Lambda_0$, let $\Phi^{\Lambda \Lambda_0}({\underline}{\Phi})$ be a solution of the equation $${\underline}{\Phi}=\left(\Phi - \mathbf{C}^{\Lambda \Lambda_0} \frac{\delta L^{\Lambda \Lambda_0}}{\delta \Phi}\right)\Big|_{\Phi^{\Lambda\Lambda_0}({\underline}{\Phi})}. \label{eq_phi}$$ The reduced effective action is $${\mathsf{\Gamma}}^{\Lambda \Lambda_0}({\underline}{\Phi}):= \Big(L^{\Lambda \Lambda_0}(\Phi) - \frac{1}{2} \langle \frac{\delta L^{\Lambda \Lambda_0}}{\delta \Phi},\hat{\mathbf{1}} \mathbf{C}^{\Lambda \Lambda_0} \frac{\delta L^{\Lambda \Lambda_0}}{\delta \Phi}\rangle \Big) \Big|_{\Phi^{\Lambda\Lambda_0}({\underline}{\Phi})}.\label{eq_17g}$$ From definitions \[def\_gamma\] and \[def\_gamme\] it follows that for $0<\Lambda< \Lambda_0$ $$\begin{aligned} {\mathsf{\Gamma}}^{\Lambda\Lambda_0}({\underline}{\Phi})=&\Gamma^{\Lambda\Lambda_0}({\underline}{\Phi}) -\frac{1}{2}\langle {\underline}{\Phi}, \mathbf{C}^{-1}_{\Lambda\Lambda_0} {\underline}{\Phi} \rangle,\label{eq_17g2}\\ \Phi^{\Lambda\Lambda_0}({\underline}{\Phi})=&\hat{\mathbf{1}}_{\bar{c}} \mathbf{C}^{\Lambda \Lambda_0}K^{\Lambda\Lambda_0}({\underline}{\Phi}),\\ {\mathsf{\Gamma}}^{\Lambda \Lambda_0}({\underline}{\Phi})=& \Big(L^{\Lambda \Lambda_0}(\Phi) - \frac{1}{2} \langle ({\underline}{\Phi}-\Phi),\mathbf{C}^{-1}_{\Lambda \Lambda_0} ({\underline}{\Phi}-\Phi)\rangle \Big) \Big|_{\Phi^{\Lambda\Lambda_0}({\underline}{\Phi})}. \label{eq_gamme}\end{aligned}$$ Using equation we see that, before the replacement, the rhs of , as a function of $\Phi$ for fixed ${\underline}{\Phi}$, has an extremum at $\Phi=\Phi^{\Lambda \Lambda_0}({\underline}{\Phi})$. Applying $\partial_\Lambda$ to equation , substituting $\dot{L}$ with the rhs of and using the property of extremum we obtain $$\dot{{\mathsf{\Gamma}}}^{\Lambda \Lambda_0}({\underline}{\Phi})= \frac{\hbar}{2} \langle \frac{\delta}{\delta \Phi}, \hat{\mathbf{1}} \dot{\mathbf{C}}^{\Lambda \Lambda_0} \frac{\delta}{\delta \Phi} \rangle L^{\Lambda \Lambda_0} \Big|_{\Phi^{\Lambda\Lambda_0}({\underline}{\Phi})}\,.$$ Defining $$\begin{aligned} W^{\Lambda \Lambda_0}_{q,-k}&:=(2\pi)^4 \frac{\delta^2 W^{\Lambda \Lambda_0}}{\delta K(-q) \delta K(k)}\,,&\Gamma^{\Lambda \Lambda_0}_{q,-k}&:=(2\pi)^4 \frac{\delta^2 \Gamma^{\Lambda \Lambda_0}}{\delta {\underline}{\Phi}(-q) \delta {\underline}{\Phi}(k)}\,,\end{aligned}$$ it is easy to see that $$\int d^4 k \, W^{\Lambda \Lambda_0}_{q,-k} \, \hat{\mathbf{1}}_{\bar{c}}\, \Gamma^{\Lambda \Lambda_0}_{k,-p} \, \hat{\mathbf{1}}_c=\delta(q-p)\,.$$ This implies that, using similar notations, $$L^{\Lambda \Lambda_0}_{q,-p} = \int d^4 k \, {\mathsf{\Gamma}}^{\Lambda \Lambda_0}_{q,-k} \left(\delta(k-p)+\hat{\mathbf{1}} \mathbf{C}^{\Lambda \Lambda_0}(p) {\mathsf{\Gamma}}^{\Lambda \Lambda_0}_{-p,k}\right)^{-1}\,.$$ Eventually, one obtains the FE for the functional ${\mathsf{\Gamma}}$ and also for the functional ${\mathsf{\Gamma}}_\chi$ with one operator insertion of source $\chi$ $$\begin{aligned} \dot{{\mathsf{\Gamma}}}&=\frac{\hbar}{2} \langle \dot{\mathbf{C}} \, \delta_\phi\delta_{\bar{\phi}}{\mathsf{\Gamma}},\left(1+\hat{\mathbf{1}}\mathbf{C}\, \delta_{\bar{\phi}} \delta_\phi {\mathsf{\Gamma}}\right)^{-1} \rangle\,,\label{eq_fl}\\ \delta_\chi\dot{{\mathsf{\Gamma}}}&=\frac{\hbar}{2} \langle \dot{\mathbf{C}}\left(1+ \delta_{\bar{\phi}} \delta_\phi{\mathsf{\Gamma}}\, \mathbf{C}\hat{\mathbf{1}} \right)^{-1} \delta_\chi \delta_{\bar{\phi}} \delta_\phi {\mathsf{\Gamma}}, \left(1+\hat{\mathbf{1}}\mathbf{C} \, \delta_{\bar{\phi}} \delta_\phi {\mathsf{\Gamma}}\right)^{-1} \rangle\,, \label{eq_fl1}\end{aligned}$$ where $\phi$, $\bar{\phi} \in \{{\underline}{A},{\underline}{B}, {\underline}{c},{\underline{\bar{c}}}\}$ and we omit appropriate sums over field labels. Generalization to $\Vec{\chi}=(\chi_1,...,\chi_{n_\chi})$ with $n_{\chi} > 1$ is straightforward. The FE for ${\mathsf{\Gamma}}$ in modern form has been introduced in [@Wet2; @bam2; @mor; @kks2; @komu]. Flow equations with composite operator insertions have been introduced in [@keko]. The mass dimension of a vertex function $\Gamma^{\Vec{\phi};w}_{\Vec{\chi}}(\Vec{p})$ with ${\textsc{n}}$ fields $\phi_i \in \{A,B,c,\bar{c}\}$, $n_{\chi}$ insertions of sources $\chi_i$ and $\|w\|$ momentum derivatives is $d:=4-\sum^{n_\chi}_i [\chi_i]- \sum^{{\textsc{n}}}_i [\phi_i] - \|w\|$, where $[F]$ stands for the mass dimension of $F$ in position space. We say that such a term is **irrelevant** if $d<0$, as for example $\Gamma^{AAc;w}_\gamma$ for $[\gamma]=2$, and **relevant** otherwise. Furthermore, we call a relevant term **marginal** if $d=0$, as for example $\Gamma^{AAAA}$, or **strictly relevant** if $d>0$.\ Expanding in formal power series in $\hbar$ we have $${\mathsf{\Gamma}}^{\Lambda \Lambda_0}({\underline}{\Phi})=\sum^{\infty}_{l=0} \hbar^l {\mathsf{\Gamma}}^{\Lambda \Lambda_0}_l({\underline}{\Phi})\,.$$ We also note that the FE and admit an inductive structure in the loop number. This property allows us to prove statements by induction, first establishing them at tree-level, then proving that if they hold up to loop order $l-1\geqslant 0$ they are also valid at order $l$.\ The proposition that follows proves that vertex functions ${\mathsf{\Gamma}}$ involving $B$ fields do vanish. We use the notation $B\Vec{\phi}$ to denote sequences of field labels with $\phi_i\in\{A,B,c,\bar{c}\}$. Assume vanishing renormalization conditions for all relevant terms with at least one $B$ field: $$\begin{aligned} {\mathsf{\Gamma}}^{B\Vec{\phi};0\Lambda_0;w}_l(\Vec{q})&=0\,, \label{eq_16cdt3}\end{aligned}$$ where $\Vec{q}$ is nonexceptional for marginal terms and vanishing otherwise. Then, for all $B\Vec{\phi}$, $l$, $w$, $\Vec{p}$, and $0<\Lambda\leqslant \Lambda_0$, $${\mathsf{\Gamma}}^{B\Vec{\phi};\Lambda\Lambda_0;w}_l(\Vec{p}) = 0\,.$$ Rank-2 marginal terms are tensors with two vector indices, say $\mu$ and $\nu$, and can be decomposed in the basis $\{ \delta_{\mu\nu}, q_{i\mu} q_{j\nu} \}$, where $q_i,q_j$ range over all momenta in $\vec{q}$ other than $q_0$. For rank-2 marginal terms only the coefficient of $\delta_{\mu\nu}$ is set to zero in . #### Proof We prove the statement by induction, increasing in the loop order, $l-1\mapsto l$. Given $l$, we proceed by descending from $w_{max}$ in the number of derivatives, $\|w\| \mapsto \|w\|-1$. For fixed $l$ and $w$, all possible terms ${\mathsf{\Gamma}}^{B\Vec{\phi};w}_l$ are considered. By construction, for fixed $l$ and $B\Vec{\phi}$, the inductive scheme deals first with the irrelevant terms and continues, if they exist, with the marginal terms, followed by more and more relevant terms. The identity ${\mathsf{\Gamma}}^{B\Vec{\phi};\Lambda\Lambda_0;w}_0(\Vec{p})=0$ follows from the definition of ${\mathsf{\Gamma}}$. Assume that the statement of the theorem holds up to loop order $l-1\geqslant 0$. It follows that at order $l$ the rhs of the FE for vertex functions with $B$ fields vanishes. Using the FE we integrate the irrelevant terms from $\Lambda_0$ downward to arbitrary $\Lambda>0$. The boundary conditions ${\mathsf{\Gamma}}^{B\Vec{\phi};\Lambda_0\Lambda_0;w}_l(\Vec{p})=0$ and the vanishing of the rhs of the FE imply that all irrelevant terms with $B$ fields vanish at loop order $l$. The boundary conditions ${\mathsf{\Gamma}}^{B\Vec{\phi}_r;0\Lambda_0;w}_l(\Vec{q})=0$ and the vanishing of the rhs of the FE imply that all marginal terms vanish at their renormalization point for arbitrary $\Lambda>0$. The derivatives wrt momenta of marginal terms are irrelevant terms. Consequently, we conclude that the marginal ${\mathsf{\Gamma}}^{B\Vec{\phi};\Lambda \Lambda_0;w}_l(\Vec{p})$ vanish for all $\Vec{p}$. Similar arguments hold for all strictly relevant terms.$\blacksquare$\ In the following we will always adopt the renormalization conditions . Consequently, counterterms involving $B$ fields are not generated. Let us denote by $\tilde{W}$, $\tilde{Z}$ the functional $W$, $Z$ with $b$ set to zero. $$\begin{aligned} \tilde{W}(j,\bar{\eta},\eta)&:=W(j,0,\bar{\eta},\eta),&\tilde{Z}(j,\bar{\eta},\eta)&:=Z(j,0,\bar{\eta},\eta)\,. \label{eq_tilda}\end{aligned}$$ The covariance matrix $\tilde{\mathbf{C}}$ is obtained from $\mathbf{C}$ by removing the column and row which correspond to $b$. $$W^{\Lambda\Lambda_0}(j,b,\bar{\eta},\eta)=\frac{1}{2\xi}\langle b,b \rangle + \tilde{W}^{\Lambda\Lambda_0}(j-i\frac{1}{\xi}\partial b,\bar{\eta},\eta)\,. \label{eq_w}$$ #### Proof Using the definition of the partition function $Z^{\Lambda\Lambda_0}$ one computes $$\begin{aligned} Z^{\Lambda\Lambda_0}(j+i\frac{1}{\xi}\partial b,b,\bar{\eta},\eta)&=e^{- \frac{1}{\hbar}\mathbb{L}^{\Lambda_0\Lambda_0}}\chi^{\Lambda\Lambda_0}(ij,ib,i\bar{\eta},i\eta),\\ \mathbb{L}^{\Lambda_0\Lambda_0}&:=L^{\Lambda_0\Lambda_0}(\frac{\delta}{\delta j},\frac{\delta}{\delta \bar{\eta}},\frac{\delta}{\delta \eta})\,.\end{aligned}$$ From definition it follows $$\begin{aligned} Z^{\Lambda\Lambda_0}(j+i\frac{1}{\xi}\partial b,b,\bar{\eta},\eta)&=e^{\frac{1}{2 \hbar \xi} \langle b,b \rangle }e^{- \frac{1}{\hbar}\mathbb{L}^{\Lambda_0\Lambda_0}}e^{-\frac{1}{\hbar}\langle \bar{\eta}, S^{\Lambda\Lambda_0} \eta \rangle + \frac{1}{2\hbar}\langle j, C^{\Lambda\Lambda_0} j \rangle}\,.\end{aligned}$$ Observing that the expression multiplying $e^{\frac{1}{2 \hbar \xi} \langle b,b \rangle }$ is $\tilde{Z}^{\Lambda\Lambda_0}(j,\bar{\eta},\eta)$ we obtain $$\begin{aligned} Z^{\Lambda\Lambda_0}(j,b,\bar{\eta},\eta)&=e^{\frac{1}{2 \hbar \xi} \langle b,b \rangle }\tilde{Z}^{\Lambda\Lambda_0}(j-i\frac{1}{\xi}\partial b,\bar{\eta},\eta)\,.\end{aligned}$$ Taking the logarithm finishes the proof. $\blacksquare$\ Substitution of $W^{\Lambda\Lambda_0}$ (\[eq\_w\]) into the definition of $\Gamma^{\Lambda\Lambda_0}$ and integration by parts give $$\begin{aligned} \Gamma^{\Lambda\Lambda_0}({\underline}{A},{\underline}{B},{\underline{\bar{c}}},{\underline}{c})&=\langle b, {\underline}{B}-i\frac{1}{\xi}\partial {\underline}{A} - \frac{1}{2} b \rangle + \tilde{\Gamma}^{\Lambda\Lambda_0}({\underline}{A},{\underline{\bar{c}}},{\underline}{c}),\\ \tilde{\Gamma}^{\Lambda\Lambda_0}({\underline}{A},{\underline{\bar{c}}},{\underline}{c})&:=\langle j, {\underline}{A} \rangle + \langle \bar{\eta}, {\underline}{c} \rangle + \langle {\underline{\bar{c}}}, \eta \rangle - \tilde{W}^{\Lambda\Lambda_0}(j,\bar{\eta},\eta)\Big|_{K^{\Lambda \Lambda_0}({\underline}{\Phi})}\,. \label{eq_legendre_w}\end{aligned}$$ From definition \[def\_gamma\] it follows that $b=\xi {\underline}{B} -i \partial {\underline}{A}$. Consequently, the above expression becomes $$\begin{aligned} \Gamma^{\Lambda\Lambda_0}({\underline}{A},{\underline}{B},{\underline{\bar{c}}},{\underline}{c})&=\frac{1}{2\xi} \langle \xi{\underline}{B}-i\partial {\underline}{A}, \xi{\underline}{B}-i\partial {\underline}{A}\rangle + \tilde{\Gamma}^{\Lambda\Lambda_0}({\underline}{A},{\underline{\bar{c}}},{\underline}{c})\,.\label{eq_23gamma}\end{aligned}$$ Differentiation wrt ${\underline}{A}$ yields an important identity: $$\begin{aligned} \frac{\delta \Gamma^{\Lambda\Lambda_0}}{\delta {\underline}{A}^a_\mu}&=i\partial_\mu({\underline}{B}^a-i\frac{1}{\xi}\partial {\underline}{A}^a)+\frac{\delta \tilde{\Gamma}^{\Lambda\Lambda_0}}{\delta {\underline}{A}^a_\mu}\,. \label{eq_partial_A}\end{aligned}$$ For $\tilde{\Phi}:=(A^a_\mu,c^a,\bar{c}^a)$ the functional $\tilde{{\mathsf{\Gamma}}}^{\Lambda \Lambda_0}(\tilde{{\underline}{\Phi}})$ is defined from $L^{\Lambda \Lambda_0}(\tilde{\Phi})$ in analogy with . For ${0<\Lambda<\Lambda_0}$, it follows that $$\tilde{\Gamma}^{\Lambda \Lambda_0}(\tilde{{\underline}{\Phi}})=\tilde{{\mathsf{\Gamma}}}^{\Lambda \Lambda_0}(\tilde{{\underline}{\Phi}}) + \frac{1}{2}\langle \tilde{{\underline}{\Phi}}, \tilde{\mathbf{C}}^{-1}_{\Lambda \Lambda_0}\, \tilde{{\underline}{\Phi}} \rangle.\label{eq_17g3}$$ Substitution and into yields ${\mathsf{\Gamma}}^{\Lambda \Lambda_0}({\underline}{\Phi})=\tilde{{\mathsf{\Gamma}}}^{\Lambda \Lambda_0}(\tilde{{\underline}{\Phi}})$. Note also that at $\Lambda=\Lambda_0$ we have ${\underline}{\Phi}=\Phi$ and &\^[\_0 \_0]{}()=L\^[\_0 \_0]{}()=L\^[\_0 \_0]{}()=\^[\_0 \_0]{}()=d\^4 x (\^[\_0 \_0]{}\_0 + \^[\_0 \_0]{}\_[ct]{}).&\[eq\_27j\] Violated Slavnov–Taylor identities {#sec_vsti} ---------------------------------- We are working in a framework where gauge invariance is broken already in the classical lagrangian due to the gauge fixing term. It has then been realized that invariance of the lagrangian under the BRST transformations ensures the gauge invariance of physical quantities to be calculated from the theory [@niel; @pig]. On the level of correlation functions (Green’s functions in the relativistic theory) this invariance leads to a system of identities between different correlation functions which are called Slavnov Taylor identities (STI) [@st; @st2; @taylor]. These identities may be used to argue that physical quantities obtained from these functions, as for example the pole of the propagators for all physical fields of the Standard Model, are gauge invariant [@gg]. In our framework gauge invariance is also violated in an even more serious way by the presence of the regulators in , . We want to show that for a suitable class of renormalization conditions, which does not restrict the freedom in fixing the physical coupling constant and the normalization of the fields, gauge invariance can be recovered in the renormalized theory. This means we want to show that the STI hold once we take the limits $\Lambda \to 0$ and $\Lambda_0 \to \infty$. The first step is then to write a system of violated STI suitable for our subsequent analysis of their restoration. To do so we thus analyse the behavior of the regularized generating functionals of the correlation functions under BRST transformations. The infinitesimal BRST transformations can be generated by composite operator insertions for which we also have a freedom of normalization, as encoded by the constants $R_i$ introduced below [@komu]. We derive the *violated* STI setting $\Lambda=0$. Remember the comments in the beginning of section \[sec\_gf\] as regards the implications of this choice for the generating functionals on the IR side. We consider the functional $Z^{0\Lambda_0}_{vst}$ defined with the modified Lagrangian density $$\begin{aligned} \mathcal{L}^{\Lambda_0\Lambda_0}_{vst}&:=\mathcal{L}^{\Lambda_0\Lambda_0} + \gamma \psi^{\Lambda_0} + \omega \Omega^{\Lambda_0},&L^{\Lambda_0\Lambda_0}_{vst}&:=\int d^4 x \, \mathcal{L}^{\Lambda_0\Lambda_0}_{vst},\label{eq_17vst}\end{aligned}$$ where $\gamma$, $\omega$ are external sources, and $$\begin{aligned} \psi^{\Lambda_0}&:=R^{\Lambda_0}_1 \partial c -igR^{\Lambda_0}_2[A,c],&\Omega^{\Lambda_0}&:=\frac{1}{2i}gR^{\Lambda_0}_3\{c,c\},&R^{\Lambda_0}_i&=1+O(\hbar)\,. \label{eq_21brs}\end{aligned}$$ The requirement that at zero loop order $\psi^{\Lambda_0}$ and $\Omega^{\Lambda_0}$ correspond to the classical BRST variation implies that the constants $R^{\Lambda_0}_i$ are equal to one at tree level. At higher orders we admit counterterms for the BRST transformation [@komu]. Performing the change of variables $\Phi \mapsto \Phi + \delta_\epsilon \Phi$ we obtain the identity $$\begin{aligned} -\hbar \int \delta_\epsilon (d\mu_{0\Lambda_0}e^{- \frac{1}{\hbar} L^{\Lambda_0\Lambda_0}_{vst}})\,e^{\frac{1}{\hbar}K \cdot \Phi }=\int d\mu_{0\Lambda_0}e^{- \frac{1}{\hbar} L^{\Lambda_0\Lambda_0}_{vst} + \frac{1}{\hbar}K \cdot \Phi } K\cdot \delta_\epsilon (\Phi),\label{eq_17delta}\end{aligned}$$ where $$\begin{aligned} \delta_\epsilon A&:=\epsilon \, \sigma_{0\Lambda_0} * \psi^{\Lambda_0},&\delta_\epsilon c&:=-\epsilon \, \sigma_{0\Lambda_0} * \Omega^{\Lambda_0},&\delta_\epsilon \bar{c}&:=\epsilon \, \sigma_{0\Lambda_0} * iB\,\label{eq_20brs}.\end{aligned}$$ Using formula for an infinitesimal change of variables and substituting the variation $\delta_\epsilon \Phi$ with its explicit form we have $$\begin{aligned} -\hbar\, \delta_\epsilon(d\mu_{0 \Lambda_0}(\Phi))=&d\mu_{0 \Lambda_0}(\Phi)\,\epsilon \, I^{\Lambda_0}_1(\Phi)\,,\\ -\hbar\, \delta_\epsilon(e^{-\frac{1}{\hbar}L^{\Lambda_0 \Lambda_0}_{vst}(\Phi)} )=&e^{-\frac{1}{\hbar}L^{\Lambda_0 \Lambda_0}_{vst}(\Phi)} \epsilon \, I^{\Lambda_0}_2(\Phi)\,,\end{aligned}$$ where $$\begin{aligned} I^{\Lambda_0}_1(\Phi):=&\langle A,C^{-1} \psi^{\Lambda_0}\rangle-\langle \bar{c}, S^{-1} \Omega^{\Lambda_0}\rangle - i \langle B, S^{-1} c\rangle\nonumber\\ &+i\langle\partial(B-i\frac{1}{\xi}\partial A), \sigma_{0\Lambda_0} * \psi^{\Lambda_0} \rangle\,,\\ I^{\Lambda_0}_2(\Phi):=&\langle \sigma_{0\Lambda_0} * \frac{\delta L^{\Lambda_0 \Lambda_0}}{\delta A}, \psi^{\Lambda_0} \rangle - \langle \sigma_{0\Lambda_0} * \frac{\delta L^{\Lambda_0 \Lambda_0}}{\delta c}, \Omega^{\Lambda_0} \rangle\nonumber\\ &+i\langle B,\sigma_{0\Lambda_0} * \frac{\delta L^{\Lambda_0 \Lambda_0}}{\delta \bar{c}}\rangle + \langle \gamma, Q^{\Lambda_0}_{\rho \gamma} \rangle + \langle \omega, Q^{\Lambda_0}_{\rho\omega} \rangle \,,\\ Q^{\Lambda_0}_{\rho \gamma^a_\mu}:=& g(R^{\Lambda_0}_3-R^{\Lambda_0}_2)\epsilon^{abs} (\sigma_{0\Lambda_0}*\psi^{\Lambda_0})^b_\mu c^s \nonumber\\ &+gR^{\Lambda_0}_1R^{\Lambda_0}_3\epsilon^{abs}( c^s (\sigma_{0\Lambda_0}*\partial_\mu c^b) - \sigma_{0\Lambda_0}* (c^s \partial_\mu c^b)) \nonumber\\ &+g^2R^{\Lambda_0}_2R^{\Lambda_0}_3\epsilon^{ase}\epsilon^{ebt}((\sigma_{0\Lambda_0}* (A^b_\mu c^t)) c^s - A^b_\mu (\sigma_{0\Lambda_0}* (c^t c^s)))\,,\label{eq_11rg}\\ Q^{\Lambda_0}_{\rho\omega^a}:=& (gR^{\Lambda_0}_3)^2 c^s (\sigma_{0\Lambda_0} * (c^a c^s))\,.\label{eq_11rw}\end{aligned}$$ The terms $Q^{\Lambda_0}_{\rho \gamma}$ and $Q^{\Lambda_0}_{\rho\omega}$ originate from the variations $\delta_\epsilon(\gamma \psi^{\Lambda_0})$, $\delta_\epsilon(\omega \Omega^{\Lambda_0})$, see . Substituting $B \mapsto \frac{1}{\xi}(b + i \partial A)$, see , in both terms $I^{\Lambda_0}_i$ we get $$\int d\mu_{0 \Lambda_0}\, I^{\Lambda_0}_i \, e^{-\frac{1}{\hbar}L^{\Lambda_0 \Lambda_0}_{vst} + \frac{1}{\hbar}K \cdot \Phi} = \int d\mu_{0 \Lambda_0}\, J^{\Lambda_0}_i\, e^{-\frac{1}{\hbar}L^{\Lambda_0 \Lambda_0}_{vst} + \frac{1}{\hbar}K \cdot \Phi},$$ where $$\begin{aligned} J^{\Lambda_0}_1(\Phi):=&\langle A,C^{-1} \psi^{\Lambda_0}\rangle-\langle \bar{c}, S^{-1} \Omega^{\Lambda_0}\rangle + \frac{1}{\xi} \langle \partial A, S^{-1} c\rangle\nonumber\\ &-i\frac{1}{\xi}\langle b,S^{-1} c + \sigma_{0\Lambda_0} * \partial \psi^{\Lambda_0} \rangle\,,\\ J^{\Lambda_0}_2(\Phi):=&\langle \sigma_{0\Lambda_0} * \frac{\delta L^{\Lambda_0 \Lambda_0}}{\delta A},\psi^{\Lambda_0}\rangle - \langle \sigma_{0\Lambda_0} * \frac{\delta L^{\Lambda_0 \Lambda_0}}{\delta c},\Omega^{\Lambda_0}\rangle\nonumber\\ &+i\frac{1}{\xi}\langle b + i\partial A, \sigma_{0\Lambda_0} * \frac{\delta L^{\Lambda_0 \Lambda_0}}{\delta \bar{c}}\rangle + \langle \gamma, Q^{\Lambda_0}_{\rho \gamma} \rangle + \langle \omega, Q^{\Lambda_0}_{\rho\omega} \rangle\,.\end{aligned}$$ Introducing the operators $Q^{\Lambda_0}_\rho$ and $Q^{\Lambda_0}_{\beta}$, $$\begin{aligned} Q^{\Lambda_0}_\rho:=&\frac{\delta L^{\Lambda_0\Lambda_0}}{\delta A_\mu}\sigma_{0\Lambda_0}*\psi^{\Lambda_0}_\mu - \frac{\delta L^{\Lambda_0\Lambda_0}}{\delta c}\sigma_{0\Lambda_0}*\Omega^{\Lambda_0} - \frac{1}{\xi} \frac{\delta L^{\Lambda_0 \Lambda_0}}{\delta \bar{c}} \sigma_{0\Lambda_0}* \partial A \nonumber\\ &+A C^{-1} \psi^{\Lambda_0} - \bar{c} S^{-1}\Omega^{\Lambda_0} + \frac{1}{\xi} \partial A S^{-1} c\,,\label{eq_328a}\\ Q^{\Lambda_0}_{\beta}:=&\sigma_{0\Lambda_0}* \Big(\frac{\delta L^{\Lambda_0\Lambda_0}}{\delta \bar{c}} - \partial \psi^{\Lambda_0}\Big) - S^{-1}c\,,\label{eq_10a}\end{aligned}$$ we have $$J^{\Lambda_0}_1 + J^{\Lambda_0}_2= i \frac{1}{\xi}\langle b, Q^{\Lambda_0}_{\beta} \rangle + \langle \gamma, Q^{\Lambda_0}_{\rho \gamma} \rangle + \langle \omega, Q^{\Lambda_0}_{\rho\omega} \rangle + \int d^4x \, Q^{\Lambda_0}_\rho(x) \,.$$ We may now express the lhs of  as $$\epsilon \int d \mu_{0 \Lambda_0} (J^{\Lambda_0}_1 + J^{\Lambda_0}_2) \, e^{-\frac{1}{\hbar}L^{\Lambda_0 \Lambda_0}_{vst} + \frac{1}{\hbar} K \cdot \Phi} =-\epsilon \, \mathcal{D} Z^{0 \Lambda_0}_{aux}\Big|_{\rho,\beta=0},\label{eq_20l}$$ where $$\begin{aligned} \mathcal{D}:=&\int d^4x \, \hbar \frac{\delta}{\delta \rho(x)} \,+ i\frac{1}{\xi}\langle b, \hbar\frac{\delta}{\delta \beta} \rangle,\label{eq_20d}\\ Z^{0\Lambda_0}_{aux}:=&\int d\mu_{0\Lambda_0}(\Phi)\,e^{- \frac{1}{\hbar} L^{\Lambda_0\Lambda_0}_{aux}+\frac{1}{\hbar}K \cdot \Phi },\\ \mathcal{L}^{\Lambda_0\Lambda_0}_{aux}:=&\mathcal{L}^{\Lambda_0\Lambda_0}_{vst} +\rho Q^{\Lambda_0}_\rho + \rho \gamma^a_\mu Q^{\Lambda_0}_{\rho \gamma^a_\mu} + \rho \omega^a Q^{\Lambda_0}_{\rho \omega^a}+ \beta Q^{\Lambda_0}_{\beta}\,.\label{eq_17aux}\end{aligned}$$ Defining $$\begin{aligned} \mathcal{S}:=&\langle j,\sigma_{0\Lambda_0} * \hbar \frac{\delta}{\delta \gamma} \rangle + \langle \bar{\eta},\sigma_{0\Lambda_0}* \hbar \frac{\delta}{\delta \omega} \rangle - i\langle \sigma_{0\Lambda_0} * \hbar \frac{\delta}{\delta b},\eta \rangle,\label{eq_20s}\end{aligned}$$ we write the rhs of in the following form $$\int d\mu_{0\Lambda_0}\,e^{-\frac{1}{\hbar}L^{\Lambda_0 \Lambda_0}_{vst} + \frac{1}{\hbar} K \cdot \Phi} K \cdot \delta_\epsilon \Phi =-\epsilon \, \mathcal{S}Z^{0 \Lambda_0}_{vst}=-\epsilon \,\mathcal{S}Z^{0 \Lambda_0}_{aux}\Big|_{\rho,\beta=0}.\label{eq_20r}$$ Using equations and we write identity as follows $$\mathcal{D} Z^{0\Lambda_0}_{aux}\Big|_{\rho,\beta=0} = \mathcal{S} Z^{0\Lambda_0}_{aux}\Big|_{\rho,\beta=0}\,.\label{eq_21z}$$ Then we perform the Legendre transform on the auxiliary functional $W_{aux}:=\hbar \log Z_{aux}$. This defines the functional $\Gamma_{aux}$, from which are obtained $\tilde{\Gamma}_{aux}$, see , and the functionals $$\begin{aligned} \tilde{{\mathsf{\Gamma}}}^{0\Lambda_0}_{\chi}&:=\frac{\delta \tilde{\Gamma}^{0\Lambda_0}_{aux}}{\delta \chi}\Big|_{\chi=0},&\tilde{{\mathsf{\Gamma}}}^{0\Lambda_0}_1&:= \int d^4 x \, \tilde{\Gamma}^{0\Lambda_0}_{\rho(x)}\,,\label{eq_11ga}\end{aligned}$$ where $\chi \in \{\rho,\beta\}$ and $\gamma$, $\omega$ are arbitrary. Equation then gives $$\tilde{{\mathsf{\Gamma}}}^{0\Lambda_0}_1 + \langle (i{\underline}{B}+\frac{1}{\xi}\partial {\underline}{A}),\tilde{{\mathsf{\Gamma}}}^{0\Lambda_0}_\beta\rangle =\frac{1}{2}\mathcal{S}{{{\mkern2mu\underline{\mkern-2mu \Gamma\mkern-2mu}\mkern2mu }}}^{0 \Lambda_0}, \label{eq_vst}$$ where $$\begin{aligned} {{{\mkern2mu\underline{\mkern-2mu \Gamma\mkern-2mu}\mkern2mu }}}^{0 \Lambda_0}&:=i\langle {\underline}{B},\bar{\omega} \rangle + \tilde{{{{\mkern2mu\underline{\mkern-2mu \Gamma\mkern-2mu}\mkern2mu }}}}^{0 \Lambda_0},&\tilde{{{{\mkern2mu\underline{\mkern-2mu \Gamma\mkern-2mu}\mkern2mu }}}}^{0 \Lambda_0}&:= \tilde{\Gamma}^{0 \Lambda_0} + \frac{1}{2\xi}\langle {\underline}{A}, \partial \partial \, {\underline}{A} \rangle\,,\end{aligned}$$ and $\mathcal{S}:=\mathcal{S}_{\tilde{c}} + \mathcal{S}_A - \mathcal{S}_c$, with $$\begin{aligned} \mathcal{S}_A&:=\langle \frac{\delta {{{\mkern2mu\underline{\mkern-2mu \Gamma\mkern-2mu}\mkern2mu }}}^{0 \Lambda_0}}{\delta {\underline}{A}},\sigma_{0\Lambda_0}*\frac{\delta}{\delta \gamma}\rangle + \langle \frac{\delta {{{\mkern2mu\underline{\mkern-2mu \Gamma\mkern-2mu}\mkern2mu }}}^{0\Lambda_0}}{\delta \gamma},\sigma_{0\Lambda_0}*\frac{\delta}{\delta {\underline}{A}}\rangle,\\ \mathcal{S}_{\tilde{c}}&:=\langle \frac{\delta {{{\mkern2mu\underline{\mkern-2mu \Gamma\mkern-2mu}\mkern2mu }}}^{0 \Lambda_0}}{\delta \tilde{c}},\sigma_{0\Lambda_0}*\frac{\delta}{\delta \bar{\omega}} \rangle + \langle \frac{\delta {{{\mkern2mu\underline{\mkern-2mu \Gamma\mkern-2mu}\mkern2mu }}}^{0 \Lambda_0}}{\delta \bar{\omega}},\sigma_{0\Lambda_0}* \frac{\delta}{\delta \tilde{c}} \rangle,\\ \mathcal{S}_c&:=\langle \frac{\delta {{{\mkern2mu\underline{\mkern-2mu \Gamma\mkern-2mu}\mkern2mu }}}^{0\Lambda_0}}{\delta {\underline}{c}},\sigma_{0\Lambda_0}*\frac{\delta}{\delta \omega}\rangle + \langle \frac{\delta {{{\mkern2mu\underline{\mkern-2mu \Gamma\mkern-2mu}\mkern2mu }}}^{0\Lambda_0}}{\delta \omega},\sigma_{0\Lambda_0}*\frac{\delta}{\delta {\underline}{c}}\rangle\,,\\ \frac{\delta}{\delta \tilde{c}}&:=\frac{\delta }{\delta {\underline{\bar{c}}}} - \partial \frac{\delta }{\delta \gamma}\,.\label{eq_409s}\end{aligned}$$ We rewrite equation  in the following form $$\begin{aligned} \langle i{\underline}{B},\tilde{{\mathsf{\Gamma}}}^{0\Lambda_0}_\beta \rangle &=\frac{1}{2}\mathcal{S}_{\tilde{c}} {{{\mkern2mu\underline{\mkern-2mu \Gamma\mkern-2mu}\mkern2mu }}}^{0\Lambda_0}=\langle i{\underline}{B},\sigma_{0\Lambda_0}*\frac{\delta}{\delta \tilde{c}} \tilde{{{{\mkern2mu\underline{\mkern-2mu \Gamma\mkern-2mu}\mkern2mu }}}}^{0\Lambda_0}\rangle,\label{eq_407c}\\ \tilde{{\mathrm{F}}}^{0 \Lambda_0}_1&=\frac{1}{2}\tilde{S}\tilde{{{{\mkern2mu\underline{\mkern-2mu \Gamma\mkern-2mu}\mkern2mu }}}}^{0\Lambda_0},\label{eq_407b}\end{aligned}$$ where $$\begin{aligned} \tilde{S}&:=\mathcal{S}_A - \mathcal{S}_c,&\tilde{{\mathrm{F}}}^{0 \Lambda_0}_1:=\tilde{{\mathsf{\Gamma}}}^{0 \Lambda_0}_1 + \frac{1}{\xi}\langle \partial A, \tilde{{\mathsf{\Gamma}}}^{0\Lambda_0}_\beta \rangle\,.\label{eq_407a}\end{aligned}$$ The introduction of the functional $\tilde{{{{\mkern2mu\underline{\mkern-2mu \Gamma\mkern-2mu}\mkern2mu }}}}$ leads to relation and to the consistency condition given in below. They are important in the analysis of the renormalization conditions for $\tilde{{\mathsf{\Gamma}}}^{0\Lambda_0;\Vec{\phi}}_1$, see section \[sec\_sti\].\ An algebraic computation shows that $$\begin{aligned} (\mathcal{S}_i\mathcal{S}_j + \mathcal{S}_j\mathcal{S}_i){{{\mkern2mu\underline{\mkern-2mu \Gamma\mkern-2mu}\mkern2mu }}}^{0\Lambda_0}&=0,& \forall i,j &\in \{A,c,\tilde{c}\}\,.\label{eq_88gr} \end{aligned}$$ Consequently $\tilde{S}^2 \tilde{{{{\mkern2mu\underline{\mkern-2mu \Gamma\mkern-2mu}\mkern2mu }}}}^{0\Lambda_0}=0$. Thus application of the operator $\tilde{S}$ to equation  yields $$\tilde{S}\tilde{{\mathrm{F}}}^{0\Lambda_0}_1=0.\label{eq_cohom}$$ Using again , we also have $$\frac{1}{2}(\tilde{S}\mathcal{S}_{\tilde{c}}+\mathcal{S}_{\tilde{c}}\tilde{S}){{{\mkern2mu\underline{\mkern-2mu \Gamma\mkern-2mu}\mkern2mu }}}^{0 \Lambda_0}=0,\quad\mbox{and thus}\quad \tilde{S} \tilde{{\mathsf{\Gamma}}}_\beta + \sigma_{0\Lambda_0}*\frac{\delta}{\delta \tilde{c}} \tilde{{\mathrm{F}}}^{0\Lambda_0}_1 =0\,.\label{eq_404a}$$ Finally we set $\gamma,\omega=0$ in , to get the AGE and the STI: $$\begin{aligned} \tilde{{\mathsf{\Gamma}}}^{0\Lambda_0}_\beta=&\sigma_{0\Lambda_0}* \Big(\frac{\delta \tilde{{{{\mkern2mu\underline{\mkern-2mu \Gamma\mkern-2mu}\mkern2mu }}}}^{0 \Lambda_0}}{\delta {\underline{\bar{c}}}} - \partial \tilde{{{{\mkern2mu\underline{\mkern-2mu \Gamma\mkern-2mu}\mkern2mu }}}}^{0\Lambda_0}_\gamma \Big)&&(AGE)\,, \label{eq_21gh}\\ \tilde{{\mathrm{F}}}^{0 \Lambda_0}_1=&\langle \frac{\delta \tilde{{{{\mkern2mu\underline{\mkern-2mu \Gamma\mkern-2mu}\mkern2mu }}}}^{0\Lambda_0}}{\delta {\underline}{A}},\sigma_{0\Lambda_0}*\tilde{{{{\mkern2mu\underline{\mkern-2mu \Gamma\mkern-2mu}\mkern2mu }}}}^{0\Lambda_0}_\gamma\rangle -\langle \frac{\delta \tilde{{{{\mkern2mu\underline{\mkern-2mu \Gamma\mkern-2mu}\mkern2mu }}}}^{0\Lambda_0} }{\delta {\underline}{c}},\sigma_{0\Lambda_0}*\tilde{{{{\mkern2mu\underline{\mkern-2mu \Gamma\mkern-2mu}\mkern2mu }}}}^{0\Lambda_0}_\omega \rangle&&(STI). \label{eq_21st}\end{aligned}$$ The goal is to show that $\tilde{{\mathsf{\Gamma}}}^{0\infty}_{\beta}=0$ and $\tilde{{\mathsf{\Gamma}}}^{0\infty}_{1}=0$, in the sense of theorems \[thm\_3\], \[thm\_4\]. In appendix \[sec\_rr\] we give a version of these two equations valid for $0 < \Lambda < \Lambda_0$. ### The Antighost Equation In this section we extend the AGE for arbitrary $\Lambda \in [0,\Lambda_0]$. In notations , , and with $\mathcal{L}^{\Lambda_0 \Lambda_0}_{aux}$ given in we have $$-\tilde{W}^{\Lambda \Lambda_0}_{\beta}\tilde{Z}^{\Lambda \Lambda_0}_{vst}=-\hbar \frac{\delta}{\delta \beta} \tilde{Z}^{\Lambda \Lambda_0}_{aux}\Big|_{\rho,\beta=0}=\int d \tilde{\mu} \, Q^{\Lambda_0}_\beta e^{-\frac{1}{\hbar} L^{\Lambda_0 \Lambda_0}_{vst} + \frac{1}{\hbar} \tilde{K} \cdot \tilde{\Phi}}\,,\label{eq_325a}$$ where $d \tilde{\mu}:=d \mu_{\Lambda \Lambda_0}(\tilde{\Phi})$, $\tilde{\Phi}:=(A,c,\bar{c})$, $\tilde{K}:=(j,\bar{\eta},\eta)$. Integration by parts, see , yields $$\int d \tilde{\mu} \, \left(\frac{\delta L^{\Lambda_0 \Lambda_0}_{vst}}{\delta \bar{c}} - S^{-1}_{\Lambda \Lambda_0} c - \eta \right)e^{-\frac{1}{\hbar} L^{\Lambda_0 \Lambda_0}_{vst} + \frac{1}{\hbar} \tilde{K} \cdot \tilde{\Phi}}=0.\label{eq_325b}$$ From definition we also have $$\frac{\delta L^{\Lambda_0 \Lambda_0}_{vst}}{\delta \bar{c}}=\frac{\delta L^{\Lambda_0 \Lambda_0}}{\delta \bar{c}}\,.\label{eq_325c}$$ Substituting $Q_{\beta}$ from into , then using , and putting $\gamma,\omega=0$, we obtain $$-\tilde{W}^{\Lambda \Lambda_0}_{\beta}=\sigma_{0 \Lambda_0}* \Big(S^{-1}_{\Lambda \Lambda_0} \frac{\delta \tilde{W}^{\Lambda \Lambda_0}}{\delta \bar{\eta}} + \eta + \partial \tilde{W}^{\Lambda \Lambda_0}_\gamma \Big) - S^{-1}\frac{\delta \tilde{W}^{\Lambda \Lambda_0}}{\delta \bar{\eta}}\,.$$ Finally we perform the Legendre transform $$\begin{aligned} \tilde{{\mathsf{\Gamma}}}^{\Lambda \Lambda_0}_{\beta}&=\sigma_{0 \Lambda_0}* \Big(S^{-1}_{\Lambda \Lambda_0}{\underline}{c} + \frac{\delta \tilde{\Gamma}^{\Lambda \Lambda_0}}{\delta {\underline{\bar{c}}}} - \partial \tilde{{\mathsf{\Gamma}}}^{\Lambda \Lambda_0}_\gamma \Big) - S^{-1}{\underline}{c}\nonumber\\ &=\sigma_{0 \Lambda_0}* \Big(\frac{\delta \tilde{{\mathsf{\Gamma}}}^{\Lambda \Lambda_0}}{\delta {\underline{\bar{c}}}} - \partial \tilde{{\mathsf{\Gamma}}}^{\Lambda \Lambda_0}_\gamma \Big) - S^{-1}{\underline}{c} \,.\label{eq_20ge}\end{aligned}$$ One can also prove identity by noting that its lhs and the ${\mathsf{\Gamma}}$-functionals on the rhs obey FEs of form . Moreover, identity implies that both sides have the same boundary condition at $\Lambda=0$. Momentum bounds {#sec_21bnd} =============== From now on we use the following conventions: - $0<\Lambda\leqslant\Lambda_0$, unless otherwise stated. - $M$ is a fixed mass parameter such that $0<M\leqslant \Lambda_0$. - We omit the tilda for all vertex functions and insertions, for example $\tilde{\Gamma}^{\Vec{\phi};w} \mapsto \Gamma^{\Vec{\phi};w}$, $\tilde{{\mathsf{\Gamma}}}^{\Vec{\phi};w} \mapsto {\mathsf{\Gamma}}^{\Vec{\phi};w}$, $\tilde{\Gamma}^{\Vec{\phi};w}_\chi \mapsto \Gamma^{\Vec{\phi};w}_\chi$. - We use $A$, $c$, $\bar{c}$ instead of ${\underline}{A}$, ${\underline}{c}$, ${\underline{\bar{c}}}$, respectively. - A tensor monomial is a tensor product of Kronecker $\delta$’s and momentum variables in $p:=(p_1,...,p_{n-1})$, for example $\delta_{\mu \nu}p_{1\rho}p_{2\sigma}$. Let $\{\delta^s p^k\}$ be the set of all monomials being a product of $s$ Kronecker $\delta$’s and $k$ momenta $p_i$ and let $\{\delta^s p^k\}_r$ be the union of the sets $\{\delta^s p^k\}$ such that $r$ equals the rank of the monomials: $r=2s+k$. For example $\{\delta^2\}_4=\{\delta_{\mu \nu} \delta_{\rho \sigma},\delta_{\mu \sigma} \delta_{\rho \nu},\delta_{\mu \rho} \delta_{\nu \sigma}\}$. - For $\Vec{p} \in \mathbb{P}_n$ we define the $\eta$-function by $$\eta(\Vec{p}):= \min_{S \in \wp_{n-1} \backslash \{\emptyset\}}(|\sum_{i \in S} p_i|,M)\,.$$ where $\wp_{n-1}$ denotes the power set of $[n-1]$ (the sum does not include $p_0=-\sum_1^{n-1}p_i$). A momentum configuration $\Vec{p}$ is nonexceptional iff $ \eta(\Vec{p}) \neq 0$ and exceptional otherwise. - For a fixed constant $c$ such that $0<c<1$ we define $$\mathbb{M}_n:=\{\Vec{p} \in \mathbb{P}_n: \eta(\Vec{p})>c M \mbox{ and } p^2_i\leqslant M^2\; \forall i \in [n-1]\}\label{eq_2ren}\,.$$ Every $\Vec{p} \in \mathbb{M}_n$ is nonexceptional. - $\forall n\geqslant 2$, a momentum configuration $\Vec{p}\in \mathbb{M}_n$ is **symmetric** iff $\Vec{p} \in \mathbb{M}^s_n$, $$\mathbb{M}^s_n:=\{\Vec{p} \in \mathbb{M}_n: p_i p_j=\frac{M^2}{n-1}(n\,\delta_{ij}-1)\; \forall i,j \in [n-1]\}\,.\label{eq_9sym}$$ - $\forall n \geqslant 3$, a momentum configuration $\Vec{p} \in \mathbb{M}_n$ is **coplanar** iff $\Vec{p}\in \mathbb{M}^{cp}_n$, $$\mathbb{M}^{cp}_n:=\{\Vec{p} \in \mathbb{M}_n: \mathrm{dim}(\mathrm{span}(p_0,...,p_{n-1}))=2 \}\,.\label{eq_9cp}$$ - In the following, a renormalization point is denoted by $\Vec{q}\in\mathbb{P}_n$. See appendix \[sec\_ren\] for the list of all relevant terms and their renormalization points. Weighted trees -------------- The bounds on the vertex functions presented in section \[sec\_thms\] are expressed in terms of sets of weighted trees that are introduced by definitions \[def\_tree0\], \[def\_tree\] below. As seen from , to each edge $e$ of a weighted tree is associated a factor $(|p_e|+\Lambda)^{-\theta(e)}$, $p_e$ being the momentum traversing the edge and $\theta(e)$ being the $\theta$-weight of the edge, expressed as a sum of the $\rho$ and $\sigma$-weights of the edge, see . The relation expresses the fact that the total $\theta$-weight of a tree is in agreement with power counting. Nonvanishing $\sigma$-weights are introduced in order to define viable tree bounds for momentum derived vertex functions. The definition of the $\sigma$-weight is inspired by how momentum derivatives are distributed along a tree, taking care of momentum conservation. Before giving the definition of the weighted trees we set up some necessary notations. - A tree $\tau$ is a connected graph with no cycles. The sets of vertices and edges of a tree $\tau$ are denoted respectively by $V(\tau)$ and $E(\tau)$, or shortly $V$, $E$. In the following, the terms “edge“ and “line“ are equivalent. - Let $V_m$ be the set of vertices of valence $m$. Then, $V=\bigcup_{m \geqslant 1} V_m$. - Let $E_1$ be the set of edges incident to vertices of valence 1. In other words, $E_1$ is the set of external edges. - Each tree has a bijection $\psi:\{0,...,n-1\} \to V_1$, $i \mapsto v_i$ and a sequence of $n$ field labels $\Vec{\varphi}=(\varphi_0,...,\varphi_{n-1})$, $\varphi_i \in \{A,c,\bar{c},\gamma,\omega,\beta\}$ and $n=|V_1|$. The field label $\varphi_i$ defines the type of the vertex $v_i \in V_1$. Let $V_{\varphi} \subseteq V_1$ be the set of all vertices of type $\varphi$, for example $V_A$. Furthermore, let $E_\varphi \subseteq E_1$ be the set of all edges incident to vertices in $V_\varphi$, for example $E_{\varkappa}$ with $\varkappa \in \{\gamma,\omega\}$. When needed, the edges are labeled by integers and the vertices by symbols. The edge incident to a vertex $v_i \in V_1$ has the same index $i$. As an example, for the tree above we have:\ --------------------------------------------- ----------------------- ------------------------------ ---------------- $V=\{c_0,A_1,A_2, \bar{c}_3,u, u^\prime\}$, $V_3=\{u^\prime,u\}$, $V_1=\{c_0,A_1,A_2, \bar{c}_3\}$, $V_A=\{A_1,A_2\}$, $V_{\bar{c}}=\{\bar{c}_3\}$, $V_c=\{c_0\}$, $E_1=\{0,1,2,3\}$, $E_A=\{1,2\}$, $E_{\bar{c}}=\{3\}$, $E_{c}=\{0\}$. --------------------------------------------- ----------------------- ------------------------------ ---------------- - Recalling the definition of $\mathbb{P}_n$ in , for every edge $e \in E$ and vertex $v \in V_1$, the momentum assignments $p_e$, $p_v$ are functions from the set $\mathbb{P}_n$ to $\mathbb{R}^4$, with $n=|V_1|$, defined by the following construction: - label the vertices in $V_1$ by means of $\psi: i \mapsto v_i$ and set $p_{v_i}(\Vec{p}):=p_i$, - apply momentum conservation to all vertices to get $p_e(\Vec{p})$. We use similar notations for multi-indices: $w_v:=w_{\psi^{-1}(v)}$ for $w \in \mathbb{W}_n$ and $v \in V_1$. Given the momentum assignments, a set-valued function $K$ on $E$ is defined by $$K_e:=\{v \in V_1 \backslash \{v_0\}: \frac{\partial p_e}{\partial p_v} \neq 0 \}.$$ For the tree given above we have $p_{c_0}=-(p_{A_1}+ p_{A_2} + p_{\bar{c}_3})$ and $$\begin{aligned} K_0&=\{A_1,A_2,\bar{c}_3\},&K_1&=\{A_1\},&K_2&=\{A_2\},\\ K_3&=\{\bar{c}_3\},&K_4&=\{A_1,A_2\}\,.\end{aligned}$$ Some additional structure is needed, always in view of the bounds. - The vertices in $V_3$ are additionally labeled either as “**regular**” ($\bullet$) or as “**hollow**” ($\circ$). The sets of regular and hollow vertices are respectively denoted by $V_\bullet$ and $V_\circ$, hence $V_3=V_\bullet \cup V_\circ$. In terms of our bounds, regular vertices do change the $\rho$-weight of incident edges, while hollow vertices do not, see definition \[def\_rho\]. We use hollow vertices for the bounds on 3-point functions and in the proof of the theorems, see for example section \[sec\_junc\] on the junction of weighted trees and definition \[def\_fgm\]. - Edges carry zero or more labels “\*”. Edges are referred to as “**\*-edges**” if they have one or more labels “\*”, and as “**regular edges**” otherwise. The set of all \*-edges is denoted by $E_*$. The \*-edges play a special role in our bounds, because to each $e\in E_*$ is associated a supplementary factor $|p_e|+\Lambda$, see and theorem \[thm\_4\]. \[def\_tree0\] Let be given a sequence of $n\geqslant 3$ field labels, $\Vec{\varphi}=(\varphi_0,...,\varphi_{n-1})$, with $\varphi_i \in \{A,c,\bar{c},\gamma,\omega,\beta\}$. Let $\mathbb{T}_{\Vec{\varphi}}$ denote the set of all trees that satisfy the following rules: - There is a bijection $\psi:\{0,...,n-1\} \to V_1$. Each $v_i \in V_1$ has type $\varphi_i$. - $ V= V_1 \cup V_3$. - If $n=3$ then $V_3=V_\circ$. - $|E_*| \in \mathbb{N}$. \[def\_tree\] In the notations of definition \[def\_tree0\], let $\mathcal{T}^{(s)}_{\Vec{\varphi}}$ denote the set of all trees in $\mathbb{T}_{\Vec{\varphi}}$ with total number of labels “\*” equal to $s$ and such that $V_3=V_\bullet$ whenever $n>3$. For shortness we set $\mathcal{T}_{\Vec{\varphi}} :=\mathcal{T}_{\Vec{\varphi}}^{(0)}$ and $\mathcal{T}_{1\Vec{\varphi}}:=\mathcal{T}_{\Vec{\varphi}}^{(1)}$. As an example, below we show two trees, $\tau_3 \in \mathcal{T}_{1c A A}$ and $\tau_7 \in \mathcal{T}_{\beta c c \bar{c} A A A}$.\ \[def\_rho\] Fix a tree from $\mathbb{T}_{\Vec{\varphi}}$. A $\rho$-weight is a function $\rho:E \to \{0,1,2\}$ with the following properties: 1. $\forall e \in E_1$, $\rho(e)=0$. 2. There exists a map $\chi: V_\bullet \to E \backslash E_1$ such that - if $\chi(v)=e$, then $e$ is incident to $v$, - $ \forall e \in E \backslash E_1$, $\rho(e)=2- \left\vert{\chi^{-1}(\{e\})}\right\vert.$ \[def\_sigma\] Let be given a tree from $\mathbb{T}_{\Vec{\varphi}}$ and $w \in \mathbb{W}_n$, with $n=|V_1|$. A $\sigma$-weight is a function $\sigma: E \to \mathbb{N}$ defined by $$\sigma(e):=\sum_{v \in V_1} \sigma_v(e),$$ where $(\sigma_v: E \to \mathbb{N})_{v \in V_1}$ is a family of functions such that $$\begin{aligned} \sum \limits_{e \in E} \sigma_v(e)&=w_v\;,&\sigma_v(e)&=0\mbox{ if } v \not \in K_e\;.\end{aligned}$$ By definition , $w_{0}=0$ for every $w \in \mathbb{W}_n$. Hence, $\sigma_{v_0}(e)=0$ $\forall e \in E$. Let be given a tree $\tau \in \mathbb{T}_{\Vec{\varphi}}$ and $w \in \mathbb{W}_n$, with $n=|V_1|$. A $\theta$-weight is a function $\theta:E \to \mathbb{N}$ defined by $$\theta(e):=\rho(e)+\sigma(e),\label{eq_16theta}$$ where $\rho$ and $\sigma$ are a $\rho$-weight and a $\sigma$-weight corresponding to $w$, respectively. The pair $(\tau, \theta)$ is a weighted tree. The total $\theta$-weight of $(\tau,\theta)$ is $$\theta(\tau):=\sum_{e \in E} \theta(e).$$ The set of all $\theta$-weights corresponding to given $\tau$ and $w$ is denoted by $\Theta^w_\tau$. For every tree $\tau \in \mathcal{T}^{(s)}_{\Vec{\varphi}}$ with $n\geqslant 4$ the total $\theta$-weight is given by the formula $$\theta(\tau)=n + \|w\| - 4\,.\label{eq_409a}$$ This relation follows from definitions \[def\_tree\], \[def\_rho\], \[def\_sigma\], which give the sum rule $\sum_{e \in E} \theta(e)=\|w\|+2|E\setminus E_1|-|V_3|$, and from the relations $|E\setminus E_1|-|V_3|+1=0$ and $|V_3|=n-2$. As an example we consider three trees $\tau_1,\tau_2,\tau_3 \in \mathcal{T}_{AAAA}$. We give three different weights $\theta_a$, $\theta_b$, $\theta_c$, which all correspond to the derivative wrt the momentum $p_1$, literally $w_1=1$ and $w=(0,1,0,0)$. We find a family of weighted trees $\{(\tau_i,\theta): \theta \in \Theta^w_{\tau_i}\}_{i \in \{1,2,3\}}$, where $$\begin{aligned} \Theta^w_{\tau_3}=\Theta^w_{\tau_1}&=\{\theta_a,\theta_b,\theta_c\},&\Theta^w_{\tau_2}&=\{\theta_a,\theta_c\}.\end{aligned}$$\ $e \in E$ $\theta_a$ $\theta_b$ $\theta_c$ ----------- ------------ ------------ ------------ 0 1 0 0 1 0 0 1 2 0 0 0 3 0 0 0 4 0 1 0 Theorems {#sec_thms} -------- We always assume that the renormalization constants are independent of $\Lambda_0$ (though weakly $\Lambda_0$-dependent “renormalization constants” can also be accommodated for, see [@kks]). From now on we denote $\Vec{\varkappa}:=(\varkappa_1,....,\varkappa_{n_\varkappa})$ with $\varkappa_i \in \{\gamma, \omega\}$, $n_{\varkappa} \geqslant 0$. \[rc1\]We impose on all strictly relevant terms vanishing renormalization conditions at zero momentum and $\Lambda=0$: $$\begin{aligned} {\mathsf{\Gamma}}^{0\Lambda_0;\Vec{\phi};w}_{\Vec{\varkappa}}(0)&=0,&\mbox{if}\quad 2n_\varkappa +{\textsc{n}}+\|w\|<4\,. \label{eq_16cdt2}\end{aligned}$$ \[rc2\] On the following marginal terms we impose renormalization conditions at zero momentum and $\Lambda=M$: $$\begin{aligned} {\mathsf{\Gamma}}^{M\Lambda_0;c\bar{c} c \bar{c}}(0)&=0,&{\mathsf{\Gamma}}^{M\Lambda_0;c\bar{c} A A}(0)&=0,&\partial_A{\mathsf{\Gamma}}^{M\Lambda_0;c \bar{c} A}(0)&=0\,, \label{eq_16cdt1}\end{aligned}$$ for the notation see . Remark that Bose–Fermi symmetry and translation invariance imply that $\partial_A{\mathsf{\Gamma}}^{M\Lambda_0;c \bar{c} A}(0)=0$ iff $\partial_c{\mathsf{\Gamma}}^{M\Lambda_0;A \bar{c} c}(0)=0$. To prove proposition \[prop\_barc\] and theorem \[thm\_1\] all remaining marginal renormalization constants are chosen at $\Lambda=0$ arbitrarily but in agreement with the global symmetries of the regularized theory: SU(2), Euclidean isometries ISO(4), ghost number conservation. For instance, all renormalization conditions must comply with the vanishing of the ghost number violating functions, like ${\mathsf{\Gamma}}^{\Lambda,\Lambda_0;cccc}$ or ${\mathsf{\Gamma}}^{\Lambda,\Lambda_0}_{(\varkappa_0,\varkappa_1)}$ for $\varkappa_i \in \{\gamma,\omega\}$. The list of the remaining marginal renormalization constants follows literally from and appendix \[sec\_gamma2\]. \[prop\_barc\]Assume the validity of hypotheses \[rc1\] and \[rc2\]. For all sequences of ${\textsc{n}}\geqslant 3$ field labels in $\{A,c,\bar{c}\}$ with $\phi_{{\textsc{n}}-1}=\bar{c}$, denoted by $\Vec{\phi}\bar{c}$, all $w=(w',0)\in\mathbb{W}_{{\textsc{n}}}$, all $(\Vec{p},0)\in\mathbb{P}_{{\textsc{n}}}$, and all positive $\Lambda$, $\Lambda_0$ s.t. ${\max(\Lambda, M) \leqslant \Lambda_0}$, $${\mathsf{\Gamma}}^{\Lambda\Lambda_0;\Vec{\phi} \bar{c};w}(\Vec{p},0)=0\,.\label{eq_anti}$$ Note that in the momentum of the indicated antighost $\bar{c}$ vanishes, and there is no derivative wrt this momentum. #### Proof We prove the statement by induction, increasing in the loop order, $l-1\mapsto l$. Given $l$, we proceed by descending from $w_{max}$ in the number of derivatives, $\|w\| \mapsto \|w\|-1$. For fixed $l$ and $w$, all possible terms ${\mathsf{\Gamma}}^{\Vec{\phi}\bar{c};w}_l$ are considered. By construction, for fixed $l$ and $\Vec{\phi}\bar{c}$, the inductive scheme deals first with the irrelevant terms and continues, if they exist, with the marginal terms, followed by more and more relevant terms. Since the momentum of the antighost has been assumed to vanish, the statement holds at loop order $l=0$. The validity of the statement for all loop orders smaller than $l$ implies that $\dot{{\mathsf{\Gamma}}}^{\Lambda\Lambda_0;\Vec{\phi} \bar{c};w}_l(\Vec{p},0)=0$. The irrelevant terms have vanishing boundary conditions, hence ${\mathsf{\Gamma}}^{\Lambda_0\Lambda_0;\Vec{\phi} \bar{c};w}_l(\Vec{p},0)=0$. Integrating the FE from $\Lambda_0$ downwards to arbitrary $\Lambda>0$, we get ${\mathsf{\Gamma}}^{\Lambda \Lambda_0;\Vec{\phi} \bar{c};w}_l(\Vec{p},0)=0$ for the irrelevant terms. Next we consider the marginal terms. Since the corresponding irrelevant terms have already been shown to vanish at vanishing antighost momentum, we use the Taylor formula to extend to arbitrary momenta $(\Vec{p},0)$, still preserving the vanishing antighost momentum. Then, we integrate the FE from $M$ to arbitrary $\Lambda>0$, which completes the proof that ${\mathsf{\Gamma}}^{\Lambda \Lambda_0;\Vec{\phi} \bar{c};w}_l(\Vec{p},0)=0$ for marginal terms. Similar arguments hold for all the strictly relevant terms ${\mathsf{\Gamma}}^{\Vec{\phi} \bar{c};w}_l$.$\blacksquare$ The following counterterms vanish, see , $$\begin{aligned} r^{0,\bar{c} c A A}_{1}&=0,&r^{0,\bar{c} c \bar{c} c}&=0,&r^{0,\bar{c} c A A}_{2}&=0,&r^{0,A \bar{c} c}_2&=0\label{eq_21r1}.\end{aligned}$$ #### Proof Using we have, for all $p_2,p_3\in\mathbb{R}^4$ and $\Lambda = \Lambda_0$ (omitted), $$\begin{aligned} {\mathsf{\Gamma}}^{c \bar{c} c \bar{c} }(0,p_2,p_3)&=0,&{\mathsf{\Gamma}}^{c \bar{c} AA }(0,p_2,p_3)&=0,&\partial_{p_2} {\mathsf{\Gamma}}^{A \bar{c} c}(0,p_2)&=0\,.\label{vangas}\end{aligned}$$ Recall and .$\blacksquare$ For all $X \in \{\beta,1\}$, $\Vec{\varkappa}$, $\Vec{\phi}\bar{c}$, $w=(w',0)$, $\Vec{p}$, and all positive $\Lambda$, $\Lambda_0$ s.t. $\max(\Lambda, M) \leqslant \Lambda_0$: $${\mathsf{\Gamma}}^{\Lambda\Lambda_0;\Vec{\phi} \bar{c};w}_{X\Vec{\varkappa}}(\Vec{p},0)=0 \label{eq_21anti}.$$ #### Proof It follows from the definitions of the inserted functions given in , , that at tree level ${\mathsf{\Gamma}}^{\Lambda \Lambda_0;\Vec{\phi} \bar{c};w}_{X\Vec{\varkappa};l=0}(\Vec{p},0)=0$. Then using equation one shows that for all these terms we have vanishing boundary conditions, ${\mathsf{\Gamma}}^{\Lambda_0\Lambda_0;\Vec{\phi} \bar{c};w}_{X\Vec{\varkappa};l}(\Vec{p},0)=0$ $\forall l$. Assuming that the statement is true at the loop order $l-1\geqslant 0$, by induction in $l$, using and integrating the FE from $\Lambda_0$ to arbitrary $\Lambda$ one shows that it holds at the loop order $l$.$\blacksquare$ From now on, for simplicity of notation we write $\mathcal{P}^{(k)}_{s}$ to denote polynomials with nonnegative coefficients and degree $s$ where the superscript is a label to make one polynomial different from another. We define: $$\begin{aligned} P^{\lambda_1 \lambda_2}_{s}(\Vec{p})&:=\mathcal{P}^{(0)}_s\Big(\log_+ \frac{\max(|\Vec{p}|,M)}{\lambda_1 + \eta(\Vec{p})}\Big) + \mathcal{P}^{(1)}_s\Big(\log_{+} \frac{\lambda_2}{M}\Big)\,,\label{eq_1p}\\ P^{\lambda}_{s}(\Vec{p})&:=P^{\lambda \lambda}_{s}(\Vec{p})\,,\\ \Pi^{\lambda}_{\tau,\theta}(\Vec{p})&:= \prod \limits_{e \in E} \left(\lambda + |p_e|\right)^{-\theta(e)}\,,\label{eq_pi}\\ Q^{\Lambda;w}_{\tau}(\Vec{p})&:=\frac{\prod \limits_{e \in E_*}(\Lambda + |p_e|)}{\prod \limits_{e \in E_\varkappa}(\Lambda + |p_e|)}\left\{\begin{matrix}\inf \limits_{i \in \mathbb{I}} \sum \limits_{\theta \in \Theta^{w^\prime(i)}_{\tau}} \Pi^{\Lambda}_{\tau,\theta}(\Vec{p}),&|V_1|=3\,,\label{eq_16bnd}\\ \sum \limits_{\theta \in \Theta^w_\tau} \Pi^{\Lambda}_{\tau,\theta}(\Vec{p}),&otherwise\,,\end{matrix}\right.\end{aligned}$$ where $\tau \in \mathbb{T}_{\Vec{\varphi}}$, $w^\prime(i)$ is obtained from $w$ by diminishing $w_i$ by one unit, and, for nonvanishing $w$, $\mathbb{I}:=\{i: w_i>0\}$. The following sets are also used in theorems \[thm\_1\]–\[thm\_4\]: $$\begin{aligned} \mathbb{Y}_n^{+}&:=\{(\Lambda,\Lambda_0):0<\Lambda\leqslant\Lambda_0\text{~and~}\Lambda_0\geqslant M\}\times \mathbb{P}_n\,, \\ \mathbb{Y}_n&:= \mathbb{Y}_n^{+}\;\bigcup\;\{(0,\Lambda_0):\Lambda_0\geqslant M\}\times \{\vec{p}\in\mathbb{P}_n:\eta(\vec{p})\neq0\}.\end{aligned}$$ \[thm\_1\] There exists a collection of regular vertex functions ${\mathsf{\Gamma}}^{\Vec{\phi}}_{\Vec{\varkappa};l}$ on $\mathbb{Y}_{{\textsc{n}}+n_\varkappa}^+$, complying with the global symmetries of the theory, satisfying the FE and the renormalization conditions given by hypotheses \[rc1\] and \[rc2\], and with irrelevant terms vanishing at $\Lambda=\Lambda_0$. Furthermore, for all $\Vec{\phi}$, $\Vec{\varkappa}$, all $l\in\mathbb{N}$, $w \in \mathbb{W}_{{\textsc{n}}+n_\varkappa}$, the following bounds hold on $\mathbb{Y}_{{\textsc{n}}+n_\varkappa}^+$: a) : $d\geqslant 0$ or ${\textsc{n}}+n_\varkappa=2$ $$|{\mathsf{\Gamma}}^{\Lambda\Lambda_0;\Vec{\phi};w}_{\Vec{\varkappa};l}(\Vec{p})| \leqslant (\Lambda + |\Vec{p}|)^d P^{\Lambda}_r(\Vec{p})\,,\\$$ b) : $d<0$ $$|{\mathsf{\Gamma}}^{\Lambda\Lambda_0;\Vec{\phi};w}_{\Vec{\varkappa};l}(\Vec{p})| \leqslant \sum \limits_{\tau \in \mathcal{T}_{\Vec{\varkappa}\Vec{\phi}}} Q^{\Lambda;w}_{\tau}(\Vec{p}) \; P^{\Lambda}_r(\Vec{p})\,.$$ Here $d:=4-2n_\varkappa-{\textsc{n}}-\|w\|$. If $l=0$ then $r:=0$, otherwise $r$ stands for $r(d,l)$. $$r(d,l):=\left\{\begin{matrix}2l,&d \geqslant 0\,,\\2l-1,&d<0\,.\end{matrix}\right.$$ Theorem \[thm\_1\] shows in particular that the functions ${\mathsf{\Gamma}}^{\Lambda \Lambda_0;\Vec{\phi};w}_{\Vec{\varkappa}}$ are bounded uniformly in $\Lambda_0$. To prove convergence in the limit $\Lambda_0 \to \infty$ we establish the following bounds for their derivatives wrt $\Lambda_0$. \[thm\_2\]Let be given a collection of vertex functions ${\mathsf{\Gamma}}^{\Vec{\phi}}_{\Vec{\varkappa};l}$ as in theorem \[thm\_1\]. Then, for all $\Vec{\phi}$, $\Vec{\varkappa}$, all $l\in\mathbb{N}$, $w \in \mathbb{W}_{{\textsc{n}}+n_\varkappa}$, the following bounds hold on $\mathbb{Y}_{{\textsc{n}}+n_\varkappa}^+$: a) : $d \geqslant 0$ or ${\textsc{n}}+n_\varkappa=2$ $$|\partial_{\Lambda_0} {\mathsf{\Gamma}}^{\Lambda\Lambda_0;\Vec{\phi};w}_{\Vec{\varkappa};l}(\Vec{p})| \leqslant \frac{\Lambda + M+|\Vec{p}|}{\Lambda^2_0} (\Lambda + |p|)^d P^{\Lambda\Lambda_0}_{r}(\Vec{p})\,,$$ b) : $d<0$ $$|\partial_{\Lambda_0}{\mathsf{\Gamma}}^{\Lambda\Lambda_0;\Vec{\phi};w}_{\Vec{\varkappa};l}(\Vec{p})| \leqslant \frac{\Lambda + M+|\Vec{p}|}{\Lambda^2_0} \sum \limits_{\tau \in \mathcal{T}_{\Vec{\varkappa}\Vec{\phi}}} Q^{\Lambda;w}_{\tau}(\Vec{p}) \, P^{\Lambda\Lambda_0}_{r}(\Vec{p})\,.$$ See theorem \[thm\_1\] for the definition of $d$ and $r$. Convergence of the limit $\Lambda \to 0^+$ of the terms ${\mathsf{\Gamma}}^{\Lambda\Lambda_0;\Vec{\phi};w}_{\Vec{\varkappa}}(\Vec{p})$, $\partial_{\Lambda_0}{\mathsf{\Gamma}}^{\Lambda\Lambda_0;\Vec{\phi};w}_{\Vec{\varkappa}}(\Vec{p})$ when $\Vec{p}$ is nonexceptional (or $d>0$) follows from the Cauchy criterion $$|\partial^k_{\Lambda_0} {\mathsf{\Gamma}}^{\Lambda\Lambda_0;\Vec{\phi};w}_{\Vec{\varkappa}}(\Vec{p})-\partial^k_{\Lambda_0} {\mathsf{\Gamma}}^{\Lambda^\prime\Lambda_0;\Vec{\phi};w}_{\Vec{\varkappa}}(\Vec{p})|\leqslant \int \limits^{\Lambda^\prime}_{\Lambda} d \lambda \, |\partial_{\lambda}\partial^k_{\Lambda_0} {\mathsf{\Gamma}}^{\lambda\Lambda_0;\Vec{\phi};w}_{\Vec{\varkappa}}(\Vec{p})|\,,$$ and the bounds from theorems \[thm\_1\], \[thm\_2\]. Convergence of the limit $\Lambda_0 \to \infty$ of the terms ${\mathsf{\Gamma}}^{0\Lambda_0;\Vec{\phi};w}_{\Vec{\varkappa}}(\Vec{p})$ when $\Vec{p}$ is nonexceptional (or $d>0$) follows from Cauchy criterion and the bounds from theorem \[thm\_2\], $$|{\mathsf{\Gamma}}^{0\Lambda_0;\Vec{\phi};w}_{\Vec{\varkappa}}(\Vec{p})-{\mathsf{\Gamma}}^{0\Lambda^\prime_0;\Vec{\phi};w}_{\Vec{\varkappa}}(\Vec{p})|\leqslant \int \limits^{\Lambda^\prime_0}_{\Lambda_0} d \lambda_0 \, |\partial_{\lambda_0}{\mathsf{\Gamma}}^{0\lambda_0;\Vec{\phi};w}_{\Vec{\varkappa}}(\Vec{p})|\,.$$ In the following we will consider the functions ${\mathsf{\Gamma}}^{0\Lambda_0; \Vec{\phi};w}_{1\Vec{\varkappa}}$, ${\mathsf{\Gamma}}^{0\Lambda_0;\Vec{\phi};w}_{\beta\Vec{\varkappa}}$ which appear on the lhs respectively of the ST identities  and of the AGE . The goal of theorems \[thm\_3\], \[thm\_4\] is to show that ${\mathsf{\Gamma}}^{0\Lambda_0;\Vec{\phi};w}_{1\Vec{\varkappa}}$ and ${\mathsf{\Gamma}}^{0\Lambda_0;\Vec{\phi};w}_{\beta\Vec{\varkappa}}$ vanish in the limit $\Lambda_0 \to \infty$, which restores the STI and AGE. The renormalization conditions for these functions at $\Lambda=0$ are obtained from the rhs of the STI and AGE. In section \[sec\_sti\] we show that the required boundary marginal terms ${\mathsf{\Gamma}}^{0\Lambda_0;\Vec{\phi};w}_{1 \Vec{\varkappa}}$, ${\mathsf{\Gamma}}^{0\Lambda_0;\Vec{\phi};w}_{\beta \Vec{\varkappa}}$ satisfy the bounds of the theorems under the conditions specified in hypothesis \[rc3\]. \[rc3\] We allow $R^{AAA}$, $r^{AA}_1$, $r^{\bar{c}c}$ to be chosen arbitrarily but the remaining marginal renormalization constants must satisfy a set of equations: $R_1$, $R_2$, $R_3$, $r^{AA}_2$, $R^{A \bar{c} c}_1$, $R^{AAAA}_{1,2}$, see appendix \[sec\_gamma2\] and in appendix \[sec\_gamma\] for notations. For shortness we also introduce the following definition $$\begin{aligned} F^{\Lambda\Lambda_0}_s(\Vec{p})&:=\frac{M+|\Vec{p}| + \Lambda}{\Lambda_0}\Big(1+\Big(\frac{|\Vec{p}|}{\Lambda_0}\Big)^{w_{max}}\Big)\mathcal{P}^{(2)}_{s}\Big(\frac{|\Vec{p}|}{\Lambda + M} \Big)\,.\end{aligned}$$ \[thm\_3\]Let be given a collection of vertex functions ${\mathsf{\Gamma}}^{\Vec{\phi}}_{\Vec{\varkappa};l}$, regular on $\mathbb{Y}_{{\textsc{n}}+n_\varkappa}$, complying with the global symmetries of the theory, satisfying the hypotheses \[rc1\], \[rc2\], \[rc3\] and the bounds of theorems \[thm\_1\],\[thm\_2\]. Let ${\mathsf{\Gamma}}^{\Vec{\phi}}_{\beta,\Vec{\varkappa};l}$ be a collection of vertex functions with one insertion of the operator $Q_\beta^{\Lambda_0}$ , regular on $\mathbb{Y}_{1+{\textsc{n}}+n_\varkappa}$, complying with the global symmetries of the theory, satisfying the FE, and s.t. the AGE holds. Then, for all $\Vec{\phi}$, $\Vec{\varkappa}$, all $l\in\mathbb{N}$, $w \in \mathbb{W}_{1+{\textsc{n}}+n_\varkappa}$, the following bounds hold on $\mathbb{Y}_{1+{\textsc{n}}+n_\varkappa}$: a) : $d \geqslant 0$ or ${\textsc{n}}+n_\varkappa=1$ $$|{\mathsf{\Gamma}}^{\Lambda\Lambda_0;\Vec{\phi};w}_{\beta\Vec{\varkappa};l}(\Vec{p})| \leqslant (\Lambda + |\Vec{p}|)^{d} F^{\Lambda\Lambda_0}_{s_\beta}(\Vec{p})P^{\Lambda\Lambda_0}_{r_\beta}(\Vec{p})\,,$$ b) : $d<0$ $$|{\mathsf{\Gamma}}^{\Lambda\Lambda_0;\Vec{\phi};w}_{\beta \Vec{\varkappa};l}(\Vec{p})| \leqslant \sum \limits_{\tau \in \mathcal{T}_{\beta \Vec{\varkappa} \Vec{\phi}}} Q^{\Lambda;w}_{\tau}(\Vec{p}) \, F^{\Lambda\Lambda_0}_{s_\beta}(\Vec{p})P^{\Lambda\Lambda_0}_{r_\beta}(\Vec{p})\,.$$ Here $d:=3-2n_\varkappa-{\textsc{n}}-\|w\|$ and $s_\beta:=0$. If $l=0$ then $r_\beta:=0$ otherwise $r_\beta$ stands for $r_\beta(d,l)$. $$r_\beta(d,l):=\left\{\begin{matrix}2l,&d\geqslant 0\,,\\2l-1,&d<0\,.\end{matrix}\right.$$ We note that the 1-point vertex function with integrated insertion  vanishes, ${\mathsf{\Gamma}}^{\Lambda \Lambda_0;c;w}_1=0$, e.g. due to SU(2) symmetry. \[thm\_4\]Let be given a collection of vertex functions ${\mathsf{\Gamma}}^{\Vec{\phi}}_{\Vec{\varkappa};l}$, ${\mathsf{\Gamma}}^{\Vec{\phi}}_{\beta\Vec{\varkappa};l}$ as in theorem \[thm\_3\]. Let ${\mathsf{\Gamma}}^{\Vec{\phi}}_{1,\Vec{\varkappa};l}$ be a collection of vertex functions with one integrated insertion of the appropriate operator among $Q_\rho^{\Lambda_0}$ , $Q_{\rho\gamma}^{\Lambda_0}$ , $Q_{\rho\omega}^{\Lambda_0}$ , regular on $\mathbb{Y}_{{\textsc{n}}+n_\varkappa}$, complying with the global symmetries of the theory, and satisfying the FE. Assume that the STI and consistency conditions and do hold. Then, for all $\Vec{\phi}$, $\Vec{\varkappa}$, all $l\in\mathbb{N}$, $w \in \mathbb{W}_{{\textsc{n}}+n_\varkappa}$, the following bounds hold on $\mathbb{Y}_{{\textsc{n}}+n_\varkappa}$: a) : $d>0$ or ${\textsc{n}}+n_\varkappa=2$ $$|{\mathsf{\Gamma}}^{\Lambda\Lambda_0;\Vec{\phi};w}_{1\Vec{\varkappa};l}(\Vec{p})| \leqslant (\Lambda + |\Vec{p}|)^d F^{\Lambda\Lambda_0}_{s_1}(\Vec{p})P^{\Lambda\Lambda_0}_{r_1}(\Vec{p})\,,$$ b) : $d \leqslant 0$ $$|{\mathsf{\Gamma}}^{\Lambda\Lambda_0;\Vec{\phi};w}_{1\Vec{\varkappa};l}(\Vec{p})| \leqslant \sum \limits_{\tau \in \mathcal{T}_{1\Vec{\varkappa}\Vec{\phi}}} Q^{\Lambda;w}_{\tau}(\Vec{p}) \, F^{\Lambda\Lambda_0}_{s_1}(\Vec{p})P^{\Lambda\Lambda_0}_{r_1}(\Vec{p})\,.$$ Here $d:=5-2n_\varkappa-{\textsc{n}}-\|w\|$. If $l=0$ then $r_1:=0$, $s_1:=0$ otherwise $r_1$, $s_1$ stand respectively for $r_1(d,l)$, $s_1(d,l)$. $$\begin{aligned} r_1(d,l)&:=\left\{\begin{matrix}3l,&d> 0\,,\\3l-1,&d= 0\,,\\3l-2,&d< 0\,,\end{matrix}\right.&s_1(d,l)&:=\left\{\begin{matrix}l,&d \geqslant 0\,,\\l-1,&d < 0\,.\end{matrix}\right.\end{aligned}$$ Proof of Theorems \[thm\_1\]–\[thm\_4\] {#sec_327b} ======================================= In this section we will prove theorems \[thm\_1\], \[thm\_2\], \[thm\_3\] and \[thm\_4\] in this order. We proceed by induction in the loop order $l$. We first verify that they hold at tree level $l=0$. Afterwards we assume that they hold true up to loop order $l-1\geqslant 0$, and we will verify the induction step from $l-1$ to $l$. Put $D_X:=4$ for all vertex functions ${\mathsf{\Gamma}}^{\Vec{\phi}}_{\Vec{\varkappa}}$. For all inserted functions ${\mathsf{\Gamma}}^{\Vec{\phi};w}_{X \Vec{\varkappa}}$ with $X \in \{\beta,1\}$ and $\|w\| \leqslant w_{max}$ let $$\begin{aligned} D_X&:=\left\{\begin{matrix}3,&X=\beta\,,\\5,&X=1\,,\end{matrix}\right.&d_X&:=D_X - 2n_\varkappa -{\textsc{n}}-\|w\|\,.\end{aligned}$$ Note that at zero loop order ${\mathsf{\Gamma}}^{\Lambda \Lambda_0}_{l=0}={\mathsf{\Gamma}}^{\Lambda_0 \Lambda_0}_{l=0}$. Using , and the definition of $\mathcal{L}_{aux}$ in  one finds that in momentum space $$\begin{aligned} {\mathsf{\Gamma}}^{\Lambda_0 \Lambda_0;\Vec{\phi}}_1&=Q^{\Lambda_0;\Vec{\phi}}_{\rho(0)},&{\mathsf{\Gamma}}^{\Lambda_0 \Lambda_0;\Vec{\phi}}_{1\varkappa}&=Q^{\Lambda_0;\Vec{\phi}}_{\rho(0)\varkappa},&{\mathsf{\Gamma}}^{\Lambda_0 \Lambda_0;\Vec{\phi}}_\beta&=Q^{\Lambda_0;\Vec{\phi}}_\beta\,,\end{aligned}$$ where the momentum variable corresponding to the source $\rho$ is set to zero. Everywhere in the following $Q^{\Lambda_0}_{\rho}$ will stand for $Q^{\Lambda_0}_{\rho(0)}$. From definition of $Q^{\Lambda_0}_\rho$ in it follows that the vertex functions $Q^{\Lambda_0;\Vec{\phi}}_{\rho;l=0}$ with ${\textsc{n}}=2$ vanish. The nonvanishing functions $Q^{\Lambda_0;\Vec{\phi}}_{\rho;l=0}$, $Q^{\Lambda_0;\Vec{\phi}}_{\beta;l=0}$ have the form $h_s(p,q)(1-\sigma_{0\Lambda_0}(p^2))$ where $h_s$ is a homogeneous tensor polynomial of degree $s\leqslant 2$ in the momentum variables $p,q \in \mathbb{R}^4$ which depends at most linearly on the momentum $q$. From the definitions of $Q^{\Lambda_0}_{\rho\omega}$ and $Q^{\Lambda_0}_{\rho\gamma}$ we obtain that $Q^{\Lambda_0;\Vec{\phi}}_{\rho\varkappa;l=0}$ has the form $h_s(p)(\sigma_{0\Lambda_0}((p+q)^2)-\sigma_{0\Lambda_0}(p^2))$ with $s\leqslant 1$. For $\|w\|\leqslant s$ (relevant terms) using inequalities , ,  we have $$|{\mathsf{\Gamma}}^{\Lambda \Lambda_0;\Vec{\phi};w}_{X \Vec{\varkappa};l=0}(\Vec{p})|\leqslant c\,\Lambda^{s-\|w\|}_0 \Big(\frac{|\Vec{p}|}{\Lambda_0}\Big)^{s+1-\|w\|} \leqslant c\, \frac{|\Vec{p}|}{\Lambda_0} (|\Vec{p}| + \Lambda)^{s-\|w\|}\,.$$ For $\|w\|>s$ (irrelevant terms) the same inequalities yield $$\begin{aligned} |{\mathsf{\Gamma}}^{\Lambda \Lambda_0;\Vec{\phi};w}_{X \Vec{\varkappa};l=0}(\Vec{p})|&\leqslant c \,\Lambda^{s-\|w\|}_0 \Big(\frac{|\Vec{p}|}{\Lambda_0} + 1\Big)\nonumber\\ &\leqslant c\,(|\Vec{p}|+\Lambda)^{s-\|w\|}\,\frac{|\Vec{p}|+\Lambda}{\Lambda_0} \Big(\frac{|\Vec{p}|}{\Lambda_0}+1\Big)^{\|w\|-s}\,.\end{aligned}$$ Since $s-\|w\|$ is the dimension $d_X$ we have the following bounds $$\begin{aligned} |{\mathsf{\Gamma}}^{\Lambda \Lambda_0;\Vec{\phi};w}_{X \Vec{\varkappa};l=0}(\Vec{p})|&\leqslant (\Lambda + |\Vec{p}|)^{d_X} F^{\Lambda \Lambda_0}_0(\Vec{p}),&X & \in \{\beta,1\}.\end{aligned}$$ Thus the statements of theorems \[thm\_1\]–\[thm\_4\] hold at loop number $l=0$. The proof proceeds by induction on $l$ and on the number of derivatives $\|w\|$, ascending in $l$ and, for fixed $l$, descending in $\|w\|$ from $w_{max}$ to 0. Chains of vertex functions -------------------------- \[def\_split\]A division in $m$ parts of a finite set $\mathbb{I}$ is a sequence $S:=(s_j)_{j \in [m]}$ of $m$ disjoint sets $s_j \subseteq \mathbb{I}$, possibly empty, such that $\bigcup_{j \in [m]} s_{j}=\mathbb{I}$. An ordered partition is a division with all $s_j$ nonempty. Given a division $S$ as above stated, a division of a sequence $\Vec{\Psi}=(\Psi_i)_{i \in \mathbb{I}}$ is the sequence of elements $\Vec{\Psi}_j := (\Psi_i)_{i \in s_j}$, with $j\in[m]$. \[def\_chain\] Let $S$ be a division in $m$ parts of a finite set $\mathbb{I}$, and $\Vec{\Psi}=(\Psi_i)_{i\in \mathbb{I}}$ be a sequence of labels $\Psi_i\in\{A,c,\bar{c},\gamma,\omega,\beta,1\}$. Denote by $(\Vec{\Psi}_j)$ the division of $\Vec{\Psi}$ induced by $S$. A **chain** of vertex functions is then defined by the expression $$\mathcal{F}^{\zeta_1 \Vec{\Psi} \bar{\zeta}_{m}}_{S}:= {\mathsf{\Gamma}}^{\zeta_1 \Vec{\Psi}_1 \bar{\zeta}_1} \prod \limits^{m}_{j=2} \mathbf{C}_{\zeta_j \bar{\zeta}_{j-1}} {\mathsf{\Gamma}}^{\zeta_j \Vec{\Psi}_j \bar{\zeta}_j}\,,$$ where the repeated field labels $\zeta_j,\bar{\zeta}_j$ belong to $\{A, c, \bar{c}\}$ and are summed over (as usual). Using this definition the FE has the form $$\dot{{\mathsf{\Gamma}}}^{\Vec{\Psi}}= \frac{\hbar}{2} \sum \limits_{S} (-)^{\pi_a}\langle \dot{C} \mathcal{F}^{A \Vec{\Psi} A}_S + \dot{S}(\mathcal{F}^{\bar{c} \Vec{\Psi} c}_S - \mathcal{F}^{c \Vec{\Psi} \bar{c}}_S) \rangle\,.$$ The sum above runs over all possible divisions of $[0: n-1]$, $n$ being the number of components of $\Vec{\Psi}$. The symbol $\pi_a$ denotes the number of transpositions $mod\, 2$ of the anticommuting variables $\{c,\bar{c},\beta,\gamma, 1\}$ in the permutation $i \mapsto \pi(i)$ such that $(\Psi_{\pi(0)},...,\Psi_{\pi(n-1)})=\Vec{\Psi}_{1}\oplus...\oplus\Vec{\Psi}_{m}$, where $(a_1,...,a_p) \oplus (a_{p+1},...,a_q)=(a_1,...,a_q)$. A preliminary step toward the proof of theorem \[thm\_1\] is to bound $\partial^w (\mathbf{C}_{\zeta \bar{\zeta}}{\mathsf{\Gamma}}^{\zeta \bar{\zeta}})$ with $\|w\| \leqslant w_{max}$. For all $0<k<l$, $0 \leqslant w \leqslant w_{max}$, $p \in \mathbb{R}^4$ $$\Big|\Big(\prod \limits^w_{i=0} \frac{\partial}{\partial p_{\mu_i}} \Big)\left( {\mathsf{\Gamma}}^{\zeta \bar{\zeta};\Lambda\Lambda_0}_{k}(p) \mathbf{C}^{\Lambda \Lambda_0}_{\zeta \bar{\zeta}}(p) \right)\Big| \leqslant \frac{P^{\Lambda}_{2k}}{(|p|+\Lambda)^w}\,.\label{eq_16gc}$$ #### Proof Using inequality we see that $$|\partial^w ({\mathsf{\Gamma}}^{\zeta \bar{\zeta};\Lambda\Lambda_0}_{k} \mathbf{C}^{\Lambda \Lambda_0}_{\zeta \bar{\zeta}})| \leqslant \sum \limits^{w}_{w_1=0}\frac{w!}{w_1! (w-w_1)!}|\partial^{w_1} {\mathsf{\Gamma}}^{\zeta \bar{\zeta};\Lambda\Lambda_0}_{k}| |\partial^{w-w_1} \mathbf{C}^{\Lambda \Lambda_0}_{\zeta \bar{\zeta}}|\,.$$ Setting $w_2=w-w_1$ it follows from and the bounds of theorem \[thm\_1\] already proved inductively for $k<l$ that $$\frac{1}{w_2!}|\partial^{w_1} {\mathsf{\Gamma}}^{\zeta \bar{\zeta};\Lambda\Lambda_0}_{k}||\partial^{w_2} \mathbf{C}^{\Lambda \Lambda_0}_{\zeta \bar{\zeta}}|\leqslant \frac{c_\xi d^{w_2} P^{\Lambda}_{2k}}{(|p|+\Lambda)^{w_1+w_2}}\,.$$ Because $w\leqslant w_{max}$ the constants $c_\xi$, $d^{w_2}$ may be absorbed in $P^{\Lambda}_{2k}$. $\blacksquare$ \[def\_bchain\]Let be given 1. a sequence $\Vec{\Psi}=(\Psi_i)_{i \in \mathbb{I}}$ as in Definition \[def\_chain\]; 2. an ordered partition $S=(s_j)_{j \in [m]}$ of $\mathbb{I}$; 3. the sequences of field labels $(\zeta_j)_{j \in [m]}$, $(\bar{\zeta}_j)_{j \in [m]}$; 4. a multi-index $w \in \mathbb{W}_n$ and a sequence $\textsf{w}:=(\textsf{w}_j)_{j \in [m]}$ such that $\textsf{w}_j \in \mathbb{W}_n$ and $\sum_{j \in [m]} \textsf{w}_{j}=w$. Then, we define a **reduced chain** of vertex functions as $${\mathsf{\Gamma}}^{\zeta_1 \Vec{\Psi}_1 \bar{\zeta}_1;\textsf{w}_1} \prod \limits^{m}_{j=2} \mathbf{C}_{\zeta_j \bar{\zeta}_{j-1}} {\mathsf{\Gamma}}^{\zeta_j \Vec{\Psi}_j \bar{\zeta}_j;\textsf{w}_j},\label{eq_bchain}$$ where ${\mathsf{\Gamma}}^{\zeta \Vec{\Psi}_{j} \bar{\zeta};\textsf{w}_j}=\partial^{\textsf{w}_j}({\mathsf{\Gamma}}^{\zeta \Vec{\Psi}_{j} \bar{\zeta}})$ are derivatives wrt the external momenta appearing in ${\dot{{\mathsf{\Gamma}}}}^{\Vec{\Psi}}$, and the sequences $(\zeta_j)_{j \in [m]}$, $(\bar{\zeta}_j)_{j \in [m]}$ are **fixed**. Introducing the auxiliary quantities $\mathring{\Upsilon}:=(\Vec{\Upsilon}_j)_{j \in [m]}$ and $\Vec{\Upsilon}_j:=\zeta_j \Vec{\Psi}_j \bar{\zeta}_{j}$, we denote by $\mathcal{S}^{\mathring{\Upsilon};\textsf{w}}$ or, with some abuse of notations, by $\mathcal{S}^{\zeta_1 \Vec{\Psi} \bar{\zeta}_m;\textsf{w}}_S$. The adjective “reduced” indicates that the chains contain neither ${\mathsf{\Gamma}}^{AA}$, ${\mathsf{\Gamma}}^{c \bar{c}}$ nor derivatives applied to the propagators $\mathbf{C}$. It follows from inequalities , and theorems \[thm\_1\]–\[thm\_4\] proved in loop order $l-1$ that there exists a common bound for the terms ${\mathsf{\Gamma}}^{\zeta \Vec{\Psi}\bar{\zeta};w_1 + w_2+ w_3}_{l_1 + l_2}\mathbf{C}$ and ${\mathsf{\Gamma}}^{\zeta \Vec{\Psi}\bar{\zeta};w_1}_{l_1}\partial^{w_2}(\mathbf{C}{\mathsf{\Gamma}}^{\zeta \bar{\zeta}}_{l_2})\partial^{w_3} \mathbf{C}$. This property basically follows from an explicit check of the degree of the polynomials in the bounds and from the fact that, for $w_2,w_3$ supported on $p$, a factor $(|p|+\Lambda)^{-\|w_2+w_3\|}$ times a weighted tree $(\tau,\theta)$ in the bound of ${\mathsf{\Gamma}}^{\zeta \Vec{\Psi}\bar{\zeta};w_1}_{l_1}$ may be bounded by the weighted tree $(\tau,\theta')$ in the bound of ${\mathsf{\Gamma}}^{\zeta \Vec{\Psi}\bar{\zeta};w_1 + w_2+ w_3}_{l_1 + l_2}$, with $\theta'$ equal to $\theta$ everywhere but on the edge $e\in E_1$ carrying momentum $p$, for which $\theta'(e)=\theta(e)+\|w_2+w_3\|$. Hence to bound $\dot{{\mathsf{\Gamma}}}^{\Vec{\Psi}}_{l}$ it is enough to consider a loop integral with a reduced chain $$\int \frac{d^4k}{(2\pi)^4} \; \dot{\mathbf{C}}_{\zeta \bar{\zeta}}(k) \, \mathcal{S}^{\zeta \Vec{\Psi} \bar{\zeta};\textsf{w}}_{S;l-1}(k,\Vec{p}_S,-k), \label{eq_iloop}$$ where $\Vec{p}_{s_j}=(p_i)_{ i\in s_j}$. As an example we give in Appendix \[ex\_AAcc\] the complete list of chains for ${\mathsf{\Gamma}}^{AA c \bar{c}}$. The appellative “reduced” may be omitted in the following, since it is always clear from the context whether a chain is reduced or not. Junction of weighted trees {#sec_junc} -------------------------- Given a reduced chain $\mathcal{S}^{\mathring{\Upsilon};\textsf{w}}$ we define its amplitude $\hat{\mathcal{S}}^{\mathring{\Upsilon};\textsf{w}}$ by substituting the vertex functions and propagators with their corresponding bounds taken from theorems \[thm\_1\]–\[thm\_4\] and inequalities . Recalling that $\Vec{\Upsilon}_j:=\zeta_j \Vec{\Psi}_j \bar{\zeta}_{j}$, we then set $$\begin{aligned} \hat{\mathcal{S}}^{\Lambda;\mathring{\Upsilon};\textsf{w}}&:=\hat{S}^{\Lambda;\Vec{\Upsilon}_1;\textsf{w}_1} \prod \limits^{m}_{j=2} \frac{1}{ (\Lambda + |p_{\zeta_j}|)^2} \hat{S}^{\Lambda;\Vec{\Upsilon}_j;\textsf{w}_j}\label{eq_tchain} \\ \hat{S}^{\Lambda;\Vec{\Upsilon}_j;w}&:=\left\{\begin{matrix}(\Lambda + |\Vec{p}|)^{d_X},&case\ a,\\\sum \limits_{\tau \in \mathcal{T}_{\Vec{\Upsilon}_j}} Q^{\Lambda;w}_{\tau},&case\ b,\end{matrix}\right. \,. \label{eq_hatp}\end{aligned}$$ Here the cases $a$ and $b$ refer to the respective parts in theorems \[thm\_1\]–\[thm\_4\]. The tree structure of the bound is spoiled if there exists an interval $\mathbb{J}^a:=[j^a:j^a+m^a-1] \subset [m]$ such that all $\hat{S}^{\Lambda;\Vec{\Upsilon}_j;w}$ for $j\in\mathbb{J}^a$ correspond to a strictly relevant contribution, associated to the cases $a$ in the theorems. A workaround for this difficulty will start with the following definition. For every tree $\tau \in \mathbb{T}_{\Vec{\varphi}}$, set $E_{1;v}:=\{e \in E_1:e \mbox{ incident to } v\}$. \[def\_fgm\] Let $\Vec{\varphi}$ be an arbitrary sequence with $\varphi_i \in \{A,c,\bar{c},\beta,\gamma,\omega\}$. A tree $f \in \mathbb{T}_{\Vec{\varphi}}$ is a **fragment** if - there exists $s \in \{0,1\}$ s.t.  $|V_\circ|+s$ equals the total number of ”\*” labels, - $\forall v \in V_\circ$: $|E_{1;v}|\geqslant 2$ and $E_{1;v}\cap E_* \not= \emptyset$. The set of all such fragments is denoted by $\mathcal{F}^{(s)}_{\Vec{\varphi}}$. Moreover we set $\mathcal{F}_{\Vec{\varphi}}:=\mathcal{F}^{(0)}_{\Vec{\varphi}}$ and $\mathcal{F}_{1\Vec{\varphi}}:=\mathcal{F}^{(1)}_{\Vec{\varphi}}$. For each $s\in\{0,1\}$, $\mathcal{T}^{(s)}_{\Vec{\varphi}} \subset \mathcal{F}^{(s)}_{\Vec{\varphi}} \subset \mathbb{T}_{\Vec{\varphi}}$. Let us now state a useful property which relies on the above definition. Let be given a subsequence $\mathbb{J}^a:=[j^a:j^a+m^a-1] \subset [m]$ with any number of elements $m^a$. Let $S$ and $\Vec{\Psi}$ be as in definition \[def\_bchain\]. Define the following restrictions: $S^a=S|_{\mathbb{J}^a}$, $\textsf{w}^a=\textsf{w}|_{\mathbb{J}^a}$. Set $w^a:=\sum_{j \in \mathbb{J}^a}\textsf{w}_j$ and $\Vec{\Psi}^a:=(\Psi_i)$ with $i\in \cup_{j \in \mathbb{J}^a} \;s_j$. Then, there exists a set of fragments $\mathcal{F}_{\zeta \Vec{\Psi}^a \bar{\zeta}}$ such that $$\begin{aligned} \hat{\mathcal{S}}^{\Lambda;\zeta \Vec{\Psi}^a \bar{\zeta};\textsf{w}^a}_{S^a} &\leqslant 2^{\frac{|S_a|}{2}} \sum \limits_{f \in \mathcal{F}_{\zeta \Vec{\Psi}^a \bar{\zeta}}} Q^{\Lambda;w^a}_{f}\,. \label{eq_26f}\end{aligned}$$ Here we give a proof of for an example, generalisation is clear. Consider an amplitude $\hat{\mathcal{S}}^{\zeta_1 A^4 \bar{\zeta}_4;w}$ composed of four elements $\hat{S}^{\zeta_j A \bar{\zeta}_j;w}$ where $j \in \{1,...,4\}$ and $w=0$. First let us define the following set of fragments $\mathcal{F}$ (here $s=0$):\ -- -- -- -- \ In each fragment the \*-edge corresponds to a factor $\Lambda + |p_e|$ in the corresponding amplitude $Q^{\Lambda;w=0}_{f \in \mathcal{F}}$. One shows the following bound: $$\mbox{\begin{minipage}[c][48pt]{164pt}\tiny\centering\PandocStartInclude{pics/chain3_AA.tex}\PandocEndInclude{input}{1196}{80}\end{minipage}}\leqslant 2^{\frac{4}{2}} \sum \limits_{f \in \mathcal{F}} Q^{\Lambda;w=0}_{f}\,,$$ or more explicitly: $$\frac{\prod \limits_{v \in V_3} (\Lambda + |\Vec{p}_v|)}{ \prod \limits_{e \in E \backslash E_1} (\Lambda + |p_e|)^2} \leqslant 2^{\frac{|V_3|}{2}} \sum \limits_{\{\chi\}} \frac{\prod \limits_{v \in V_3} (\Lambda + |p_{\chi(v)}|)}{\prod \limits_{e \in E \backslash E_1} (\Lambda + |p_e|)^2}\,.\label{704a}$$ Here the $\oplus$-vertices stand each for a corresponding $\hat{S}^{\zeta_j A \bar{\zeta}_j;w=0}$, see ; the set of $\oplus$-vertices is identified with $V_3$ in ; $\Vec{p}_v$ indicates the set of incoming momenta of the vertex $v$; $p_e$ denotes the momentum corresponding to an edge $e$; the sum runs over the set of functions $\chi:V_3 \to E\backslash E_1 \cup \{e_0,e_5\}$ which map every vertex $v \in V_3$ to an edge incident to $v$; the $|C|$’s stand for the usual bounds on the corresponding propagators. Let be given a sequence of fragments $\Vec{f}:=(f_1,...,f_{m'})$ with $f_j\in\mathcal{F}_{\zeta \Vec{\Psi}_j \bar{\zeta}}$ and a sequence $\textsf{w}^\prime=(\textsf{w}^\prime_j)_{j \in [m^\prime]}$ with $\textsf{w}^\prime_j \in \mathbb{W}_n$. We define the amplitude $\hat{\mathcal{Q}}^{\Lambda;\textsf{w}^\prime}_{\Vec{f}}$ by $$\hat{\mathcal{Q}}^{\Lambda;\textsf{w}^\prime}_{\Vec{f}}:=\,Q^{\Lambda;\textsf{w}^\prime_1}_{f_1} \prod \limits^{m'}_{j=2} \frac{1}{(\Lambda + |p_{\zeta_j}|)^2} Q^{\Lambda;\textsf{w}^\prime_j}_{f_j}\,.$$ Given an amplitude $\hat{\mathcal{Q}}^{\Lambda;\textsf{w}^\prime}_{\Vec{f}}$ as above there exists a fragment $f \in \mathcal{F}_{\zeta \Vec{\Psi}\bar{\zeta}}$ such that $$\hat{\mathcal{Q}}^{\Lambda;\textsf{w}^\prime}_{\Vec{f}} \leqslant Q^{\Lambda;\textsf{w}^\prime}_{f}\,. \label{eq_fgm}$$ #### Proof We proceed by induction in $m'$. If there are no joints, $m'=1$, the statement is evident. Assume it is true for some $m'-1 \geqslant 0$ and consider a sequence of $m'$ fragments. Let $v_l,v_r \in V_3$ be the left and right vertices of a joint. Recall definition in . - $v_l,v_r \in V_3\backslash V_\circ$. $$\begin{aligned} \mbox{\begin{minipage}[c][32pt]{112pt}\centering\tiny\PandocStartInclude{pics/junc_AA_1.tex}\PandocEndInclude{input}{1217}{80}\end{minipage}}&\leqslant\mbox{\begin{minipage}[c][32pt]{48pt}\centering\tiny\PandocStartInclude{pics/junc_AA_2.tex}\PandocEndInclude{input}{1217}{183}\end{minipage}}\\ \Pi^{\Lambda}_{f_l,\theta_l} \;\frac{1}{(\Lambda + |p|)^2} \;\Pi^{\Lambda}_{f_r,\theta_r}&= \Pi^{\Lambda}_{f,\theta}\end{aligned}$$ Here the corresponding external edges have been merged together to form a new internal edge with $\theta$-weight equal to 2. The $\theta$-weight of all other edges is unchanged. - $v_l \in V_3\backslash V_\circ,v_r \in V_\circ$ or vice versa. $$\begin{aligned} \mbox{\begin{minipage}[c][38pt]{112pt}\centering\tiny\PandocStartInclude{pics/junc_AA_3.tex}\PandocEndInclude{input}{1223}{80}\end{minipage}}&\leqslant\mbox{\begin{minipage}[c][38pt]{48pt}\centering\tiny\PandocStartInclude{pics/junc_AA_4.tex}\PandocEndInclude{input}{1223}{183}\end{minipage}}\\ \Pi^{\Lambda}_{f_l,\theta_l} \;\frac{1}{(\Lambda +|p|)^2} \;(\Lambda + |p|) \Pi^{\Lambda}_{f_r,\theta_r}&=\Pi^{\Lambda}_{f,\theta}\\ \mbox{\begin{minipage}[c][38pt]{112pt}\centering\tiny\PandocStartInclude{pics/junc_AA_6.tex}\PandocEndInclude{input}{1225}{80}\end{minipage}}&\leqslant\mbox{\begin{minipage}[c][38pt]{48pt}\centering\tiny\PandocStartInclude{pics/junc_AA_7.tex}\PandocEndInclude{input}{1225}{183}\end{minipage}}\\ \Pi^{\Lambda}_{f_l,\theta_l}\; \frac{1}{(\Lambda +|p|)^2}\; (\Lambda + |p_*|) \Pi^{\Lambda}_{f_r,\theta_r} &=(\Lambda + |p_*|) \Pi^{\Lambda}_{f,\theta}\end{aligned}$$ - $v_l,v_r \in V_\circ$. $$\begin{aligned} \mbox{\begin{minipage}[c][40pt]{112pt}\centering\tiny\PandocStartInclude{pics/junc_AA_8.tex}\PandocEndInclude{input}{1230}{80}\end{minipage}}&\leqslant\mbox{\begin{minipage}[c][40pt]{48pt}\centering\tiny\PandocStartInclude{pics/junc_AA_9.tex}\PandocEndInclude{input}{1230}{183}\end{minipage}}\\ \Pi^{\Lambda}_{f_l,\theta_l} (\Lambda + |p|)\;\frac{1}{(\Lambda +|p|)^2}\; (\Lambda + |p_*|) \Pi^{\Lambda}_{f_r,\theta_r} &=(\Lambda + |p_*|) \Pi^{\Lambda}_{f,\theta}\\ \mbox{\begin{minipage}[c][40pt]{112pt}\tiny\centering\PandocStartInclude{pics/junc_AA_10.tex}\PandocEndInclude{input}{1232}{81}\end{minipage}}&\leqslant\mbox{\begin{minipage}[c][40pt]{48pt}\centering\tiny\PandocStartInclude{pics/junc_AA_11.tex}\PandocEndInclude{input}{1232}{185}\end{minipage}}\\ \Pi^{\Lambda}_{f_l,\theta_l} (\Lambda + |p|)\;\frac{1}{(\Lambda +|p|)^2} \;(\Lambda + |p|) \Pi^{\Lambda}_{f_r,\theta_r} &=\Pi^{\Lambda}_{f,\theta}\end{aligned}$$ Hence, by merging two fragments we can decrease the number of joints by one and then apply the induction hypothesis. $\blacksquare$\ In the simpler context of $\phi^4$ theory, a completely explicit description of the junction of trees can be found in [@ric]. According to equations , the loop integral in is bounded by the following expression $$\int \dot{\mathbf{C}}^{\Lambda\Lambda_0}_{\zeta \bar{\zeta}} \hat{\mathcal{S}}^{\Lambda;\mathring{\Upsilon};\textsf{w}} \, \mathfrak{P}^{\Lambda}_{l-1} \leqslant \Lambda \,\hat{\mathcal{S}}^{\Lambda;\mathring{\Upsilon};\textsf{w}} \, \mathfrak{P}^{\Lambda}_{l-1}\Big|_{p_{\zeta}, p_{\bar{\zeta}}=0}\,,\label{eq_8l}$$ $$\mathfrak{P}^{\Lambda}_{l}:=\left\{\begin{matrix}F^{\Lambda \Lambda_0}_{l} P^{\Lambda \Lambda_0}_{3l},&X=1,\\F^{\Lambda \Lambda_0}_0 P^{\Lambda \Lambda_0}_{2l},&X=\beta,\\P^{\Lambda\Lambda}_{2l},&otherwise,\end{matrix}\right.\label{ric_11}$$ which follows directly from the definition of $r$, $r_X$, $s_X$ given in theorems \[thm\_1\]–\[thm\_4\]. For any function $f$ depending on the variable $p_\zeta$, and other variables which we need not specify, we define $\mathcal{R}_{\zeta}: f(...,p_\zeta,...) \mapsto f(...,0,...)$ where $\mathcal{R}$ stands for restriction. Then we set $\mathcal{R}_{\zeta \bar{\zeta}}:=\mathcal{R}_{\zeta} \circ \mathcal{R}_{\bar{\zeta}}$. For an amplitude $\hat{\mathcal{S}}^{\Lambda;\zeta \Vec{\Psi}\bar{\zeta};\textsf{w}}_S$ there exists a set of trees such that $$\mathcal{R}_{\zeta \bar{\zeta}}(\hat{\mathcal{S}}^{\Lambda;\zeta \Vec{\Psi}\bar{\zeta};\textsf{w}}_S) \leqslant 2^{\frac{|S|}{2}} \sum_{\tau \in \mathcal{T}_{\zeta\Vec{\Psi}\bar{\zeta}}} \mathcal{R}_{\zeta \bar{\zeta}}(Q^{\Lambda;w}_\tau)\,.\label{eq_8r}$$ #### Proof Using definition and inequalities , we obtain that $\hat{\mathcal{S}}^{\Lambda;\zeta \Vec{\Psi}\bar{\zeta};\textsf{w}}_S$ is bounded by a sum of fragment amplitudes. It remains to show that for a fragment $f\in \mathcal{F}_{\zeta \Vec{\Psi}\bar{\zeta}}$ there exists a tree $\tau \in \mathcal{T}_{\zeta \Vec{\Psi}\bar{\zeta}}$ such that $\mathcal{R}_{\zeta \bar{\zeta}}(Q^{\Lambda;w}_{f}) \leqslant \mathcal{R}_{\zeta \bar{\zeta}}(Q^{\Lambda;w}_\tau)$. We denote by a double bar line the edges $\zeta,\bar{\zeta}$ and consider the case when $f \not \in \mathcal{T}_{\zeta \Vec{\Psi}\bar{\zeta}}$. Based on the inequality $\frac{\Lambda}{\Lambda + |p_\phi|}\leqslant 1$ we find $$\mbox{\begin{minipage}[c][48pt]{96pt}\centering\tiny\PandocStartInclude{pics/fgm_1.tex}\PandocEndInclude{input}{1257}{75}\end{minipage}}\leqslant\mbox{\begin{minipage}[c][48pt]{96pt}\centering\tiny\PandocStartInclude{pics/fgm_2.tex}\PandocEndInclude{input}{1257}{173}\end{minipage}}$$ $\blacksquare$\ At any loop order $l^\prime < l$ using the bound of theorem \[thm\_1\] and the inequality $$1\leqslant \frac{1}{\Lambda + |p_\varkappa|}\left((\Lambda + |p_1|) + (\Lambda + |p_2|)\right),$$ where $p_{\varkappa}+p_1 + p_2=0$, one realizes that for the marginal vertex function ${\mathsf{\Gamma}}^{\Lambda\Lambda_0;\phi_1\phi_2}_{\varkappa;l^\prime}$ the following inequality holds $$|{\mathsf{\Gamma}}^{\Lambda\Lambda_0;\phi_1\phi_2}_{\varkappa;l^\prime}(\Vec{p})|\leqslant \frac{1}{\Lambda + |p_\varkappa|} \Big(\mbox{\begin{minipage}[c][26pt]{64pt}\tiny\centering\PandocStartInclude{pics/junc_gamma_2.tex}\PandocEndInclude{input}{1266}{201}\end{minipage}}+\mbox{\begin{minipage}[c][26pt]{64pt}\tiny\centering\PandocStartInclude{pics/junc_gamma_3.tex}\PandocEndInclude{input}{1266}{298}\end{minipage}}\Big) P^\Lambda_{r_\varkappa(l^\prime)}\,.$$ Similarly, substituting the relevant terms ${\mathsf{\Gamma}}^{\Lambda\Lambda_0;\Vec{\phi};w}_{\beta;l^\prime}$, ${\textsc{n}}+\|w\|\leqslant 3$, at loop number $l^\prime<l$ with the bounds of theorem \[thm\_3\] we have $$\begin{aligned} |{\mathsf{\Gamma}}^{\Lambda\Lambda_0;\phi_1 \phi_2 \phi_3}_{\beta;l^\prime}|&\leqslant\mbox{\begin{minipage}[c][32pt]{64pt}\centering\tiny\PandocStartInclude{pics/junc_theta_0.tex}\PandocEndInclude{input}{1270}{157}\end{minipage}}F^{\Lambda\Lambda_0}_{0}P^{\Lambda\Lambda_0}_{r_\beta(l^\prime)}\,,\\ |{\mathsf{\Gamma}}^{\Lambda\Lambda_0;\phi_1 \phi_2}_{\beta;l^\prime}|&\leqslant\Big(\mbox{\begin{minipage}[c][32pt]{64pt}\tiny\centering\PandocStartInclude{pics/junc_theta_1.tex}\PandocEndInclude{input}{1271}{155}\end{minipage}}+\mbox{\begin{minipage}[c][32pt]{64pt}\tiny\centering\PandocStartInclude{pics/junc_theta_2.tex}\PandocEndInclude{input}{1271}{252}\end{minipage}}\Big)F^{\Lambda\Lambda_0}_{0}P^{\Lambda\Lambda_0}_{r_\beta(l^\prime)}\,,\\ |\partial {\mathsf{\Gamma}}^{\Lambda\Lambda_0;\phi_1 \phi_2}_{\beta;l^\prime}|&\leqslant\mbox{\begin{minipage}[c][32pt]{64pt}\tiny\centering\PandocStartInclude{pics/junc_theta_3.tex}\PandocEndInclude{input}{1272}{159}\end{minipage}}F^{\Lambda\Lambda_0}_{0}P^{\Lambda\Lambda_0}_{r_\beta(l^\prime)}\,.\end{aligned}$$ Furthermore using the bounds of theorem \[thm\_4\] which are assumed to be true for any loop order $l^\prime<l$, we obtain the following inequalities for strictly relevant terms ${\mathsf{\Gamma}}^{\Lambda\Lambda_0 \Vec{\phi};w}_{1;l^\prime}$, ${\textsc{n}}+\|w\|<5$, $$\begin{aligned} |{\mathsf{\Gamma}}^{\Lambda\Lambda_0;\phi_0 \phi_1 \phi_2 \phi_3}_{1;l^\prime}|&\leqslant\Big(\mbox{\begin{minipage}[c][32pt]{64pt}\tiny\centering\PandocStartInclude{pics/junc_gamma1_phi4_a.tex}\PandocEndInclude{input}{1275}{184}\end{minipage}}+\mbox{\begin{minipage}[c][32pt]{64pt}\tiny\centering\PandocStartInclude{pics/junc_gamma1_phi4_b.tex}\PandocEndInclude{input}{1275}{287}\end{minipage}}+...\Big)F^{\Lambda\Lambda_0}_{s_1(l^\prime)}P^{\Lambda\Lambda_0}_{r_1(l^\prime)},\\ |{\mathsf{\Gamma}}^{\Lambda\Lambda_0;\phi_0 \phi_1 \phi_2}_{1;l^\prime}|&\leqslant\Big(\mbox{\begin{minipage}[c][32pt]{64pt}\tiny\centering\PandocStartInclude{pics/junc_gamma1_phi3_a.tex}\PandocEndInclude{input}{1276}{164}\end{minipage}}+\mbox{\begin{minipage}[c][32pt]{64pt}\tiny\centering\PandocStartInclude{pics/junc_gamma1_phi3_b.tex}\PandocEndInclude{input}{1276}{267}\end{minipage}}+...\Big)F^{\Lambda\Lambda_0}_{s_1(l^\prime)}P^{\Lambda\Lambda_0}_{r_1(l^\prime)},\\ |\partial{\mathsf{\Gamma}}^{\Lambda\Lambda_0;\phi_0 \phi_1 \phi_2}_{1;l^\prime}|&\leqslant\Big(\mbox{\begin{minipage}[c][32pt]{64pt}\tiny\centering\PandocStartInclude{pics/junc_gamma1_phi3_c.tex}\PandocEndInclude{input}{1277}{172}\end{minipage}}+\mbox{\begin{minipage}[c][32pt]{64pt}\tiny\centering\PandocStartInclude{pics/junc_gamma1_phi3_d.tex}\PandocEndInclude{input}{1277}{275}\end{minipage}}\Big)F^{\Lambda\Lambda_0}_{s_1(l^\prime)}P^{\Lambda\Lambda_0}_{r_1(l^\prime)},\\ |{\mathsf{\Gamma}}^{\Lambda\Lambda_0;\phi_0 \phi_1;w}_{1;l^\prime}\mathbf{C}^{\Lambda \Lambda_0}|&\leqslant \frac{\Lambda + |p|}{(\Lambda + |p|)^{\|w\|}}F^{\Lambda\Lambda_0}_{s_1(l^\prime)}P^{\Lambda\Lambda_0}_{r_1(l^\prime)}\,,\end{aligned}$$ where the dots stand for omitted fragments obtained by permuting the “\*” over the external edges. Substituting the bounds for $\Gamma^{\Vec{\phi};w}_{1\varkappa}$ with those for $\partial_{\bar{c}} \Gamma^{\bar{c}\Vec{\phi};w}_{1}$ yields a similar result for the relevant terms $\Gamma^{\Vec{\phi};w}_{1,\Vec{\varkappa}}$ with $n_\varkappa>0$. Consequently the amplitude $\hat{S}^{\Lambda;\Vec{\phi};w}_{1\Vec{\varkappa}}$ is bounded by a sum of amplitudes $Q^{\Lambda;w}_{f}$ with $f \in \mathcal{F}_{1 \Vec{\varphi}}$. Irrelevant terms ---------------- The irrelevant terms at arbitrary $0<\Lambda\leqslant \Lambda_0$ are reconstructed by using the FE: $${\mathsf{\Gamma}}^{\Lambda \Lambda_0;\Vec{\phi};w}_{X\Vec{\varkappa};l}(\Vec{p})={\mathsf{\Gamma}}^{\Lambda_0 \Lambda_0;\Vec{\phi};w}_{X\Vec{\varkappa};l}(\Vec{p}) + \int \limits^{\Lambda}_{\Lambda_0} d \lambda \, \dot{{\mathsf{\Gamma}}}^{\lambda \Lambda_0;\Vec{\phi};w}_{X\Vec{\varkappa};l}(\Vec{p})\,.$$ At $\Lambda=\Lambda_0$ for all loop orders $l > 0$, for all $n_\varkappa \geqslant 0$ and $d_X < 0$ we have $$|{\mathsf{\Gamma}}^{\Lambda_0 \Lambda_0;\Vec{\phi};w}_{X\Vec{\varkappa};l}(\Vec{p})|\leqslant \left\{\begin{matrix}(|\Vec{p}|+\Lambda_0)^{d_X} F^{\Lambda_0 \Lambda_0}_{s_X}P^{\Lambda_0 \Lambda_0}_{r_X}&X \in \{\beta,1\},\\0&otherwise,\end{matrix}\right.$$ where the upper inequality is obtained using the bounds on relevant terms from theorem \[thm\_1\]. To integrate the FE from the boundary $\Lambda_0$ to $\Lambda$ we substitute the chains $\mathcal{S}^{\zeta \phi_0 \phi_1 \bar{\zeta}}$, $\mathcal{S}^{\zeta \varkappa \phi \bar{\zeta}}$ with the bounds given in theorem \[thm\_1\] and use inequalities , . Eventually we get $$\begin{aligned} |{\mathsf{\Gamma}}^{\Lambda\Lambda_0;\phi_0 \phi_1;w}_l(p)|&\leqslant (\Lambda+|p|)^{2-\|w\|} P^{\Lambda}_{2l-1},&\|w\|&> 2\,,\\ |{\mathsf{\Gamma}}^{\Lambda\Lambda_0;\phi;w}_{\varkappa;l}(p)|&\leqslant (\Lambda+|p|)^{1-\|w\|} P^{\Lambda}_{2l-1},&\|w\|&> 1\,.\end{aligned}$$ In a similar way substitution of $\mathcal{S}^{\zeta \beta \phi \bar{\zeta}}$, $\mathcal{S}^{\zeta \phi_0 \phi_1 \bar{\zeta}}_1$ with the bounds of theorems \[thm\_3\], \[thm\_4\] and then integration from $\Lambda_0$ to $\Lambda$, using , give $$\begin{aligned} |{\mathsf{\Gamma}}^{\Lambda \Lambda_0;\phi;w}_{\beta;l}(p)-{\mathsf{\Gamma}}^{\Lambda_0 \Lambda_0;\phi;w}_{\beta;l}(p)|&<\frac{F^{\Lambda\Lambda_0}_0 P^{\Lambda \Lambda_0}_{2l-1}}{(\Lambda + |p|)^{\|w\|-2}},&\|w\|&>2\,,\\ |{\mathsf{\Gamma}}^{\Lambda \Lambda_0;\phi_0 \phi_1;w}_{1;l}(p)-{\mathsf{\Gamma}}^{\Lambda_0 \Lambda_0;\phi_0 \phi_1;w}_{1;l}(p)|&<\frac{F^{\Lambda\Lambda_0}_{l-1} P^{\Lambda \Lambda_0}_{3l-2}}{(\Lambda + |p|)^{\|w\|-3}},&\|w\|&>3\,.\end{aligned}$$ \[prop\_r\]Let $\tau_i \in \mathcal{T}_{\zeta \bar{\zeta} \Vec{\phi}}$, $\phi_j \in \{A,c,\bar{c},\gamma,\omega\}$, $|V_1|>4$, $k \in \mathbb{N}$ and $\theta(\tau_i)>2$. Then $\exists \tau_f \in \mathcal{T}_{\Vec{\phi}}$ such that $$\int \limits^{\Lambda_0}_\Lambda d\lambda \, \lambda \; \mathcal{R}_{\zeta \bar{\zeta}}(Q^{\lambda;w}_{\tau_i}) \; P^{\lambda}_{k}|_{\eta=0} \leqslant Q^{\Lambda;w}_{\tau_f} P^{\Lambda}_{k+1}. \label{eq_16irr}$$ #### Proof Denote by $v,\bar{v} \in V \backslash V_1$ the vertices incident to $e_\zeta, e_{\bar{\zeta}} \in E_1$. First we assume that $v \neq \bar{v}$. The result of the restriction $\mathcal{R}_\zeta(Q^{\lambda;w}_{\tau_i})$ is the amplitude of the tree $\tau_i$ with two edges $e_1=\{u_1,v\}$, $e_2=\{u_2,v\}$ carrying opposite momenta. Here $\{u,v\}$ denotes the edge which links the vertices $u$, $v$. Without restriction we assume that $e_1=\chi(v)$. Furthermore we define a subtree $\tau^\prime$ of the initial tree $\tau_i$ by the substitution $e_2 \mapsto \{u_1,u_2\}$ such that it does not include the vertices $v$, $\zeta$ nor the edges $e_1$, $e_\zeta$. $\rho(e_1)=1$.\ The identity $p_{e_1}=-p_{e_2}$ implies $$\lambda \frac{1}{(\lambda + |p_{e_1}|)^{\theta(e_1)}} \leqslant \frac{1}{(\lambda + |p_{e_1}|)^{\sigma(e_1)}}=\frac{1}{(\lambda + |p_{e_2}|)^{\sigma(e_1)}}.$$ Consequently, $\lambda \mathcal{R}_\zeta (Q^{\lambda;w}_{\tau_i})$ is bounded by the amplitude of the subtree $\tau^\prime $ with $\sigma^\prime(e_2)=\sigma(e_2)+\sigma(e_1)$. $\rho(e_1)=0$ and $\rho(e_2) > 0$. $$\lambda \frac{1}{(\lambda + |p_{e_2}|)^{\theta(e_2)}} \leqslant \frac{1}{(\lambda + |p_{e_2}|)^{\theta(e_2)-1}}.$$ Thus, $\lambda \mathcal{R}_\zeta(Q^{\lambda;w}_{\tau_i})$ is bounded by the amplitude of the subtree $\tau^\prime$ with $\sigma^\prime(e_2)=\sigma(e_2)+\sigma(e_1)$, $\rho^\prime(e_2)=\rho(e_2)-1$ and $\chi^\prime(u_1)=e_2$. $\rho(e_1)=0$ and $\rho(e_2)=0$.\ This implies $e_2 \in E_1$. Because $|V_1|>4$ the vertex $u_1$ is incident to an edge $e_u \in E \backslash E_1$ such that $\rho(e_u)>0$. $$\lambda \frac{1}{(\lambda + |p_{e_u}|)^{\theta(e_u)}} \leqslant \frac{1}{(\lambda + |p_{e_u}|)^{\theta(e_u) - 1}}.$$ It follows that $\lambda \mathcal{R}_\zeta Q^{\lambda;w}_{\tau_i}$ is bounded by the amplitude of the subtree $\tau^\prime$ with $\sigma^\prime(e_2)=\sigma(e_2)+\sigma(e_1)$, $\rho^\prime(e_u)=\rho(e_u)-1$ and $\chi^\prime(u_1)=e_u$. For the subtree $\tau^\prime$ the total weight satisfies $\theta(\tau^\prime) > 1$. Consequently, we repeat the above reduction by substituting $\lambda$-multiplication with integration. Then using inequalities , we have $$\int \limits^{\Lambda_0}_\Lambda \frac{d \lambda \, P^{\lambda}_{k}|_{\eta=0}}{(\lambda + |p_a|)(\lambda + |p_b|)} \leqslant \frac{P^{\Lambda}_{k+1}}{\Lambda + |p_a| + |p_b|}\leqslant \frac{P^{\Lambda}_{k+1}}{\Lambda + |\hat{p}|}\,, \label{eq_irr}$$ where $\hat{p} \in \{p_a,p_b,p_a + p_b\}$. Here $p_a$ denotes the momentum of the edge used for the reduction, and $\hat{p}=p_b$ is the momentum of an arbitrary edge with nonvanishing $\theta$-weight of the final tree $\tau_f$. Because of the previous edge reduction we can reach the threshold $|V^\prime_1|=4$. In this case we can choose freely $\hat{p}$ to equal $p_a$, $p_b$ or $p_a + p_b$ thus keeping the $\sigma$-weight on the corresponding edge, see and . The case $v=\bar{v}$ follows directly from , . To see this we denote by $e \in E \backslash E_1$ the edge incident to $v$ and introduce a vertex $u\in V_3$ adjacent to $v$. Hence, $\chi(v)=e=\{v,u\}$. If $\tilde{\theta}$, $\theta$ are two $\theta$-weights where the only difference is that $\chi(u)=e$ for the first and $\chi(u) \not = e$ for the second, then $$\Pi^{\lambda}_{\tau_i,\tilde{\theta}}\leqslant \Pi^\lambda_{\tau_i,\theta}=\frac{1}{\lambda} \frac{1}{\lambda + p_{e \in E(\tau_f)}}\Pi^\lambda_{\tau_f,\theta}\,.$$ The final tree $\tau_f$ is a subtree of $\tau_i$ which does not include vertices $\zeta$,$\bar{\zeta}$,$v$ nor the corresponding edges. Moreover, $\theta(\tau_i)>2 \implies \theta(\tau_f)>0$.$\blacksquare$ Inequality can be applied to bound $\mathcal{S}^{\Lambda \Lambda_0;\Vec{\varkappa}\Vec{\phi};w}$. In this case $\tau_i \in \mathcal{T}_{\zeta \bar{\zeta}\Vec{\varkappa} \Vec{\phi}}$ and the total weight satisfies $\theta(\tau_i)>2-n_\varkappa$. But in theorem \[thm\_1\] for each edge $e_\varkappa \in E_\varkappa$ we have $(\Lambda + |p_\varkappa|)$ as a denominator which is equivalent to an additional $\sigma$-weight of the edge $e_\varkappa$. An effective tree $\tilde{\tau}$ with $\tilde{\sigma}(e_\varkappa)=\sigma(e_\varkappa)+1$ has $\theta(\tilde{\tau})>2$ and satisfies the conditions of proposition \[prop\_r\]. Using a similar inequality follows for the irrelevant functions ${\mathsf{\Gamma}}^{\Lambda \Lambda_0;\Vec{\phi};w}_{\beta\Vec{\varkappa};l}$. For any $\tau_i \in \mathcal{T}_{\zeta\bar{\zeta} \beta\Vec{\varkappa}\Vec{\phi}}$ there exists $\tau_f \in \mathcal{T}_{\beta\Vec{\varkappa}\Vec{\phi}}$ such that $$\int \limits^{\Lambda_0}_\Lambda d\lambda \, \lambda \; \mathcal{R}_{\zeta \bar{\zeta}}(Q^{\lambda;w}_{\tau_i}) \; F^{\lambda \Lambda_0}_0 P^{\lambda \Lambda_0}_{2(l-1)}|_{\eta=0} \leqslant Q^{\Lambda;w}_{\tau_f} F^{\Lambda \Lambda_0}_0 P^{\Lambda\Lambda_0}_{2l-1}\,.$$ Before application of to the irrelevant terms ${\mathsf{\Gamma}}^{\Lambda \Lambda_0;\Vec{\phi};w}_{1\Vec{\varkappa};l}$ we need a minor change in $$\int \limits^{\Lambda_0}_\Lambda d \lambda \left(\frac{\lambda + |p_*|}{\lambda + |p_e|}\right)\frac{F^{\lambda\Lambda_0}_{s} P^{\lambda \Lambda_0}_{k}|_{\eta=0}}{(\lambda + |p_a|)(\lambda + |p_b|)} \leqslant \left(\frac{\Lambda + |p^\prime_*|}{\Lambda + |p_e|}\right) \frac{F^{\Lambda\Lambda_0}_{s} P^{\Lambda\Lambda_0}_{k+1}}{\Lambda + |p_b|}\,,$$ where $p^\prime_*$ is one out of $p_*$ or $p_e$ in such way that $|p^\prime_*|:=\max(|p_*|,|p_e|)$. If $p_*=p_e$ the label “\*” is moved to edge $e$. Finally, $\forall \tau_i \in \mathcal{T}_{1\zeta\bar{\zeta} \Vec{\varkappa}\Vec{\phi}}$ $$\int \limits^{\Lambda_0}_\Lambda d\lambda \, \lambda \; \mathcal{R}_{\zeta \bar{\zeta}}(Q^{\lambda;w}_{\tau_i}) \; F^{\lambda \Lambda_0}_{l-1} P^{\lambda \Lambda_0}_{3(l-1)}|_{\eta=0} \leqslant \sum \limits_{\tau \in \mathcal{T}_{1\Vec{\varkappa}\Vec{\phi}}} Q^{\Lambda;w}_{\tau} F^{\Lambda \Lambda_0}_{l-1}P^{\Lambda\Lambda_0}_{3l-2}\,.$$ Marginal terms -------------- Let ${\mathsf{\Gamma}}^{\Lambda\Lambda_0;\Vec{\phi};w}_{\Vec{\varkappa};l}$ denote a marginal term, $n_\varkappa \leqslant 1$. Then $\forall \Lambda<\eta(\Vec{p})$ $$|{\mathsf{\Gamma}}^{\Lambda\Lambda_0;\Vec{\phi};w}_{\Vec{\varkappa};l}(\Vec{p})-{\mathsf{\Gamma}}^{\eta \Lambda_0;\Vec{\phi};w}_{\Vec{\varkappa};l}(\Vec{p})|\leqslant P^{\Lambda}_{2l-2}(\Vec{p}).\label{eq_flw0}$$ #### Proof Note that in theorem \[thm\_1\] for all $l^\prime < l$ the bounds for ${\mathsf{\Gamma}}^{\Lambda\Lambda_0;\Vec{\phi};w}_{\varkappa;l^\prime}$ are more restrictive than the ones for $\partial_{\bar{c}}{\mathsf{\Gamma}}^{\Lambda\Lambda_0;\bar{c}\Vec{\phi};w}_{l^\prime}$. So we will only treat the case $n_\varkappa=0$ explicitly. In this case we integrate the FE from $\eta$ to $\Lambda$ $$\begin{aligned} |{\mathsf{\Gamma}}^{\Lambda\Lambda_0;\Vec{\phi};w}_{l}(\Vec{p})-{\mathsf{\Gamma}}^{\eta \Lambda_0;\Vec{\phi};w}_{l}(\Vec{p})|\leqslant& \int \limits^{\eta}_\Lambda d\lambda | \dot{{\mathsf{\Gamma}}}^{\lambda\Lambda_0;\Vec{\phi}}_{l}(\Vec{p})|\nonumber \\ \leqslant& \sum \limits_{\tau \in \mathcal{T}_{\zeta \bar{\zeta} \Vec{\phi}}}\int \limits^{\eta}_\Lambda d\lambda \, \lambda \, \mathcal{R}_{\zeta \bar{\zeta}}(Q^{\lambda;w}_{\tau}) \, P^{\lambda}_{2l-2}|_{\eta_\tau=0},\end{aligned}$$ where $\theta(\tau)=2$. Denoting by $q_a$, $q_b$ the momenta of the edges with nonvanishing $\theta$-weight and using equations , for each term of the sum we obtain $$\int \limits^{\eta}_\Lambda d\lambda \, \frac{\lambda \, P^{\lambda}_{k}|_{\eta_\tau=0}}{(\lambda + |q_a|)(\lambda + |q_b|)} \leqslant \int \limits^{\eta}_\Lambda d\lambda \, \frac{P^{\lambda}_{k}|_{\eta_\tau=0}}{\lambda + |q_b|} \leqslant P^{\Lambda}_{k},\quad k=2l-2.$$ $\blacksquare$ ### ${\mathsf{\Gamma}}^{c\bar{c}AA}$ and ${\mathsf{\Gamma}}^{c\bar{c}c\bar{c}}$ The renormalization condition is ${\mathsf{\Gamma}}^{M\Lambda_0;\Vec{\phi}}_l(0)=0$. For $p_2,p_3 \in \mathbb{R}^4$ and $\Lambda^\prime:=\max(\Lambda,\eta(\Vec{p}))$ equation gives ${\mathsf{\Gamma}}^{\Lambda^\prime \Lambda_0;\Vec{\phi}}_l(0_{\bar{c}},p_2,p_3)=0$, where the subscript $\bar{c}$ indicates the momentum of the antighost. $$|{\mathsf{\Gamma}}^{\Lambda^\prime \Lambda_0;\Vec{\phi}}_l(\Vec{p})|\leqslant \int \limits^1_0 dt \, |p_{\bar{c}}|\,|\partial_{\bar{c}} {\mathsf{\Gamma}}^{\Lambda^\prime \Lambda_0;\Vec{\phi}}_l(tp_{\bar{c}},p_2,p_3)|\,.$$ Substituting $|\partial_{\bar{c}} {\mathsf{\Gamma}}^{\Lambda^\prime \Lambda_0;\Vec{\phi}}_l|$ with the bound of theorem \[thm\_1\] and using inequality  we obtain $$|{\mathsf{\Gamma}}^{\Lambda^\prime \Lambda_0;\Vec{\phi}}_l(\Vec{p})| \leqslant \Big(1+\log_+\frac{|\Vec{p}|}{\Lambda^\prime + \eta}\Big)P^{\Lambda^\prime\Lambda^\prime}_{2l-1}(\Vec{p})\Big|_{\eta=0} \leqslant P^{\Lambda}_{2l}(\Vec{p})\,.$$ If $\Lambda^\prime=\Lambda$ the proof is finished. Otherwise we use . ### $\partial_A {\mathsf{\Gamma}}^{c\bar{c}A}$ The renormalization condition is $\partial_A {\mathsf{\Gamma}}^{M\Lambda_0;c \bar{c} A}_l(0)=0$. For $p_A \in \mathbb{R}^4$ and $\Lambda^\prime=\max(\Lambda,\eta(\Vec{p}))$ equation gives $\partial_A {\mathsf{\Gamma}}^{\Lambda^\prime \Lambda_0;c \bar{c} A}_l(0,p_A)=0$. $$|\partial_A {\mathsf{\Gamma}}^{\Lambda^\prime \Lambda_0;c\bar{c}A}_l(\Vec{p})|\leqslant \int \limits^1_0 dt \, |p_{\bar{c}}|\, |\partial_{\bar{c}} \partial_A {\mathsf{\Gamma}}^{\Lambda^\prime \Lambda_0;c \bar{c} A}_l(tp_{\bar{c}},p_A)|\,.$$ We substitute $|\partial_{\bar{c}} \partial_A {\mathsf{\Gamma}}^{\Lambda^\prime \Lambda_0;c \bar{c} A}_l(\Vec{p}_t)|$ with the bound from theorem \[thm\_1\] with the choice $w^\prime(2)=(0,1,0)$, see after for the definition of $w^\prime$. Then $$Q^{\Lambda^\prime;(0,1,1)}_{c\bar{c}A} \in \left\{\frac{1}{\Lambda^\prime + t|p_{\bar{c}}|},\frac{1}{\Lambda^\prime + |tp_{\bar{c}} + p_A|}\right\}\,,$$ and using we obtain the inequality $$|\partial_A {\mathsf{\Gamma}}^{\Lambda^\prime \Lambda_0;c\bar{c}A}_l(\Vec{p})|\leqslant P^{\Lambda^\prime\Lambda^\prime}_{2l}(\Vec{p})\leqslant P^{\Lambda}_{2l}(\Vec{p})\,.$$ If $\Lambda^\prime=\Lambda$ the proof is finished. If not we use . ### Renormalization at $\Lambda=0$ {#sec_20r} First we consider the marginal terms ${\mathsf{\Gamma}}^{c c}_{\omega;l}$, ${\mathsf{\Gamma}}^{c A}_{\gamma;l}$, ${\mathsf{\Gamma}}^{AAAA}_l$, $\partial_{\bar{c}} {\mathsf{\Gamma}}^{c\bar{c}A}_l$, $\partial {\mathsf{\Gamma}}^{AAA}_l$, $\partial\partial {\mathsf{\Gamma}}^{AA}_l$, $\partial \partial {\mathsf{\Gamma}}^{c \bar{c}}_l$, here denoted by ${\mathsf{\Gamma}}^{\Vec{\phi};w}_{\Vec{\varkappa};l}$. The marginal terms ${\mathsf{\Gamma}}^{\Vec{\phi}}_{1\Vec{\varkappa};l}$ and ${\mathsf{\Gamma}}^{\Vec{\phi}}_{\beta;l}$ will be discussed later. Let $\{\delta^s{\underline}{e}\}_r$ be a basis at the renormalization point $\Vec{q}$. We define ${\mathsf{\Gamma}}^{\Vec{\phi};w}_{\Vec{\varkappa};l}(\Vec{q})$ in the following way: $${\mathsf{\Gamma}}^{0 \Lambda_0;\Vec{\phi};w}_{\Vec{\varkappa};l}(\Vec{q}):=\sum \limits_{t \in \{\delta^s\}_r} r_t\, t + \sum \limits_{t \alpha \in \{\delta^s {\underline}{e}^{k>0}\}_r} \zeta_t \,t\,,$$ where the coefficients $r_t$ are fixed by the renormalization conditions, see appendix \[sec\_ren\] and hypothesis \[rc3\], and the remaining coefficients $\zeta_t$ are defined using lemma \[lem\_pr\]. Then, from the bounds on the irrelevant terms and lemma \[lem\_rn\], see , it follows that ${\mathsf{\Gamma}}^{\Vec{\phi};w}_{\Vec{\varkappa};l}(\Vec{q})$ complies with theorem \[thm\_1\] at loop order $l$. Let $\Lambda^\prime:=\max(\Lambda,\eta(\Vec{p}))$. It is easy to verify the following inequalities $$\begin{aligned} \int \limits^{\Lambda^\prime}_0 d\lambda \frac{\log^k_+ \frac{M}{\lambda}}{\lambda + M} &< k! + \log_+ \frac{\Lambda^\prime}{M},& \int \limits^{\Lambda^\prime}_0 d\lambda \frac{\log^k_+ \frac{\lambda}{M}}{\lambda +M}&<2\Big(1+ \log^{k+1}_+ \frac{\Lambda^\prime}{M}\Big)\,.\label{eq_int01}\end{aligned}$$ Recalling , we obtain the following bound $$|\dot{{\mathsf{\Gamma}}}^{\lambda\Lambda_0;\Vec{\phi};w}_{\Vec{\varkappa};l}(\Vec{q})|\leqslant \lambda\frac{\mathcal{P}^{(0)}_{2l-2}(\log_+ \frac{M}{\lambda})+\mathcal{P}^{(1)}_{2l-2}(\log_+(\frac{\lambda}{M}))}{(\lambda + M)^2}\,.\label{eq_int02}$$ This implies that $$|{\mathsf{\Gamma}}^{\Lambda^\prime \Lambda_0;\Vec{\phi};w}_{\Vec{\varkappa};l}(\Vec{q})-{\mathsf{\Gamma}}^{0\Lambda_0;\Vec{\phi};w}_{\Vec{\varkappa};l}(\Vec{q})|\leqslant \int \limits^{\Lambda^\prime}_{0} d\lambda \; |\dot{{\mathsf{\Gamma}}}^{\lambda\Lambda_0;\Vec{\phi};w}_{\Vec{\varkappa};l}(\Vec{q})| \leqslant \mathcal{P}^{(1)}_{2l-1}\Big(\log_+ \frac{\Lambda}{M}\Big).$$ Using inequality we get $$\begin{aligned} |{\mathsf{\Gamma}}^{\Lambda^\prime\Lambda_0;\Vec{\phi};w}_{\Vec{\varkappa};l}(0)-{\mathsf{\Gamma}}^{\Lambda^\prime\Lambda_0;\Vec{\phi};w}_{\Vec{\varkappa};l}(\Vec{q})|&\leqslant \sum^{n-1}_{j=1} I_j(\Vec{q})\leqslant P^{\Lambda^\prime}_{2l}(0),\\ |{\mathsf{\Gamma}}^{\Lambda^\prime\Lambda_0;\Vec{\phi};w}_{\Vec{\varkappa};l}(\Vec{p})-{\mathsf{\Gamma}}^{\Lambda^\prime\Lambda_0;\Vec{\phi};w}_{\Vec{\varkappa};l}(0)|&\leqslant \sum^{n-1}_{j=1} I_j(\Vec{p})\leqslant P^{\Lambda}_{2l}(\Vec{p}),\end{aligned}$$ where $I_j(\Vec{q})$ is the interpolation along the vector $q_j$, $$I_j(\Vec{q})=\int \limits^1_0 dt \, |q_j|\, |\partial_j {\mathsf{\Gamma}}^{\Lambda^\prime\Lambda_0;\Vec{\phi};w}_{\Vec{\varkappa};l}(\sum^{j-1}_{i=1}\Vec{q}_i+t\Vec{q}_j)|. \label{eq_path}$$ Here to each vector $q_i\in \mathbb{R}^4$ is associated $\Vec{q}_i\in \mathbb{P}_n$ whose components are $(\Vec{q}_i)_{k}=-q_i \delta_{k,0} + q_i \delta_{k,i}$. Once again we have substituted $|\partial_j {\mathsf{\Gamma}}^{M\Lambda_0;\Vec{\phi};w}_{\Vec{\varkappa};l}|$ with the bound from theorem \[thm\_1\]. If $\Lambda=\Lambda^\prime$ we stop here. If not we use . The remaining marginal terms ${\mathsf{\Gamma}}^{\Lambda \Lambda_0;\Vec{\phi};w}_{X \Vec{\varkappa};l}$ with $X \in \{\beta,1\}$ can be treated similarly. Note that the bound from theorem \[thm\_1\] for terms of the type $\partial_{\bar{c}} {\mathsf{\Gamma}}^{\bar{c} \Vec{\phi};w}_{1;l}$ is the same as the one for the corresponding terms ${\mathsf{\Gamma}}^{\Vec{\phi};w}_{1\varkappa;l}$. Consequently the proof of the bounds for the marginal terms ${\mathsf{\Gamma}}^{ccA}_{1\gamma;l}$, ${\mathsf{\Gamma}}^{ccc}_{1\omega;l}$, $\partial {\mathsf{\Gamma}}^{cc}_{1\gamma;l}$ is the same as the proof for respectively $\partial_{\bar{c}} {\mathsf{\Gamma}}^{\bar{c}ccA}_{1;l}$, $\partial \partial_{\bar{c}} {\mathsf{\Gamma}}^{\bar{c}cc}_{1;l}$ which we shall consider now. Let us denote by ${\mathsf{\Gamma}}^{0\Lambda_0;\Vec{\phi};w}_{X;l}$ any marginal term without insertions $\gamma$ or $\omega$, and by $\Vec{q}$ the corresponding renormalization point as given in appendix \[sec\_ren\]. We now anticipate the important fact that the relevant renormalization constants comply with the bounds, which will be proven in section \[sec\_sti\]. Then using the bounds on irrelevant terms from theorems \[thm\_3\], \[thm\_4\] and lemma \[lem\_rn\], see , we obtain $$|{\mathsf{\Gamma}}^{0\Lambda_0;\Vec{\phi};w}_{X;l}(\Vec{q})|\leqslant \frac{M}{\Lambda_0} \mathcal{P}^{(1)}_{r_X}\Big(\log_+ \frac{\Lambda_0}{M}\Big)\,,$$ in agreement with theorems \[thm\_3\], \[thm\_4\]. Set $\Lambda^\prime:=\max(\Lambda,M)$. To integrate the FE from $0$ to $\Lambda^\prime$ we substitute the chain of vertex functions with the trees from theorems \[thm\_3\], \[thm\_4\]. Then we have $$|{\mathsf{\Gamma}}^{\Lambda^\prime \Lambda_0;\Vec{\phi};w}_{X,l}(\Vec{q})-{\mathsf{\Gamma}}^{0\Lambda_0;\Vec{\phi};w}_{X,l}(\Vec{q})| \leqslant \frac{\Lambda + M}{\Lambda_0}\mathcal{P}^{(1)}_{r_X}\Big(\log_+ \frac{\Lambda_0}{M}\Big).\label{eq_413a}$$ Using inequality we obtain $$\begin{aligned} |{\mathsf{\Gamma}}^{\Lambda^\prime\Lambda_0;\Vec{\phi};w}_{X;l}(0)-{\mathsf{\Gamma}}^{\Lambda^\prime\Lambda_0;\Vec{\phi};w}_{X;l}(\Vec{q})|&\leqslant \frac{\Lambda + M}{\Lambda_0}P^{\Lambda^\prime \Lambda_0}_{r_X}(0),\label{eq_413b}\\ |{\mathsf{\Gamma}}^{\Lambda^\prime\Lambda_0;\Vec{\phi};w}_{X;l}(0)-{\mathsf{\Gamma}}^{\Lambda^\prime\Lambda_0;\Vec{\phi};w}_{X;l}(\Vec{p})|&\leqslant F^{\Lambda \Lambda_0}_{s_X}(\Vec{p})P^{\Lambda \Lambda_0}_{r_X}(\Vec{p}).\label{eq_413c}\end{aligned}$$ Note here that as compared to the case $X=\beta$, in the case $X=1$ there appears an additional factor $$\frac{\Lambda^\prime + |p_*|}{\Lambda^\prime + |p_a|}\leqslant 1+2\frac{|\Vec{p}|}{\Lambda + M}\,,$$ in the tree bounds from theorem \[thm\_4\]. This factor leads to the polynominal $\mathcal{P}^{(2)}_{s_1}$ in the bounds of theorem \[thm\_4\]. If $\Lambda=\Lambda^\prime$ the proof is finished. If not we integrate downwards using the FE, substitute the chain with the tree bound from theorems \[thm\_3\], \[thm\_4\] and use  to get $$\begin{aligned} |{\mathsf{\Gamma}}^{\Lambda \Lambda_0;\Vec{\phi};w}_{\beta;l}(\Vec{p})-{\mathsf{\Gamma}}^{M \Lambda_0;\Vec{\phi};w}_{\beta;l}(\Vec{p})|&\leqslant F^{\Lambda \Lambda_0}_{0}(\Vec{p}) P^{\Lambda \Lambda_0}_{2l-1}(\Vec{p})\,,\\ |{\mathsf{\Gamma}}^{\Lambda \Lambda_0;\Vec{\phi};w}_{1;l}(0)-{\mathsf{\Gamma}}^{M\Lambda_0;\Vec{\phi};w}_{1;l}(\Vec{p})|&\leqslant \sum \limits_{\tau \in \mathcal{T}_{1\Vec{\phi}}} Q^{\Lambda;w}_\tau\, F^{\Lambda \Lambda_0}_{l-1} P^{\Lambda \Lambda_0}_{3l-2}(\Vec{p})\,.\end{aligned}$$ Strictly relevant terms ----------------------- If $n_\varkappa=0$ the notation ${\mathsf{\Gamma}}^{\Vec{\phi};w}_{\Vec{\varkappa};l}$ stands for ${\mathsf{\Gamma}}^{c\bar{c}A}_l$, ${\mathsf{\Gamma}}^{AAA}_l$, $\partial {\mathsf{\Gamma}}^{AA}_l$, $\partial {\mathsf{\Gamma}}^{c \bar{c}}_l$. In the case $n_\varkappa=1$, it stands for ${\mathsf{\Gamma}}^c_{\gamma;l}$. Moreover we impose ${\mathsf{\Gamma}}^{0\Lambda_0\Vec{\phi};w}_{\Vec{\varkappa};l}(0)=0$ and denote by $\Vec{p}$ arbitrary momenta with corresponding $\eta(\Vec{p})$. We integrate the FE upwards from 0 to $\Lambda$ and substitute $|\dot{{\mathsf{\Gamma}}}^{\lambda\Lambda_0\Vec{\phi};w}_{\Vec{\varkappa};l}|$ with the tree bound of theorem \[thm\_1\]. Then $$|{\mathsf{\Gamma}}^{\Lambda\Lambda_0\Vec{\phi};w}_{\Vec{\varkappa};l}(0)|\leqslant \int \limits^\Lambda_0 d\lambda \; |\dot{{\mathsf{\Gamma}}}^{\lambda\Lambda_0\Vec{\phi};w}_{\Vec{\varkappa};l}(0)|\leqslant \int \limits^{\Lambda}_0 d\lambda \, \lambda^{d-1}P^{\lambda \Lambda}_{2l-2}(0)\leqslant \int \limits^{\Lambda + \eta}_0 d\lambda \, \lambda^{d-1}P^{\lambda \Lambda}_{2l-2}(0)\,,$$ where $d>0$. Inequality then gives $$|{\mathsf{\Gamma}}^{\Lambda\Lambda_0\Vec{\phi};w}_{\Vec{\varkappa};l}(0)|\leqslant (\Lambda + \eta)^d P^{\Lambda+\eta\;\Lambda}_{2l-2}(0). \label{eq_111}$$ Substituting $\partial {\mathsf{\Gamma}}^{\Lambda\Lambda_0\Vec{\phi};w}_{\Vec{\varkappa};l}$ with the bound from theorem \[thm\_1\] we obtain $$\begin{aligned} |{\mathsf{\Gamma}}^{\Lambda\Lambda_0\Vec{\phi};w}_{\Vec{\varkappa};l}(\Vec{p})-{\mathsf{\Gamma}}^{\Lambda\Lambda_0\Vec{\phi};w}_{\Vec{\varkappa};l}(0)|\leqslant& \int \limits^1_0 dt \, |p_i| |\partial_i {\mathsf{\Gamma}}^{\Lambda\Lambda_0\Vec{\phi};w}_{\Vec{\varkappa};l}(t\Vec{p})|\nonumber\\ \leqslant& |\Vec{p}| \int \limits^1_0 dt \, (\Lambda + t|\Vec{p}|)^{d-1} P^{\Lambda}_{2l}(t\Vec{p}) \leqslant I+I^\prime,\label{eq_31228}\end{aligned}$$ where $$\begin{aligned} I:=& |\Vec{p}| (\Lambda + |\Vec{p}|)^{d-1}\int \limits^1_0 dt \, \mathcal{P}^{(0)}_{2l}(\log_+ \frac{\max(M, |\Vec{p}|)}{\Lambda + t\eta(\Vec{p})}),\\ I^\prime:=&(\Lambda + |\Vec{p}|)^{d} \, \mathcal{P}^{(1)}_{2l}(\log_+ \frac{\Lambda}{M}).\end{aligned}$$ The calculations given in yield $$I\leqslant |\Vec{p}|(\Lambda + |\Vec{p}|)^{d-1} \, \mathcal{P}^{(0)}_{2l}(\log_+ \frac{\max(M, |\Vec{p}|)}{\Lambda + \eta(\Vec{p})}).$$ This implies $$|{\mathsf{\Gamma}}^{\Lambda\Lambda_0\Vec{\phi};w}_{\Vec{\varkappa};l}(\Vec{p})-{\mathsf{\Gamma}}^{\Lambda\Lambda_0\Vec{\phi};w}_{\Vec{\varkappa};l}(0)| \leqslant (\Lambda + |\Vec{p}|)^{d}P^{\Lambda}_{2l}(\Vec{p}). \label{eq_112}$$ Combining and proves the bounds of theorem \[thm\_1\] to loop order $l$. ### $ \partial {\mathsf{\Gamma}}^{\phi^3}_1$, ${\mathsf{\Gamma}}^{\phi^4}_1$ and ${\mathsf{\Gamma}}^{cc}_{1\gamma}$ The goal of this section is to explain the expression for the polynomial degree $r_1$. Subsequently we denote by ${\mathsf{\Gamma}}^{\Vec{\phi};w}_1$ the following terms: $\partial {\mathsf{\Gamma}}^{c AA}_1$, $\partial {\mathsf{\Gamma}}^{c c \bar{c}}_1$, ${\mathsf{\Gamma}}^{c AAA}_1$, ${\mathsf{\Gamma}}^{c Ac \bar{c}}_1$ and we impose vanishing renormalization conditions at the origin, ${\mathsf{\Gamma}}^{0 \Lambda_0;\Vec{\phi};w;}_1(0)=0$. From the bounds of theorem \[thm\_4\] one realises that the analysis for $\Gamma^{cc}_{1\gamma}$ is similar to $\partial_{\bar{c}}{\mathsf{\Gamma}}^{cc \bar{c}}_1$. For an arbitrary $\Vec{p}\in \mathbb{P}_n$ let $\Lambda^\prime:=\max(\Lambda,\eta(\Vec{p}))$. Using , $$\begin{aligned} |{\mathsf{\Gamma}}^{\Lambda^\prime \Lambda_0\Vec{\phi};w}_{1;l}(0)|\leqslant&(\Lambda + \eta) \, \frac{\Lambda + M}{\Lambda_0}\, P^{\Lambda+\eta\;\Lambda_0}_{3(l-1)}(0).\end{aligned}$$ We extend ${\mathsf{\Gamma}}^{\Lambda^\prime \Lambda_0;\Vec{\phi};w}_{1;l}$ from 0 to $\Vec{p}$ using the usual path given in equation (with $\Vec{q} \mapsto \Vec{p}$), where we substitute $\partial {\mathsf{\Gamma}}^{\Lambda^\prime \Lambda_0\Vec{\phi};w}_{1;l}$ with the tree bound of theorem \[thm\_4\]. The rational factor in this bound makes these terms different from other strictly relevant terms, for example $\partial \partial {\mathsf{\Gamma}}^{c A}_{1}$, ${\mathsf{\Gamma}}^{cA}_{\beta}$ or $\partial {\mathsf{\Gamma}}^{c}_{\beta}$. Then using inequality we have $$|{\mathsf{\Gamma}}^{\Lambda^\prime \Lambda_0;\Vec{\phi};w}_{1;l}(\Vec{p})-{\mathsf{\Gamma}}^{\Lambda^\prime \Lambda_0;\Vec{\phi};w}_{1;l}(0)|\leqslant (\Lambda + |\Vec{p}|)\Big(1 + \log_+ \frac{|\Vec{p}|}{\Lambda + \eta}\Big)F^{\Lambda \Lambda_0}_l(\Vec{p})\, P^{\Lambda \Lambda_0}_{3l-1}(\Vec{p}).$$ If $\Lambda=\Lambda^\prime$ the proof is finished. If not we repeat the steps used to show integrating the FE from $\eta$ to $\Lambda$ and using $$|{\mathsf{\Gamma}}^{\Lambda \Lambda_0;\Vec{\phi};w}_{1;l}(\Vec{p})-{\mathsf{\Gamma}}^{\eta \Lambda_0;\Vec{\phi};w}_{1;l}(\Vec{p})|\leqslant |\Vec{p}|\, F^{\Lambda \Lambda_0}_{l-1}(\Vec{p})\, P^{\Lambda \Lambda_0}_{3l-2}(\Vec{p}).$$ ### $ {\mathsf{\Gamma}}^{\Vec{\phi};w}_\beta$, ${\mathsf{\Gamma}}^{\phi^2;w}_1$ and $ {\mathsf{\Gamma}}^{\phi^3}_1$ In this section we briefly discuss the remaining strictly relevant terms ${\mathsf{\Gamma}}^{\Vec{\phi};w}_\beta$, ${\mathsf{\Gamma}}^{\phi^3}_1$ and ${\mathsf{\Gamma}}^{\phi^2;w}_1$ denoting all of them by ${\mathsf{\Gamma}}^{\Vec{\phi};w}_X$ with $X \in \{\beta,1\}$. We impose renormalization conditions ${\mathsf{\Gamma}}^{0 \Lambda_0;\Vec{\phi};w}_X(0)=0$ and integrate the FE from 0 to $\Lambda$. We use the bounds of theorems \[thm\_3\],\[thm\_4\] and then for an arbitrary $\Vec{p}$ extend the integration up to $\Lambda + \eta(\Vec{p})$ $$|{\mathsf{\Gamma}}^{\Lambda \Lambda_0;\Vec{\phi};w}_X(0)|\leqslant (\Lambda+\eta)^{d_X}\frac{\Lambda+M}{\Lambda_0}P^{\Lambda+\eta \;\Lambda_0}_{r_X}(0)\,.$$ Integration along the path given in equation and using yields $$|{\mathsf{\Gamma}}^{\Lambda \Lambda_0;\Vec{\phi};w}_X(\Vec{p})-{\mathsf{\Gamma}}^{\Lambda \Lambda_0;\Vec{\phi};w}_X(0)|\leqslant (\Lambda + |\Vec{p}|)^{d_X} F_{s_X}(\Vec{p}) P^{\Lambda \Lambda_0}_{s_X}(\Vec{p}).$$ Convergence {#sec_conv} ----------- We first prove the bounds for $\partial_{\Lambda_0}{\mathsf{\Gamma}}^{\Vec{\phi}}$. Then we proceed with the other functions $\partial_{\Lambda_0} {\mathsf{\Gamma}}^{\Vec{\phi}}_{\Vec{\varkappa}}$ ascending in the number of insertions $n_\varkappa$. We use the same inductive scheme as before, based on the FE. We start with the irrelevant terms integrating the FE from $\Lambda_0$ to $\Lambda$, using the boundary conditions ${\mathsf{\Gamma}}^{\Lambda_0 \Lambda_0;\Vec{\phi};w}_l=0$ and applying the derivative wrt $\Lambda_0$. $$\partial_{\Lambda_0} {\mathsf{\Gamma}}^{\Lambda \Lambda_0;\Vec{\phi};w}_{\Vec{\varkappa};l}= - \dot{{\mathsf{\Gamma}}}^{\Lambda_0 \Lambda_0;\Vec{\phi};w}_{\Vec{\varkappa};l} + \int \limits^\Lambda_{\Lambda_0} d \lambda \,\partial_{\Lambda_0} \dot{{\mathsf{\Gamma}}}^{\lambda \Lambda_0;\Vec{\phi};w}_{\Vec{\varkappa};l}.$$ To bound the first term of the expression we substitute into the FE the irrelevant tree bound $$|\dot{{\mathsf{\Gamma}}}^{\Lambda_0 \Lambda_0;\Vec{\phi};w}_{\Vec{\varkappa};l}|\leqslant \frac{1}{\Lambda_0}\sum_{\tau \in \mathcal{T}_{\Vec{\varkappa}\Vec{\phi}}} Q^{\Lambda_0;w}_\tau P^{\Lambda_0 \Lambda_0}_{2(l-1)}.$$ If $n_\varkappa=0$ then $\tau \in \mathcal{T}_{\Vec{\phi}}$ and $\theta(\tau) >0$. Consequently, recalling $$\Pi^{\Lambda_0}_{\tau,\theta}(\Vec{p}) \leqslant \frac{\Lambda + |\Vec{p}|}{\Lambda_0}\Pi^{\Lambda}_{\tau,\theta}(\Vec{p}), \quad\mbox{and thus}\quad Q^{\Lambda_0;w}_\tau \leqslant \frac{\Lambda + |\Vec{p}|}{\Lambda_0} Q^{\Lambda;w}_\tau.$$ Otherwise, the denominator $\Lambda_0 + |p_e|$ with $e_\varkappa \in E_\varkappa$ gives the inequality $$\frac{1}{\Lambda_0 + |p_\varkappa|}\leqslant \frac{\Lambda + |\Vec{p}|}{\Lambda_0} \frac{1}{\Lambda + |p_\varkappa|}.$$ In both cases this yields $$|\dot{{\mathsf{\Gamma}}}^{\Lambda_0 \Lambda_0;\Vec{\phi};w}_{\Vec{\varkappa};l}|\leqslant \frac{\Lambda + |\Vec{p}|}{\Lambda^2_0} \sum_{\tau \in \mathcal{T}_{\Vec{\varkappa}\Vec{\phi}}} Q^{\Lambda;w}_\tau P^{\Lambda \Lambda_0}_{2(l-1)}.$$ To analyse the second term we apply $\partial^w_p\partial_{\Lambda_0}$ to the chain of vertex functions given in definition \[def\_chain\]. This gives a chain with the element $\partial_{\Lambda_0}((\partial^{w_1}_p \mathbf{C}){\mathsf{\Gamma}}^{;w_2}_{l^\prime})$ $l^\prime < l$ which we bound using , and theorem \[thm\_2\]. $$|\partial_{\Lambda_0}((\partial^{w_1}_p \mathbf{C}){\mathsf{\Gamma}}^{\lambda \Lambda_0;\zeta\Vec{\phi}\bar{\zeta};w_2}_{\Vec{\varkappa};l^\prime})|\leqslant \,|\partial_{\Lambda_0}\partial^{w_1}_p\mathbf{C}|\,|{\mathsf{\Gamma}}^{\lambda \Lambda_0;\zeta\Vec{\phi}\bar{\zeta};w_2}_{\Vec{\varkappa};l^\prime}|+|\partial^{w_1}_p\mathbf{C}|\,|\partial_{\Lambda_0}{\mathsf{\Gamma}}^{\lambda \Lambda_0;\zeta\Vec{\phi}\bar{\zeta};w_2}_{\Vec{\varkappa};l^\prime}|$$ $$\leqslant \frac{c}{(\lambda+|p_{\zeta}|)^{2 + \|w_1\|}} \Big(\frac{\lambda + |p_{\zeta}| }{\Lambda^2_0}|{\mathsf{\Gamma}}^{\lambda \Lambda_0;\zeta\Vec{\phi}\bar{\zeta};w_2}_{\Vec{\varkappa};l^\prime}|+|\partial_{\Lambda_0}{\mathsf{\Gamma}}^{\lambda \Lambda_0;\zeta\Vec{\phi}\bar{\zeta};w_2}_{\Vec{\varkappa};l^\prime}|\Big)$$ $$\leqslant \frac{1}{(\lambda+|p_\zeta|)^{2}} \frac{\lambda +M+ |\Vec{p}| }{\Lambda^2_0}\sum \limits_{\tau \in \mathcal{T}_{\Vec{\varkappa}\zeta \Vec{\phi} \bar{\zeta}}} Q^{\lambda; w_2 + w_1}_\tau P^{\lambda \Lambda_0}_{r(l^\prime)}(\Vec{p}) \,.$$ We proceed as for the proof of inequality substituting expression with the integral $$\int \limits^{\Lambda_0}_\Lambda d\lambda \frac{(\lambda + M+|\Vec{p}|) P^{\lambda\Lambda_0}_{2l-2}|_{\eta_\tau=0}}{\Lambda^2_0(\lambda+|p_a|)(\lambda + |p_b|)}\leqslant \frac{\Lambda + M+ |\Vec{p}|}{\Lambda^2_0}\frac{1}{\Lambda + |\hat{p}|}P^{\Lambda\Lambda_0}_{2l-1}(\Vec{p}),$$ see the explanation after . If $\tau \in \mathcal{T}_{\varkappa\phi \phi}$ we always choose $\hat{p}=p_\varkappa$ in order to preserve the factor $\Lambda + |p_\varkappa|$ in the denominator of $Q^{\Lambda}_{\tau}$. For the marginal terms we shall integrate the FE upwards from 0 to $\Lambda$. For the terms with the antighost we use renormalization conditions $$\begin{aligned} {\mathsf{\Gamma}}^{M\Lambda_0;c\bar{c}AA}_l(0)&=0,&{\mathsf{\Gamma}}^{M\Lambda_0;c\bar{c} c \bar{c}}_l(0)&=0,&\partial_A {\mathsf{\Gamma}}^{M\Lambda_0;c \bar{c}A}_l(0)&=0.\end{aligned}$$ Using equation we obtain that for these terms at $\Lambda^\prime=\max(\Lambda,\eta(\Vec{p}))$ $$\begin{aligned} &|\partial_{\Lambda_0}{\mathsf{\Gamma}}^{\Lambda^\prime\Lambda_0;\Vec{\phi};w}_l(\Vec{p})|\leqslant\int \limits^1_0 dt \, |p_{\bar{c}}|\, |\partial_{\Lambda_0} \partial_{\bar{c}}{\mathsf{\Gamma}}^{\Lambda^\prime\Lambda_0;\Vec{\phi};w}_l(tp_{\bar{c}},...)|\nonumber \\ &\quad \leqslant \frac{\Lambda^\prime + M+|\Vec{p}|}{\Lambda^2_0} \sum_{\tau \in \mathcal{T}_{\Vec{\phi}}} \int \limits^1_0 dt \, |p_{\bar{c}}|\, Q^{\Lambda^\prime;w+1_{\bar{c}}}_{\tau}P^{\Lambda^\prime\Lambda_0}_{2l-1}\leqslant \frac{\Lambda + M+|\Vec{p}|}{\Lambda^2_0} P^{\Lambda\Lambda_0}_{2l}\,,\end{aligned}$$ where we have substituted $\partial_{\Lambda_0} \partial_{\bar{c}} {\mathsf{\Gamma}}^{\Vec{\phi};w}_l$ with the bound of theorem \[thm\_2\] and applied inequality . The remaining marginal terms ${\mathsf{\Gamma}}^{AAAA}$, $\partial_{\bar{c}}{\mathsf{\Gamma}}^{c\bar{c}A}$, ${\mathsf{\Gamma}}^{\phi \phi}_{\varkappa}$, $\partial \partial {\mathsf{\Gamma}}^{\phi \phi}$, $\partial {\mathsf{\Gamma}}^{c}_\gamma$, are renormalized at $\Lambda=0$ and nonvanishing momentum $\Vec{q}$, chosen in $\mathbb{M}^{s}_n$ in all cases but ${\mathsf{\Gamma}}^{AAAA}$ for which $\Vec{q}\in\mathbb{M}^{cp}_4$. See appendix \[sec\_ren\] for the list of all relevant terms and their renormalization points. Since the renormalization constants are independent of $\Lambda_0$, their derivative wrt $\Lambda_0$ vanishes: it follows that the coefficients of $\delta$-tensors in the decomposition of $\partial_{\Lambda_0}{\mathsf{\Gamma}}^{0\Lambda_0;\Vec{\phi};w}_{\Vec{\varkappa};l}(\Vec{q})$ vanish. Hence using lemma \[lem\_rn\] and the bounds on irrelevant terms we have $$|\partial_{\Lambda_0}{\mathsf{\Gamma}}^{0\Lambda_0;\Vec{\phi};w}_{\Vec{\varkappa};l}(\Vec{q})|\leqslant \frac{M}{\Lambda^2_0}\mathcal{P}^{(1)}_{2l-1}(\log_+ \frac{\Lambda_0}{M}).$$ We integrate the FE from 0 to $\Lambda^\prime$ and substitute the chain with the tree bound. Using inequalities , it is easy to get the following bound $$|\partial_{\Lambda_0}{\mathsf{\Gamma}}^{\Lambda^\prime\Lambda_0;\Vec{\phi};w}_{\Vec{\varkappa};l}(\Vec{q}) - \partial_{\Lambda_0}{\mathsf{\Gamma}}^{0\Lambda_0;\Vec{\phi};w}_{\Vec{\varkappa};l}(\Vec{q})|\leqslant \frac{\Lambda + M}{\Lambda^2_0}\mathcal{P}^{(1)}_{2l-1}(\log_+ \frac{\Lambda_0}{M}).$$ Integrating back and forth along the path given in equation , substituting the irrelevant term $\partial_{\Lambda_0}\partial {\mathsf{\Gamma}}^{\Lambda^\prime \Lambda_0 \Vec{\phi};w}_{\Vec{\varkappa};l}$ with its bounds and using inequality  we obtain $$\begin{aligned} |\partial_{\Lambda_0}{\mathsf{\Gamma}}^{\Lambda^\prime\Lambda_0;\Vec{\phi};w}_{\Vec{\varkappa};l}(0)-\partial_{\Lambda_0}{\mathsf{\Gamma}}^{\Lambda^\prime\Lambda_0;\Vec{\phi};w}_{\Vec{\varkappa};l}(\Vec{q})|&\leqslant \frac{\Lambda + M}{\Lambda^2_0}P^{\Lambda^\prime \Lambda_0}_{2l}(0),\\ |\partial_{\Lambda_0} {\mathsf{\Gamma}}^{\Lambda^\prime\Lambda_0;\Vec{\phi};w}_{\Vec{\varkappa};l}(\Vec{p})-\partial_{\Lambda_0}{\mathsf{\Gamma}}^{\Lambda^\prime\Lambda_0;\Vec{\phi};w}_{\Vec{\varkappa};l}(0)|&\leqslant \frac{\Lambda + M + |\Vec{p}|}{\Lambda^2_0}P^{\Lambda \Lambda_0}_{2l}(\Vec{p}).\end{aligned}$$ If $\Lambda=\Lambda^\prime$ the proof of the bounds on the marginal terms is complete. Otherwise we integrate the FE downwards from $\eta$ to $\Lambda$ and repeat the arguments given to prove inequality with a minor change in the integrand $$\int \limits^{\eta}_\Lambda d\lambda \, \frac{\lambda + M +|\Vec{p}|}{\Lambda^2_0}\frac{\lambda \, P^{\lambda \Lambda_0}_{k}|_{\eta_\tau=0}}{(\lambda + |p_a|)(\lambda + |p_b|)} \leqslant \frac{M + |\Vec{p}|}{\Lambda^2_0}\int \limits^{\eta}_\Lambda d\lambda \, \frac{2P^{\lambda \Lambda_0}_{k}|_{\eta_\tau=0}}{\lambda + |p_b|}.$$ For the strictly relevant terms we integrate the FE from 0 to $\Lambda$ substituting the vertex functions and propagators with their bounds and extending the upper limit of integration to $\Lambda + \eta$ $$|\partial_{\Lambda_0}{\mathsf{\Gamma}}^{\Lambda\Lambda_0\Vec{\phi};w}_{\Vec{\varkappa};l}(0)|\leqslant \frac{\Lambda+M}{\Lambda^2_0}\int \limits^{\Lambda + \eta}_0 d\lambda \, \lambda^{d-1}P^{\lambda\Lambda_0}_{2l-2}(0),$$ where $d>0$. Using inequality we obtain $$|\partial_{\Lambda_0}{\mathsf{\Gamma}}^{\Lambda\Lambda_0\Vec{\phi};w}_{\Vec{\varkappa};l} (0)|\leqslant \frac{\Lambda+M}{\Lambda^2_0} (\Lambda + \eta)^{d} P^{\Lambda+\eta \;\Lambda_0}_{2l-2}(0).$$ To extend to momentum $\Vec{p}$ we proceed as in , the only change being an additional factor of $\frac{\Lambda+M +| \Vec{p}|}{\Lambda^2_0}$. Restoration of the STI {#sec_sti} ====================== As mentioned before theorems \[thm\_3\], \[thm\_4\] we now consider all nontrivial marginal terms $\Gamma^{\Lambda \Lambda_0;\Vec{\phi};w}_{X\Vec{\varkappa};l}$ with $X \in \{\beta, 1\}$ at $\Lambda=0$. We want to show that these terms verify the bounds of theorems \[thm\_3\], \[thm\_4\]. In this section, since $\Lambda=0$, we will omit the parameters $\Lambda$, $\Lambda_0$ in the notations wherever this is not ambiguous, i.e. we write ${\mathsf{\Gamma}}^{\Vec{\phi}}$ for ${\mathsf{\Gamma}}^{0\Lambda_0;\Vec{\phi}}$. The subsequent relations are obtained by projecting the AGE and the STI on the respective monomial in the fields (for example, $c$ in section \[gh\_c\]) to read off the lhs from the rhs, taken at the renormalization point. We will establish appropriate relations in order to make the coefficients of the $\delta$-tensors wrt the monomial basis at the renormalization point satisfy the bounds of these two theorems. In this analysis we make particular use of the consistency conditions, see \[407s\]. In section \[sec\_329r\] we prove the existence of a solution for the above mentioned system of relations that does not depend on the UV cutoff. In the remaining sections we treat the different marginal terms one by one. ### Smallness relations It is helful to introduce the notion of *small* terms, which vanish in the limit $\Lambda_0 \to \infty$. For fixed loop order $l$ and $X \in \{\beta, 1\}$, a homogeneous function $f(\Vec{p},M,\Lambda_0)$ of mass dimension $[f]$ is said *small* on a subset $Y \subset \mathbb{M}_n$, and denoted by $f \stackrel{X,Y,l}{\sim} 0$, if for all $w \in \mathbb{W}_n$ with $\|w\|\leqslant [f]$, there exists a polynomial $\mathcal{P}^{(1)}_{r_X}$ of degree $r_X([f]-\|w\|,l)$, see theorems \[thm\_3\] and \[thm\_4\], such that the following bound holds for all $\Lambda_0 \geqslant M$ and all $\Vec{p} \in Y$, see : $$|\partial^w_p f (\Vec{p},M,\Lambda_0)| \leqslant \frac{M^{1+ [f] - \|w\|}}{\Lambda_0}\mathcal{P}^{(1)}_{r_X}(\log_+ \frac{\Lambda_0}{M}).\label{eq_1s}$$ Furthermore, $f \stackrel{X,Y,l}{\sim} g$ iff $[f]=[g]$ and $f-g \stackrel{X,Y,l}{\sim} 0$. Because both relations $\stackrel{1,Y,l}{\sim}$ and $\stackrel{\beta,Y,l}{\sim}$ only differ by the degree of polynomials we have $f \stackrel{\beta,Y,l}{\sim} g \implies f \stackrel{1,Y,l}{\sim} g$. Since the loop order $l$ and the renormalization point $\Vec{q}$ are evident from the context, we write: $$\begin{aligned} f \stackrel{Y}\sim g&\quad\mbox{for}\quad f \stackrel{1,Y,l}{\sim} g,& f \stackrel{\beta,Y}{\sim} g&\quad\mbox{for}\quad f \stackrel{\beta,Y,l}{\sim} g,\label{eq_88sim}\\ f \sim g&\quad\mbox{for}\quad f \stackrel{1,\{\Vec{q}\}}{\sim} g,& f {\stackrel{\beta}{\sim}}g &\quad\mbox{for}\quad f \stackrel{\beta,\{\Vec{q}\}}{\sim} g. \end{aligned}$$ Theorem \[thm\_1\] implies that for every vertex function ${\mathsf{\Gamma}}^{\Vec{\phi}}_{\Vec{\varkappa}}(\Vec{p})$ there exists a constant $c$ such that $\forall w\in\mathbb{W}_n$, $\forall \Vec{p} \in \mathbb{M}_n$, $\forall \Lambda_0 \geqslant M$ $$|\partial^w {\mathsf{\Gamma}}^{\Vec{\phi}}_{\Vec{\varkappa}}(\Vec{p})| \leqslant c M^{4 - 2n_\varkappa - {\textsc{n}}- \|w\|}\,.$$ Using also that $$|\partial^w(\sigma_{0\Lambda_0}-1)|< c_w \, \frac{1}{\Lambda^w_0}\,,\quad |\sigma_{0\Lambda_0}-1|< c_0\,\frac{M}{\Lambda_0}\,,$$ the terms on rhs of the STI and the AGE satisfy the relations: $$\begin{aligned} {\mathsf{\Gamma}}^{\Vec{\phi}_1;w_1} \partial^{w_2} (\sigma_{0\Lambda_0} {\mathsf{\Gamma}}^{\Vec{\phi}_2}_{\varkappa})&\stackrel{\beta,Y}{\sim}{\mathsf{\Gamma}}^{\Vec{\phi}_1;w_1} {\mathsf{\Gamma}}^{\Vec{\phi}_2;w_2}_{\varkappa}\,,\\ \partial^{w} (\sigma_{0\Lambda_0} {\mathsf{\Gamma}}^{\Vec{\phi}})&\stackrel{\beta,Y}{\sim}{\mathsf{\Gamma}}^{\Vec{\phi};w}\,,\\ \partial^{w} (p \sigma_{0\Lambda_0} {\mathsf{\Gamma}}^{\Vec{\phi}}_\gamma)&\stackrel{\beta,Y}{\sim}p {\mathsf{\Gamma}}^{\Vec{\phi};w}_{\gamma}\,.\end{aligned}$$ This fact will be useful in the calculations underlying the following sections. ### The functional ${\mathrm{F}}_{1, rel}$ {#sec_327w} In this section we introduce the notation for the renormalization constants for the functional ${\mathrm{F}}_1$ . For this purpose we define the auxiliary functional $$\begin{aligned} {\mathrm{F}}^{\Lambda_0}_{1,rel}:=&\epsilon^{dab} \langle U^{\gamma cc}_\sigma| \check{\gamma}^d_\sigma \check{c}^a \check{c}^b\rangle + \epsilon^{sab} \epsilon^{sde} u^{\gamma Acc} \langle \check{\gamma}^a_\sigma \check{A}^b_\sigma \check{c}^d \check{c}^e\rangle\nonumber\\ &+ \langle U^{cA}_\sigma |\check{c}^d\check{A}^d_\sigma\rangle + \epsilon^{dab} \langle U^{\bar{c}cc} |\check{\bar{c}}^d\check{c}^a \check{c}^b\rangle + \epsilon^{dab} \langle U^{cAA}_{\mu \nu} |\check{c}^d\check{A}^a_\mu\check{A}^b_\nu\rangle\nonumber\\ &+\langle U^{cAAA}_{\rho \mu \nu} |\check{c}^d\check{A}^d_\rho\check{A}^a_\mu\check{A}^a_\nu \rangle + \langle U^{\bar{c}ccA}_{1;\mu} |\check{\bar{c}}^b \check{c}^b\check{c}^a \check{A}^a_\mu \rangle \nonumber\\ &+ \langle U^{\bar{c}ccA}_{2;\mu} |\check{\bar{c}}^a\check{c}^b\check{c}^b \check{A}^a_\mu \rangle + \epsilon^{bed} u^{c\bar{c}cAA}_{1} \langle \check{c}^b\check{\bar{c}}^e\check{c}^d \check{A}^a_\mu \check{A}^a_\mu \rangle \nonumber\\ &+ \epsilon^{bed} u^{c\bar{c}cAA}_{2} \langle \check{c}^b\check{\bar{c}}^a\check{c}^d \check{A}^a_\mu \check{A}^e_\mu \rangle + \epsilon^{bed} u^{c\bar{c}cAA}_{3} \langle \check{c}^a\check{\bar{c}}^e\check{c}^d \check{A}^a_\mu \check{A}^b_\mu \rangle,\end{aligned}$$ where $$\begin{aligned} U^{\gamma cc}_\sigma(l,p,q):=&i(p+q)_\sigma u^{\gamma cc},\\ U^{cA}_\sigma(l,p):=&ip_\sigma p^2 u^{cA},\\ U^{\bar{c}cc}(l,p,q):=&(p^2 + q^2)u^{\bar{c}cc}_1 + 2pq u^{\bar{c}cc}_2,\\ U^{cAA}_{\mu \nu}(l,p,q):=&(p_\mu p_\nu - q_\mu q_\nu)u^{cAA}_1 + \delta_{\mu \nu} (p^2-q^2)u^{cAA}_2,\\ U^{cAAA}_{\rho \mu \nu}(l,k,p,q):=&i(u^{cAAA}_1 k_\rho + u^{cAAA}_2 p_\rho+u^{cAAA}_2q_\rho) \delta_{\mu \nu}\nonumber\\ &+i(u^{cAAA}_3 k_\nu + u^{cAAA}_4 q_\nu + u^{cAAA}_5 p_\nu)\delta_{\rho \mu} \nonumber\\ &+i(u^{cAAA}_3 k_\mu + u^{cAAA}_4 p_\mu + u^{cAAA}_5 q_\mu)\delta_{\rho \nu},\label{eq_ucAAA}\\ U^{\bar{c}ccA}_{1;\mu}(l,k,q,p):=& ip_\mu u^{\bar{c}ccA}_1 + ik_\mu u^{\bar{c}ccA}_2 + iq_\mu u^{\bar{c}ccA}_3,\\ U^{\bar{c}ccA}_{2;\mu}(l,k,q,p):=&i(k_\mu - q_\mu)u^{\bar{c}ccA}_4\,,\end{aligned}$$ and the $u$’s are functions of $\Lambda_0$ defined by the marginal renormalization conditions $${\mathrm{F}}^{0\Lambda_0;\Vec{\phi};w}_{1;\Vec{\varkappa}}(\Vec{q})={\mathrm{F}}^{\Lambda_0;\Vec{\phi};w}_{1,rel;\Vec{\varkappa}}(\Vec{q})+\sum_{t \in \{\delta^s {\underline}{e}^{k>0}\}_r} \zeta^{\Lambda_0}_t\, t\,.$$ Here $2n_\varkappa+ {\textsc{n}}+\|w\|=5$, $\Vec{q}$ is the renormalization point defined in section \[sec\_ren\], ${\underline}{e}=(e_i)_{i \in [m]}$ is an orthogonal basis for the linear span of $\Vec{q}$, $r$ is the tensor rank of ${\mathrm{F}}^{0\Lambda_0;\Vec{\phi};w}_{1}(\Vec{q})$. The $\zeta^{\Lambda_0}_t$ are the uniquely defined coefficients of tensors $t$. Note that we implicitly set to zero all constants associated to strictly relevant renormalization conditions for ${\mathrm{F}}_1$. These constants are not needed because, thanks to hypothesis \[rc1\], the RHS of the STI and AGE at the current loop order vanish at zero momenta. The renormalization points $\vec{q}$ are chosen in agreement with the hypotheses of lemma \[lem\_rn\]. From lemma \[lem\_rn\], theorem \[thm\_3\], and the irrelevant bounds of theorem \[thm\_4\], for the marginal terms one has $${\mathrm{F}}^{0\Lambda_0;\Vec{\phi};w}_{1;\Vec{\varkappa}}(\Vec{q})\sim {\mathrm{F}}^{\Lambda_0;\Vec{\phi};w}_{1,rel;\Vec{\varkappa}}(\Vec{q}).$$ ### Consistency conditions {#407s} Here we establish the consistency conditions implied by the nilpotency, see and . Below we will rely on the validity of theorem \[thm\_4\] at loop orders $l’<l$ for all terms and at the current order $l$ only for irrelevant terms: these properties are true in our inductive scheme. Recall definitions , and . Using the AGE , the bounds of theorems \[thm\_1\]-\[thm\_3\], and we get $$\Big(S {\mathsf{\Gamma}}_{\beta}\Big)^{\Vec{\phi}}\,\stackrel{\mathbb{M}_n}{\sim}\, 0, \quad\mbox{and thus}\quad \Big(\frac{\delta}{\delta \tilde{c}} {\mathrm{F}}_{1}\Big)^{\Vec{\phi}}\, \stackrel{\mathbb{M}_n}{\sim}\, 0\,,\label{eq_88gh}$$ where $\Vec{\phi}=(\phi_1,...,\phi_{n-1})$ and $\phi_i \in \{A,c,\bar{c}\}$. Equation , theorems \[thm\_1\], \[thm\_2\], and the bounds of theorem \[thm\_4\] for irrelevant terms yield $$\begin{aligned} \Big(\frac{\delta}{\delta \tilde{c}} {\mathrm{F}}_{1,rel}\Big)^{\Vec{\phi}}& \stackrel{\mathbb{M}_n}{\sim}0,&\Vec{\phi} &\in \{(c,c), (c,c,A)\}\,.\label{eq_26gh}\end{aligned}$$ See section \[sec\_327w\] for the definition of ${\mathrm{F}}_{1,rel}$ and of the constants $u^\Phi$. In section \[st\_cccAA\] it will be shown that $u^{c\bar{c}cAA}_{1,2,3} \sim 0$. Equation  then gives $$\begin{aligned} u^{\gamma c c}&\sim u^{\bar{c}cc}_1 \sim u^{\bar{c}cc}_2,&-2u^{\gamma A cc}& \sim u^{\bar{c}ccA}_1 \sim u^{\bar{c}ccA}_2 \sim u^{\bar{c}ccA}_3,&u^{\bar{c}ccA}_4& \sim 0. \label{eq_cccA}\end{aligned}$$ Let us exploit to obtain more constraints on the renormalization constants $u^\Phi$. At loop order $l$ $$S {\mathrm{F}}_{1;l}=S_0 {\mathrm{F}}_{1;l} + \sum_{l^{\prime}<l} S_{l-l^{\prime}} {\mathrm{F}}_{1;l^{\prime}}.$$ By induction $(S_{l-l^{\prime}} {\mathrm{F}}_{1;l^{\prime}})^{\Vec{\phi}}\stackrel{\mathbb{M}_n}{\sim} 0$ for all loop orders $l^\prime < l$. Then equation implies that $$(S_0 {\mathrm{F}}_{1;l})^{\Vec{\phi}} \, \stackrel{\mathbb{M}_{n}}{\sim} \, 0,\label{eq_404b}$$ where $$\begin{aligned} S_0=&\langle \tilde{\delta}_{A^d_\sigma}{{{\mkern2mu\underline{\mkern-2mu \Gamma\mkern-2mu}\mkern2mu }}}_0,\sigma_{0 \Lambda_0}\tilde{\delta}_{\gamma^d_\sigma} \rangle + \langle \tilde{\delta}_{\gamma^d_\sigma}{{{\mkern2mu\underline{\mkern-2mu \Gamma\mkern-2mu}\mkern2mu }}}_0,\sigma_{0 \Lambda_0}\tilde{\delta}_{A^d_\sigma} \rangle\nonumber\\ &- \langle \tilde{\delta}_{c^d}{{{\mkern2mu\underline{\mkern-2mu \Gamma\mkern-2mu}\mkern2mu }}}_0,\sigma_{0 \Lambda_0}\tilde{\delta}_{ \omega^d} \rangle - \langle \tilde{\delta}_{\omega^d}{{{\mkern2mu\underline{\mkern-2mu \Gamma\mkern-2mu}\mkern2mu }}}_0,\sigma_{0 \Lambda_0}\tilde{\delta}_{c^d} \rangle,\label{eq_404s}\end{aligned}$$ and recalling notation for $\tilde{\delta}_\phi$ from $$\begin{aligned} \tilde{\delta}_{A^d_\sigma(q)}{{{\mkern2mu\underline{\mkern-2mu \Gamma\mkern-2mu}\mkern2mu }}}_0=&g\epsilon^{adb} (\langle \check{\gamma}^a_\sigma \check{c}^b;q\rangle + \langle i p_\sigma \check{\bar{c}}^a(p) \check{c}^b;q\rangle) + A^d_{\sigma^\prime}(q)(\delta_{\sigma^\prime \sigma} q^2-q_{\sigma^\prime} q_\sigma)\sigma^{-1}_{0 \Lambda_0}(q^2) \nonumber\\ &+3 \epsilon^{dab} \langle F^{AAA}_{\sigma \mu \nu}(q,\cdot,\cdot)|\check{A}^a_\mu \check{A}^b_\nu;q\rangle + 4R^{AAAA}_{\sigma \rho \mu \nu} \langle \check{A}^d_\rho \check{A}^a_\mu \check{A}^a_\nu;q\rangle,\\ \tilde{\delta}_{\gamma^d_\sigma(q)}{{{\mkern2mu\underline{\mkern-2mu \Gamma\mkern-2mu}\mkern2mu }}}_0=& iq_\sigma c^d(q)+ g\epsilon^{dab} \langle \check{A}^a_\sigma \check{c}^b;q \rangle,\\ \tilde{\delta}_{c^d(q)}{{{\mkern2mu\underline{\mkern-2mu \Gamma\mkern-2mu}\mkern2mu }}}_0=&g\epsilon^{dab}\langle i p_\mu \check{\bar{c}}^a(p)\check{A}^b_\mu + \check{\gamma}^a_\mu \check{A}^b_\mu + \check{\omega}^b \check{c}^a;q\rangle - iq_\mu \gamma^d_\mu(q)-\bar{c}^d(q) q^2 \sigma^{-1}_{0 \Lambda_0}(q^2),\\ \tilde{\delta}_{\omega^d(q)}{{{\mkern2mu\underline{\mkern-2mu \Gamma\mkern-2mu}\mkern2mu }}}_0=& \frac{1}{2} g\epsilon^{dab} \langle \check{c}^a\check{c}^b;q \rangle.\end{aligned}$$ Here the notation $\langle i p_\sigma \check{\bar{c}}^a(p) \check{c}^b;q\rangle$ corresponds to $\langle \phi_1 \phi_2 ;q\rangle$ with $\phi_1(p)= i p_\sigma\check{\bar{c}}^a(p)$, $\phi_2=\check{c}^b$.\ For all $\Vec{\phi}$, $\Vec{\varkappa}$ and $w$ such that ${\textsc{n}}+ 2n_\varkappa +\|w\|=6$ we have $$(S_0 {\mathrm{F}}_{1;l})^{\Vec{\phi};w}_{\Vec{\varkappa}}=(S_0 {\mathrm{F}}_{1,rel;l})^{\Vec{\phi};w}_{\Vec{\varkappa}} + \sum_{\pi} (-)^{\pi_a} S^{\Vec{\phi}_1;w_1}_{0,\Vec{\varkappa}_1} \Delta^{\Vec{\phi_2};w_2}_{\Vec{\varkappa}_2;l},\label{S0F1}$$ where $\Delta^{\Lambda \Lambda_0}_l:={\mathrm{F}}^{\Lambda\Lambda_0}_{1;l}-{\mathrm{F}}^{\Lambda_0}_{1,rel;l}\,$, the sum runs over the permutations $\pi=(\pi_\phi,\pi_\varkappa,\pi_w)$ such that $\Vec{\phi}_{\pi_\phi}=\Vec{\phi}_1\oplus \Vec{\phi}_2$, $\Vec{\varkappa}_{\pi_\varkappa}=\Vec{\varkappa}_1\oplus \Vec{\varkappa}_2$, $w_{\pi_w}=w_1 + w_2$, and $\pi_a$ is the number of transpositions $mod \, 2$ of anticommuting variables in the permutation $\pi$. Using , for the terms in the sum on the rhs of  we have $$\begin{aligned} |S^{\Vec{\phi}_1;w_1}_{0,\Vec{\varkappa}_1}\Delta^{\Vec{\phi_2};w_2}_{\Vec{\varkappa}_2;l}|&\leqslant |{{{\mkern2mu\underline{\mkern-2mu \Gamma\mkern-2mu}\mkern2mu }}}^{A\Vec{\phi}_1;w^{\prime}_1}_{\Vec{\varkappa}_1;0} ,\sigma^{w^{\prime \prime}_1}_{0 \Lambda_0}\Delta^{\Vec{\phi_2};w_2}_{\Vec{\varkappa}_2\gamma;l}| + |{{{\mkern2mu\underline{\mkern-2mu \Gamma\mkern-2mu}\mkern2mu }}}^{c\Vec{\phi}_1;w^{\prime}_1}_{\Vec{\varkappa}_1;0} ,\sigma^{w^{\prime \prime}_1}_{0 \Lambda_0}\Delta^{\Vec{\phi_2};w_2}_{\Vec{\varkappa}_2\omega;l}|\nonumber\\ &+|{{{\mkern2mu\underline{\mkern-2mu \Gamma\mkern-2mu}\mkern2mu }}}^{\Vec{\phi}_1;w^{\prime}_1}_{\Vec{\varkappa}_1\gamma;0} ,\sigma^{w^{\prime \prime}_1}_{0 \Lambda_0}\Delta^{A\Vec{\phi_2};w_2}_{\Vec{\varkappa}_2;l}| + |{{{\mkern2mu\underline{\mkern-2mu \Gamma\mkern-2mu}\mkern2mu }}}^{\Vec{\phi}_1;w^{\prime}_1}_{\Vec{\varkappa}_1\omega;0} ,\sigma^{w^{\prime \prime}_1}\Delta^{c\Vec{\phi_2};w_2}_{\Vec{\varkappa}_2;l}|.\label{eq_405a}\end{aligned}$$ Let us show that the lhs of is *small* on $\mathbb{M}_n$. Using section \[sec\_327w\] and the bounds on irrelevant terms of theorems \[thm\_3\], \[thm\_4\] we see that for all marginal terms $$\Delta^{\Vec{\phi};w}_{\Vec{\varkappa};l}\sim 0,\quad\mbox{and thus}\quad\Delta^{\Vec{\phi};w}_{\Vec{\varkappa};l}\, \stackrel{\mathbb{M}_n}{\sim} \,0.\label{eq_405b}$$ The relation on the rhs can be obtained by adapting the interpolation in equations -. Define ${\textsc{n}}_i:=|\Vec{\phi}_i|$ and $n_{\varkappa_i}:=|\Vec{\varkappa}_i|$. Consider the sum of the first and second term on the rhs. If $2n_{\varkappa_2}+{\textsc{n}}_2+\|w_2\|\geqslant 3$ then the bounds of theorems \[thm\_1\], \[thm\_3\], \[thm\_4\] and imply that the sum is *small*. On the other hand if $2n_{\varkappa_1}+{\textsc{n}}_1+\|w_1\|>3$ then $\|w^{\prime \prime}_1\|>0$ and the bounds of theorems \[thm\_1\], \[thm\_3\] also give that the sum is *small*. The analysis of the sum of the third and fourth term on the rhs is similar. If $2n_{\varkappa_2}+{\textsc{n}}_2+\|w_2\|\geqslant 4$ then the bounds of theorems \[thm\_1\], \[thm\_3\], \[thm\_4\] and imply that the sum is *small*. If $2n_{\varkappa_1}+{\textsc{n}}_1+\|w_1\|> 2$ then $\|w^{\prime \prime}_1\|>0$ and using the bounds of theorems \[thm\_1\], \[thm\_3\] we obtain again that the sum is *small*. It follows that the lhs of is *small*. This fact and imply that $(S_0 {\mathrm{F}}_{1,rel;l})^{\Vec{\phi};w} \stackrel{\mathbb{M}_{n}}{\sim} 0$ for all marginal terms, which leads to the following equations $$\begin{aligned} g u^{\gamma cc} & \sim - u^{\gamma A cc},&u^{cAA}_1 + u^{cAA}_2&\sim \frac{gu^{cA}}{2}, \label{eq_cAA}\\ u^{cAAA}_1&\sim u^{cAAA}_2,&u^{cAAA}_3\sim u^{cAAA}_4&\sim u^{cAAA}_5. \label{eq_cAAA}\end{aligned}$$ ### Existence of a constant solution {#sec_329r} By our convention (which is the standard one) the renormalization constants that are solutions of the relations listed in \[rc3\] are supposed not to depend on $\Lambda_0$. We give here a proof of this property, which is not evident because these relations contain nontrivial functions of $\Lambda_0$, here denoted by $\zeta^{\Lambda_0}_{\Phi}$. The relations corresponding to the marginal terms ${\mathsf{\Gamma}}^{\Vec{\phi};w}_{1 \Vec{\varkappa}}$, ${\mathsf{\Gamma}}^{\Vec{\phi};w}_{\beta \Vec{\varkappa}}$ have respectively the general form $$\begin{aligned} c^\Phi + \varrho \, C^\Phi_1 \varrho + \zeta^{\Lambda_0}_{\Phi} &\sim 0,& \zeta^{\Lambda_0}_{\Phi}&:= \varrho \, C^\Phi_2 \zeta^{\Lambda_0} + \zeta^{\Lambda_0} C^\Phi_3 \zeta^{\Lambda_0}\,,\\ c^\Phi + V^\Phi_1 \varrho + \zeta^{\Lambda_0}_{\Phi} &{\stackrel{\beta}{\sim}}0,& \zeta^{\Lambda_0}_{\Phi}&:= V^\Phi_2 \zeta^{\Lambda_0}\,.\end{aligned}$$ Here $\varrho=(r^{\phi...}_i, R_i, \Sigma^{AA}_{L}, \Sigma^{AA}_{T}, \Sigma^{\bar{c}c} )$ denotes the relevant terms for vertex functions, see for the list of $r^{\phi...}_i$ and appendixes \[sec\_gamma\], \[sec\_gamma2\] for the remaining terms. The sequence $\zeta^{\Lambda_0}$ stands for the irrelevant terms listed in appendixes \[sec\_gamma\], \[sec\_gamma2\] and for the derivative of $r^{\phi...}_i$, $R_i$ wrt scalar products of momenta. Finally, $c^\Phi$ is a constant, $V^\Phi_{1,2}$ are constant vectors, and $C^\Phi_{1,2,3}$ are constant matrices. At loop order $l$, the terms $\zeta^{\Lambda_0}_{\Phi,l}$ depend only on $\varrho_{l'}$ of loop order $l'<l$: this property holds because each $\zeta^{\Lambda_0}_{\Phi,l}$ is at least linear in the $\zeta^{\Lambda_0}$ and because all the $\zeta^{\Lambda_0}_{l=0}$ vanish. Moreover, at order $l$ for each relation we have a distinct renormalization constant. Consequently, the aforementioned relations have a solution. The existence of a solution $\varrho_l$ independent of $\Lambda_0$ follows immediately if the limit $\lim_{\Lambda_0\to\infty}\zeta^{\Lambda_0}_{\Phi,l}$ exists: in this case it is enough to choose a solution of the following equations $$\begin{aligned} c^\Phi + \varrho \, C^\Phi_1 \varrho + \zeta^{\infty}_{\Phi} &= 0,&c^\Phi + V^\Phi_1 \varrho + \zeta^{\infty}_{\Phi} &= 0.\end{aligned}$$ The convergence of $\zeta^{\Lambda_0}_{\Phi,l}$ relies on the validity of the bounds of theorem \[thm\_2\] up to order $l$ for all irrelevant terms ${\mathsf{\Gamma}}^{\Vec{\phi};w}_{\Vec{\varkappa}}$ and up to order $l-1$ for all the relevant ones. This property holds because in our inductive scheme at fixed loop order the irrelevant terms are treated before the relevant ones. ${\mathsf{\Gamma}}^{c}_\beta$ {#gh_c} ----------------------------- The renormalization point is $\Vec{q}=(-\bar{q},\bar{q})\in \mathbb{M}^s_2$, see . $$\begin{aligned} {\mathsf{\Gamma}}^{c^a}_{\beta^b}(p)&{\stackrel{\beta}{\sim}}\sigma_{0 \Lambda_0}(p^2)\,\Gamma^{c^a \bar{c}^b}(p)+i p_\mu \Gamma^{c^a}_{\gamma^b_\mu}(p) {\stackrel{\beta}{\sim}}-\delta^{ab} f(p^2)\,,\\ f(x)&:= x \left(1+\Sigma^{\bar{c}c}(x) - R_1(x)\right). \label{eq_331d}\end{aligned}$$ For the marginal term we obtain $$-\frac{\delta^{ab}}{3} \, {\mathsf{\Gamma}}^{c^a;p_\mu p_\nu}_{\beta^b}(p) {\stackrel{\beta}{\sim}}2\delta_{\mu \nu} f^\prime(p^2) + 4p_\mu p_\nu f^{\prime \prime}(p^2) \,.$$ The coefficient of $\delta_{\mu \nu}$ is *small* at the renormalization point iff $$\begin{aligned} f^\prime=1-r^{\bar{c}c} - R_1 - \zeta^{\Lambda_0}_{\beta c} &{\stackrel{\beta}{\sim}}0,& \zeta^{\Lambda_0}_{\beta c}(p^2)&:=p^2\frac{\partial R_1(p^2)}{\partial p^2}\,. \label{eq_gh_c}\end{aligned}$$ This gives the renormalization condition for $R_1$. ${\mathsf{\Gamma}}^{cA}_\beta$ ------------------------------ The renormalization point is $\Vec{q}=(\bar{k},\bar{p},\bar{q})\in \mathbb{M}^s_3$, see . With $k=-p-q$, $$\begin{aligned} {\mathsf{\Gamma}}^{c^a A^d_\mu}_{\beta^b}(p,q)&{\stackrel{\beta}{\sim}}\Gamma^{A^d_\mu c^a \bar{c}^b}(p,k)-ik_\rho \Gamma^{c^a A^d_\mu}_{\gamma^b_\rho}(p,q)\\ &=i\epsilon^{dab} \mathcal{I}^{0\Lambda_0}_\mu(p,q)\,,\\ \mathcal{I}^{0\Lambda_0}_\mu(p,q)&=k_\mu R^{A\bar{c}c}_1(k,p)+p_\mu r^{A\bar{c}c}_2(k,p) -gk_\rho F^{\gamma A c}_{\rho \mu}(q,p)\,.\end{aligned}$$ Let $\Delta^{\Lambda\Lambda_0}_{\mu\nu}:=\mathcal{I}^{\Lambda\Lambda_0;p_\nu}_\mu-\mathcal{I}^{\Lambda\Lambda_0;q_\nu}_\mu$. At zero external momenta and $\Lambda=M$ we have $\Delta^{M\Lambda_0}_{\mu \nu}(0)=0$. Then using the bounds of theorem \[thm\_3\] we get $$|\Delta^{M\Lambda_0}_{\mu \nu}(\Vec{q})|\leqslant \int \limits^1_0 dt\,M |\partial \partial \Gamma^{M\Lambda_0;c A}_{\beta}| {\stackrel{\beta}{\sim}}0 \,.$$ The term $\Delta^{\Lambda\Lambda_0}_{\mu \nu}$ obeys the FE, see . It remains to integrate the FE from 0 to $\Lambda$ and use inequality to obtain $$\Delta^{0\Lambda_0}_{\mu \nu}(\Vec{q})-\Delta^{M\Lambda_0}_{\mu \nu}(\Vec{q}){\stackrel{\beta}{\sim}}0,\quad\mbox{and thus}\quad\Delta^{0\Lambda_0}_{\mu\nu}(\Vec{q}) {\stackrel{\beta}{\sim}}0\,.$$ Hence in the monomial basis $\{\delta^s Q^k\}_2$ with $Q=(\bar{p},\bar{q})$ the $\delta$-component of $\mathcal{I}^{A\bar{c}c;q_\nu}_\mu$ is *small* at the renormalization point if the following condition holds $$R^{A\bar{c}c}_1 - gR_2-\zeta^{\Lambda_0}_{\beta cA} {\stackrel{\beta}{\sim}}0\,.\label{eq_gh_cA}$$ This gives the renormalization condition for $R^{A\bar{c}c}_1$. ${\mathsf{\Gamma}}^{cAA}_\beta$ and ${\mathsf{\Gamma}}^{cc\bar{c}}_\beta$ ------------------------------------------------------------------------- The renormalization point is $\Vec{q}=(\bar{k},\bar{l},\bar{p},\bar{q})\in \mathbb{M}^s_4$, see . $$\begin{aligned} {\mathsf{\Gamma}}^{c^a A^t_\mu A^s_\nu}_{\beta^b}(l,p,q)&{\stackrel{\beta}{\sim}}\Gamma^{c^a \bar{c}^b A^t_\mu A^s_\nu}(k,p,q)-ik_\rho \Gamma^{c^a A^t_\mu A^s_\nu}_{\gamma^b_\rho}(l,p,q)\\ {\mathsf{\Gamma}}^{c^a c^t \bar{c}^s}_{\beta^b}(l,p,q)&{\stackrel{\beta}{\sim}}\Gamma^{c^a \bar{c}^b c^t \bar{c}^s}(k,p,q)-ik_\rho \Gamma^{c^a c^t \bar{c}^s}_{\gamma^b_\rho}(l,p,q)\end{aligned}$$ At $\Lambda=M$ it follows from property  that these terms vanish at zero momenta. Denoting the renormalization point by $\Vec{q}$, using the bounds of theorem \[thm\_3\] and integrating the FE from $M$ to 0 we obtain $$\begin{aligned} |{\mathsf{\Gamma}}^{M\Lambda_0;\Vec{\phi}}_{\beta}(\Vec{q})|\leqslant& \int \limits^1_0 dt\,|\Vec{q}| |\partial \Gamma^{M\Lambda_0;\Vec{\phi}}_{\beta}(t\Vec{q})|{\stackrel{\beta}{\sim}}0\,,\\ |{\mathsf{\Gamma}}^{0\Lambda_0;\Vec{\phi}}_{\beta}(\Vec{q})-{\mathsf{\Gamma}}^{M\Lambda_0;\Vec{\phi}}_{\beta}(\Vec{q})|\leqslant& \frac{M}{\Lambda_0}\mathcal{P}^{(1)}_{2(l-1)}(\log_+\frac{\Lambda_0}{M}){\stackrel{\beta}{\sim}}0\,.\end{aligned}$$ ${\mathsf{\Gamma}}^{c \bar{c} c \bar{c} c}_1$ and ${\mathsf{\Gamma}}^{c A A A A}_1$ {#st_cAAAA} ------------------------------------------------------------------------------------- These functions do not have nonvanishing marginal terms: $$\begin{aligned} \epsilon^{dab}\langle \check{c}^d\check{A}^a_\mu\check{A}^b_\nu\check{A}^s_\mu\check{A}^s_\nu\rangle&=0,&\epsilon^{dab}\langle \check{c}^d\check{\bar{c}}^s\check{c}^a\check{\bar{c}}^s\check{c}^b\rangle&=0\,.\end{aligned}$$ ${\mathsf{\Gamma}}_1^{\bar{c} c c A A}$ {#st_cccAA} --------------------------------------- From equation it follows that for $\Lambda=M$ the function vanishes if the antighost momentum is zero. Using the bounds of theorem \[thm\_4\] first we obtain at the renormalization point $|{\mathsf{\Gamma}}_1^{M \Lambda_0;\bar{c} c c A A}(\Vec{q})| \sim 0$ where $\Vec{q} \in \mathbb{M}^s_5$, and then integrating the FE from $M$ to 0 we show that the term is *small* at $\Lambda=0$. ${\mathsf{\Gamma}}^{c A}_1$ --------------------------- The renormalization point is $\Vec{q}=(-\bar{q},\bar{q})\in \mathbb{M}^s_2$, see . $$\begin{aligned} {\mathsf{\Gamma}}^{c^a A^b_\mu}_1(p)& \sim i \delta^{ab} F^{AA}_{T;\mu\nu}(p) R_1 p_\nu=i \delta^{ab} p_{\mu} f(p^2)\,,\\ f(x)&:=\frac{1}{\xi} xR_1(x)\Sigma^{AA}_L(x)\,.\end{aligned}$$ The marginal term satisfies $${\mathsf{\Gamma}}^{c^a A^b;ppp}_1(p) \sim i\delta^{ab}\Big(2 f^\prime(p^2)(\Sigma_{t \in \{\delta^2\}_4} t) + 4f^{\prime \prime}(p^2) (\Sigma_{t \in \{\delta pp\}_4} t) + 8 pppp f^{\prime \prime \prime}(p^2)\Big)\,.$$ For the coefficient of $\delta$-tensors we have $$\xi f^\prime(p^2)=R_1(p^2) \Big(\Sigma^{AA}_L(p^2) + p^2 \frac{\partial \Sigma^{AA}_L(p^2)}{\partial p^2}\Big) + p^2 \Sigma^{AA}_L(p^2) \frac{\partial R_1(p^2)}{\partial p^2}\,.$$ Recalling the definition of $r^{AA}_{1,2}$ in appendix \[sec\_gamma\], $$r^{AA}_2(p^2) + r^{AA}_1(p^2) + p^2 \frac{\partial r^{AA}_2(p^2)}{\partial p^2}= \frac{1}{\xi}\Big(\Sigma^{AA}_L(p^2) + p^2 \frac{\partial \Sigma^{AA}_L(p^2)}{\partial p^2}\Big)\,.$$ We then obtain the following sufficient condition $$u^{cA} \sim 0 \Longleftrightarrow R_1(r^{AA}_2 + r^{AA}_1) + \zeta^{\Lambda_0}_{cA} \sim 0\,,\label{eq_st_cA}$$ where $$\zeta^{\Lambda_0}_{cA}(p^2):=p^2 \Big(R_1(p^2)\frac{\partial r^{AA}_2(p^2)}{\partial p^2} + \frac{1}{\xi}\Sigma^{AA}_L(p^2)\frac{\partial R_1(p^2)}{\partial p^2}\Big)\,.$$ See section \[sec\_327w\] for the definition of $u^{cA}$. Relation gives us the renormalization condition for $r^{AA}_2$. ${\mathsf{\Gamma}}^{c A A}_1$ ----------------------------- The renormalization point is $\Vec{q}=(\bar{k},\bar{p},\bar{q})\in \mathbb{M}^s_3$, see . With $k=-p-q$, $$\begin{aligned} {\mathsf{\Gamma}}^{c^s A^a_\mu A^b_\nu}_1(p,q) \sim &\Gamma^{A^t_\rho A^a_\mu A^b_\nu}(p,q)\Gamma^{c^s}_{\gamma^t_\rho}(k)+\sum \limits_{Z_2} F^{AA}_{T;\mu \rho}(p)\Gamma^{c^s A^b_\nu}_{\gamma^a_\rho}(k,q)\,,\label{eq_29cAA}\end{aligned}$$ where the sum $\sum_{Z_2}$ runs over all cyclic permutations of $\{(\mu,p,a),(\nu,q,b)\}$. The marginal terms are: ${\mathsf{\Gamma}}^{c A A;pp}_1$, ${\mathsf{\Gamma}}^{c A A;pq}_1$, ${\mathsf{\Gamma}}^{c A A;qq}_1$. Using equation we see that $u^{cAA}_2 \sim 0 \implies u^{cAA}_1 \sim 0$. Acting with $\partial_p\partial_p$ on both sides of  we obtain $$\begin{aligned} u^{cAA}_2 &\sim 0&& \Longleftrightarrow &gR_2\left(1+r^{AA}_1\right) - 2R_1R^{AAA} + \zeta^{\Lambda_0}_{cAA} &\sim 0\,.\label{eq_st_cAA}\end{aligned}$$ This gives the renormalization condition for $R_2$. ${\mathsf{\Gamma}}_1^{\bar{c} c c A}$ and ${\mathsf{\Gamma}}^{c c A}_{1 \gamma}$ -------------------------------------------------------------------------------- The renormalization point is $\Vec{q}=(\bar{l},\bar{k},\bar{q},\bar{p}) \in \mathbb{M}^s_4$, see . With $l=-k-q-p$, $$\begin{aligned} {\mathsf{\Gamma}}_1^{\bar{c}^ac^bc^dA^s_\mu}(k,q,p)\sim&\, \Gamma^{c^d \bar{c}^a A^s_\mu A^t_\rho}(l,p,k)\Gamma^{c^b}_{\gamma^t_\rho}(k)-\Gamma^{c^b \bar{c}^a A^s_\mu A^t_\rho}(l,p,q)\Gamma^{c^d}_{\gamma^t_\rho}(q) \nonumber\\ &+\Gamma^{A^t_\rho c^d \bar{c}^a}(q,l)\Gamma^{A^s_\mu c^b}_{\gamma^t_\rho}(p,k) -\Gamma^{A^t_\rho c^b \bar{c}^a}(k,l)\Gamma^{A^s_\mu c^d}_{\gamma^t_\rho}(p,q) \nonumber\\ &+F^{AA}_{T;\mu \rho}(p)\Gamma^{ c^b c^d \bar{c}^a}_{\gamma^s_\rho}(k,q,l)+\Gamma^{c^t \bar{c}^a A^s_\mu}(l,p)\Gamma^{c^b c^d}_{\omega^t}(k,q) \nonumber\\ &+\Gamma^{c^t \bar{c}^a}(l)\sigma_{0 \Lambda_0}(l^2)\Gamma^{c^b c^d A^s_\mu}_{\omega^t}(k,q,p)\,.\end{aligned}$$ From equation it follows that $u^{\bar{c}ccA}_4\sim 0$ and $$u^{\bar{c}ccA}_1 \sim 0 \implies u^{\gamma A cc}\sim 0, u^{\bar{c}ccA}_2\sim 0, u^{\bar{c}ccA}_3 \sim 0\,.$$ Consequently, we need only one condition $$\begin{aligned} u^{\bar{c}ccA}_1& \sim 0&& \Longleftrightarrow &g(R_2-R_3) R^{A\bar{c}c}_1 + \zeta^{\Lambda_0}_{\bar{c}ccA} &\sim 0\,.\label{eq_st_cccA}\end{aligned}$$ This gives the renormalization condition for $R_3$. ${\mathsf{\Gamma}}_1^{\bar{c} c c}$ and ${\mathsf{\Gamma}}^{c c}_{1\gamma}$ {#st_ccc} --------------------------------------------------------------------------- From equations , we have $$u^{\bar{c}ccA}_1 \sim 0 \implies u^{\gamma Acc} \sim 0 \implies u^{\gamma cc} \sim 0 \implies u^{\bar{c}cc}_{i \in \{1,2\}}\sim 0.$$ Consequently, the marginal contribution to the functions is *small*. ${\mathsf{\Gamma}}_1^{c A A A}$ {#st_cAAA} ------------------------------- The renormalization point is $\Vec{q}=(\bar{l},\bar{k},\bar{q},\bar{p}) \in \mathbb{M}^{cp}_4$, see . With $l=-k-q-p$, $$\begin{aligned} {\mathsf{\Gamma}}_1^{c^s A^t_\mu A^b_\nu A^d_\rho}(p,q,k)\sim&\Gamma^{A^a_\sigma A^t_\mu A^b_\nu A^d_\rho}(p,q,k)\Gamma^{c^s}_{\gamma^a_\sigma}(l) \nonumber\\ &+\sum \limits_{Z_3}\Gamma^{A^a_\sigma A^t_\mu A^b_\nu}(p,q)\Gamma^{A^d_\rho c^s}_{\gamma^a_\sigma}(k,l) \nonumber\\ &+\sum \limits_{Z_3} F^{AA}_{T;\rho \alpha}(k) \Gamma^{c^s A^t_\mu A^b_\nu}_{\gamma^d_\alpha}(l,p,q),\end{aligned}$$ where $F^{AA}_T$ is defined in , and the sum $\sum_{Z_3}$ runs over all cyclic permutations of $\{(d,\rho,k),(b,\nu,q),(t,\mu,p)\}$. From it follows that we need two equations $$\begin{aligned} u^{cAAA}_1&\sim 0&& \Longleftrightarrow& 8R_1R^{AAAA}_2-4gR_2R^{AAA} + \zeta^{\Lambda_0}_{cAAA,1}& \sim 0\,,\\ u^{cAAA}_3&\sim 0&& \Longleftrightarrow& 4R_1R^{AAAA}_1+2gR_2R^{AAA} + \zeta^{\Lambda_0}_{cAAA,3}& \sim 0\,.\label{eq_st_cAAA}\end{aligned}$$ These equations give the renormalization conditions for $R^{AAAA}_{1,2}$. Acknowledgements ================ We want to thank the reviewer for careful reading of the paper, useful remarks and criticism to which we paid serious attention. A.N. Efremov is indepted to the Institute for Theoretical Physics of the University of Leipzig whose facilities and financial support were used during the work on the final version of the manuscript. We are especially greatful to S. Hollands for the time he devoted to study this work, numerous discussions and warm hospitality. We also thank H. Gies from the University of Jena for his interest in our paper, valuable remarks and questions. Properties of Gaussian measures {#sec_gauss} =============================== In the following $d\nu=d\nu_{C}(A)d\nu_{S}(c,\bar{c})$ is the measure given in , $$\begin{gathered} d\nu_{C_1+C_2}(A)f(A)=d\nu_{C_1}(A_1)d\nu_{C_2}(A_2)f(A_1+A_2),\\ d\nu_{t C}(A)f(A) = d\nu_C(A)f(t^{\frac{1}{2}} A),\\ d\nu_C(A-\delta A)=d\nu_C(A)e^{-\frac{1}{2\hbar} \langle \delta A, C^{-1}_{\Lambda \Lambda_0} \delta A \rangle}e^{\frac{1}{\hbar}\langle A, C^{-1}_{\Lambda \Lambda_0} \delta A \rangle},\\ \frac{d}{d\Lambda}d\nu_{C}f(A)=\frac{1}{2}d\nu_{C}\langle \frac{\delta}{\delta A}, \hbar\dot C^{\Lambda \Lambda_0} \frac{\delta}{\delta A} \rangle f(A),\\ d \nu_C\, G(A) \left(AC^{-1}_{\Lambda \Lambda_0}-\hbar\frac{\delta }{\delta A} \right) F(A) =d \nu_C \left( \hbar \frac{\delta}{\delta A} G(A) \right) F(A). \label{eq_parts}\end{gathered}$$ When integrating over Grassmann variables one obtains $$\begin{gathered} d\nu_{S_1+S_2}(c,\bar{c})f(\bar{c},c)=d\nu_{S_1}(c_1,\bar{c}_1)d\nu_{S_2}(c_2,\bar{c}_2)f(\bar{c}_1+\bar{c}_2,c_1+c_2),\\ d\nu_{t S}(c,\bar{c})f(\bar{c},c) = d\nu_S(c,\bar{c})f(t^{\frac{1}{2}} \bar{c},t^{\frac{1}{2}} c),\\ d\nu_S(c-\delta c,\bar{c}-\delta \bar{c})=d\nu(c,\bar{c})e^{\frac{1}{\hbar}\langle \delta \bar{c},S^{-1}_{\Lambda \Lambda_0} \delta c \rangle}e^{-\frac{1}{\hbar}(\langle \bar{c},S^{-1}_{\Lambda \Lambda_0}\delta c \rangle + \langle \delta \bar{c},S^{-1}_{\Lambda \Lambda_0}c \rangle)},\\ \frac{d}{d\Lambda}d\nu_{S}f(\bar{c},c)=d\nu_{S}\langle \frac{\delta}{\delta c}, \hbar \dot{S}^{\Lambda \Lambda_0} \frac{\delta}{\delta \bar{c}} \rangle f(\bar{c},c),\\ d\nu_{S}\,\tilde{G}(\bar{c},c) \left(-\bar{c}S^{-1}_{\Lambda \Lambda_0} + \hbar \frac{\delta}{\delta c}\right) \tilde{F}(\bar{c},c)=d\nu_{S} \left(\hbar \frac{\delta_R}{\delta c} \tilde{G}(\bar{c},c) \right) \tilde{F}(\bar{c},c),\label{eq_parts2a}\\ d\nu_{S}\, \tilde{G}(\bar{c},c) \left(S^{-1}_{\Lambda \Lambda_0}c + \hbar \frac{\delta}{\delta \bar{c}}\right) \tilde{F}(\bar{c},c) = d\nu_{S} \left( \hbar\frac{\delta_R}{\delta \bar{c}} \tilde{G}(\bar{c},c) \right) \tilde{F}(\bar{c},c),\label{eq_parts2b}\end{gathered}$$ where right functional derivatives are distinguished from left ones by the label $R$. Properties , , are proved for $$\begin{aligned} G=&e^{\frac{i}{\hbar}\langle j,A \rangle},&F&=e^{\frac{i}{\hbar}\langle j^\prime,A \rangle},&\tilde{G}&=e^{\frac{i}{\hbar}(\langle \bar{c},\eta \rangle + \langle \bar{\eta}, c \rangle)},&\tilde{F}&=e^{\frac{i}{\hbar}(\langle \bar{c}, \eta^\prime \rangle + \langle \bar{\eta}^\prime, c \rangle)}\,,\end{aligned}$$ and extended to polynomials in the fields by functional differentiation. Chains of vertex functions {#ex_AAcc} ========================== For the purpose of example we give the complete list of reduced chains which appear in the loop integrals for $\dot{\Gamma}^{AAc\bar{c};w}$, together with the corresponding "dotted” propagators. The external fields are underlined. Moreover, $\sum_{i=0}^{k-1} w_i=w$, $k$ being the number of vertex functions in each chain. $$\begin{array}{ll} \dot{C}\Gamma^{A {\underline}{A} A;w_0}C\Gamma^{A {\underline}{A} A;w_1} C\Gamma^{A {\underline}{c} \bar{c};w_2}S\Gamma^{c {\underline{\bar{c}}} A;w_3},&\dot{C}\Gamma^{AA {\underline}{A} {\underline}{A} {\underline}{c} {\underline{\bar{c}}};w},\\ \dot{S}\Gamma^{c{\underline}{A}\bar{c};w_0}S\Gamma^{c {\underline}{A} \bar{c};w_1} S\Gamma^{Ac{\underline{\bar{c}}};w_2}C\Gamma^{{\underline}{c} \bar{c}A;w_3},&\dot{S}\Gamma^{c \bar{c} {\underline}{A} {\underline}{A} {\underline}{c} {\underline{\bar{c}}} ;w},\\ \dot{C}\Gamma^{A {\underline}{A} {\underline}{A} A;w_0} C\Gamma^{A {\underline}{c} \bar{c};w_1}S\Gamma^{c {\underline{\bar{c}}} A;w_2},&\dot{C}\Gamma^{A A {\underline}{A} {\underline}{c} {\underline{\bar{c}}};w_0} C \Gamma^{A {\underline}{A} A;w_1},\\ \dot{S}\Gamma^{c{\underline}{A} {\underline}{A} \bar{c};w_0}S\Gamma^{Ac{\underline{\bar{c}}};w_1}C\Gamma^{{\underline}{c} \bar{c}A;w_2},&\dot{S}\Gamma^{{\underline}{A} {\underline}{c} {\underline{\bar{c}}} c \bar{c};w_0} S \Gamma^{{\underline}{A} c \bar{c};w_1},\\ \dot{C}\Gamma^{A {\underline}{A} {\underline}{c} \bar{c};w_0} S\Gamma^{{\underline}{A} c \bar{c};w_1}S\Gamma^{A c {\underline{\bar{c}}};w_2},&\dot{C}\Gamma^{A {\underline}{A} {\underline}{A} c {\underline{\bar{c}}};w_0} S\Gamma^{A {\underline}{c} \bar{c};w_1},\\ \dot{S}\Gamma^{\bar{c} {\underline}{A} {\underline}{c} A;w_0}C\Gamma^{A {\underline}{A} A;w_1}C\Gamma^{A c {\underline{\bar{c}}} ;w_2},&\dot{S}\Gamma^{A {\underline}{A} {\underline}{A} c {\underline{\bar{c}}} ;w_0} C\Gamma^{A {\underline}{c} \bar{c};w_1},\\ \dot{C}\Gamma^{A {\underline}{A} {\underline{\bar{c}}} c;w_0} S\Gamma^{A {\underline}{c} \bar{c};w_1}C\Gamma^{A {\underline}{A} A;w_2},&\dot{C}\Gamma^{A {\underline}{A} {\underline}{A} {\underline}{c} \bar{c};w_0} S\Gamma^{A c {\underline{\bar{c}}};w_1},\\ \dot{S}\Gamma^{c {\underline}{A} {\underline{\bar{c}}} A;w_0}C\Gamma^{A {\underline}{c} \bar{c};w_1}S\Gamma^{{\underline}{A} c \bar{c} ;w_2},&\dot{S}\Gamma^{A {\underline}{A} {\underline}{A} {\underline}{c} \bar{c};w_0} C\Gamma^{A c {\underline{\bar{c}}};w_1},\\ \dot{C}\Gamma^{A {\underline}{c} {\underline{\bar{c}}} A;w_0} C\Gamma^{A {\underline}{A} A;w_1}C\Gamma^{A {\underline}{A} A;w_2},&\dot{C}\Gamma^{A {\underline}{A} c {\underline{\bar{c}}};w_0} S\Gamma^{A {\underline}{A} {\underline}{c} \bar{c};w_1},\\ \dot{S}\Gamma^{c {\underline}{c} {\underline{\bar{c}}} \bar{c};w_0}S\Gamma^{{\underline}{A} c \bar{c};w_1}S\Gamma^{{\underline}{A} c \bar{c} ;w_2},&\dot{S}\Gamma^{A {\underline}{A} c {\underline{\bar{c}}};w_0}C\Gamma^{A {\underline}{A} {\underline}{c} \bar{c} ;w_1},\\ \dot{C}\Gamma^{A {\underline}{A} {\underline}{A} A;w_0} C\Gamma^{A {\underline}{c} {\underline{\bar{c}}} A;w_1},&\dot{S}\Gamma^{{\underline}{A} {\underline}{A} c \bar{c};w_0}S\Gamma^{c \bar{c} {\underline}{c} {\underline{\bar{c}}} ;w_1}.\\ \end{array}$$ Tensors {#sec_t} ======= For the definition of the tensor monomial sets $\{\delta^s q^n\}$, $\{\delta^s q^n\}_r$ see beginning of page . \[lem\_monom\]Let $q=(q_1,...,q_m)$ where $q_i \in \mathbb{R}^D$ are $m \in \mathbb{N}$ linearly independent vectors. Then the tensor monomials $\{\delta^s q^n\}_r$ of positive rank $r=2s+n\leqslant 2(D-m)+1$ are linearly independent, $$\sum \limits_{t \in \{\delta^s q^n\}_r} c_t \, t=0\implies c_t=0,\;\forall t. \label{eq_monom}$$ #### Proof Observe that, for $m,r,D\in\mathbb{N}$, the inequality $r\leqslant 2(D-m)+1$ is equivalent to $m+s\leqslant D$ for all $s,n\in\mathbb{N}$ such that $r=n+2s$. Let $I:=\{1,...,r\}$. Let $\mathcal{P}_{s}$ be the set of all divisions of the set $I$ in $m+s$ pairwise-disjoint, possibly-empty sets, $$I=\big(\cup_{j=1}^m V_j \big)\;\bigcup\;\big(\cup_{k=1}^{s}S_k\big),$$ such that $S_k=\{s_k^1,s_k^2\}$, $s_k^1<s_k^2$, and $\min{S_1}<...<\min{S_s}$. There is a bijection that maps a division $(V_j,S_k)\in\mathcal{P}_{s}$ to a tensor monomial $t\in\{\delta^s q^n\}$, constructed by the relation $$t_{\mu_1...\mu_r}= \prod_{j=1}^m \prod_{v_j\in V_j} q_{j;\mu_{v_j}} \prod_{k=1}^s \delta_{\mu_{s_k^1}\mu_{s_k^2}}.$$ Let us first prove the statement of the lemma for orthonormal $q_j$. In an appropriate basis of $\mathbb{R}^D$, their components are $$\begin{aligned} \label{qkcomp} q_{j;\mu}&=\delta_{j\mu},& j\in \{1,...,m\},\quad \mu \in \{1,...,D\}.\end{aligned}$$ Let us assume that $\sum_{t} c_t\, t=0$, with $t\in \{\delta^s q^n\}_r$. We will proceed by proving inductively that $c_{t}=0$ for all $t\in \{\delta^s q^n\}$, from $s=D-m$ down to $s=0$. Fix $\bar{s}\le D-m$ and assume that $c_t=0$ for all $t$ involving more than $\bar{s}$ Kronecker’s tensors (which is vacuously true for $\bar{s}=D-m$, due to the rank constraint). Let us prove that $c_{\bar{t}}=0$ for an arbitrary $\bar{t}\in\{\delta^{\bar{s}} q^{\bar{n}}\}$, which is associated to a division $(\bar{V}_j,\bar{S}_k)\in\mathcal{P}_{\bar{s}}$. Fix the values of the indices $\bar{\mu}_i$, with $i\in I$, by $$\begin{aligned} \bar{\mu}_i&=\left \{ \begin{matrix} j,&\mbox{if }\exists j \mbox{ such that }i\in \bar{V}_j,\\ m+k,&\mbox{if } \exists k \mbox{ such that } i \in \bar{S}_k,\end{matrix} \right. \label{eq_17idx}\end{aligned}$$ Note that this choice is possible because $m+\bar{s}\leqslant D$. It is enough to show that whenever $t_{\bar{\mu}_1...\bar{\mu}_r}\neq0$ for $t\in\{\delta^s q^n\}$ and $s\leqslant\bar{s}$ (i.e. $n\geqslant \bar{n}$) then $s=\bar{s}$ and $t=\bar{t}$: in fact this property, the inductive hypothesis, and the vanishing of the sum $\sum_t c_t\, t$ imply that $c_{\bar{t}}=0$. To prove the aforementioned property, introduce the division $(V_j, S_k)\in\mathcal{P}_{s}$ defining the tensor $t$ and, using , correspondingly write $$\label{tNotVan} 0\neq t_{{\bar{\mu}}_1...{\bar{\mu}}_r}= \prod_{j=1}^m \prod_{v_j\in V_j} \delta_{j{\bar{\mu}}_{v_j}} \prod_{k=1}^s \delta_{{\bar{\mu}}_{s_k^1}{\bar{\mu}}_{s_k^2}}.$$ Relations and imply that $V_j\subseteq \bar{V}_j$ for all $j$, which, together with the inductive condition $n\geqslant\bar{n}$, leads to $n=\bar{n}$ and, because the rank $r$ is fixed, to $s=\bar{s}$. Relations , , and $s=\bar{s}$ imply that there is an injective map $f:I\to I$ such that $S_j=\bar{S}_{f(j)}$. By definition of the $S_j$, it then follows that $\min\bar{S}_{f(1)}<...<\min\bar{S}_{f(1)}$: this is only possible if $f$ is the identity, which concludes the first part of the proof. Let us now prove the statement for $m$ linearly independent vectors $p_1,...,p_m$. The sum $\sum_{t} c_t\, t=0$, with $t\in\{\delta^s p^n\}_r$, may be rewritten as $$\sum_{2s+n=r}\,\sum_{1\leqslant k_1,...,k_n\leqslant m}\,\sum_{\pi}\, c_{k_1,...,k_n;\pi} \, \prod^n_{j=1} (p_{k_j})_{\mu^\pi_{j}}\,\prod^{s}_{j'=1}\delta_{\mu^\pi_{m+2j'-1}\mu^\pi_{m+2j'}} =0,$$ where $\mu^\pi_j:=\mu_{\pi(j)}$ and the sum over $\pi$ runs over the right coset of permutation groups $S_r\backslash(S_n {\times} S_s {\times} S^s_2 )$. Expressing the $p_k$ in terms of $m$ orthonormal vectors $q_{k^\prime}$, $p_k=A_{k k^\prime}q_{k^\prime}$, gives a tensor transformation leading to the coefficients in the $\{\delta^s q^n\}_r$ basis: $$c^{\prime}_{k^\prime_1,...,k^\prime_n;\pi}=c_{k_1,...,k_n;\pi}\prod^n_{j=1} A_{k_j k^\prime_j}.$$ The validity of equation for the $q_k$ implies that $c^{\prime}_{k^\prime_1,...,k^\prime_n;\pi}=0$, which, by invertibility of the matrix $A \in GL(m,\mathbb{R})$, gives $c_{k_1,...,k_n;\pi}=0$. $\blacksquare$\ Note that for $m=n_\phi-1$ and $D=4$, the condition for linear independence of the monomials in $\{\delta^s q^k\}_r$ reads $$r \leqslant 2(4-(n_\phi-1))+1=11-2n_\phi.$$ \[lem\_monom2\] Let $q=(q_1,...,q_m)$ with $q_i\in\mathbb{R}^D$ and $m$ nonnegative integer. The tensor monomials $\{\delta^s q^n\}_r$ of positive rank $r\geqslant 2(D-m+1)$ are linearly dependent. #### Proof The signature of a permutation $\pi$ is denoted by $(-)^\pi$. - The Gram matrix $g:= g(t_1,\dots,t_k)$ of $k$ tensors $t_i$ of equal rank is defined by its components: $g_{ij}:=(t_i,t_j)$ for all $i,j\in[k]$. The scalar product is $O(D)$-invariant: $(\underline{R}\,t,\underline{R}\,u)=(t,u)$ for all tensors $t,u$ of rank $r$ and all $\underline{R}\,:=R\otimes\cdots\otimes R$ ($r$ times) with $R\in O(D)$. As a consequence, $g(\underline{R}\,t_1,\dots,\underline{R}\,t_k) =g(t_1,\dots,t_k)$ for $k$ tensors of rank $r$. It is a well-known fact that tensors of equal rank $t_1,\dots,t_k$ are linearly independent iff their Gram matrix $g(t_1,\dots,t_k)$ is invertible. It follows that $t_1,\dots,t_k$ are linearly independent iff $\underline{R}\,t_1,\dots,\underline{R}\,t_k$ are linearly independent for some $\underline{R}$ as above stated. - We assume that $m>0$ because whenever $m=0$ for each $s\geqslant D+1$ one has \[ldep-relation2ter\] \_[S\_s]{} (-)\^ \_[\_1\_[(1)]{}]{} \_[\_s\_[(s)]{}]{}=0. - We assume that $q_1,\dots,q_m$ are linearly independent because otherwise there exist $c_i$ not all vanishing such that $\sum_{i\in[m]}c_i\, q_i=0$, which, for every $r>0$, yields \[ldep-relation3\] \_[i\_1,…,i\_r]{} c\_[i\_1]{}c\_[i\_r]{}q\_[i\_1]{}q\_[i\_r]{}=0. - If the tensors $\{\delta^s q^n\}_r$ are linearly dependent then the tensors $\{\delta^s q^n\}_{r'}$ are linearly dependent, for every $r'> r$. Proof: linear dependence of the tensors $\{\delta^s q^n\}_r$ yields $$\sum_{t\in \{\delta^s q^n\}_r} c_t\, t=0\,,$$ where not all $c_t$ vanish. Linear independence of $q_1,\dots,q_m$ implies that $q_1\neq0$. The proof is concluded by applying $r'-r$ times the tensorial product $\otimes q_1$, which gives $$\sum_{t\in \{\delta^s q^n\}_r} c_t\,t\otimes (q_1)^{\otimes r'-r}=0\,.$$ - For $q_1,\dots, q_D$ linearly independent the statement of the lemma follows from the previous fact and a well-known relation involving the Gram matrix $g:=g(q_1,\dots,q_D)$: \[ldep-Kron=\] \_= \_[i,j]{}q\_[i,]{} (g\^[-1]{})\_[ij]{}q\_[j,]{}. - It is then enough to prove the statement of the lemma for ${r=2(D-m+1)}$, ${0<m<D}$, and $q_1,\dots,q_m$ linearly independent and such that $q_i^\mu=0$ for all ${\mu\in[m+1: D]}$ and ${i\in[m]}$. There exist $q_{m+1},\dots,q_D$ orthonormal vectors such that $q_i^\mu=\delta_i^\mu$ for all ${\mu\in[D]}$ and $i\in[m+1 : D]$. Relation then gives \[ldep-deltaperp\] \_= \_[i,j]{}q\_[i,]{} (g\_[[[/-6mu/]{}]{}]{}\^[-1]{})\_[ij]{}q\_[j,]{} +\^\_, where $g_{{{\scriptscriptstyle/\mkern-6mu/}}}:=g(q_1,\dots,q_m)$ and $\delta^\perp_{\mu\nu} :=\sum_{i\in [m+1 : D]} q_{i,\mu}q_{i,\nu}$. For $s=D-m+1$, one has \[ldep-relation\] \_[S\_[s]{}]{} (-)\^ \^\_[\_[m+1]{}\_[m+(1)]{}]{} \^\_[\_[m+s]{}\_[m+(s)]{}]{}=0. Combining and concludes the proof of the lemma. $\blacksquare$\ The following lemma states a necessary condition for a regular, $O(4)$-invariant tensor field. \[lem\_pr\]Let $f({\underline}{y})$ be a regular, $O(4)$-invariant tensor field of rank $r$ where ${\underline}{y}:=(y_1,...,y_m)$ with $y_j \in \mathbb{R}^4$. Assume that the tensor monomials $\{\delta^s{\underline}{y}^k\}_{r}$ as well as $\{\delta^s{\underline}{y}^k\}_{r+1}$ are linearly independent pointwise for all ${\underline}{y} \in O$ where $O$ is some open set. Then on $O$ we have $$\begin{aligned} f&=\sum \limits_{t \in \{\delta^s{\underline}{y}^k\}_{r}} f_{t}\, t\,,&\partial_j f&=\sum \limits_{t^\prime \in \{\delta^s{\underline}{y}^k\}_{r+1}} f_{j,t^\prime}\, t^\prime\,. \end{aligned}$$ Furthermore for every $t \in \{\delta^s {\underline}{y}^{k>0}\}_r$ there exist $j$ and $t^\prime \in \{\delta^s {\underline}{y}^k\}_{r+1}$ such that $f_t=f_{j,t^\prime}$ on $O$. #### Proof For shortness we consider only the case $m=2$. We have $$f=\sum \limits_{t \in \{\delta^s\}_{r}} u_{t}\, t + \sum_{t \in \{\delta^s{\underline}{y}^{k>0}\}_{r}} \zeta_t \, t,\label{eq_415a}$$ where $u_t$, $\zeta_t$ are regular functions of the scalar parameters $\mathbb{X}=\{\tfrac{1}{2}y^2_1,\tfrac{1}{2}y^2_2,y_1y_2\}$. Apply the operator $\partial_j$ to both sides of . The Leibniz rule gives $$\begin{aligned} \partial_{j} (u_{t} \delta^s) &= \sum \limits_{x \in \mathbb{X}} (\partial_x u_{t})\, \delta^s \, \partial_j x,\label{eq_415b}\\ \partial_{j} (\zeta_{t} \delta^s{\underline}{y}^k) &= \sum \limits_{x \in \mathbb{X}} (\partial_x \zeta_{t})\, \delta^s \, {\underline}{y}^k \partial_j x \;+ \zeta_{t} \delta^s \, \partial_j {\underline}{y}^k,\label{eq_415c}\end{aligned}$$ where $\partial_j x\in\{0,y_1,y_2\}$, ${\underline}{y}^k:=y_{i_1}... y_{i_k}$ with $i_l \in \{1,2\}$, and $$\partial_j{\underline}{y}^k=\partial_j \prod^k_{l=1} y_{i_l} = \sum^k_{q=1} \delta_{j,i_q}\, \prod^k_{\substack{l=1\\l\neq q}} y_{i_l}.$$ The nonvanishing tensor monomials arising at a given $j\in\{1,2\}$ from the rhs of and have rank $r+1$, are linearly independent by assumption, and are pairwise different. Each coefficient $\zeta_{t}$ of the tensor $\delta^s y_{i_1}...y_j...y_{i_k}$ in the decomposition appears also as the coefficient of $\delta^s y_{i_1}...\delta ...y_{i_k}$ in the decomposition for $\partial_j f$. $\blacksquare$ Lemma \[lem\_rn\] below relies on lemma \[lem\_monom\], which shows that, for $m$ linearly independent vectors $\underline{e}=(e_1,\cdots,e_{m})$, the relation $r+1\leqslant 9-2m$ is a sufficient condition for the linear independence of the tensor monomials $\{\delta^s\underline{e}^k\}_{r}$ and $\{\delta^s\underline{e}^k\}_{r+1}$, see lemma \[lem\_monom2\] for a necessary condition. The renormalization points in appendix \[sec\_ren\] are chosen to comply with the aforementioned relation. The proof of lemma \[lem\_rn\] is in the same spirit as the one of the preceding lemma \[lem\_pr\]. \[lem\_rn\]Let $F$ be a regular, $O(4)$-invariant tensor field of rank $r\in \{2,4\}$ on $\mathbb{P}_n$. Let be given $\Vec{q} \in \mathbb{M}_n$ and $m\geqslant 2$ linearly independent vectors $\underline{e}=(e_1,\cdots,e_{m})$, such that $\mathrm{span}(\Vec{q})=\mathrm{span}({\underline}{e})$. Assume that $r+1\leqslant 9-2m$. By lemma \[lem\_monom\] there exist unique coefficients $F_t$ such that $F(\Vec{q})=\Sigma_{t \in \{\delta^s \underline{e}^k\}_r}\, F_t\, t$. Furthermore, $$\begin{aligned} |F(\Vec{q})|&\leqslant c \max\Big( \big(|F_t|\big)_{t \in \{\delta^s \}_r},(M|\partial_k F (\Vec{q})|)_{k\in[n-1]}\Big),\label{eq_73} \\ \big|\sum_{t \in \{\delta^s \underline{e}^{k>0}\}_r}\,F_t\,t\big|& \leqslant c\, \max\Big( (M|\partial_k F (\Vec{q})|)_{k\in[n-1]}\Big)\,. \label{eq_73bis}\end{aligned}$$ The bounds hold with the same constant $c$ for all $F$ of equal rank. #### Proof The coefficients $\big(|F_t|\big)_{t \in \{\delta^s \}_r}$ in the basis $\{\delta^s {\underline}{e}^k\}_r$ do not depend on the choice of the vectors ${\underline}{e}$. Hence it is enough to prove in the case when $e_ie_j=M^2\delta_{ij}$. For simplicity we assume that $m=2$, the extension to other $m$ being clear.\ By hypothesis, there exists a $(n-1)\times 2$ matrix $L$ such that $q_k=L_{ki} e_i$ and $$|L|:= \sqrt{L_{ki} L_{ki}}=\frac{1}{M}\sqrt{L_{ki} L_{kj}e_ie_j}=\frac{1}{M}\Big(\sum^{n-1}_{k=1} q^2_k\Big)^{\frac{1}{2}}\leqslant (n-1)^{\frac{1}{2}}.\label{eq_26a}$$ Denote by $\mathbb{E}\subset{\mathbb{R}}^4$ the linear span of the vectors $q_0,..,q_{n-1}$. The matrix $L$ induces a linear map ${L:\mathbb{E}^2 \to \mathbb{R}^{4(n-1)}}$, ${\underline}{y}\mapsto L{\underline}{y}$, where ${\underline}{y}=(y_1,y_2)$ and $(L{\underline}{y})_k=L_{ki}y_i$. We also define an auxiliary function on $\mathbb{E}^2$: $f({\underline}{y}):=F(L{\underline}{y})$. Setting $\partial_{y_i}:=\partial/\partial y_i$ and $\partial_k:=\partial/\partial p_k$, the Cauchy–Schwarz inequality and imply that, $$\label{eqdyGa} \textstyle{\sum}_{i=1}^2|\partial_{y_i} f({\underline}{e})|^2\leqslant |L|^2 \; \sum_{k=1}^{n-1}|\partial_k F(\Vec{q})|^2 \leqslant (n-1)^2 \max \limits_k (|\partial_k F(\Vec{q})|^2).$$ For all $y_1,y_2 \in \mathbb{E}$, denote as usual by $\{\delta^s {\underline}{y}^k\}$ the set of all monomials being a tensor product of $k$ vectors in ${\underline}{y}=(y_1,y_2)$ and of $s$ Kronecker tensors, and by $\{\delta^s {\underline}{y}^k\}_r$ the union of all $\{\delta^s {\underline}{y}^k\}$ such that $2s+k=r$. By , whenever $r\leqslant 5$ and $y_1,y_2$ are linearly independent, the elements of $\{\delta^s {\underline}{y}^k\}_r$ are linearly independent. In this case, we label the tensor monomials by fixing a family of disjoint sets $A_{k,s}$ and a family of bijections $\alpha\mapsto t_\alpha$ from each $A_{k,s}$ to $\{\delta^s {\underline}{y}^k\}$. Furthermore, we define the auxiliary sets $$\begin{aligned} A^r&:=\bigcup \limits_{2s+k=r}A_{k,s}\,, &A^r_+&:=\bigcup \limits_{2s+k=r,\,k\geqslant 1}A_{k,s}\,, &A^r_0&:=A^r\setminus A^r_+\,.\label{eq_415d}\end{aligned}$$ For $r \in \{2,4\}$ and ${\underline}{y}=(y_1,y_2)$ in an open neighborhood of ${\underline}{e}$ (in $\mathbb{E}^2$) for which $y_1,y_2$ are linearly independent, we write the following tensor decomposition: $$f({\underline}{y})=\sum \limits_{\alpha \in A^r_0} u_{\alpha} t_\alpha + \sum_{\alpha\in A^r_+} \zeta_{\alpha} t_\alpha,\label{eq_24g}$$ where $u_\alpha$, $\zeta_\alpha$ are regular functions of the scalar parameters $\mathbb{X}=\{\tfrac{1}{2}y^2_1,\tfrac{1}{2}y^2_2,y_1y_2\}$. Evaluating at ${\underline}{y}={\underline}{e}$ and using the general fact that $|t|=2^s M^k$ for ${t\in\{\delta^s {\underline}{e}^k\}}$ we obtain $$|f({\underline}{e})|\leqslant \sum_{\alpha \in A^r_0} 2^s\, |u_{\alpha}| +\sum_{\alpha \in A^r_+} |\zeta_{\alpha}|\,2^s M^k.\label{eq_26gao}$$ We now want to prove the existence of a constant $c_1>0$ such that $$\sum_{\alpha \in A^r_+} |\zeta_{\alpha}|\,2^s M^k \;\leqslant\; c_1 M \Big(\sum_{i\in\{1,2\}}|\partial_{y_i}f({\underline}{e})|^2\Big)^{1/2}. \label{eq_24bnd}$$ Apply the operator $\partial_{y_i}$ to both sides of . The Leibniz rule gives $$\begin{aligned} \partial_{y_i} (u_{\alpha} \delta^s) &= \sum \limits_{x \in \mathbb{X}} (\partial_x u_{\alpha})\, \delta^s \, \partial_{y_i} x,\label{eq_24t0}\\ \partial_{y_i} (\zeta_{\alpha} \delta^sy^k) &= \sum \limits_{x \in \mathbb{X}} (\partial_x \zeta_{\alpha})\, \delta^s \, y^k \partial_{y_i} x \;+ \zeta_{\alpha} \delta^s \, \partial_{y_i}y^k,\label{eq_24t1}\end{aligned}$$ where $\partial_{y_i} x\in\{0,y_1,y_2\}$, $y^k=y_{i_1}... y_{i_k}$ with $i_l \in \{1,2\}$, and $$\label{eq_24dyn} \partial_{y_i}y^k=\partial_{y_i} \prod^k_{l=1} y_{i_l} = \sum^k_{j=1} \delta_{i,i_j}\, \prod^k_{\substack{l=1\\l\neq j}} y_{i_l}.$$ Fix ${\underline}{y}={\underline}{e}$. The nonvanishing tensor monomials arising at a given $i\in\{1,2\}$ from the rhs of and have rank $r+1\in\{3,5\}$, are pairwise different, and are a subset of the tensor monomials in $A^{r+1}$, themselves linearly independent by . Denote by $B^{r+1}_i\subset A^{r+1}_+$ the subset labeling the monomials of type $\delta^s\delta_{i,i_j}\prod_{l\neq j}y_{i_l}$ arising from and at the given $i$. By construction, we can define the maps ${\pi_i:B^{r+1}_i\to A^{r}_+}$ with $i\in\{1,2\}$ such that:\ i) if $t_{\pi_i(\beta)}\in\{\delta^s {\underline}{e}^k\}$ then $t_{\beta}\in\{\delta^{s+1} {\underline}{e}^{k-1}\}$ (in this case, $|t_{\pi_i(\beta)}|=2^sM^k$ and $|t_{\beta}|=2^{s+1}M^{k-1}$);\ ii) for each $\beta\in B^{r+1}_i$ the coefficient of $t_\beta$ in and that of $t_{\pi_i(\beta)}$ in are the same, namely $\zeta_{\pi_i(\beta)}$;\ iii) $\pi_1(B^{r+1}_1)\cup\pi_2(B^{r+1}_2)=A^{r}_+$.\ The following bound holds at ${\underline}{y}={\underline}{e}$ for every tensor $\Psi=\sum_\beta \Psi_\beta t_\beta/|t_\beta|$ with $\beta\in A^{r'}$ and $r'\leqslant5$, and for every nonempty $B\subset A^{r'}$: $$|\Psi|^2=(\Psi,\Psi)=\Psi_\beta^*\, G_{\beta \beta^\prime} \,\Psi_{\beta^\prime}\geqslant \lambda_\mathrm{1}\sum_{\beta \in A^{r'}} |\Psi_\beta|^2 \geqslant \lambda_\mathrm{1} \sum_{\beta \in B} |\Psi_\beta|^2,\label{eq_26gram}$$ where $\lambda_\mathrm{1}>0$ is the smallest eigenvalue of the Gram matrix of components $G_{\beta\beta^\prime}:=(t_\beta,t_{\beta'})/(|t_\beta||t_{\beta'}|)$, which is positive definite by . Application of to $\Psi_i:=\partial_{y_i} f({\underline}{e})$ for each $i\in\{1,2\}$, with $r'=r+1$ and $B=B^{r+1}_i$, gives $$\sum_{i\in\{1,2\}}|\partial_{y_i} f({\underline}{e})|^2 \geqslant \lambda_\mathrm{1} \sum\limits_{\substack{i\in\{1,2\}\\\beta \in B^{r+1}_i}} |\zeta_{\pi_i(\beta)}|^2 |t_\beta|^2 \geqslant \frac{4\lambda_\mathrm{1}}{M^2} \sum_{\alpha \in A^{r}_+} |\zeta_\alpha|^2 \,(2^{s} M^{k})^2 ,\label{sumpaf}$$ from which follows the bound .\ Inequalities , and lead to the bounds , , with a constant $$c=\max\Big(2^{\frac{r}{2}}|A_0|, (n-1)\sqrt{\frac{|A^r_+|}{4\lambda_1}}\Big)\,.$$ $\blacksquare$ Basic estimates {#sec_b} =============== Let $0\leqslant q \leqslant p \leqslant P$, $k \in \mathbb{N}$. Then $\exists C_k>0$ such that $$\int \limits^{+\infty}_\Lambda \frac{d\lambda \; \log^k_+ \frac{P}{\lambda}}{(\lambda + p) (\lambda + q)}\leqslant C_k\frac{1+\log^{k+1}_+ \frac{P}{\Lambda + q}}{\Lambda + p + q}.\label{eq_twin}$$ #### Proof Let $I^k$ be the left hand side of the inequality and $$I^k_{[a,b]}:=\int \limits^{b}_a \frac{d\lambda \; }{(\lambda + p) (\lambda + q)}\log^k_+ \frac{P}{\lambda}.$$ We begin with the case $k \geqslant 1$. - $\Lambda > P$, $I^k=0$. - $q \leqslant \Lambda \leqslant P$, $I^k=I^k_{[\Lambda,P]}$. $$\begin{aligned} I^k\leqslant& \frac{1}{\Lambda + p} \int \limits^{P}_\Lambda \frac{d \lambda}{\lambda + q}\log^k \frac{2P}{\lambda + q}\leqslant \frac{2}{k+1}\frac{1}{\Lambda + p+q}\log^{k+1} \frac{2P}{\Lambda + q}\nonumber\\ &\leqslant \frac{A_k}{\Lambda + p+q} (\log^{k+1}_+ \frac{P}{\Lambda + q} + 1),\end{aligned}$$ where $$A_k:=\frac{2 (\log 2 + 1)^{k+1}}{k+1},$$ and we have used the inequality $$\frac{\log_+ x + \log 2}{(1+\log^n_+ x)^{\frac{1}{n}}}<1 + \log 2.$$ - $\Lambda<q \leqslant P$, $I^k=I^k_{[\Lambda,q]} + I^k_{[q,P]}$. $$\begin{aligned} I^k_{[q,P]}\leqslant& \frac{1}{\Lambda + p} \int \limits^{P}_q \frac{d \lambda}{\lambda + q} \log^k \frac{2P}{\lambda+ q}\leqslant \frac{1}{\Lambda + p} \int \limits^{P}_\Lambda \frac{d \lambda}{\lambda + q} \log^k \frac{2P}{\lambda+ q}\nonumber\\ &\leqslant \frac{A_k}{\Lambda + p+q} (\log^{k+1}_+ \frac{P}{\Lambda + q} + 1),\\ I^k_{[\Lambda,q]}\leqslant& \frac{1}{(\Lambda + p)(\Lambda+q)} \int \limits^q_0 d \lambda \;\log^k \frac{P}{\lambda}< \frac{q k!}{(\Lambda + p)(\Lambda+q)} \sum \limits^k_{j=0} \frac{1}{j!}\log^j \frac{P}{q}\nonumber\\ &< \frac{e k!}{\Lambda + p}( \log^k \frac{2P}{\Lambda +q} + 1)< \frac{2 e k!}{\Lambda + p + q}( \log^k_+ \frac{P}{\Lambda +q} + 2).\end{aligned}$$ It remains to consider the case $k=0$. - $\Lambda > p$. This implies $2(\lambda + q)>\lambda+p +2q> \lambda+p$, $\forall \lambda>\Lambda$. $$I^0= 2 \int \limits^{+\infty}_\Lambda \frac{d \lambda}{(\lambda+p)^2}<\frac{4}{\Lambda + p + q}.$$ - $\Lambda \leqslant p$, $I^0=I^0_{[\Lambda,p]} + I^0_{[p,+\infty]}$. $$\begin{aligned} I^0_{[p,+\infty]}<&2 \int \limits^{+\infty}_p \frac{d \lambda}{(\lambda+p)^2}<\frac{1}{p}<\frac{3}{\Lambda +p+q},\\ I^0_{[\Lambda,p]}<&\frac{1}{\Lambda+p}\int \limits^{p}_\Lambda \frac{d \lambda}{\lambda+q}<\frac{2}{\Lambda + p + q} \log\frac{2P}{\Lambda + q}\nonumber\\ &<\frac{2}{\Lambda + p + q}(\log 2+ \log_+ \frac{P}{\Lambda + q}).\end{aligned}$$ $\blacksquare$ Let $0 \leqslant q \leqslant p$, $\eta:=\min(M,q)$, $k \in \mathbb{N}$. Then $\exists C_k>0$ such that $$\int \limits^{+\infty}_\Lambda \frac{d\lambda \; \log^k_+ \frac{\lambda}{M}}{(\lambda + p) (\lambda + q)} \leqslant C_k \frac{1+ \log^{k+1}_+ \frac{p}{\Lambda + \eta}+ \log^{k+1}_+ \frac{\Lambda}{M}}{\Lambda + p + q}. \label{eq_twin2}$$ #### Proof Denote by $I^k$ the lhs of equation . If $k=0$ then the inequality follows from with $P=p$. Let $k \geqslant 1$ and $\mu=\max(\Lambda,M)$. - $p \leqslant \Lambda$. $$I^k \leqslant \int \limits^{+\infty}_{\mu} d \lambda \frac{2\log^k \frac{\lambda}{M}}{(\lambda + p)^2} \leqslant \int \limits^{+\infty}_{\mu} d \lambda f^\prime(\lambda)\leqslant 2^{k+2}k!\sqrt{e} \frac{1 + \log^k_+ \frac{\Lambda}{M}}{\Lambda + p + q},$$ $$f(\lambda):=-\frac{k!2^{k+1}}{\lambda + p}\sum^k_{j=0} \frac{\log^k \frac{\lambda}{M}}{j!2^j}.$$ - $\Lambda<p$. $$J^k:=\int \limits^p_\Lambda d\lambda \frac{\log^k_+ \frac{\lambda}{M}}{\lambda + q},\quad I^k \leqslant \frac{2J^k}{\Lambda + p + q} + \int \limits^{+\infty}_{\mu} d\lambda \frac{2 \log^k_+ \frac{\lambda}{M}}{(\lambda + p)^2}.$$ The integral on the rhs of $I^k$ is exactly the same as in the case $p\leqslant \Lambda$. For the integral $J^k$ using the inequality $$\log_+\frac{\lambda}{M}\leqslant \log 2 + \log_+ \frac{\lambda}{\Lambda+M} + \log_+ \frac{\Lambda}{M},$$ we have $$J^k<3^{k+1} \left(1 + \log^{k+1}_+ \frac{p}{\Lambda+\eta} + \log^{k+1}_+ \frac{\Lambda}{M}\right).$$ $\blacksquare$ For $0<\Lambda<\eta\leqslant M$, $k \in \mathbb{N}$ $$\int \limits^{\eta}_\Lambda d\lambda \frac{1}{\lambda + \eta} \log^k_+ \frac{\lambda}{M} < 1 . \label{eq_int3}$$ #### Proof Denote by $I^k$ the lhs of equation . If $k>0$ then $I^k=0$. It remains to consider the integral $$\int \limits^{\eta}_\Lambda \frac{d \lambda}{\lambda + \eta} = \log\frac{2\eta}{\Lambda +\eta} < \log 2.$$ $\blacksquare$ Let $0<\Lambda<\eta\leqslant P$, $k \in \mathbb{N}$. Then $\exists C_k>0$ such that $$\int \limits^{\eta}_\Lambda \frac{d\lambda }{\lambda + \eta} \log^k_+ \frac{P}{\lambda} < C_k\Big(1 + \log^k_+ \frac{P}{\Lambda + \eta}\Big). \label{eq_int4}$$ #### Proof Denote by $I^k_{[\Lambda,\eta]}$ the lhs of equation . $$I^k_{[\Lambda,\eta]} < I^k_{[0,\eta]}<\frac{1}{\eta}\int \limits^\eta_0 d\lambda\, \log^k_+ \frac{P}{\lambda} < 2k! e^2 \Big( 1 + \log^k_+ \frac{P}{\eta +\Lambda} \Big).$$ $\blacksquare$ Let $0<\Lambda< M \leqslant P$, $0 \leqslant \eta \leqslant M$, $k \in \mathbb{N}$. Then $\exists C_k>0$ such that $$\int \limits^{M}_\Lambda \frac{d\lambda }{\lambda + \eta} \log^k_+ \frac{P}{\lambda} < C_k\Big(1 + \log^{k+1}_+ \frac{P}{\Lambda + \eta}\Big). \label{eq_int4b}$$ #### Proof Denote by $I^k_{[\Lambda,M]}$ the lhs of equation . If $\Lambda \geqslant \eta$ then $$I^k_{[\Lambda,M]} \leqslant \int \limits^M_\Lambda \frac{d\lambda }{\lambda + \eta} \log^k_+ \frac{2P}{\lambda + \eta} \leqslant \frac{2^{k+1}}{k+1} \Big( 1 + \log^{k+1}_+ \frac{P}{\Lambda + \eta}\Big).$$ If $\eta \geqslant \Lambda$ then $I^k < I^k_{[0,\eta]} + I^k_{[\eta,M]}$ where $$\begin{aligned} I^k_{[0,\eta]}&=\int \limits^\eta_0 \frac{d\lambda}{\lambda + \eta} \log^k_+ \frac{P}{\lambda} < 2k! e^2 \Big(1 + \log^k_+ \frac{P}{\eta + \Lambda}\Big),\\ I^k_{[\eta,M]}&=\int \limits^M_\eta \frac{d\lambda}{\lambda + \eta} \log^k_+ \frac{2P}{\lambda + \eta}<\frac{2^{k+1}}{k+1} \Big( 1 + \log^{k+1}_+ \frac{P}{\Lambda + \eta}\Big).\end{aligned}$$ $\blacksquare$ For $q \geqslant 0$, $k \in \mathbb{N}$, $P>0$ there exists a constant $C_k>0$ such that $$\int \limits^{\Lambda_0}_\Lambda \frac{d\lambda }{\lambda + q} \log^k_+ \frac{P}{\lambda} < C_k\Big(1 + \log^{k+1}_+ \frac{P}{\Lambda + q} + \log_+ \frac{\Lambda_0}{\Lambda + q}\Big). \label{eq_int4a}$$ #### Proof Denote by $I^k_{[\Lambda,\Lambda_0]}$ the lhs of equation . If $k=0$ then $$I^0_{[\Lambda,\Lambda_0]}< 1 + \log_+ \frac{\Lambda_0}{\Lambda + q}.$$ For $k>0$ and $\Lambda \geqslant q$ $$I^k_{[\Lambda,\Lambda_0]}< \frac{2^{k+1}}{k+1}\Big(1 +\log^{k+1}_+ \frac{P}{\Lambda + q}\Big).$$ If $k>0$ and $\Lambda<q$ then $I^k_{[\Lambda,\Lambda_0]}< I^k_{[0,q]} + I^k_{[q,\Lambda_0]}$ where $$\begin{aligned} I^k_{[0,q]}&<\frac{1}{q}\int \limits^q_0 d\lambda \,\log^k_+\frac{P}{\lambda}< k!e^2\Big( 1 + \log^{k+1}_+ \frac{P}{\Lambda +q}\Big),\\ I^k_{[q,\Lambda_0]}&<\frac{2^{k+1}}{k+1}\Big(1 +\log^{k+1}_+ \frac{P}{\Lambda + q}\Big).\end{aligned}$$ $\blacksquare$ Let $a,d>0$, $b\geqslant 0$ and $m,k \in \mathbb{N}$. Then $\exists C_{k,m}>0$ such that $$\int \limits^{a}_0 dx \; x^m \log^k_+ \frac{d}{b + x} \leqslant C_{k,m}a^{m+1}\Big(1 + \log^k_+ \frac{d}{a+b}\Big). \label{eq_int5}$$ #### Proof By direct calculations it is easy to show that $$\begin{aligned} x^m \log^k_+ \frac{d}{b + x}&\leqslant f^\prime,&f&:=\frac{k!x^{m+1}}{(m+1)^{k+1}}\sum^k_{j=0} \frac{(m+1)^j}{j!} \log^j_+ \frac{d}{b+x}.\end{aligned}$$ Consequently, the lhs of equation is bounded above by $f(a)$, $$\begin{aligned} f(a)=&\frac{k!}{(m+1)^{k+1}}a^{m+1} \sum^k_{j=0} \frac{(m+1)^j}{j!}\log^j_+ \frac{d}{a+b}\nonumber\\ &\leqslant \frac{e^{m+1} k!}{(m+1)^{k+1}} a^{m+1} \Big(1+\log^k_+ \frac{d}{a+b}\Big).\end{aligned}$$ $\blacksquare$ Let $p,q \in \mathbb{R}^4$, $\Lambda^\prime>0$, $\Lambda^\prime \geqslant \eta \geqslant 0$. Then $$\int \limits^{1}_0 dt \; \frac{|p|}{\Lambda^\prime + |tp + q|} \leqslant 2\Big(\log 4 + \log_+ \frac{|p|}{\Lambda^\prime + \eta}\Big).\label{eq_int7}$$ #### Proof Let $I_{[0,1]}$ denote the lhs of equation . There exists $t_1 \in [0,1]$ such that $|tp + q|\geqslant |p||t-t_1|$ for all $t \in [0,1]$. (To prove this fact: write $q=-t_q\, p+ q^\perp$, with $p.q^\perp=0$, in such a way that $|t\,p-q|\geqslant|t-t_q|\,|p|$; then set $t_1:=0$ for $t_q<0$, $t_1:=1$ for $t_q>1$, and $t_1=t_q$ otherwise.) $$\begin{aligned} I_{[0,1]}=&I_{[0,t_1]}+I_{[t_1,1]}\leqslant 2 \log\frac{\Lambda^\prime+|p|}{\Lambda^\prime}\leqslant 2 \log 4\max(1,\frac{|p|}{\Lambda^\prime+\eta})\nonumber\\ &\leqslant 2 \Big(\log 4 + \log_+\frac{|p|}{\Lambda^\prime + \eta}\Big).\end{aligned}$$ $\blacksquare$ Let $r>0$, $w \in \mathbb{N}$ and $x,y \in \mathbb{R}^4$ $$\frac{e^{-r x^2}}{(1+|x-y|)^w}\leqslant \frac{w!\max(2,1+\frac{1}{2r})^w}{(1+|y|)^w}.\label{eq_frac}$$ #### Proof Choosing the Cartesian coordinate system such that one of the basis vectors $e_L$ is along $y$ we have $$\begin{aligned} \frac{e^{-r x^2}}{(1+|x-y|)^w}&\leqslant f(t),&f(t):=\frac{e^{-r t^2}}{(1+|t-t_0|)^w}.\end{aligned}$$ where $t$, $t_0$ are the longitudinal coordinates, $x=x_T + t e_L$ and $y=t_0 e_L$, $t_0=|y|$. If $t\geqslant t_0$ then $f$ is strictly decreasing. If $t \leqslant t_0$ then $f(t)=g(t)$ where $$\begin{aligned} g(t)&:=\frac{e^{-r t^2}}{(t_1-t)^w},&t_1&:=1+t_0.\end{aligned}$$ If $t < t_1$ then g is either increasing or has a local maximum at $t_-$. $$\begin{aligned} g^\prime(t) &\leqslant 0 \mbox{ if } (\Delta \geqslant 0) \wedge (t_- \leqslant t \leqslant t_+),&t_{\mp}&:=\frac{t_1 \mp \sqrt{\Delta}}{2},\\g^\prime(t) &\geqslant 0 \mbox{ otherwise },& \Delta&:=t^2_1-2\frac{w}{r}.\end{aligned}$$ Consequently, $f(t) \leqslant \max(g(t_-),g(t_0))$ where $$\begin{aligned} g(t_-)&=\frac{e^{-rt^2_-}}{t^w_+}\leqslant \frac{2^w}{t^w_1}=\frac{2^w}{(1+t_0)^w},&g(t_0)&=e^{-rt^2_0}\leqslant\frac{w!(1+\frac{1}{2r})^w}{(1+t_0)^w}.\end{aligned}$$ $\blacksquare$ Let $r>0$. There is a constant $C$ such that $$e^{-r\frac{s^2}{\Lambda^2}}\log_+ \frac{\max(M,\sqrt{p^2 + s^2})}{\Lambda}< C+\frac{1}{2}\log_+\frac{1}{r}+\log_+ \frac{\max(M,p)}{\Lambda}. \label{eq_plog}$$ #### Proof Using the following inequality $$\frac{\max(M,\sqrt{p^2 + s^2})}{\Lambda}\leqslant 2\max(1,\frac{s}{\Lambda})\max(1,\frac{\max(M,p)}{\Lambda}),$$ we bound the lhs of the statement by $$\log_+\frac{\max(M,p)}{\Lambda} + \log 2 + \frac{1}{2}\log_+\frac{1}{r} + \frac{1}{2}e^{-z}\log_+ z,\quad z:=r\frac{s^2}{\Lambda^2}.$$ The inequality $e^{-z}\log_+z < e^{-1}$ finishes the proof.$\blacksquare$ Let $x,y,m \geqslant 0$, $\mathcal{P}^{(0)}(x)=\sum \limits^n_{k=0} c_k x^k$ a polynomial of the degree $n$, $\log_m x:=\log_+ \max(x,m)$. Then there exist polynomials $\mathcal{P}^{(1)}$, $\mathcal{P}^{(2)}$ of the degree $n$ such that $$\mathcal{P}^{(0)}(\log_m \sqrt{y^2 + x^2} )\leqslant \mathcal{P}^{(1)}(\log_m y) + \mathcal{P}^{(2)}(\log_m x).$$ #### Proof Substitution of the inequalities $$\max(\sqrt{y^2+x^2},m) \leqslant \max(y + x,m) \leqslant \max(y,m) + \max(x,m),$$ $$\max(a+b,1) \leqslant \max(a,1) + \max(b,1) \leqslant 2\max(a,1)\max(b,1),$$ into the definition $\log_+ a := \log \max(a,1)$ yields $$\log_m \sqrt{y^2 + x^2} \leqslant \log_m y + \log_m x+\log2.$$ This gives $$\mathcal{P}^{(0)}(\log_m \sqrt{x^2 + y^2}) \leqslant \sum \limits^n_{k=0} c_k 3^k \left( \log^k_m y + \log^k_m x + 1 \right).$$ $\blacksquare$ \[lem\_402a\]For a fixed $s \in \mathbb{N}$ there exists a constant $c$ such that $\forall u \leqslant w_{max}$ and $\forall x \in\mathbb{R}^4$ $$|\partial^u \left( x^{\otimes s}(1-e^{-x^4})\right)| \leqslant c \, \left\{\begin{matrix} |x|^{s + 1 -u},&u \leqslant s ,\\ 1, & otherwise. \end{matrix}\right.\label{eq_402t}$$ #### Proof First we consider the case $u\leqslant s$ $$|\partial^u (x^{\otimes s}(1-e^{-x^4}))|\leqslant|\partial^{u} x^{\otimes s}|(1-e^{-x^4}) + \sum_{u_2>0} \frac{u! 4^{u_2}}{u_1! u_2!}|\partial^{u_1} x^{\otimes s+ 3u_2}|\, e^{-x^4}.\label{eq_402a}$$ For the derivatives on the right we have $$|\partial^{u_1} x^{\otimes s+ 3u_2}|\leqslant (s+3u_2)! \,\frac{|x|^{s-u+1}}{(s-u+1)!}\frac{|x|^{4u_2-1}}{(4u_2-1)!}\,.\label{eq_402b}$$ Furthermore, $\forall k \in \mathbb{N}$ $$\frac{|x|^k}{k!}e^{-x^4} < \sum^{\infty}_{n=0} \frac{|x|^n}{n!}e^{-x^4}=e^{-x^4+|x|}<e.\label{eq_402c}$$ Noting that $1-e^{-x^4} \leqslant |x|$, we obtain the bound in the case $u\leqslant s$ $$\frac{s!|x|^{s+1-u}}{(s-u)!} + \frac{e |x|^{s+1-u}}{(s-u+1)!} \sum_{u_2>0} \frac{u! 4^{u_2} (s+3u_2)!}{u_1! u_2!}\leqslant c |x|^{s+1-u}\,.$$ For $u>s$ using we have $$\sum \frac{u! 4^{u_2}(s+3u_2)!}{u_1! u_2! }\frac{|x|^{s+3u_2-u_1} }{(s+3u_2-u_1)!}\,e^{-x^4} \leqslant (s+3u)! 5^{u} e \leqslant c\,.$$ $\blacksquare$ For a fixed $s \in \mathbb{N}\backslash\{0\}$ there exists a constant $c$ such that for all $u,v \leqslant w_{max}$ and all $x=(x_1,x_2)$ with $x_i \in\mathbb{R}^4$ $$|\partial^v_2 \partial^u_1 \left(x_2 \otimes x^{\otimes s-1}_1 (1-e^{-x^4_1})\right)| \leqslant c \, \left\{\begin{matrix} |x|^{s + 1 -u-v},&u+v \leqslant s ,\\ |x|+1, & otherwise. \end{matrix}\right. \label{eq_331b}$$ #### Proof First let $v \in \{0,1\}$. Using lemma \[lem\_402a\] we obtain for $0\leqslant u \leqslant s$ $$\begin{aligned} |\partial^u_1 (x_2 \otimes x^{\otimes s-1}_1(1-e^{-x^4_1}))|&\leqslant c |x|^{s+1-u}\,,\\ |\partial_2 \partial^u_1 (x_2 \otimes x^{\otimes s-1}_1(1-e^{-x^4_1}))|&\leqslant c |x|^{s-u}\,,\end{aligned}$$ and for $u> s$ we also have $$\begin{aligned} |\partial^u_1 (x_2 \otimes x^{\otimes s-1}_1(1-e^{-x^4_1}))|&\leqslant c|x|\,,\\ |\partial_2 \partial^u_1 (x_2 \otimes x^{\otimes s-1}_1(1-e^{-x^4_1}))|&\leqslant c\,.\end{aligned}$$ Finally, for $v>1$ $\partial^v_2 \partial^u_1 (x_2 \otimes x^{\otimes s-1}_1(1-e^{-x^4_1}))=0$.$\blacksquare$ For $s \in \{0,1\}$ there exists a constant $c$ such that for all $u,v \leqslant w_{max}$ and all $x=(x_1,x_2)$ with $x_i \in\mathbb{R}^4$ $$|\partial^v_2 \partial^u_1 \left(x^{\otimes s}_1(e^{-x^4_1}-e^{-(x_1 +x_2)^4})\right)| \leqslant c \, \left\{\begin{matrix} |x|^{s+1-u-v},&u+v\leqslant s,\\ |x|+1, & otherwise. \end{matrix}\right.\label{eq_402s}$$ #### Proof First let $s=0$. For $u=v=0$ put $y=x_1 +x_2$ and assume $|x_1| \leqslant |y|\leqslant 1$ $$e^{-x^4_1}-e^{-y^4} =e^{-x^4_1}(1-e^{-(y^4-x^4_1)})\leqslant y^4-x^4_1\leqslant |y|\leqslant 2|x|\,.\label{eq_402d}$$ Inequality implies that $$\partial^w e^{-x^4}< e (3w)!5^4.\label{eq_402f}$$ Consequently for $u+v>0$ $$|\partial^v_2 \partial^u_1 (e^{-x^4_1}-e^{-(x_1 +x_2)^4})| \leqslant |\partial^u_1 e^{-x^4_1}|+ |\partial^v_2 \partial^u_1 e^{-(x_1 +x_2)^4})| \leqslant c_2\,.$$ Finally, we consider the case $s=1$. Using and we get $$|x_1(e^{-x^4_1}-e^{-(x_1 + x_2)^4})| \leqslant 2|x|^2\,,$$ $$|\partial_2 (x_1(e^{-x^4_1}-e^{-{x_1 + x_2}^4}))| \leqslant |x||\partial_2 (e^{-x^4_1}-e^{-(x_1 + x_2)^4})| \leqslant c_3|x|\,,$$ $$|\partial_1 (x_1(e^{-x^4_1}-e^{-(x_1 + x_2)^4}))| \leqslant c_4|x|\,,$$ and for $u+v>1$ $$|\partial^u_1 \partial^v_2 (x_1(e^{-x^4_1}-e^{-(x_1 + x_2)^4}))| \leqslant c_5 + |x|c_6\,.$$ $\blacksquare$ Let $0 \leqslant \beta \leqslant 1$, $ x \geqslant 0$ and $$f(x):=\frac{1}{x} (e^{-\beta x}-e^{-x})$$ Then $\forall w \in \mathbb{N}$ $$|\partial^w f(x)| < e \frac{w!}{(1 + x)^{w+1}}. \label{eq_f}$$ #### Proof We have an identity $$\begin{aligned} \partial^w f(x)&=(-1)^w g_w(x),&g_w(x):= \int \limits^1_\beta d \gamma \, \gamma^w e^{-\gamma x},\end{aligned}$$ It follows $$\begin{aligned} 0\leqslant g_w(x) \leqslant& \int \limits^1_0 d \gamma \, \gamma^w e^{-\gamma (x+1)}e^{\gamma}<e \int \limits^1_0 d \gamma \, \gamma^w e^{-\gamma (x+1)} \nonumber\\ &<e \frac{1}{(x+1)^{w+1}}\int \limits^\infty_0 d z \, z^w e^{-z}=e \frac{\Gamma(w+1)}{(x+1)^{w+1}}.\end{aligned}$$ $\blacksquare$ Let $0 \leqslant \beta \leqslant 1$, $x \geqslant 0$ and $$h(x):=\frac{1}{x} (e^{-\beta x^2}-e^{-x^2})$$ Then $\forall w \in \mathbb{N}$, $\forall C>1$ $$|\partial^w h(x)|< w!e(C+1) \frac{(2 \sqrt{e} C)^w}{(1+x)^{w+1}}. \label{eq_h}$$ #### Proof $$\begin{aligned} |\partial^w h(x)|&\leqslant |x\partial^w f(x^2)| + |w\partial^{w-1} f(x^2)|,&f(x^2)&:=\frac{1}{x}h(x). \end{aligned}$$ Using an auxiliary variable $y$ $$\begin{aligned} \frac{\partial^w}{(\partial x)^w} f(x^2)&=\left(\frac{\partial}{\partial x}+2x\frac{\partial}{\partial y}\right)^w f(y)|_{y=x^2}\nonumber \\ &=\sum \limits^{2k\leqslant w}_{k=0} \frac{w! (2k-1)!!}{(w-2k)! (2k)!}\left(2 \frac{\partial}{\partial y}\right)^k \left(2x \frac{\partial}{\partial y}\right)^{w-2k} f(y)|_{y=x^2} \nonumber\\ &=\sum \limits^{2k\leqslant w}_{k=0} \frac{w!}{(w-2k)! k!}(2x)^{w-2k} \left(\frac{\partial}{\partial y}\right)^{w-k} f(y)|_{y=x^2}.\end{aligned}$$ Equation gives $$|x \partial^w f(x^2)|< w! 2^w e \sum \limits^{2k\leqslant w}_{k=0} \frac{1}{k!}\frac{(w-k)!}{(w-2k)!}2^{-2k}\frac{x^{w-2k+1}}{(1+x^2)^{w-k+1}}.$$ With an arbitrary constant $C>1$ we have $$\begin{aligned} \frac{x}{1+x^2}& < \frac{C}{1+x},&(1+x^2)&\geqslant \frac{1}{2}(1+x)^2,&\frac{(w-k)!}{(w-2k)!}&\leqslant w^k.\end{aligned}$$ Consequently, $$|x \partial^w f(x^2)|< \frac{w!Ce (2C)^w}{(1+x)^{w+1}} \sum \limits^{2k\leqslant w}_{k=0} \frac{1}{k!} \left(\frac{w}{2C^2}\right)^k< \frac{w!Ce (2 \sqrt{e} C)^w}{(1+x)^{w+1}}.$$ Similarly, we obtain $$|w \partial^{w-1} f(x^2)|< \frac{2 w! e (2 \sqrt{e} C)^{w-1}}{(1+x)^{w+1}}.$$ $\blacksquare$ Let $f(p^2)$ be a scalar function. Then $$|\prod \limits^{w}_{i=1} \frac{\partial}{\partial p_{\mu_i}} f(p^2)| \leqslant 2^w \sum \limits^{2k \leqslant w}_{k=0} \frac{w!}{(w-2k)! k!} |p|^{w-2k} |\left(\frac{\partial}{\partial p^2}\right)^{w-k} f(p^2)|.\label{eq_df}$$ #### Proof With the aid of an auxiliary variable $y$ $$\prod \limits^w_{i=1} \frac{\partial}{\partial p_{\mu_i}}f(p^2)=\prod \limits^w_{i=1} \left( \frac{\partial}{\partial p_{\mu_i}} + 2p_{\mu_i} \frac{\partial}{\partial y}\right)f(y)|_{y=p^2}.$$ A partial derivative wrt $p_{\mu}$ contributes only if it can be paired with $2p_\mu$ term. Consequently, we can compute the right hand side by considering the possible pairs, $$(\frac{\partial}{\partial p_{\mu_i}} + 2p_{\mu_i} \frac{\partial}{\partial y})(\frac{\partial}{\partial p_{\mu_j}} + 2p_{\mu_j} \frac{\partial}{\partial y}) = 2 \delta_{\mu_i \mu_j}\frac{\partial}{\partial y} + 2p_{\mu_i} 2p_{\mu_j}\left(\frac{\partial}{\partial y}\right)^2.$$ It gives $$\sum \limits^{2k \leqslant w}_{k=0, \pi} \frac{1}{(w-2k)!(2k)!} \sum \limits_{\sigma,\pi^\prime:=\sigma \pi} \frac{1}{2^k k!} \prod \limits^k_{i=1}\left( 2 \delta_{\pi^\prime_i \pi^\prime_{i+1}} \frac{\partial}{\partial y}\right) \prod \limits^w_{i=2k+1}\left(2p_{\pi_i} \frac{\partial}{\partial y}\right),$$ where the outer and inner sums run over $w!$, $(2k)!$ permutations, respectively. Using the inequality $|A_{\Vec{\mu}}B_{\Vec{\nu}}| \leqslant |A_{\Vec{\mu}}| |B_{\Vec{\nu}}|$ we obtain the upper bound. $\blacksquare$ Let $C>1$ $$|\prod \limits^{w}_{i=1} \frac{\partial}{\partial p_{\mu_i}} S^{\Lambda\Lambda_0}(p)|<2 w! e(C+1)\frac{(2^2 e^{\frac{3}{2}} C^2)^w}{(\Lambda+|p|)^{w+2}}.\label{eq_dS}$$ #### Proof We change the variable $x_\mu =p_\mu /\Lambda$ $$\prod \limits^{w}_{i=1} \frac{\partial}{\partial p_{\mu_i}} S^{\Lambda\Lambda_0}(p) = \frac{1}{\Lambda^{2w+2}} \prod \limits^{w}_{i=1} \frac{\partial}{\partial x_{\mu_i}} h(x^2).$$ Equations , yield $$|\partial^{w} h(x^2)|<2^w e(C+1) \sum \limits^{2k\leqslant w}_{k=0} \frac{w!}{k!}\frac{(w-k)!}{(w-2k)!}\frac{x^{w-2k}(2\sqrt{e}C)^{w-k}}{(1+x^2)^{w-k+1}}.$$ $\blacksquare$ Let $C>1$ $$\begin{aligned} |\prod \limits^{w}_{i=1} \frac{\partial}{\partial p_{\mu_i}} C^{\Lambda\Lambda_0}_{\mu \nu}(p)|<2^3 w! e(C+1 + (\xi+1)(C^2 +3))\frac{(2^2 e^{\frac{3}{2}} C^2)^w}{(\Lambda+|p|)^{w+2}}.\label{eq_dC}\end{aligned}$$ #### Proof $$|\partial^w C^{\Lambda\Lambda_0}_{\mu \nu}(p)| \leqslant 4 |\partial^w S^{\Lambda\Lambda_0}(p)| + |\xi -1||\partial^w \frac{p_\mu p_\nu}{p^2}S^{\Lambda\Lambda_0}(p)|.$$ The first term is bounded in . Using $x_\mu =p_\mu /\Lambda$ we have an upper bound for the last term $$x^2 |\partial^w f(x^4)|+2^3 w|x||\partial^{w-1}f(x^4)|+4^2 w(w-1)|\partial^{w-2}f(x^4)|,$$ where $f(x^4)$ is the same as in . Equation \[eq\_f\] gives $$\begin{aligned} |\partial^w f(x^2)|&< \frac{2 w! e (2 \sqrt{e} C)^w}{(1+x)^{w+2}},&|\partial^w f(x^4)|&< \frac{2^3 w! e (2^2 e^{\frac{3}{4}} C^2)^w}{(1+x)^{w+4}}.\end{aligned}$$ Consequently, $$|\partial^w x_\mu x_\nu f(x^4)|<2^3 (C^2 + 3)ew! \frac{(2^2 C^2 e^{\frac{3}{4}})^w}{(1+x)^{w+2}}.$$ $\blacksquare$ For all $p \in \mathbb{R}^4$ there exists a constant $C$ such that $$|\dot{\mathbf{C}}^{\Lambda\Lambda_0}(p)| \leqslant C\frac{1}{\Lambda^3} e^{-\frac{p^2}{\Lambda^2}}.\label{eq_dS2}$$ #### Proof Using the inequality $xe^{-x^2}\leqslant x e^{1-2x}=xe^{1-x}e^{-x}\leqslant e^{-x}$ we obtain $$\begin{aligned} |\dot{S}^{\Lambda\Lambda_0}(p)|&=4\frac{1}{\Lambda^3}xe^{-x^2}\leqslant \frac{4}{\Lambda^3}e^{-x} ,&x:=\frac{p^2}{\Lambda^2},\\ |\dot{C}^{\Lambda\Lambda_0}_{\mu \nu}(p)| &\leqslant 4\frac{1 + |\xi-1|}{\Lambda^3}xe^{-x^2}.\end{aligned}$$ $\blacksquare$ For all $p \in \mathbb{R}^4$ there exists a constant $C$ such that $$|\partial^w_p \partial_{\Lambda_0} \mathbf{C}^{\Lambda\Lambda_0}(p)| \leqslant C\frac{1}{\Lambda_0(\Lambda_0 + |p|)^{2 + \|w\|}}.\label{eq_dS3}$$ #### Proof Let $f=e^{-x^4} x x$ then using $|\partial^w_x|g||\leqslant |\partial^w_x g|$ we have $$\begin{aligned} |\partial^w_p \partial_{\Lambda_0} C^{\Lambda\Lambda_0}(p)|\leqslant&\frac{4(2+|\xi-1|)|\partial^w_x f|}{\Lambda^{\|w\|+3}_0},&|\partial^w_p \partial_{\Lambda_0} S^{\Lambda\Lambda_0}(p)|\leqslant&\frac{4|\partial^w_x f|}{\Lambda^{\|w\|+3}_0},\end{aligned}$$ where introducing $C_1=2^{\frac{3}{2}w_{max}}(4w_{max}+2)!$ $$|\partial^w_x f|\leqslant C_1 e^{-x^4}(|x|^{3\|w\| +2}+1)\leqslant eC_1 e^{-2x^2}(|x|^{3\|w\| +2}+1).$$ Then it is easy to see that for $0\leqslant m\leqslant 3w_{max}+2$ $$\max(e^{-x^2}(x^m+1))\leqslant \max(e^{-x^2}x^m) + 1\leqslant e^{\frac{m}{2}(\log \frac{m}{2} -1)} + 1\leqslant C_2.$$ To go further we need the following inequality for all $y \geqslant 0$ and $k \in \mathbb{N}$ $$\begin{aligned} g_k(y)&\leqslant k!,&g_k(y)&:=(1+y)^k e^{-y}, \label{eq_21dS3}\end{aligned}$$ which is obtained looking for the maximum $\bar{y}_k=k-1$ and using $g_0(\bar{y}_0)=1$. $$g_k(\bar{y}_k)=kg_{k-1}(\bar{y}_k)\leqslant k g_{k-1}(\bar{y}_{k-1}) \implies g_k(y) \leqslant g_k(\bar{y}_k)\leqslant k!$$ Defining $C_3:=e^2 C_1 C_2$ and using inequality we have $$|\partial^w_x f|\leqslant C_3 e^{-2x} \leqslant C_3\frac{2!}{(1 + |x|)^2} \frac{\|w\|!}{(1+x)^{\|w\|}}\leqslant \frac{2\,C_3 \,w_{max}!}{(1 + |x|)^{2+\|w\|}}.$$ $\blacksquare$ Let $\mathcal{C}^{\Lambda\Lambda_0}$ be one of the propagators $S^{\Lambda\Lambda_0}$ or $C^{\Lambda\Lambda_0}_{\mu \nu}$, as defined in , . There are positive constants $c_0$, $c_1$, $d$ such that for all $w \in \mathbb{N}$, $p \in \mathbb{R}^4$, $0<\Lambda \leqslant \Lambda_0$, and with $c_\xi:=c_0 + \xi c_1$, $$\begin{aligned} |\Big(\prod \limits^{w}_{i=1} \frac{\partial}{\partial p_{\mu_i}} \Big)\mathcal{C}^{\Lambda\Lambda_0}(p)|&<\frac{w!\, d^w c_\xi}{(|p|+\Lambda)^{w+2}},&|\dot{\mathcal{C}}^{\Lambda\Lambda_0}(p)|&<\frac{c_\xi}{\Lambda^3}e^{- \frac{p^2}{\Lambda^2} }\,. \label{eq_16C}\end{aligned}$$ #### Proof The statement follows from , , .$\blacksquare$ The functional $\Gamma_{n\leqslant 4}$ {#sec_gamma} ====================================== We expand the generating functional $\tilde{\Gamma}^{0\Lambda_0}({\underline}{A}, {\underline}{c}, {\underline{\bar{c}}})$ of as formal series in ${\underline}{A}, {\underline}{c}, {\underline{\bar{c}}}$. As usual, we adopt the shorthand notation $\Gamma^{0\Lambda_0}(A,\bar{c},c)$ for $\tilde{\Gamma}^{0\Lambda_0}({\underline}{A}, {\underline}{c}, {\underline{\bar{c}}})$. $$\begin{aligned} \Gamma^{0\Lambda_0}&=\Gamma^{0\Lambda_0}_{n\leqslant 4} + \Gamma^{0\Lambda_0}_{n\geqslant 5},&\Gamma^{0\Lambda_0}_{n\leqslant 4}&:=\sum \limits_{n=1}^4 \Gamma^{0\Lambda_0}_n,\end{aligned}$$ where $n$ counts the number of fields. The functionals $\Gamma^{0\Lambda_0}_n$ with $n\leqslant4$ contain both relevant and irrelevant terms. We assume hypothesis \[rc1\]. In general the tensors $\zeta^{\Vec{\phi}}_{\mu_1...\mu_r}$ appearing in the form factors $F^{\Vec{\phi}}_{\mu_1...\mu_r}(\Vec{p})$ are elements of $\mathrm{span}(\{\delta^s {\underline}{p}^{k>0}\}_{r>0})$ where ${\underline}{p}=(p_1,...,p_{n-1})$. 1. [**One-point function**]{}\ There are no local terms that preserve Euclidean invariance and global SU(2) symmetry. It follows that $\Gamma_1=0$. 2. [**Two-point functions**]{} $$\begin{aligned} \Gamma^{0\Lambda_0}_2&= \frac{1}{2}\langle F^{AA}_{\mu\nu}\check{A}^a_\mu \check{A}^a_\nu \rangle + \langle F^{\bar{c}c} \check{\bar{c}}^a\check{c}^a \rangle ,\\ F^{AA}_{\mu\nu}(p)&:=(\delta_{\mu\nu}p^2-p_\mu p_\nu)(\sigma^{-1}_{0 \Lambda_0}(p^2)+\Sigma^{AA}_T(p^2)) \nonumber \\ &+\frac{1}{\xi}p_\mu p_\nu(\sigma^{-1}_{0 \Lambda_0}(p^2)+\Sigma^{AA}_L(p^2)),\\ F^{\bar{c}c}(p)&:=-p^2(\sigma^{-1}_{0 \Lambda_0}(p^2) + \Sigma^{\bar{c}c}(p^2))\,.\end{aligned}$$ We assume that the form factors $\Sigma^{AA}$ and $\Sigma^{\bar{c}c}$ include all loop corrections. Note that for the functional ${{{\mkern2mu\underline{\mkern-2mu \Gamma\mkern-2mu}\mkern2mu }}}^{0\Lambda_0}_2$ we have $$\begin{aligned} {{{\mkern2mu\underline{\mkern-2mu \Gamma\mkern-2mu}\mkern2mu }}}^{0\Lambda_0}_2&= \frac{1}{2}\langle {{\mkern2mu\underline{\mkern-2mu F\mkern-2mu}\mkern2mu }}^{AA}_{\mu\nu}\check{A}^a_\mu \check{A}^a_\nu \rangle + \langle F^{\bar{c}c} \check{\bar{c}}^a\check{c}^a \rangle ,\\ {{\mkern2mu\underline{\mkern-2mu F\mkern-2mu}\mkern2mu }}^{AA}_{\mu\nu}(p)&:=F^{AA}_{\mu\nu}(p) - \frac{1}{2\xi}p_\mu p_\nu\,.\end{aligned}$$ With $p^2=M^2$ substitution of the above definitions into the expressions $\sigma_{0 \Lambda_0} F^{AA}$, $\sigma_{0 \Lambda_0} F^{\bar{c}c}$ appearing in AGE and STI gives $$\sigma_{0 \Lambda_0}(p^2)\,F^{\bar{c}c}(p){\stackrel{\beta}{\sim}}-p^2(1 + \Sigma^{\bar{c}c}(p^2)),$$ $$\sigma_{0 \Lambda_0}(p^2)\,{{\mkern2mu\underline{\mkern-2mu F\mkern-2mu}\mkern2mu }}^{AA}_{\mu\nu}(p){\stackrel{\beta}{\sim}}F^{A A}_{T;\mu \nu}(p)\,,$$ $$F^{A A}_{T;\mu \nu}(p):=(\delta_{\mu \nu}p^2 - p_\mu p_\nu)(1+\Sigma^{AA}_T(p^2))+\frac{1}{\xi} p_\mu p_\nu \Sigma^{AA}_L(p^2). \label{def_aa_t}$$ Using for the functional ${\mathsf{\Gamma}}^{0\Lambda_0}_2$ we have $$\begin{aligned} {\mathsf{\Gamma}}^{0\Lambda_0}_2&= \frac{1}{2}\langle \mathsf{F}^{AA}_{\mu\nu}\check{A}^a_\mu \check{A}^a_\nu \rangle + \langle \mathsf{F}^{\bar{c}c} \check{\bar{c}}^a\check{c}^a \rangle ,\\ \mathsf{F}^{AA}_{\mu\nu}(p)&:=(\delta_{\mu\nu}p^2-p_\mu p_\nu)\Sigma^{AA}_T(p^2)+\frac{1}{\xi}p_\mu p_\nu\Sigma^{AA}_L(p^2)\,,\\ \mathsf{F}^{\bar{c}c}(p)&:=-p^2\Sigma^{\bar{c}c}(p^2)\,.\end{aligned}$$ For marginal terms we obtain $$\begin{aligned} \mathsf{F}^{AA;p_\rho p_\sigma}_{\mu\nu}(p)=&2\delta_{\mu \nu}\delta_{\rho \sigma} r^{AA}_1 + 2(\delta_{\mu \rho} \delta_{\nu \sigma} + \delta_{\nu \rho} \delta_{\mu \sigma}) r^{AA}_2+\zeta^{AA}_{\mu \nu \rho \sigma} \;,\\ \mathsf{F}^{\bar{c}c;p_\rho p_\sigma}(p)=&2\delta_{\rho \sigma}r^{\bar{c}c}(p^2) +\zeta^{\bar{c}c}_{\rho \sigma}\;,\\ r^{AA}_1(p^2):=&\Sigma^{AA}_T(p^2) + p^2 \frac{\partial \Sigma^{AA}_T(p^2)}{\partial p^2}\,,\\ r^{AA}_2(p^2):=&\frac{1}{\xi}\Sigma^{AA}_L(p^2) - \Sigma^{AA}_T(p^2)\,,\\ r^{\bar{c}c}(p^2):=&-\Sigma^{\bar{c}c}(p^2) - p^2 \frac{\partial \Sigma^{\bar{c}c}(p^2)}{\partial p^2}\,.\end{aligned}$$ 3. [**Three-point functions**]{} $$\begin{aligned} \Gamma^{0\Lambda_0}_3=&\langle \epsilon_{abd} F^{AAA}_{\rho\mu\nu} \check{A}^a_\rho \check{A}^b_\mu \check{A}^d_\nu \rangle + \langle \epsilon_{adb} F^{A \bar{c} c}_\mu \check{A}^a_\mu \check{\bar{c}}^b \check{c}^d \rangle,\\ F^{A \bar{c} c}_\mu(k,p,q):=&ip_\mu R^{A\bar{c}c}_1(p,q) + iq_\mu r^{A\bar{c}c}_2(p,q),\\ R^{A\bar{c}c}_1(p,q):=&g+r^{A\bar{c}c}_1(p,q),\\ F^{AAA}_{\rho\mu\nu}(k,p,q):=&i\delta_{\mu\nu}(p_\rho - q_\rho)R^{AAA}(p,q) + i\delta_{\mu\nu}k_\rho \zeta^{AAA}_-(p,q) \nonumber\\ &+i\zeta^{AAA}_{\rho \mu \nu}(p,q),\\ R^{AAA}(p,q):=&\frac{1}{2}g + r^{AAA}(p,q).\end{aligned}$$ Here $R^{AAA}(p,q)$ is a symmetric function whereas $\zeta^{AAA}_-(p,q)$ is antisymmetric. 4. [**Four-point functions**]{} $$\begin{aligned} \Gamma^{0\Lambda_0}_4=&\langle F^{AAAA}_{\sigma \rho \mu \nu} \check{A}^b_\sigma \check{A}^b_\rho \check{A}^a_\mu \check{A}^a_\nu +F^{\bar{c}cAA}_{1, \mu \nu} \check{\bar{c}}^b \check{c}^b \check{A}^a_\mu \check{A}^a_\nu \nonumber \\ &+F^{\bar{c}cAA}_{2, \mu \nu} \check{\bar{c}}^a \check{c}^b \check{A}^a_\mu \check{A}^b_\nu +r^{\bar{c}c\bar{c}c} \check{\bar{c}}^b \check{c}^b \check{\bar{c}}^a \check{c}^a \rangle,\\ F^{\bar{c}cAA}_{n, \mu \nu}:=&\delta_{\mu \nu} r^{\bar{c}cAA}_n +\zeta^{\bar{c}cAA}_{n, \mu \nu},\\ F^{AAAA}_{\sigma \rho \mu \nu}:=&R^{AAAA}_{\sigma \rho \mu \nu}+\zeta^{AAAA}_{\sigma \rho \mu \nu},\\ R^{AAAA}_{\sigma \rho \mu \nu}:=&\frac{1}{2}(\delta_{\mu \rho} \delta_{\nu \sigma}+ \delta_{\mu \sigma} \delta_{\rho \nu})R^{AAAA}_1 +\delta_{\mu \nu} \delta_{\sigma \rho} R^{AAAA}_2 \nonumber\\ &+\frac{1}{2}(\delta_{\mu \rho} \delta_{\nu \sigma} - \delta_{\mu \sigma} \delta_{\rho \nu})\zeta^{AAAA}_-,\\ R^{AAAA}_1:=&-\frac{g^2}{4}+r^{AAAA}_1,\\ R^{AAAA}_2:=&\frac{g^2}{4}+r^{AAAA}_2\end{aligned}$$ Here the terms $$\begin{matrix} r^{AA}_1,&r^{AA}_2,&r^{\bar{c}c},&R^{A\bar{c}c}_1,&r^{A\bar{c}c}_2,&R^{AAA},\\ R^{AAAA}_1,&R^{AAAA}_2,&r^{\bar{c}cAA}_1,&r^{\bar{c}cAA}_2,&r^{\bar{c}c\bar{c}c},&\label{eq_11r} \end{matrix}$$ are scalar functions of momenta, $\Lambda_0$, and $M$. All 11 renormalization constants are fine-tuned by imposing appropriate renormalization conditions. The functionals ${\mathsf{\Gamma}}_{\gamma; n\leqslant 2}$ and ${\mathsf{\Gamma}}_{\omega;n\leqslant 2}$ {#sec_gamma2} ======================================================================================================== With ${\mathsf{\Gamma}}^{0\Lambda_0}_{\varkappa}(p):=\tilde{\delta}_{\varkappa(p)} {\mathsf{\Gamma}}^{0\Lambda_0}|_{\varkappa=0}$, $$\begin{aligned} {\mathsf{\Gamma}}^{0\Lambda_0}_{\gamma^a_\mu; n\leqslant 2}(p)=&R_1 ip_\mu c^a(p)+ g\epsilon^{abs} \langle F^{\gamma A c}_{\mu \nu}| \check{A}^b_\nu\check{c}^s;p \rangle,\\ F^{\gamma A c}_{\mu \nu}(k,q):=&\delta_{\mu \nu}R_2 + \zeta^{\gamma A c}_{\mu \nu}(k,q),\\ \zeta^{\gamma A c}_{\mu \nu}(k,q):=&k_\mu q_\nu \zeta^{\gamma A c}_1 + k_\nu q_\mu\zeta^{\gamma A c}_2 +k_\mu k_\nu \zeta^{\gamma A c}_3 + q_\mu q_\nu \zeta^{\gamma A c}_4,\\ {\mathsf{\Gamma}}^{0\Lambda_0}_{\omega^a; n\leqslant 2}(p)=& \frac{1}{2} g\epsilon^{abs} \langle R_3 |\check{c}^b \check{c}^s;p \rangle.\end{aligned}$$ Here $R_1$, $R_2$, $R_3$ are scalar functions of momenta, $\Lambda_0$, and $M$. Violated STI for $\Lambda>0$ {#sec_rr} ============================ In this section we present, omitting the details of the calculation, the extension of some results of section \[sec\_vsti\] to arbitrary $0< \Lambda < \Lambda_0$. $$\begin{aligned} \tilde{{\mathsf{\Gamma}}}^{\Lambda \Lambda_0}_\beta=&\sigma_{0\Lambda_0}*\Big(\frac{\delta \tilde{{\digamma}}^{\Lambda \Lambda_0}}{\delta {\underline{\bar{c}}}} - \partial \frac{\delta \tilde{{\digamma}}^{\Lambda \Lambda_0}}{\delta \gamma}\Big)\,,\label{628a}\\ \tilde{{\mathsf{\Gamma}}}^{\Lambda \Lambda_0}_1=&\langle\frac{\delta \tilde{{\digamma}}^{\Lambda \Lambda_0}}{\delta {\underline}{A}},\sigma_{0\Lambda_0}*\frac{\delta \tilde{{\digamma}}^{\Lambda \Lambda_0}}{\delta \gamma}\rangle-\langle\frac{\delta \tilde{{\digamma}}^{\Lambda \Lambda_0}}{\delta {\underline}{c}},\sigma_{0\Lambda_0}*\frac{\delta \tilde{{\digamma}}^{\Lambda \Lambda_0}}{\delta \omega}\rangle \nonumber\\ & - \frac{1}{\xi}\langle \partial {\underline}{A},\sigma_{0\Lambda_0}*\frac{\delta \tilde{{\digamma}}^{\Lambda \Lambda_0}}{\delta {\underline{\bar{c}}}}\rangle + \hbar\tilde{\Delta}^{\Lambda \Lambda_0}\,,\label{628b}\\ \tilde{{\digamma}}^{\Lambda \Lambda_0}:=&\tilde{{\mathsf{\Gamma}}}^{\Lambda \Lambda_0} + \frac{1}{2}\langle \tilde{{\underline}{\Phi}}, \tilde{\mathbf{C}}^{-1}_{0 \Lambda_0}\tilde{{\underline}{\Phi}}\rangle\,,\\ \tilde{\Delta}^{\Lambda \Lambda_0}:=&\langle (\sigma_{\Lambda},0,0)\Big(1+\frac{\delta^2 \tilde{{\mathsf{\Gamma}}}^{\Lambda \Lambda_0}}{\delta \tilde{{\underline}{\Phi}} \delta \tilde{{\underline}{\Phi}}} \hat{\mathbf{1}}\tilde{\mathbf{C}}^{\Lambda \Lambda_0}\Big)^{-1}\frac{\delta^2 \tilde{{\mathsf{\Gamma}}}^{\Lambda \Lambda_0}}{\delta \tilde{{\underline}{\Phi}} \delta \gamma }\rangle \nonumber\\ &+\langle (0,\sigma_{\Lambda},0)\Big(1+\frac{\delta^2 \tilde{{\mathsf{\Gamma}}}^{\Lambda \Lambda_0}}{\delta \tilde{{\underline}{\Phi}} \delta \tilde{{\underline}{\Phi}}} \hat{\mathbf{1}}\tilde{\mathbf{C}}^{\Lambda \Lambda_0}\Big)^{-1}\frac{\delta^2 \tilde{{\mathsf{\Gamma}}}^{\Lambda \Lambda_0}}{\delta \tilde{{\underline}{\Phi}}\delta \omega }\rangle \nonumber\\ &-\langle (\sigma_{\Lambda}\partial,0,0)\Big(1+\frac{\delta^2 \tilde{{\mathsf{\Gamma}}}^{\Lambda \Lambda_0}}{\delta \tilde{{\underline}{\Phi}} \delta \tilde{{\underline}{\Phi}}} \hat{\mathbf{1}}\tilde{\mathbf{C}}^{\Lambda \Lambda_0}\Big)^{-1}\frac{\delta {\underline}{c}}{\delta \tilde{{\underline}{\Phi}}}\rangle\,.\end{aligned}$$ Note that in relations and there still appears a convolution with $\sigma_{0\Lambda_0}$ since we have chosen to define the regularized BRST transformation to include a convolution with this function, see . Using the bounds of theorem \[thm\_1\] one can show that $\lim \limits_{\Lambda \to 0} \Delta^{\Lambda \Lambda_0}=0$ at nonexceptional momenta. It follows that $\lim \limits_{\Lambda \to 0} \tilde{{\mathsf{\Gamma}}}^{\Lambda \Lambda_0}_1=\tilde{{\mathsf{\Gamma}}}^{0 \Lambda_0}_1$ at nonexceptional momenta. More information can be found in [@thesis]. List of the renormalization points {#sec_ren} ================================== [cc]{} $X$ $[X]$ $n_X$ $r_{\mathrm{m}}$ $r_X$ ren.p. ---------------------------------------------------- ------- ------- ------------------ --------- ------------------------------------------- $\partial\partial{\mathsf{\Gamma}}^{\bar{c}c}$ 0 2 7 0 $\Lambda=0,\;\vec{q}\in \mathbb{M}^s_2$ $\partial{\mathsf{\Gamma}}_\gamma^{c}$ 0 2 7 2 $\Lambda=0,\;\vec{q}\in\mathbb{M}^{s}_2$ $\partial\partial{\mathsf{\Gamma}}^{AA}$ 0 2 7 4 $\Lambda=0,\;\vec{q}\in\mathbb{M}^{s}_2$ ${\mathsf{\Gamma}}_\omega^{cc}$ 0 3 5 0 $\Lambda=0,\;\vec{q}\in\mathbb{M}^{s}_3$ $\partial_{A}{\mathsf{\Gamma}}^{c\bar{c}A}$ 0 3 5 2 $\Lambda=M,\;\vec{q}=0$ $\partial_{\bar{c}} {\mathsf{\Gamma}}^{c\bar{c}A}$ 0 3 5 2 $\Lambda=0,\;\vec{q} \in \mathbb{M}^s_3$ ${\mathsf{\Gamma}}_\gamma^{cA}$ 0 3 5 2 $\Lambda=0,\;\vec{q}\in\mathbb{M}^{s}_3$ ${\mathsf{\Gamma}}_\gamma^{AA}$ 0 3 5 3 $\Lambda=0,\;\vec{q}\in\mathbb{M}^{s}_3$ $\partial{\mathsf{\Gamma}}^{AAA}$ 0 3 5 4 $\Lambda=0,\;\vec{q}\in\mathbb{M}^{s}_3$ ${\mathsf{\Gamma}}^{\bar{c}c\bar{c}c}$ 0 4 3 0 $\Lambda=M,\;\vec{q}=0$ ${\mathsf{\Gamma}}^{\bar{c}cAA}$ 0 4 3 2 $\Lambda=M,\;\vec{q}=0$ ${\mathsf{\Gamma}}^{AAAA}$ 0 4 3 4${}^*$ $\Lambda=0,\;\vec{q}\in\mathbb{M}^{cp}_4$ $\partial\partial{\mathsf{\Gamma}}_{\beta}^{c}$ 0 2 7 2 $\Lambda=0,\;\vec{q}\in\mathbb{M}^{s}_2$ $\partial\partial\partial{\mathsf{\Gamma}}_1^{cA}$ 0 2 7 4 $\Lambda=0,\;\vec{q}\in\mathbb{M}^{s}_2$ $\partial\partial{\mathsf{\Gamma}}_1^{c\bar{c}c}$ 0 3 5 2 $\Lambda=0,\;\vec{q}\in\mathbb{M}^{s}_3$ $\partial{\mathsf{\Gamma}}_{1,\gamma}^{cc}$ 0 3 5 2 $\Lambda=0,\;\vec{q}\in\mathbb{M}^{s}_3$ $\partial\partial{\mathsf{\Gamma}}_1^{cAA}$ 0 3 5 4 $\Lambda=0,\;\vec{q}\in\mathbb{M}^{s}_3$ ${\mathsf{\Gamma}}_{\beta}^{c\bar{c}c}$ 0 4 3 0 $\Lambda=0,\;\vec{q}\in\mathbb{M}^{s}_4$ ${\mathsf{\Gamma}}_{1,\omega}^{ccc}$ 0 4 3 0 $\Lambda=0,\;\vec{q}\in\mathbb{M}^{s}_4$ ${\mathsf{\Gamma}}_{\beta}^{cAA}$ 0 4 3 2 $\Lambda=0,\;\vec{q}\in\mathbb{M}^{s}_4$ $\partial{\mathsf{\Gamma}}_1^{c\bar{c}cA}$ 0 4 3 2 $\Lambda=0,\;\vec{q}\in\mathbb{M}^{s}_4$ ${\mathsf{\Gamma}}_{1,\gamma}^{ccA}$ 0 4 3 2 $\Lambda=0,\;\vec{q}\in\mathbb{M}^{s}_4$ $\partial{\mathsf{\Gamma}}_1^{cAAA}$ 0 4 3 4${}^*$ $\Lambda=0,\;\vec{q}\in\mathbb{M}^{cp}_4$ ${\mathsf{\Gamma}}_1^{c\bar{c}c\bar{c}c}$ 0 5 1 0 $\Lambda=0,\;\vec{q}\in\mathbb{M}^{s}_5$ ${\mathsf{\Gamma}}_1^{c\bar{c}cAA}$ 0 5 1 $2^*$ $\Lambda=0,\;\vec{q}\in\mathbb{M}^{s}_5$ ${\mathsf{\Gamma}}_1^{cAAAA}$ 0 5 1 4${}^*$ $\Lambda=0,\;\vec{q}\in\mathbb{M}^{cp}_5$ & $X$ $[X]$ $n_X$ $r_X$ ren.p. -------------------------------------------- ------- ------- ------- ------------------------- $\partial{\mathsf{\Gamma}}^{\bar{c}c}$ 1 2 1 $\Lambda=0,\;\vec{q}=0$ $\partial{\mathsf{\Gamma}}^{AA}$ 1 2 3 $\Lambda=0,\;\vec{q}=0$ ${\mathsf{\Gamma}}^{\bar{c}cA}$ 1 3 1 $\Lambda=0,\;\vec{q}=0$ ${\mathsf{\Gamma}}^{AAA}$ 1 3 3 $\Lambda=0,\;\vec{q}=0$ ${\mathsf{\Gamma}}^{\bar{c}c}$ 2 2 0 $\Lambda=0,\;\vec{q}=0$ ${\mathsf{\Gamma}}^{AA}$ 2 2 2 $\Lambda=0,\;\vec{q}=0$ $\partial{\mathsf{\Gamma}}_{\beta}^{c}$ 1 2 1 $\Lambda=0,\;\vec{q}=0$ $\partial\partial{\mathsf{\Gamma}}_1^{cA}$ 1 2 3 $\Lambda=0,\;\vec{q}=0$ ${\mathsf{\Gamma}}_{\beta}^{cA}$ 1 3 1 $\Lambda=0,\;\vec{q}=0$ $\partial{\mathsf{\Gamma}}_1^{c\bar{c}c}$ 1 3 1 $\Lambda=0,\;\vec{q}=0$ ${\mathsf{\Gamma}}_{1,\gamma}^{cc}$ 1 3 1 $\Lambda=0,\;\vec{q}=0$ $\partial{\mathsf{\Gamma}}_1^{cAA}$ 1 3 3 $\Lambda=0,\;\vec{q}=0$ ${\mathsf{\Gamma}}_1^{c\bar{c}cA}$ 1 4 1 $\Lambda=0,\;\vec{q}=0$ ${\mathsf{\Gamma}}_1^{cAAA}$ 1 4 3 $\Lambda=0,\;\vec{q}=0$ ${\mathsf{\Gamma}}_{\beta}^{c}$ 2 2 0 $\Lambda=0,\;\vec{q}=0$ $\partial{\mathsf{\Gamma}}_1^{cA}$ 2 2 2 $\Lambda=0,\;\vec{q}=0$ ${\mathsf{\Gamma}}_1^{c\bar{c}c}$ 2 3 0 $\Lambda=0,\;\vec{q}=0$ ${\mathsf{\Gamma}}_1^{cAA}$ 2 3 2 $\Lambda=0,\;\vec{q}=0$ ${\mathsf{\Gamma}}_1^{cA}$ 3 2 1 $\Lambda=0,\;\vec{q}=0$ List of all terms preserving the global symmetries, with an arbitrary number of $\gamma,\omega$ insertions, and with at most one $\beta$ or $1$ insertion (not both). Notation: $[X]$ is the mass dimension of $X$ (reduced Fourier transform); $r_X$ is the tensor rank; $n_X$ is the total number of fields and sources (not including $1$); $\partial$ stands for a momentum derivative; “ren.p.” stands for “renormalization point”. A $*$ in the rank entry means that the condition $r_{\mathrm{m}}:= 11-2n_X\ge r+1$ is violated for a term $X$: as stated in Lemma \[lem\_monom\] the tensor monomials $\{\delta^s\underline{q}^k\}_{r+1}$ are not linearly independent for $\vec{q}=(q_0,\underline{q})\in\mathbb{M}^{s}_{n_X}$, hence they are not suitable as a basis for the form-factor decomposition of $\partial X$. See lemma \[lem\_rn\] and sections \[st\_cAAAA\], \[st\_cccAA\], \[st\_cAAA\]. List of insertions {#sec_op} ================== [ |Sc|Sc|Sc|Sc|]{} X & \[X\]& $\mathrm{gh}(X)$& def\ $\psi_\mu^a$ & $2$&$1$&\ $\gamma_\mu^a$ & $2$&$-1$&\ $\Omega^a$ &$2$&$2$&\ $\omega^a$ &$2$&$-2$&\ $Q_\rho$ & $5 $&$1$&\ $Q_{\rho\gamma}$ & $3 $&$2$&\ $Q_{\rho\omega}$ & $3 $&$3$&\ $\rho$& $-1$&$-1$&\ $Q_{\beta}$ &$3 $& $1$&\ ${\beta}$&$1$& $-1$&\ List of operators and sources, and their quantum numbers. Notation: $[X]$ stands for the mass dimension of $X$ in position space; the ghost charge of the ghost field is $\mathrm{gh}(c):=1$.
--- abstract: 'In the present work we assess the capabilities of neural networks to predict temporally evolving turbulent flows. In particular, we use the nine-equation shear flow model by Moehlis [*et al.*]{} \[New J. Phys. [**6**]{}, 56 (2004)\] to generate training data for two types of neural networks: the multilayer perceptron (MLP) and the long short-term memory (LSTM) network. We tested a number of neural network architectures by varying the number of layers, number of units per layer, dimension of the input, weight initialization and activation functions in order to obtain the best configurations for flow prediction. Due to its ability to exploit the sequential nature of the data, the LSTM network outperformed the MLP. The LSTM led to excellent predictions of turbulence statistics (with relative errors of $0.45\%$ and $2.49\%$ in mean and fluctuating quantities, respectively) and of the dynamical behavior of the system (characterized by Poincaré maps and Lyapunov exponents). future applications aimed at developing accurate and efficient data-driven subgrid-scale models for large-eddy simulations of more complex wall-bounded turbulent flows, including channels and developing boundary layers.' author: - 'P. A. Srinivasan' - - 'H. Azizpour' - 'P. Schlatter' - 'R. Vinuesa' bibliography: - 'ml\_bib.bib' title: Predictions of turbulent shear flows using deep neural networks --- Artificial neural networks are computational frameworks for learning specific tasks solely from examples, and are a popular method within the general area of machine learning. They were inspired by their biological counterparts, which are organized as hierarchical layers of neurons as discovered by Hubel and Wiesel.[@hubel_wiesel] Although in existence for several decades, the recent success of deep neural networks(DNNs) can be attributed to the increased computational power (mainly through graphics processing units or GPU) and the generation of large-scale datasets which benefit the training of the overparametrized DNNs. As discussed in the recent review by Duraisamy [*et al.*]{},[@duraisamy_et_al] there have been several studies aimed at using machine learning to model turbulent flows, especially in the context of Reynolds-averaged Navier–Stokes (RANS) simulations.[@ling_et_al; @wu_et_al] Other applications to modelling the near-wall region of turbulent flows were reported by Milano and Koumoutsakos,[@milano_koumoutsakos] while Lapeyre [*et al.*]{}[@lapeyre_et_al] and Beck [*et al.*]{}[@beck_et_al] have documented the possibility of using machine learning in designing subgrid-scale (SGS) models for large-eddy simulation (LES). Machine learning has also been applied to flow control,[@kim_nn; @gautier_et_al] development of low-dimensional models,[@shimizu_kawahara] generation of inflow conditions[@fukami_et_al] or structure identification in two-dimensional decaying turbulence,[@jimenez_ml] and as discussed by Kutz[@kutz] DNNs will progressively be more widely used in fluid mechanics in the coming years. In this study we assess the feasibility of using DNNs to predict the temporal dynamics of simple turbulent flows. If such predictions are satisfactory, then DNNs could be employed for instance as SGS models in LES of more complex cases. As a simplified yet fairly complete model for wall-bounded turbulence, we consider the nine-equation model of a shear flow between infinite parallel free-slip walls subjected to a sinusoidal body force developed by Moehlis [*et al.*]{}[@moehlis_et_al] In this model, nine Fourier modes $\mathbf{u}_{j}(\mathbf{x})$ represent the mean profile, streamwise vortices, streaks and their instabilities, as well as the coupling among them. The instantaneous velocity fields in spatial coordinates $\mathbf{x}$ and time $t$ are then given by superposition of these nine modes as: $$\textcolor{black}{\mathbf{\tilde{u}}(\mathbf{x},t) = \sum_{j=1}^9 a_{j}(t) \mathbf{u}_j(\mathbf{x}).}$$ Through Galerkin projection, a system of nine ordinary differential equations (ODEs) for the nine mode amplitudes $a_{j}(t)$ is obtained. In this work, we considered a Reynolds number based on channel full height $2h$ and laminar velocity $U_{0}$ at a distance of $h/2$ from the top wall equal to $Re=400$, and we used $U_{0}$ and $h$ as velocity and length scales, . The ODE model was then employed to produce time series of the nine amplitudes, to be used as training and validation datasets. Note that, as discussed by Moehlis [*et al.*]{},[@moehlis_et_al] the lifetime of the turbulent state is highly sensitive to the initial conditions and domain size. We produced over 10,000 different turbulent time series with a time span of 4,000 time units (where the time unit is defined in terms of $U_{0}$ and $h$), and considered a constant time step between samples of 1 time unit. The following initial conditions were employed for the Fourier modes: $(a_{1}^{0},a_{2}^{0},a_{3}^{0},a_{5}^{0},a_{6}^{0},a_{7}^{0},a_{8}^{0},a_{9}^{0}) = (1,0.07066,-0.07076,0,0,0,0,0)$, as in the work by Kim[@kim_thesis] (here the superscript 0 denotes a value at $t=0$). The value of $a_{4}^{0}$ was randomly perturbed around 0, and if the generated series reached a fixed point or a periodic orbit then it was discarded. We considered a domain with $L_{x}=4 \pi$, $L_{y}=2$ and $L_{z}=2 \pi$, where $x$, $y$ and $z$ are the streamwise, wall-normal and spanwise coordinates. In this study we assess flow predictions obtained with , and use the machine learning software framework developed by Google Research called TensorFlow.[@tensor_flow] First, we consider multilayer perceptrons (MLPs),[@rumelhart1985learning] which are the most basic type of artificial neural network. As shown schematically in Figure \[fig\_MLP\], they consist of two or more layers of nodes (also called neurons), with each node connected to all nodes in the preceding and succeeding layers. Neural network training is based on the back-propagation algorithm,[@rumelhart_et_al] which operates layer by layer and updates the parameters in each layer’s nodes to improve the predictions on the training data If $X$ and $\Psi$ are input and output spaces, and each pair of vectors $(\pmb{\chi},\pmb{\psi})$ is a training example or sample, the objective of the neural network is to find the mapping ${f: X \rightarrow \Psi}$ using the training set such that a loss function $L(f(\pmb{\chi}); \pmb{\psi})$ is minimized. For an MLP, the mapping $f(\pmb{\chi})$ can be decomposed into a sequence of simple transfer functions *i.e.* linear matrix transform followed by an element-wise nonlinear function called activation function (*e.g.* sigmoidal). Figure \[fig\_MLP\] summarizes the basic structure of an MLP, where $l$ is the number of hidden layers, $n$ the number of neurons per hidden layer (which in our case will be constant for all the hidden layers) and $p$ is the number of previous values used to predict the next one. The evaluation of $\pmb{\zeta} = f(\pmb{\chi})$ is done using Algorithm \[algo\_mlp\], which involves the following parameters for each layer $i$: $g_{i}$ are the activation functions (which filter the contributing units in each layer), whereas the weight matrices $\mathbf{W}_i$ and biases $\mathbf{b}_i$ perform a linear transformation from one hidden layer space to the next. The goal of training an MLP is to determine $\mathbf{W}_i$ and $\mathbf{b}_i$ using training data, for the given activation functions. The back-propagation algorithm used to update the weights and biases involves the use of gradient descent, for which a so-called learning rate (step size) needs to be defined. The learning rate can be manually set to a constant value or allowed to decay over time, since an excessively large learning rate may prevent convergence. For all the training runs in this work, a variant of the stochastic gradient descent algorithm called adaptive moment estimation (Adam)[@kingma_ba] is used. Adam has an adaptive learning rate method which is commonly used to train deep networks. set $\pmb{\zeta}_{0} \leftarrow \pmb{\chi}$ return $\pmb{\zeta} \leftarrow \pmb{\zeta}_{l+1}$ The updates for the weights and biases in the algorithm depend on the choice of the loss function, and a standard choice for regression problems is the mean-squared-error. Since training a neural network is essentially fitting a mathematical function to data, it inherently has a risk of overfitting. An overfitted network is one that has achieved very low losses and performs well on the training set, but performs poorly on new data. To overcome this problem, we use a regularization technique which penalizes too large parameters based on their $L_2$ norm. It is also common to set aside a small portion of the training set as a validation set which will not be used during training. While training, one complete pass through the training set is called an epoch. At the end of each epoch, the model is evaluated using the validation set and a validation loss is calculated using the loss function. A common method to avoid overfitting is called early stopping, whereby the training is stopped when the validation loss starts increasing. Other relevant decisions to be made when training deep learning models are the strategy for weight initialization and the choice of activation functions. Since the MLP architecture has a large number of weights, the training algorithm seeks to find a local minimum in a high-dimensional space, and the rate of convergence of this search, among other factors, depends on the initial values of these weights. Although randomly selecting initial weights based on a uniform distribution is a natural choice, there are a few other options that may perform better. For instance, since training a neural network involves a forward pass coupled with back-propagation, it is generally better to prevent successive enlargement or shrinkage of layer outputs. Hence, initializations based on a suitable normal distribution perform better than those based on uniform distributions. A truncated normal distribution, which bounds the random variable both above and below, is a commonly used distribution for most learning tasks. Other variations based on the normal distribution, showing better convergence properties than the truncated normal, were proposed by Glorot and Bengio[@glorot_bengio] and He [*et al.*]{}[@he_et_al] Regarding the activation function, the most widely used is the hyperbolic tangent, although other alternatives (see for instance the work by LeCun [*et al.*]{}[@lecun_et_al]) such as the rectified linear unit (ReLU),[@nair2010rectified] the Leaky ReLU[@maas2013rectifier] and the exponential linear unit (eLU)[@clevert2015fast] are also reported in the literature. For the predictions of the near-wall turbulence model using MLPs, we consider the mean-squared-error as loss function, and define the validation loss as: $$\textcolor{black}{L(f(\pmb{\chi});\pmb{\psi})=\frac{1}{2m} \sum_{j=1}^{m} \left | \psi^{j}-f(\chi^{j}) \right |^{2},}$$ We use $20\%$ of the training data as a validation set, and consider early stopping to avoid overfitting. The various initialization strategies were compared in the study by Srinivasan,[@premsrinivasan] and the one by Glorot and Bengio[@glorot_bengio] was observed to outperform all the others. Therefore, that specific initialization is employed in this study. Srinivasan[@premsrinivasan] also compared the various activation functions, and the hyperbolic tangent provided the best agreement with reference data, therefore it was employed for the MLP predictions reported below. The next step to design an MLP architecture is to decide the number of hidden layers $l$, the number of units per hidden layer $n$ and the dimensionality of the input $d$, which is the number of predicted quantities (9 in our case) multiplied by the number of previous values from each amplitude $a_{j}$ used to predict the next one, denoted by $p$. $$\label{epsilon_eq} \textcolor{black}{\varepsilon_{1}=\frac{1}{(N_{s}-p) a_{1,{\rm lam}} } \sum_{j=p+1}^{N_{s}} \left | a_{1,{\rm tra}}^{j}-a_{1,{\rm pred}}^{j} \right |,}$$ ------------ ------- ------- ------- ------- ------- $p$ $l=1$ $l=2$ $l=3$ $l=4$ $l=5$ \[3pt\] 10 23.34 33.66 25.37 26.01 23.81 50 28.28 20.59 17.66 16.77 22.95 100 18.97 13.58 12.76 17.17 14.42 500 65.21 3.32 6.80 1.43 3.09 ------------ ------- ------- ------- ------- ------- \[table\_epsilon\] After using 1,000 datasets (each of them consisting of a time series spanning 4,000 time units) to train the MLP, the predicted mode amplitudes were employed to reconstruct the velocity fields. These fields were used to calculate the mean velocity profile $\overline{u}(y)$ and the streamwise velocity fluctuations $\overline{u^{2}}(y)$, based on averaging over the two periodic directions $x$ and $z$, in time over the $4,000$ time units of a time series, and then performing an ensemble average over 500 different time series, which were sufficient to ensure statistical convergence.[@premsrinivasan] These profiles were compared with reference statistics obtained from the Moehlis [*et al.*]{}[@moehlis_et_al] model, also based on 500 time series which had not been seen by the MLP during training. We define the relative error between the model and the MLP prediction (denoted by the subindices ‘mod’ and ‘pred’, respectively) for the mean flow as: $$\textcolor{black}{E_{\overline{u}}=\frac{1}{2\ {\rm max}(\overline{u}_{{\rm mod}})} \int_{-1}^{1} \left | \overline{u}_{{\rm mod}}-\overline{u}_{{\rm pred}} \right | {\rm d}y,}$$ where the normalization with the maximum of $\overline{u}$ is introduced to avoid spurious error estimates close to the centerline where the velocity is 0. This error is defined analogously for $\overline{u^{2}}$, and both are reported in Table \[table\_mlp\] for each of the architectures, together with the validation loss defined above. These results show that in general MLPs with higher numbers of hidden layers $l$ lead to better predictions of the amplitudes and also of the turbulence statistics, whereas increasing $n$ does not always lead to better predictions. The best MLP architecture among the ones under study was MLP4, with 5 hidden layers and a total of 90 neurons per layer. Although the agreement with the reference mean velocity profile obtained with MLP4 is acceptable (with a relative error of $3.21\%$), the streamwise velocity fluctuation profile exhibits a much larger deviation with $18.61\%$ relative error. Given the relatively large size of the network, and particularly the high input dimension, we explored alternative DNNs which are able to better exploit the sequential nature of the data as discussed below. -------------- ----- ----- -------------------------- ------------------------------ ----------------------- Architecture $l$ $n$ $E_{\overline{u}}$$[\%]$ $E_{\overline{u^{2}}}$$[\%]$ Validation Loss \[3pt\] MLP1 4 45 1.84 24.91 $3.96 \times 10^{-5}$ MLP2 3 90 10.96 36.16 $4.38 \times 10^{-5}$ MLP3 4 90 7.00 29.04 $3.90 \times 10^{-5}$ MLP4 5 90 3.21 18.61 $3.84 \times 10^{-5}$ MLP5 4 180 5.87 27.85 $3.99 \times 10^{-5}$ -------------- ----- ----- -------------------------- ------------------------------ ----------------------- : Summary of MLP architectures and their performance using 1,000 training datasets. \[table\_mlp\] An alternative to the MLP is the recurrent neural network (RNN), which is a type of neural network specifically suited to learn from sequential data such as time series. These networks have a much closer resemblance to biological neural networks, since they also learn the temporal dynamics of the input sequence. As opposed to MLPs, RNNs map a sequence of data points to another sequence using recurrent connections in time as outlined in Algorithm \[algo\_rnn\]. A standard RNN is a neural network containing a single hidden layer with a feedback loop. The output of the hidden layer in the previous time instance is fed back into the hidden layer along with the current input. An RNN can be thought of as a very deep MLP with one layer for each instance in the input sequence where all layers share the parameters. This sequential feeding of the input data makes the temporal relationship apparent to the network. An RNN is parametrized by three weight matrices ($\mathbf{W}_{h\chi}$, $\mathbf{W}_{hh}$ and $\mathbf{W}_{\zeta h}$) and two biases ($\mathbf{b}_h$ and $\mathbf{b}_{\zeta}$). In Algorithm \[algo\_rnn\], $g_h$ and $g_{\zeta}$ are the hidden and output activation functions, respectively. The loss function of an RNN for a single training example is the sum over $p$ time steps of the squared error. On the other hand, the weights and biases are computed by back-propagation through time.[@werbos] RNNs are particularly difficult to train especially for sequences with long-range temporal dependencies. Since the back-propagation algorithm uses the chain rule to compute the weight corrections, small gradients in the later layers restrict the rate of learning of earlier layers. For long sequences, this gradient can gradually vanish, a fact that prevents the neural network from learning any further. This is called the problem of vanishing gradients. Long short-term memory (LSTM) networks developed by Hochreiter and Schmidhuber[@hochreiter_schmidhuber] use a gating mechanism to actively control the dynamics of the recurrent connections and thereby mitigate the vanishing gradient issue. Thanks to the same gating mechanism, LSTMs can also model longer temporal dependencies than standard RNNs. The LSTM will be the RNN employed in this study. set $\mathbf{h}_0 \leftarrow 0$ The impact of the considered LSTM architecture on the computed turbulence statistics is summarized in Table \[table\_lstm\], where the relative errors $E_{\overline{u}}$ and $E_{\overline{u^{2}}}$ are calculated as above based on ensembles of 500 time series spanning 4,000 time units each. Note that the LSTM1 and LSTM3 architectures have a single layer, while in LSTM2 we consider a second one, and in all the cases we have 90 units per layer. In this first assessment we only considered 100 datasets for training, and it can be observed how even using much less training data, leads to relative errors lower than MLP4 using $p=10$, [*i.e.*]{} an input dimension 50 times lower than that in all the MLPs. --------------- ----- --------- --------- -------------------- -------------------------- ------------------------------ ----------------------------------------- Architecture $p$ $n_{1}$ $n_{2}$ Number of datasets $E_{\overline{u}}$$[\%]$ $E_{\overline{u^{2}}}$$[\%]$ Validation Loss \[3pt\] LSTM1 10 90 – 100 2.36 14.73 $2.0 \times 10^{-8}$ LSTM1 10 90 – 1,000 0.83 3.44 $8.5 \times 10^{-9}$ LSTM1 10 90 – 10,000 0.45 2.49 $5.2 \times 10^{-9}$ LSTM2 10 90 90 100 1.94 6.82 $2.4 \times 10^{-8}$ LSTM3 25 90 – 100 $\textcolor{black}{7.4} \times 10^{-8}$ --------------- ----- --------- --------- -------------------- -------------------------- ------------------------------ ----------------------------------------- : Summary of LSTM architectures and their performance using different numbers of training datasets. \[table\_lstm\] Furthermore, we also assessed the effect of increasing the number of datasets used for training, and calculated the deviations with respect to the reference turbulence statistics as summarized in Table \[table\_lstm\]. The data on this Table allows a direct comparison between the performance of the LSTM1 and MLP4 architectures, both trained with 1,000 datasets: while the LSTM network leads to relative errors of $0.83\%$ and $3.44\%$ in the mean and the fluctuations, the MLP significantly higher errors of $3.21\%$ and $18.61\%$, respectively. The LSTM1 results are in even better agreement with the reference statistics when 10,000 training datasets are employed for training, highlight the excellent predicting capabilities of the LSTM network, One of the most relevant features of the nine-equation model by Moehlis [*et al.*]{}[@moehlis_et_al] is the fact that it contains the features present in the near-wall cycle of wall-bounded turbulence,[@hamilton_et_al] namely the streamwise vortices, the streaks, their instabilities and the nonlinear interactions among the various structures. Instantaneous velocity fields obtained from the LSTM1 architecture are shown in Figure \[fig\_field\], where it can be observed that the network exhibits the streamwise vortices convecting near-wall fluid towards the opposing wall, forming high- and low-speed streaks when the flow is lifted up and pulled down towards the wall, respectively. These streaks become unstable due to a sinusoidal modulation in the spanwise direction, which is visible on the vertical midplane. The close resemblance between the LSTM flow fields and the ones reported by Moehlis [*et al.*]{}[@moehlis_et_al] indicates that the LSTM network reproduces the underlying physics in the nine-equation model, a fact that further justifies the excellent agreement of the turbulence statistics. ![Instantaneous velocity fields reconstructed from the mode amplitudes for the LSTM1 architecture. The colored contours represent the velocity component perpendicular to the visualized plane, and the vectors the velocity components in the plane. The left panel is averaged in the streamwise direction, and the right panel shows the flow in the vertical midplane.[]{data-label="fig_field"}](fig_3_lstm_3a){width="49.00000%"} ![[]{data-label="fig_vort"}](vorticity_plot.pdf){width="95.00000%"} It is important to note that although the LSTM1 network yields excellent predictions of the turbulence statistics and the predicted fields exhibit the most relevant flow structures, it shows deviations in the instantaneous predictions of the mode amplitudes. ![[]{data-label="fig_amp"}](amp_t4000_4){width="100.00000%"} define the relative error in the prediction of the amplitude $a_{1}(t)$ in a single time series consisting of $N_{s}$ samples as: $$\label{e_eq} \textcolor{black}{e_{1}=\frac{1}{ (N_{s}-p) a_{1,{\rm lam}} } \sum_{j=p+1}^{N_{s}} \left | a_{1,{\rm mod}}^{j}-a_{1,{\rm pred}}^{j} \right |,}$$ where also here ‘mod’ and ‘pred’ denote data from the nine-equation model and predictions from the NN, respectively, and $a_{1,{\rm lam}}=1$ is the value for laminar flow. Despite the very low validation loss obtained when considering the LSTM1 network trained with 10,000 datasets ($5.2 \times 10^{-9}$), the relative error in the instantaneous prediction of $a_{1}$ was relatively high ($e_{1}=13.08\%$), and those of the other modes were of comparable magnitude. The reason for this discrepancy lies in the nature of the nine-equation model, which is a dynamical system in which very small perturbations lead to completely different temporal evolutions of the modes. Thus, even if the network is able to reproduce the behavior of the flow very accurately, the instantaneous values differ due to the chaotic nature of the system. In order to further analyze the quality of the predictions, we assessed whether the dynamic behavior of the predicted flow was consistent with the reference model. To this end, we first consider the Poincaré map defined as the intersection of the flow state with the hyperplane $a_{2}=0$ on the $a_{1}-a_{3}$ space (with the additional condition ${\rm d}a_{2}/{\rm d}t <0$). This Poincaré map is a lower-dimensional representation of the system which allows to assess whether the correlation between amplitudes $a_{1}$ and $a_{3}$ (which are the amplitudes of the modes containing the base laminar profile and the streamwise vortices[@moehlis_et_al]) is adequately reproduced by the LSTM network. In Figure \[fig\_dynamical\] (left) we show the probability density function (pdf) of the Poincaré maps constructed from the 500 time series obtained from the LSTM1 prediction and the reference nine-equation model. The similarity between both maps shows that the LSTM network effectively captures the correlation between both amplitudes, suggesting that the interaction between modes $\mathbf{u}_{1}$ and $\mathbf{u}_{3}$ is adequately represented by the network. Additionally, we analyzed the rate of separation among trajectories in the nine-equation model and the LSTM predictions through their Lyapunov exponents. Given two time series 1 and 2, we define the separation of these trajectories as the Euclidean norm in nine-dimensional space: $$\textcolor{black}{\left | \delta \mathbf{A}(t) \right | = \left [ \sum_{i=1}^{9} \left (a_{i,1}(t)-a_{i,2}(t) \right )^{2} \right ]^{1/2},}$$ and denote the separation at $t=t_{0}$ as $ | \delta \mathbf{A}_{0} |$. The initial divergence of both trajectories can be assumed as: $$\textcolor{black}{\left | \delta \mathbf{A}(t') \right | = \exp (\lambda t') \left | \delta \mathbf{A}_{0} \right |,}$$ where $\lambda$ is the so-called Lyapunov exponent and $t'=t-t_{0}$. Here we introduce a perturbation with norm $| \delta \mathbf{A}_{0} | =10^{-6}$ (which corresponds to the order of magnitude of the accuracy of the LSTM1 architecture) at $t_{0}=500$, where all the coefficients are perturbed, and analyze its divergence with respect to the unperturbed trajectory. In Figure \[fig\_dynamical\] (right) we show the evolution of $ | \delta \mathbf{A}(t) |$ with time for both the nine-equation model and the LSTM prediction, ensemble-averaged over 10 initial conditions. The rates of divergence from both reference and prediction are very similar, and the estimated values of $\lambda$ are almost identical: 0.0296 for the reference and 0.0264 for the LSTM. After the initial period of divergence, both the nine-equation model and the network saturate at $t \simeq 1,000$. This again supports the claim that the LSTM prediction reproduces the dynamical behavior of the original nine-equation model, and the discrepancy in instantaneous predictions is due to the chaotic nature of the system. In this study we have illustrated the potential of neural networks to predict the temporal dynamics of . We used the nine-equation model by Moehlis [*et al.*]{}[@moehlis_et_al] to train a number of MLP and LSTM networks, using several architectures, and obtained flow predictions from all of them. While both types of neural network are able to capture the most relevant flow structures present in the flow, the LSTM yields excellent agreement with the reference data, both in terms of turbulence statistics and dynamical behavior of the system. Note that we employed the Moehlis [*et al.*]{}[@moehlis_et_al] model of the near-wall cycle due to its simplicity and flexibility to produce large amounts of training data, thus facilitating the training and testing of the various network architectures. After this initial assessment of the suitability of using neural networks to predict simplified turbulent flows, the next step will be to extend the present machine-learning framework to cases where the governing equations are *not* known, *e.g.* the LES equations arising due to non-trivial (spatial and temporal) filtering, or the generation of (instantaneous) inflow and boundary conditions. Using a standard workstation (Intel(R) Core(TM) i7-4930K CPU at 3.4 GHz), the LSTM1 training with 10,000 time series required around 70 hours. After training the network, it takes around 12 minutes to produce 500 time series, which is the amount of data required to obtain converged statistics. On the other hand, on the same workstation it takes around 6 minutes to produce the same amount of data by integrating the nine-equation model by Moehlis [*et al.*]{}[@moehlis_et_al] Thus, after the initial investment required to train the network, the computer time necessary to predict the flow is only around twice as large as that of resolving a nine-equation model. Furthermore, neural networks, once trained, can be accurately summarized into smaller and more computationally efficient networks which will reduce the time complexity of prediction significantly.[@hubara2017quantized] Consequently, we consider that the computational cost of evaluating the neural network is sufficiently low to constitute an efficient alternative for predicting instantaneous variables as *e.g.* in SGS models. Since the network designs discussed here can also be employed for the prediction of other types of flows, this study may serve as a guideline for the most suitable strategies to perform such predictions. Other natural extensions of the present work include the use of deep neural networks to generate inflow conditions for spatially developing flows (see the study by Jarrin [*et al.*]{}[@jarrin_et_al]), and also to set off-wall boundary conditions in high-Reynolds-number wall-bounded turbulence simulations (as discussed in the work by Mizuno and Jiménez[@mizuno_jimenez]). Due to the fact that a significant fraction of the computational cost is employed to resolve the near-wall region, the use of adequate off-wall boundary conditions may allow to achieve very high Reynolds numbers in turbulence simulations. These two applications highlight the Note that all the TensorFlow setups employed in this work are available online.[@codes] The authors thank Martin Lellep (University of Marburg) for helpful comments on the original manuscript. We also acknowledge the funding provided by the Swedish e-Science Research Centre (SeRC) and the Knut and Alice Wallenberg (KAW) Foundation. Part of the analysis was performed on resources provided by the Swedish National Infrastructure for Computing (SNIC) at PDC and HPC2N.
--- abstract: 'The generation of optical beams with multiple, mutually-coherent orbital-angular-momentum (OAM) modes using phase gratings is analyzed from the perspective of energy distribution and radial mode composition. We show that phase gratings designed with equally-weighted Laguerre-Gauss (LG) modes will generate beams with uneven energy distribution among OAM components. This unwanted outcome cannot be corrected by adjusting the width of the illuminating beam. We propose a way to design phase gratings that will produce a uniform energy distribution among the constituent OAM states after illumination, while minimizing the content of high radial modes. This method is based on a generalized definition for the LG modes that takes advantage of the freedom to select their radial scales.' author: - | Jaime Cisternas, Jaime A. Anguita, Gustavo Funes\ Facultad de Ingeniería y Ciencias Aplicadas, Universidad de los Andes,\ Mons. Alvaro del Portillo 12455, Las Condes, 7620001 Santiago, Chile\ Millennium Institute for Research in Optics (MIRO), Chile title: Enforcing energy balance in coherently superimposed optical vortices --- Introduction ============ Spatial modes with orbital angular momentum (OAM) form an important class of transverse modes and have drawn significant attention for their capability of carrying information on their theoretically unbound state space and the availability of several methods of generation and detection of optical vortices [@Allen1992; @Gibson2004; @YaoPadgett2011; @Wang2012; @Wang2015]. Data aggregation could be achieved in unguided optical communications —from optical interconnects to free-space communications— by using OAM-carrying optical modes in a multiplexing arrangement [@Gibson2004; @Anguita2008; @Wang2012; @Sun2016], in which each state carries the information of a single channel. Data aggregation is also possible through signal modulation in the OAM-state space, by superimposing two or more states, drawn from a defined set, to create a multi-dimensional information symbol [@Rop2012; @Anguita2014; @Djordjevic2017]. Multi-dimensional OAM modulation could be made more robust than OAM multiplexing in the presence of channel distortions, by choosing OAM state combinations (and thus, information symbols) to increase the minimum distance of the set if the characteristics of the communication channel are measured and taken into account. This is particularly meaningful in the context of quantum or classical communications over long, unguided channels in terrestrial and earth-to-satellite links. Several techniques have been proposed to create optical vortices, including mode transformation, astigmatic mode conversion, spiral phase-plates, amplitude and phase gratings, and computer-generated holograms [@Heckenberg1992; @Beijersbergen1993; @Sueda2004; @Guo2005; @Gonzalez2006; @Ando2009; @Berkhout2010; @YaoPadgett2011; @Ruffato2017]. The most common generation method used in current experimental demonstrations consists on passing a zero-order Gaussian beam through a forked grating etched on a transparent material or programmed onto a reflective spatial-light modulator (SLM) [@Wang2012]. If the diffractive element contains a single phase dislocation with state $\ell$, the emerging beam (in the first diffraction order) is not a pure state —due to the amplitude ambiguity of the diffractive element—, and may be expressed as a composition of Laguerre-Gauss (LG) modes with orbital state $\ell$ and different radial states $p$. The emerging beam will be sensitive to the initial conditions of the illuminating beam (i.e., to its diameter and curvature). The electric field of LG modes have a complex rotating phase $\exp(i \ell \phi)$ and an amplitude that depends on the generalized Laguerre polynomial $L_p^\ell (\cdot)$. The latter creates an additional dependence on the radial index $p$ [@Allen1992] which cannot be fully specified in phase-only gratings, thus creating intensity profiles that appear as a collection of concentric rings [@Dennis2009]. The physical meaning of this ‘forgotten’ radial number $p$ has been the subject of recent research [@Karimi2014; @Plick2015]. Appropriate procedures for measuring the LG spectrum of a light beam have been proposed [@Qassim2014; @Geneaux2017]. But equally forgotten were the radial scales of the LG modes and the possibility of freely adjusting the scale of each orthogonal subspace of OAM states. Energy balance among OAM states is critical to achieve good signal-to-noise ratio on each dimension forming the basis of an OAM-based signal modulation scheme in an optical communication link. In this work, we show that in coherently superimposed OAM states using a diffractive element, the emerging energy is not evenly distributed over the constituent states if the element imposing the superposition is designed by adding the electric fields of LG components with uniform amplitude weights and equal design waists. By analyzing the components of the emerging electric field using numerical propagation simulations, we propose and evaluate two better grating designs: one based on a recipe, and another based on a simple minimization method. We foresee applications of multi-vortex beams with uniform energy distribution among component OAM states in optical communications, quantum cryptography, and optical manipulation. ![Examples of two-state OAM superpositions, using (a) $\ell=\{1,3\}$, and (b) $\ell=\{2,6\}$. Each case shows a grating generated with equal design waists $w_0^\mathrm{ref}=1$ mm (left-most picture), a far-field intensity with incident beam waist $w_{i}=1$ mm (central picture), and a second far-field intensity with incident beam waist $w_{i}=2$ mm (right-most picture). The sides of the square sections of the diffraction-grating are $5.12$ mm long.[]{data-label="fig:intpatt1"}](fig1a.pdf){width="65.00000%"} \(a) $\ell=\{1,3\}$\ ![Examples of two-state OAM superpositions, using (a) $\ell=\{1,3\}$, and (b) $\ell=\{2,6\}$. Each case shows a grating generated with equal design waists $w_0^\mathrm{ref}=1$ mm (left-most picture), a far-field intensity with incident beam waist $w_{i}=1$ mm (central picture), and a second far-field intensity with incident beam waist $w_{i}=2$ mm (right-most picture). The sides of the square sections of the diffraction-grating are $5.12$ mm long.[]{data-label="fig:intpatt1"}](fig1b.pdf){width="65.00000%"} \(b) $\ell=\{2,6\}$ Generalized LG basis for orbital state superpositions ===================================================== We refer to a coherent OAM state superposition as the coherent addition of two or more optical vortices’ electric fields, each with distinct, integer OAM state, sharing the same optical axis. Unlike incoherent OAM superpositions, whose intensity patterns are concentric rings, the intensity profile of a coherent OAM superposition is not ring-shaped, and may take a great variety of distributions.. A conventional recipe to design a grating capable of diffracting a two-state OAM superposition is the following: extract the phase term of the electric field resulting from the addition of the complex amplitudes of two LG modes that use the same zero-order Gaussian waist diameter [@Anguita2014]. This bidimensional phase pattern can be used as a digital hologram if programmed to a SLM. Examples of two-state phase patterns are the forked gratings in Fig. \[fig:intpatt1\], where the gray scale represents phase values from $0$ to $2\pi$ radians. A forked grating will show one or more off-center singularities, depending on the orbital states of the composition. For two states, the number of off-center singularities is equal to the absolute value of the orbital numbers’ difference. This rule assumes that the design modes have the same zero-order waist. A detailed description follows. Other techniques utilize the grating depth for controlling the amplitude of the beam with phase-only SLMs at the expense of a large fraction of laser power being diffracted towards unwanted orders [@Clark2016]. Here we restrict our study to *blazed* gratings that basically diffract a single order. Now most of the ideas of our approach, such as the combinations of LG modes of different radial scales, can be combined with schemes that modify the amplitude. Let $\Phi_{\ell,p}(r,\phi,z;w_\ell)$ be the electric field —in cylindrical coordinates $r$, $\phi$, and $z$— of a LG mode of orbital number $\ell$ (an integer which we also refer to as OAM state or topological charge), radial order $p$ (a nonnegative integer), and beam waist $w_\ell$. At a propagation distance $z$ from the beam waist, the electric field is given by $$\begin{aligned} \label{eq:LGfield} \Phi_{\ell,p}(r,\phi,z;w_\ell)& \triangleq \sqrt{\frac{2p!}{\pi(p+|\ell|)!}} \,\frac{1}{w(\ell;z)} \left(\frac{r\sqrt{2}}{w(\ell;z)}\right)^{|\ell|} L_p^{|\ell|} \left[\frac{2r^2}{w^2(\ell;z)}\right] \exp\left[\frac{-r^2}{w^2(\ell;z)}\right] \nonumber \\ &\quad \times \exp(-i\ell\phi) \exp\left[\frac{-ikr^2 z}{2(z^2+z_R^2)}\right] \exp\left[i(2p+|\ell|+1)\tan^{-1}\frac{z}{z_R}\right] \:,\end{aligned}$$ where $w(\ell;z)=w_\ell \sqrt{1+(z/z_R)^2}$ is the beam waist at distance $z$; $L_p^\ell(\cdot)$ designates the generalized Laguerre polynomial; $z_R=\pi w_\ell^2/\lambda$ is the Rayleigh range; $\lambda$ is the optical wavelength; and $k=2\pi/\lambda$ is the propagation constant. In the previous definition one could also introduce an arbitrary translation $z_\ell$ along the propagation axis. Orthogonality of generalized LG modes ------------------------------------- The *traditional* LG basis is defined using $w_\ell=w_0$, that is, a fixed spatial scaling for all modes. It is well known that traditional LG modes are mutually orthogonal for distinct, integer values of $\ell$ and $p$. That is, $$\label{eq:orthogon} \langle \Phi_{\ell_1,p_1}(r,\phi,z; w_0) , \Phi_{\ell_2,p_2}(r,\phi,z; w_0) \rangle = \delta_{\ell_1,\ell_2} \delta_{p_1,p_2}$$ for any fixed value of $z$, where $\langle\cdot,\cdot\rangle$ is the inner product defined as the integration in the transverse plane, $$\label{eq:innerprod} \langle \zeta(r,\phi) , \varphi(r,\phi) \rangle \triangleq \int_0^{2\pi} \int_0^\infty \zeta (r,\phi) \overline{\varphi} (r,\phi)\, r\, dr\, d\phi.$$ Now the *new* LG basis is defined by freely choosing the waists $w_{\ell}$ for each $\ell$. Orthogonality between modes in the new basis is maintained. That is, $$\label{eq:orthogon2} \langle \Phi_{\ell_1,p_1}(r,\phi,z;w_{\ell_1}),\Phi_{\ell_2,p_2}(r,\phi,z;w_{\ell_2})\rangle = \delta_{\ell_1,\ell_2} \delta_{p_1,p_2}$$ This can be verified considering two cases: (i) if the two modes have the same angular momenta $\ell_1=\ell_2$ (and, by construction, $w_{\ell_1}=w_{\ell_2}$), orthogonality holds if $p_1 \ne p_2$, due to the orthogonality between the generalized Laguerre polynomials $L_{p_1}^\ell$ and $L_{p_2}^\ell$; (ii) if the angular momenta are different $\ell_1 \ne \ell_2$, the inner product of the fields is zero, because the angular integration of $\exp[-i(\ell_1-\ell_2)\phi]$ vanishes, regardless of the choice of $w_{\ell_1}, w_{\ell_2}$, $p_1, p_2$. In the Appendices we review in more detail this derivation. In contrast, Vallone [@Vallone2017] studied the loss of orthogonality between two LG modes of equal state $\ell$ but *different* waists. The following inner product was considered: $\langle \Phi_{\ell,p_1}(r,\phi,z;w_1),\Phi_{\ell,p_2}(r,\phi,z;w_2)\rangle$, and expressed in terms of a hypergeometric polynomial. LG bases constructed with a set of distinct values of $w_0, w_1, w_{-1}, w_2, w_{-2},\ldots$ will span the same space of superpositions of aligned vortices (as explained in the Appendices), even though their field components —for any given $\ell$— had different waists. Both traditional and new LG bases satisfy the *completeness* requirement: any superposition of aligned vortices can be written as a finite or converging sum of LG modes, regardless of the set of $w_\ell$. However, choosing a ‘good’ set can make convergence faster (i.e., with fewer terms). The influence of basis scale on mode spectrum was also studied in [@Schulze2012; @Vallone2017]. We take advantage of the degrees of freedom provided by the beam waists $w_\ell$ in two ways: (i) in decomposing the reference field $u^\mathrm{ref}$ used to generate the grating and (ii) in decomposing the coherent superposition $u^\mathrm{s}$ generated by the grating, as illuminated by a zero-order Gaussian beam. ![Examples of two-state OAM superpositions, using (a) $\ell=\{1,3\}$, and (b) $\ell=\{2,6\}$. Each case shows a grating generated with equal design radii $r_\ell^\mathrm{ref}$ (left-most picture), a far-field intensity with incident beam waist $w_{i}=1$ mm (central picture), and a second far-field intensity with incident beam waist $w_{i}=2$ mm (right-most picture). In (a) we used waists $w_1^\mathrm{ref}=1$ mm, $w_3^\mathrm{ref}=0.577$ mm, and in (b) waists $w_2^\mathrm{ref}=1$ mm, $w_6^\mathrm{ref}=0.577$ mm. The sides of the square sections of the diffraction-grating are $5.12$ mm long.[]{data-label="fig:intpatt2"}](fig2a.pdf){width="65.00000%"} \(a) $\ell=\{1,3\}$\ ![Examples of two-state OAM superpositions, using (a) $\ell=\{1,3\}$, and (b) $\ell=\{2,6\}$. Each case shows a grating generated with equal design radii $r_\ell^\mathrm{ref}$ (left-most picture), a far-field intensity with incident beam waist $w_{i}=1$ mm (central picture), and a second far-field intensity with incident beam waist $w_{i}=2$ mm (right-most picture). In (a) we used waists $w_1^\mathrm{ref}=1$ mm, $w_3^\mathrm{ref}=0.577$ mm, and in (b) waists $w_2^\mathrm{ref}=1$ mm, $w_6^\mathrm{ref}=0.577$ mm. The sides of the square sections of the diffraction-grating are $5.12$ mm long.[]{data-label="fig:intpatt2"}](fig2b.pdf){width="65.00000%"} \(b) $\ell=\{2,6\}$ Reference field for the OAM superposition ----------------------------------------- The reference complex field to be used as the interference pattern may be expressed as $$\label{eq:reference} u^\mathrm{ref}(r,\phi) = \sum_{\ell \in \mathbb{Z}} c_{\ell}^\mathrm{ref} \Phi_{\ell,0}(r,\phi; w_\ell^\mathrm{ref}),$$ in which we have chosen modes with $p=0$, $z=0$, and the set $\{c_{\ell}^\mathrm{ref}\}$ are arbitrary complex amplitude coefficients. The constituent LG fields $\Phi_{\ell,0}$ have distinct beam waists $w_{\ell}^\mathrm{ref}$. Choosing a set of distinct values for $\{w_\ell^\mathrm{ref}\}$ modifies the position of dislocations in the phase pattern and the energy distribution in the diffracted beam, even if $\{c_{\ell}^\mathrm{ref}\}$ were all equal. This is apparent by comparing the examples of superpositions $\{1,3\}$ and $\{2,6\}$ of Figs. \[fig:intpatt1\] and \[fig:intpatt2\]: equal design radius \[$w_\ell^\mathrm{ref}=1$ mm in Fig. \[fig:intpatt1\]\] show a clearer separation of dislocations than the case with different design radius \[$w_\ell^\mathrm{ref}=\{1,0.577\}$ mm in Fig. \[fig:intpatt2\]\]. The radius of the illuminating beam also impacts the diffracted mode: a simple observation of the gratings in both figures reveals that a certain minimum beam radius is required to ensure the illumination of all phase dislocations. If a certain LG superposition $u^\mathrm{s}$ is sought after at the far-field, a reference field $u^\mathrm{ref}$ and a proper incident beam radius must be chosen. As it will be shown later, the mode decomposition of $u^\mathrm{ref}$ and $u^\mathrm{s}$ are usually quite different, and the waists $\{w_\ell\}$ used for analysis may be quite different from the waists $\{ w_\ell^\mathrm{ref} \}$ of the reference field. Diffracted field of the OAM superposition ----------------------------------------- The light field $u^s$ that emerges from the grating may be expressed as a superposition of mutually-orthogonal LG modes. That is, $$\label{eq:decomposition} u^\mathrm{s}(r,\phi,z) = \sum_{\ell \in \mathbb{Z}} \sum_{p \in \mathbb{Z}^\ast} c_{\ell,p}(w_\ell) \Phi_{\ell,p}(r,\phi,z; w_\ell),$$ where the complex coefficient $c_{\ell,p}(w_\ell)$ —for each $\ell$ and $p$ in the sum— is given by the scalar projection $$\label{eq:c_coeff} c_{\ell,p}(w_{\ell}) = \langle u^\mathrm{s}(r,\phi,z), \Phi_{\ell,p}(r,\phi,z; w_\ell) \rangle,$$ using definition given in Eq. (\[eq:innerprod\]). The projection coefficients will not depend on $z$, as they are fully determined at the grating’s exit surface. Thus, OAM components may be analyzed right after the grating on the corresponding diffraction order, even though the beam’s intensity had not developed to its far-field distribution. The fraction of energy observed on each LG component mode, represented by $|c_{\ell,p}(w_\ell)|^2$, will depend on the choice of the waists $w_\ell$. From the stand point of an analyzing device that observes $u^{s}$ without [*a-priori*]{} information, the choice of $w_\ell$ is arbitrary [@Schulze2012]. An example of this important —and somehow, counterintuitive— characteristic, is described below.\ [*Example 1*]{} We numerically compare the energy distribution among OAM states $\ell=\{1,3\}$ in a beam $u^\mathrm{s}$ generated by two different digital forked gratings. The gratings are $2048\times 2048$ pixels wide, with spatial resolution $\Delta x=\Delta y=10$ $\mu$m, average fringe pitch $\Lambda=250$ $\mu$m and wavelength $\lambda=660$ nm. 1. Base case: a phase grating is designed with a reference $u^\mathrm{ref}_b$ using waists $w_1^\mathrm{ref}=w_3^\mathrm{ref}=1$ mm and equal amplitude weights, $c_1^\mathrm{ref}=c_3^\mathrm{ref}=1$. 2. A second grating with a reference $u^\mathrm{ref}_r$, in which $w_1^\mathrm{ref}=1$ mm. Waists $w_3^\mathrm{ref}$ is chosen such that the radius $r_\ell^\mathrm{ref} \triangleq w_\ell^\mathrm{ref}\sqrt{|\ell|/2}$ [@Padgett2015] satisfies the condition $r_1^\mathrm{ref}=r_3^\mathrm{ref}$. That is, the LG components would have intensity rings of equal radii, if they were generated independently. Both gratings are illuminated using a beam width $w_\mathrm{i}=1$ mm and only the first diffraction order is analyzed. Fig. \[fig:radial\] depicts the measured fraction of energy $|c_{\ell,0}(w_\ell)|^2$ (using Eq. (\[eq:c\_coeff\])) on the lowest radial order ($p=0$) as a function of $w_\ell$, for each $\ell$. Designs (i) and (ii) correspond to Fig. \[fig:radial\] (a) and Fig. \[fig:radial\] (b), respectively. ![Dependence of $|c_{\ell,0}|^2$ on the choice of waist $w_\ell$ in the observed superposition $u^\mathrm{s}$, for a grating programmed with states $\ell=\{1,3\}$. (a) $u^\mathrm{ref}$ uses equal design waists $w_1^\mathrm{ref}=w_3^\mathrm{ref}= 1$ mm. (b) $u^\mathrm{ref}$ uses waists $w_1^\mathrm{ref}=1\ \mathrm{mm}, w_3^\mathrm{ref}= 0.577\ \mathrm{mm}$. In both cases, the illuminating Gaussian beam has radius $w_\mathrm{i}=1$ mm.[]{data-label="fig:radial"}](fig3a.pdf "fig:"){width="60.00000%"}\ (a)\ ![Dependence of $|c_{\ell,0}|^2$ on the choice of waist $w_\ell$ in the observed superposition $u^\mathrm{s}$, for a grating programmed with states $\ell=\{1,3\}$. (a) $u^\mathrm{ref}$ uses equal design waists $w_1^\mathrm{ref}=w_3^\mathrm{ref}= 1$ mm. (b) $u^\mathrm{ref}$ uses waists $w_1^\mathrm{ref}=1\ \mathrm{mm}, w_3^\mathrm{ref}= 0.577\ \mathrm{mm}$. In both cases, the illuminating Gaussian beam has radius $w_\mathrm{i}=1$ mm.[]{data-label="fig:radial"}](fig3b.pdf "fig:"){width="60.00000%"}\ (b) For both design cases, the measured energy varies significantly with $w_\ell$ and features a maximum. For the base case \[Fig. \[fig:radial\] (a)\], a maximum of $|c_{1,0}|^2=0.811$ occurs at $w_1=0.62$ mm for $\ell=1$, and a maximum of $|c_{3,0}|^2=0.101$ at $w_3=0.83$ mm for $\ell=3$. This is obviously an unwanted outcome, as one component in the superposition has roughly 8 times the energy of the other. For the design with equal $r^\mathrm{ref}$ \[Fig. \[fig:radial\] (b)\], maxima occur at $w_1=0.710$ and at $w_3=0.50$ mm, respectively, and with a significantly better energy distribution. The values for all maxima and the values of $w_\ell$ at which they occur are listed in Table \[table:firstexample\]. ------------------------ ------- --------------- ---- ------- --------------- $w_1$ $|c_{1,0}|^2$    $w_3$ $|c_{3,0}|^2$ base case 0.620 0.811 0.83 0.101 equal $r^\mathrm{ref}$ 0.710 0.337 0.50 0.395 ------------------------ ------- --------------- ---- ------- --------------- : *Example 1*. Fraction of energy on each diffracted OAM component (using Eq.(\[eq:c\_coeff\])) at their maximum values and their corresponding $w_\ell$ (given in mm).[]{data-label="table:firstexample"} The generated light beam acquires the waist not from the grating but from the spatial distribution of the illuminating mode, and the width follows $w_\ell = r_\mathrm{rms} \sqrt{2/(|\ell|+1)}$ [@Padgett2015]. For a superposition of modes, the widths will depend on the location of phase singularities in the grating. Using a different width on the illumination beam will produce a different set of plots, but the behavior described above prevails.\ [*Example 2*]{} A coherent superposition of three OAM states, namely, $\ell=\{2,6,10\}$, is made using the same numerical conditions of the first example. The gratings are defined as follows: 1. Base case. Reference $u^\mathrm{ref}_b$ uses $w_2^\mathrm{ref}=w_6^\mathrm{ref}=w_{10}^\mathrm{ref}=1$ mm and equal amplitude weights. The grating is illuminated with a Guassian beam with $w_\mathrm{i}=1$ mm. 2. Reference $u^\mathrm{ref}_r$ uses $w_2^\mathrm{ref}=1$ mm and satisfies the condition $r_2^\mathrm{ref}=r_6^\mathrm{ref}=r_{10}^\mathrm{ref}$, as defined in *Example 1*. Beam width $w_\mathrm{i}=1$ mm is used for illumination. Energy distribution and waists are detailed in Table \[table:secondexample\]. Again, the equal-radii recipe gives a large improvement in energy distribution over the base case, although a significant contrast between the components still remains. ------------------------ ------------------ --------------- --- ------------------ --------------- --- --------------------- ---------------- $w^\mathrm{s}_2$ $|c_{2,0}|^2$   $w^\mathrm{s}_6$ $|c_{6,0}|^2$   $w^\mathrm{s}_{10}$ $|c_{10,0}|^2$ base case 0.57 0.835 0.76 0.0126 0.85 0.0005 equal $r^\mathrm{ref}$ 0.41 0.573 0.46 0.0603 0.41 0.149 ------------------------ ------------------ --------------- --- ------------------ --------------- --- --------------------- ---------------- : *Example 2*. Distribution of energy in a three-state superposition and the corresponding optimal waists (in mm).[]{data-label="table:secondexample"} Examples 1 and 2 show that to evaluate the merits of different phase gratings, it is key to identify the scales $w_\ell$ of the generated OAM states and define the analysis LG basis accordingly so most of the energy is concentrated on the lowest-order radial mode for every $\ell$. This choice is also used to seek optimal design waists to equalize the energy of the coherent OAM superposition, as it is described in the following section. Grating design for equal-energy superpositions ============================================== Acknowledging the impact of the observation basis on energy distribution, an independent metric needs to be used to analyze the generated modes. We define the *eigen waist* of mode $\ell$ present in the diffracted beam $u^\mathrm{s}$ as $$\label{eq:properw} w_\ell^\mathrm{s} \triangleq \mathrm{arg} \max_{w_\ell \in \mathbb{R}^+} |c_{\ell,0}(w_\ell)|^2,$$ that is, we select the value of $w_\ell$ that delivers the maximum contribution of $|c_{\ell,0}(w_\ell)|^2$ for each orbital mode in $u^\mathrm{s}$, as computed with Eq. (\[eq:c\_coeff\]). This choice is justified by the indetermination of the basis that analyzes $u^\mathrm{s}$ in terms of $p$. A similar approach was used in Ref. [@Vallone2017] but with a different goal in mind: to optimize the expansion of a generic beam in LG modes. By adjusting the waists $\{w_\ell^\mathrm{ref}\}$ and the amplitudes $\{c_\ell^\mathrm{ref}\}$ of the superimposed orbital components of the grating —as described by Eq. (\[eq:reference\])— one can design an ‘optimal’ grating, in the sense that the diffracted beam satisfies the following criteria. Our goal is to balance the energy distribution among OAM states in the diffracted beam, while preserving similar eigen waists $w^s_\ell$, so that all constituent orbital modes be diffracted at a similar rate. The latter prevents increasing the optics size at the analyzer. We propose the following optimization problem for a two-state superposition: $$\begin{aligned}\label{eq:firstmin} & \underset{w_{\ell_1}^\mathrm{ref}, w_{\ell_2}^\mathrm{ref}, c_{\ell_1}^\mathrm{ref}, c_{\ell_2}^\mathrm{ref}} {\text{minimize}} \,\,\, (|c_{\ell_1,0}| -|c_{\ell_2,0}|)^2 + \eta (w_{\ell_1}^s- w_{\ell_2}^s)^2 \\ & \text{subject to:}\\ & (i) \,u^\mathrm{ref}=c_{\ell_1}^\mathrm{ref} \Phi_{\ell_1,0}(\cdot; w_{\ell_1}^\mathrm{ref}) + c_{\ell_2}^\mathrm{ref} \Phi_{\ell_2,0}(\cdot; w_{\ell_2}^\mathrm{ref}) \\ & (ii)\text{ $u^\mathrm{s}$ as generated by grating built from $u^\mathrm{ref}$} \\ & (iii)\text{ $w^\mathrm{s}_{\ell_1},w^\mathrm{s}_{\ell_2},c_{\ell_1,0},c_{\ell_2,0}$ extracted from $u^\mathrm{s}$.} \end{aligned}$$ The first term of the objective function seeks to balance energy among states and the second seeks to maintain similar diffraction angles (to constrain the optics). The constant $\eta>0$ is used as a normalization factor, to account for the differences in magnitude of $w_\ell^s$ and $c_{\ell,0}$. We set $\eta=10^5$ in the examples presented in the article. A smaller value of $\eta$ would produce a closer similarity between the energies of the modes, but at the cost of a larger difference between their waists. Similarly, for $N$ states one can pose the following problem: $$\begin{aligned}\label{eq:secondmin} & \underset{ \{ w_{\ell_k}^\mathrm{ref}, c_{\ell_k}^\mathrm{ref} \}_{k=1\ldots N}} {\text{minimize}} \sigma( \{ |c_{\ell_k,0}| \}_{k=1\ldots N} ) + \eta \sigma( \{ w_{\ell_k}^s \}_{k=1\ldots N} ) \\ & \text{subject to: } (i)\ u^\mathrm{ref}=\sum\nolimits^{N}_{k=1} c_{\ell_k}^\mathrm{ref} \Phi_{\ell_k,0}(\cdot; w_{\ell_k}^\mathrm{ref}) \\ & (ii)\text{ $u^\mathrm{s}$ as generated by grating built from $u^\mathrm{ref}$,} \\ & (iii)\text{ $\{w^\mathrm{s}_{\ell_k},c_{\ell_k,0} \}$ extracted from $u^\mathrm{s}$,}\\ &\text{where $\sigma(\{x_k\})$ is the variance of numbers $x_k$}. \end{aligned}$$ Superp. $w^\mathrm{ref}_{\ell_1}$ $c^\mathrm{ref}_{\ell_1}$ $w^\mathrm{ref}_{\ell_2}$ $c^\mathrm{ref}_{\ell_2}$ $w^\mathrm{ref}_{\ell_3}$ $c^\mathrm{ref}_{\ell_3}$ $w^\mathrm{ref}_{\ell_4}$ $c^\mathrm{ref}_{\ell_4}$ ------------- --------------------------- --------------------------- --------------------------- --------------------------- --------------------------- --------------------------- --------------------------- --------------------------- {1,3} 0.95 1.0 0.58 1.0 {2,6} 0.79 1.0 0.41 0.86 {2,6,10} 0.62 1.0 0.32 0.97 0.36 1.15 {1,5,9} 0.11 1.0 0.41 0.98 0.37 1.07 {1,-2,4,-5} 1.0 1.0 0.72 1.08 0.50 0.93 0.47 0.98 : Grating’s optimal design parameters $\{w^\mathrm{ref}_\ell\}$ (in mm) and $\{c_\ell^\mathrm{ref}\}$ to reach a balanced state energy in $u^\mathrm{s}$, assuming $w_i=1$ mm.[]{data-label="table:optimal_grating"} Superp. $w^\mathrm{s}_{\ell_1}$ $|c_{\ell_1,0}|^2$ $w^\mathrm{s}_{\ell_2}$ $|c_{\ell_2,0}|^2$ $w^\mathrm{s}_{\ell_3}$ $|c_{\ell_3,0}|^2$ $w^\mathrm{s}_{\ell_4}$ $|c_{\ell_4,0}|^2$ ------------- ------------------------- -------------------- ------------------------- -------------------- ------------------------- -------------------- ------------------------- -------------------- {1,3} 0.51 0.61 0.51 0.61 {2,6} 0.31 0.59 0.37 0.57 {2,6,10} 0.26 0.48 0.27 0.48 0.38 0.48 {1,5,9} 0.87 0.47 0.32 0.45 0.37 0.46 {1,-2,4,-5} 0.25 0.41 0.49 0.39 0.41 0.41 0.48 0.39 : Eigen waists and energies obtained from the optimal grating designs of Table \[table:optimal\_grating\], illuminated with $w_i=1$ mm.[]{data-label="table:optimal_energy"} We solve Eqs. (\[eq:firstmin\]) and (\[eq:secondmin\]) with the Nelder-Mead algorithm implemented in SciPy, a Python library. Although the functional assumes a rather simple expression, the connection between the design parameters and the beam features involves the numerical propagation of a beam into the far field. The Nelder-Mead algorithm [@NelderMead] is a general-purpose multi-variable minimization scheme that does not require the specification of partial derivatives and is provided in standard numerical libraries. Other similar algorithms should work as well. In our numerical experiments we used a variety of selections for $\{ \ell_i\}_{i=1 \ldots N}$. In these experiences 20 iterations were enough for reaching convergence with a relative tolerance of $10^{-5}$. Digital gratings are generated using a fixed resolution of $\Delta x=10$ $\mu$m and grating pitch $\Lambda=250$ $\mu$m. We fix the illumination beam waist to $w_i=1$ mm. The optimal parameters found for several combination of states are listed in Table [\[table:optimal\_grating\]]{}. Table [\[table:optimal\_energy\]]{} presents the resulting eigen waists and energy fractions, as produced by the optimal gratings whose parameters are given in Table [\[table:optimal\_grating\]]{}. In all cases evaluated, the minimization reached a solution that produced a diffracted beam $u^\mathrm{s}$ with a good compromise between energy balance and similarity of waists. In contrast to the results of Tables \[table:firstexample\] and \[table:secondexample\], the balance of energy at each eigen waist is almost perfect with the proposed optimization method. In some of the examples, the ‘optimal’ energies of the modes are not exactly equal. This is a result of using the functionals (\[eq:firstmin\]) and (\[eq:secondmin\]) that combine the similarity of the energies and the similarity of the waists. The optimal superposition will show a compromise that can be fine-tuned by adjusting the parameter $\eta$. OAM states with negative sign may be added to the superpositions presented in Table [\[table:optimal\_grating\]]{} by simply using the waists and amplitudes of their positive counterparts. No further optimization is required. Figure \[fig:best\_gratings\] shows four sample intensity profiles as seen at the far field of $u^\mathrm{s}$ for superpositions $\{1,3\}$, $\{2,6\}$, $\{1,5,9\}$ and $\{1,-2,4,-5\}$ with balanced energy. As a practical note, in any given system with predefined states, acquiring knowledge of the optimal $u^\mathrm{ref}$ for the grating and the eigen waists for the analyzer (e.g., the receiver in a communication system) is a one-time operation, and therefore, complexity is not an issue. In this work our first motivation was to enhance the balance of energies of the OAM modes but keeping the waists also similar. This method is presented as a proof of principle: specific applications of OAM superpositions may lead to other requirements that can be mathematically represented in other definitions of the minimization functional (\[eq:secondmin\]). [cc]{} ![image](fig4a.pdf){width="45.00000%"} & ![image](fig4b.pdf){width="45.00000%"}\ (a) $\ell=\{1,3\}$ & (b) $\ell=\{2,6\}$\ \ ![image](fig4c.pdf){width="45.00000%"} & ![image](fig4d.pdf){width="45.00000%"}\ (c) $\ell=\{1,5,9\}$ & (d) $\ell=\{1,-2,4,-5\}$ Conclusions =========== In this work we investigated the coherent superpositions of two or more coaxial LG modes for use in multi-dimensional OAM-based modulation for optical communications. Particularly, we focused on the question of how to construct a coherent OAM superposition in which the constituent modes have equally distributed energy. This energy balance is critical to achieve good signal-to-noise ratio on each dimension that forms the basis of an OAM-based modulation scheme in an optical communication link. To answer this question, we generalized the definition of a LG basis, and then proposed three ways of manipulating the new degrees of freedom to create composite modes: (i) the basic scheme, that consists of a simple addition of the modes and creating a fork pattern with equal reference waists; (ii) the equal-radii scheme, that uses distinct reference waists in each constituent mode such that the radii of the rings are equal for each OAM mode; and (iii) an optimization scheme that selects the values of the reference waists to enforce energy balance on the diffracted beams. We have shown that the first approach is flawed since it directs most of the energy into one state. The equal-radii recipe provides a moderate improvement in energy equalization. The optimization scheme provides an efficient algorithm that, for a given set of OAM states, constructs a phase grating that enforces a balanced energy distribution between the desired OAM modes. Our future work involves an evaluation of the optimal gratings in a laboratory experiment. Orthogonality and completeness of a generalized LG basis ======================================================== As we will see the generalized LG modes defined in Eq. (\[eq:LGfield\]) are orthogonal to each other under the inner product: $$\label{eq:innerprodapp} \langle \zeta(r,\phi) , \varphi(r,\phi) \rangle \triangleq \int_0^{2\pi} \int_0^\infty \zeta (r,\phi) \overline{\varphi} (r,\phi)\, r\, dr\, d\phi.$$ This definition satisfies the standard properties: it is linear in the first argument; antilinear in the second; swapping of arguments is equivalent to complex conjugation; and it defines a norm: $$\label{eq:norm} || \zeta(r,\phi) ||^2 \triangleq \langle \zeta(r,\phi) , \zeta(r,\phi) \rangle$$ that is always positive unless $\zeta(r,\phi)$ vanishes everywhere. To prove the generalized orthogonality (Eq. \[eq:orthogon2\]), let us introduce the definition: $$\label{eq:simplif} \varphi_{\ell,p}(r;w_\ell) \triangleq \sqrt{\frac{4p!}{(p+|\ell|)!}} \,\frac{1}{w_\ell} \left(\frac{r\sqrt{2}}{w_\ell}\right)^{|\ell|} L_p^{|\ell|} \left[\frac{2r^2}{w^2_\ell}\right] \exp\left[\frac{-r^2}{w^2_\ell}\right]$$ so every LG mode (we use $z=0$ here for simplicity) can be written as a product of a radial term and an angular term: $$\Phi_{\ell,p}(r,\phi,0;w_\ell) = \varphi_{\ell,p}(r; w_\ell)\ \frac{1}{\sqrt{2\pi}}\exp(-i\ell\phi)$$ The properties of each factor are standard results of Sturm-Liouville theory that we summarize here. Complex exponentials (for integers $\ell,\ell_1,\ell_2$) verify: $$\label{eq:orth1} \int_{-\pi}^\pi e^{-i\ell_1 \phi} e^{i\ell_2 \phi}\ d\phi = 2 \pi \delta_{\ell_1 \ell_2}$$ and: $$\label{eq:comp1} \sum_{\ell \in \mathbb{Z}} e^{i\ell \phi} e^{-i\ell \phi'} = 2 \pi \delta(\phi-\phi')$$ Generalized Laguerre polynomials (for nonnegative integers $\ell$ and $p_1,p_2$) verify: $$\label{eq:orth2} \int_0^\infty e^{-x} x^\ell L_{p_1}^\ell (x) L_{p_2}^\ell(x) dx = \frac{(p_1+\ell)!}{p_1!} \delta_{p_1 p_2}$$ and $$\label{eq:comp2} \sum_{p \in \mathbb{Z}^\ast} e^{-x/2} x^{\ell/2} L_{p}^\ell (x) e^{-y/2} y^{\ell/2} L_{p}^\ell(y) = \frac{(p+\ell)!}{p!} \delta(x-y)$$ These last two expressions can be rewritten using the definition of $\varphi_{\ell,p}(r;w_\ell)$ and the change of variable $x=2r^2/w_\ell^2$: $$\label{eq:orth3} \int_0^\infty \varphi_{\ell,p_1}(r;w_\ell) \varphi_{\ell,p_2}(r;w_\ell)\, r\, dr = \delta_{p_1 p_2}$$ and $$\label{eq:comp3} \sum_{p \in \mathbb{Z}^\ast} \varphi_{\ell,p}(r;w_\ell)\ \varphi_{\ell,p}(r';w_\ell) = \frac{1}{r'} \delta(r-r')$$ where the scale $w_\ell$ is completely arbitrary. The generalized orthogonality Eq. (\[eq:orthogon2\]) now follows from Eqs. (\[eq:orth1\]) and (\[eq:orth3\]). The relevance of the completeness relations Eqs. (\[eq:comp1\]) and (\[eq:comp3\]) can be explained by a simple argument. For any function $f(r,\phi)$: $$\begin{aligned} f(r,\phi) &= \int_{-\pi}^\pi \int_0^\infty f(r',\phi') \delta(r-r')\delta(\phi-\phi')\, dr'\, d\phi' \nonumber \\ &= \sum_{\ell,p} \langle f(r',\phi'), \Phi_{\ell,p}(r',\phi';w_\ell) \rangle\ \Phi_{\ell,p}(r,\phi;w_\ell)\end{aligned}$$ So if one can construct a delta function by adding products of LG modes, then one can express any superposition of aligned vortices as a sum of LG modes. Equivalence between LG bases ============================ Considering two LG bases, one defined with a single scale $w$, and the other defined with $w_0, w_1$, $w_{-1}, w_2, w_{-2},\ldots$, we can write for any function $f$: $$f(r,\phi) = \sum_{\ell,p} a_{\ell,p} \Phi_{\ell,p} (r,\phi;w) = \sum_{\ell,p} b_{\ell,p} \Phi_{\ell,p} (r,\phi;w_\ell)$$ and it is possible to connect the coefficients $a$ and $b$ by a linear transformation: $$b_{\ell,p} = \sum_{p'} c^\ell_{p,p'} a_{\ell,p'}$$ where the elements of the infinite transformation matrix: $$c^\ell_{p,p'} = \langle \Phi_{\ell,p'} (r,\phi;w), \Phi_{\ell,p} (r,\phi;w_\ell) \rangle$$ There is no connection between coefficients with different $\ell$: the change of basis is *reducible*. Ref. [@Vallone2017] provides a closed expression for $c^\ell_{p,p'}$ in terms of the hypergeometric polynomial ${}_2 F_1$ with arguments that involve $\ell, p, p'$ and the ratio $w/w_\ell$. The $L^2$-norm of the function $f$ can be computed as the sum of energies of the LG modes in either basis: $$|| f||^2 = \langle f,f \rangle = \sum_{\ell,p} |a_{\ell,p}|^2 = \sum_{\ell,p} |b_{\ell,p}|^2$$ Furthermore, if one introduces the projections (for any integer $\ell$): $$f_\ell (r) \triangleq \int_{-\pi}^{\pi} \frac{1}{\sqrt{2\pi}} \exp(i \ell \phi) f(r,\phi)\, d\phi$$ one can easily show: $$f_\ell (r) = \sum_{p} a_{\ell,p} \varphi_{\ell,p}(r; w) = \sum_{p} b_{\ell,p} \varphi_{\ell,p}(r;w_\ell)$$ and infer that the energy of each projection does not depend on the selection of the waists: $$\begin{aligned} ||f_\ell ||^2 &=& \int_0^\infty f_\ell (r) \overline{f}_\ell (r)\, r\, dr \\ &=& \sum_{p} |a_{\ell,p}|^2 = \sum_{p} |b_{\ell,p}|^2\end{aligned}$$ and $$||f||^2 = \sum_\ell ||f_\ell ||^2$$ These properties provide a mathematical framework for the freedom of scales of the LG modes. 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--- abstract: 'Let $G\subset\GL(V)$ be a complex reductive group. Let $G''$ denote $\{\phi\in\GL(V)\mid p\circ\phi=p\text{ for all }p\in\C[V]^G\}$. We show that, in general, $G''=G$. In case $G$ is the adjoint group of a simple Lie algebra $\lieg$, we show that $G''$ is an order 2 extension of $G$. We also calculate $G''$ for all representations of ${\operatorname{SL}}_2$.' address: | Department of Mathematics\ Brandeis University\ Waltham, MA 02454-9110 author: - 'Gerald W. Schwarz' title: Linear maps preserving invariants --- Introduction ============ Our base field is $\C$, the field of complex numbers. Let $G\subset\GL(V)$ be a reductive group. Let $G'=\{\phi\in\GL(V)\mid p\circ \phi=p\text{ for all }p\in\C[V]^G\}$. Several authors have studied the problem of determining $G'$. If $G$ is finite, then one easily sees that $G'=G$. Solomon [@Sol1; @Sol2] has classified many triples consisting of reductive groups $H\subset G$ and a $G$-module $V$ such that $\C(V)^H=\C(V)^G$ (rational invariant functions). If $G$ and $H$ are semisimple, then this is the same thing as finding triples where we have equality of the polynomial invariants: $\C[V]^H=\C[V]^G$. We show that for “general” faithful $G$-modules $V$ we have that $G=G'$. We also compute $G'$ for all representations of ${\operatorname{SL}}_2$. First we study the case that $G$ is the adjoint group of a simple Lie algebra $\lieg$. Our interest in this case is due to the paper of Ra[ï]{}s [@Rais1] where the question of determining $G'$ is raised. The case that $\lieg=\lie{sl}_n$ was also settled by him [@Rais2], where it is shown that $G'/G$ is generated by the mapping $\lie{sl}_n\ni X \mapsto X^t$ where $X^t$ denotes the transpose of $X$. In §2 we show that, in general, $G'/G$ is generated by the element $-\psi$ where $\psi\colon\lieg\to\lieg$ is a certain automorphism of $\lieg$ of order 2. In the case of $\lie{sl}_n$, $\psi(X)=-X^t$, so that our result reproduces that of Ra[ï]{}s. The computation of $G'$ for $\lieg$ semisimple follows easily from the case that $\lieg$ is simple. In §3 we prove our result that $G=G'$ for general $G$ and general $G$-modules $V$. In §4 we consider representations of ${\operatorname{SL}}_2$. We thank M.  Ra[ï]{}s for bringing his paper [@Rais1] to our attention and we thank him and D. Wehlau for help and advice. The adjoint case ================ Let $\lieg$ be a semisimple Lie algebra, so we have $\lieg=\lieg_1\oplus\dots\oplus\lieg_r$ where the $\lieg_i$ are simple ideals. Let $\phi\in G'$. Then $\phi(\lieg_i)=\lieg_i$ for all $i$, and $\phi|_{\lieg_i}=\pm\sigma_i$ where $\sigma_i$ is an automorphism of $\lieg_i$. By a theorem of Dixmier [@Dix79] we know that the Lie algebra of $G'$ is $\ad(\lieg)\subset\lie{gl}(\lieg)$. Thus $\phi$ acts on $\ad(\lieg)\simeq\lieg$ via an automorphism $\sigma$ where $\phi\circ\ad X\circ \phi\inv=\ad\sigma(X)$ for $X\in\lieg$. Since $\phi$ induces the identity on $\C[\lieg]^G$, so does $\sigma$, and it follows that $\sigma=\prod_i\sigma_i$ where $\sigma_i\in{\operatorname{Aut}}(\lieg_i)$, $i=1,\dots,r$. By Schur’s lemma, $\phi\circ\sigma\inv$ restricted to $\lieg_i$ is multiplication by some scalar $\lambda_i\in\C^*$, $i=1,\dots,r$. Since ${\operatorname{Aut}}(\lieg_i)$ and $G'$ preserve the invariant of degree 2 corresponding to the Killing form on each $\lieg_i$ we must have that $\lambda_i=\pm 1$, $i=1,\dots,r$. From now on we assume that $\lieg$ is simple. Let $\sigma\in{\operatorname{Aut}}(\lieg)$. Then we know that, up to multiplication by an element of $G={\operatorname{Aut}}(\lieg)^0$, we can arrange that $\sigma$ preserves a fixed Cartan subalgebra $\liet\subset \lieg$. Thus we may assume that $\phi$ preserves $\liet$. Let $T$ denote the corresponding maximal torus of $G$. We may modify $\phi$ by an element of $G$ so that $\phi$ is the identity on $\liet$. By Chevalley’s theorem, restriction to $\liet$ gives an isomorphism of $\C[\lieg]^G$ with $\C[\liet]^W$ where $W$ is the Weyl group of $\lieg$. Thus the restriction of $\phi$ to $\liet$ coincides with an element of $W$, where every element of $W$ is the restriction of an element of $G$ stabilizing $\liet$. Thus we may assume that $\phi$ is the identity on $\liet$. Let $\Phi$ be the set of roots and $\Phi^+$ a choice of positive roots. Let $\Pi$ denote the set of simple roots. Since $\phi=\pm\sigma$ is the identity on $\liet$, $\sigma(x)=c_\sigma x$ for all $x\in\liet$ where $c_\sigma=\pm 1$. Hence either $\sigma$ sends each $\lieg_\alpha$ to itself or it sends each $\lieg_\alpha$ to $\lieg_{-\alpha}$, $\alpha\in \Phi$. Choose nonzero elements $x_\alpha\in\lieg_\alpha$, $\alpha\in\Pi$, and choose elements $y_\alpha\in\lieg_{-\alpha}$ such that $(x_\alpha,y_\alpha,[x_\alpha,y_\alpha])$ is an $\lie{sl}_2$-triple. Let $\psi$ denote the unique order 2 automorphism of $\lieg$ such that $\psi(x)=-x$, $x\in\liet$ and $\psi(x_\alpha)=-y_\alpha$, $\alpha\in\Pi$ (see [@Hum72 14.3]). \[prop:csigma\] 1. If $c_\sigma=1$, then $\sigma$ is inner. 2. If $c_\sigma=-1$, then $\sigma$ differs from $\psi$ by an element of $\Ad(T)$. If $c_\sigma=1$, then $\sigma(x_\alpha)=c_\alpha x_\alpha$, $c_\alpha\in\C$, $\alpha\in\Pi$. There is a $t\in T$ such that $\Ad(t)(x_\alpha)=c_\alpha x_\alpha$, $\alpha\in \Pi$. It follows that $\sigma=\Ad(t)\in G$. If $c_\sigma=-1$, we can modify $\sigma$ by an element of $T$ so that it becomes $\psi$. \[prop:inner\] Let $\lieg$ be simple. Then the following are equivalent. 1. Every representation of $\lieg$ is self-dual.\[selfdual\] 2. The automorphism $\psi$ is inner. \[inner\] 3. The generators of $\C[\lieg]^G$ have even degree.\[evendegree\] 4. $\lieg$ is of the following type: \[type\] 1. $\mathsf{B_n}$, $n\geq 1$, 2. $\mathsf{C_n}$, $n\geq 3$, 3. $\mathsf{D_{2n}}$, $n\geq 2$, 4. $\mathsf{E_7}$, $\mathsf{E_8}$, $\mathsf{F_4}$ or $\mathsf{G_2}$. The equivalence of , and is well-known. Now given a highest weight vector $\lambda$ of $\lieg$, the highest weight vector of the corresponding dual representation $V(\lambda)^*$ is $-\rho(\lambda)$ where $\rho$ is the unique element of the Weyl group $W$ which sends $\Phi^+$ to $\Phi^-$ ([@Hum72 §21, Exercise 6]. Suppose that we have . Then, since $\psi$ is inner and it normalizes $\liet$, it gives an element of $W$, namely $\rho$, so that $V(\lambda)^*\simeq V(\lambda)$ for all $\lambda$ and holds. Conversely, if holds, then $-\rho$ is the identity on the set of weights, hence $\rho(\alpha)=-\alpha$ for all $\alpha\in\Phi$. It follows that $\rho\circ\psi$ is an automorphism of $\lieg$ which is the identity on $\liet$ and sends $\lieg_\alpha$ to $\lieg_\alpha$ for all $\alpha$. Then $\rho\circ\psi\in\Ad(T)$ so that $\psi$ is inner. \[thm:main\] The group $G'/G$ has order $2$, generated by $-\psi$. If $\phi=\sigma\in{\operatorname{Aut}}(\lieg)$, then Proposition \[prop:csigma\] shows that $\phi=\sigma\in G$. If $\phi=-\sigma$, then by Proposition \[prop:csigma\] we may assume that $\phi=-\psi$. Now $-\psi$ induces an automorphism of $\C[\lieg]^G$ and $-\psi$ is the identity on $\liet$. Hence Chevalley’s theorem shows that $-\psi\in G'$ and we know that $-\psi$ generates $G'/G$. Moreover, $-\psi$ is not in ${\operatorname{Aut}}(\lieg)$, so that $-\psi\not\in G$. \[cor:main\] Suppose that $\psi$ is inner. Then $G'/G$ is generated by multiplication by $-1$. We leave it to the reader to formulate versions of Theorem \[thm:main\] and Corollary \[cor:main\] for the semisimple case. The general case ================ We have a finite dimensional vector space $V$and $G$ is a reductive subgroup of $\GL(V)$. Let $G':=\{\phi\in\GL(V)\mid p\circ\phi=p \text{ for all }p\in\C[V]^G\}$. We show that, “in general,” we have $G'=G$. Let $U$ denote the subset of $V$ consisting of closed $G$-orbits with trivial stabilizer. It follows from Luna’s slice theorem [@Luna] that $U$ is open in $V$. \[thm:G’=G\] Suppose that $V\setminus U$ is of codimension $2$ in $V$. Then $G'=G$. Let $\phi\in G'$ and let $x\in U$. Then $\phi(x)=\psi(x)\cdot x$ where $\psi\colon U\to G$ is a well-defined morphism. Since $G$ is affine, we may consider $\psi$ as a mapping from $U\to G\subset\C^n$ for some $n$ where $G$ is Zariski closed in $\C^n$. Our condition on the codimension of $V\setminus U$ guarantees that each component of $\psi$ is a regular function on $V$, hence $\psi$ extends to a morphism defined on all of $V$, with image in $G$. Now let $x\in U$. Then $$\phi(x)=\lim_{t\to 0}\phi(tx)/t=\lim_{t\to 0}\psi(tx)tx/t=\psi(0)(x).$$ Thus $\phi$ is just the action of $\psi(0)\in G$, so $G'=G$. Representations of SL$_2$ ========================== As an illustration, we consider representations of $G={\operatorname{SL}}_2$ or $G={\operatorname{SO}}_3$. We only consider representations with no nonzero fixed subspace. We let $R_j$ denote the irreducible representation of dimension $j+1$, $j\geq 0$, and $kR_j$ denotes the direct sum of $k$ copies of $R_j$, $k\geq 1$. When we have a representation only containing copies of $R_j$ for $j$ even, then we are considering representations of ${\operatorname{SO}}_3$. From [@Sch95 11.9] we know that all representations of $G$ satisfy the hypotheses of Theorem \[thm:G’=G\] except for the following cases, where we compute $G'$. 1. For $R_1$ we have $G'=\GL_2$, for $2R_1$ we have $G'={\operatorname{O}}_4$ and for $3R_1$ we have $G'=G$. 2. For $R_2$ we have $G'={\operatorname{O}}_3$ and for $2R_2$ we have $G'={\operatorname{O}}_3$. (Here $G={\operatorname{SO}}_3$.) 3. For $R_2\oplus R_1$ we have $G'=\{g'\in\GL_2\mid \det(g')=\pm 1\}$. 4. For $R_3$ the group $G'$ is the same as in case (3). 5. For $R_4$ we have $G'=G={\operatorname{SO}}_3$. Most of the calculations are easy, we mention some details for some of the non obvious cases. Suppose that our representation is $R_4$, which has generating invariants of degrees 2 and 3. The Lie algebra $\lieg'$ acts irreducibly on $R_4$, hence it is the sum of a center and a semisimple Lie algebra [@Jac62 Ch. II, Theorem 11]. Clearly we cannot have a nontrivial center, so that $\lieg'$ is semisimple. Now a case by case check of the possibilities forces $\lieg'=\lieg$. Suppose that $g'\in G'\setminus G$. Then conjugation by $g'$ gives an inner automorphism of $G$, hence we can correct $g'$ by an element of $G$ so that $g'$ commutes with $G$. Thus $g'$ acts on $R_4$ as a scalar. But to preserve the invariants the scalar must be 1. Thus we have $G'=G$. Similar considerations give that $\lieg'=\lieg$ in case (4), so that $G'/G$ is generated by scalar multiplication by $ i$ (since the generating invariant of $R_3$ has degree 4), which shows that $G'$ is as claimed. In case (3), one sees that $\lieg'=\lieg$, so that generators of $G'/ G$ act as scalars on $R_2$ and $R_1$. Now generators of the invariants have degrees $(2,0)$ and $(1,2)$ so that $G'/G$ is generated by an element which is multiplication by $-1$ on $R_2$ and multiplication by $i$ on $R_1$. Hence $G'$ is as claimed. \[2\][ [\#2](http://www.ams.org/mathscinet-getitem?mr=#1) ]{}\[2\][\#2]{} [HWD06b]{} J. Dixmier, *Champs de vecteurs adjoints sur les groupes et algèbres de Lie semi-simples*, J. Reine Angew. Math. 309 (1979), 183–190. J.E. Humphreys, *Introduction to Lie Algebras and Representation Theory*, Graduate Texts in Mathematics **9**, Springer-Verlag, New York, 1972. N. Jacobson, *Lie algebras*, Interscience Publishers, New York, 1962. D. Luna, *Slices étales*, Bull. Soc. Math. France, mémoire **33** (1973), 81–105. M. Ra[ï]{}s, *Distributions homogénes sur des espaces de matrices*, Bull. Soc. Math. France, mémoire **30** (1972). , *Notes sur la notion d’invariant caractéristique*, http://arxiv.org/abs/0707.0782v1. G.W. Schwarz, *Lifting differential operators from orbit spaces*, Ann. Sci. Ecole Norm. Sup. **28** (1995), 253–306. S. Solomon, *Irreducible linear group-subgroup pairs with the same invariants*, J. Lie Theory **15** (2005), no. 1, 105–123. , *Orthogonal linear group-subgroup pairs with the same invariants*, J. of Alg. **299** (2006), no. 2, 623–647.
--- abstract: 'We experimentally observed the Mollow quintuplet (MQ) in $F=3/2$ hyperfine structure state of $^{3}\textrm{He}$ atoms. The metastability-exchange collisions (MECs) transfer the Mollow Triplet (MT) from the ground states of $^{3}\textrm{He}$ atoms to the metastable states, and the MQ is demonstrated by four Zeeman levels of $F=3/2$ hyperfine states with linearly polarized light. The similar effect also achieves in the mixture cell of $^{3}\textrm{He}$ and $^{4}\textrm{He}$.' author: - Yuanzhi Zhan - Xiang Peng - Sheng Li - Liang Zhang - Jingbiao Chen - Hong Guo nocite: '[@*]' title: 'Observation of Mollow quintuplet in $F=3/2$ hyperfine structure state of $^{3}\textrm{He}$ atomic cell' --- The interaction between light and materials is an important research area of quantum system. The Fermi Golden Rule describes the weak coupling between the light and atoms, which reveals the transition between the energy eigenstates of atoms [@fermi1950]. Mollow triplet (MT) is described as that two-level atoms are driven by strong coherent field in free space [@mollow1969power], which reveals the change from singlet spectrum to the triplet. MT of resonant light scattering was achieved in the atomic beam of sodium [@wu1975investigation], quantum dot [@fischer2016self], silicon vacancy of diamond [@zhou2017coherent], and superconducting circuits [@baur2009measurement]. The ratio of center peak and sidebands of MT is influenced by coherence [@nathan2015theory] or collisions [@khoa2016mollow]. The quintuplet spectrum has been reported in three different systems. The first one is three-level atoms coupled with two high-intensity resonant laser beam sharing a common level [@cohen1977simultaneous], the second one is two-level atoms coupled with bimodality cavity [@Nha2000resonance], and the third one is the resonance fluorescence spectrum of quantum dots system [@vam2009spin]. The so-called Mollow quintuplet (MQ) is modeled by summing Mollow triplets (MTs) of two natural excitons polarized in orthogonal direction of quantum dot with the same laser [@rong2013mollow], and furthermore two modified MTs can induce the Mollow nonuplets (MNs) in two coupled quantum dots system at strong exchange regime [@angelatos2015entanglement]. In this letter we observe the MQ in $F=3/2$ hyperfine structure state of $^{3}\textrm{He}$ atoms by detection with linearly polarized light, and the experimental results indicate to the alignment effect with a different physical process from that mentioned before. We use a RF field to drive the ground state $1 ^1S_0$, and the MT is induced in $m_I=+1/2$ and $m_I=-1/2$ states, according to the energy-level diagram of $^{3}\textrm{He}$ atoms shown in Fig. \[fig:energy\]. The metastability-exchange collisions (MECs) can transfer the MT from the ground states to $F=1/2$ metastable states [@zhan2018observation], and the transferred MT also induced the further MQ in $F=3/2$ metastable states. The experimental results reveal that the MQ effect is related the higher order spin quantum number state and the detection with linearly polarized light. ![\[fig:energy\] The energy-level diagram of $^{3}\textrm{He}$ atoms (not in scale). $1^{1}S_{0}$, $2^{3}S_{1}$, $2^{3}P_{0}$ are the ground state, metastable state, excited state, respectively. The metastable state contains two hyperfine states, $F=1/2$ and $F=3/2$, each of which is split into 2 or 4 Zeeman sublevels in a static magnetic field $\bf{B_0}$. The blue line indicates the metastability exchange collisions (MECs) between $1^{1}S_{0}$ and $2^{3}S_{1}$. The interval depends on the gyromagnetic ratio, i.e. $\gamma_{\rm g}=$ 3.2 kHz/G for ground state, $\gamma_\mu=$ 3.8 MHz/G and $\gamma_{\mu'}=$ 1.9 MHz/G for $F=1/2$, $F=3/2$, respectively [@colegrove1963polarization]. The green line indicates the RF discharge to generate $2^{3}S_{1}$ atoms. The red line indicates the optical transition $\textrm{C}_8$ between the $2^{3}S_{1}$ and $2^{3}P_{0}$ states with the vacuum wavelength 1083.353 nm, which is used for optical pumping and probing [@nacher1985optical]. ](fig_energy.pdf){width="40.00000%"} ![\[fig:experiment\] The experimental setup for the MQ measurement in $^{3}\textrm{He}$ atoms. PBS: Polarization Beam Splitter, QWP: Quarter Wave Plate, HWP: Half Wave Plate, BE: Beam Expander, BT: Beam Trap, PD: Photo Detector. ](fig_experiment.pdf){width="45.00000%"} The experimental setup is shown in Fig. \[fig:experiment\] and we use the two lasers to pump and detect $F=3/2$ hyperfine states of metastable states, respectively. Both the pump (laser 1) and the probe (laser 2) beams are from laser source (NKT Photonics Y10), and the pump beam is power-enhanced by the a laser amplifier (LEA Photonics MLXX-EYFA-CW-SLM-P-TKS). The pump (probe) beam propagates along $z$ ($x$) axis and has the power of 15 W (0.3 mW) with $1/e^2$ waist diameter of about 20 mm (1 mm). The pump beam keeps the circularly polarized before entering $^{3}\textrm{He}$ atomic cell, and the probe beam is circularly or linearly polarized to observe the orientation and alignment effect of the $F=3/2$ metastable states. The home-made pure $^{3}\textrm{He}$ (pressure: 0.6 Torr) cylindrical atomic cell (size: $\phi$50$\times$L70 mm$^3$ ) is located in the seven-layer magnetic shield, and is excited by a radio-frequency power source (50 MHz, 0.8 W) to continuously discharge and generate the metastable-state atoms. The solenoid generates the static magnetic field $B_0$ along $z$, and a set of helmholtz coil generates the oscillating magnetic field $B_{\rm M}$ along $y$. The digital processing system includes the PXI-4461 and PXI-4462 (resolution: 24-Bit, sampling rate: 204.8 kS/s) of the National Instruments, which is used for controlling the helmholtz coil and signal acquisition. ![\[fig:onoffpump\] The frequency spectrum by detection with different optical polarization and different hyperfine structure states. Figures in first (second) row are detected by $C_8$ ($C_9$) line, and figures in left (right) column are detected by circularly (linearly) polarized light. The black squares are the experimental data, and the red lines are the fitting curves. ](fig_onoffpump.pdf){width="45.00000%"} Figure \[fig:onoffpump\] manifests the MT and MQ spectra by detection of different optical polarization and different hyperfine structure states. The first row of the Fig. \[fig:onoffpump\] shows the detection with $C_8$ line, while the second row shows the $C_9$ ones. The pumping beam keeps the circularly polarized light, and the Fig. \[fig:onoffpump\](a.1 and b.1) shows the circularly polarized probe while Fig. \[fig:onoffpump\](a.2 and b.2) shows the linearly polarized probe. Comparing each figure in Fig. \[fig:onoffpump\], the MT appears by the detection with circularly polarized light no matter which hyperfine structure state is probed, while the MQ appears by the linearly polarized probe and only in $F=3/2$ hyperfine structure state. Notice that the center frequency of the MT and MQ signal is the Larmor frequency of the ground state. We have reported a MT signal of ground state can be transferred through metastability exchange collisions (MECs) to the metastable states, which reveals that the signal of Fig. \[fig:onoffpump\] (b.1) is the transferred MT of ground state. The different polarized probe beam indicates that the orientation and alignment effect of the metastable states atoms may be related to the MT and MQ signal, respectively. ![image](fig_direction.pdf){width="90.00000%"} The circularly polarized pump beam (in z axis) and linearly polarized probe beam (in x axis) continuously interact with the atoms, and the frequency of oscillating magnetic field (in y axis) is set as $\omega =\omega_{\textrm{g}}$. The direction of the polarization is rotated in y-z plane shown in Fig. \[fig:direction\](a), and the parameter $\theta$ is defined as the angle between the polarization and y axis. Changing $\theta$, the MQ signals will disappear at $\theta=0, 90$ (perpendicular or parallel the static magnetic field $B_0$) shown in Fig. \[fig:direction\](b). The detailed relationship between MQ signal and direction of linearly polarized light is shown in Fig. \[fig:direction\](c), which shows the amplitude of MQ each peaks is changed simultaneously with $\theta$, and the maximum of the amplitude appears at the condition $\theta = 45, 135$. Focusing on the amplitude of center peak, the relationship with $\theta$ is shown in Fig. \[fig:direction\](d). The experimental data (black squares) are corresponding to the function $\rm sin$$(2\theta)$. ![\[fig:MTC8C9\] Dependence of FFT spectrum on changing the amplitude (a) and frequency (b) of the oscillating magnetic field for experiment. The amplitude of oscillating magnetic field in is about 3000 nT, and the frequency of the oscillating magnetic field is at about 1.30 kHz. ](fig_MTC8C9.pdf){width="45.00000%"} With changing both the amplitude and frequency of oscillating magnetic field at the condition $\theta = 45$, the MQ signals are shown in Fig. \[fig:MTC8C9\]. The interval of each peak is the Rabi frequency $\Omega_R$, and the first order sidebands and second order sidebands appear together with changing the amplitude of oscillating magnetic field. $\Omega_R$ approximately behaves linearly with changing the amplitude of the resonant oscillating magnetic field $B_M$, which accords with $\Omega_R =(1/2)\gamma_{0}B_{\textrm{M}}$ and indicates a method of measuring the amplitude for the oscillating magnetic field. Figure \[fig:MTC8C9\](b) shows the center peak and the first sidebands changing like MT signal with changing frequency $\omega_M$ of oscillating field, and the second order sidebands only appears at the $\omega_M \approx \omega_L$. The small sidebands near center peak appearing at the far detuning condition still need further researches. Depending on the experimental results, an possible explanation of the Mollow quintuplet manifests here. The physical picture of MT has been described as the dressed atom [@cohenbook], and the MT transfer from the ground state to $F=1/2$ metastable state has been reported [@zhan2018observation]. The oscillating magnetic field drives the ground-state coherence of two Zeeman energy eigenstates or bare states $\vert$$\uparrow$$\rangle _{\textrm{g}}$ and $\vert$$\downarrow$$\rangle _{\textrm{g}}$. The new energy eigenstates or a series of dressed states are formed as the superposition states $\vert$$+$,$N$$\rangle_{\textrm{g}}$$=(1/\sqrt{2})$$($$\vert$$\downarrow$,$n$$\rangle_{\textrm{g}}$$+$$\vert$$\uparrow$,$n-1$$\rangle_{\textrm{g}}$$)$ and $\vert$$-$,$N$$\rangle_{\textrm{g}}$$=(1/\sqrt{2})$$($$\vert$$\downarrow$,$n$$\rangle_{\textrm{g}}$$-$$\vert$$\uparrow$,$n-1$$\rangle_{\textrm{g}}$$)$, where $n$ is the quantum number of oscillating field, $\vert$$\uparrow$,$n$$\rangle_{\textrm{g}}$ ($\vert$$\downarrow$,$n$$\rangle_{\textrm{g}}$) is the direct product state of atoms and the oscillating field and $N$ is the total number of the excitations in the system. Note that the energy interval $\hbar \Omega_R = (1/2)\hbar\gamma_{\textrm{g}}B_{\textrm{M}}$, where $B_{\textrm{M}}$ is the amplitude of the oscillating magnetic field. The MT can be transferred from the ground state to the metastable states has been reported, which reveals that the MECs between the dressed ground state and the undressed metastable states is a linear effect of the first order tensors of the density matrix $J_{+}$ and $J_{-}$. Furthermore, the MECs between the dressed ground state and the dressed metastable states create higher order nonlinear effect, and the MQ is the second order tensors of the density matrix like $J_{+}J_{z}$ and $J_{-}J_{z}$. The second order tensors $J_{+}J_{z}$ and $J_{-}J_{z}$ include two transitions, which are between $\Delta N = 1$ and $\Delta N = 0$ dressed states. The two transitions happen continuously, like that the Rabi oscillation modulates the MT, which change the frequencies of MT to the $\omega_{+}\pm \Omega_{\rm R}$, $\omega_{g}\pm \Omega_{\rm R}$ and $\omega_{-}\pm \Omega_{\rm R}$. The spectrum of the effect contain five frequencies $\omega_{g}+2\Omega_{\rm R}$, $\omega_{g}+\Omega_{\rm R}$, $\omega_{g}$, $\omega_{g}-\Omega_{\rm R}$ and $\omega_{g}-2\Omega_{\rm R}$ called Mollow quintuplet. In order to verify the explanation, the experimental results of $^{3}\textrm{He}$-$^{4}\textrm{He}$ hybrid cell is show in Fig. \[fig:D0\]. The metastable state of $^{4}\textrm{He}$ has three Zeeman states whose alignment effect also can be detected by linearly polarized light, and the spectrum gives a MQ signal. ![\[fig:D0\] The frequency spectrum of detection by different optical polarization with $D_0$ line of $^{4}\textrm{He}$. The detections by circular polarized light and linearly polarized light are shown in (a) and (b), respectively. The black squares are the experimental data, and the red lines are the fitting curves. ](fig_D0.pdf){width="45.00000%"} We have observed the MQ signal in $F=3/2$ hyperfine structure state of $^{3}\textrm{He}$ atoms by detecting with linearly polarized light tuned to $C_9$ line. Comparing the MT and MQ signal at different polarization and center frequency of probe light, we give an possible explanation for MQ is related to the alignment of $^{3}\textrm{He}$ atoms, and the experimental results of $^{3}\textrm{He}$-$^{4}\textrm{He}$ hybrid cell give more evidence for the explanation. The difference between the MT and MQ signal reveals the demand of a new model for the MECs. The existing models of quintuplet spectrum are three levels system coupled two strong laser, two levels system coupled bimodality cavity, and two orthogonal polarization coupled one strong laser, but they cannot describe our experiments. We only give a phenomenological description to explain the generation of MQ, and the further theory needs more research. The frequency interval of the sidebands is linear with the amplitude of the resonant oscillating magnetic field, which satisfies $\hbar \Omega_R = (1/2)\hbar\gamma_{g}B_{\textrm{M}}$, and indicates the possible method of measuring the amplitude of the oscillating magnetic field.\ **Funding.** This project is supported by National Natural Science Foundation of China (61571018, 61531003, 91436210); National key research and development program.\ **Acknowledgment.** We thank for W.L., H.W. and H.d.W with experimental and technical assistance. [99]{} E. Fermi, Nuclear Physics, Notes compiled by J. Orear, A. H. Rosenfeld, and R. A. Schluter (The University of Chicago Press, Chicago, 1950). B. R. Mollow, Phys. Rev. $\bf{188}$, 1969 (1969). F. Y. Wu, R. E. Grove, and S. Ezekiel, Phys. Rev. Lett. $\bf{35}$, 1426 (1975). K. A. Fischer, K. Müller, A. Rundquist, T. 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--- abstract: 'We compute the rings $H^*(N;\mathbb{F}_2)$ for $N$ a closed $\mathbb{S}ol^3$-manifold, and then determine the Borsuk-Ulam indices $BU(N,\phi)$ with $\phi\not=0$ in $H^1(N;\mathbb{F}_2)$.' address: | School of Mathematics and Statistics\ University of Sydney, NSW 2006\ Australia author: - 'J.A.Hillman' title: 'The $\mathbb{F}_2$-cohomology rings of $\mathbb{S}ol^3$-manifolds' --- The Borsuk-Ulam Theorem states that any continuous function $f:S^n\to\mathbb{R}^n$ takes the same value at some antipodal pair of points. This may be put in a broader context as follows. Let $N$ be an $n$-manifold and let $N_\phi$ be the double cover associated to an epimorphism $\phi:\pi\to{Z/2Z}$. Let $t_\phi$ be the covering involution. The [*Borsuk-Ulam index*]{} $BU(N,\phi)$ is the maximal value of $k$ such that for all maps $f:N_\phi\to\mathbb{R}^k$ there is an $x\in{N_\phi}$ with $f(x)=f(t_\phi(x))$. Then the Borsuk-Ulam Theorem is equivalent to the assertion that $BU(RP^n,\alpha)=n$, where $\alpha:\pi_1(RP^n)\to{Z/2Z}$ is the canonical epimorphism. In low dimensions this invariant may be determined cohomologically, and is known for many pairs $(N,\phi)$, with $N$ a Seifert fibred 3-manifold, including all those with geometry $\mathbb{E}^3$, $\mathbb{S}^3$, $\mathbb{S}^2\times\mathbb{E}^1$, $\mathbb{N}il^3$ or $\mathbb{H}^2\times\mathbb{E}^1$ [@GHZ; @BGHZ]. Here we shall determine this invariant for all such pairs with $N$ a closed $\mathbb{S}ol^3$-manifold. This follows easily once we know the mod-2 cohomology rings of such manifolds. We compute these using Poincaré duality and elementary properties of cup-product in the low-degree cohomology of groups. (Our approach can also be applied to $\mathbb{E}^3$- and $\mathbb{N}il^3$-manifolds.) I would like to thank the organizers of the XVIII Encontro Brasiliero de Topologia for the invitation to their meeting in Aguas de Lindoias, in July 2012, which lead to this work, and S.T.Martins, for sending me a copy of his PhD thesis [@Ma12]. The use of the extraspecial 2-group $E$ (introduced before Lemma 6 below) was prompted by the work of J.F.Carlson on the cohomology rings of 2-groups [@jfc]. $\mathbb{S}ol^3$-manifolds and their groups =========================================== Let $M$ be a closed $\mathbb{S}ol^3$-manifold. Then $\pi=\pi_1(M)$ has an unique maximal abelian normal subgroup $\sqrt\pi$, which is free abelian of rank 2. (This subgroup is in fact the Hirsch-Plotkin radical [@Ro] of $\pi$.) The quotient $\pi/\sqrt\pi$ is virtually $\mathbb{Z}$ (i.e., has two ends), and so is an extension of $\mathbb{Z}$ or $D_\infty=Z/2Z*Z/2Z$ by a finite normal subgroup. The preimage of this finite normal subgroup is torsion-free, and so is either $\mathbb{Z}^2$ or $\mathbb{Z}\rtimes_{-1}\mathbb{Z}$ (the Klein bottle group). Since $Out(\mathbb{Z}\rtimes_{-1}\mathbb{Z})$ is finite and $\pi$ is not virtually abelian, this preimage must be $\sqrt\pi$. Hence $\pi/\sqrt\pi\cong\mathbb{Z}$ or $D_\infty$. Suppose first that $\pi/\sqrt\pi\cong\mathbb{Z}$. Then $M$ is the mapping torus of a self-homeomorphism of $T=S^1\times{S^1}$, and $\pi\cong\mathbb{Z}^2\rtimes_\Theta\mathbb{Z}$, where $\Theta=\left(\begin{smallmatrix} a&c\\ b&d \end{smallmatrix}\right)\in{GL}(2,\mathbb{Z})$. Thus $\pi$ has a presentation $$\langle{t,x,y}\mid{txt^{-1}=x^ay^b},~tyt^{-1}=x^cy^d,~xy=yx\rangle.$$ Let $\varepsilon=\det\Theta=\pm1$ and $\tau=tr\Theta=a+d$. Then $M$ is orientable if and only if $\varepsilon=1$, in which case $|\tau|>2$, since $\pi$ is not virtually nilpotent. Let $\theta$ be a root of ${\det(\Theta-XI_2)}=X^2-\tau{X}+\varepsilon$, the characteristic polynomial of $\Theta$. Then $\theta$ is a unit in the quadratic number field $\mathbb{Q}[\theta]$, and $\sqrt\pi$ is isomorphic to an ideal $I$ in the ring $\mathbb{Z}[\theta]$. (The latter may not be the full ring of integers in $\mathbb{Q}[\theta]$!) There is a converse. Let $[I]$ denote the isomorphism class of the ideal $I$. The Galois involution of the quadratic field $\mathbb{Q}[\theta]$ acts on the ring $\mathbb{Z}[\theta]$, since $\bar\theta=\tau-\theta\in\mathbb{Z}[\theta]$, and hence acts on the set of ideal classes. Let $\alpha$ be a quadratic algebraic unit which is not a root of unity, and let $J$ be a nonzero ideal in $\mathbb{Z}[\alpha]$. Let $A$ be the automorphism of $J\cong\mathbb{Z}^2$ given by left multiplication by $\alpha$, and let $\pi=J\rtimes_A\mathbb{Z}$. Then 1. $\pi$ is a $\mathbb{S}ol^3$-group; 2. the groups corresponding to two such pairs $(\alpha,J)$ and $(\beta,K)$ are isomorphic if and only if either $\beta=\alpha$ or $\alpha^{-1}$ and $[K]=[J]$, or $\beta=\bar\alpha$ or $\bar\alpha^{-1}$ and $[K]=\overline{[J]}$; 3. given $\alpha$, the number of isomorphism classes of such groups $\pi$ is finite. The group $\pi$ is the fundamental group of the mapping torus of a self-homeomorphism of $T$. If $\alpha$ is not a root of unity then this is a $\mathbb{S}ol^3$-manifold. Let $\pi=\langle{J,t}|tjt^{-1}=\alpha{j}~\forall{j}\in{J}\rangle$ and $\widetilde\pi=\langle{k,\widetilde{t}}| \widetilde{t}j\widetilde{t}^{-1}=\beta{k}~\forall{k}\in{K}\rangle$ be two such groups. An isomorphism $f:\pi\cong\widetilde\pi$ restricts to an isomorphism $f_J:J=\sqrt\pi\cong\sqrt{\widetilde\pi}=K$. Hence it induces an isomorphism $\pi/\sqrt\pi\cong\widetilde\pi/\sqrt{\widetilde\pi}$, and so $f(t)=\widetilde{t}^\eta{k}$, for some $\eta=\pm1$ and $k\in{K}$. We may assume that $f(t)=\widetilde{t}$, after replacing $\beta$ by $\beta^{-1}$, if necessary. The characteristic polynomials of the automorphism of $J$ and $K$ induced by conjugation by $t$ and $\widetilde{t}$ (respectively) must then agree. Thus either $\beta=\alpha$ and $f_J$ is an isomorphism of $\mathbb{Z}[\alpha]$-modules, or $\beta=\bar\alpha$ and $\overline{f_J}:J\to\overline{K}$ is an an isomorphism of $\mathbb{Z}[\alpha]$-modules. The converse is similarly straightforward. The group $\pi$ is determined up to a finite ambiguity by $\alpha$ (equivalently, by the polynomial $t^2-\tau{t}+\varepsilon$), since $\mathbb{Z}[\alpha]$ has finitely many ideal classes, by the Jordan-Zassenhaus Theorem. If $\pi/\sqrt\pi\cong{D_\infty}$ then $\pi\cong{B*_TC}$, where $B$ and $C$ are torsion-free, $T\cong\mathbb{Z}^2$ and $[B:T]=[C:T]=2$. Thus $M$ is the union of two twisted $I$-bundles. Since $\beta_1(\pi;\mathbb{Q})=0$ and $\chi(M)=0$, $M$ is orientable, and so $B$ and $C$ must be copies of the Klein bottle group. Hence $M$ is the union of two copies of the mapping cylinder of the double cover of the Klein bottle. The double cover of $M$ corresponding to the preimage of $\sqrt{D_\infty}$ in $\pi$ is a mapping torus. In particular, $\pi$ has a presentation $$\langle u,v,y,z\mid uyu^{-1}=y^{-1}\!,~vzv^{-1}=z^{-1}\!,~yz=zy,~v^2=u^{2a}y^b,~ z=u^{2c}y^d \rangle,$$ where $\left(\begin{smallmatrix} a&c\\ b&d \end{smallmatrix}\right)\in{GL}(2,\mathbb{Z})$ corresponds to the identification of $\sqrt{C}$ with $T=\sqrt{B}$. This presentation simplifies immediately to $$\langle u,v,y\mid uyu^{-1}=y^{-1},~v^2=u^{2a}y^b,~ vu^{2c}y^dv^{-1}=u^{-2c}y^{-d} \rangle.$$ Hence $\pi^{ab}\cong{Z/4cZ}\oplus{Z/4Z}$ if $b$ is odd, and $\pi^{ab}\cong{Z/4cZ}\oplus(Z/2Z)^2$ if $b$ is even. Let $x=u^2$. Then conjugation by $uv$ acts on $\langle{x,y}\rangle\cong\mathbb{Z}^2$ via $\Psi=\eta\left(\begin{smallmatrix} ad+bc&2ac\\ 2bd&ad+bc \end{smallmatrix}\right)$, where $\eta=ad-bc=\pm1$. Hence $\det\Psi=1$, $\Psi\equiv{I_2}$ [*mod*]{} $(2)$ and $tr\Psi\equiv2$ [*mod*]{} $(4)$. (These conditions are not independent, for if $\Psi=I_2+2N$ then $tr\Psi=2+2trN$ and $\det\Psi\equiv1+2trN$ [*mod*]{} (4), so $trN$ is even and $tr\Psi\equiv2$ [*mod*]{} $(4)$ if also $\det\Psi=1$.) Moreover, $abcd\not=0$, since $M$ is not flat. Conversely, any $\left(\begin{smallmatrix} a&c\\ b&d \end{smallmatrix}\right)\in{GL}(2,\mathbb{Z})$ with $abcd\not=0$ gives rise to such a $\mathbb{S}ol^3$-manifold, for then $|tr\Psi|=2|ad+bc|\geq6$. Moreover, suppose $P=\left(\begin{smallmatrix} 2k+1&2m\\ 2n&2k+1 \end{smallmatrix}\right)\in{SL(2,\mathbb{Z})}$, where $mn\not=0$. Then $k(k+1)=mn$, and so we may write $m=m_1m_2$ and $n=n_1n_2$, with $k=m_1n_1$ and $k+1=m_2n_2$. The $\mathbb{S}ol^3$-rational homology sphere corresponding to $\left(\begin{smallmatrix} m_1&-m_2\\ -n_2&n_1 \end{smallmatrix}\right)\in{GL}(2,\mathbb{Z})$, is doubly covered by the mapping torus asociated to $P$. The above matrix calculations show that a quadratic unit $\alpha$ is realized by such a $\mathbb{S}ol^3$-manifold if and only if $\alpha\bar\alpha=1$, $|\alpha+\bar\alpha|>2$ and $\alpha+\bar\alpha\equiv2$ [*mod*]{} $(4)$. Determining the possible ideal classes represented by $\sqrt\pi$ is more complicated. Let $\alpha$ be a quadratic unit which is not a root of unity, and let $J$ be a nonzero ideal in $\mathbb{Z}[\alpha]$. Let $A$ be the automorphism of $J\cong\mathbb{Z}^2$ given by left multiplication by $\alpha$, and let $\kappa=J\rtimes_A\mathbb{Z}$. Then $\kappa$ is a subgroup of index $2$ in a $\mathbb{S}ol^3$-group $\pi$ with $\pi/\sqrt\pi\cong{D_\infty}$ $\Leftrightarrow\,\alpha\bar\alpha=1$, $\alpha\equiv1$ [*mod*]{} $2\mathbb{Z}[\alpha]$ and there are $\lambda,\mu\not=0\in\mathbb{Z}[\alpha]$ and $v,w\in{J}$ such that $\lambda\overline{J}=\mu{J}$, $\lambda\bar{v}=\mu{v}$ and $\lambda\bar{w}=\bar\alpha\mu{w}$, but $\bar\lambda{v}\not=\bar\lambda{j}+\bar\mu\bar{j}$ and $\bar\lambda{w}\not=\bar\lambda{j}+\alpha\bar\mu\bar{j}$ for any $j\in{J}$. Given $\alpha$, the number of isomorphism classes of such groups $\pi$ is finite. Suppose that $\pi=\langle\kappa,u\rangle$ with $\pi/\sqrt\pi\cong{D_\infty}$ and $[\pi:\kappa]=2$, and that $t\in\kappa$ generates $\kappa$ [*mod*]{} $\sqrt\pi$. Then $t^{-1}$ is conjugate to $t$, and so $A$ and $A^{-1}$ have the same characteristic polynomial. Since $trA\not=0$, $\alpha\bar\alpha=\det{A}=1$. Let $B(j)=uju^{-1}$ and $f(j)=\overline{B(j)}$, for all $j\in{J}$. Then $B$ is an isomorphism of groups and $f:J\to\overline{J}$ is an isomorphism of $\mathbb{Z}[\alpha]$-modules. Let $v=u^2$ and $w=(tu)^2$. Then $B^2=(AB)^2=I$, $Bv=v$ and $ABw=w$. Since $A$ has infinite order, $B\not=I$, and so $\det{B}=-1$. Moreover, $B\equiv{AB}\equiv{I_2}$ [*mod*]{} (2), since $\langle{J,u}\rangle$ and $\langle{J,tu}\rangle\cong\pi_1(Kb)$. Therefore $A\equiv{I_2}$ [*mod*]{} (2) also, and so $\alpha\equiv1$ [*mod*]{} $2\mathbb{Z}[\alpha]$. Since $\pi$ is torsion free, $(uj)^2$ and $(tuj)^2$ are nontrivial, for all $j\in{J}$. Equivalently, $v\not\in(I+B)J$ and $w\not\in(I+AB)J$. The isomorphism $f$ extends to an automorphism $f_\mathbb{Q}=id_\mathbb{Q}\otimes{f}$ of $\mathbb{Q}[\alpha]$, as a vector space over itself. We may write $f_\mathbb{Q}(1)=\frac\mu\lambda$, for some nonzero $\lambda,\mu\in\mathbb{Z}[\alpha]$. (Note that $\frac{\mu\bar\mu}{\lambda\bar\lambda}= \det{f_\mathbb{Q}}=-\det{B}=1$.) Then $\lambda{f(j)}=\mu{j}$, for all $j\in{J}$, since $\mathbb{Z}[\alpha]$ is an integral domain. The linear conditions on $v$ and $w$ become $\lambda\bar{v}=\mu{v}$ and $\lambda\bar{w}=\bar\alpha\mu{w}$, while $\bar\lambda{v}\not=\bar\lambda{j}+\bar\mu\bar{j}$ and $\bar\lambda{w}\not=\bar\lambda{j}+\alpha\bar\mu\bar{j}$ for any $j\in{J}$. Conversely, suppose that these conditions hold. Let $Bj=\overline{\frac\mu\lambda{j}}$, for all $j\in{J}$, and let $\pi$ be the group with presentation $$\langle\kappa,u|u^2=v,~utu^{-1}=t^{-1}wv^{-1}, ~uju^{-1}=Bj~\forall{j}\in{J}\rangle.$$ Then $\pi$ is torsion free and has $\kappa$ as a subgroup of index 2. and so is a $\mathbb{S}ol^3$-group. Clearly $\pi/\sqrt\kappa\cong{D_\infty}$, and so $\sqrt\kappa\leq\sqrt\pi\leq\kappa$. Hence $\sqrt\pi=\sqrt\kappa$ and $\pi/\sqrt\pi\cong{D_\infty}$. Since $\kappa$ has trivial centre the extensions of $Z/2Z$ by $\kappa$ are determined by the image in $Out(\kappa)$ of the action of $Z/2Z$ on $\kappa$. Since There are finitely many groups $\kappa$ realizing $\alpha$, by Theorem 1, and $Out(\kappa)$ is finite, by Theorem 8.10 of [@Hi], there are finitely many such groups $\pi$. In particular, the ideal class $[J]$ must be fixed by the Galois involution. For example, if $\alpha\bar\alpha=1$ and $\alpha\equiv1$ [*mod*]{} $2\mathbb{Z}[\alpha]$ then $J=\mathbb{Z}[\alpha]$, $v=1$ and $w=\alpha$ satisfy the other conditions, with $\lambda=\mu=1$. Note that if $\alpha$ is a quadratic unit such that $\alpha\bar\alpha=1$ and $\delta=\alpha-1\in2\mathbb{Z}[\alpha]$ then $\bar\delta=-\alpha^{-1}\delta\in2\mathbb{Z}[\alpha]$ also, and so $\alpha+\bar\alpha=2-\delta\bar\delta\equiv2$ [*mod*]{} (4). (This is equivalent to an earlier matrix argument.) Every subgroup of finite index in $\pi$ can be generated by three elements, while proper subgroups of infinite index need at most two generators. If a nontrivial normal subgroup $N$ has infinite index in $\pi$ then it has Hirsch length $\leq2$. Hence it is abelian, and so has finite index in $\sqrt\pi$. Thus proper quotients of a $\mathbb{S}ol^3$-group $\pi$ either have two ends or are finite. the mod-$2$ cohomology ring =========================== Martins has constructed an explicit free resolution $P_*\to\mathbb{Z}$ of the augmentation $\mathbb{Z}[\pi]$-module, and a diagonal approximation $\Delta:P_*\to{P_*\otimes{P_*}}$, which he used to compute the integral and mod-$p$ cohomology rings for semidirect products $\pi\cong\mathbb{Z}^2\rtimes_\Theta\mathbb{Z}$ with $\Theta\in{GL(2,\mathbb{Z})}$ [@Ma12]. We shall take a somewhat different approach, first computing cup products into $H^2(\pi;\mathbb{F}_2)$ and then using Poincaré duality. Our strategy in determing relations in $H^2(\pi;\mathbb{F}_2)$ shall be to use restrictions to subgroups (such as $\sqrt\pi$) and epimorphisms to quotient groups (such as $\pi/\sqrt\pi$ or small finite 2-groups), with known cohomology rings. We shall usually write $H_*(X)$ and $H^*(X)$ for the homology and cohomology of a space or group $X$, with coefficients $\mathbb{F}_2$, and denote the cup-product by juxtaposition. In each case considered below, the given generators for a group $G$ represent a basis for $H_1(G)$, and we shall use the corresponding Kronecker dual bases for $H^1(G)=Hom(H_1(G),\mathbb{F}_2)$. Let $w=w_1(\pi)$. Then $w\alpha\beta=\alpha^2\beta+\alpha\beta^2$, for all $\alpha, \beta\in{H^1}(\pi)$. In particular, if $w=0$ then $\alpha^2\beta=\alpha\beta^2$ and $(\alpha+\beta)^3=\alpha^3+\beta^3$. The first assertion follows from the Wu relation $Sq^1z=w\cup{z}$ for all $z\in{H}^{n-1}(X)$, which holds for any $PD_n$-complex $X$. The second follows easily. If $G$ is a group let $X^n(G)=\langle{g^n}|g\in{G}\rangle$ be the subgroup generated by all $n^{th}$ powers. The next lemma is a refinement of Theorem 2 of [@Hi87] (which is restated here as part (1) of the lemma). Let $G$ be a group, and $\rho,\phi,\psi\in{H^1}(G)$. Let $K=\mathrm{Ker}(\rho)$ and $L=K\cap\mathrm{Ker}(\phi)$. Then 1. the kernel of cup product from the symmetric product $\odot^2H^1(G)$ to $H^2(G)$ is the dual of $X^2(G)/X^4(G)[G,X^2(G)]$; 2. the canonical projections induce isomorphisms $H^1(G/X^2(K))\cong{H^1}(G/X^2(L))\cong{H^1}(G/X^4(G))\cong{H^1}(G)$; 3. $\rho\phi=0$ in $H^2(G)\Leftrightarrow \rho\phi=0$ in $H^2(G/X^2(K))$; 4. $\phi^2=\rho\phi+\rho\psi$ in $H^2(G) \Leftrightarrow\phi^2=\rho\phi+\rho\psi$ in $H^2(G/X^2(L))$. Part (1) is Theorem 2 of [@Hi87], while part (2) is clear. If $\phi\psi=0$ in $H^2(G)$ then there is a 1-cochain $F:G\to\mathbb{F}_2$ such that $\phi(g)\psi(h)=\delta{F}(g,h)=F(gh)+F(g)+F(h)$, for all $g,h\in{G}$. Part (3) follows easily, since $F$ restricts to a homomorphism on $K$, and is constant on cosets of $X^2(K)$. Part (4) is similar. In many of the cases considered here, the coefficients in the linear relations determining the kernel of cup product may be found by restricting to 2-generator subgroups. However, this is not always enough to determine the triple products in $H^3(\pi)$. Let $\{T,Y\}$ be the basis for $H^1(D_8)$ corresponding to the presentation $D_8=\langle{t,y}|t^2=y^4=1,~tyt^{-1}=y^{-1}\rangle$. Then $(T+Y)Y=0$ in $H^2(D_8)$. Let $D_\infty$ have the presentation $\langle{u,v}|u^2=v^2=1\rangle$, and let $U,V$ be the dual basis for $H^1(D_\infty)$. Then $H^*(D_\infty)=\mathbb{F}_2[U,V]/(UV)$. Let $f:D_\infty\to{D_8}$ be the epimorphism given by $f(u)=t$ and $f(v)=ty$. Then $f$ induces an isomorphism $D_\infty/X^4(D_\infty)\cong{D_8}$, so $H^2(f)$ is injective. Since $f^*U=T+Y$ and $f^*V=Y$, we see that $(T+Y)Y=0$ in $H^2(D_8)$. Let $E$ be the “almost extraspecial" 2-group with presentation $$\langle{t,u,v}\mid{t^2=1},~u^2=v^2,~tut^{-1}=u^{-1},~tv=vt,~uv=vu\rangle.$$ Let $\{T,U,V\}$ be the basis for $H^1(E)$ corresponding to the above presentation. Then $TU+U^2+V^2=0$ in $H^2(E)$. Since $X^2(E)\cong{Z/2Z}$, the kernel of cup product from $\odot^2H^1(G)$ to $H^2(G)$ has dimension 1 [@Hi87]. Thus there is an unique nontrivial linear relation $aT^2+bU^2+cV^2+dTU +eTV +fUV=0$ in $H^2(E)$. The coefficients can be determined by restriction to the subgroups $\langle{t}\rangle\cong{Z/4Z}$, $\langle{t,u}\rangle\cong{D_8}$, $\langle{t,v}\rangle\cong{Z/4Z}\oplus{Z/2Z}$, and $\langle{u,v}\rangle\cong{Z/4Z}\oplus{Z/2Z}$. mapping tori ============= Suppose that $\pi\cong\mathbb{Z}^2\rtimes_\Theta\mathbb{Z}$, where $\Theta=\left(\begin{smallmatrix} a&c\\ b&d \end{smallmatrix}\right)\in{GL}(2,\mathbb{Z})$. Let $\varepsilon=ad-bc=\pm1$ and $\tau=a+d$. Let $\Delta_1=\det(\Theta-I_2)=1-\tau+\varepsilon$ and $\Delta_2=(a-1,b,c,d-1)$ be the elementary divisors of $\Theta-I_2$. Then $\Delta^2_2$ divides $\Delta_1$, and $$\pi^{ab}\cong\mathbb{Z}\oplus {Z/(\Delta_1/\Delta_2)Z}\oplus{Z/\Delta_2Z}.$$ Let $\beta=\beta_1(\pi;\mathbb{F}_2)$. Then $1\leq\beta\leq3$, and $\beta_2(\pi;\mathbb{F}_2)=\beta$, by Poincaré duality. Let $\rho:\pi\to{Z/2Z}$ be the unique epimorphimorphism which factors through $\pi/\sqrt\pi\cong\mathbb{Z}$. If $\pi$ is non-orientable then $\rho=w_1(M)$, and $K=\pi^+$, the maximal orientable subgroup of $\pi$. 1\. If $\tau$ is odd then $\Delta_1$ is odd and $\pi^{ab}\cong\mathbb{Z}\oplus{odd}$. In this case $\rho$ is the unique epimorphism from $\pi$ to ${Z/2Z}$, and $$H^*(\pi)\cong\mathbb{F}_2[\rho,\Xi]/(\rho^2,\Xi^2),$$ where $\Xi$ has degree 2, by Poincaré duality. 2\. If $\tau\equiv\varepsilon-1$ [*mod*]{} $(4)$ then $\pi^{ab}\cong\mathbb{Z}\oplus{Z/2Z}\oplus{odd}$, and $\beta=2$. Hence $H^1(\pi)=\langle{\rho,\sigma}\rangle$, where $\sigma$ does not factor through $Z/4Z$. Moreover, if $G=\pi/X^4(\pi)$ then $X^2(G)\cong(Z/2Z)^2$ is central in $G$. Thus $\rho^2=\rho\sigma=0$, by Lemma 4, while $\sigma^2\not=0$. Hence $H^2(\pi)=\langle\sigma^2,\Xi\rangle$, for some $\Xi$ of degree 2. Duality then implies that $\sigma^3=\rho\Xi\not=0$. We may assume also that $\sigma\Xi=0$, and so $$H^*(\pi)\cong\mathbb{F}_2[\rho,\sigma,\Xi]/ (\rho^2,\rho\sigma,\sigma\Xi,\rho\Xi+\sigma^3,\Xi^2).$$ 3\. If $\tau\equiv\varepsilon+1$ [*mod*]{} (4) and $\Delta_2$ is odd then $\pi^{ab}\cong\mathbb{Z}\oplus{Z/2^kZ}\oplus{odd}$, for some $k\geq2$. Hence $H^1(\pi)=\langle\rho,\sigma\rangle$, where $\sigma^2=\rho^2=0$. Since $\rho\sigma=0$, by the nondegeneracy of Poincaré duality, $$H^*(\pi)\cong\mathbb{F}_2[\rho,\sigma,\Xi,\Omega]/ (\rho^2,\rho\sigma,\sigma^2,\rho\Omega,\sigma\Xi,\rho\Xi+\sigma\Omega, \Xi^2,\Omega^2,\Xi\Omega),$$ where $\Xi$ and $\Omega$ have degree 2. In all the remaining cases $\beta=3$. For if $\tau\equiv\varepsilon+1$ [*mod*]{} $(4)$ and $\Delta_2$ is even then $a$ and $d$ are odd and $b$ and $c$ are even. Hence $\Delta_1=2^kq$ and $\Delta_2=2^\ell{q'}$, where $0<\ell\leq\frac{k}2$ and $q, q'$ are odd. In this case $\pi^{ab}\cong\mathbb{Z}\oplus{Z/2^{k-\ell}Z}\oplus{Z/2^\ell{Z}}\oplus{odd}$, so the images of $\{t,x,y\}$ form a basis for $H_1(\pi)$. Let $\{\rho,\sigma,\psi\}$ be the dual basis, so that $$\sigma(x)=\psi(y)=1\quad\mathrm{and}\quad \sigma(t)=\sigma(y)=\psi(t)=\psi(x)=0.$$ If $G=\pi/X^4(\pi)$ then $X^2(G)=\langle{t^2,x^2,y^2}\rangle\cong(Z/2Z)^3$ is central in $G$, so the kernel of cup product from $\odot^2H^1(\pi)$ to $H^2(\pi)$ has rank 3. It then follows from Poincaré duality that $H^*(\pi)$ is generated as a ring by $H^1(\pi)$. In each case, $\rho\sigma^2=\rho\rho\sigma=0$ and $\rho\psi^2=\rho\rho\psi=0$, by Lemma 3. Hence $\rho\sigma\psi\not=0$, by the nondegeneracy of Poincaré duality. It then follows easily that $\rho\sigma$, $\rho\psi$ and $\sigma\psi$ are linearly independent, and so form a basis for $H^2(\pi)$. We may write $$\sigma^2=m\rho\sigma+n\rho\psi+p\sigma\psi\quad \mathrm{ and}\quad\psi^2=q\rho\sigma+r\rho\psi+s\sigma\psi,$$ for some $m,\dots,s$. On restricting to $\sqrt\pi$, we see that $p=s=0$, since $\sigma^2|_{\sqrt\pi}=\psi^2|_{\sqrt\pi}=0$ and $\rho|_{\sqrt\pi}=0$, while $\sigma\psi|_{\sqrt\pi}\not=0$. Since $\rho\sigma^2=\rho^2\sigma=\rho\psi^2=\rho^2\psi=0$, taking cup products with $\sigma$ and $\psi$ gives $$\sigma^3=n\rho\sigma\psi,\quad\sigma^2\psi=m\rho\sigma\psi, \quad\psi^3=q\rho\sigma\psi\quad\mathrm{and}\quad\sigma\psi^2=r\rho\sigma\psi.$$ 4\. If $\ell\geq2$ then $a\equiv{d}\equiv1$ and $b,c\equiv0$ [*mod*]{} (4), so $\varepsilon\equiv1$ [*mod*]{} (4) also, i.e., $\pi$ is orientable. In this case $\sigma^2=\psi^2=\rho^2=0$, and so $$H^*(\pi)\cong\mathbb{F}_2[\rho,\sigma,\psi]/(\rho^2,\sigma^2,\psi^2).$$ Suppose now that $\ell=1$. 5\. If $\pi$ is orientable and $\Delta_1\equiv0$ [*mod*]{} (8) we may assume that one of $\sigma$, $\psi$ or $\sigma+\psi$ factors through $Z/4Z$. Thus either $\sigma^2=0$, $\psi^2=0$ or $\sigma^2=\psi^2$. We may assume that $\sigma^2\not=0$. Then $\rho\sigma^2=\rho^2\sigma=0$ and $\psi\sigma^2=\psi^2\sigma=0$, and so $\sigma^3\not=0$, by the nonsingularity of Poincaré duality. Hence $$H^*(\pi)\cong\mathbb{F}_2[\rho,\sigma,\psi]/(\rho^2,\rho\psi+\sigma^2,\psi^2).$$ In this case we see that $\phi^3=0\Leftrightarrow\phi^2=0$. If $\pi$ is orientable and $\Delta_1\equiv4$ [*mod*]{} (8) then $\tau\equiv6$ [*mod*]{} (8) and $a,d$ are odd, and so $a\equiv{d}$ [*mod*]{} (4). In this case $\psi^2\not=0$ and $(\sigma+\psi)^2\not=0$ also, and so $\sigma^2=m\rho\sigma+n\rho\psi$ and $\psi^2=q\rho\sigma+r\rho\psi$ are linearly independent. Hence $mr+nq=1$ in $\mathbb{F}_2$. Since $w=0$, $\sigma^2\psi=\sigma\psi^2$ and so $m=r$. 6\. Suppose first that $a\equiv1$ [*mod*]{} (4). Then $bc\equiv4$ [*mod*]{} (8), and so $b\equiv{c}\equiv2$ [*mod*]{} (4). Let $L_\phi=\mathrm{Ker}(\rho)\cap\mathrm{Ker}(\phi)$. Then $\pi/X^2(L_\phi)$ has a presentation $$\langle {t,x,y}\mid t^4=x^4=y^2=1,~tx=xt,~tyt^{-1}=x^2y,~xy=yx\rangle.$$ Let $J=\langle{t,x}\rangle\cong(Z/4Z)^2$. Then $\sigma^2|_J=\rho\psi|_J=0$, while $\rho\sigma|_J\not=0$. Applying part (3) of Lemma 4, we see that $m=0$, and so $\sigma^2=\rho\psi$ and $\psi^2=\rho\sigma$. (Note, however, that Lemma 4 does [*not*]{} assert that the relation $\psi^2=q\rho\sigma+r\rho\psi$ also holds in $\pi/X^2(L_\phi)$! For this, we could use $L_\psi=\mathrm{Ker}(\rho)\cap\mathrm{Ker}(\psi)$ instead.) Hence $$H^*(\pi)\cong\mathbb{F}_2[\rho,\sigma,\psi]/(\rho^2,\rho\psi+\sigma^2, \rho\sigma+\psi^2).$$ In particular, $\sigma^3=\psi^3=(\rho+\sigma)^3=(\rho+\psi)^3\not=0$. If $a\equiv-1$ [*mod*]{} (4) then $bc\equiv0$ [*mod*]{} (8). If, say, $b\equiv2$ [*mod*]{} (4) (so $c\equiv0$ [*mod*]{} (4)) then the change of basis $x'=x$, $y'=xy$ reduces this case to the one just considered. In terms of the given basis, we have $$H^*(\pi)\cong\mathbb{F}_2[\rho,\sigma,\psi]/(\rho^2, \rho\sigma+\sigma^2,\rho\psi+\sigma^2+\psi^2).$$ In this case $\sigma^3\not=0$, but $\psi^3=0$. A similar result holds if $b\equiv0$ [*mod*]{} (4) and $c\equiv2$ [*mod*]{} (4). 7\. If, however, $a\equiv-1$ [*mod*]{} (4) and $b\equiv{c}\equiv0$ [*mod*]{} (4) then $\pi/X^4(\pi)$ has a presentation $$\langle {t,x,y}\mid t^4=x^4=y^4=1,~txt^{-1}=x^{-1},~tyt^{-1}=y^{-1}, ~xy=yx\rangle.$$ In this case $J=\langle{t,x}\rangle$ is non-abelian, and $\sigma^2|_J\not=0$, while $\rho\psi|_J=0$. Hence we must have $m=r=1$. It is clear from the symmetry of the presentation for $\pi/X^4(\pi)$ that we must also have $n=q$ in this case, and so $n=q=0$. Thus $$H^*(\pi)\cong\mathbb{F}_2[\rho,\sigma,\psi]/(\rho^2, \rho\sigma+\sigma^2,\rho\psi+\psi^2).$$ We now find that $\phi^3=0$ for all $\phi\in{H^1}(\pi)$. If $\ell=1$ and $M$ is non-orientable then $a$ and $d$ are odd, and $\Delta_1=-a-d\equiv0$ [*mod*]{} (4). In this case $\rho=w_1(M)$, and so $\sigma^2\psi+\sigma\psi^2=\rho\sigma\psi\not=0$, by Lemma 3. After swapping $x$ and $y$, if necessary, we may assume that $a\equiv1$ [*mod*]{} (4). 8\. If $bc\equiv0$ [*mod*]{} (8) then, after a further change of basis of the form $x'=x, y'=xy$ or $x'=xy, y'=y$, if necessary, we may assume that $b\equiv{c}\equiv0$ [*mod*]{} (4). Then $\sigma^2=0$, and $\pi/\langle\langle{t^2,x,y^4}\rangle\rangle\cong{D_8}$, so $(\rho+\psi)\psi=0$ also. Hence $$H^*(\pi)\cong\mathbb{F}_2[\rho,\sigma,\psi]/ (\rho^2,\sigma^2,\rho\psi+\psi^2).$$ In particular, $(\sigma+\psi)^3=(\rho+\sigma+\psi)^3\not=0$, and all other classes have cube 0. In terms of the given bases, the other cases are: If ${b}\equiv0$ and $c\equiv2$ [*mod*]{} (4) then $$H^*(\pi)\cong\mathbb{F}_2[\rho,\sigma,\psi]/ (\rho^2,\sigma^2+\psi^2,\rho\psi+\psi^2,\sigma^2\psi).$$ Here $\sigma^3=(\rho+\sigma)^3\not=0$ and all other classes have cube 0. If ${b}\equiv2$ and ${c}\equiv0$ [*mod*]{} (4) then $$H^*(\pi)\cong\mathbb{F}_2[\rho,\sigma,\psi]/(\rho^2, \sigma^2,\psi^2+\rho\sigma+\rho\psi).$$ Here $\psi^3=(\rho+\psi)^3\not=0$ and all other classes have cube 0. 9\. If ${b}\equiv{c}\equiv2$ [*mod*]{} (4) then $\sigma^2$ and $\psi^2$ are linearly independent. There are three distinct epimorphisms from $\pi$ to the almost extraspecial group $E$, given by $f(x)=u^{-1}v), f(y)=u$; $g(x)=v,g(y)=uv^{-1}$; and $h(x)=v, h(y)=u$. Using these epimorphisms to pull back the relation given in Lemma 5, we find that $$H^*(\pi)\cong\mathbb{F}_2[\rho,\sigma,\psi]/(\rho^2, \sigma^2+\rho\psi,\psi^2+\rho\sigma+\rho\psi).$$ In particular, every epimorphism $\phi\not=\rho$ has nonzero cube. unions of twisted $I$-bundles ============================= Suppose that $\pi/\sqrt\pi\cong{D_\infty}$. Then $\pi$ is orientable, and has a presentation $$\langle u,v,y\mid uyu^{-1}=y^{-1},~v^2=u^{2a}y^b,~ vu^{2c}y^dv^{-1}=u^{-2c}y^{-d} \rangle,$$ where $ad-bc=\pm1$ and $abcd\not=0$. Let $B=\langle{u,y}\rangle$ and $C=\langle{v,u^{2c}y^d}\rangle$. If $b$ is odd then $\pi^{ab}\cong{Z/4cZ}\oplus{Z/4Z}$, where the summands are generated by $u$ and $u^{-a}v$, respectively. Let $U(u)=V(v)=1$, $U(v)=a$ and $V(u)=0$. Then $$H^*(\pi)\cong\mathbb{F}_2[U,V,\Xi,\Omega]/ (U^2,UV,V^2,U\Xi+V\Omega,\Xi^2,\Omega^2,\Xi\Omega),$$ where $\Xi$ and $\Omega$ have degree 2. If $b$ is even then $\pi^{ab}\cong{Z/4cZ}\oplus(Z/2Z)^2$ and the images of $u,v$ and $y$ represent a basis for $H_1(\pi)$. Let $\{U,V,Y\}\in{H^1}(\pi)$ be the dual basis. Then $U^2$, $V^2$ and $Y^2$ are all nonzero, but $W=U+V$ lifts to a homomorphism from $\pi$ to $Z/4Z$, and so $W^2=0$. Hence $U^2=V^2$. Since $U$ and $V$ are induced from classes in $H^1(D_\infty)$ we have $UV=0$. We also have $UY|_B=Y^2|_B$ and $VY|_C=Y^2|_C$, while $U|_C,V|_B,U^2|_B$ and $V^2|_C$ are all 0. Suppose that $pU^2+qY^2+rUY+sVY=0$ in $H^2(\pi)$. On restricting to the subgroups $B$ and $C$, we find that $q+r=q+s=0$. Since $U^2\not=0$ we must have $q=r=s=1$. Multiplying by $U$ and $V$, we find that $UY^2+U^2Y=0$ and $VY^2+V^2Y=0$. Poincaré duality for $\pi$ now implies that $\{U^2,Y^2,UY\}$ is a basis for $H^2(\pi)$, while $UY^2=U^2Y=VY^2$ generates $H^3(\pi)$. We see also that $U^3=U^2V=UV^2=V^3=(U+V)^3=0$, while $(U+Y)^3=(V+Y)^3=(U+V+Y)^3=Y^3$. Suppose first that $b\equiv0$ [*mod*]{} (4). Then $G=\pi/\langle\langle{uv,u^2,y^4}\rangle\rangle\cong{D_8}$. Hence $(U+V+Y)Y=0$ in $H^3(\pi)$. It follows easily that $Y^3=0$, and so all cubes are 0 in $H^3(\pi)$. If $b\equiv2$ [*mod*]{} (4) then $\pi/\langle\langle{u^2,(uv)^2,v^4,y^4}\rangle\rangle$ has a presentation $$\langle{u,v,y}\mid{u^2=(uv)^2=v^4=1},~uyu^{-1}=vyv^{-1}=y^{-1},~v^2=y^2\rangle$$ Hence there is an epimorphism $f:\pi\to{E}$, given by $f(u)=t$, $f(v)=u$ and $f(y)=u^{-1}t^{-1}v$. Since $f^*T=U+Y$, $f^*U=V+Y$, $f^*V=Y$ and $UV=0$, it follows from Lemma 6 that $UY+VY+V^2+Y^2=0$ in $H^2(\pi)$. Multiplying by $Y$, we find that $UY^2+Y^3=0$ and so $Y^3\not=0$. In this case, only the cubes induced from $H^*(\pi/\sqrt\pi)$ are zero. the borsuk-ulam index ===================== We may identify an epimorphism $\phi$ with a nonzero class in $H^1(N;\mathbb{F}_2)$. Then $BU(N,\phi)=1\Leftrightarrow\phi$ lifts to an integral class $\Phi\in{H^1(N;\mathbb{Z})}$, while $BU(N,\phi)=n\Leftrightarrow\phi^n\not=0$ in $H^n(N;\mathbb{F}_2)$ In general, $1\leq{BU(N,\phi)}\leq{n}$. See [@GHZ]. When $n=3$ the remaining possibility is that $BU(M,\phi)=2\Leftrightarrow\phi^2=0$ but $\phi$ is not the reduction of an integral class. Suppose first that $\pi/\sqrt\pi\cong\mathbb{Z}$. Then the following results are immediate from §3. 1\. If $\rho:\pi\to{Z/2Z}$ is the unique epimorphism which factors through $\pi/\sqrt\pi\cong\mathbb{Z}$ then $BU(M,\rho)=1$. 2\. If $\tau\equiv\varepsilon-1$ [*mod*]{} $(4)$ then $BU(M,\phi)=3$ for all $\phi\not=\rho$. 3\. If $\tau\equiv\varepsilon+1$ [*mod*]{} (4) and either $\Delta_2$ is odd or $a\equiv{d}\equiv1$ [*mod*]{} (4) and $b,c$ are divisible by 4, then $BU(M,\phi)=2$ for all $\phi\not=\rho$. 4\. If $\varepsilon=1$, $\Delta_1\equiv0$ [*mod*]{} (8) and $\Delta_2\equiv2$ [*mod*]{} (4) then $BU(M,\phi)=2$ for the two epimorphisms $\phi\not=\rho$ such that $\phi^2=0$ (i.e, that factor through $Z/4Z$) and $BU(M,\phi)=3$ for the four such that $\phi^2\not=0$. 5\. If $\varepsilon=1$, $\Delta_1\equiv4$ [*mod*]{} (8) and $\Theta\equiv-I_2$ [*mod*]{} (4) then $BU(M,\phi)=2$ for all $\phi\not=\rho$. 6\. If $\varepsilon=1$ and $\Delta_1\equiv4$ [*mod*]{} (8), but $\Theta\not\equiv-I_2$ [*mod*]{} (4), then $BU(M,\phi)=2$ for the two epimorphisms $\phi\not=\rho$ such that $\phi^2=0$ and $BU(M,\phi)=3$ for the four such that $\phi^2\not=0$. 7\. If $\varepsilon=-1$, $\tau\equiv0$ [*mod*]{} (4), $\Delta_2\equiv2$ [*mod*]{} (4) and $bc\equiv0$ [*mod*]{} (8) then $BU(M,\phi)=2$ for the four epimorphisms $\phi\not=\rho$ such that $\phi^3=0$ and $BU(M,\phi)=3$ for the two such that $\phi^3\not=0$. 8\. If $\varepsilon=-1$, $\tau\equiv0$ [*mod*]{} (4), $\Delta_2\equiv2$ [*mod*]{} (4) and $bc\equiv4$ [*mod*]{} (8) then $BU(M,\phi)=3$ for all $\phi\not=\rho$. Suppose now that $\pi/\sqrt\pi\cong{D_\infty}$. Then the following results are immediate from §4. 9\. If $\pi^{ab}\cong{Z/4cZ}\oplus{Z/4Z}$ then $BU(M,\phi)=2$ for all $\phi$. 10\. If $\pi^{ab}\cong{Z/4cZ}\oplus(Z/2Z)^2$ and $b\equiv0$ [*mod*]{} (4) then $BU(M,\phi)=2$ for all $\phi$. 11\. If $\pi^{ab}\cong{Z/4cZ}\oplus(Z/2Z)^2$ and $b\equiv2$ [*mod*]{} (4) then $BU(M,\phi)=2$ for epimorphisms $\phi$ which factors through $\pi/\sqrt\pi$, while $BU(M,\phi)=3$ otherwise. other geometries ================ We remark finally that similar arguments may be used to determine the $\mathbb{F}_2$-cohomology rings and Borsuk-Ulam invariants for pairs $(N,\phi)$ with $N$ a closed $\mathbb{E}^3$- or $\mathbb{N}il^3$-manifold. These manifolds are all Seifert fibred over flat 2-orbifolds. Since they have been covered in [@BGHZ], we shall confine ourselves to some brief observations. The ten closed flat 3-manifolds may be easily treated individually. The only one admitting a class $\phi$ with $\phi^3\not=0$ has group $G_4$, with holonomy $Z/4Z$ and abelianization $\mathbb{Z}\oplus{Z/2Z}$. Thus $H^1(\pi)=\langle{T,X}\rangle$, where $T^2=0$ and $X^2\not=0$. We may deduce that $TX=0$ also, by mapping $G_4$ onto $D_8$. It follows easily that $$H^*(G_4)\cong\mathbb{F}_2[T,X,\Omega]/(T^2,TX,X\Omega, T\Omega+X^3,\Omega^2),$$ where $\Omega$ has degree 2. (Thus $X^3=(T+X)^3\not=0$. These classes correspond to the two epimorphisms without integral lifts.) The possible Seifert bases $B$ of closed $\mathbb{N}il^3$-manifolds are the seven flat 2-orbifolds with no reflector curves: $B=T$, $Kb$, $S(2,2,2,2)$, $S(2,4,4)$, $S(2,3,6)$, $S(3,3,3)$ or $P(2,2)$. Let $\beta=\pi_1^{orb}(B)$ be the orbifold fundamental group of the base. Then $\pi^{ab}$ is an extension of $\beta^{ab}$ by a finite cyclic group $Z/qZ$, if the base is orientable ($B\not=Kb$ or $P(2,2)$), and by $Z/(2,q)Z$ otherwise. The ring $H^*(\pi)$ depends only on the base $B$ and the residue of $q$ [*mod*]{} (4). If $B=T$ or $Kb$ then $\pi\cong\mathbb{Z}^2\rtimes_\Theta\mathbb{Z}$, for some $\Theta\in{GL(2,\mathbb{Z})}$. These are in fact the cases requiring most effort. In all other cases $\pi^{ab}$ is finite, and the projection of $\pi$ onto $\beta$ induces an isomorphism $H_1(\pi)\cong{H_1(\beta)}$. When $B=S(2,3,6)$ or $S(3,3,3)$ this group is cyclic. (In particular, such $\mathbb{N}il^3$-manifolds are neither mapping tori nor unions of twisted $I$-bundles.) When $B=S(2,4,4)$ we have $\pi/X^4(\pi)\cong\beta/X^4(\beta)\cong{G_4}/X^4(G_4)$. The cases of $S(2,2,2,2)$ and $P(2,2)$ are related to those of the flat 3-manifolds $G_2$ and $B_4$, respectively. The Borsuk-Ulam Theorem and its applications and extensions are treated in detail in the book [@Mat]. [99]{} Gonçalves, D.L., Hayat, C., Zvengrowski, P. The Borsuk-Ulam theorem for manifolds, with applications to dimensions two and three, in [*Group actions and homogeneous spaces*]{}, Fak. Mat. Fyziky Inform. Univ. Komenského, Bratislava (2010), 9–28. Bauval, A., Gonçalves, D.L., Hayat, C. and Zvengrowski, P. The Borsuk-Ulam Theorem for double coverings of Seifert manifolds, Proceedings of the Brazilian-Polish Topology meeting, July 2012. Carlson, J.F. Cohomology of 2-groups, www.math.uga.edu/ jfs/groups2/ Hillman, J.A. The kernel of integral cup product, J. Austral. Math. Soc. 43 (1987), 10–15. Hillman, J.A. [*Four-Manifolds, Geometries and Knots*]{}, GT Monographs 5, Geometry and Topology Publications (2002). Latest revision: seehttp://www.maths.usyd.edu.au/u/jonh/ . Martins, S.T. [*Aproximaçiones da diagonal e anéis da cohomologia dos grupos fundamentales das superfícies, de fibrados de toros e de certos grupos virtualmente cíclicos,*]{} PhD thesis, Universidade de São Paulo (2012). Matoušek, J. (with A.Björner and G.M.Ziegler) [*Using the Borsuk-Ulam Theorem. Lectures on topological methods in combinatorics and geometry*]{}, Universitext, Springer-Verlag, Berlin – Heidelberg – New York (2003). Robinson, D.J.S. [*A Course in the Theory of Groups*]{}, Graduate Texts in Mathematics 80, Springer-Verlag, Berlin - Heidelberg - New York (1982).
--- abstract: 'It is shown that the classical description of pair production effect is possible, i.e. one can describe pair production without a reference to quantum principles. Pair production appears at statistical description of stochastic relativistic particles. There is a special force field which is responsible for pair production. This field is a reasonable consequence of quantum stochasticity. Consideration of quantum systems as stochastic systems and statistical description of them generates hydrodynamic interpretation of quantum phenomena. In the collision problem the hydrodynamic interpretation appears to be alternative to the conventional interpretation, based on quantum principles and on consideration of the wave function as a principal object of dynamics. Hydrodynamic interpretation leads to such a statement of the collision problem, which is alternative to the conventional S-matrix theory.' author: - 'Yuri A. Rylov' date: | Institute for Problems in Mechanics, Russian Academy of Sciences\ 101-1,Vernadskii Ave., Moscow, 119526, Russia\ email: rylov@ipmnet.ru\ Web site: [$http://rsfq1.physics.sunysb.edu/\symbol{126}rylov/yrylov.htm$]{}\ or mirror Web site: [$http://gasdyn-ipm.ipmnet.ru/\symbol{126}rylov/yrylov.htm$]{} title: 'Classical description of pair production. ' --- Introduction ============ Pair production is a specific quantum effect. It is described conventionally in terms of quantum theory. We do not know how to describe the pair production effects in classical terms. But if the classical description of the pair production were possible, it would be very useful. Indeed, at high energies of particles their de Broglie wave length $\lambda _{\mathrm{B}}=\hbar /p$ becomes to be very small, and description of high energy particles becomes semiclassical or even classical. The classical description is simpler, than the quantum one, because it uses less information and admits a simpler interpretation. Unfortunately, we cannot realize advantages of classical description, because at high energies the pair production processes are dominating, but we do not know how to describe them in classical terms. In general, we know about the pair production only that it appears at collision of high energy particles, and it is described only in terms of quantum theory. But what agents are responsible for the pair production remains to be seen. In this paper we are going to investigate what agents are responsible for the pair production and to show that a classical description of pair production effects is possible. Speaking about classical description, we mean only that *principles of quantum mechanics are not used* at such a description. Classical description does not mean a classical approximation of quantum description. Our classical description is an exact description. In particular, it describes quantum effects, using quantum constant $\hbar $, which appears as a constant describing the space-time properties (but not as an attribute of quantum principles). In our consideration we use model conceptions of quantum phenomena (MCQP), which is a new stage in investigation of microcosm. MCQP relates to conventional quantum theory (axiomatic conception of quantum phenomena (ACQP)) in the same way, as the statistical physics relates to thermodynamics. The main difference between MCQP and ACQP in investigation methods which are used. Methods of MCQP are more subtle and effective, than those of ACQP. One can see this from the next table $ \begin{array}{cc} \text{ACQP} & \text{MCQP} \\ \begin{array}{c} \text{Combination of nonrelativistic } \\ \text{quantum technique with } \\ \text{principles of relativity} \end{array} & \begin{array}{c} \text{Consequent relativistic description } \\ \text{at all stages} \end{array} \\ \begin{array}{c} \text{1. Quantization: procedure on } \\ \text{the conceptual level:} \\ \mathbf{p}\rightarrow -i\hbar \mathbf{\nabla }\;\;\;\text{etc. } \end{array} & \begin{array}{c} \text{1. Dynamic quantization: relativistic } \\ \text{procedure on the dynamic level} \\ m^{2}\rightarrow m_{\mathrm{eff}}^{2}=m^{2}+\frac{\hbar ^{2}}{c^{2}}\left( \kappa _{l}\kappa ^{l}+\partial _{l}\kappa ^{l}\right) \end{array} \\ \begin{array}{c} \text{2. Transition to classical description:} \\ \text{procedure on conceptual level} \\ \hbar \rightarrow 0\qquad \psi \rightarrow \left( x,p\right) \end{array} & \begin{array}{c} \text{2. Dynamic disquantization: relativistic} \\ \text{ procedure on dynamic level} \\ \partial ^{k}\rightarrow \frac{j^{k}j^{l}}{j_{s}j^{s}}\partial _{l} \end{array} \\ \begin{array}{c} \text{3. Interpretation in terms of wave} \\ \text{function }\psi \end{array} & \begin{array}{c} \text{3. Interpretation in terms of statistical} \\ \text{average world lines (WL)} \\ \frac{dx^{i}}{d\tau }=j^{i}\left( x\right) ,\;\;\; \\ j^{k}=-\frac{i\hbar }{2}\left( \psi ^{\ast }\partial ^{k}\psi -\partial ^{k}\psi ^{\ast }\cdot \psi \right) \end{array} \\ \begin{array}{c} \text{4. One kind of measurement, as } \\ \text{far as only one statistical average } \\ \text{object }\left\langle \mathcal{S}\right\rangle \text{ is considered. It is } \\ \text{referred to as quantum system} \end{array} & \begin{array}{c} \text{4.Two kinds of measurement, because } \\ \text{two kinds of objects (individual }\mathcal{S}_{\mathrm{st}}\text{ } \\ \text{and statistical average }\left\langle \mathcal{S}\right\rangle \text{) are } \\ \text{considered} \end{array} \\ \begin{array}{c} \text{5. Additional hypotheses are used} \\ \text{(QM principles) } \end{array} & \begin{array}{c} \text{5. \textit{No additional hypotheses are used}} \end{array} \end{array} $ In this paper we develop technique of MCQP and apply methods of MCQP for investigation of the pair production process. This technique is based on consideration of quantum mechanics as a special case of the stochastic system dynamics [@R2002]. The stochastic system dynamics is suitable for description of any stochastic systems (not only quantum), and because of this it cannot use quantum principles and does not use them. All description is produced in classical (non-quantum) terms, although description in terms of wave function is also possible as a special case of a description. Such an approach is very convenient, because it gives a very simple interpretation of quantum effects in terms of classical dynamics of particles and fluids. The main distinction between the quantum mechanics and dynamics of stochastic systems lies in their relation to the wave function. In the quantum mechanics the wave function $\psi $ is a fundamental object, whose properties are defined by quantum principles. As a result the interpretation of all objects which is constructed on the base of the wave function is rather obscure, because the meaning of the wave function itself is obscure. In the stochastic system dynamics the wave function is simply a method of description of any nondissipative fluid [@R99]. In this case the meaning of all objects constructed on the base of the wave function and the meaning of the wave function itself are obtained rather simply in terms of classical description of a fluid (continuous medium). Connection between the fluid and the Schrödinger equation is known since the beginning of the quantum mechanics construction [@M26; @B26]. In after years many authors developed this interplay known as hydrodynamic interpretation of quantum mechanics [@B52; @T52; @T53; @JZ63; @JZ64; @HZ69; @B73; @BH89; @H93]. But this interpretation was founded ultimately on the wave function as a fundamental object of dynamics. It cannot go outside the framework of quantum principles, because the connection between the hydrodynamic interpretation and the quantum mechanics was one-way connection. One could obtain the irrotational fluid flow from the dynamic equation for the wave function (Schrödinger equation), but one did not know how to transform dynamic equations for a fluid to the dynamic equation for a wave function. In other words, we did not know how to describe rotational fluid flow in terms of the wave function. In terms of the wave function we could describe only irrotational fluid flow. To describe arbitrary fluid flow in terms of a wave function, one needs to integrate conventional dynamic equations for a fluid (Euler equations). Indeed, the Schrödinger equation $$i\hbar \frac{\partial \psi }{\partial t}+\frac{\hbar ^{2}}{2m}\mathbf{\nabla }^{2}\psi =0 \label{a0.1}$$ may be reduced to the hydrodynamic equations for the variables $\rho ,\mathbf{v} $, describing the fluid state. Substituting $\psi =\sqrt{\rho }\exp \left( i\hbar \varphi \right) $ in (\[a0.1\]) and separating real and imaginary parts of the equation, we obtain expressions for time derivatives $\partial _{0}\rho $ and $\partial _{0}\varphi $. To obtain expression for the time derivative $\partial _{0}\mathbf{v}$ of the velocity $\mathbf{v=}\frac{\hbar }{m}\mathbf{\nabla }\varphi $, we need to differentiate dynamic equation for $\partial _{0}\varphi $, forming combination $\partial _{0}\mathbf{v=\nabla }\left( \frac{\hbar }{m}\partial _{0}\varphi \right) $. The reverse transition from hydrodynamic equations to dynamic equations for the wave function needs a general integration of hydrodynamic equations. This integration is simple in the partial case of irrotational flow, but it is a rather complicated mathematical problem in the general case, when a result of integration has to contain three arbitrary functions of three arguments. Without producing this integration one cannot derive description of a fluid in terms of the wave function, and one cannot manipulate dynamic equations, transforming them from representation in terms of $\rho $, $\mathbf{v}$ to representation in terms of wave function and back. This problem has not been solved for years. It had been solved in the end of eighties, and the first application of this integration can be found in [@R89]. Systematical application of this integration for description of quantum phenomena began in 1995 [@R95; @R995]. The problem of general integration of four hydrodynamic Euler equations $$\begin{aligned} \partial _{0}\rho +\mathbf{\nabla }\left( \rho \mathbf{v}\right) &=&0 \label{a0.2} \\ \partial _{0}\mathbf{v+}\left( \mathbf{v\nabla }\right) \mathbf{v} &=&\mathbf{-}\frac{1}{\rho }\mathbf{\nabla }p,\qquad p=p\left( \rho ,\mathbf{\nabla }\rho \right) \label{a0.3}\end{aligned}$$ seems to be hopeless. It is really so, if the Euler system (\[a0.2\]), (\[a0.3\]) is considered to be a complete system of dynamic equations. In fact, the Euler equations (\[a0.2\]), (\[a0.3\]) do not form a complete system of dynamic equations, because it does not describe motion of fluid particles along their trajectories. To obtain the complete system of dynamic equations, we should add to the Euler system so called Lin constraints [@L63] $$\partial _{0}\mathbf{\xi }+\left( \mathbf{v\nabla }\right) \mathbf{\xi }=0 \label{a0.4}$$ where $\mathbf{\xi }=\mathbf{\xi }\left( t,\mathbf{x}\right) =\left\{ \xi _{1},\xi _{2},\xi _{3}\right\} $ are three independent integrals of dynamic equations $$\frac{d\mathbf{x}}{dt}=\mathbf{v}\left( t,\mathbf{x}\right) ,$$ describing motion of fluid particles along their trajectories. Seven equations (\[a0.2\]) – (\[a0.4\]) form the complete system of dynamic equations, whereas four Euler equations (\[a0.2\]), (\[a0.3\]) form only a closed subsystem of the complete system of dynamic equations. The wave function is expressed via hydrodynamic potentials $\mathbf{\xi }=\left\{ \xi _{1},\xi _{2},\xi _{3}\right\} $**,** which are known also as Clebsch potentials [@C57; @C59]. In general case of arbitrary fluid flow in three-dimensional space the complex wave function $\psi $ has two complex components $\psi _{1}$, $\psi _{2}$ (or three independent real components) $$\psi =\left( \begin{array}{c} \psi _{1} \\ \psi _{2} \end{array} \right) =\left( \begin{array}{c} \sqrt{\rho }e^{i\varphi }u_{1}\left( \mathbf{\xi }\right) \\ \sqrt{\rho }e^{i\varphi }u_{2}\left( \mathbf{\xi }\right) \end{array} \right) ,\qquad \left| u_{1}\right| ^{2}+\left| u_{2}\right| ^{2}=1 \label{a0.5}$$ It is impossible to obtain general solution of the Euler system (\[a0.2\]), (\[a0.3\]), but one can partially integrate the complete system (\[a0.2\]) – (\[a0.4\]), reducing its order to four dynamic equations for the wave function (\[a0.5\]). Practically it means that one integrates dynamic equations (\[a0.4\]), where the function $\mathbf{v}\left( t,\mathbf{x}\right) $ is determined implicitly by equations (\[a0.2\]), (\[a0.3\]). Such an integration and reduction of the order of the complete system of dynamic equations appears to be possible, because the system (\[a0.2\]) – (\[a0.4\]) has the symmetry group, connected with transformations of the Clebsch potentials $$\xi _{\alpha }\rightarrow \tilde{\xi}_{\alpha }=\tilde{\xi}_{\alpha }\left( \mathbf{\xi }\right) ,\qquad \alpha =1,2,3,\qquad \frac{\partial \left( \tilde{\xi}_{1},\tilde{\xi}_{2},\tilde{\xi}_{3}\right) }{\partial \left( \xi _{1},\xi _{2},\xi _{3}\right) }\neq 0 \label{a0.6}$$ Being a finite function of Clebsch potentials $\mathbf{\xi }$**,** the wave function contains information on motion of fluid particles along their trajectories. This information cannot be obtained from the Euler equations (\[a0.2\]), (\[a0.3\]) without using Lin constraints (\[a0.4\]). Dependence on $\mathbf{\xi }$ takes place only in the case, when the wave function has more, than one component. In the case of the Schrödinger equation for the spinless particle the wave function has only one component $\psi =\sqrt{\rho }e^{i\varphi }u_{1}\left( \mathbf{\xi }\right) $, where in virtue of the second equation (\[a0.5\]) $u_{1}\left( \mathbf{\xi }\right) =1$, and the wave function does not depend on $\mathbf{\xi }$. See details in [@R999; @R99]. After this integration it became clear, that the wave function is only a method of the fluid description. Application of the wave function as a fundamental object of quantum mechanics is too restrictive, and its meaning is obscure, if it is determined axiomatically by quantum principles. The meaning of the wave function becomes clear, when it is considered to be a method of the fluid description. Two possible conceptions of the quantum phenomena are described by the following scheme $$\begin{tabular}{llllll} $ \begin{array}{c} \text{axiomatic} \\ \text{conception} \end{array} :$ & $ \begin{array}{c} \text{quantum} \\ \text{principles} \end{array} $ & $\Rightarrow $ & $ \begin{array}{c} \text{wave } \\ \text{function} \end{array} $ & $\Rightarrow $ & $ \begin{array}{c} \text{hydrodynamic} \\ \text{interpretation} \end{array} $ \\ & & & & & \\ $ \begin{array}{c} \text{model } \\ \text{conception} \end{array} :$ & $ \begin{array}{c} \text{statistical} \\ \text{description} \end{array} $ & $\Rightarrow $ & $ \begin{array}{c} \text{fluid} \\ \text{dynamics} \end{array} $ & $\Rightarrow $ & $ \begin{array}{c} \text{wave } \\ \text{function} \end{array} $\end{tabular}$$ Quantum principles are foundation of the conventional quantum theory, and axiomatic properties of the wave function form a foundation of the hydrodynamic interpretation, i.e. the wave function is primary and hydrodynamic interpretation is secondary. In the model description of quantum phenomena the quantum principles are replaced by the statistical description, which is connected closely with the fluid dynamics, and fluid dynamics is a basis for introduction of the wave function, i.e. the fluid dynamics is primary and the wave function is secondary. Let $\mathcal{S}_{\mathrm{st}}$ be stochastic particle, whose state $X$ is described by variables $\left\{ \mathbf{x},\frac{d\mathbf{x}}{dt}\right\} $, where $\mathbf{x}$ is the particle position. Evolution of the particle state is stochastic and there exist no dynamic equations for $\mathcal{S}_{\mathrm{st}}$. Evolution of the state of $\mathcal{S}_{\mathrm{st}}$ contains both regular and stochastic components. To separate the regular evolution components, we consider a set (statistical ensemble) $\mathcal{E}\left[ \mathcal{S}_{\mathrm{st}}\right] $ of many independent identical stochastic particles $\mathcal{S}_{\mathrm{st}}$. All stochastic particles $\mathcal{S}_{\mathrm{st}}$ start from the same initial state. It means that all $\mathcal{S}_{\mathrm{st}}$ are prepared in the same way. If the number $N$ of $\mathcal{S}_{\mathrm{st}}$ is very large, the stochastic elements of evolution compensate each other, but regular ones are accumulated. In the limit $N\rightarrow \infty $ the statistical ensemble $\mathcal{E}\left[ \mathcal{S}_{\mathrm{st}}\right] $ turns to a dynamic system, whose state evolves according to some dynamic equations. The stochastic particle $\mathcal{S}_{\mathrm{st}}$ has six degrees of freedom, whereas the statistical ensemble $\mathcal{E}\left[ \mathcal{S}_{\mathrm{st}}\right] $ of stochastic particles $\mathcal{S}_{\mathrm{st}}$ has infinite number of freedom degrees. $\mathcal{E}\left[ \mathcal{S}_{\mathrm{st}}\right] $ is a continuous dynamic system of the type of a fluid (continuous medium). If the statistical ensemble $\mathcal{E}\left[ \mathcal{S}_{\mathrm{st}}\right] $ is normalized to one particle, it turns to the statistical average particle $\left\langle \mathcal{S}_{\mathrm{st}}\right\rangle $ (See details in [@R02]), which is a kind of a fluid. The action for $\left\langle \mathcal{S}_{\mathrm{st}}\right\rangle $, or for $\mathcal{E}\left[ \mathcal{S}_{\mathrm{st}}\right] $ can be reduced to the form of a continuous system $\mathcal{S}_{\mathrm{red}}\left[ \mathcal{S}_{\mathrm{d}}\right] $, which consists of many deterministic particles $\mathcal{S}_{\mathrm{d}}$, interacting between themselves. Type of this interaction depends on the character of stochasticity of the particle $\mathcal{S}_{\mathrm{st}}$. Let the statistical ensemble $\mathcal{E}_{\mathrm{d}}\left[ \mathcal{S}_{\mathrm{d}}\right] $ of deterministic classical particles $\mathcal{S}_{\mathrm{d}}$ be described by the action $\mathcal{A}_{\mathcal{E}_{\mathrm{d}}\left[ \mathcal{S}_{\mathrm{d}}\left( P\right) \right] }$, where $P$ are parameters describing $\mathcal{S}_{\mathrm{d}}$ (for instance, mass, charge). Let under influence of some stochastic agent the deterministic particle $\mathcal{S}_{\mathrm{d}}$ turn to a stochastic particle $\mathcal{S}_{\mathrm{st}}$. The action $\mathcal{A}_{\mathcal{E}_{\mathrm{st}}\left[ \mathcal{S}_{\mathrm{st}}\right] }$ for the statistical ensemble $\mathcal{E}_{\mathrm{st}}\left[ \mathcal{S}_{\mathrm{st}}\right] $ is reduced to the action $\mathcal{A}_{\mathcal{S}_{\mathrm{red}}\left[ \mathcal{S}_{\mathrm{d}}\right] }=\mathcal{A}_{\mathcal{E}_{\mathrm{st}}\left[ \mathcal{S}_{\mathrm{st}}\right] }$ for some set $\mathcal{S}_{\mathrm{red}}\left[ \mathcal{S}_{\mathrm{d}}\right] $ of identical interacting deterministic particles $\mathcal{S}_{\mathrm{d}}$. The action $\mathcal{A}_{\mathcal{S}_{\mathrm{red}}\left[ \mathcal{S}_{\mathrm{d}}\right] }$ as a functional of $\mathcal{S}_{\mathrm{d}}$ has the form $\mathcal{A}_{\mathcal{E}_{\mathrm{d}}\left[ \mathcal{S}_{\mathrm{d}}\left( P_{\mathrm{eff}}\right) \right] }$, where parameters $P_{\mathrm{eff}}$ are parameters $P$ of the deterministic particle $\mathcal{S}_{\mathrm{d}}$, averaged over the statistical ensemble, and this averaging describes interaction of particles $\mathcal{S}_{\mathrm{d}}$ in the set $\mathcal{S}_{\mathrm{red}}\left[ \mathcal{S}_{\mathrm{d}}\right] $. It means that $$\mathcal{A}_{\mathcal{E}_{\mathrm{st}}\left[ \mathcal{S}_{\mathrm{st}}\right] }=\mathcal{A}_{\mathcal{S}_{\mathrm{red}}\left[ \mathcal{S}_{\mathrm{d}}\left( P\right) \right] }=\mathcal{A}_{\mathcal{E}_{\mathrm{d}}\left[ \mathcal{S}_{\mathrm{d}}\left( P_{\mathrm{eff}}\right) \right] } \label{a0.6a}$$ In other words, stochasticity of particles $\mathcal{S}_{\mathrm{st}}$ in the ensemble $\mathcal{E}_{\mathrm{st}}\left[ \mathcal{S}_{\mathrm{st}}\right] $ is replaced by interaction of $\mathcal{S}_{\mathrm{d}}$ in $\mathcal{S}_{\mathrm{red}}\left[ \mathcal{S}_{\mathrm{d}}\right] $, and this interaction is described by a change $$P\rightarrow P_{\mathrm{eff}} \label{a0.6b}$$ in the action $\mathcal{A}_{\mathcal{E}_{\mathrm{d}}\left[ \mathcal{S}_{\mathrm{d}}\left( P\right) \right] }$. How to determine the change (\[a0.6b\])? At this stage of investigation the change (\[a0.6b\]) is phenomenological. In general, the form of the change (\[a0.6b\]) is determined by the properties (character) of stochasticity. But we do not know how to describe the type of stochasticity and their properties. Then the form of change (\[a0.6b\]) labels the type of stochasticity, and the change (\[a0.6b\]) may be considered to be a method of the stochasticity description. Before the further discussion let us consider an example of quantum stochasticity. Let $\mathcal{S}_{\mathrm{d}}$ be a free deterministic nonrelativistic particle of the mass $m$. The action for $\mathcal{S}_{\mathrm{d}}$ has the form $$\mathcal{S}_{\mathrm{d}}:\qquad \mathcal{A}\left[ \mathbf{x}\right] =\int L\left( \mathbf{x,}\frac{d\mathbf{x}}{dt}\right) dt, \label{a0.7}$$ $$L\left( \mathbf{x,}\frac{d\mathbf{x}}{dt}\right) =-mc^{2}+\frac{m}{2}\left( \frac{d\mathbf{x}}{dt}\right) ^{2} \label{a0.8}$$ where $\mathbf{x}=\mathbf{x}\left( t\right) $ and $c$ is the speed of the light. The action for the ensemble $\mathcal{E}_{\mathrm{d}}\left[ \mathcal{S}_{\mathrm{d}}\right] $ of free deterministic particles $\mathcal{S}_{\mathrm{d}}$ has the form $$\mathcal{E}_{\mathrm{d}}\left[ \mathcal{S}_{\mathrm{d}}\right] :\qquad \mathcal{A}_{\mathcal{E}_{\mathrm{d}}\left[ \mathcal{S}_{\mathrm{d}}\left( m\right) \right] }\left[ \mathbf{x}\right] =\int L\left( \mathbf{x,}\frac{d\mathbf{x}}{dt}\right) dtd\mathbf{\xi }, \label{a0.9}$$ where the Lagrangian function has the same form (\[a0.8\]), but $\mathbf{x}=\mathbf{x}\left( t,\mathbf{\xi }\right) $, $\mathbf{\xi }=\left\{ \xi _{1},\xi _{2},\xi _{3}\right\} $. Here variables $\mathbf{\xi }$ label the particles $\mathcal{S}_{\mathrm{d}}$ of the statistical ensemble $\mathcal{E}_{\mathrm{d}}\left[ \mathcal{S}_{\mathrm{d}}\right] $. The action (\[a0.9\]) describes some fluid without pressure. In the case of a free noncharged particle the only parameter of the particle is its mass $m$. At the quantum stochasticity the change (\[a0.6b\]) has the form $$m\rightarrow m_{\mathrm{eff}}=m\left( 1-\frac{\mathbf{u}^{2}}{2c^{2}}+\frac{\hbar }{2mc^{2}}\mathbf{\nabla u}\right) \label{a0.9a}$$ where $\mathbf{u}=\mathbf{u}\left( t,\mathbf{x}\right) $ is the mean value of the stochastic velocity component. Quantum constant $\hbar $ appears here as a coupling constant, describing connection between the regular and stochastic components of particle motion. The velocity $\mathbf{u}$ is supposed to be small with respect to $c$. Then we must make the change (\[a0.9a\]) only in the first term of (\[a0.8\]), because the same change in the second term give the quantity of the order $O\left( c^{-2}\right) $. After change (\[a0.9a\]) the action (\[a0.9\]) turns to the action $$\mathcal{E}_{\mathrm{st}}\left[ \mathcal{S}_{\mathrm{st}}\right] :\qquad \mathcal{A}_{\mathcal{E}_{\mathrm{d}}\left[ \mathcal{S}_{\mathrm{d}}\left( m_{\mathrm{eff}}\right) \right] }\left[ \mathbf{x,u}\right] =\int L\left( \mathbf{x,}\frac{d\mathbf{x}}{dt}\right) +L_{\mathrm{st}}\left( \mathbf{u},\mathbf{\nabla u}\right) dtd\mathbf{\xi }, \label{a0.10}$$ $$L_{\mathrm{st}}\left( \mathbf{u},\mathbf{\nabla u}\right) =\frac{m}{2}\mathbf{u}^{2}-\frac{\hbar }{2}\mathbf{\nabla u} \label{a0.11}$$ where $\mathbf{x}=\mathbf{x}\left( t,\mathbf{\xi }\right) $, $\mathbf{\xi }=\left\{ \xi _{1},\xi _{2},\xi _{3}\right\} $, but $\mathbf{u}=\mathbf{u}\left( t,\mathbf{x}\right) $ is a function of $t,\mathbf{x}$. The action (\[a0.10\]) also describes a fluid, but now it is a fluid with a pressure and its irrotational flow is described by the Schrödinger equation [@R02; @R002]. The action (\[a0.10\]) admits a simple interpretation. The derivative $d\mathbf{x/}dt$ describes the mean velocity of a fluid particle which does not coincide with the velocity of stochastic particle $\mathcal{S}_{\mathrm{st}}$, as well as the velocity of a gas particle does not coincide with the velocity of a gas molecule. The gas molecule has stochastic velocity component. Likewise, the stochastic particle $\mathcal{S}_{\mathrm{st}}$ has the stochastic velocity component. The mean value of this stochastic component is described by the variable $\mathbf{u}$. For a stochastic Brownian particle the mean value of the stochastic component velocity is described by the relation $$\mathbf{u}_{\mathrm{Br}}=-D\mathbf{\nabla }\ln \rho _{\mathrm{Br}} \label{a0.12}$$ where $D$ is the diffusion coefficient, and $\rho _{\mathrm{Br}}$ is the density of randomly wandering Brownian particles. Variation of action (\[a0.10\]) with respect to $\mathbf{u}$ gives the same result $$\mathbf{u=}-\frac{\hbar }{2m}\mathbf{\nabla }\ln \rho ,\qquad \rho =\frac{\partial \left( \xi _{1},\xi _{2},\xi _{3}\right) }{\partial \left( x^{1},x^{2},x^{3}\right) } \label{a0.13}$$ where $\hbar /2m$ may be regarded as the diffusion coefficient. To obtain the relation (\[a0.13\]), we need to rewrite (\[a0.10\]) in the form of integral over variables $\mathbf{x}$**,** because $\mathbf{u}$ is a function of $\mathbf{x}$ (but not of $\mathbf{\xi })$. $$\int L_{\mathrm{st}}\left( \mathbf{u},\mathbf{\nabla u}\right) dtd\mathbf{\xi =}\int \left( \frac{m\mathbf{u}^{2}}{2}-\frac{\hbar }{2}\mathbf{\nabla u}\right) \rho dtd\mathbf{x},\qquad \rho =\frac{\partial \left( \xi _{1},\xi _{2},\xi _{3}\right) }{\partial \left( x^{1},x^{2},x^{3}\right) } \label{a0.14}$$ Then variation of (\[a0.14\]) with respect to $\mathbf{u}$ gives the relation (\[a0.13\]). The nature of stochastic velocities $\mathbf{u}$ and $\mathbf{u}_{\mathrm{Br}}$ is similar. Both velocities $\mathbf{u}$ and $\mathbf{u}_{\mathrm{Br}}$ are a result of random wandering. However, dynamic equations are different, because dissipative Brownian particle obeys the Aristotelian dynamics, whereas conservative quantum particle $\mathcal{S}_{\mathrm{st}}$ obeys the Newtonian dynamics. The state of the statistical average Brownian particle $\left\langle \mathcal{S}_{\mathrm{Br}}\right\rangle $ is described by the variable $\rho _{\mathrm{Br}}$. According to (\[a0.12\]) the velocity $\mathbf{u}_{\mathrm{Br}}$ is a function of the state $\rho _{\mathrm{Br}}$, and dynamic equation for the state $\rho _{\mathrm{Br}}$ has the form of the continuity equation $$\frac{d\rho _{\mathrm{Br}}}{\partial t}+\mathbf{\nabla }\left( \rho _{\mathrm{Br}}\mathbf{u}_{\mathrm{Br}}\right) =0$$ The state of the statistical average particle $\left\langle \mathcal{S}_{\mathrm{st}}\right\rangle $ is described by the variables $\rho ,\frac{d\mathbf{x}}{dt},\mathbf{u}$, and the energy $m\mathbf{u}^{2}/2$ of the stochastic component must be taken into account in the Lagrangian function. The first term $\frac{m}{2}\mathbf{u}^{2}$ of $L_{\mathrm{st}}$ is the kinetic energy of the stochastic component. But this term does not contain time derivative. It depends only on $\mathbf{x}$. It means that it acts on the particle motion as a potential energy. Indeed, variation of (\[a0.10\]) with respect to $\mathbf{x}$ gives dynamic equation $$m\frac{d^{2}\mathbf{x}}{dt^{2}}=-\mathbf{\nabla }U\left( \mathbf{x}\right) \mathbf{,\qquad }U\left( \mathbf{x}\right) =\left( \frac{m\mathbf{u}^{2}}{2}-\frac{\hbar }{2}\mathbf{\nabla u}\right)$$ The second term in (\[a0.14\]) ensures connection between $\mathbf{u}$ and collective variable $\rho $. For application to relativistic particle the relation (\[a0.9a\]) should be rewritten in the form $$m^{2}\rightarrow m_{\mathrm{eff}}^{2}=m^{2}\left( 1-\frac{\mathbf{u}^{2}}{c^{2}}+\frac{\hbar }{mc^{2}}\mathbf{\nabla u}\right) \label{a0.15}$$ Besides we should additionally replace nonrelativistic expressions $\mathbf{u}^{2}$ and $\mathbf{\nabla u}$ by relativistic ones $-g_{ik}u^{i}u^{k}$ and $\partial _{i}u^{i}$ respectively, where $u^{i}$ is the 4-vector, describing the mean value of the stochastic component of velocity. Then the action (\[a0.8\]), (\[a0.10\]), (\[a0.11\]) transforms to the relativistic form $$\mathcal{A}\left[ x,\kappa \right] =-\int mcK\sqrt{g_{ik}\dot{x}^{i}\dot{x}^{k}}d\xi _{0}d\mathbf{\xi ,\qquad }K\mathbf{=}\sqrt{1+\frac{\hbar ^{2}}{m^{2}c^{2}}\left( \kappa ^{l}\kappa _{l}+\partial _{l}\kappa ^{l}\right) } \label{a0.16}$$ where $x=\left\{ x^{i}\right\} ,$ $i=0,1,2,3$ is a function of variables $\xi =\left\{ \xi _{0},\mathbf{\xi }\right\} =\left\{ \xi _{i}\right\} ,$ $\;i=0,1,2,3$ $\dot{x}^{i}\equiv dx^{i}/d\xi _{0}$. Variables $\kappa =\left\{ \kappa ^{i}\right\} ,$ $\;i=0,1,2,3$ are functions of $x$. The variables $\kappa ^{l}$ are connected with the variables $u^{l}$ by means of relation $$u^{l}=\frac{\hbar }{m}\kappa ^{l},\qquad l=0,1,2,3 \label{a0.17}$$ As well as the nonrelativistic velocity $\mathbf{u}$**,** the vector field $\kappa _{l}=g_{lk}\kappa ^{k}$ has a potential $\kappa $ and can be represented in the form $$\kappa _{l}=\partial _{l}\kappa =\frac{1}{2}\partial _{l}\ln \rho , \label{a0.18}$$ where $$\rho =\frac{J\sqrt{\dot{x}_{s}\dot{x}^{s}}}{\rho _{0}mcK}=\frac{\sqrt{j_{s}j^{s}}}{\rho _{0}mcK},\qquad j^{k}=J\dot{x}^{k}\equiv \frac{\partial J}{\partial \xi _{0,k}}\equiv \frac{\partial \left( x^{k},\xi _{1},\xi _{2},\xi _{3}\right) }{\partial \left( x^{0},x^{1},x^{2},x^{3}\right) }, \label{a0.19}$$ $$J=\frac{\partial \left( \xi _{0},\xi _{1},\xi _{2},\xi _{3}\right) }{\partial \left( x^{0},x^{1},x^{2},x^{3}\right) }=\det \left\| \xi _{i,k}\right\| ,\qquad \xi _{i,k}=\partial _{k}\xi _{i},\qquad i,k=0,1,2,3 \label{a0.20}$$ $\rho _{0}=$const. Result (\[a0.18\]) is obtained by variation of (\[a0.16\]) with respect to $\kappa ^{l}$ in the same way, as equation (\[a0.13\]) was derived from (\[a0.10\]). The action (\[a0.16\]) describes some relativistic fluid, whose particles interact via the self-consistent vector field $\kappa ^{l}$. Properties of this fluid were investigated in [@R98]. It has been shown that irrotational flow of this fluid is described by the Klein-Gordon equation. Deterministic relativistic particles $\mathcal{S}_{\mathrm{d}}$ of the set (fluid) $\mathcal{S}_{\mathrm{red}}\left[ \mathcal{S}_{\mathrm{d}}\right] $ interact between themselves via self-consistent vector field $\kappa ^{l}$, $i=0,1,2,3$. We shall refer to the field $\kappa ^{l}$ and its potential $\kappa $, defined by the relation (\[a0.17\]) as $\kappa $-field. We shall show that the $\kappa $-field has some unusual properties. In particular, the $\kappa $-field can produce pairs. The field producing pairs ========================= To show that the $\kappa $-field enables to produce pairs, we shall consider the expression $\frac{\hbar ^{2}}{m^{2}c^{2}}\left( \kappa ^{l}\kappa _{l}+\partial _{l}\kappa ^{l}\right) $ constructed of the $\kappa $-field as some given external field $f$. We consider motion of a charged deterministic particle in the given electromagnetic field $A_{i}$ and some given scalar field $f$, changing the particle mass. The action for the particle has the form of the type (\[a0.16\]) $$\mathcal{A}\left[ q\right] =\int L\left( q,\dot{q}\right) d\tau ,\qquad L=-\sqrt{m^{2}c^{2}\left( 1+f\left( q\right) \right) g_{ik}\dot{q}^{i}\dot{q}^{k}}-\frac{e}{c}A_{k}\dot{q}^{k} \label{a1.1}$$ where relations $x^{i}=q^{i}\left( \tau \right) ,$ $i=0,1,2,3$ describe the world line of the particle, and $\dot{q}^{k}\equiv dq^{i}/d\tau $. The quantities $A_{k}=A_{k}\left( q\right) ,$ $k=0,1,2,3$ are given electromagnetic potentials, and $f=f\left( q\right) $ is some given field, changing the effective particle mass $m_{\mathrm{eff}}=m\sqrt{\left( 1+f\left( q\right) \right) }$. The canonical momentum $p_{k}$ is defined by the relation $$p_{k}=\frac{\partial L}{\partial \dot{q}^{k}}=-\frac{mcKg_{ki}\dot{q}^{i}}{\sqrt{g_{ls}\dot{q}^{l}\dot{q}^{s}}}-\frac{e}{c}A_{k},\qquad K=\sqrt{\left( 1+f\left( q\right) \right) } \label{a1.2}$$ Dynamic equations have the form $$\frac{dp_{k}}{d\tau }=-mc\sqrt{g_{ik}\dot{q}^{i}\dot{q}^{k}}\frac{\partial K}{\partial q^{k}}-\frac{e}{c}\frac{\partial A_{i}}{\partial q^{k}}\dot{q}^{i} \label{a1.3}$$ The action (\[a1.1\]), as well as dynamic equations (\[a1.3\]), (\[a1.2\]) are invariant with respect to a transformation of the world line parametrization $$\tau \rightarrow \tilde{\tau}=\tilde{\tau}\left( \tau \right) \label{a1.4}$$ provided $\partial \tilde{\tau}/\partial \tau >0$. If $\partial \tilde{\tau}/\partial \tau <0$ the dynamic equations (\[a1.3\]), (\[a1.2\]) stop to be invariant with respect to transformation (\[a1.4\]) of the world line parametrization. The parametrization of the world line is a method of the world line description, and correctly written dynamic equations have not to depend on the method of description. Let us modify the action (\[a1.1\]), introducing orientation $\mathbf{\varepsilon }$ of the world line. The orientation of the world line is a unit vector $\mathbf{\varepsilon }$, tangent to the world line. Component $\varepsilon $ of orientation $\mathbf{\varepsilon }$ is the projection of the vector $\mathbf{\varepsilon }$ on to the vector $dq^{i}/d\tau $$$\varepsilon =\frac{g_{ik}\varepsilon ^{i}\dot{q}^{k}}{\sqrt{g_{ls}\dot{q}^{l}\dot{q}^{s}}}=\pm 1 \label{a1.4a}$$ At the parametrization transformation (\[a1.4\]) the component $\varepsilon $ of orientation $\mathbf{\varepsilon }$ transforms as follows $$\varepsilon \rightarrow \tilde{\varepsilon}=\varepsilon \mathrm{sgn}\left( \partial \tilde{\tau}/\partial \tau \right) ,\qquad \mathrm{sgn}\left( x\right) =\frac{x}{\left| x\right| } \label{a1.5}$$ Component $\varepsilon $ of orientation $\mathbf{\varepsilon }$ is invariant with respect to coordinate transformation as it follows from (\[a1.4a\]). Let us rewrite the relations (\[a1.1\]) – (\[a1.3\]) in the form, which is invariant with respect to the arbitrary transformation (\[a1.4\]) $$\mathcal{A}\left[ q\right] =\int\limits_{\min \left( \tau ^{\prime },\tau ^{\prime \prime }\right) }^{\max \left( \tau ^{\prime },\tau ^{\prime \prime }\right) }L\left( q,\dot{q}\right) d\tau ,\qquad L=-\sqrt{m^{2}c^{2}\left( 1+f\left( q\right) \right) g_{ik}\dot{q}^{i}\dot{q}^{k}}-\frac{\varepsilon e}{c}A_{k}\dot{q}^{k} \label{a1.6}$$ where $\tau ^{\prime }$ and $\tau ^{\prime \prime }$ are values of the parameter $\tau $ at the ends of the integration interval. The particle momentum is defined $$p_{k}=\frac{\partial L}{\partial \dot{q}^{k}}=-\frac{mcKg_{ki}\dot{q}^{i}}{\sqrt{g_{ls}\dot{q}^{l}\dot{q}^{s}}}-\frac{\varepsilon e}{c}A_{k},\qquad K=\sqrt{\left( 1+f\left( q\right) \right) } \label{a1.7}$$ $$\frac{dp_{k}}{d\tau }=-mc\sqrt{g_{ik}\dot{q}^{i}\dot{q}^{k}}\frac{\partial K}{\partial q^{k}}-\frac{\varepsilon e}{c}\frac{\partial A_{i}}{\partial q^{k}}\dot{q}^{i} \label{a1.8}$$ Note that now the momentum (\[a1.7\]) and dynamic equations (\[a1.8\]) are invariant with respect to arbitrary transformation (\[a1.4\]). One can see from (\[a1.7\]), that the vector $$\dot{q}_{k}=\sqrt{\frac{g_{ls}\dot{q}^{l}\dot{q}^{s}}{1+f\left( q\right) }}\frac{\left( p_{k}+\frac{\varepsilon e}{c}A_{k}\right) }{mc} \label{a1.10}$$ becomes to be spacelike $\left( g_{ls}\dot{q}^{l}\dot{q}^{s}<0\right) $, if $f\left( q\right) <-1$, because only in this case the expression under radical in (\[a1.10\]) is real. The Jacobi-Hamilton equation for the action (\[a1.6\]) has the form $$g^{ik}\left( \frac{\partial S}{\partial q^{i}}+\frac{\varepsilon e}{c}A_{i}\right) \left( \frac{\partial S}{\partial q^{k}}+\frac{\varepsilon e}{c}A_{k}\right) =m^{2}c^{2}\left( 1+f\left( q\right) \right) \label{a1.9}$$ Let us consider solution of the Hamilton-Jacobi equation for the two-dimensional space-time $\left( t,x\right) $, when $A_{i}=0$, and $f=f\left( t\right) $ is a function of only time $t$. In this case the solution of equation (\[a1.9\]) has the form $$S\left( t,x,p_{0}\right) =p_{0}x+\int\limits_{0}^{t}c\sqrt{m^{2}c^{2}\left( 1+f\left( t\right) \right) +p_{0}^{2}}dt+C\mathbf{,\qquad }p_{0},C=\text{const} \label{a1.11}$$ and solution of dynamic equations takes the form $$\frac{\partial S\left( t,x,p_{0}\right) }{\partial p_{0}}-x_{0}=x-x_{0}+\int\limits_{0}^{t}\frac{p_{0}cdt}{\sqrt{m^{2}c^{2}\left( 1+f\left( t\right) \right) +p_{0}^{2}}}=0,\qquad x_{0}=\text{const} \label{a1.12}$$ Let for example $$f\left( t\right) =\left\{ \begin{array}{lll} 0 & \text{if} & t<0 \\ -\frac{V^{2}}{m^{2}c^{4}t_{0}^{2}}t\left( t-t_{0}\right) & \text{if} & 0<t<t_{0} \\ 0 & \text{if} & t_{0}<t \end{array} \right. ,\qquad t_{0},V=\text{const} \label{a1.13}$$ The solution (\[a1.12\]) takes the form $$x=\left\{ \begin{array}{lll} x_{0}-\frac{p_{0}c^{2}}{E}t & \text{if} & t<0 \\ x_{0}-\int\limits_{0}^{t}\frac{p_{0}cdt}{\sqrt{E^{2}-V^{2}t\left( t-t_{0}\right) /t_{0}^{2}}} & \text{if} & 0<t<t_{0} \\ x_{1}+\alpha \frac{p_{0}c^{2}}{E}\left( t-t_{0}\right) & \text{if} & t_{0}<t \end{array} \right. ,\qquad E=c\sqrt{m^{2}c^{2}+p_{0}^{2}} \label{a1.14}$$ where $\alpha =\pm 1$. Sign of $\alpha $ and the constant $x_{1}$ are determined from the continuity condition of the world line at $t=t_{0}$. The solution (\[a1.14\]) has different form, depending on the sign of the constant $4E^{2}-V^{2}$. If $4E^{2}>V^{2}$, the solution (\[a1.14\]) takes the form $$x=\left\{ \begin{array}{ll} x_{0}-\frac{p_{0}c^{2}}{E}t & \text{if }t<0 \\ x_{0}-\frac{p_{0}c^{2}t_{0}}{V}\arcsin \frac{2V\left( \sqrt{E^{2}t_{0}^{2}-V^{2}t\left( t-t_{0}\right) }-E\left( t_{0}-2t\right) \right) }{t_{0}\left( 4E^{2}+V^{2}\right) } & \text{if }0<t<t_{0} \\ x_{0}-p_{0}c^{2}\frac{t_{0}}{V}\arcsin \frac{4EV}{4E^{2}+V^{2}}-\frac{p_{0}c^{2}}{E}\left( t-t_{0}\right) & \text{if }t_{0}<t \end{array} \right. ,\qquad E^{2}>V^{2}/4 \label{a1.15}$$ In the case, when $4E^{2}<V^{2}$, the world line is reflected from the region $\Omega _{\mathrm{fb}}$ of the space-time determined by the condition $0<t<t_{0}$, and the coordinate $x$ is not a single-valued function of the time $t$. In this case we use parametric representation for the solution (\[a1.14\]). We have $$x=\left\{ \begin{array}{lll} x_{0}-\frac{p_{0}c^{2}t_{0}}{2E}\left( 1-A\cosh \tau \right) & \text{if} & \tau <-\tau _{0} \\ x_{0}-\frac{p_{0}c^{2}t_{0}}{V}\left( \tau +\tau _{0}\right) & \text{if} & -\tau _{0}<\tau <\tau _{0} \\ x_{0}-\frac{2p_{0}t_{0}}{V}\tau _{0}+\frac{p_{0}c^{2}t_{0}}{2E}\left( 1-A\cosh \tau \right) & \text{if} & \tau _{0}<\tau \end{array} \right. \label{a1.17}$$ $$t=\frac{t_{0}}{2}\left( 1-A\cosh \tau \right) \label{a1.18}$$ where $$A=\sqrt{1-\frac{4E^{2}}{V^{2}}},\qquad \tau _{0}=\func{arccosh}\frac{1}{A}=\func{arccosh}\frac{1}{\sqrt{1-\frac{4E^{2}}{V^{2}}}} \label{a1.19}$$ The solution (\[a1.17\]), (\[a1.18\]) describes annihilation of particle and antiparticle with the energy $E<V/2$ in the region $0<t<t_{0}$. Solution, describing the particle-antiparticle generation, has the form $$x=\left\{ \begin{array}{lll} x_{0}-\frac{p_{0}c^{2}t_{0}}{2E}\left( A\cosh \tau -1\right) & \text{if} & \tau <-\tau _{0} \\ x_{0}+\frac{p_{0}c^{2}t_{0}}{V}\left( \tau +\tau _{0}\right) & \text{if} & -\tau _{0}<\tau <\tau _{0} \\ x_{0}+\frac{2p_{0}t_{0}}{V}\tau _{0}+\frac{p_{0}c^{2}t_{0}}{2E}\left( A\cosh \tau -1\right) & \text{if} & \tau _{0}<\tau \end{array} \right. \label{a1.20}$$ $$t=\frac{t_{0}}{2}\left( A\cosh \tau -1\right) \label{a1.21}$$ where parameters $A,\tau _{0}$ are defined by the relation (\[a1.19\]), and the relation $2E<V$ takes place. In both cases (\[a1.17\]) and (\[a1.20\]) at $\left| t\right| \rightarrow \infty $ the world line has two branches, which can be approximated by the relations $$x=x_{0}+vt_{1}\pm v\left( t-t_{1}\right) ,\qquad t_{1}=t_{0}\frac{E}{V} \label{a1.22}$$ where $v=-\frac{p_{0}c^{2}}{E}$ is the particle velocity, and $v=\frac{p_{0}c^{2}}{E}$ is the antiparticle velocity. Thus, the region $\Omega _{\mathrm{fb}}$ with the $\kappa $-field reflects world lines of particles with the energy $E<V/2$. If the particle energy $E>V/2$, the particle passes the region $\Omega _{\mathrm{fb}}$ through. The world line penetrates into forbidden region $\Omega _{\mathrm{fb}}$ the deeper the larger is its energy. Connection between energy-momentum and canonical momentum ========================================================= It is common practice to think that in the absence of electromagnetic field the canonical momentum (\[a1.7\]) coincides with the particle energy-momentum $P_{i}$. Then the particle energy may be negative, as far as the canonical momentum component $p_{0}$ may be positive and negative. Existence of negative particle energy is very undesirable, because it leads to many very strange corollaries. In reality, the energy-momentum  $P_{i}$, $\;i=0,1,2,3$ and canonical momentum $p_{i}$, $\;i=0,1,2,3$ are different quantities which can coincide only in the case, when there are only particles, or only antiparticles. They are defined differently. The canonical momentum $p_{i}$ is defined as the quantity canonically conjugate to coordinate $x^{i}$ of the particle position by means of the relation (\[a1.7\]). The energy-momentum $P_{i}$ is defined by the relation $$P_{i}=\int\limits_{V}T_{i}^{0}d\mathbf{x,\qquad }i=0,1,2,3 \label{a2.1}$$ where $T_{i}^{0}=g_{ik}T^{k0}$ are components of the energy-momentum tensor, and $V$ is the 3-volume around the particle, whose energy-momentum is determined. The energy-momentum tensor $T^{ik}$ is defined as a source of the gravitational field, i.e. by means of the relation $$T^{ik}\left( x\right) =\frac{\delta \mathcal{A}}{\delta g_{ik\left( x\right) }}=-\frac{2}{\sqrt{-g}}\frac{\partial }{\partial g_{ik}\left( x\right) }\left( \sqrt{-g}\mathcal{L}\right) \label{a2.2}$$ where $\mathcal{A}$ is the action (\[a1.6\]), written in the arbitrary curvilinear coordinate system in the form $$\mathcal{A}\left[ q^{i},g_{ik},A_{i}\right] =\int \mathcal{L}\left( q^{i},g_{ik},A_{i}\right) \sqrt{-g}d^{4}x \label{a2.3}$$ and $g=\det \left| \left| g_{ik}\right| \right| .$ In the same way the electric charge of a particle is defined by the relation $$Q=\int\limits_{V}j_{\mathrm{c}}^{0}d\mathbf{x,} \label{a2.4}$$ where $j_{\mathrm{c}}^{i}$, $i=0,1,2,3$ is electric current density, defined as the source of the electromagnetic field, i.e. by the relation $$j_{\mathrm{c}}^{i}\left( x\right) =c\frac{\delta \mathcal{A}}{\delta A_{i}\left( x\right) }=\frac{c}{\sqrt{-g}}\frac{\partial }{\partial A_{i}\left( x\right) }\left( \sqrt{-g}\mathcal{L}\right) \label{a2.5}$$ To obtain expressions for the quantities $j_{\mathrm{c}}^{l}$, $T^{ik}$, let us write the action (\[a1.6\]) in the arbitrary curvilinear coordinate system and represent it in the form (\[a2.3\]). One obtains $$\mathcal{A}=\int d^{4}x\int\limits_{\min \left( \tau ^{\prime },\tau ^{\prime \prime }\right) }^{\max \left( \tau ^{\prime },\tau ^{\prime \prime }\right) }L\left( q,\dot{q}\right) \delta ^{\left( 4\right) }\left( x-q\left( \tau \right) \right) d\tau , \label{a2.6}$$ where $$\delta ^{\left( 4\right) }\left( x-q\left( \tau \right) \right) =\delta \left( x^{0}-q^{0}\left( \tau \right) \right) \delta \left( \mathbf{x}-\mathbf{q}\left( \tau \right) \right) =\prod_{i=0}^{i=3}\delta \left( x^{i}-q^{i}\left( \tau \right) \right) \label{a2.7}$$ Integrating (\[a2.6\]) over $\tau $ and using $\delta $-function $\delta \left( x^{0}-q^{0}\left( \tau \right) \right) $, one obtains $$\mathcal{A}=\int d^{4}x\sum_{j}\frac{L\left( q\left( \tau _{j}\right) ,\dot{q}\left( \tau _{j}\right) \right) }{\left| \dot{q}^{0}\left( \tau _{j}\right) \right| }\delta \left( \mathbf{x}-\mathbf{q}\left( \tau _{j}\right) \right) , \label{a2.8}$$ where $\tau _{j}=\tau _{j}\left( x^{0}\right) $ are roots of the equation $$x^{0}-q^{0}\left( \tau _{j}\right) =0 \label{a2.9}$$ Comparing relations (\[a2.8\]) and (\[a2.3\]), we conclude $$\begin{aligned} \mathcal{L}\sqrt{-g} &=&\sum_{j}\frac{L\left( q\left( \tau _{j}\right) ,\dot{q}\left( \tau _{j}\right) \right) }{\left| \dot{q}^{0}\left( \tau _{j}\right) \right| }\delta \left( \mathbf{x}-\mathbf{q}\left( \tau _{j}\right) \right) \nonumber \\ &=&\sum_{j}\left( -\sqrt{m^{2}c^{2}\left( 1+f\left( q\left( \tau _{j}\right) \right) \right) g_{ik}\dot{q}^{i}\left( \tau _{j}\right) \dot{q}^{k}\left( \tau _{j}\right) }-\frac{\varepsilon e}{c}A_{k}\dot{q}^{k}\left( \tau _{j}\right) \right) \nonumber \\ &&\times \frac{\delta \left( \mathbf{x}-\mathbf{q}\left( \tau _{j}\right) \right) }{\left| \dot{q}^{0}\left( \tau _{j}\right) \right| } \label{a2.10}\end{aligned}$$ Now let us substitute relation (\[a2.10\]) in the relations (\[a2.2\]), (\[a2.5\]) and set  $A_{i}=0$, $\;g_{ik}=$diag$\left\{ c^{2},-1,-1,-1\right\} $. After calculations we obtain relations for the energy-momentum tensor and the current density $$T^{ik}\left( x^{0},\mathbf{x}\right) =\sum_{j}\frac{mc\sqrt{1+f\left( q\left( \tau _{j}\right) \right) }\dot{q}^{i}\left( \tau _{j}\right) \dot{q}^{k}\left( \tau _{j}\right) }{c\sqrt{g_{ls}\dot{q}^{l}\left( \tau _{j}\right) \dot{q}^{s}\left( \tau _{j}\right) }}\frac{\delta \left( \mathbf{x}-\mathbf{q}\left( \tau _{j}\right) \right) }{\left| \dot{q}^{0}\left( \tau _{j}\right) \right| } \label{a2.11}$$ $$j^{k}_{\mathrm{c}}=-\sum_{j}\varepsilon e\dot{q}^{k}\left( \tau _{j}\right) \frac{\delta \left( \mathbf{x}-\mathbf{q}\left( \tau _{j}\right) \right) }{\left| \dot{q}^{0}\left( \tau _{j}\right) \right| } \label{a2.12}$$ where $\tau _{j}=\tau _{j}\left( x^{0}\right) $. Substituting relations (\[a2.11\]), (\[a2.12\]) in relations (\[a2.1\]), (\[a2.4\]), we obtain $$\begin{aligned} P_{i} &=&\sum_{j}\frac{mc\sqrt{1+f\left( q\left( \tau _{j}\right) \right) }\dot{q}_{i}\left( \tau _{j}\right) \dot{q}^{0}\left( \tau _{j}\right) }{\left| \dot{q}^{0}\left( \tau _{j}\right) \right| \sqrt{g_{ls}\dot{q}^{l}\left( \tau _{j}\right) \dot{q}^{s}\left( \tau _{j}\right) }} \nonumber \\ &=&-\sum_{j}\frac{p_{i}\left( \tau _{j}\right) \dot{q}^{0}\left( \tau _{j}\right) }{\left| \dot{q}^{0}\left( \tau _{j}\right) \right| }=-\sum_{j}p_{i}\left( \tau _{j}\right) \mathrm{sgn}\left( \dot{q}^{0}\left( \tau _{j}\right) \right) \nonumber \\ &=&\sum_{j}p_{i}\left( \tau _{j}\right) \mathrm{sgn}\left( p_{0}\left( \tau _{j}\right) \right) ,\qquad i=0,1,2,3 \label{a2.13}\end{aligned}$$ $$Q=-\sum_{j}\varepsilon e\cdot \mathrm{sgn}\left( \dot{q}^{0}\left( \tau _{j}\right) \right) =\sum_{j}\varepsilon e\cdot \mathrm{sgn}\left( p_{0}\left( \tau _{j}\right) \right) \label{a2.14}$$ In particular, we obtain for the energy $E=P_{0}$$$P_{0}=E=\sum_{j}p_{0}\left( \tau _{j}\right) \mathrm{sgn}\left( p_{0}\left( \tau _{j}\right) \right) =\sum_{j}\left| p_{0}\left( \tau _{j}\right) \right| \label{a2.15}$$ In the case, when $f\left( q\right) +1<0$ in some regions of the space-time, the equation (\[a2.9\]) may have several roots. Each of roots corresponds to some intersection of the world line with the plane $t=$const. Any of such intersections describes either particle, when sgn$\left( \dot{q}^{0}\right) =-$sgn$\left( p_{0}\right) >0$, or antiparticle, when sgn$\left( \dot{q}^{0}\right) =-$sgn$\left( p_{0}\right) <0.$ According to (\[a2.14\]) the electric charge $Q_{\mathrm{p}}$ of a particle is equal to $-\varepsilon e$, whereas the electric charge $Q_{\mathrm{a}}$ of an antiparticle is equal to $\varepsilon e$. Thus, the particle and antiparticle have electric charges of different signs, because the sign of $p_{0}$ is different for particle and antiparticle. According to (\[a2.15\]) the energy is positive for a particle as well as for an antiparticle, although the sign of the temporal component $p_{0}$ of the canonical momentum is different for particle and antiparticle. Such a situation seems to be very reasonable. It can be interpreted as follows. Let the term ”SWL” (section of world line) mean the collective concept with respect to concepts ”particle” and ”antiparticle”. The particle and antiparticle distinguish in the sign of the temporal component $p_{0}$ of the canonical momentum and in the sign of the temporal component $-K\dot{q}^{0}\left( g_{ls}\dot{q}^{l}\dot{q}^{s}\right) ^{-1/2}$ of the velocity 4-vector. Flux of the world lines through some spacelike three-dimensional area $dS_{i}$ for particles and the flux of the world lines through the same area $dS_{i}$ for antiparticles have different signs. This fact may be interpreted in the sense that a particle is one SWL, whereas an antiparticle is minus one SWL. If SWL has the canonical momentum $p_{i}$, then the number $n$ of SWLs is $n=-$sgn$\left( p_{0}\right) $. The relations (\[a2.13\]) – (\[a2.15\]) can be written in the form $$\begin{aligned} Q &=&-\sum_{j}\varepsilon en\left( \tau _{j}\right) ,\qquad E=-\sum_{j}p_{0}\left( \tau _{j}\right) n\left( \tau _{j}\right) \label{a2.16} \\ P_{i} &=&-\sum_{j}p_{i}\left( \tau _{j}\right) n\left( \tau _{j}\right) ,\qquad i=0,1,2,3,\qquad n=-\mathrm{sgn}\left( p_{0}\right) \label{a2.17}\end{aligned}$$ This is reasonable formulas for additive quantities, when, the additive quantity $a$ of several objects (SWLs) is a sum of additive quantities $a$ of each object (SWL) multiplied by the number $n$ of objects (SWLs). It is unusual only that the number $n$ of objects (SWLs) may be negative. Peculiarities of description in terms of world lines ==================================================== From viewpoint of the consistent relativity theory a world line is a real physical object, whereas a particle and an antiparticle (SWLs) are attributes of the world line, which appear as a result of intersection of the world line with the plane $t=$const. We shall use a special term ‘WL’ for the world line, considered to be a real physical object. In the nonrelativistic theory a real physical object is a particle (or antiparticle), whereas the world line is an attribute of a particle (its history). Particle and antiparticle in the nonrelativistic theory (or in inconsistent relativistic theory) are two different objects, having different parameters. The particle is described by the following parameters $\left\{ m,e\right\} $, where $m$ is a mass, and $e$ is a charge constant. The antiparticle is described by the parameters $\left\{ m,-e\right\} $. Particle and antiparticle are described by different dynamic systems, because these systems have different parameters. Note that at such a description the component $\varepsilon $ of orientation $\mathbf{\varepsilon }$ is not a parameter of dynamic system. It is a method of description of SWL, because $\varepsilon $ changes at the change $\tau \rightarrow -\tau $ of the world line parametrization. At nonrelativistic approach one cannot describe pair production classically, because one cannot connect fragments of world lines of two different objects (particle and antiparticle). In the framework of classical theory consideration of a particle and an antiparticle as two different objects is incompatible. In the quantum theory, where the particle world line may not exist, it is possible to introduce special operators, describing production and annihilation of a particle (antiparticle). In quantum theory pair generation is a corollary of dynamic equations, and one can imagine such a situation (at proper choice of dynamic equations), when particles and antiparticles are generated separately. In the classical theory the pair generation of particles and antiparticles is described on the deeper (descriptive) level. Choice of dynamic equations cannot change anything. Nevertheless, let us start from the consistent relativistic theory, where SWLs are attributes of WL, and imagine that, describing motion of SWLs classically, we divide the world line $\mathcal{L}$ into fragments $\mathcal{L}_{i} $, $i=1,2...$in such a way, that each fragment $\mathcal{L}_{i}$ describe either a particle, or an antiparticle. To make this, it is sufficient to make the following transformation for fragments $\mathcal{L}_{i}$, describing antiparticle. $$e\rightarrow -e,\qquad \tau \rightarrow -\tau ,\qquad \varepsilon \rightarrow \varepsilon \label{a3.1}$$ According to (\[a2.13\]), (\[a2.14\]) this transformation changes neither energy-momentum vector $P_{i}$, nor the charge $Q$ of SWL, but transformation (\[a3.1\]) changes the sign of the canonical momentum for antiparticles and does not change it for particles. As a result the temporal component $p_{0}$ of particles and antiparticles becomes negative, and one can identify the energy $E$ of particles and antiparticles with the time component $-p_{0}$ of their canonical momentum, i.e. $E=-p_{0}$. Conventionally such an identification is always produced, to satisfy so called causality principle. Temporal component $-p_{0}$ of canonical momentum is considered to be an energy of both particle and antiparticle. Using such an identification, it is always supposed that $dt/d\tau >0$, and one may ignore concept of orientation. The action (\[a1.6\]) and dynamic equations (\[a1.8\]) are invariant with respect to transformation (\[a3.1\]), whereas the canonical momentum (\[a1.7\]) changes its sign. As a result all fragments $\mathcal{L}_{i}$ of $\mathcal{L}$ have the same sgn$\left( p_{0}\right) $, and $E=-p_{0}$ along the whole $\mathcal{L}$. At the same time different fragments $\mathcal{L}_{i}$ of $\mathcal{L}$ (particles and antiparticles) are described as different dynamic systems with parameters $\left\{ m,e\right\} $ and $\left\{ m,-e\right\} $ respectively. Then interval between any two points $P$ and $P^{\prime }$, belonging to the same fragment $\mathcal{L}_{i}$, is timelike (we neglect the short spacelike segments of $\mathcal{L}_{i}$ near its ends). If interval between two points $P_{1}$ and $P_{2}$ is spacelike, then $P_{1}\in \mathcal{L}_{i}$ and $P_{2}\in \mathcal{L}_{k}$, $\;i\neq k$. The points belong to different dynamic systems and the Poisson brackets between dynamic variables at the points $P_{1}$ and $P_{2}$ vanish $$\left\{ u\left( P_{1}\right) ,v\left( P_{2}\right) \right\} =0,\qquad \left( x_{\left( \mathrm{1}\right) }^{i}-x_{\left( \mathrm{2}\right) }^{i}\right) \left( x_{\left( \mathrm{1}\right) i}-x_{\left( \mathrm{2}\right) i}\right) <0 \label{a2.18}$$ where $u\left( P_{1}\right) $ and $v\left( P_{2}\right) $ are any dynamic variables at the point $P_{1}$ and at the point $P_{2}$ respectively. The quantities $x_{\left( \mathrm{1}\right) }^{i}$ and $x_{\left( \mathrm{2}\right) }^{i}$ are coordinates of points $P_{1}$ and $P_{2}$ respectively. In the quantum field theory, where fragmentation of the total world line $\mathcal{L}$ is used, the relation (\[a2.18\]) takes the form $$\left[ u\left( x_{1}\right) ,v\left( x_{2}\right) \right] _{-}=0,\qquad \left( x_{1}-x_{2}\right) ^{2}<0 \label{a2.19}$$ where $\left[ ...\right] _{-}$ means a commutator of two operators, describing dynamic variables $u$ and $v$ at points $x_{1}$ and $x_{2}$ respectively. The relation (\[a2.19\]) is known as the causality principle. Interpretation of this principle is such. Events at the points $P_{1}$ and $P_{2}$ are independent, because no interaction can propagate faster, than the speed of the light. If the world line $\mathcal{L}$ is considered to be a whole object (WL), described by dynamic equations (\[a1.7\]), (\[a1.8\]) and parameter $\tau $, changing monotonically along $\mathcal{L}$, there are points $P_{1}$ and $P_{2}$, separated by spacelike interval and belonging to $\mathcal{L}$. Of course, it is possible only, if there are pair production and pair annihilation. For instance, in the example of the second section (see (\[a1.22\])) the points with coordinates $P_{1}=\left( t,x\right) $ and $P_{2}=\left( t,2\left( x_{0}+vt_{1}\right) -x\right) $ are separated by spacelike interval $2i\left| x-x_{0}-vt_{1}\right| $. According to (\[a1.22\]) they both belong to the same world line. As far as the state $\left( x_{2},p_{2}\right) $ of dynamic system at the point $P_{2}$ is a function of its state $\left( x_{1},p_{1}\right) $ at the point $P_{1}$, the relation (\[a2.18\]) cannot be fulfilled for all functions $u$ and $v$ of dynamic variables. It means that the causality principle in the form (\[a2.18\]) is violated. Events at points $P_{1}$ and $P_{2}$ appear to be dependent, when the particle and antiparticle are described as different states of total WL (but not as separate dynamic systems). Does it mean that the description in terms of WL is incompatible with the causality principle? No, because description in terms of one WL is only a part of the complete description of the particle-antiparticle collision. Indeed, let us imagine that we prepare high energy particle and antiparticle at the points $P_{1}$ and $P_{2}$, separated by spacelike interval. Let these SWLs move to meet each other. Then there are the following possibilities: (1) the particle and antiparticle annihilate (description in terms of one WL), (2) particle and antiparticle do not collide (description in terms of two WLs), (3) particle and antiparticle collide and generates $n$ pairs additionally (description in terms of $n+1$ WLs). Thus, the complete description of particle-antiparticle collision is described statistically by means of a sum of $k$-WL descriptions $k=1,2,...$ The causality principle can be applied only to the complete description. Description in terms of one WL is a special part of the complete description, and this partial description may contain a correlation between the points, separated by a spacelike interval, because the fact that the particle and antiparticle annihilate supposes some correlation between the states at the points $P_{1}$ and $P_{2}$, separated by a spacelike interval. This correlation for one-WL description means that not all Poison brackets (\[a2.18\]) do vanish. This correlation does not mean a violation of the causality principle, because one-WL description does not pretend to a complete description of the particle-antiparticle collision. The conventional description in terms of particles and antiparticles pretend to a complete description, and fulfillment of the causality principle is essential for the complete description. Understanding that the complete description is a very complicated problem, we divide the problem into parts (one WL, two WLs,...) and try to solve any part separately. Our strategy of investigation reminds the investigation of interaction of nonrelativistic particles, where the number of particles conserves. Then the whole investigation can be separated into one-particle problem, two-particle problem, three-particle problem,...These problems are solved in series. In the relativistic case the number of WLs is conserved, and the whole problem can be separated into parts, corresponding to fixed number of WLs. Some ideas of such an approach one can find in [@R94], but details need further development. In any case a description of the pair production in terms of one WL does not contradict to the causality principle, because this description does not pretend to a complete description of the collision process. In the modern relativistic quantum field theory a description is produced in terms of particles and antiparticles (but not in terms of WLs). This statement follows from identification of the energy operator $E$ with the Hamilton operator $\ H=p_{0}$ which is defined as an evolution operator. Formally this identification follows from relations $$i\hbar \partial _{k}u=\left[ u,P_{k}\right] _{-},\qquad P_{k}=\int T_{k}^{0}d\mathbf{x},\qquad k=0,1,2,3 \label{a3.6}$$ where $u$ is an operator of arbitrary dynamic variable, $\left[ ...\right] _{-}$ denotes commutator, and $P_{k}$ is the energy-momentum operator of the dynamic system. At such a description the causality principle is valid, because description is carried out in terms of particles and antiparticles (but not in terms of WLs). A difficulty of such an approach is a consideration of indefinite number of nonconserved objects (particles and antiparticles). This leads a perturbative description of the pair production processes. Conceptual defect of conventional description with a use of identification (\[a3.6\]) is incompatibility of the relations (\[a3.6\]) with the dynamic equations. Restrictions (\[a3.6\]) are compatible with dynamic equations, obtained at the canonical secondary quantization of linear equation $$\hbar ^{2}\partial _{i}\partial ^{i}\psi +m^{2}c^{2}\psi =0 \label{a3.8}$$ But they stop to be compatible with dynamic equations, obtained at the canonical secondary quantization of nonlinear equation $$\hbar ^{2}\partial _{i}\partial ^{i}\psi +m^{2}c^{2}\psi =g\psi ^{\ast }\psi \psi ,\qquad g=\text{const} \label{a3.9}$$ The fact is that the secondary quantization of (\[a3.9\]) can be produced without imposition of constraints (\[a3.6\]). Such a quantization is produced in terms of WLs [@R001]. It does not use the perturbation theory and admits to obtain solution of the scattering problem without cut off at $t\rightarrow \pm \infty $. If the secondary quantization can be carried out without imposition of (\[a3.6\]), one should test whether the relations (\[a3.6\]) are compatible with dynamic equations. The test shows that the constraints (\[a3.6\]) are compatible with the dynamic equations only for linear equation (\[a3.8\]), but they are incompatible with (\[a3.9\]) [@R001]. After imposition of constraints (\[a3.6\]) the problem becomes overdetermined. The overdetermined problem is inconsistent, and one can obtain practically any desired result for such a statement of the problem. It is quite reasonable that the causality principle (\[a2.19\]) is not fulfilled at the quantization in terms of WLs, because it is a corollary of the constraints (\[a3.6\]). Secondary quantization in terms WLs [@R001] does not describe pair production. It does not describe even scattering of a particle on antiparticle. But at the same time this quantization is consistent and well defined from the formal viewpoint. What does it mean? It means only that the nonlinear self-action term in (\[a3.9\]) does not describe pair production, because the pair production is apparently a more subtle effect. But why conventionally at quantization [@GJ68; @GJ70; @GJ970; @GJ72] of (\[a3.9\]) one obtains pair production? The answer is simple. Because of overdetermination of the problem, that makes the problem inconsistent. Practically, it looks as follows. Imposition of the constraint (\[a3.9\]) leads to fragmentation of the total world line into particles and antiparticles. After scattering one needs to produce defragmentation and collect fragments into the total world lines. Process of defragmentation is imperfect, because of inconsistent perturbative description. Reminders of imperfect defragmentation imitate pair production. At the conventional approach any nonlinear term leads to pair production, whereas classical description of the second section shows that the field, producing pairs must have rather specific properties. Failure of the paper [@R001] in description of pair production and inconsistency of conventional quantization in terms of particles and antiparticles show that one needs to investigate properties of the $\kappa $-field, producing pairs, more carefully, trying to understand what factor is responsible for pair production. Transformation of dynamic equations =================================== Let us add to the action (\[a0.16\]) the term describing interaction with the electromagnetic field and write it in the form $$\begin{aligned} \mathcal{A}\left[ x,\kappa \right] &=&\int \left\{ -mcK\sqrt{g_{ik}\dot{x}^{i}\dot{x}^{k}}-\frac{e}{c}A_{k}\dot{x}^{k}\right\} d^{4}\xi ,\qquad d^{4}\xi =d\xi _{0}d\mathbf{\xi ,} \label{A.1} \\ K &=&\sqrt{1+\lambda ^{2}\left( \kappa _{l}\kappa ^{l}+\partial _{l}\kappa ^{l}\right) },\qquad \lambda =\frac{\hbar }{mc} \label{A.2}\end{aligned}$$ where $x=\left\{ x^{i}\left( \xi _{0},\mathbf{\xi }\right) \right\} ,\;$ $i=0,1,2,3$ are dependent variables. $\xi =\left\{ \xi _{0},\mathbf{\xi }\right\} =\left\{ \xi _{k}\right\} ,\;\;k=0,1,2,3$ are independent variables, and $\dot{x}^{i}\equiv dx^{i}/d\xi _{0}.$ The quantities $\kappa ^{l}=\left\{ \kappa ^{l}\left( x\right) \right\} ,\;$ $l=0,1,2,3$ are dependent variables, describing stochastic component of the particle motion, $A_{k}=\left\{ A_{k}\left( x\right) \right\} ,\;\;k=0,1,2,3$ is the potential of electromagnetic field. We shall refer to the dynamic system, described by the action (\[A.1\]), (\[A.2\]) as $\mathcal{S}_{\mathrm{KG}}$, because irrotational flow of $\mathcal{S}_{\mathrm{KG}}$ is described by the Klein-Gordon equation [@R98]. We present here this transformation to the Klein-Gordon form, because we shall use intermediate stages of this transformation for further investigations. Let us consider variables $\xi =\xi \left( x\right) $ in (\[A.1\]) as dependent variables and variables $x$ as independent variables. Let the Jacobian $$J=\frac{\partial \left( \xi _{0},\xi _{1},\xi _{2},\xi _{3}\right) }{\partial \left( x^{0},x^{1},x^{2},x^{3}\right) }=\det \left| \left| \xi _{i,k}\right| \right| ,\qquad \xi _{i,k}\equiv \partial _{k}\xi _{i},\qquad i,k=0,1,2,3 \label{A.3}$$ be considered to be a multilinear function of $\xi _{i,k}$. Then $$d^{4}\xi =Jd^{4}x,\qquad \dot{x}^{i}\equiv \frac{dx^{i}}{d\xi _{0}}\equiv \frac{\partial \left( x^{i},\xi _{1},\xi _{2},\xi _{3}\right) }{\partial \left( \xi _{0},\xi _{1},\xi _{2},\xi _{3}\right) }=J^{-1}\frac{\partial J}{\partial \xi _{0,i}} \label{A.4}$$ After transformation to dependent variables $\xi $ the action (\[A.1\]) takes the form $$\mathcal{A}\left[ \xi ,\kappa \right] =\int \left\{ -mcK\sqrt{g_{ik}\frac{\partial J}{\partial \xi _{0,i}}\frac{\partial J}{\partial \xi _{0,k}}}-\frac{e}{c}A_{k}\frac{\partial J}{\partial \xi _{0,k}}\right\} d^{4}x\mathbf{,} \label{A.5}$$ Let us introduce new variables $$j^{k}=\frac{\partial J}{\partial \xi _{0,k}},\qquad k=0,1,2,3 \label{A.6}$$ by means of Lagrange multipliers $p_{k}$$$\mathcal{A}\left[ \xi ,\kappa ,j,p\right] =\int \left\{ -mcK\sqrt{g_{ik}j^{i}j^{k}}-\frac{e}{c}A_{k}j^{k}+p_{k}\left( \frac{\partial J}{\partial \xi _{0,k}}-j^{k}\right) \right\} d^{4}x\mathbf{,} \label{A.7}$$ Variation with respect to $\xi _{i}$ gives $$\frac{\delta \mathcal{A}}{\delta \xi _{i}}=-\partial _{l}\left( p_{k}\frac{\partial ^{2}J}{\partial \xi _{0,k}\partial \xi _{i,l}}\right) =0,\qquad i=0,1,2,3 \label{A.8}$$ Using identities $$\frac{\partial ^{2}J}{\partial \xi _{0,k}\partial \xi _{i,l}}\equiv J^{-1}\left( \frac{\partial J}{\partial \xi _{0,k}}\frac{\partial J}{\partial \xi _{i,l}}-\frac{\partial J}{\partial \xi _{0,l}}\frac{\partial J}{\partial \xi _{i,k}}\right) \label{A.9}$$ $$\frac{\partial J}{\partial \xi _{i,l}}\xi _{k,l}\equiv J\delta _{k}^{i},\qquad \partial _{l}\frac{\partial ^{2}J}{\partial \xi _{0,k}\partial \xi _{i,l}}\equiv 0 \label{A.10}$$ one can test by direct substitution that the general solution of linear equations (\[A.8\]) has the form $$p_{k}=b_{0}\left( \partial _{k}\varphi +g^{\alpha }\left( \mathbf{\xi }\right) \partial _{k}\xi _{\alpha }\right) ,\qquad k=0,1,2,3 \label{A.11}$$ where $b_{0}\neq 0$ is a constant, $g^{\alpha }\left( \mathbf{\xi }\right) ,\;\;\alpha =1,2,3$ are arbitrary functions of $\mathbf{\xi =}\left\{ \xi _{1},\xi _{2},\xi _{3}\right\} $, and $\varphi $ is the dynamic variable $\xi _{0}$, which stops to be fictitious. Let us substitute (\[A.11\]) in (\[A.7\]). The term of the form $\partial J/\partial \xi _{0,k}\partial _{k}\varphi $ is reduced to Jacobian and does not contribute to dynamic equation. The terms of the form $\xi _{\alpha ,k}\partial J/\partial \xi _{0,k}$ vanish due to identities (\[A.10\]). We obtain $$\mathcal{A}\left[ \varphi ,\mathbf{\xi },\kappa ,j\right] =\int \left\{ -mcK\sqrt{g_{ik}j^{i}j^{k}}-j^{k}\pi _{k}\right\} d^{4}x\mathbf{,} \label{A.12}$$ where quantities $\pi _{k}$ are determined by the relations $$\pi _{k}=b_{0}\left( \partial _{k}\varphi +g^{\alpha }\left( \mathbf{\xi }\right) \partial _{k}\xi _{\alpha }\right) +\frac{e}{c}A_{k},\qquad k=0,1,2,3 \label{A.12b}$$ Integration of (\[A.8\]) in the form (\[A.11\]) is that integration which was mentioned in introduction as a conceptual operation which admits to introduce a wave function. Note that coefficients in the system (\[A.8\]) of equations for $p_{k}$ are constructed of minors of the Jacobian (\[A.3\]). It is the circumstance that admits to produce a general integration. Variation of (\[A.12\]) with respect to $\kappa ^{l}$ gives $$\frac{\delta \mathcal{A}}{\delta \kappa ^{l}}=-\frac{\lambda ^{2}mc\sqrt{g_{ik}j^{i}j^{k}}}{K}\kappa _{l}+\partial _{l}\frac{\lambda ^{2}mc\sqrt{g_{ik}j^{i}j^{k}}}{2K}=0 \label{A.12a}$$ It can be written in the form $$\kappa _{l}=\partial _{l}\kappa =\frac{1}{2}\partial _{l}\ln \rho ,\qquad e^{2\kappa }=\frac{\rho }{\rho _{0}}\equiv \frac{\sqrt{j_{s}j^{s}}}{\rho _{0}mcK}, \label{A.13}$$ where $\rho _{0}=$const is the integration constant. Substituting (\[A.2\]) in (\[A.13\]), we obtain dynamic equation for $\kappa $$$\hbar ^{2}\left( \partial _{l}\kappa \cdot \partial ^{l}\kappa +\partial _{l}\partial ^{l}\kappa \right) =\frac{e^{-4\kappa }j_{s}j^{s}}{\rho _{0}^{2}}-m^{2}c^{2} \label{A.14}$$ Variation of (\[A.12\]) with respect to $j^{k}$ gives $$\pi _{k}=-\frac{mcKj_{k}}{\sqrt{g_{ls}j^{l}j^{s}}} \label{A.16}$$ or $$\pi _{k}g^{kl}\pi _{l}=m^{2}c^{2}K^{2} \label{A.15}$$ Substituting the second equation (\[A.13\]) in (\[A.16\]), we obtain $$j_{k}=-\rho _{0}e^{2\kappa }\pi _{k}, \label{A.17}$$ Now we eliminate the variables $j^{k}$ from the action (\[A.12\]), using relation (\[A.17\]) and (\[A.13\]). We obtain $$\mathcal{A}\left[ \varphi ,\mathbf{\xi },\kappa \right] =\int \rho _{0}e^{2\kappa }\left\{ -m^{2}c^{2}K^{2}+\pi ^{k}\pi _{k}\right\} d^{4}x\mathbf{,} \label{A.28}$$ where $\pi _{k}$ is determined by the relation (\[A.12b\]). Using expression (\[A.2\]) for $K,$ the first term of the action (\[A.28\]) can be transformed as follows. $$\begin{aligned} -m^{2}c^{2}e^{2\kappa }K^{2} &=&-m^{2}c^{2}e^{2\kappa }\left( 1+\lambda ^{2}\left( \partial _{l}\kappa \partial ^{l}\kappa +\partial _{l}\partial ^{l}\kappa \right) \right) \\ &=&-m^{2}c^{2}e^{2\kappa }+\hbar ^{2}e^{2\kappa }\partial _{l}\kappa \partial ^{l}\kappa -\frac{\hbar ^{2}}{2}\partial _{l}\partial ^{l}e^{2\kappa }\end{aligned}$$ Let us take into account that the last term has the form of divergence. It does not contribute to dynamic equations and can be omitted. Omitting this term, we obtain $$\mathcal{A}\left[ \varphi ,\mathbf{\xi },\kappa \right] =\int \rho _{0}e^{2\kappa }\left\{ -m^{2}c^{2}+\hbar ^{2}\partial _{l}\kappa \partial ^{l}\kappa +\pi ^{k}\pi _{k}\right\} d^{4}x\mathbf{,} \label{A.29}$$ Instead of dynamic variables $\varphi ,\mathbf{\xi },\kappa $ we introduce $n $-component complex function $$\psi =\left\{ \psi _{\alpha }\right\} =\left\{ \sqrt{\rho }e^{i\varphi }u_{\alpha }\left( \mathbf{\xi }\right) \right\} =\left\{ \sqrt{\rho _{0}}e^{\kappa +i\varphi }u_{\alpha }\left( \mathbf{\xi }\right) \right\} ,\qquad \alpha =1,2,...n \label{A.30}$$ Here $u_{\alpha }$ are functions of only $\mathbf{\xi }=\left\{ \xi _{1},\xi _{2},\xi _{3}\right\} $, having the following properties $$\sum\limits_{\alpha =1}^{\alpha =n}u_{\alpha }^{\ast }u_{\alpha }=1,\qquad -\frac{i}{2}\sum\limits_{\alpha =1}^{\alpha =n}\left( u_{\alpha }^{\ast }\frac{\partial u_{\alpha }}{\partial \xi _{\beta }}-\frac{\partial u_{\alpha }^{\ast }}{\partial \xi _{\beta }}u_{\alpha }\right) =g^{\beta }\left( \mathbf{\xi }\right) \label{A.31}$$ where ($^{\ast }$) denotes complex conjugation. The number $n$ of components of the wave function $\psi $ is chosen in such a way, that equations (\[A.31\]) have a solution. Then we obtain $$\begin{aligned} \psi ^{\ast }\psi &\equiv &\sum\limits_{\alpha =1}^{\alpha =n}\psi _{\alpha }^{\ast }\psi _{\alpha }=\rho =\rho _{0}e^{2\kappa },\qquad \partial _{l}\kappa =\frac{\partial _{l}\left( \psi ^{\ast }\psi \right) }{2\psi ^{\ast }\psi } \label{A.33} \\ \pi _{k} &=&-\frac{ib_{0}\left( \psi ^{\ast }\partial _{k}\psi -\partial _{k}\psi ^{\ast }\cdot \psi \right) }{2\psi ^{\ast }\psi }+\frac{e}{c}A_{k},\qquad k=0,1,2,3 \label{A.34}\end{aligned}$$ Substituting relations (\[A.33\]), (\[A.34\]) in (\[A.29\]), we obtain the action, written in terms of the wave function $\psi $$$\begin{aligned} \mathcal{A}\left[ \psi ,\psi ^{\ast }\right] &=&\int \left\{ \left[ \frac{ib_{0}\left( \psi ^{\ast }\partial _{k}\psi -\partial _{k}\psi ^{\ast }\cdot \psi \right) }{2\psi ^{\ast }\psi }-\frac{e}{c}A_{k}\right] \left[ \frac{ib_{0}\left( \psi ^{\ast }\partial ^{k}\psi -\partial ^{k}\psi ^{\ast }\cdot \psi \right) }{2\psi ^{\ast }\psi }-\frac{e}{c}A^{k}\right] \right. \nonumber \\ &&+\left. \hbar ^{2}\frac{\partial _{l}\left( \psi ^{\ast }\psi \right) \partial ^{l}\left( \psi ^{\ast }\psi \right) }{4\left( \psi ^{\ast }\psi \right) ^{2}}-m^{2}c^{2}\right\} \psi ^{\ast }\psi d^{4}x \label{A.35}\end{aligned}$$ Let us use the identity $$\begin{aligned} &&\frac{\left( \psi ^{\ast }\partial _{l}\psi -\partial _{l}\psi ^{\ast }\cdot \psi \right) \left( \psi ^{\ast }\partial ^{l}\psi -\partial ^{l}\psi ^{\ast }\cdot \psi \right) }{4\psi ^{\ast }\psi }+\partial _{l}\psi ^{\ast }\partial ^{l}\psi \nonumber \\ &\equiv &\frac{\partial _{l}\left( \psi ^{\ast }\psi \right) \partial ^{l}\left( \psi ^{\ast }\psi \right) }{4\psi ^{\ast }\psi }+\frac{g^{ls}}{2}\psi ^{\ast }\psi \sum\limits_{\alpha ,\beta =1}^{\alpha ,\beta =n}Q_{\alpha \beta ,l}^{\ast }Q_{\alpha \beta ,s} \label{A.36}\end{aligned}$$ where $$Q_{\alpha \beta ,l}=\frac{1}{\psi ^{\ast }\psi }\left| \begin{array}{cc} \psi _{\alpha } & \psi _{\beta } \\ \partial _{l}\psi _{\alpha } & \partial _{l}\psi _{\beta } \end{array} \right| ,\qquad Q_{\alpha \beta ,l}^{\ast }=\frac{1}{\psi ^{\ast }\psi }\left| \begin{array}{cc} \psi _{\alpha }^{\ast } & \psi _{\beta }^{\ast } \\ \partial _{l}\psi _{\alpha }^{\ast } & \partial _{l}\psi _{\beta }^{\ast } \end{array} \right| \label{A.37}$$ Then we obtain $$\begin{aligned} \mathcal{A}\left[ \psi ,\psi ^{\ast }\right] &=&\int \left\{ \left( ib_{0}\partial _{k}+\frac{e}{c}A_{k}\right) \psi ^{\ast }\left( -ib_{0}\partial ^{k}+\frac{e}{c}A^{k}\right) \psi +\frac{b_{0}^{2}}{2}\sum\limits_{\alpha ,\beta =1}^{\alpha ,\beta =n}g^{ls}Q_{\alpha \beta ,l}Q_{\alpha \beta ,s}^{\ast }\psi ^{\ast }\psi \right. \nonumber \\ &&\left. -m^{2}c^{2}\psi ^{\ast }\psi +\left( \hbar ^{2}-b_{0}^{2}\right) \frac{\partial _{l}\left( \psi ^{\ast }\psi \right) \partial ^{l}\left( \psi ^{\ast }\psi \right) }{4\psi ^{\ast }\psi }\right\} d^{4}x \label{A.38}\end{aligned}$$ Let us consider the case of irrotational flow, when $g^{\alpha }\left( \mathbf{\xi }\right) =0$ and the function $\psi $ has only one component. It follows from (\[A.37\]), that $Q_{\alpha \beta ,l}=0$, and only the last term in (\[A.38\]) is not bilinear with respect to $\psi ,\psi ^{\ast }$. The constant $b_{0}$ is an arbitrary integration constant. One may set $b_{0}=\hbar $. Then we obtain instead of (\[A.38\]) $$\mathcal{A}\left[ \psi ,\psi ^{\ast }\right] =\int \left\{ \left( i\hbar \partial _{k}+\frac{e}{c}A_{k}\right) \psi ^{\ast }\left( -i\hbar \partial ^{k}+\frac{e}{c}A^{k}\right) \psi -m^{2}c^{2}\psi ^{\ast }\psi \right\} d^{4}x \label{A.39}$$ Variation of the action (\[A.39\]) with respect to $\psi ^{\ast }$ generates the Klein-Gordon equation $$\left( -i\hbar \partial _{k}+\frac{e}{c}A_{k}\right) \left( -i\hbar \partial ^{k}+\frac{e}{c}A^{k}\right) \psi -m^{2}c^{2}\psi =0 \label{A.40}$$ Thus, description in terms of the Klein-Gordon equation is a special case of the stochastic system description by means of the action (\[A.1\]), (\[A.2\]). In the case, when the fluid flow is rotational, and the wave function $\psi $ is two-component, the identity (\[A.36\]) takes the form $$\begin{aligned} &&\frac{\left( \psi ^{\ast }\partial _{l}\psi -\partial _{l}\psi ^{\ast }\cdot \psi \right) \left( \psi ^{\ast }\partial ^{l}\psi -\partial ^{l}\psi ^{\ast }\cdot \psi \right) }{4\rho }-\frac{\left( \partial _{l}\rho \right) \left( \partial ^{l}\rho \right) }{4\rho } \nonumber \\ &\equiv &-\partial _{l}\psi ^{\ast }\partial ^{l}\psi +\frac{1}{4}\left( \partial _{l}s_{\alpha }\right) \left( \partial ^{l}s_{\alpha }\right) \rho \label{A.41}\end{aligned}$$ where 3-vector $\mathbf{s=}\left\{ s_{1},s_{2},s_{3},\right\} $ is defined by the relation $$\rho =\psi ^{\ast }\psi ,\qquad s_{\alpha }=\frac{\psi ^{\ast }\sigma _{\alpha }\psi }{\rho },\qquad \alpha =1,2,3 \label{A.42}$$ $$\psi =\left( _{\psi _{2}}^{\psi _{1}}\right) ,\qquad \psi ^{\ast }=\left( \psi _{1}^{\ast },\psi _{2}^{\ast }\right) , \label{A.43}$$ and Pauli matrices $\mathbf{\sigma }=\left\{ \sigma _{1},\sigma _{2},\sigma _{3}\right\} $ have the form $$\sigma _{1}=\left( \begin{array}{cc} 0 & 1 \\ 1 & 0 \end{array} \right) ,\qquad \sigma _{2}=\left( \begin{array}{cc} 0 & -i \\ i & 0 \end{array} \right) ,\qquad \sigma _{1}=\left( \begin{array}{cc} 1 & 0 \\ 0 & -1 \end{array} \right) \label{A.44}$$ Note that 3-vectors $\mathbf{s}$ and $\mathbf{\sigma }$ are vectors in the space $V_{\xi }$ of the Clebsch potentials $\mathbf{\xi }=\left\{ \xi _{1},\xi _{2},\xi _{3}\right\} $ and transform as vectors at the transformations (\[a0.6\]) of Clebsch potentials $\mathbf{\xi }$. The wave function $\psi $ is a spinor in the space $V_{\xi }$. The quantities $\mathbf{s}$, $\mathbf{\sigma }$ and $\psi $ are scalars in the 3-space $V_{x} $ of usual coordinates $\mathbf{x}=\left\{ x^{1},x^{2},x^{3}\right\} $. In general, transformations of Clebsch potentials $\mathbf{\xi }$ and those of coordinates $\mathbf{x}$ are independent. However, the action (\[A.35\]) does not contain any reference to the Clebsch potentials $\mathbf{\xi }$ and transformations (\[a0.6\]) of $\mathbf{\xi }$. If we consider only linear transformations of space coordinates $\mathbf{x}$$$x^{\alpha }\rightarrow \tilde{x}^{\alpha }=b^{\alpha }+\omega _{.\beta }^{\alpha }x^{\beta },\qquad \alpha =1,2,3 \label{A.44a}$$ nothing prevents from accompanying any transformation (\[A.44a\]) with the similar transformation $$\xi _{\alpha }\rightarrow \tilde{\xi}_{\alpha }=b^{\alpha }+\omega _{.\beta }^{\alpha }\xi _{\beta },\qquad \alpha =1,2,3 \label{A.44b}$$ of Clebsch potentials $\mathbf{\xi }$. The formulas for linear transformation of vectors and spinors in $V_{x}$ do not contain the coordinates $\mathbf{x}$ explicitly, and one can consider vectors and spinors in $V_{\xi }$ as vectors and spinors in $V_{x}$, provided we consider linear transformations (\[A.44a\]), (\[A.44b\]) always together. Using identity (\[A.41\]), we obtain from (\[A.35\]) $$\mathcal{A}\left[ \psi ,\psi ^{\ast }\right] =\int \left\{ \left( i\hbar \partial _{k}+\frac{e}{c}A_{k}\right) \psi ^{\ast }\left( -i\hbar \partial ^{k}+\frac{e}{c}A^{k}\right) \psi -m^{2}c^{2}\rho -\frac{\hbar ^{2}}{4}\left( \partial _{l}s_{\alpha }\right) \left( \partial ^{l}s_{\alpha }\right) \rho \right\} d^{4}x \label{A.45}$$ Dynamic equation, generated by the action (\[A.45\]), has the form $$\begin{aligned} &&\left( -i\hbar \partial _{k}+\frac{e}{c}A_{k}\right) \left( -i\hbar \partial ^{k}+\frac{e}{c}A^{k}\right) \psi -\left( m^{2}c^{2}+\frac{\hbar ^{2}}{4}\left( \partial _{l}s_{\alpha }\right) \left( \partial ^{l}s_{\alpha }\right) \right) \psi \nonumber \\ &=&\hbar ^{2}\frac{\partial _{l}\left( \rho \partial ^{l}s_{\alpha }\right) }{2\rho }\left( \sigma _{\alpha }-s_{\alpha }\right) \psi \label{A.46}\end{aligned}$$ The gradient of the unit 3-vector $\mathbf{s}=\left\{ s_{1},s_{2},s_{3}\right\} $ describes rotational component of the fluid flow. If $\mathbf{s}=$const the dynamic equation (\[A.46\]) turns to the conventional Klein-Gordon equation (\[A.40\]). Curl of the vector field $\pi _{k}$, determined by the relation (\[A.34\]), is expressed only via derivatives of the unit 3-vector $\mathbf{s}$. To show this, let us represent the wave function (\[A.30\]) in the form $$\psi =\sqrt{\rho }e^{i\varphi }\left( \mathbf{n\sigma }\right) \chi ,\qquad \psi ^{\ast }=\sqrt{\rho }e^{-i\varphi }\chi ^{\ast }\left( \mathbf{\sigma n}\right) ,\qquad \mathbf{n}^{2}=1,\qquad \chi ^{\ast }\chi =1 \label{A.47}$$ where $\mathbf{n}=\left\{ n_{1},n_{2},n_{3}\right\} $ is some unit 3-vector, $\chi =\left( _{\chi _{2}}^{\chi _{1}}\right) $, $\chi ^{\ast }=\left( \chi _{1}^{\ast },\chi _{2}^{\ast }\right) $ are constant two-component quantities, and $\mathbf{\sigma }=\left\{ \sigma _{1},\sigma _{2},\sigma _{3}\right\} $ are Pauli matrices (\[A.44\]). The unit vector $\mathbf{s}$ and the unit vector $\mathbf{n}$ are connected by means of the relations $$\mathbf{s}=2\mathbf{n}\left( \mathbf{nz}\right) -\mathbf{z},\qquad \mathbf{n}=\frac{\mathbf{s}+\mathbf{z}}{\sqrt{2\left( 1+\left( \mathbf{sz}\right) \right) }} \label{A.48}$$ where $\mathbf{z}$ is a constant unit vector defined by the relation $$\mathbf{z}=\chi ^{\ast }\mathbf{\sigma }\chi ,\qquad \mathbf{z}^{2}=\chi ^{\ast }\chi =1 \label{A.49}$$ All 3-vectors $\mathbf{n}$, $\mathbf{s}$, $\mathbf{z}$ are vectors in $V_{\xi }$. Let us substitute the relation (\[A.47\]) into expression $\partial _{l}\pi _{k}-\partial _{k}\pi _{l}$ for the curl of the vector field $\pi _{k}$ defined by the relation (\[A.34\]). Then gradually reducing powers of $\sigma $ by means of the identity $$\sigma _{\alpha }\sigma _{\beta }\equiv \delta _{\alpha \beta }+i\varepsilon _{\alpha \beta \gamma }\sigma _{\gamma },\qquad \alpha ,\beta =1,2,3 \label{A.50}$$ where $\varepsilon _{\alpha \beta \gamma }$ is the Levi-Chivita pseudotensor ($\varepsilon _{123}=1$), we obtain after calculations $$\begin{aligned} \pi _{k} &=&-\frac{ib_{0}\left( \psi ^{\ast }\partial _{k}\psi -\partial _{k}\psi ^{\ast }\cdot \psi \right) }{2\psi ^{\ast }\psi }+\frac{e}{c}A_{k} \nonumber \\ &=&b_{0}\left( \partial _{k}\varphi +\varepsilon _{\alpha \beta \gamma }n_{\alpha }\partial _{k}n_{\beta }z_{\gamma }\right) +\frac{e}{c}A_{k} \qquad k=0,1,2,3 \label{A.51}\end{aligned}$$ $$\partial _{k}\pi _{l}-\partial _{l}\pi _{k}=-4b_{0}\left[ \partial _{k}\mathbf{n}\times \partial _{l}\mathbf{n}\right] \mathbf{z}+\frac{e}{c}\left( \partial _{k}A_{l}-\partial _{l}A_{k}\right) ,\qquad k,l=0,1,2,3 \label{A.52}$$ The relation (\[A.52\]) may be expressed also via the 3-vector $\mathbf{s}$, provided we use the formulae (\[A.48\]). Note that the two-component form of the wave function can describe irrotational flow. For instance, if $\psi =\left( _{\psi _{1}}^{\psi _{1}}\right) $, $s_{1}=1,$ $s_{2}=s_{3}=0$, the dynamic equation (\[A.46\]) reduces to the form (\[A.40\]), and curl of $\pi _{k}$, defined by (\[A.52\]) vanishes. Interpretation of solutions of the free Klein-Gordon equation ============================================================= Let us write the action (\[A.12\]), (\[A.2\]) in the arbitrary curvilinear coordinate system. We obtain for dynamic system $\mathcal{S}_{\mathrm{KG}}$ $$\begin{aligned} \mathcal{A}\left[ \varphi ,\mathbf{\xi },\kappa ,j\right] &=&\int \left\{ -mcK\sqrt{g_{ik}j^{i}j^{k}}-j^{k}\pi _{k}\right\} \sqrt{-g}d^{4}x\mathbf{,} \label{a4.1} \\ K &=&\sqrt{1+\lambda ^{2}\left( g_{ik}\kappa ^{i}\kappa ^{k}+\frac{1}{\sqrt{-g}}\partial _{l}\left( \sqrt{-g}\kappa ^{l}\right) \right) },\qquad \lambda =\frac{\hbar }{mc} \label{a4.2}\end{aligned}$$ $$g=\det \left| \left| g_{ik}\right| \right| \label{a4.2a}$$ where quantities $\pi _{k}$ are determined by the relations (\[A.12b\]) $$\pi _{k}=b_{0}\left( \partial _{k}\varphi +g^{\alpha }\left( \mathbf{\xi }\right) \partial _{k}\xi _{\alpha }\right) +\frac{e}{c}A_{k},\qquad k=0,1,2,3 \label{a4.3}$$ Trying to interpret dynamic system $\mathcal{S}_{\mathrm{KG}}$ described by the action (\[a4.1\]) – (\[a4.3\]), we shall *use only dynamic considerations*. This means that $\mathcal{S}_{\mathrm{KG}}$ is considered to be a fluid, and world lines of the fluid particles are interpreted as average world lines of stochastic particles. We consider $\mathcal{S}_{\mathrm{KG}}$ as a fluid and interpret its motion in terms of hydrodynamics. We ignore interpretation, based on quantum principles, because this interpretation in terms of the wave function is purely empirical. It has been tested only for nonrelativistic quantum phenomena. In the relativistic case there are problems for interpretation of expressions constructed on the basis of the wave function. From viewpoint of hydrodynamic interpretation the wave function is only a very special method of description of $\mathcal{S}_{\mathrm{KG}}$. This method is very convenient for solution of dynamic equations, because they are linear in terms of wave function, but it is not the best method for interpretation of the fluid motion. We shall interpret dynamic system, following the behaviour of such physical quantities as the current $j^{i}$ and the energy-momentum tensor $T^{ik}$. The energy-momentum tensor for $\mathcal{S}_{\mathrm{KG}}$ is obtained by means of relation (\[a2.2\]). Applying the relation (\[a2.2\]) to the action (\[a4.1\]) – (\[a4.3\]) and taking into account that the Lagrangian density vanishes due to (\[A.16\]), we obtain $$\begin{aligned} T^{ik} &=&\frac{mcKj^{i}j^{k}}{\sqrt{g_{ls}j^{l}j^{s}}}+\lambda ^{2}\frac{mc\sqrt{g_{ls}j^{l}j^{s}}}{K}\kappa ^{i}\kappa ^{k}+\lambda ^{2}\frac{mc\sqrt{g_{ls}j^{l}j^{s}}}{K}\left( -\frac{1}{2\left( \sqrt{-g}\right) }g^{ik}\partial _{l}\left( \sqrt{-g}\kappa ^{l}\right) \right) \\ &&-\lambda ^{2}\left( \frac{1}{2}g^{ik}\kappa ^{j}\right) \partial _{j}\frac{mc\sqrt{g_{ls}j^{l}j^{s}}}{K}\end{aligned}$$ We use the relation (\[A.13\]) $$e^{2\kappa }=\frac{\rho }{\rho _{0}}\equiv \frac{\sqrt{j_{s}j^{s}}}{\rho _{0}mcK}. \label{a4.8}$$ Returning to the inertial coordinate system, where $\sqrt{-g}=c$, we obtain $$T^{ik}=\frac{j^{i}j^{k}}{\rho }+\hbar ^{2}\rho \kappa ^{i}\kappa ^{k}-\frac{\hbar ^{2}}{2}g^{ik}\partial _{l}\left( \rho \kappa ^{l}\right) \label{a4.7}$$ or $$\begin{aligned} T^{ik} &=&\frac{j^{i}j^{k}+j_{\mathrm{st}}^{i}j_{\mathrm{st}}^{k}}{\rho }-\frac{\hbar }{2}g^{ik}\partial _{l}j_{\mathrm{st}}^{l}, \label{a4.10} \\ j_{\mathrm{st}}^{k} &=&\hbar \rho \kappa ^{k}=\frac{\hbar }{2}\partial ^{k}\rho ,\qquad \rho =\rho _{0}e^{2\kappa }=\frac{\sqrt{j^{l}j_{l}}}{mcK},\qquad \kappa ^{i}=g^{ik}\partial _{k}\kappa \label{a4.10a}\end{aligned}$$ The energy-momentum tensor (\[a4.10\]) looks as the energy-momentum tensor of two fluids. Flux of one fluid is described by the 4-vector $j^{i}$, whereas the flux of other fluid is described by the 4-vector $j_{\mathrm{st}}^{i}=\hbar \rho \kappa ^{i}=\hbar g^{ik}\rho \partial _{k}\kappa $. In the first fluid there is no pressure. The pressure $p$ in the second fluid is described by the expression $$p=\frac{\hbar }{2}\partial _{l}j_{\mathrm{st}}^{l}$$ The two fluids interact via common factor $\rho =\rho _{0}e^{2\kappa }$. Eigenvectors $u_{i}$ and eigenvalues $\mu $ of the energy-momentum tensor are defined by the relations $$T^{ik}u_{k}=\mu g^{ik}u_{k},\qquad u_{k}=\alpha j_{k}+\beta j_{\mathrm{st}k} \label{a4.11}$$ Calculation gives $$\mu _{1,2}=-\frac{\hbar }{2}\partial _{l}j_{\mathrm{st}}^{l}+\frac{1}{2\rho }\left( j_{l}j^{l}+j_{\mathrm{st}l}j_{\mathrm{st}}^{l}\pm \sqrt{\left( j^{l}j_{l}-\left( j_{\mathrm{st}}^{l}j_{\mathrm{st}l}\right) \right) ^{2}+4\left( j_{\mathrm{st}}^{l}j_{l}\right) ^{2}}\right) \label{a4.12}$$ $$\beta _{1,2}=\frac{\left( j_{l}j^{l}-j_{\mathrm{st}l}j_{\mathrm{st}}^{l}\pm \sqrt{\left( j^{l}j_{l}-\left( j_{\mathrm{st}}^{l}j_{\mathrm{st}l}\right) \right) ^{2}+4\left( j_{\mathrm{st}}^{l}j_{l}\right) ^{2}}\right) }{2j_{l}j_{\mathrm{st}}^{l}}\alpha \label{a4.12a}$$ There is a very simple and interesting case, when one of fluids vanishes and $$j^{i}\equiv 0,\qquad i=0,1,2,3. \label{a4.14}$$ As it follows from (\[A.14\]), in this case the scalar $e^{\kappa }$ satisfies the free Klein-Gordon equation $$\left( \hbar ^{2}\partial _{l}\partial ^{l}+m^{2}c^{2}\right) e^{\kappa }=0 \label{a4.15}$$ and $$K=\sqrt{1+\lambda ^{2}e^{-\kappa }\partial _{l}\partial ^{l}e^{\kappa }}=0 \label{a4.16}$$ On the other side, in this case the wave function (\[A.30\]) is one-component, and it satisfies the Klein-Gordon equation (\[A.40\]). If the condition (\[a4.14\]) is satisfied, the wave function has the form $\psi =e^{\kappa +i\varphi }$, $\varphi =$const. Equations (\[a4.15\]) and (\[A.40\]) are compatible, provided $A_{i}=0$. Under conditions (\[a4.14\]) and (\[a4.16\]) the density $\rho $ (\[a4.8\]) and the energy-momentum tensor (\[a4.10\]) become indefinite. Besides, $\kappa $ is to be real and $e^{\kappa }\geq 0$. To obtain the canonical energy-momentum tensor for the irrotational flow, we consider the action of the form (\[A.39\]) with additional condition $A_{i}=0$, where $\psi $ is the one-component wave function. The action has the form $$\mathcal{A}\left[ \psi ,\psi ^{\ast }\right] =\int \left\{ \hbar ^{2}\partial _{k}\psi ^{\ast }\cdot \partial ^{k}\psi -m^{2}c^{2}\psi ^{\ast }\psi \right\} d^{4}x \label{a4.17}$$ Then canonical energy-momentum tensor $$T_{\mathrm{c}}\,_{k}^{l}=\hbar ^{2}\left( \partial _{k}\psi ^{\ast }\cdot \partial ^{l}\psi +\partial ^{l}\psi ^{\ast }\cdot \partial _{k}\psi \right) -\delta _{k}^{l}\left( \hbar ^{2}\partial _{k}\psi ^{\ast }\cdot \partial ^{k}\psi -m^{2}c^{2}\psi ^{\ast }\psi \right) \label{a4.18}$$ coincides with the relation (\[a4.10\]). Indeed, let us substitute in (\[a4.18\]) the expression $$\partial _{k}\psi =\sqrt{\rho _{0}}e^{\kappa }\partial _{k}\kappa -\frac{i}{\hbar \sqrt{\rho _{0}}}e^{-\kappa }j_{k}=\sqrt{\rho }\partial _{k}\kappa -\frac{i}{\hbar \sqrt{\rho }}j_{k} \label{a4.19a}$$ which is valid for the irrotational flow. $$T_{\mathrm{c}kl}=\frac{j_{l}j_{k}}{\rho }+\hbar ^{2}\rho \partial _{l}\kappa \cdot \partial _{k}\kappa -\rho _{0}e^{2\kappa }g_{kl}\left( \hbar ^{2}\partial _{s}\kappa \cdot \partial ^{s}\kappa +\frac{1}{\rho _{0}^{2}}e^{-4\kappa }j_{s}j^{s}-m^{2}c^{2}\right)$$ Taking into account dynamic equation (\[A.14\]), we eliminate two last terms $$T_{\mathrm{c}kl}=\frac{j_{l}j_{k}}{\rho }+\hbar ^{2}\rho \partial _{l}\kappa \cdot \partial _{k}\kappa -g_{kl}\frac{\hbar ^{2}}{2}\partial _{s}\partial ^{s}e^{2\kappa } \label{a4.19b}$$ and the relations (\[a4.7\]) and (\[a4.19b\]) coincide. Solution of equation (\[a4.15\]), which is stationary in the whole space-time, is $e^{\kappa }=e^{\kappa \left( \mathbf{x}\right) }$. It satisfies the equation $$\left( \hbar ^{2}\partial _{\alpha }\partial _{\alpha }-m^{2}c^{2}\right) e^{\kappa \left( \mathbf{x}\right) }=0$$ Solutions of this equation have the form $$\psi _{\mathbf{k}}=e^{\kappa \left( \mathbf{x}\right) }=B_{\mathbf{k}}\cos \left( \frac{\mathbf{kx}}{\hbar }+\phi _{\mathbf{k}}\right) ,\qquad \mathbf{k}^{2}=m^{2}c^{2},\qquad B_{\mathbf{k}}=\text{const} \label{a4.20}$$ None of these solutions satisfies the condition $\psi \geq 0$ except for $\psi =0$. For this solution $$T^{00}=0,\qquad T^{0\alpha }=0,$$ and the solution $\psi =0$ may be interpreted as a vacuum state. Possibility of construction of a linear combination of solutions (\[a4.20\]) which would be positive in the whole space-time seems to be problematic, although it is possible in small regions $\Omega _{\mathrm{v}}$ of the space-time. At the boundaries of $\Omega _{\mathrm{v}}$ the solution $\psi $ becomes negative and $\kappa $ becomes to be complex. It corresponds to $j^{k}\neq 0$, i.e. to appearance of SWLs (particles and antiparticles). For solutions $e^{\kappa }$, which are linear combinations of solutions (\[a4.20\]), $T^{00}\geq 0$, and these solutions may be interpreted as excited states of vacuum inside the region $\Omega _{\mathrm{v}}$. At the boundaries of $\Omega _{\mathrm{v}}$ pairs of SWLs appear. In general, the pairs are virtual in the sense that for each of SWL $p_{i}p^{i}\neq m^{2}c^{2}$. Another case, when there is only one fluid, described by the vector $j^{k}$ and $j_{\mathrm{st}}^{l}=0$ corresponds to description of statistical ensemble of relativistic classical (deterministic) particles. Interpretation of $\protect\kappa $-field in the Klein-Gordon dynamic system ============================================================================ Let us consider solutions of the free Klein-Gordon equation ($A_{k}=0$,  $k=0,1,2,3$). For simplicity of calculations we set inside this paragraph $\hbar =c=1$. The general solution of the free Klein-Gordon equation has the form $$\begin{aligned} \psi \left( t,\mathbf{x}\right) &=&\left( 2\pi \right) ^{-3/2}\int e^{-i\varepsilon _{k}E\left( \mathbf{k}\right) t+i\mathbf{kx}}\frac{a\left( K\right) }{\sqrt{2E\left( \mathbf{k}\right) }}dK,\qquad \label{a7.1} \\ K &=&\left\{ \varepsilon _{k},\mathbf{k}\right\} ,\qquad E\left( \mathbf{k}\right) =\left| \sqrt{m^{2}+\mathbf{k}^{2}}\right| ,\qquad \mathbf{kx}\equiv -k_{\alpha }x^{\alpha }\equiv k_{\alpha }x_{\alpha } \label{a7.2}\end{aligned}$$ where $a\left( K\right) $ is a function of $K=\left\{ \varepsilon _{k},\mathbf{k}\right\} $, $\;\mathbf{k}=\left\{ k_{1},k_{2},k_{3}\right\} $,$\;\;\varepsilon _{k}=\pm 1$ and $$\int \left( .\right) dK\equiv \sum\limits_{\varepsilon _{k}=\pm 1}\int \left( .\right) d\mathbf{k}\equiv \sum\limits_{\varepsilon _{k}=\pm 1}\iiint_{-\infty }^{+\infty }\left( .\right) dk_{1}dk_{2}dk_{3} \label{a7.3}$$ Let us set $$a_{P}\left( K\right) =a_{\varepsilon _{p},\mathbf{p}}\left( K\right) =A\sqrt{2E\left( \mathbf{k}\right) }\exp \left( -\frac{\left( \mathbf{k}-\mathbf{p}\right) ^{2}}{2\Delta }\right) \delta _{\varepsilon _{k},\varepsilon _{p}},\qquad A=\text{const} \label{a7.4}$$ The quantities $P=\left\{ \varepsilon _{p},\mathbf{p}\right\} $ and $\Delta $ are parameters of the solution. We suppose $$\sqrt{\Delta }\ll \lambda ^{-1}=m,\qquad \hbar =c=1 \label{a7.6}$$ Under the condition (\[a7.6\]) the following approximate expression for the wave function is obtained $$\psi _{P}\left( t,\mathbf{x}\right) =\frac{B\Delta ^{3/2}}{\left( 1+i\varepsilon _{k}\Delta t\left( \frac{3}{2E}+\frac{\mathbf{v}^{2}}{2}\right) \right) }\exp \left( -\frac{\Delta \left( \mathbf{x}-\mathbf{v}t\right) ^{2}}{2}-i\varepsilon _{p}E\left( \mathbf{p}\right) t+i\mathbf{px}\right)$$ where $$\mathbf{v}=\frac{\varepsilon _{p}\mathbf{p}}{E\left( \mathbf{p}\right) },\qquad B=B^{\ast }=\text{const} \label{a7.7}$$ At $\left| t\right| \ll E/\Delta $ we shall use a simpler expression for the wave function $$\psi _{P}\left( t,\mathbf{x}\right) =B\Delta ^{3/2}\exp \left( -\frac{\Delta \left( \mathbf{x}-\mathbf{v}t\right) ^{2}}{2}-i\varepsilon _{p}E\left( \mathbf{p}\right) t+i\mathbf{px}\right) \label{a7.8}$$ In the space-time region $\left| \mathbf{x}-\mathbf{v}t\right| <\Delta ^{-1/2}$ we obtain the following values for the quantities $j^{k}$, $\kappa ^{k}$, $T_{k}^{0}$ $$\begin{array}{lll} j^{0}=\left( \varepsilon _{p}E\left( \mathbf{p}\right) +O\left( \Delta \right) \right) \rho & \kappa ^{0}=\Delta \left( \mathbf{x}-\mathbf{v}t\right) \mathbf{v} & T_{0}^{0}=\left( E^{2}\left( \mathbf{p}\right) +O\left( \Delta \right) \right) \rho \\ j^{\alpha }=\left( p_{\alpha }+O\left( \Delta \right) \right) \rho & \kappa ^{\alpha }=\Delta \left( x^{\alpha }-v^{\alpha }t\right) & T_{\alpha }^{0}=\left( \varepsilon _{p}E\left( \mathbf{p}\right) p_{a}+O\left( \Delta \right) \right) \rho \\ \rho =\psi _{P}^{\ast }\psi _{P} & m_{\mathrm{eff}}^{2}=m^{2}+O\left( \Delta \right) & T_{\alpha }^{\beta }=\left( p_{\alpha }p^{\beta }+O\left( \Delta \right) \right) \rho \end{array} \label{a7.9}$$ where $$\psi _{P}^{\ast }\psi _{P}=\rho _{0}e^{2\kappa }=B^{2}\Delta ^{3}\exp \left( -\Delta \left( \mathbf{x}-\mathbf{v}t\right) ^{2}\right) \label{a7.11}$$ Interpretation from hydrodynamic viewpoint looks as follows. We assume that the state $P=\left\{ 1,\mathbf{p}\right\} $ with $\varepsilon _{p}=1$ corresponds to a particle, and the state $P=\left\{ -1,\mathbf{p}\right\} $ with $\varepsilon _{p}=-1$ corresponds to an antiparticle, because as it follows from (\[a7.9\]) the flux density $j^{0}$ has different sign in the two states. The total number $N$ of SWLs is $$N_{P}=\varepsilon _{p}E\left( \mathbf{p}\right) I,\qquad I=\int \psi _{P}^{\ast }\psi _{P}d\mathbf{x}=B^{2}\left( \pi \Delta \right) ^{3/2} \label{a7.12}$$ The total energy-momentum vector $$P_{k}=\left\{ E^{2}\left( \mathbf{p}\right) I,E\left( \mathbf{p}\right) \mathbf{p}I\right\} \label{a7.13}$$ does not depend on orientation $\varepsilon _{p}$, whereas the energy per one SWL $$\frac{P_{0}}{N_{P}}=\frac{E_{P}}{N_{P}}=\varepsilon _{p}E\left( \mathbf{p}\right) \label{a7.14}$$ depends on $\varepsilon _{p}$. It is negative for $\varepsilon _{p}=-1$, because the number $N_{P}$ of SWLs is negative. Thus, the state $\psi _{P}$ describes a fluid, consisting of a cloud of $N_{P}$ SWLs, moving with velocity $\mathbf{v}=\mathbf{p}/E\left( \mathbf{p}\right) +O\left( \Delta \right) $. The pressure in the fluid is small (of the order $O\left( \Delta \right) $), and one can speak on freely moving SWLs (particles or antiparticles). From viewpoint of quantum principles the state $\psi _{\left\{ 1,\mathbf{p}\right\} }$ (for particles) is interpreted similarly. But for interpretation of the state $\psi _{\left\{ -1,\mathbf{p}\right\} }$ (for antiparticles) there are problems [@FV58], connected with negative value of the quantity $E_{P}/N_{P}$ (energy per SWL). Now let us consider the state $\psi =\psi _{\left\{ 1,\mathbf{p}\right\} }+\psi _{\left\{ 1,-\mathbf{p}\right\} }$. In the conventional quantum mechanics the wave function is a fundamental object of theory. The wave functions $\psi _{\left\{ 1,\mathbf{p}\right\} }$ and $\psi _{\left\{ 1,-\mathbf{p}\right\} }$ evolve independently and this fact is interpreted in the sense that the particles described by $\psi _{\left\{ 1,\mathbf{p}\right\} }$ and the particles described by $\psi _{\left\{ 1,-\mathbf{p}\right\} }$ do not interact. Thus, from viewpoint of quantum principles the state $\psi $ describes two clouds of particles, moving with velocities $\mathbf{v}$ and $-\mathbf{v}$ one through another without interaction. From hydrodynamic viewpoint the state $\psi =\psi _{\left\{ 1,\mathbf{p}\right\} }+\psi _{\left\{ 1,-\mathbf{p}\right\} }$ describes rather complicated picture of two colliding clouds of particles. According to the hydrodynamic interpretation the world lines of particles in the two-dimensional space-time are defined by the equation $$\frac{dx}{dt}=\frac{j^{1}\left( t,x\right) }{j^{0}\left( t,x\right) }=\frac{\left( p\sinh \left( 2\Delta xvt\right) \right) -\Delta vt\left( \sin \left( 2px\right) \right) }{E\left( \cosh \left( 2\Delta xvt\right) +\cos \left( 2px\right) \right) +\Delta xv\left( \sin \left( 2px\right) \right) }+O\left( \Delta ^{2}\right) \label{a7.17}$$ Shape of world lines, obtained by numerical calculations for $-2/\sqrt{\Delta }<t<2/\sqrt{\Delta }$ is shown in figure 1. One can see, that after collision of two clouds the particles of clouds reflect and move in the opposite direction. This situation correlates with values of the energy-momentum tensor and other hydrodynamic quantities, calculated in the space-time region $x^{2},t^{2}\le \Delta ^{-1}$, where the two clouds overlap. These quantities are calculated to within $\Delta ^{0}$ $$\begin{array}{lll} j^{0}=2CE\left( p\right) \cos ^{2}\left( px\right) & \kappa _{0}=0 & T^{00}=2CE^{2}\left( p\right) \cos ^{2}\left( px\right) \\ j_{1}=0 & \kappa _{1}=-p\tan \left( px\right) & T^{01}=0 \\ \rho =2C\cos ^{2}\left( px\right) & m_{\mathrm{eff}}^{2}=E^{2}\left( p\right) & T^{11}=2Cp^{2}\sin ^{2}\left( px\right) \end{array} \label{a7.15}$$ where $$C=B^{2}\exp \left( -\Delta \left( x^{2}+v^{2}t^{2}\right) \right) \label{a7.16}$$ The tensor energy-momentum component $T^{00}=j^{0}j^{0}/\rho $ is determined by the 4-vector $j^{k}$, whereas the pressure in the fluid is determined by the component $T^{11}=\rho \kappa ^{1}\kappa ^{1}$. If the particles of two clouds do not interact, the energy-momentum tensor is a sum of energy-momentum tensors for each cloud separately. It has the form $$T^{00}=2CE^{2}\left( p\right) ,\qquad T^{01}=0,\qquad T^{11}=2Cp^{2} \label{a7.16a}$$ and this expression distinguishes from (\[a7.15\]) Let us consider the state $\psi =\psi _{\left\{ \varepsilon _{p},\mathbf{p}\right\} }+\psi _{\left\{ -\varepsilon _{p},\mathbf{p}\right\} }$. From viewpoint of quantum principles the state $\psi $ describes a cloud of particles and a cloud of antiparticles, moving with velocities $\mathbf{v}$ and $-\mathbf{v}$ one through another without interaction. From viewpoint of hydrodynamic interpretation the state $\psi _{\left\{ 1,\mathbf{p}\right\} }+\psi _{\left\{ -1,\mathbf{p}\right\} }$ describes a complicated picture of interacting particles and antiparticles. In the process of interaction the pair production and the pair annihilation take place. The process of interaction is balanced, and finally we see a cloud of particles and a cloud of antiparticles escaping one from another, as if they have passed one through another without interaction. In the two-dimensional space-time the mean world lines are described by the equation $$\frac{dx}{dt}=\frac{j^{1}}{j^{0}}=\frac{p\cosh \left( 2\Delta xvt\right) +p\cos \left( 2Et\right) -\Delta vt\sin \left( 2Et\right) }{E\sinh \left( 2\Delta xvt\right) +\Delta xv\sin \left( 2Et\right) }$$ Shape of world lines obtained by numerical calculations for $-2/\sqrt{\Delta }<x<2/\sqrt{\Delta }$ is shown in figures 2. In the state $\psi _{\left\{ 1,\mathbf{p}\right\} }+\psi _{\left\{ -1,\mathbf{p}\right\} }$ colliding particles and antiparticles annihilate, their energy transforms to energy of the $\kappa $-field. Thereafter the $\kappa $-field generate SWL pairs and energy of the $\kappa $-field turns to energy of escaping particles and antiparticles. Distribution of space-time regions, where effective mass $m_{\mathrm{eff}}$ is imaginary is shown in figure 3. Described interaction of particles and antiparticles correlates with values of the energy-momentum tensor and other hydrodynamic quantities, calculated in the space-time region $x^{2},t^{2}\le \Delta ^{-1}$, where the two clouds overlap. These quantities are calculated to within $\Delta ^{0}$ $$\begin{array}{lll} j_{0}=0 & \kappa _{0}=-E\tan \left( Et\right) & T^{00}=2E^{2}C\sin ^{2}\left( Et\right) \\ j_{1}=-2pC\cos ^{2}\left( Et\right) & \kappa _{1}=0 & T^{01}=0 \\ \rho =2C\cos ^{2}\left( Et\right) & m_{\mathrm{eff}}^{2}=-p^{2} & T^{11}=2p^{2}C\cos ^{2}\left( Et\right) \end{array} \label{a7.20}$$ where $C$ is determined by the relation (\[a7.16\]). Our investigation of solutions of the free Klein-Gordon equation is a preliminary consideration, which is necessary for correct mathematical formulation of the collision problem and of the problem pair production as a special case of the collision problem. Two different interpretations of the states $\psi _{\left\{ \varepsilon _{p},\mathbf{p}\right\} }+\psi _{\left\{ \varepsilon _{p},-\mathbf{p}\right\} }$ and $\psi _{\left\{ 1,\mathbf{p}\right\} }+\psi _{\left\{ -1,\mathbf{p}\right\} }$ generate two different statements of the collision problem. The conventional interpretation based on the wave function as a fundamental object of dynamics leads to the S-matrix theory, where detailed description of dynamic processes inside the collision space-time region is supposed to be impossible. For solution of these problems the energy-momentum conservation law is used mainly. Alternative approach based on sequential consideration of fluid dynamics and hydrodynamic quantities leads to another statement of the collision problem. Comparison of these two different statements of the collision problem is possible only at the detailed presentation of the alternative statement of the collision problem. Unfortunately, such a presentation cannot be made in the present paper. We restrict our consideration to the problem of two elastic balls collision, which reminds to some extent the collision problem of the microparticle collision. At the head-on collision of two identical elastic balls they interchange their momenta, and the whole situation may be described as one ball passes through another one without interaction. It is a very simple interpretation, but it is not effective, because it can be used only for head-on collision. If the impact parameter does not vanish, or the balls are not identical, we are forced to introduce interaction between the balls. If we assume that the identical balls do not interact at the head-on collision, it is rather difficult to understand, why they begin to interact with non-vanishing impact parameter. One needs to invent a special interaction for explanation of non-head-on collision. Is it possible to construct such a theory of two balls collision, which is based on the supposition that the balls do not interact at the head-on collision? Yes it is possible, but such a theory is founded mainly on the energy-momentum conservation law, and it does not fit for explanation of inelastic collisions, where detailed investigation of interaction between the balls is necessary. The approach, when the head-on collision of two identical elastic balls is considered to be a strong interaction of two balls, which leads to an interchange of momenta, seems to be more reasonable, because it explains freely, why result of collision is not reduced to the interchange of momenta at the collision with non-vanishing impact parameter. At such an approach the interchange of momenta is explained by balanced interaction at the head-on collision. If the impact parameter does not vanish, this balance is violated. Although explanation of the balanced interaction is more complicated, but application of this interaction to the case of unbalanced interaction appears to be simpler. Something like this we have in the case of two alternative approaches to the microparticle collision problem. Concluding remarks ================== Thus, quantum system is a special case of stochastic system, and quantum phenomena can be considered to be a result of some quantum stochasticity, generated by the space-time properties [@R91]. Pair production is a natural result of the sequential relativistic description of this stochasticity. The statistical description of quantum phenomena generates hydrodynamic interpretation of quantum phenomena, which does not coincide, in general, with the interpretation, based on application of quantum principles, and the interpretation appears to be essential for statement of the microparticle collision problem. [99]{} Yu.A. Rylov, ”Non-Riemannian model of space-time, responsible for quantum effects” . *J.Math. Phys*. **32**, 2092-2098, (1991). Yu. A. Rylov, ”Dynamics of stochastic systems and peculiarities of measurements in them”. (In preparation, available at http:// arXiv.org /abs/physics /0210003). 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--- author: - Sreekanth Malladi bibliography: - 'UNIF10-eptcs.bib' title: Disabling equational theories in unification for cryptographic protocol analysis through tagging ---
--- abstract: 'We present the results of a study of simulated Giant Molecular Clouds (GMCs) formed in a Milky Way-type galactic disk with a flat rotation curve. This simulation, which does not include star formation or feedback, produces clouds with masses ranging between $10^4 {\rm M}_{\odot}$ and $10^7 {\rm M}_{\odot}$. We compare our simulated cloud population to two observational surveys; The Boston University- Five College Radio Astronomy Observatory Galactic Ring Survey and the BIMA All-Disk Survey of M33. An analysis of the global cloud properties as well as a comparison of Larson’s scaling relations is carried out. We find that simulated cloud properties agree well with the observed cloud properties, with the closest agreement occurring between the clouds at comparable resolution in M33. Our clouds are highly filamentary - a property that derives both from their formation due to gravitational instability in the sheared galactic environment, as well as to cloud- cloud gravitational encounters. We also find that the rate at which potentially star forming gas accumulates within dense regions - wherein $n_{\rm thresh} \ge 10^4$cm$^{-3}$ - is 3$\%$ per 10 Myr, in clouds of roughly $10^6 M_{\odot} $. This suggests that star formation rates in observed clouds are related to the rates at which gas can be accumulated into dense subregions within GMCs via filamentary flows. The most internally well-resolved clouds are chosen for listing in a catalogue of simulated GMCs; the first of its kind. The catalogued clouds are available as an extracted data set from the global simulation.' author: - 'Samantha M. Benincasa, Elizabeth J. Tasker, Ralph E. Pudritz, and James Wadsley' title: 'Giant Molecular Cloud Formation in Disk Galaxies: Characterizing Simulated versus Observed Cloud Catalogues' --- Introduction ============ Stars form in the secluded depths of the densest cores in cold interstellar clouds. These extended structures of molecular hydrogen and dust are galactic wombs for their stellar populations and it is their environments that control the creation rate and properties of the new stars. Therefore, in order to explore any question regarding star formation, we must ultimately account for the properties of these stellar cradles that will be their birth place. The molecular hydrogen clouds are collectively referred to as the Giant Molecular Clouds (GMCs). The average gas density of these structures is a few hundred atoms per ${\rm cm}^3$, with stars forming within a star cluster in cores that exceed densities of ${\rm n_{H}} > 10^4 \ {\rm cm}^{-3}$ [@Lada2010]. While the cores themselves are clearly undergoing gravitational collapse, the gravitational binding of the cloud as a whole is more debatable, with both observations and simulations disagreeing as to which side of the virial line a cloud is likely to sit [@hopkins; @Dobbs_virial; @Tasker2011; @Tasker2009; @Hirota2011; @Heyer2009; @McKee2007; @Rosolowsky2007]. One of the main reasons why the binding of the cloud is not clear is that the cloud’s formation is not a straight forward process. The galactic environment in which a GMC forms is rife with instabilities and shocks that move through a multiphase, turbulent medium that shapes, stirs and buffets the cloud. This means that attempts to explore the star forming environment assuming a simple distribution of gas may fall far short of the true internal properties of a GMC formed in the above conditions. This has implications for numerical studies on the scale of star forming cores. To what extent then, do the local properties of star formation within highly bound subregions such as dense cores and clumps follow from the larger scale internal dynamics of the clouds in which such entities are formed? The answer to this question is particularly important for numerical studies since the highest resolutions available to chart the formation and evolution of clouds and their substructure are barely adequate to spatially resolve the largest, cluster forming clumps - let alone the details of individual star forming cores. Despite the complexity of their formation process, many properties of the GMCs are well described by observations, both in the empirical relations relating mass, size and velocity dispersion  [@larson1981] and in global observational studies of GMC populations in the Milky Way [@Heyer2009; @duval], and nearby galaxies such as M33 [@M33], IC342 [@Hirota2011], M64 [@Rosolowsky2005], M31 [@Rosolowsky2007] and the LMC [@Kawamura2009]. These provide handles to quantitatively assess whether a GMC created in simulations is a realistic star formation environment. If the object passes these tests, then it can be used confidently to study the initial conditions for the star formation process. One of the major goals of this paper is to create a catalogue of GMCs that agree with the observational data that can be used as a comparison tool for observers and theorists alike and can potentially form the starting point for further higher resolution studies in star formation. For this, we will focus on two observational surveys in particular; the Boston University- Five College Radio Astronomy Observatory Galactic Ring Survey [@Heyer2009] and the BIMA All-Disk Survey of M33 [@M33]. Another important application of our simulations is to measure the rate at which dense, star forming gas develops within our most massive (i.e., best spatially resolved ) GMCs without the complicating effects of stellar feedback. Numerical Studies of GMCs ------------------------- There are two main theories concerning the initial formation of GMCs; the ‘top-down’ model wherein clouds are formed as a result of global instabilities in the sheared, self-gravitating interstellar medium of the galactic disk and the ‘bottom-up’ scenario where clouds are formed by agglomeration or coagulation [@McKee2007 and references therein]. The actual physical mechanism may involve a number of processes, but due to time-scale restraints, the former ‘top-down’ method has been more widely investigated. It is this approach that will be addressed in this paper. The ‘top-down’ category can be further split by asking what causes the gathering of gas into a GMC. For this, there are also two main candidates; buoyancy and self-gravity. The former of these centres around the interplay between gravity (which wants to compress the gas in the galactic mid-plane) and the magnetic field (which acts to gather gas at the galactic footpoints of rising loops of magnetic field) in the Parker instability. Specifically the Parker instability produces the lower mass clouds of a few 10$^5$ M$_{\odot}$ (where the gas is not very self gravitating), while self gravity produces the higher mass clouds (a few 10$^6$ M$_{\odot}$), where magnetic fields have little effect[@elmegreen1982]. The added compression need not necessarily be due to a spiral wave, since the sheared galactic medium can also act with weak magnetic fields to produce magneto-rotational instabilities (MRI) in the galactic disk [@sellwood1999]. Such MRI instabilities have also been shown to seed GMC growth in local-scale simulations [@kim]. Once initiated, the Toomre instability of the self-gravitating sheared gas was found to rapidly build massive GMCs, reaching $\gtrsim 10^7\,{\rm M}_\odot$ . The self-gravity, top-down formation has been explored on both global and local scales. On the local scale, colliding flows have been used to create the clouds. These are counted in the ‘top-down’ camp since the flows themselves can be driven by gravitational instabilities such as spiral shocks or cloud-cloud collisions, although may also have other sources such as supernovae or stellar winds. [@Heitsch2008] and [@banerjee] have performed studies of this kind, where clouds form at the interface of two colliding flows of gas, naturally accounting for the turbulent nature of the GMCs. [@superbubble] utilised the idea of colliding wind-blown superbubbles, finding both cold gas clumps and filaments formed at the interface. On the global scale, the entire galactic disk is included in these simulations. The disk becomes gravitationally unstable via the Toomre instability and fragments to form the GMCs [@Tasker2011; @Dobbs_virial; @Bournaud2010; @Tasker2009; @Agertz2009; @Dobbs_agg]. This scenario has the advantage of naturally including the effects of shear and also interactions between neighbouring clouds, which drives the cloud’s internal turbulence [@Dobbs_virial] and may play an important role in the star formation process [@Tasker2011; @Tasker2009; @Tan2000]. Once formed, the clouds grow by accretion and cloud-cloud collisions. Star formation is among the mechanisms that would destroy or shred such clouds. In the absence of star formation, @Tasker2009 found good agreement between the properties of the created GMC population and those from observations, including mass, radius, velocity dispersion and rotation. They also found a high cloud collision rate, with clouds merging approximately every $1/5$th of an orbital period. Such a result suggests that cloud collisions are an important component of achieving realistic simulated GMCs, emphasising the importance of including the global environment in their formation. Simulations on this scale that have followed the GMC formation process through to include the production of stars and their resultant feedback, find that these internal processes can dictate the GMC properties at later times. In the presence of a spiral potential, @Dobbs_virial found that without feedback, clouds tend to be spherical and bound while they become unbound when feedback is included. This result was repeated by @hopkins who compared the impact of different forms of feedback and found that while no feedback produced very tightly bound clumps, there was a wide variation when feedback was included, with larger clouds tending to be more bound. Overall, the top-down formation mechanism for GMCs has proved to be highly successful in numerical simulations, both with and without the inclusion of magnetic fields and stellar feedback. However, which roles dominate during the star formation process and cloud properties such as their gravitational boundedness remain an active topic of discussion. The success of the Tasker & Tan (2009) model, despite its simplicity in the absence of feedback or magnetic fields, suggests that gravity may be the controlling force in the GMCÕs early evolution. With this in mind, we adopt a similar model (although with a few notable changes laid out in Section ¤2) to explore the pre-star formation and evolution of GMCs through a top-down, self-gravitating mechanism in the absence of star formation and localised feedback. This allows us to both study the interactions and formation of GMCs in a purely dynamical sense and measure the rate at which dense, potentially star forming gas, develops within the clouds. ![image](f1.eps){width="180mm" height="55mm"} Observational Surveys of GMCs ----------------------------- We compare our simulated clouds with two major observational surveys. The first of these is the Boston University - Five College Radio Astronomy Observatory Galactic Ring Survey (BU-FCRAO) [@Heyer2009; @duval] (hereafter, GRS), which uses the tracer $^{13}$CO (J = 1 $\rightarrow$ 0) . This survey re-examined the clouds found in the seminal @solomon study of the GMCs in the galactic ring. These new results have an increased resolution of $0.2$pc at a distance of $1$kpc and utilised the CLUMPFIND automated algorithm[@clumpfind] to identify the clouds within the sample. Our second comparison survey is the BIMA All-Disk survey, which uses the tracer $^{13}$CO (J = 1 $\rightarrow$ 0) and looks at extragalactic clouds in M33[@M33]. The inclusion of this second survey has two major advantages: the first is that comparisons with both galactic and extragalactic clouds allows more general conclusions to be drawn about the nature of the GMCs. Secondly, this survey has a spatial resolution of $20$pc, which is close to our own resolution limit (see Section §2), making it the more comparable data set. Additional support for our top-down, gravitational fragmentation model is found in their paper, with @M33 suggesting that the clouds in M33 are most likely to have been formed through instabilities, rather than agglomeration and further suggest that the properties of the GMCs are determined by the mass of the cloud. In this paper we present an analysis of clouds harvested from a simulation of a Milky Way-type disk. In Section §2, we discuss the simulation and method of GMC analysis. Section §3 looks at the structure of the GMCs and in Sections §4 and §5 we compare the cloud properties to two observational surveys of clouds in the Milky Way and M33. Finally in Section §6, we present a numerical catalogue of these GMCs and propose that these can be used as realistic initial conditions for higher resolution studies of massive star and star cluster formation. Section §7 details our conclusions. Numerical Methods ================= The clouds analysed in this paper are harvested from an isolated global galaxy disk simulation of a Milky Way-type galaxy. The simulation was performed using *Enzo*; a three-dimensional adaptive mesh refinement hydrodynamics code [@enzo1; @enzo2; @enzo3]. The disk is contained in a three-dimensional box of side length 32 kpc with a root grid of $128^3$ and five possible levels of refinement. This results in a limiting resolution of 7.8 pc. The disk itself is kept at a minimum resolution of 31 pc, with three levels of refinement permanently placed around that volume. This ensures the disk is stable in the absence of cooling. *Enzo* evolves the gas using a three-dimensional version of the *Zeus* hydrodynamics algorithm [@zeus]. This routine uses an artificial viscosity term to represent shocks, where the variable associated with this, the quadratic artificial viscosity, was set to 2.0 (the default value). We include a radially dependent photoelectric heating term described in detail in @Tasker2011, but no other star formation or localised feedback. Radiative cooling is allowed down to $300$K, a value that corresponds to a minimum velocity dispersion of $1.8$km/s. This limit crudely allows for processes that are below our resolution limit, such as the internal turbulence of the cloud on scales below $7.8$pc and pressure from magnetic fields. The cooling follows the analytical curve of @Sarazin1987 to $T = 10^4$K and the extension to $300$K from @Rosen1995. In addition to this first simulation, a second simulation was completed wherein a pressure floor was imposed. This floor acted in such way that when the Jeans length became less than four times the cell size (and therefore the simulation became unresolved according to the @Truelove1997 criterion), an additional pressure term was added that caused the gas to follow a polytrope with an adiabatic index of $\gamma = 2$. This brought the collapse to a halt and prevented the creation of isolated single cells with extremely high density that are not resolved by the simulation. In fact, the inclusion of this term produced identical results for the cloud bulk properties of mass, radius and surface density and only affected the velocity dispersion due to the effect of the increased pressure on the cloud’s internal dynamics. As this pressure injection is a numerical addition, the resultant velocity dispersion did not hold physical meaning. Since we do ultimately expect the gas within the cloud to form a dense core during star formation, we have opted to present the first simulation without the pressure floor in this paper, but note that the internal dynamics of the cloud are poorly resolved at this resolution. The disk is initially borderline gravitationally stable, fragmenting as the gas cools to form dense clouds of gas that we identify as the giant molecular clouds. The initial disk mass is 6.5$\times10^9 M_{\odot}$. Full details of the initial set-up of the simulation are given in [@Tasker2009; @Tasker2011]. We developed a special procedure to create clouds with high spatial resolution. In difference to the previous simulations of [@Tasker2009; @Tasker2011], this galactic simulation includes a rotating frame of reference at a radius of 6 kpc. Gas at this radius does not move with respect to the grid, minimizing the artificial numerical support that occurs from circular motion over a Cartesian mesh. Gas to either side of the co-rotating radius moves in opposite directions. We therefore concentrate our analysis to within 1 kpc of the co-rotating radius, from $r > 5.5$kpc - $r < 6.5$kpc, and focus our refinement over this region, allowing it to incrementally decrease outside these radial points. This technique allows us to gain considerably more realistic spatial resolution for clouds that would otherwise become more bloated through such artificial numerical spreading. ![A slice through the z-plane of a typical cloud. The image size is $300$pc and overlaid with a contour to show regions within our number density threshold of $100$cm$^{-3}$, which would be tagged as belonging to the cloud. The image is centred on cloud 1323, however, 5 other clouds also fall into the field of view (positions are marked with stars). Velocity field vectors have also been overlaid to show the bulk motion of the gas. Note that the gas flows along the large-scale filamentary structure of the cloud and gathers in several dense regions (of order $10^4 cm^{-3}$). Image rendered using yt [@yt][]{data-label="fig:mod_velocity"}](f2.eps){width="85mm" height="70mm"} Clouds in our simulations are identified using a friends-of-friends cell finding algorithm. This process identifies clouds based on a density threshold, which we selected to be $n_{\rm thresh}$ = $100 {\rm cm}^{-3}$. This value was selected based on a review of the observational catalogues of Milky Way GMC’s [@rathborne_clump; @duval], as being a representative value for the average density of these objects. The algorithm initially identifies peak cells in the simulation which have a density higher than $n_{\rm thresh}$ and exceed the density values of their neighbouring cells. If two peak cells are closer than a set separation length, $l_s = 4\times \Delta x$, where $\Delta x$ is the minimum cell size of 7.8pc, then they are tagged together as the same cloud. Neighbouring cells whose density also exceeds $n_{\rm thresh}$ are then assigned to the cloud via a friends-of-friends scheme, whereby the code steps through the neighbours of each cell assigned to the cloud to search for more cloud components. A more detailed description of this process can be found in [@Tasker2009]. Clouds in this analysis are harvested from the torus enclosing the co-rotating radius, between $5.5$ to $6.5$kpc from the galactic center, where as discussed, the artificial numerical effects will be minimized. We also limit our analysis to clouds that have at least three cells in each dimension; a minimum of 27 cells in total. This ensures our objects have our best resolution and are suitable candidates for the simulated catalogue. The full criteria for the catalogue are discussed in Section §6. Structure of the Clouds {#sec:structure} ======================= Figure \[fig:zoom\_scale\] shows density projections of the face-on disk at different scales. From left to right, the first panel shows a 20kpc spread of the entire disk. The co-rotation radius is clearly visible as the position of the most resolved clouds, which have access to all refinement levels and are unhindered by artificial support from motion over the grid. To aid the eye, white rings have been overlaid over our analysis region at disk radii of 5.5kpc and 6.5kpc. The center panel zooms into a 5kpc portion of the region of co-rotation. Here again it is clear that the most resolved clouds are found in the region of co-rotation. The last 1 kpc panel zooms in on two clouds tagged by the cloud finder algorithm, which are centrally positioned in the image. The differences between the structures well outside the high-resolution, co-rotating region are due to differences in resolution and numerical diffusion. The co-rotation technique dramatically reduces numerical diffusion due to advection errors. The combined effect is roughly equivalent to convolving the high resolution data with a 50 pc Gaussian filter. As shown in Figure \[fig:zoom\_scale\], the highly resolved chains of clouds connected by thin filaments between 5.5 and 6.5 kpc become single thick filaments at low resolution (near 7 kpc) and the cloud complexes become large clouds with the same total mass. A $300$pc wide slice through a typical cloud identified by the algorithm is shown in Figure \[fig:mod\_velocity\]. The cloud is elongated, with clear filamentary structures created through tidal interactions with nearby clouds. It is also flattened, having most of its extent in the xy-plane. The contour line shows cells that would be tagged by the algorithm at the density threshold of $100$cm$^{-3}$. We can see from the overlaid velocity fields that most of the gas in the cloud and surrounding interstellar medium is flowing into the densest knot of the structure, within which would lie the gravitationally collapsing core. The gas flow is strongly directed along the filaments, towards the dense regions within the GMC. Shown in Figure \[fig:evol\] is a density projection of a 2 kpc portion of the galactic disk, shown at four different times in the simulation; 130, 150, 200 and 240Myr. Our analysis is completed for the clouds in the disk at 240 Myr, after one complete orbital period for the edge of our main region at $r = 6.5$kpc. These frames allow us to see the evolutionary paths that our clouds follow. Early on at 130Myr, the structures formed are very small and almost bead-like. As times passes, these smaller objects grow larger through processes of collision and gas accretion. Once more massive, the continuing interactions between the clouds lead to spiral tails of lower density gas being stripped from the outer layers. By 200 Myr the clouds have taken on the more expected filamentary nature shown in Figure \[fig:mod\_velocity\]. ![Surface density maps of the same portion of the disk taken at 4 different time steps. Each of the images span 2 kpc.[]{data-label="fig:evol"}](f3.eps) Analysis of Global Cloud Properties {#sec:bulk} =================================== ----------------------------------------------- ----------------------------------------------- ![image](f4a.eps){width="70mm" height="55mm"} ![image](f4b.eps){width="70mm" height="55mm"} ![image](f4c.eps){width="70mm" height="55mm"} ![image](f4d.eps){width="70mm" height="55mm"} ![image](f4e.eps){width="70mm" height="55mm"} ![image](f4f.eps){width="70mm" height="55mm"} ----------------------------------------------- ----------------------------------------------- Figure \[fig:histograms\] presents the bulk physical properties of the simulated GMCs at $t = 240$Myr. The six histograms show the results for the cloud mass, surface density, velocity dispersion, virial mass and virial parameter. The dashed overlaid lines show results from the observational data sets in the GRS Milky Way survey (blue dashed) and the M33 survey (red dot-dash). The top left histogram in Figure \[fig:histograms\] shows the cloud mass as calculated by the total mass in all cells associated with the cloud via our cloud finder scheme described in Section §2. This cloud mass is compared to the derived CO mass of the GRS Milky Way and M33 GMCs. Comparing our result with the two observational data sets, we see the peak in our distribution agrees well with the M33 peak mass at $M_{\rm c, peak} = 10^5$M$_\odot$ . With the M33 mass range extending from $3\times10^4 $M$_{\odot}$ to $1\times 10^6$M$_{\odot}$, we find good agreement with the overall profile, with our own mass range stretching between $5.7\times10^4$M$_{\odot}$ to $2.7\times10^7$M$_{\odot}$ [@M33]. The Milky Way data set finds a slightly lower peak mass at $M_{\rm MW, peak} = 4.8\times10^{4}$M$_\odot$, reflecting the order of magnitude higher spatial resolution of the GRS Milky Way survey. However, it is worth noting that the Milky Way data cannot be quoted without some error. @Heyer2009 reports (private communication) that the masses of the Milky Way may be larger by a factor of 3 or more. If this proves to be true, our mass range is in good agreement with both data sets. While the majority of our clouds seem to agree well with the observational results, we do find a high mass tail with clouds extending to masses higher than that found in either the Milky Way or M33. This is due to the lack of star formation and feedback in the simulation, which would inhibit cloud growth [@Tasker2011]. Without such destructive mechanisms, clouds can continue to grow via accretion and mergers throughout the simulation, forming a population of old, tightly bound clouds that in reality would have formed a stellar cluster. We will discuss these objects in Section §5 and, since we do not expect to find them in actual galaxies, they are not included in the cloud catalogue. The top right histogram shows the distribution of cloud radii. We define the radius of a cloud via: $$R_A = \sqrt{\frac{A_{yz}}{\pi}}$$ where $A_{yz}$ is the projected area of a cloud in the yz-plane of the disk, an orientation chosen to agree with the typical observed angle along the plane of the disk, as in the GRS Milky Way survey. A similar definition of the radius is used for both of the comparison observational surveys. We have compared our findings using an area projected from the xy- and yz- planes and found no significant change to our cloud properties ($A_{yz}$ is used for the remainder of the analysis). Again, we find our cloud radii correspond well with the M33 data set, although our most common radius falls slightly higher than the M33 clouds; at $R_A = 20$pc, compared with $R_{\rm M33} = 10$pc. The range of radii for our clouds falls between that of the Milky Way data and M33, with a spread of $40$pc. This is a smaller range than what is found for M33, possibly due to the more uniform nature of our cloud environment since we draw clouds from a $1$kpc wide torus in the center of our disk, which is unaffected by galactic structural differences. In the second row on the left, we show the mass surface density of the clouds, which is defined as $\Sigma \equiv M_c/A_{yz}$. For this property only Milky Way data is available, which peaks at a comparable value to our clouds at $\Sigma_{\rm peak} \sim 100-200$M$_\odot/{\rm pc}^2$. Notably, we get good agreement for the surface density without any form of feedback, suggesting that internally supported turbulence in clouds may be driven by cloud interactions, rather than localized energy from star formation. The right-hand histogram on the second row of Figure \[fig:histograms\] shows the clouds’ internal velocity dispersion. The value used is the mass-weighted one-dimensional velocity dispersion about the cloud’s center of mass velocity, calculated as: $$\sigma_{1D} = \frac{1}{\sqrt{3}}\sqrt{v_x^2+v_y^2+v_z^2}.$$ Here we obtain a wide range of velocity dispersion values, upto a maximum of $14$km/s with our most common value just below $8$km/s. This peak is slightly higher than the Milky Way and M33 clouds, which find peak distribution values of $1.1$km/s and $3.6$km/s respectively and is more in keeping with the velocity dispersion of the ISM [@Kennicutt1998]. The velocity dispersion of the clouds is likely to arise from two effects. The first of these is a dispersion driven by cloud-cloud interactions. As Figure 3 shows, the cloud environment is rife with structural interactions and this affects all the clouds, regardless of their size. The second cause is for the small tail population of massive clouds seen in Figure 4. These objects have a deeper potential well which allows an increased velocity dispersion, likely to be seen as the spread of higher values in our distribution. This may decrease with the introduction of stellar feedback, without which the clouds may become more gravitationally bound for a given size than their observed counterparts. We will consider this again in Section §5. Our bottom two plots show cloud properties related to their virialisation. A measure of this is the virial parameter, $\alpha_{\rm vir}$, which is related to the ratio of kinetic to gravitational energy of a cloud. @Bertoldi1992 define this as: $$\alpha_{\rm vir} = \frac{5(c_s^2+\sigma_{\rm 1D}^2)R_A}{GM_c} \label{eq:alpha}$$ where we include both the thermal ($c_s$) and non-thermal ($\sigma_{\rm 1D}$) contributions to the cloud’s velocity dispersion. $R_A$ is the radius as calculated from the project area, as above, and $M_c$ is the cloud mass. For a uniform, spherical cloud, a value of $\alpha_{\rm vir} = 1$ implies that the kinetic energy of the cloud is half its gravitational potential, making clouds with $\alpha_{\rm vir} < 1$ dominated by gravity. If we assume that clouds are virialised objects with $\alpha_{\rm vir} = 1$, we can reverse Equation \[eq:alpha\] to get an expression for the [*virialised mass*]{}: $$M_{\rm vir} = \frac{5(c_s^2+\sigma_{\rm 1D}^2)R_A}{G}$$ This value is plotted on the left-hand histogram on the bottom line of Figure \[fig:histograms\]. Our clouds agree well with the M33 data set, although we do see a population of high mass clouds as we did for the mass distribution, due to lack of star formation to inhibit the formation of large clouds. The higher resolution Milky Way data shows a smaller mass population which we might expect to match closer if we could resolve our clouds at the same spatial limit. The right-hand plot on the bottom row shows the value of $\alpha_{\rm vir}$ for the clouds, plotted with the Milky Way data (there are no M33 values available for this quantity). Our clouds are less bound than those identified in the Milky Way: we find $15$% of clouds have $\alpha_{\rm vir} < 1$, whereas the GRS Milky Way survey finds approximately $70$%. This is in contrast to findings by @Dobbs_virial and @hopkins that a lack of feedback produces a more bound population. One explanation might be that the higher resolution observed Milky Way data is identifying bound subregions within our clouds. Alternatively, the less bound population may be destroyed by feedback effects. ![Distribution of the angle of the cloud angular momentum vector with respect to that of the galaxy at three different simulation times, $t = 140$ (red dashed), $200$ (blue dot-dash) and $240$Myr (green solid). At early times, more clouds show a prograde rotation, with $\theta < 90^\circ$, rotating in the same sense as the galaxy. As the simulation progresses, a retrograde population develops.[]{data-label="fig:orientation"}](f5.eps){width="80mm" height="60mm"} Figure \[fig:orientation\] plots the distribution of directions of cloud angular momentum vectors with respect to the galaxy. While all clouds move predominantly through the galaxy with the rotation of the disk, their rotation around their own axis may differ. Clouds with a prograde rotation have $\theta < 90^{\circ}$ and rotate in the same sense as the galaxy while clouds with a retrograde rotation have $\theta > 90^{\circ}$ and rotate in the opposite direction. In M33, @M33 found that clouds show a slight preference for aligning with the galactic rotation, with $60$% of clouds showing a prograde rotation having $\theta < 90^\circ$. At early times, the clouds in our disk are born prograde. This is shown in Figure \[fig:orientation\] by clouds at $t = 140$Myr having the highest fraction of clouds with $\theta < 45^\circ$. At later times, a retrograde population develops as new clouds feel not only the disk shear (which from angular momentum conservation must produce a prograde cloud) but also the gravitational pull of neighbouring clouds. The clouds also undergo collisions and interactions during their lifetime which may result in either a prograde or retrograde rotation. By $t = 240$Myr, $18$% of the clouds rotate retrograde. This is a smaller fraction than those observed by @M33, but in agreement with previous simulations performed by @Tasker2011. In their paper, @Tasker2011 find that the addition of photoelectric heating reduces the velocity dispersion in the disk ISM, resulting in a smaller retrograde population. This result was also noted by @Dobbs_agg. The addition of feedback is likely to affect this result, providing an additional force that can increase the disk velocity dispersion. Regardless of the exact value of the retrograde percentage, the existence of such a population strongly suggests that the cloud’s external environment plays a key role in its evolution, providing the necessary forces to create the spread of orientations observed in the GMC surveys. Mass fraction of star forming gas in clouds ------------------------------------------- ![Fraction of gas mass within the GMCs that is higher than the density thresholds, $n_{\rm thresh} = 10^4$cm$^{-3}$ and $n_{\rm thresh} = 10^5$cm$^{-3}$. $10^4$cm$^{-3}$ is the threshold at which star formation is observed to occur.[]{data-label="fig:massfrac"}](f6.eps){width="80mm"} At $100$cm$^{-3}$, the density we use to define a GMC refers to the whole cold gas cloud, but does not give us a strong handle on how much star formation we should expect. Figure \[fig:massfrac\] shows the fraction of mass contained in clouds that is higher than two density thresholds, $n_{\rm thresh} = 10^4$cm$^{-3}$ and $n_{\rm thresh} = 10^5$cm$^{-3}$, a factor of 100 and 1000 times higher than the cloud definition threshold. The value $ 10^4$cm$^{-3} $ is the density of clumps in which star formation is observed to occur [@Ginsburg2012; @Lada2010]. As simulation time progresses, the density of gas within the clouds increases. This agrees with what we saw visually in Figure \[fig:evol\]; small objects at early times gain mass through accretion and mergers, which causes the clouds to get more dense. We will also see this in the next section when we look at the evolution of the mass profile over time. This result provides interesting insights into how molecular clouds set themselves up for star formation. The Figure shows that between $t = 130 - 200$Myr, approximately 30% of the cloud mass is at densities higher than $10^4$cm$^{-3}$. At this density, stars would be expected to form with an efficiency of around 10-30% [@Lada2003]. This would give an overall cloud star formation efficiency of about 3-9%. @Krumholz2007 estimate that this number should be around $2$%, implying that our clouds convert their gas to dense cores too efficiently. This is unsurprising, however, since there is nothing to prevent the collapse within the bound region of the cloud without the inclusion of local feedback. The rate at which star forming gas appears is also very interesting. We can estimate from the approximately linear $n_{\rm thresh} = 10^4$cm$^{-3}$ curve, that the fraction of potentially star forming gas in a cloud grows by 30  % over 100 Myr, or 3 % per 10 Myr, in clouds of roughly $10^6 M_{\odot} $. This is in reasonable accord with star formation rates - suggesting that star formation rates in clouds are related to the rates at which gas can be accumulated into dense regions. A similar analysis can be performed that looks at surface, rather than volume, density thresholds. However, in this case, our result is dependent on our cloud selection scheme. Since clouds must be at least 3 cells in height, corresponding to a thickness of 23.4pc, and by definition must have a density $> 100$cm$^{-3}$, we have a sample-determined surface density threshold of $8\times 10^{21}$g/cm$^{-2}$, or a visual extinction of $A_v \simeq 10$. Since this is a result of our cloud sampling, the volume density threshold is the more revealing quantity. Characterising cloud turbulence {#sec:turb} ------------------------------- ![Characterisation of the turbulence: red line fit corresponds to $y = 0.03 x$, giving a value for the turbulence parameter $b = 0.17$, suggesting that solenoidal turbulence is the dominant form in these simulations.[]{data-label="bturb"}](f7.eps){width="80mm"} Another key to describing the internal structure of the GMCs is to investigate their turbulent energy by calculate their sonic Mach number, $\mathcal{M} \equiv <\sigma/c_s>$. For a non-magnetised gas, this value is linked to the variance in the logarithmic standard deviation, $\sigma_{\rm s} $ as: $$\sigma_s^2 = \ln(1+b^2 \mathcal{M}^2)$$ where $s = \ln(\rho/\rho_0)$ [@Kainulainen2013; @Burkhart2012; @Molina2012; @Nordlund1999]. The numerical simulations in which turbulence is driven at large-scales have found that the value of the proportionality constant, $b$, depends on the mixture of solenoidal and compressive modes of the driving. In the case of fully solenoidal (divergence-free) driving, the value of the constant is $b=1/3$, and in the case of fully compressive driving it is $b=1$[@Federrath2010; @Federrath2008]. Rearranging this equation gives: $$\exp(\sigma_s^2)-1 = b^2 \mathcal{M}^2 \label{eq:bturb}$$ which is plotted in Figure \[bturb\] with the left-hand side of Equation \[eq:bturb\] plotted on the $y$-axis and $\mathcal{M}^2$ on the $x$-axis. Since the plot is on logarithmic axes, we expect a gradient of $1$ and $(1,1)$ intercept at the value of $b$. The red line shows a $y = 0.03x$ fit, giving a value of $b = 0.17$, which is in agreement with the non-magnetized, isothermal simulations of turbulence that are driven with solenoidal forcing [@Federrath2010]. This suggests that the turbulence in the clouds of our simulations is not, on average, very compressive. This finding agrees well with recent observational measurements of the value of $b$ for the infrared dark clouds in the Milky Way; dense molecular clouds that are thought to be likely candidates for future high mass star formation. This makes them particularly good comparison points for our non-star forming GMC population. For these objects, @Kainulainen2013 find $b \sim 0.2$. Other observations of the Taurus and IC 5146 clouds suggest a slightly higher value at $b \sim 0.5$ [@Burkhart2012; @Brunt2010; @Padoan1997]. In comparing simulations with cloud observations, @Kainulainen2013b suggest that non-magnetised clouds have $b < 0.3$ and clouds in the early stage of their evolution (the closest match with our population) are inferred to be lower still. While this agreement is pleasing, it is worth noting that our value for $\sigma_s$ is restricted by the densities that can be probed by the simulation. The true cloud may have a more extended range. About a factor of 100 above our fitted line is a small population of clouds that appear to have a significantly higher value for $b$. These clouds are old, tightly bound objects that greatly exceed their Jeans mass and sit in the high mass tail in Figure 8. In reality, these objects would have already formed stars and so their presence here is a result of us not including star formation. As mentioned earlier in Section §4, these objects are excluded from the cloud catalogue since we do not expect their structure to be found in actual GMCs. Evolution of cloud properties ----------------------------- ![Evolution of the cloud mass profile during the simulation. The profile is plotted at simulation times $140, 160, 200$ and $240$Myrs.[]{data-label="fig:mass_evol"}](f8.eps){width="75mm" height="60mm"} One concern with measuring cloud properties in a numerical simulation is that they might depend on the simulation output examined. While we intentionally wait for a complete orbital period at our outer radius of interest before analyzing the cloud properties in Figure \[fig:histograms\], it is also important to quantify any time evolution in our results. Figure \[fig:mass\_evol\] shows the mass profile of the clouds at four different simulation times: 140, 160, 200 and 240Myr. The evolution of the cloud mass profile over the course of the simulation is small, with the range of cloud masses remaining approximately the same. Two small differences are seen at later times; the first and more important trend is the production of a high mass tail, with the maximum cloud mass extending from $4\times 10^6$M$_\odot$ at $t = 140$Myr to $2\times 10^7$M$_\odot$ by $t = 240$Myr. The second smaller trend is the shift in the mean cloud mass to lower values, changing from $6\times 10^5$M$_\odot$ at $t = 140$Myr to $2\times 10^5$M$_\odot$ by $t = 240$Myr. The major effect –the high mass tail– is due to the inability to destroy clouds via star formation or feedback. A small population of clouds continue to grow unabated via mergers and accretion to form a group of high mass, tightly bound clouds at the expense of lower mass clouds. The second trend is likely due to the other end of the cloud’s lifetime, where later simulation times see a population of clouds born along tidal tails and gas formed between cloud interactions. These clouds are smaller than those born out of the direct collapse of the disk. Scaling Relations ================= Larson’s empirical scaling laws relate the size, mass and velocity dispersion of molecular clouds [@larson1981]. The cloud radius and the velocity dispersion are found to obey a power-law relation of the form: $$\sigma = \sigma_{\rm pc}R^{\alpha_o}$$ where $\sigma_{\rm pc}$ and $\alpha_o$ are constants. While such a relationship is found to hold for both galactic and extra-galactic cloud populations, the exact value of these constants is debated. In his seminal paper, @larson1981 finds values of $\sigma_{pc}=1.1$ and $\alpha_o=0.38$ for clouds in the Milky Way, which @solomon remeasures at $\sigma_{pc}=0.72\pm0.07$ and $\alpha_o=0.5\pm0.05$. GMCs in M33 yields values of $\sigma_{pc}=0.72$ and $\alpha_o=0.45\pm0.02$ [@M33; @bolatto]. In Figure \[larson\_rv\], we show the line for the original Larson fit, $\sigma = 1.1R^{0.38}$, over-plotted with the values for each of our clouds in our main torus. On the y-axis, we use the combined thermal and non-thermal velocity dispersion components, $\sigma_c \equiv (c_s^2 + \sigma_{\rm 1D}^2)^{1/2}$. Our clouds lie on the fitted line, but show a large scatter over a small radius range. The small variation in radius is due to the region in which we harvest the clouds in the simulation; we intentionally take them from a uniform 1kpc thick torus in the disk and do not expect as wide a spread in cloud properties as if we had sampled through a larger region with a galactic density gradient. As discussed in Section \[sec:bulk\], clouds in our simulation have a wider velocity dispersion than those observed in the Milky Way and M33, due to lack of feedback. This also impacts our scaling relations, giving a wide scatter of velocity dispersions for a given radius. ![Larson’s size-linewidth relation, $\sigma = \sigma_{pc}R^{\alpha_o}$. The black line shows Larson’s original fit, with $\sigma = 1.1R^{0.38}$ and the individual points show our simulated clouds. []{data-label="larson_rv"}](f9.eps){width="90mm" height="75mm"} A second scaling relation can be derived by considering the GMCs to be fractal structures. The fractal dimension, $D$, can be defined as the relation between the mass and radius of the studied bodies [@mandelbrot]: $$M \propto R^D$$ For clouds in the Milky Way, the fractal dimension had been measured as $D = 2.36\pm 0.04$ [@fractalpaper; @duval]. The result for our simulated clouds is shown in Figure \[rm\_fractal\], which again shows a large scatter. ![Size-mass relation, $M = M_{pc}r^D$. The black line shows the fit to the Milky Way data in @duval, with $r = 228M^{2.36}$ and the individual points show our simulated clouds. []{data-label="rm_fractal"}](f10.eps){width="90mm" height="75mm"} A Catalogue of Simulated GMC’s ============================== [lllllllllll]{} 15323&15.24&21.92&15.96&6.0&6.85&20.7&1.07$\times 10^6$&1159.1&797.0&1.12\ 18355&11.90&10.98&16.02&6.5&8.92&22.0&2.38$\times 10^6$&1094.2&1560.8&0.89\ 6059&11.90&11.05&16.01&6.4&11.81&30.5&4.12$\times 10^6$&1028.7&1406.9&1.23\ 6505&13.45&10.36&16.00&6.2&6.59&23.3&1.04$\times 10^6$&635.1&606.6&1.21\ 651&13.28&10.32&16.01&6.3&11.84&26.4&3.22$\times 10^6$&1467.0&1464.1&1.37\ 15406&13.36&10.32&16.00&6.3&8.77&26.8&1.27$\times 10^6$&446.7&564.2&1.96\ 13683&13.39&10.26&16.01&6.3&6.84&19.7&1.36$\times 10^6$&885.5&1112.4&0.84\ 13448&15.83&10.34&15.99&5.7&8.07&24.1&1.67$\times 10^6$&805.2&913.6&1.14\ 14621&15.82&10.46&16.01&5.5&9.29&24.1&2.70$\times 10^6$&1169.0&1473.8&0.93\ 15424&16.53&9.92&16.02&6.1&11.27&22.5&3.05$\times 10^6$&1623.4&1922.9&1.11\ 1073&10.90&13.66&16.01&5.6&14.14&23.3&4.15$\times 10^6$&2097.2&2428.1&1.33\ 229&10.50&15.16&16.03&5.6&12.61&24.9&4.04$\times 10^6$&1711.4&2066.1&1.17\ 19181&10.55&15.15&16.03&5.5&8.52&26.8&2.51$\times 10^6$&709.4&1113.6&0.94\ 10870&10.42&13.02&16.00&6.3&6.23&19.2&1.16$\times 10^6$&1276.4&998.2&0.81\ 8188&20.20&12.14&16.02&5.7&8.58&26.4&2.36$\times 10^6$&778.9&1075.5&1.00\ 14865&22.12&14.38&16.02&6.3&6.21&25.3&1.30$\times 10^6$&511.1&644.3&0.94\ 1200&22.10&15.23&15.99&6.2&10.13&28.6&3.22$\times 10^6$&1147.3&1254.6&1.09\ 13523&21.56&14.93&16.02&5.7&5.45&18.2&1.06$\times 10^6$&1069.7&1020.0&0.65\ 270&21.60&15.00&16.03&5.7&7.08&22.9&1.36$\times 10^6$&1018.9&825.8&1.04\ 9668&20.75&12.61&16.02&5.8&13.00&23.3&4.27$\times 10^6$&2120.7&2496.2&1.09\ 12385&21.78&15.06&16.01&5.9&9.62&18.2&2.08$\times 10^6$&2210.9&2002.7&0.97\ 8396&21.94&15.00&16.01&6.0&9.20&20.2&2.12$\times 10^6$&1288.1&1657.1&0.97\ 1163&21.36&12.95&15.98&6.2&6.63&23.7&1.11$\times 10^6$&539.2&627.9&1.17\ 4640&21.30&12.88&15.98&6.2&6.83&21.6&1.00$\times 10^6$&677.1&685.9&1.24\ 8922&9.84&17.94&16.04&6.5&9.18&22.9&3.02$\times 10^6$&1274.0&1835.6&0.77\ 1323&9.85&17.36&16.02&6.3&11.43&29.2&1.75$\times 10^6$&664.8&653.1&2.59\ 12398&9.95&17.32&16.03&6.2&8.07&33.6&1.84$\times 10^6$&617.4&520.5&1.44\ 1838&9.98&17.29&16.01&6.2&11.15&28.9&5.40$\times 10^6$&2664.3&2058.9&0.79\ 10545&9.57&16.07&16.03&6.4&11.65&23.7&3.65$\times 10^6$&1878.0&2064.4&1.05\ 1475&21.16&18.28&16.01&5.6&12.18&29.6&4.12$\times 10^6$&1622.3&1499.9&1.26\ 17693&20.17&19.71&16.00&5.6&7.55&21.6&1.32$\times 10^6$&961.7&898.5&1.14\ 1452&22.36&16.97&16.01&6.4&8.49&27.2&1.70$\times 10^6$&819.5&734.1&1.39\ 11206&22.28&17.07&15.97&6.4&7.98&31.5&1.94$\times 10^6$&739.8&623.1&1.26\ 11692&22.00&17.61&16.02&6.2&7.12&21.6&1.45$\times 10^6$&762.8&987.4&0.93\ 6988&12.16&21.02&16.00&6.3&15.53&26.8&5.26$\times 10^6$&2057.5&2328.5&1.45\ 8434&16.09&21.73&16.02&5.7&8.51&26.8&1.69$\times 10^6$&788.9&748.8&1.39\ 19081&18.28&21.26&16.03&5.7&5.89&25.3&1.03$\times 10^6$&389.4&513.2&1.07\ 11191&18.31&21.30&16.03&5.8&11.54&23.7&2.70$\times 10^6$&1204.5&1526.0&1.39\ 16520&16.00&21.51&16.03&5.5&9.65&22.0&2.53$\times 10^6$&1534.0&1657.7&0.97\ 3358&18.93&21.67&15.99&6.4&7.86&34.1&1.93$\times 10^6$&761.7&528.1&1.33\ 8899&19.14&10.83&15.93&6.0&9.07&25.7&1.66$\times 10^6$&1228.8&800.6&1.53\ 17144&18.93&10.54&16.05&6.2&6.97&23.3&1.43$\times 10^6$&906.5&839.7&0.97\ 14354&17.64&10.13&16.06&6.1&8.96&22.5&1.76$\times 10^6$&1316.3&1107.9&1.24\ 13083&11.46&12.26&16.06&5.9&10.29&33.6&2.98$\times 10^6$&841.9&843.1&1.42\ 16541&21.67&13.86&16.04&6.1&8.23&22.5&1.97$\times 10^6$&1354.3&1238.3&0.94\ 1149&21.68&13.81&16.05&6.1&14.64&26.1&4.42$\times 10^6$&1676.3&2070.2&1.49\ 15150&11.69&20.19&16.02&6.0&7.90&30.9&1.99$\times 10^6$&619.1&665.6&1.18\ 14746&14.84&21.41&16.03&5.5&8.44&31.8&2.22$\times 10^6$&969.2&699.9&1.23\ 16961&17.00&21.58&16.00&5.7&6.56&36.3&1.72$\times 10^6$&538.1&414.8&1.13\ 1777&17.30&21.92&16.00&6.1&13.58&22.5&3.48$\times 10^6$&1989.1&2190.8&1.41\ 16128&17.21&21.86&16.07&6.0&8.31&18.7&1.32$\times 10^6$&1140.3&1198.0&1.19\ A major goal of this project has been to systematically compare our simulated giant molecular clouds with observations and compile of a catalogue of the simulated clouds for further comparison and simulation. The clouds in the catalogue are presented in Table 1. This compilation includes each cloud’s physical properties of mass ($M_c$), radius ($R_A$), average number density ($\langle n\rangle$), surface density ($\Sigma$), velocity dispersion ($\sigma_{1D}$) and virial parameter value ($\alpha_{\rm vir}$), along with their physical position in the disk. Clouds that are listed in the catalogue are considered our ‘best’ clouds in terms of their numerical resolution. To belong in this category, clouds must fulfil the following conditions: (a) be found in our torus of best resolution, within the galactic radii $5.5 < r < 6.5$kpc. (b) consist of at least 27 cells, with 3 cells in each $x, y, z$ dimension. (c) have a mass greater than $M_c > 10^6$M$_\odot$. (d) Not fall in the over-dense, highly bound cloud population shown in Figure \[bturb\]. This corresponds to existing below the line $\exp(\sigma_s^2)-1 = 2\mathcal{M}$. These criteria ensure the cloud has the best internal properties our simulation can produce. When these criteria are applied, our highest mass cloud has $M_c = 5.4\times 10^6$M$_\odot$. Our simulated clouds are available from this site (www.physics.mcmaster.ca/mcclouds) in hdf5 format. The data consists of a three dimensional uniform grid $500$pc across, with each cell $7.8$pc on the side. The projected density of the 9 most massive of these cubes is shown in Figures \[fig:clouds1\]. As we discussed in Section §3, visually our simulated clouds look similar to those observed [e.g. @rathborne_clump; @M33] with the elongated clouds shapes and a filamentary structure. This suggests that the formation and evolution mechanisms at work may be sufficient to produce the filamentary structure in the actual galaxy. Since we intentionally did not include an active star formation and feedback model in our simulation, the catalogue clouds should provide a good study for the earliest stages of the star formation process. ![image](f11.eps) Discussion and Conclusions ========================== We performed a high resolution simulation of an isolated galactic disk, which incorporated a rotating frame of reference at a galactic radius of $6$kpc. The use of the co-rotating frame improves the effective resolution by reducing the numerical dissipation. GMCs were formed via a top-down gravitational fragmentation and were analysed between galactic radii $5.5 < r < 6.5$kpc, where their limiting resolution was $7.8$pc. In addition to the study of their properties, we present a catalogue of our best simulated clouds for further study of the formation of star clusters and high mass star formation. We find that our simulated clouds compare well with observed clouds both in the Milky Way and M33, even with the absence of feedback. Our typical cloud mass is between $10^5 - 10^6$M$_\odot$, in good agreement with the M33 data which is of comparable resolution to our simulation. We find a high-mass tail of clouds with masses up to $10^7$M$_\odot$ which builds up during the simulation due to the lack of stellar feedback. This lack of feedback also causes a larger range in velocity dispersion than is observed in either galactic survey. The radius, surface density and virial mass of our clouds all agree with the observational data sets we compared against, with our clouds being marginally unbound with $\alpha_{\rm vir} \sim1$. The internal turbulence of the simulated clouds was estimated and found to be in agreement with solenoidal turbulence, in agreement with observations of clouds in the Milky Way. The fact that our simulated clouds are similar to observed ones is interesting because the cloud growth does not reach an equilibrium; clouds continue to grow with time. The cloud growth phase, we argue, does mimic reality. Molecular clouds do not live in isolation in the ISM but ought to be accreting mass from their surroundings. They should also undergo a reasonable number of collisions before their dispersal. Clouds should therefore exhibit a significant range of gravitational boundedness which is indeed observed in our simulations and in reality. Even the neglect of star formation feedback does not grossly affect cloud properties which suggests that cloud disruption need not be catastrophic. Our star formation efficiency, as estimated by the fraction of mass in clouds above a density of $n > 10^4$cm$^{-3}$, is between 3-9%, a value higher than the observationally suggested fraction, but in keeping with our lack of feedback preventing any halt to the collapse of the bound core in the cloud. We also measured the rate at which dense, potentially star forming gas accumulates within dense regions - wherein $n_{\rm thresh} \ge 10^4$cm$^{-3}$ - to be 3 % per 10 Myr, in clouds of roughly $10^6 M_{\odot} $. This again is in accord with observations of star formation rates in GMCs, and suggests that star formation rates are related to the rates at which dense gas can be accumulated into dense clumps in GMCs. The fact we were able to match many of the observed properties of the GMCs without the inclusion of star formation or feedback is proof of the importance of the galactic environment in which the GMC is born. Our results indicate that the galactic disk, with its rotational shear and interactions between neighbouring GMCs, play a vital role in the creation of the star formation environment. With this in mind, we present our cloud catalogue; a set of 51 simulated GMCs that can be used to explore the formation of stars in an environment that includes the full impact of the galaxy disk. We intend for these to be freely used by the star formation community to study this complex and fascinating topic. The authors would like to thank Helen Kirk, Mark Heyer and Jouni Kainulainen for many helpful discussions and the yt development team (yt: http://yt-project.org/) for their support during the analysis of this run. We also thank the anonymous referee for comments that helped to improve the clarity of the paper. Numerical computations were carried out on Cray XT4 at Center for Computational Astrophysics, CfCA, of the National Astronomical Observatory of Japan. Analysis was performed on machines belonging to the Shared Hierarchical Academic Research Computing Network (SHARCNET:www.sharcnet.ca) and Compute/Calcul Canada. S.M.B. was supported by McMaster University and E.J.T. would like to thank the Canadian Insitute for Theoretical Astrophysics who supported her as a CITA National Fellow. R.E.P. and J.W. were supported by NSERC Discovery Grants. Agertz, O., Lake, G., Teyssier, R., et al. 2009, , 392, 294 Banerjee, R., Vázquez-Semadeni, E., Hennebelle, P., & Klessen, R. S. 2009, , 398, 1082 Bertoldi, F., & McKee, C. F. 1992, , 395, 140 Brunt, C. M. 2010, , 513, A67 Bryan, G. L. 1999, Comput. Phys. Eng., 1, 46 Bryan, G. L. & Norman, M. L. 1997, Computational Astrophysics, 12th Kingston Meeting on Theoretical Astrophysics (ASP Conf. Ser. 123), ed. D. A. Clark & M. J. West (San Francisco, CA: ASP), 363 Bolatto, A. D, Leroy, A. 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--- abstract: 'We introduce a Green’s function method for handling radiative effects on false vacuum decay. In addition to the usual thin-wall approximation, we achieve further simplification by treating the bubble wall in the planar limit. As an application, we take the $\lambda\Phi^4$ theory, extended with $N$ additional heavier scalars, wherein we calculate analytically both the functional determinant of the quadratic fluctuations about the classical soliton configuration and the first correction to the soliton configuration itself.' author: - Björn Garbrecht - Peter Millington title: 'Green’s function method for handling radiative effects on false vacuum decay' --- Introduction ============ The association of the still recently-discovered $125\, {\rm GeV}$ scalar particle [@Aad:2012tfa; @Chatrchyan:2012ufa] with the Higgs boson of the Standard Model (SM) places the stability of the electroweak vacuum under question [@Cabibbo:1979ay; @Sher:1988mj; @Sher:1993mf; @Isidori:2001bm]. This instability, arising at an energy scale around $10^{11}\, {\rm GeV}$ [@EliasMiro:2011aa; @Degrassi:2012ry], results from the renormalization-group (RG) running of the Higgs self-coupling, whose value is driven negative by contributions dominated by top-quark loops. State-of-the-art calculations suggest that the electroweak vacuum is metastable, having a lifetime longer than the present age of the Universe and lying at the edge of the stable region [@EliasMiro:2011aa; @Degrassi:2012ry; @Alekhin:2012py; @Buttazzo:2013uya], where seemingly-small corrections may have a material impact upon predictions. The uncertainty in such predictions remains, at present, dominated by that of the top-quark pole mass [@Bezrukov:2012sa; @Masina:2012tz]. Even so, it has been suggested [@Branchina:2013jra; @Branchina:2014usa; @Branchina:2014rva; @Lalak; @Eichhorn:2015kea] that, regardless of any improved precision in the experimental determination of the latter, the presence of Planck-scale operators may weaken the claim of metastability. Nevertheless, having, as yet, no experimental evidence of additional stabilizing physics between the electroweak and Planck scales, it is provident to consider approaches to the calculation of tunneling rates that can consistently account for radiative corrections. The degree of vacuum metastability provides a strong criterion for the phenomenological viability of extensions to the SM. For example, supersymmetric scenarios can be ruled out if the electroweak symmetry-breaking vacuum decays into a color-breaking one in a timescale shorter than the age of the Universe [@Nilles:1982dy; @AlvarezGaume:1983gj; @Derendinger:1983bz; @Kusenko:1996jn; @Strumia:1996pr; @Chowdhury:2013dka; @Blinov:2013fta; @Camargo-Molina:2014pwa]. In addition, transitions between vacua can also occur at finite temperature [@Linde:1980tt; @Linde:1981zj]. In the context of early-Universe cosmology, this is of interest because the corresponding first-order phase transitions may leave behind relic gravitational waves [@Witten:1984rs; @Kosowsky:1991ua; @Caprini:2009fx]. Moreover, such phase transitions may turn out to be pivotal for generating the cosmic matter-antimatter asymmetry [@Morrissey:2012db; @Chung:2012vg]. As a consequence of these applications and the wide range of phenomenological models, there are now routine methods for computing transition rates at both vanishing and finite temperature [@Wainwright:2011kj; @Camargo-Molina:2013qva]. Vacuum transitions in scalar theories can be described in the following way [@Langer:1967ax; @Langer:1969bc; @Kobzarev:1974cp; @Coleman:1977py]. In the event that there are two non-degenerate vacua, an initially homogeneous system lying in the false vacuum will spontaneously nucleate bubbles of true vacuum, leading to the production of domain walls or “kinks.” The latter are the topological solitons that interpolate between regions of true and false vacuum. The study of these “solitary wave” solutions to non-linear equations of motion (see e.g. Ref. [@Scott:1973eg]) has a long history [@Dashen:1974cj; @Polyakov:1974ek; @Goldstone:1974gf; @Christ:1975wt; @Jackiw:1977yn; @Faddeev:1977rm], and archetypal examples of such field configurations arise in the sine-Gordon model [@Coleman:1974bu; @Dashen:1975hd] and the $\lambda \Phi^4$ theory with tachyonic mass $m^2 < 0$. The semi-classical [@Coleman:1977py] and quantum [@Callan:1977pt] descriptions of false vacuum decay in the latter theory were presented in the seminal works by Coleman and Callan (see also Ref. [@Coleman:1978ae]). Early expansion on these works included induced vacuum decay [@Affleck:1979px] and the incorporation of gravity [@Coleman:1980aw]. In order to decide whether a vacuum configuration is unstable, i.e. whether there exists a lowest-lying true vacuum, it is often necessary to account for the impact of radiative corrections. This is of particular relevance when the appearance or disappearance of minima is entirely a radiative effect [@Weinberg:1992ds], such as occurs for the Coleman-Weinberg (CW) mechanism of spontaneous symmetry breaking [@Coleman:1973jx] or in symmetry restoration at finite temperature [@Kirzhnits:1972ut; @Dolan:1973qd; @Weinberg:1974hy]. In phenomenological studies, this is commonly done by calculating the tunneling rate from the effective potential [@Jackiw:1974cv; @Cornwall:1974vz] of a *homogeneous* field configuration [@Camargo-Molina:2013qva; @Frampton:1976kf; @Frampton:1976pb], which is subsequently promoted to a space-time-dependent configuration. This practice is problematic for two reasons. Firstly, the temporal and spatial inhomogeneity of the solitonic background is not fully taken into account [@Surig:1997ne]. Secondly, in the presence of tachyonic instabilities, e.g. when there are non-convex regions in the tree-level potential, the perturbatively-calculated effective potential receives a seemingly-pathological imaginary part. The latter has been shown [@Weinberg:1987vp] to have a physical interpretation as a decay rate for an initially homogeneous field configuration (see also Ref. [@Einhorn:2007rv]). However, this subtlety can be circumvented by using constructions such as the coarse-grained effective action [@Berges:1996ja; @Berges:1996ib; @Strumia:1998nf; @Strumia:1998vd; @Strumia:1998qq; @Strumia:1999fq; @Strumia:1999fv; @Munster:2000kk] or, as is often employed in lattice simulations, the constraint effective potential [@O'Raifeartaigh:1986hi] (cf. Ref. [@Alexandre:2012ht]). In addition, within the context of the SM, the standard RG improvement of the effective potential has recently been questioned [@Gies:2014xha]. For the above reasons, it is reasonable to conclude that the use of the effective potential to calculate transition rates is neither satisfactory nor justifiable and it is desirable to consider alternative methods for determining quantum corrections, which can be applied to a wide range of models that feature vacuum decay. The first quantum corrections [@Callan:1977pt] to the tunneling rate are those arising from the functional determinant over the quadratic fluctuations about the classical soliton configuration. In the case of one-dimensional operators, these determinants may be calculated using the Gel’fand-Yaglom theorem [@Gelfand:1959nq], which may be generalized to higher dimensions in the case of radially-symmetric operators [@Baacke:1993ne; @Dunne:2006ct; @Dunne:2007rt]. General numerical techniques may then be obtained [@Baacke:2003uw; @Dunne:2005rt] for calculating tunneling rates beyond the so-called thin-wall approximation, in which the width of the bubble wall is much smaller than its radius. These approaches have also been applied to radially-separable Yang-Mills backgrounds [@Dunne:2006ac; @Dunne:2007mt] and scenarios in curved spacetime [@Dunne:2006bt]. Alternatively, as we will employ, the functional determinant may be calculated by means of the so-called heat kernel method (see e.g. Refs. [@Diakonov:1983dt; @Diakonov:1984xj; @Konoplich:1987yd; @Vassilevich:2003xt]), based upon the Schwinger proper-time representation and zeta function regularization [@Hawking:1976ja]. Previously, this approach has been used to derive approximate analytic results for the one-loop fluctuation determinant beyond the thin-wall approximation [@Munster:1999hr], as well as one-loop corrections to sphaleron rates [@Carson:1989rf; @Carson:1990jm; @Carson:1990wp]. The latter have also been calculated by direct integration of the Green’s function [@Baacke:1993jr; @Baacke:1993aj; @Baacke:1994ix; @Baacke:1994bk; @Baacke:2008zx]. Recently, it has been shown that properties of topological solitons may be studied non-perturbatively using Monte Carlo and lattice simulations by considering correlation functions directly [@Rajantie:2009bk; @Rajantie:2010tb; @Rajantie:2010fw]. Other authors have proposed methods for calculating quantum corrections based upon functional renormalization techniques [@Alexandre:2007ci]. In this article, we derive an *analytic* result for the Green’s function of the $\lambda \Phi^4$ theory in the background of the classical kink solution. Within the thin- and planar-wall approximations, we illustrate that this Green’s function may be used to determine *analytically* the leading quantum corrections to both the semi-classical bounce action and the kink solution itself, thereby allowing us to compute the tunneling rate at the two-loop level, while isolating its diagrammatic interpretation. The latter calculation is performed within the context of a toy model extended with an additional $N$ heavier scalars, where the parametric dependence on $N$ allows the identification of a concrete example in which the calculated two-loop corrections dominate over the neglected higher-loop corrections. We illustrate that the problem of calculating these radiative corrections may be reduced to one of solving one-dimensional ordinary differential equations and integrals. Thus, we anticipate that this methodical development may have numerical applications in the study of the decay rates of radiatively-generated metastable vacua, such as occur in the massless CW model [@Coleman:1973jx] or the Higgs potential of the SM. Similar methods based upon Green’s function techniques have been used previously to determine self-consistent bounce solutions *numerically* in the Hartree approximation of the pure $\lambda\Phi^4$ theory in both two and four dimensions [@Bergner:2003au; @Bergner:2003id; @Baacke:2004xk; @Baacke:2006kv]. The remainder of this article is organized as follows. In Sec. \[sec:classb\], we review the calculation, *à la* Coleman and Callan [@Coleman:1977py; @Callan:1977pt], of the classical “bounce” configuration, describing the semi-classical tunneling rate between two quasi-degenerate vacua and its first quantum corrections. In Sec. \[sec:greens\], we outline a Green’s function method for the evaluation of the functional determinant over the quantum fluctuations about the classical bounce, making comparison with existing calculations. Subsequently, in Sec. \[sec:corrb\], we illustrate that this Green’s function method may be used to calculate analytically and self-consistently the first quantum corrections to the bounce itself. In Sec. \[sec:conc\], we conclude our discussions and highlight potential applications and future directions. Finally, a number of mathematical appendices are included, outlining the technical details of the calculations summarized in Secs. \[sec:greens\] and \[sec:corrb\]. Semi-classical bounce {#sec:classb} ===================== We consider a real scalar field $\Phi \equiv \Phi(x)$, with four-dimensional Euclidean Lagrangian $\mathcal{L} = (\partial_{\mu} \Phi)^2/2 + U$ and classical potential $$\label{eq:pot} U \ = \ \frac{1}{2!} \, m_{\Phi}^2 \Phi^2 + \frac{g}{3!} \, \Phi^3 + \frac{\lambda}{4!} \, \Phi^4 + U_0 \; .$$ The mass squared is $m_{\Phi}^2 = - \, \mu^2 < 0$, $g$ is of mass dimension one, $\lambda$ is dimensionless, $U_0$ is a constant and $\partial_{\mu} \equiv \partial / \partial x_{\mu}$ denotes the derivative with respect to the Euclidean spacetime coordinate $x_{\mu} \equiv (\mathbf{x},x_4)$. Throughout, we omit spacetime and field arguments for notational convenience when no ambiguity results. ![The classical potential $U$ (left panel) and the inverted potential $-\,U$ (right panel). The arrow (right panel) indicates the trajectory of the “bounce” in imaginary time $\tau$.\[fig:potential\]](U.pdf "fig:") ![The classical potential $U$ (left panel) and the inverted potential $-\,U$ (right panel). The arrow (right panel) indicates the trajectory of the “bounce” in imaginary time $\tau$.\[fig:potential\]](Uminus.pdf "fig:") The classical potential in Eq.  has non-degenerate minima at $$\varphi \ = \ v_{\pm} \ = \ \pm \, v \, \bigg(1 + \frac{\bar{v}^2}{v^2}\bigg)^{\!\frac{1}{2}} - \bar{v} \; ,$$ as depicted in Fig. \[fig:potential\] (left panel), where we have defined $v = \sqrt{6 \mu^2 / \lambda}$ and $\bar{v} = (3 g) / (2 \lambda)$. The separation of the minima $\Delta v = v_+ - v_- = 2 d$ and the difference in their energy densities $\Delta U = U_{v_+} - U_{v-} = 2 \varepsilon$ may be written in terms of the parameters $$\begin{aligned} d &= v \, \bigg(\! 1 + \frac{\bar{v}^2}{v^2}\bigg)^{\!\!\frac{1}{2}} \! \approx v\;, \\ \frac{\varepsilon}{d} &= \frac{gv^2}{6} \bigg(\! 1 + 3 \, \frac{\bar{v}^2}{v^2}\bigg) \approx \frac{gv^2}{6} \; ,\end{aligned}$$ where the approximations are valid in the limit $v \gg \bar{v}$, i.e. $g^2 / \mu^2 \ll 8 \lambda / 3$. For $g \to 0$, $\varepsilon \to 0$ and the minima at $\varphi = \pm \, v$ become degenerate, as we would expect. Finally, the constant $U_0$ is chosen so that the potential vanishes in the false vacuum at $\varphi = + \, v$, requiring $U_0 = (\mu v/2)^{2} - g v^3 / 6$ and thus giving the barrier height to be $h = U_0 + 2 \varepsilon \approx (\mu v / 2)^2 + \varepsilon$ and $U(-\,v)=-\,gv^3/3$. The semi-classical probability for tunneling between the false ($\varphi = + \, v$) and true ($\varphi = - \, v$) vacua and its first quantum corrections were described in the seminal works by Coleman and Callan [@Coleman:1977py; @Callan:1977pt], which we now review. The classical equation of motion $$\label{eq:classeom} - \, \partial^2 \varphi + U'(\varphi) \ = \ 0\;,$$ where $'$ denotes the derivative with respect to the field $\varphi$, is analogous to that of a particle moving in a potential $- \, U$. The boundary conditions of the “bounce” are $\varphi|_{x_4 \to \pm \infty} = + \, v$ and $\dot{\varphi}|_{x_4 = 0} \ = \ 0$, where $\dot{}$ denotes the derivative with respect to $x_4$. These correspond to a particle initially at $+ \, v$ rolling through the valley in $- \, U$, reaching a turning point close to $-\,v$, before rolling back to $+ \, v$, see Fig. \[fig:potential\] (right panel). Finally, in order to ensure that the action of the bounce is finite, we require $\varphi|_{|\mathbf{x}| \to \infty} = + \, v$. Translating to four-dimensional hyperspherical coordinates, Eq.  takes the form $$\label{eq:classeomrad} - \frac{{\mathrm{d}}^2}{{\mathrm{d}}r^2} \, \varphi - \frac{3}{r} \, \frac{{\mathrm{d}}}{{\mathrm{d}}r} \, \varphi + U'(\varphi) \ = \ 0 \; ,$$ where $r^2 = \mathbf{x}^2 + x_4^2$. The boundary conditions become $\varphi|_{r \to \infty} = + \, v$ and ${\mathrm{d}}\varphi / {\mathrm{d}}r|_{r = 0} = 0$, where the latter ensures that the solution is well defined at the origin. Thus, the bounce corresponds to a four-dimensional bubble of some radius $R$, which separates the false vacuum ($\varphi = + \, v$) outside from the true vacuum inside ($\varphi = - \, v$). Analytically continuing to Minkowski spacetime ($x_4 = i x_0$), the $O(4)$ symmetry of the bounce becomes an $SO(1,3)$ symmetry, with the bubble expanding along the hyperbolic trajectory $R^2 \ = \ \mathbf{x}^2 - c^2 t^2$. The bounce action is $$\label{eq:B} B \ = \ \int \! {\mathrm{d}}^4x \; \bigg[ \frac{1}{2} \bigg( \frac{\mathrm{d\varphi}}{\mathrm{d}r} \bigg)^{\!2} + U(\varphi) \bigg] \; ,$$ which can be written as $B = B_{\rm surface} + B_{\rm vacuum}$, where $$\begin{aligned} \label{eq:Bsurface} B_{\rm surface} \ & = \ 2 \pi^2 R^3 \int_{- \, v}^{+ \, v} \! {\mathrm{d}}\varphi \; \frac{\mathrm{d\varphi}}{\mathrm{d}r} \; , \\ B_{\rm vacuum} \ & = \ 2 \pi^2 \int_{0}^{R} \! {\mathrm{d}}r \; r^3 U(- \, v) \end{aligned}$$ are the contributions from the surface tension of the bubble and the energy of the true vacuum, respectively. In writing Eq. , we have used the fact that for $\partial_{\mu} \varphi \neq 0$, i.e. for $r \sim R$, we may show that the bounce $\varphi$ satisfies the virial theorem $$\bigg(\frac{{\mathrm{d}}\varphi}{{\mathrm{d}}r}\bigg)^{\! 2} - 2 U(\varphi) \ = \ 0 \; ,$$ i.e. it is the configuration of zero total energy density. Notice that there is no contribution to the bounce action \[Eq. \] from the exterior of the bubble, since the choice of the potential Eq. , viz. $U_0$, ensures that the false vacuum has zero energy density. In the thin-wall approximation, we may safely neglect the damping term in Eq.  and the contribution from the cubic self-interaction $g\varphi^3$, as will be the case for the remainder of this article. We then obtain the well-known kink solution [@Dashen:1974cj] $$\varphi(r) \ = \ v \tanh \big[\gamma ( r - R )\big] \; ,$$ with $\gamma = \mu / \sqrt{2}$. The radius $R$ of the bubble is then obtained by extremizing the bounce action Eq. , that is by minimizing the energy difference between the surface tension of the bubble and the true vacuum. This gives $$R \ = \ \frac{12 \gamma}{g v} \; .$$ By considering the invariance of the bounce action in Eq.  under general coordinate transformations, i.e. $\varphi \to \varphi + x_{\mu} \partial_{\mu} \varphi$, we may show that $$\label{eq:fullbounceact} B \ = \ \frac{1}{2} \pi^2 R^3 \int_{- \, v}^{+ \, v} \! {\mathrm{d}}\varphi \, \frac{{\mathrm{d}}\varphi}{{\mathrm{d}}r} \; .$$ This is to say that $$B_{\rm vacuum} \ = \ - \, \frac{3}{4} \, B_{\rm surface} \; ,$$ in which case we find $$B = \ - \, \frac{1}{3} B_{\rm vacuum} \ = \ - \, \frac{\pi^2}{6} \, R^4 \, U(- \, v) \ = \ \frac{8 \pi^2 R^3 \gamma^3}{\lambda} \; .$$ The decay rate of the false vacuum, i.e. the probability per unit time for the nucleation of a bubble of true vacuum, has the generic form [@Coleman:1977py; @Callan:1977pt] $$\label{eq:tunnelprob} \varGamma \ = \ A V e^{-\,B / \hbar} \; .$$ Here, $V$ is the three-volume within which the bounce may occur, arising from integrating over the center of the bounce, and $A$ contains the quantum corrections to the classical bounce action $B$ that are the subject of the remainder of this article. The tunneling probability in Eq.  may be obtained from the path integral $$\label{eq:path} Z[0] \ = \ \int \! [ {\mathrm{d}}\Phi ] \; e^{- \, S[\Phi] / \hbar} \; ,$$ via $$\label{eq:tunnelrel} \varGamma \ = \ 2 | \mathrm{Im} \, Z[0] | / T \; ,$$ where $T$ is the Euclidean time of the bounce. In order to evaluate the functional integral over $\Phi$, we first expand around the classical bounce $\varphi$, whose equation of motion  is obtained from $$\label{eq:deltaS} \frac{\delta S[\Phi]}{\delta \Phi} \bigg|_{\Phi \, = \, \varphi} \ = \ 0 \; .$$ Writing $\Phi = \varphi + \hbar^{1/2} \hat{\Phi}$, where the factor of $\hbar^{1/2}$ is written explicitly for bookkeeping purposes, we find $$S[\Phi] \ = \ S[\varphi] + \, \frac{\hbar}{2} \! \int \! {\mathrm{d}}^4 x \; \hat{\Phi}(x) \, G^{-1}(\varphi;x) \, \hat{\Phi}(x) + \mathcal{O}(\hbar^{3/2}) \; ,$$ where $S[\varphi] \equiv B$ is the classical bounce action and $$\label{eq:Gminusdef} G^{-1}(\varphi;x) \ \equiv \ \frac{\delta^2 S[\Phi]}{\delta \Phi^2(x)}\bigg|_{\Phi \, = \, \varphi} \ = \ - \, \Delta^{\! (4)} + U''(\varphi; x) \; ,$$ in which $\Delta^{\! (4)}$ is the four-dimensional Laplacian. Before proceeding to perform the functional integration over the quadratic fluctuations about the bounce, we must consider the spectrum of the operator $G^{-1}(\varphi;x)$, which is not positive definite. By differentiating the equation of motion Eq.  with respect to $x_{\mu}$ and comparing with the eigenvalue equation $$\big( - \Delta^{(4)} + U''(\varphi) \big) \phi_{\{n\}} \ = \ \lambda_{\{n\}} \phi_{\{n\}} \; ,$$ it is straightforward to show that there exist four zero eigenmodes $\phi_{\mu} = \mathcal{N} \partial_{\mu} \varphi$, transforming as a vector of $SO(4)$ and resulting from the translational invariance of the bounce. The normalization $\mathcal{N}$ follows from Eq. , since $$\int \! {\mathrm{d}}^4 x \; \phi_{\mu}^* \phi_{\nu} \ = \ \frac{1}{4} \, \mathcal{N}^2 \delta_{\mu \nu} \! \int \! {\mathrm{d}}^4 x \; \big( \partial_{\lambda} \varphi \big)^2 \ = \ \mathcal{N}^2 B \, \delta_{\mu \nu} \; .$$ Thus, $\phi_{\mu} = B^{-1/2} \partial_{\mu} \varphi$. Differentiating Eq.  with respect to $r$ and subsequently setting $r = R$ in those terms originating from the damping term, we can show that there also exists a discrete eigenmode $\phi_0 = B^{-1/2} \partial_r \varphi$. This eigenmode transforms as a scalar of $SO(4)$, corresponding to dilatations of the classical bounce solution, and has the negative eigenvalue $$\lambda_0 \ = \ \frac{1}{B} \frac{\delta^2\!B}{\delta R^2} \ = \ - \, \frac{3}{R^2} \; .$$ It is this lowest mode that is responsible for the path integral in Eq.  obtaining the non-zero imaginary part in Eq.  [@Coleman:1987rm]. Alternatively, we may solve the eigenvalue problem directly in hyperspherical coordinates (see Appendix \[app:greens\]), by making the substitution $\phi_{\{n\}} = \phi_{nj}/r^3$.[^1] Neglecting the damping term and setting $r = R$ in the centrifugal potential, we obtain the eigenspectrum $$\label{eq:eigenspectrum} \lambda_{nj} \ = \ \gamma^2 \big( 4 - n^2 \big) + \frac{j(j + 2) - 3}{R^2} \; .$$ The radial parts of the eigenfunctions are the associated Legendre polynomials of the first kind and of order 2, i.e. $P_{2}^{n}(\varphi/v)$. Thus, demanding normalizability, the quantum number $n$ is restricted to the set $\{1,2\}$. From Eq. , we see that the negative mode corresponds to $\lambda_0 = \lambda_{20}$ ($n = 2$, $j = 0$) and the zero modes correspond to $\lambda_{21}$ ($n = 2$, $j = 1$), having degeneracy $(j + 1)^2 = 4$. The lowest two positive-definite eigenvalues are $\lambda_{10} = 2 \gamma^2 - 3 / R^2$ ($n = 1$, $j = 0$) and $\lambda_{11} = 2 \gamma^2$ ($n = 1$, $j = 1$). Thus, for $R$ large, the “continuum” of positive-definite modes begins at $\lambda_{10} \approx \lambda_{11} = 2 \gamma^2$, cf. Ref. [@Konoplich:1987yd]. In order to perform the functional integral over the five negative-semi-definite discrete modes, we expand $\hat{\Phi} = \sum_{i = 0}^{4}a_i \phi_i + \phi_+$, where $\phi_+$ comprises the continuum of positive-definite eigenmodes. The functional measure then becomes $$[ {\mathrm{d}}\Phi ] \ = \ [ {\mathrm{d}}\phi_+ ] \prod_{i = 0}^4 (2 \pi \hbar)^{- 1/2} {\mathrm{d}}a_i \; .$$ The functional integral over the four zero eigenmodes ($i = 1, \dots, 4$) is traded for an integral over the collective coordinates of the bounce [@Gervais:1974dc] (see Appendix \[app:zeromode\]) and yields a factor $$V T \bigg( \frac{B}{2 \pi \hbar} \bigg)^{\!2} \; .$$ The integral over the negative eigenmode ($i = 0$) may be performed using the method of steepest descent, giving an overall factor of $- i | \lambda_0 |^{-1/2} / 2$. Here, the overall sign is unphysical [@Callan:1977pt] and depends on the choice of analytic continuation, thereby justifying the modulus sign in Eq. . Finally, the Gaussian integral over the continuum of positive eigenmodes $\phi_+$ may be performed in the usual manner and we obtain $$\label{eq:pathinted} i Z[0] \ = \ e^{- B / \hbar} \left| \frac{\lambda_0 \, \mathrm{det}^{(5)} \, G^{-1}(\varphi)} {\frac{1}{4} (VT)^2 \big(\frac{B}{2 \pi \hbar}\big)^{\!4} (4 \gamma^2)^5 \, \mathrm{det}^{(5)} \, G^{-1}(v)} \right|^{-\tfrac{1}{2}} \; ,$$ in which $\mathrm{det}^{(5)}$, cf. Ref. [@Konoplich:1987yd], denotes the determinant calculated only over the continuum of positive-definite eigenmodes, i.e. omitting the zero and negative eigenmodes, whose contributions are included explicitly. In addition, we have normalized the determinant to that of the operator $G^{-1}(v)$, evaluated in the false vacuum. Substituting Eq.  into Eq. , we find the tunneling rate per unit volume $$\begin{aligned} \label{eq:tun} & \varGamma / V \ = \ \bigg( \frac{B}{2 \pi \hbar}\bigg)^{\!2} (2 \gamma)^5 | \lambda_0 |^{-\tfrac{1}{2}} \exp \bigg[ - \frac{1}{\hbar} \bigg( B + \hbar B^{(1)} \bigg) \bigg] \; ,\end{aligned}$$ where $$\label{eq:B1} B^{(1)} \ = \ \frac{1}{2} \, \mathrm{tr}^{(5)} \Big( \ln G^{-1}(\varphi) - \ln G^{-1}(v) \Big) \; ,$$ contains the one-loop corrections from the quadratic fluctuations around the classical bounce. Here, $\mathrm{tr}^{(5)}$ indicates that we are to trace over only the positive-definite eigenmodes. Green’s function method {#sec:greens} ======================= In this section, we outline the derivation of the Green’s function of the operator in Eq. . The technical details are included for completeness in Appendix \[app:greens\]. Subsequently, we use this Green’s function to evaluate the functional determinant in Eq.  and obtain the correction from quadratic fluctuations. In addition, we calculate analytically the tadpole contribution to the effective equation of motion and point out that this may be used to calculate the first quantum corrections to the bounce. We have the inhomogeneous Klein-Gordon equation $$\label{eq:phiKG} \big( \! - \Delta^{\! (4)} + U''(\varphi; x) \big) G(\varphi; x, x') \ = \ \delta^{(4)}(x - x') \; ,$$ where $\delta^{(4)}(x - x')$ is the four-dimensional Dirac delta function. Working in hyperspherical coordinates and writing $x_{\mu} \! \! \! {}^{(}{}'{}^{)} = r {}^{(}{}'{}^{)} \mathbf{e}_{r{}^{(}{}'{}^{)}}$, where $\mathbf{e}_{r{}^{(}{}'{}^{)}}$ are four-dimensional unit vectors, the Green’s function may be expanded as $$\label{eq:greenexp} G(\varphi; x, x') \ = \ \frac{1}{2 \pi^2} \sum_{j = 0}^{\infty}(j + 1) G_{j}(\varphi; r, r') U_j(\cos \theta) \; ,$$ where $\cos \theta = \mathbf{e}_r \cdot \mathbf{e}_{r'}$ and $U_j(z)$ are the Chebyshev polynomials of the second kind (see Appendix \[app:greens\]). The radial functions $G_j(r, r')$ satisfy the inhomogeneous equation $$\begin{aligned} \label{eq:radialdif} &\bigg[ - \frac{{\mathrm{d}}^2}{{\mathrm{d}}r^2} - \frac{3}{r} \frac{{\mathrm{d}}}{{\mathrm{d}}r} + \frac{j(j + 2)}{r^2} \nonumber \\ & \qquad \qquad + U''(r) \bigg] G_j(\varphi; r, r')\ = \ \frac{\delta(r - r')}{r'^3} \; .\end{aligned}$$ For the thin wall, we safely neglect the damping term and approximate the centrifugal term by $j(j + 2) / R^2$. For self-consistency of this approximation, we also replace the discontinuity on the rhs of Eq.  with $\delta(r - r')/R^3$. For generality of notation in what follows, it is then convenient to define $$\begin{aligned} G(u, u', m) \ & \equiv \ R^3 G_j(\varphi; r, r') \; ,\end{aligned}$$ being a function only of the normalized bounce $$\label{phi:u} u{}^{(}{}'{}^{)} \ \equiv \ \frac{\varphi(r{}^{(}{}'{}^{)})}{v} \ = \ \tanh \big[\gamma (r{}^{(}{}'{}^{)} - R)\big]$$ and the parameter $$m \ = \ 2 \bigg( 1 + \frac{j(j + 2)} {4 \gamma^2 R^2} \bigg)^{\! \tfrac{1}{2}} \; .$$ The full Green’s function may then be written $$\begin{aligned} \label{eq:greennotedef} G(\varphi; x, x') \ & \equiv \ G(u, u', \theta) \nonumber \\ & = \ \frac{1}{2 \pi^2 R^3} \sum_{j = 0}^{\infty} (j + 1) U_j(\cos \theta)G(u, u', m) \; .\end{aligned}$$ With the above approximations, the lhs of Eq.  is of Pöschl-Teller form [@Poschl:1933zz], having general solutions that may be expressed in terms of the associated Legendre functions (see Appendix \[app:greens\]). We are then able to find the full analytic solution $$\begin{aligned} \label{eq:Guupm} & G(u, u', m) \ = \ \frac{1}{2 \gamma m}\bigg[ \vartheta(u - u') \bigg(\frac{1 - u}{1 + u}\bigg)^{\! \! \tfrac{m}{2}} \! \bigg(\frac{1 + u'}{1 - u'}\bigg)^{\! \! \tfrac{m}{2}} \nonumber \\ & \quad \times \bigg( \! 1 - 3 \, \frac{(1 - u)(1 + m + u)}{(1 + m)(2 + m)} \bigg) \nonumber \\ & \quad \times \bigg( \! 1 - 3 \, \frac{(1 - u')(1 - m + u')}{(1 - m)(2 - m)} \bigg) + (u \leftrightarrow u') \bigg] \; ,\end{aligned}$$ where $\vartheta(z)$ is the generalized unit-step function. Taking the coincidence limit $u = u'$, $\theta = 0$, the local contribution to the Green’s function $G(u) \equiv G(u, u, 0)$ in Eq.  takes the form $$\label{eq:Gu} G(u) \ = \ \frac{1}{2 \pi^2 R^3} \sum_{j = 0}^{\infty} (j + 1)^2 G(u, m) \; ,$$ where $$\begin{aligned} \label{eq:Gum} & G(u, m) \ \equiv \ G(u, u, m) \nonumber \\ & \ = \ \frac{1}{2 \gamma m}\bigg[ 1 + 3 \big( 1 - u^2 \big) \sum_{n = 1}^2 \frac{(-1)^n (n - 1 - u^2) }{m^2 - n^2} \bigg] \; .\end{aligned}$$ In Eq. , the summation over $n = 1, 2$ corresponds to the contributions from the two towers of positive-definite eigenmodes of the operator $G^{-1}(\varphi; x)$, see Eq. . For $R$ large, we may approximate the summation over $j$ by an integral over a continuous variable $k \sim \frac{j + 1}{R}$ (see Appendix \[app:greens\]). In which case, we obtain $$\label{eq:Grk2} G(u) \ = \ \frac{1}{2 \pi^2} \! \int_{0}^{\infty} {\mathrm{d}}k \; k^2 \, G(u, m) \; ,$$ with $$\label{eq:continuum} m \ = \ 2 \, \bigg( 1 + \frac{k^2}{4 \gamma^2} \bigg)^{\! \tfrac{1}{2}} \; .$$ ![Schematic representation of the planar-wall approximation, illustrating the alignment of the three-dimensional hypersurface and associated coordinate system.\[fig:planar\]](planar.pdf) The continuum limit described above is entirely equivalent to the so-called planar-wall approximation. Therein, for $R$ large, we align a set of coordinates $(z_{\perp}, \mathbf{z}_{\parallel})$ with the bubble wall, as shown in Fig. \[fig:planar\]. We may then Fourier-transform with respect to the coordinates $\mathbf{z}_{\parallel}$ that lie within the three-dimensional wall, introducing a three-momentum $\mathbf{k}$, i.e. $$G(\varphi; x, x') \ = \ \! \int \! \! \frac{{\mathrm{d}}^3 \mathbf{k}}{(2 \pi)^3} \; e^{i\mathbf{k}\cdot(\mathbf{z}_{\parallel} - \mathbf{z}'_{\parallel})} \, G(\varphi; z, z', \mathbf{k}) \; ,$$ where we have let $z = z_{\perp}$ for notational convenience. The three-momentum-dependent Green’s function $G(\varphi; z, z', \mathbf{k})$ satisfies the inhomogeneous Klein-Gordon equation $$\label{eq:planwallKG} \big( - \partial_z^2 + k^2 + U''(\varphi; z) \big) G(\varphi; z, z', \mathbf{k}) \ = \ \delta(z - z') \; .$$ We may then show straightforwardly that $$G(\varphi; z, z' , \mathbf{k}) \ = \ G(u, u', m) \; ,$$ where $G(u, u', m)$ is as defined in Eq.  with $m$ given by Eq. . This planar-wall approximation is employed for the remainder of this article. Quantum-corrected bounce ------------------------ Before making use of the Green’s function calculated in the preceding section, we first derive the equation of motion for the quantum-corrected bounce. This calculation was first suggested by Goldstone and Jackiw [@Goldstone:1974gf] and, in the following sections, we will illustrate that, within the thin- and planar-wall approximations, it may be completed analytically. The one-particle irreducible (1PI) effective action [@Jackiw:1974cv] is given by the Legendre transform $$\label{effact} \Gamma[\phi] \ = \ - \, \hbar \, \ln Z[J] + \int \! {\mathrm{d}}^4 x \; J(x) \phi(x) \; ,$$ where $$\phi(x)\ =\ \hbar\,\frac{\delta \ln Z[J]}{\delta J(x)}$$ is a functional of the source $J(x)=\frac{\delta \Gamma[\phi]}{\delta \phi(x)}$ and $$\label{ZofJ} Z[J] \, = \int \! [ {\mathrm{d}}\Phi ] \; \exp \! \bigg[ - \frac{1}{\hbar} \bigg( S[\Phi] - \! \int \! {\mathrm{d}}^4 x \; J(x) \Phi(x) \bigg) \bigg] \; .$$ In order to obtain the quantum corrections to the bounce $\varphi$, we wish to evaluate the functional integral in Eq.  by expanding around the configuration $\varphi^{(1)}$, which is the solution to the quantum equation of motion $$\label{eq:gamdif} \frac{\delta \Gamma[\phi]}{\delta \phi} \bigg|_{\phi = \varphi^{(1)}} \ = \ 0 \; .$$ Here, the superscript “$(1)$” indicates that $\varphi^{(1)}$ contains the first quantum corrections to $\varphi$. It follows from Eq.  that $\varphi^{(1)}$ cannot extremize the classical action in the absence of the source $J$, i.e. $$\label{eq:actdif} \frac{\delta S[\Phi]}{\delta \Phi(x)}\bigg|_{\Phi \, = \, \varphi^{(1)}} \ = \ J(x) \ \neq \ 0 \; .$$ Writing $\Phi = \varphi^{(1)} + \hbar^{1/2} \hat{\Phi}^{(1)}$, where the factor of $\hbar^{1/2}$ is again written explicitly for bookkeeping, we proceed as in Sec. \[sec:classb\], expanding $$\begin{aligned} & S[\Phi] \ = \ S[\varphi^{(1)}] + \hbar^{1/2} \! \int \! {\mathrm{d}}^4 x \; J(x) \, \hat{\Phi}^{(1)}(x) \nonumber \\ & \quad + \frac{\hbar}{2} \int \! {\mathrm{d}}^4 x \; \hat{\Phi}^{(1)}(x) \, G^{-1}(\varphi^{(1)}; x) \, \hat{\Phi}^{(1)}(x) + \cdots \; ,\end{aligned}$$ where $$\label{eq:G1minusdef} G^{-1}(\varphi^{(1)}; x) \ \equiv \ \frac{\delta^2 S[\Phi]}{\delta \Phi^2(x)}\bigg|_{\Phi \, = \, \varphi^{(1)}} \ = \ - \, \Delta^{(4)} + U''(\varphi^{(1)}; x) \; .$$ We now write $\varphi^{(1)} = \varphi + \hbar\delta\varphi$ and expand about the quadratic fluctuations evaluated around the classical bounce $\varphi$. Thus, in performing the functional integral, we consider the same spectrum of negative and zero eigenmodes as in Sec. \[sec:classb\]. Finally, by expanding the effective action $\Gamma[\phi]$ in Eq.  around $\varphi^{(1)}=\phi-\hbar\delta\varphi$ (see Ref. [@Carrington:2004sn]), we obtain $$\begin{aligned} \label{eq:effact} & \Gamma[\varphi^{(1)}] \ = \ S[\varphi^{(1)}] + \frac{i \pi \hbar}{2}+\hbar^2B^{(2)}{}'[\varphi] \nonumber \\ & \qquad + \frac{\hbar}{2} \ln \left| \frac{\lambda_0 \, \mathrm{det}^{(5)} \, G^{-1}(\varphi)} {\frac{1}{4} (VT)^2 \big( \frac{B}{2 \pi \hbar} \big)^4 (4 \gamma^2)^5 \mathrm{det}^{(5)} \, G^{-1}(v)} \right| + \cdots \;,\end{aligned}$$ where $$\begin{aligned} B^{(2)}{}'[\varphi]\ &= \ \frac{1}{2}\int\!{\mathrm{d}}^4 x\;\delta\varphi(x)\nonumber\\&\quad\times \frac{\delta}{\delta \varphi^{(1)}(x)}\,\ln\frac{\mathrm{det}^{(5)}G^{-1}(\varphi^{(1)})}{\mathrm{det}^{(5)}G^{-1}(v)}\Bigg|_{\varphi^{(1)}=\varphi}\;.\end{aligned}$$ Functionally differentiating Eq.  with respect to $\varphi^{(1)}$, we obtain the equation of motion for the corrected bounce $$\label{eq:eom1} - \, \partial^2 \varphi^{(1)}(x) + U_{\mathrm{eff}}'(\varphi^{(1)};x) \ = \ 0 \; ,$$ where $$\label{eq:Ueffprime} U_{\mathrm{eff}}'(\varphi^{(1)}; x) \ \equiv \ U'(\varphi^{(1)}; x) + \hbar\Pi(\varphi;x) \varphi(x) \; ,$$ containing the tadpole contribution $$\label{eq:tadpole} \Pi(\varphi;x)\ =\ \frac{\lambda}{2} \, G(\varphi; x, x)\;.$$ Comparing the functional derivative of Eq.  with Eqs.  and , we see that this evaluation of the effective action is self-consistent so long as the source $$\label{eq:Jdef} J(x) \ = \ - \,\hbar\Pi(\varphi;x) \varphi(x) \; ,$$ which is, as expected, non-vanishing. We may show that the correction to the classical bounce $\delta \varphi$ satisfies the equation of motion $$\label{eq:deltaphieom} G^{-1}(\varphi; x) \, \delta \varphi(x) \ = \ - \, \Pi(\varphi;x) \varphi(x) \; .$$ The corrected bounce action $S[\varphi^{(1)}]$ contains contributions at order $\hbar^2$. Specifically, $$\begin{aligned} \label{eq:secondexp} & S[\varphi^{(1)}] \ = \ S[\varphi] \nonumber \\ & \quad + \, \frac{\hbar^2}{2} \! \int \! {\mathrm{d}}^4 x \; \delta \varphi(x) \, G^{-1}(\varphi; x) \, \delta \varphi(x) + \mathcal{O}(\hbar^{3}) \; ,\end{aligned}$$ where we have used Eqs.  and . Thus, using Eq. , we may write $$S[\varphi^{(1)}] \ = \ B + \hbar^2 B^{(2)} \; ,$$ where $$\label{eq:B2} B^{(2)} \ = \ - \, \frac{1}{2} \! \int \! {\mathrm{d}}^4 x \; \varphi(x) \Pi(\varphi;x) \delta \varphi(x) \; .$$ Hence, we obtain the tunneling rate per unit volume $$\begin{aligned} \varGamma / V \ &= \ 2 \, | \mathrm{Im} \, e^{-\Gamma[\varphi^{(1)}]/\hbar} | / (VT) \nonumber\\ &=\ \bigg( \frac{B}{2 \pi\hbar} \bigg)^{\! 2} (2 \gamma)^5 | \lambda_0 |^{- \frac{1}{2}}\nonumber \\ &\quad \times \exp \! \bigg[ - \frac{1}{\hbar} \bigg( B + \hbar B^{(1)} + \hbar^2 B^{(2)}+\hbar^2B^{(2)}{}' \bigg) \bigg] \, ,\end{aligned}$$ where $B$ is the classical bounce action; $B^{(1)}$, given in Eq. , contains the corrections from quadratic fluctations about the classical bounce; and $B^{(2)}$, given in Eq. , contains the contribution arising from the quantum corrections to the bounce itself. We note that $$B^{(2)}{}'=-2B^{(2)}\;,$$ such that the $\mathcal{O}(\hbar)$ corrections to the quadratic fluctuations flip the sign of the contribution to the bounce action from the $\mathcal{O}(\hbar)$ corrections to the bounce itself. Tadpole contribution -------------------- We will now proceed to calculate explicitly the tadpole contribution appearing in Eq. . Introducing an ultraviolet cutoff $\Lambda$, the $k$ integral can be performed in Eq. , and we obtain $$\begin{aligned} G(u) \ & = \ \frac{\gamma^2}{8 \pi^2} \bigg[ \frac{\Lambda^2}{\gamma^2} + 2 - \big( 1 - 3 u^2 \big) \ln \frac{\gamma^2}{\Lambda^2} \nonumber \\ & \qquad - \sqrt{3} \pi u^2 \big( 1 - u^2 \big) \bigg] \; .\end{aligned}$$ We choose to define the physical mass and coupling in the homogeneous non-solitonic background.[^2] The renormalization conditions are then as follows: \[eq:renormconds\] $$\begin{gathered} \frac{\partial^2 U_{\rm eff}(\varphi)}{\partial \varphi^2} \bigg|_{\varphi = v} \ = \ - \, \mu^2 + \frac{\lambda}{2} \, v^2 \ = \ 2 \mu^2 \; , \\ \frac{\partial^4 U_{\rm eff}(\varphi)}{\partial \varphi^4} \bigg|_{\varphi = v} \ = \ \lambda \; , \end{gathered}$$ where $U_{\mathrm{eff}}$ is the CW effective potential [@Coleman:1973jx]. The resulting mass and coupling counterterms are $$\begin{aligned} \delta m^2 & = - \, \frac{\lambda \gamma^2}{16 \pi^2} \bigg(\frac{\Lambda^2}{\gamma^2} - \ln \frac{\gamma^2}{\Lambda^2} - 31 \bigg) \; , \\ \delta \lambda & = - \, \frac{3 \lambda^2}{32 \pi^2} \bigg( \ln \frac{\gamma^2}{\Lambda^2} + 5 \bigg) \; .\end{aligned}$$ We then arrive at the renormalized tadpole correction $$\begin{aligned} \label{eq:phitad} & \Pi^R(u) \ = \ \frac{\lambda}{2} \, G(u) + \delta m^2 + \frac{2\gamma^2}{\lambda} \delta \lambda \, u^2 \nonumber \\ & \qquad = \ \frac{3 \lambda \gamma^2}{16 \pi^2} \bigg[6 + \big(1 - u^2\big) \bigg(5 - \frac{\pi}{\sqrt{3}} \, u^2 \bigg) \bigg] \; .\end{aligned}$$ Functional determinant ---------------------- We may calculate the traces appearing in the exponent of Eq. , which arise from the functional determinant of the operator $G^{-1}(\varphi)$ in Eq. , by using the heat kernel method (see e.g. Ref. [@Vassilevich:2003xt]). Specifically, the trace may be written in the form $$\mathrm{tr}^{(5)} \ln G^{-1}(\varphi;x) \ = \ - \int\! {\mathrm{d}}^4 x \! \int_{0}^{\infty} \! \frac{{\mathrm{d}}\tau}{\tau} \; K(\varphi; x, x | \, \tau) \; .$$ The heat kernel $K(\varphi; x, x' | \, \tau)$ is the solution to the heat-flow equation $$\label{hflow} \partial_{\tau} K(\varphi; x, x' | \, \tau) \ = \ G^{-1}(\varphi; x) K(\varphi; x, x' | \, \tau)$$ and satisfies the condition $K(\varphi; x, x' | \, 0) = \delta^{(4)}(x - x') \, $. It is convenient to work in terms of the Laplace transform of the heat kernel $$\mathcal{K}(\varphi; x, x' | s) \ = \ \int_0^{\infty}\!{\mathrm{d}}\tau \; e^{s\tau} \, K(\varphi; x, x' | \, \tau) \; ,$$ which is the solution to $$\big( - \partial^2 + s + U''(\varphi; x) \big) \mathcal{K}(\varphi; x, x' | \, s) \ = \ \delta^{(4)}(x - x') \; .$$ In the planar-wall approximation, we take $$\label{eq:LapK} \mathcal{K}(\varphi; x, x' | \, s) \ = \ \! \int \! \! \frac{{\mathrm{d}}^3\mathbf{k}}{(2\pi)^3}\;e^{i\mathbf{k}\cdot(\mathbf{z}_{\parallel}-\mathbf{z}_{\parallel}')}\, \mathcal{K}(\varphi; z, z', \mathbf{k} | \, s) \; ,$$ where $\mathcal{K}(\varphi; z, z', \mathbf{k} | \, s)$ satisfies $$\label{eq:LapKplanwall} \big( - \partial_z^2 + k^2 + s + U''(\varphi; z) \big) \mathcal{K}(\varphi; z, z', \mathbf{k} | \, s) \ = \ \delta(z - z') \; .$$ Comparing Eq.  with Eq. , we see that $\mathcal{K}(\varphi; z, z', \mathbf{k} | \, s)$ is nothing other than the Green’s function $G(u, u', m)$ in Eq.  with the replacement $k^2 \to k^2 + s$ in $m$, see Eq. . Thus, we may write $$\begin{aligned} \label{eq:B1explicit} B^{(1)} \ & = \ - \, \frac{1}{2} \! \int_{0}^{\Lambda} \! {\mathrm{d}}k \; k^2 \! \int_{0}^{\infty} \! \frac{{\mathrm{d}}\tau}{\tau} \int_{0}^{\infty} \! \! {\mathrm{d}}r \; r^3 \, \mathcal{L}^{-1}_s[\widetilde{G}(u, m)](\tau) \; ,\end{aligned}$$ where we have defined $$\label{eq:Gtilde} \widetilde{G}(u, m) \ = \ G(u, m) - G(1, m)$$ and $$\mathcal{L}^{-1}_s[\widetilde{G}(u, m)](\tau) \ = \ \int_{\mathcal{C}} \! \frac{{\mathrm{d}}s}{2\pi i} \; e^{-s\tau} \, \widetilde{G}(u, m)$$ is the inverse Laplace transform with respect to $s$, with $\mathcal{C}$ indicating the Bromwich contour. We may perform the integrals in Eq.  analytically, proceeding in order from right to left and beginning with the inverse Laplace transform. We then obtain the unrenormalized correction to the bounce action $$\begin{aligned} B^{(1)} \ = \ - \, B \, \bigg( \frac{3 \lambda}{16 \pi^2} \bigg) \bigg( \frac{\pi}{3 \sqrt{3}} + \frac{\Lambda^2}{\gamma^2} + \ln \frac{\gamma^2}{\Lambda^2} \bigg) \; .\end{aligned}$$ The technical details of the relevant integrations are included in Appendix \[app:greens\]. Adding the counterterm $$\begin{aligned} \label{eq:deltaB} \delta B^{(1)} \ & = \ \int \! {\mathrm{d}}^4 x \; \bigg( \frac{1}{2!} \delta m^2 \big( \varphi^2 - v^2 \big) + \frac{1}{4!} \delta \lambda \big( \varphi^4 - v^4 \big) \bigg) \nonumber \\ & = \ B \, \bigg( \frac{3 \lambda}{16 \pi^2} \bigg) \bigg( \frac{\Lambda^2}{\gamma^2} + \ln \frac{\gamma^2}{\Lambda^2} - 21 \bigg) \; ,\end{aligned}$$ we obtain the final renormalized result $$\begin{aligned} \label{eq:B1final} B^{(1)} \ = \ - \, B \, \bigg( \frac{3 \lambda}{16 \pi^2} \bigg) \bigg( \frac{\pi}{3 \sqrt{3}} + 21 \bigg) \; .\end{aligned}$$ In Appendix \[app:greens\], we reproduce this result by the method presented in Ref. [@Konoplich:1987yd].[^3] Radiative corrections to the bounce {#sec:corrb} =================================== We now discuss an example of the role played by loop corrections to the bounce itself. Within the perturbation expansion, one should expect that these lead to second-order corrections to the classical action of the soliton simply because the latter is evaluated for a stationary path. There are, however, important situations, in which all one-loop contributions must be resummed in order to capture the leading quantum corrections to the action. Examples include situations where the symmetry-breaking minima of the potential emerge radiatively through the CW mechanism [@Coleman:1973jx]. In the absence of a soliton, this implies that the classical solution, i.e. the homogeneous expectation value of the field, has to be found consistently by minimizing the one-loop effective potential as a [*function*]{} of the field expectation value itself. Analogously, in order to find the decay rate of the false vacuum, the bounce must be computed consistently from the one-loop effective action, which is a [*functional*]{} of the bounce itself. The methods presented in this article reduce the problem of tunneling in radiatively-generated potentials to one-dimensional ordinary differential equations and integrals. It is anticipated that it should be possible to derive numerical solutions in future work. For the purpose of illustration, however, we remain herein on the ground of analytic and perturbative approximations. In order to enhance the corrections to the bounce compared to other quantum effects that appear at second order in perturbation theory, we extend the model in Eq.  with $N$ copies of an additional scalar field $\chi$ by adding to the Lagrangian the terms $$\begin{aligned} {\cal L}\chi \ = \ \sum \limits_{i = 1}^N \left\{ \frac{1}{2!} (\partial_{\mu} \chi_i)^2 + \frac{1}{2!} m_\chi^2 \chi_i^2 + \frac{\lambda}{4} \Phi^2 \chi_i^2 \right\} \, .\end{aligned}$$ Here, we have chosen the coupling $\lambda$ to be identical to the self-coupling of $\Phi$ for the sake of simplicity in the Green’s function of the $\chi$ fields. Since $\braket{\chi_i} = 0$, the additional scalars do not impact upon the classical bounce in Sec. \[sec:classb\] or the discussion of the Green’s function in Sec. \[sec:greens\]. The Klein-Gordon equation for $\chi_i$ takes the form $$\bigg[ - \partial^2 + m_{\chi}^2 + \frac{\lambda}{2} \varphi^2 \bigg] S(\varphi; x, x') \ = \ \delta^{(4)}(x - x') \; .$$ Comparing with that of $\Phi$ in Eq. , we see that the Green’s function $S(u, u', m)$ may be obtained straightforwardly from $G(u, u', m)$ in Eq.  by making the replacement $$m \to \sqrt{6} \, \bigg( 1 + \frac{k^2 + m_{\chi}^2} {6 \gamma^2} \bigg)^{\! \tfrac{1}{2}} \; .$$ The renormalized tadpole contribution from each $\chi_i$ field, integrated over the three-momentum $\mathbf{k}$, is given by $$\label{eq:SR} \Sigma^{R}(u) \ = \ \frac{\lambda \gamma^2}{8 \pi^2} \, \frac{\gamma^2}{m_{\chi}^2} \, \big[ 72 + \big( 1 - u^2 \big) \big( 40 - 3 u^2 \big) \big] \; ,$$ where we have assumed $m_\chi^2 \gg \gamma^2$ for simplicity. The full form of $S(u)$ and the relevant counterterms are provided in Appendix \[app:Scalc\]. The renormalized correction to the classical bounce $\delta \varphi$ is governed by the equation of motion $$\begin{aligned} \bigg[ \frac{{\mathrm{d}}^2}{{\mathrm{d}}r^2} + \mu^2 - \frac{\lambda}{2} \varphi^2 \bigg] \delta \varphi \ = \ \Big( \Pi^R(u) + N \Sigma^R(u) \Big) \, \varphi \; ,\end{aligned}$$ cf. Eq. . We obtain the solution by making use of the Green’s function $G(u, u', 2) \equiv G(u, u', m)|_{k \, = \, 0}$, writing $$\begin{aligned} \label{sol:corr:Green} \delta \varphi(u) & = - \, \frac{v}{\gamma} \! \int_{-1}^1 \! {\mathrm{d}}u' \, \frac{u' G(u, u', 2)}{1 - u'^2} \, \Big( \Pi^R(u') + N \Sigma^R(u') \Big) \, ,\end{aligned}$$ where we have used Eq.  in order to substitute $\varphi$. ![The correction to the bounce $\delta \varphi$ as a function of $\gamma(r-R)$ for $N\gamma^2/m_{\chi}^2=0$ (solid), $0.5$ (dashed), $1$ (dash-dotted) and $1.5$ (dotted).\[fig:deltaphi\]](deltaphi.pdf) ![The corrected bounce $\varphi + \delta \varphi$ as a function of $\gamma(r-R)$ for $N\gamma^2/m_{\chi}^2=0$ (solid), $0.5$ (dashed), $1$ (dash-dotted) and $1.5$ (dotted). We see clearly that the impact of the tadpole correction is to broaden the bubble wall.\[fig:correctedbounce\]](phiplusdeltaphi.pdf "fig:")\ ![The corrected bounce $\varphi + \delta \varphi$ as a function of $\gamma(r-R)$ for $N\gamma^2/m_{\chi}^2=0$ (solid), $0.5$ (dashed), $1$ (dash-dotted) and $1.5$ (dotted). We see clearly that the impact of the tadpole correction is to broaden the bubble wall.\[fig:correctedbounce\]](phiplusdeltaphi2.pdf "fig:") We note at this point that $G(u, u', m)$ is singular as $k \to 0$ (or, equivalently, $m \to 2$). Nonetheless, the integral in Eq.  remains finite, since $G(u, u', m)$ is multiplied with an odd function, whereas the singularity resides in its even part. It is therefore useful to define $$G^{\rm odd}(u, u') \ \equiv \ \frac{1}{2} \Big(G(u, u', 2) - G(u, -\,u', 2) \Big) \; .$$ Within the domain $0 \leq u, u' \leq 1$, this function can be expressed as $$\begin{aligned} & G^{\rm odd}(u,u') \ = \ \vartheta(u - u') \, \frac{1}{32 \gamma} \, \frac{1 - u^2}{1 - u'^2} \bigg[ 2u' \big( 5 - 3 u'^2 \big) \nonumber \\ & \qquad + 3 \big( 1 - u'^2 \big)^2 \ln \frac{1 + u'}{1 - u'} \bigg] + ( u \leftrightarrow u') \; .\end{aligned}$$ Defining in addition $$\begin{aligned} p_0(u) \ & = \ \gamma \! \int_{-1}^{1} \! {\mathrm{d}}u' \; \frac{u'}{1 - u'^2} \, G^{\rm odd}(u, u') \nonumber \\ & = \ \frac{1 - u^2}{8} \bigg[ \frac{2u}{1 - u^2} + \ln \frac{1 + u}{1 - u} \bigg] \; , \\ p_1(u) \ & = \ \gamma \! \int_{-1}^{1} \! {\mathrm{d}}u' \; u' G^{\rm odd}(u,u') \nonumber \\ & = \ \frac{1 - u^2}{8} \ln \frac{1 + u}{1 - u} \; , \\ p_2(u) \ & = \ \gamma \! \int_{-1}^1 \! {\mathrm{d}}u' \; u'^3 G^{\rm odd}(u, u') \nonumber \\ & = \ \frac{1 - u^2}{8} \bigg[ \ln \frac{1 + u}{1 - u} - \frac{4}{3} \, u \bigg] \; , \end{aligned}$$ we find the result $$\begin{aligned} \label{bounce:correction} & \delta \varphi(u) \ = \ - \, \frac{3 \lambda v}{16 \pi^2} \bigg[ 6 \bigg( \frac{8 \gamma^2}{m_{\chi}^2} \, N + 1 \bigg) p_0(u) \nonumber \\ & \qquad + 5 \bigg( \frac{16 \gamma^2}{3 m_{\chi}^2} \, N +1 \bigg) p_1(u) - \bigg( \frac{2 \gamma^2}{m_{\chi}^2} \, N + \frac{\pi}{\sqrt{3}} \bigg) p_2(u) \bigg] \; .\end{aligned}$$ In Fig. \[fig:deltaphi\], we plot $\delta \varphi$ as a function of $\gamma(r-R)$ for a range of values of $N\gamma^2/m_{\chi}^2$. We see from Fig. \[fig:correctedbounce\], which plots the corrected bounce $\varphi+\delta\varphi$ for the same range, that the impact of this correction is to lower the height and broaden the width of the bubble wall. We note that this behaviour is in qualitative agreement with the results of the self-consistent numerical analysis in Ref. [@Bergner:2003au], presented there for the pure $\lambda\Phi^4$ theory in $1+1$ dimensions. ![\[fig:feyn:radcor\]Diagrammatic representation of various contributions to the effective action: (A) is the one-loop term $B_\chi^{(1)}$, (B) is the ${\cal O}(\lambda^2 N^2)$ contribution to $B^{(2)}$, and (C) and (D) are ${\cal O}(\lambda^2 N)$ terms. Solid lines represent the propagator $G(\varphi;x,x^\prime)$, dotted lines $S(\varphi;x,x^\prime)$. Crosses denote insertions of the bounce $\varphi$.](B2.pdf "fig:"){width="7.5cm"}\ .4cm ![\[fig:feyn:radcor\]Diagrammatic representation of various contributions to the effective action: (A) is the one-loop term $B_\chi^{(1)}$, (B) is the ${\cal O}(\lambda^2 N^2)$ contribution to $B^{(2)}$, and (C) and (D) are ${\cal O}(\lambda^2 N)$ terms. Solid lines represent the propagator $G(\varphi;x,x^\prime)$, dotted lines $S(\varphi;x,x^\prime)$. Crosses denote insertions of the bounce $\varphi$.](sunsetdaisies2.pdf "fig:"){width="4.8cm"} Substituting Eq.  into Eq. , we find the correction to the bounce action $$\begin{aligned} \label{delta:B:tree} B^{(2)}+B^{(2)}{}' \ & = \ \frac{1}{2} \! \int \! {\mathrm{d}}^4 x \; \varphi(u) \, \Big( \Pi^R(u) + N \Sigma^R(u) \Big) \, \delta \varphi(u) \nonumber \\ & = \ -\,\frac{B}{3} \bigg( \frac{3 \lambda}{16 \pi^2} \bigg)^{\! \! 2} \bigg[ \frac{291}{8} - \frac{37}{4} \, \frac{\pi}{\sqrt{3}} + \frac{5}{56} \, \frac{\pi^2}{3} \nonumber \\ & \qquad \qquad + \bigg( \frac{667}{2} - \frac{2897}{42} \, \frac{\pi}{\sqrt{3}} \bigg) \frac{\gamma^2}{m_{\chi}^2} \, N \nonumber \\ & \qquad \qquad + \frac{5829}{14} \, \frac{\gamma^4}{m_{\chi}^4} \, N^2 \bigg] \; .\end{aligned}$$ In order to obtain a finite result for Eq. , we have added to $U_0$ the correction $$\delta U_0 \ = \ \frac{9}{4} \, \bigg( \frac{3 \lambda}{16 \pi^2} \bigg)^{\! \! 2} \, \gamma^2 v^2 \, \bigg(\frac{8 \gamma^2}{m_{\chi}^2} N + 1 \bigg)^{\! \! 2} \; ,$$ ensuring that the potential continues to vanish in the false vacuum. The corrections appearing in Eq.  should be compared to the renormalized logarithm of the determinants of the Klein-Gordon operators of the $\chi$ fields in the background given by $\varphi$, which are given by $$\begin{aligned} \label{chi:oneloop} B^{(1)}_\chi = - \, B \, \bigg( \frac{3 \lambda}{16 \pi^2} \bigg) \frac{2542}{15} \, \bigg[ \frac{\gamma^2}{m_{\chi}^2} + \mathcal{O} \big( \tfrac{\gamma^4}{m_{\chi}^4} \big) \bigg] N \; .\end{aligned}$$ In comparison, the leading term in Eq.  is suppressed by a factor $\sim \lambda \mu^2 / m_\chi^2 / (16 \pi^2)$. The one-loop corrections $B^{(1)}$ and $B^{(1)}_{\chi}$ are both negative, thereby increasing the tunneling rate. It is interesting to note that, although the contribution $B^{(2)}$ to the tunneling action from the corrections to the bounce itself is positive, the net contribution of $B^{(2)}+B^{(2)}{}'$ is still negative, again increasing the tunneling rate. In Fig. \[fig:feyn:radcor\], we present a diagrammatic representation of the corrections to the bounce action. It is also useful in order to see that there appear no contributions of ${\cal O}(\lambda^2 N^2)$ relative to the bounce action $B$ in addition to those from $B^{(2)}$. In order to avoid proliferation, we only show the leading contributions in $1/N$ for a given type of diagram. At one-loop order, there is the vacuum bubble in terms of the propagator $S$ of the $\chi$ fields, Fig. \[fig:feyn:radcor\](A), which gives the contribution ${\cal O}(\lambda N)$ relative to $B$ from $B^{(1)}_\chi$ in Eq. . On substituting $\delta\varphi$ in the form of Eq.  into the action \[Eq. \], we see that the diagram corresponding to the ${\cal O}(\lambda^2 N^2)$ term in $B^{(2)}/B$ is given by Fig. \[fig:feyn:radcor\](B), where, when counting the powers of $\lambda$, one should note that each explicit insertion of $\varphi$ contributes a factor of $1/\sqrt\lambda$. Finally, at two-loop order, there are the diagrams Figs. \[fig:feyn:radcor\](C) and (D), which we do not compute but yield contributions of ${\cal O}(\lambda^2 N)$ relative to $B$. These contributions are therefore suppressed by a relative factor of $1/N$ relative to the $\mathcal{O}(\lambda^2N^2)$ in $B^{(2)}/B$, as is familiar from the standard approximation scheme known as the $1/N$ expansion [@'tHooft:1973jz]. We should remark that these arguments do not hold, of course, for the contribution to $B^{(2)}$ from the $\Phi$ tadpole, which we include here for completeness. The latter is formally the same order as other two-loop diagrams, involving only $\Phi$, that are not captured in the 1PI approximation employed here. This observation is true also of the Hartree approximation for the pure $\lambda\Phi^4$ theory analyzed numerically in Refs. [@Bergner:2003au; @Bergner:2003id; @Baacke:2004xk; @Baacke:2006kv]. Nevertheless, these additional two-loop diagrams remain subdominant compared to the $\mathcal{O}(\lambda^2 N)$ and $\mathcal{O}(\lambda^2N^2)$ contributions from the $\chi$ tadpole in Eq. . Finally, we note that approximating $\delta\varphi$ as a small perturbation to $\varphi$ , using Eq. , requires for consistency that $6 N \lambda \gamma^2/(m_\chi^2 \pi^2) \ll 1$, such that within the range of validity of present approximations, we cannot obtain $|B^{(2)}+B^{(2)}{}'|>|B^{(1)}|$. Nevertheless for large $N$, $B^{(2)}+B^{(2)}{}'$ can be the dominant two-loop contribution to the effective action. Conclusions {#sec:conc} =========== Within the context of $\lambda \Phi^4$ theory, we have described a Green’s function method for handling radiative effects on false vacuum decay. By this means and employing the thin- and planar-wall approximations, we have been able to calculate analytically and in a straightforward manner both the functional determinant of the quadratic fluctuations about the classical soliton configuration and the first correction to the configuration itself. This Green’s function method is well suited to numerical evaluation and, as a consequence, should be applicable to potentials of more general form. As such, we anticipate that it may be of particular use when the non-degeneracy of minima is purely radiatively generated. Examples of the latter include the spontaneous symmetry breaking of the massless CW model [@Coleman:1973jx] or the instability of the electroweak vacuum. Other applications might include the calculation of corrections to inflationary potentials in the time-dependent inflaton background, for instance in inflection-point or $A$-term inflation [@Allahverdi:2006iq; @Lyth:2006ec; @Bueno; @Sanchez:2006xk; @Allahverdi:2006we], which exploit the flat directions and saddle points of the MSSM potential. Furthermore, the use of Green’s functions naturally admits the introduction of finite-temperature effects or extension to non-trivial background spacetimes. Green’s functions have proved to be central objects within perturbative calculations throughout quantum field theory and it is therefore unsurprising that we find these suitable to treat solitons in $\lambda \Phi^4$ theory as well. We take this as an encouragement that further theoretically- and phenomenologically-interesting systematic results on false vacuum decay may be within reach. The authors would like to thank Jürgen Baacke, Daniel Litim and Holger Gies for helpful correspondence and discussions. The work of P.M. is supported by a University Foundation Fellowship (TUFF) from the Technische Universität München. The work of B.G. is supported by the Gottfried Wilhelm Leibniz Programme of the Deutsche Forschungsgemeinschaft (DFG). Both authors acknowledge support from the DFG cluster of excellence Origin and Structure of the Universe. Zero-mode functional measure {#app:zeromode} ============================ In order to perform the functional integration over the zero modes, we insert four copies of unity in Faddeev-Popov form [@Gervais:1974dc]: $$1 \ = \ \int \! {\mathrm{d}}y_{\mu} \; | \partial^{(y)}_{\mu} \! f(y_{\mu}) | \, \delta \big( f(y_{\mu}) \big) \; .$$ Here, $\mu$ is not summed over and $$f(y_{\mu}) \ = \ \int \! {\mathrm{d}}^4 x \; \Phi(x - y) \partial^{(x)}_{\mu} \varphi(x - y) \ = \ B^{1/2} a_{\mu} \; ,$$ where we recall that $$\Phi \ = \ \varphi + \sum_{i = 0}^4 a_i \phi_i + \phi_+ \; .$$ It follows that $$\partial_{\mu}^{(y)} \! f(y_{\mu}) \ = \ - \! \int \! {\mathrm{d}}^4 x \; \big( \partial^{(x)}_{\mu} \varphi(x - y) \big)^2 \ = \ -\,B \; ,$$ ignoring terms that are formally $\mathcal{O}(\hbar^{1/2})$. Thus, $$1 \ = \ B \! \int \! {\mathrm{d}}y_{\mu} \; \delta(B^{1/2} a_{\mu}) \ = \ B^{1/2} \! \int \! {\mathrm{d}}y_{\mu} \; \delta(a_{\mu}) \; .$$ We then have $$\begin{aligned} \int \prod_{\mu = 1}^4 (2 \pi \hbar)^{-1/2} {\mathrm{d}}a_{\mu} \ & = \ \bigg( \frac{B}{2 \pi \hbar} \bigg)^{\! 2} \! \int \! {\mathrm{d}}^4 y \prod_{\mu = 1}^4 \int \mathrm{d} a_{\mu} \; \delta(a_{\mu}) \nonumber \\ & = \ VT \bigg( \frac{B}{2 \pi \hbar} \bigg)^{\! 2} \; .\end{aligned}$$ Green’s function {#app:greens} ================ In this appendix, we include the technical details of the calculations outlined in Secs. \[sec:greens\] and \[sec:corrb\]. All functional identities used in what follows may be found in Ref. [@Abramowitz]. Expansion in hyperspherical harmonics ------------------------------------- In $d$ dimensions, the Green’s function satisfies the inhomogeneous Klein-Gordon equation $$\big( - \Delta^{(d)} + U''(\varphi) \big) G^{(d)}(\varphi; x, x') \ = \ \delta^{(d)}(x - x') \; ,$$ where $\delta^{(d)}(x - x')$ is the Dirac delta function and $\Delta^{(d)}$ is the Laplacian. Given the $O(d)$ invariance of the bounce $\varphi$, it is convenient to work in hyperspherical coordinates, in which case the Laplacian takes the form $$\Delta^{(d)} \ = \ r^{1 - d} \partial_r r^{d - 1} \partial_r + \Delta_{S^{d - 1}} \; ,$$ where $\Delta_{S^{d - 1}}$ is the Laplace-Beltrami operator on the $d - 1$ sphere. We proceed by performing a partial-wave decomposition of the Green’s function: $$G^{(d)}(\varphi; x, x') \\ = \ \sum_{j \{\ell\}} G_j(\varphi; r, r') Y^*_{j\{\ell\}}(\mathbf{e}_{r'}) Y_{j\{\ell\}}(\mathbf{e}_{r}) \; ,$$ where $x = r \mathbf{e}_r$, $x' = r' \mathbf{e}_{r'}$ and $Y_{j \{\ell\}}(\mathbf{e}_r)$ are the hyperspherical harmonics (see e.g. Ref. [@Avery]), satisfying the eigenvalue equation $$\Delta_{S^{d-1}}Y_{j\{\ell\}}\ =\ -\,j(j+d-2)Y_{j\{\ell\}}\;,$$ with $\{\ell\} = \ell_1, \; \ell_2, \; \dots, \; \ell_{d - 2}$. The hyperradial function $G_j(\varphi; r, r')$ satisfies $$\begin{aligned} \label{eq:dradialdif} & \bigg[ -\, r^{1 - d} \, \frac{{\mathrm{d}}}{{\mathrm{d}}r} \, r^{d - 1} \, \frac{{\mathrm{d}}}{{\mathrm{d}}r} + \frac{j(j + d - 2)}{r^2} + U''(\varphi) \bigg] G_j(\varphi; r, r') \nonumber \\ & \qquad = \ r'^{1 - d} \delta(r - r') \; .\end{aligned}$$ Since, for each $j$, the $\{\ell\}$ modes are degenerate, we may use the sum rule [@Avery] $$\begin{aligned} \label{eq:sumrule} & \sum_{\{\ell\}} Y^*_{j \{\ell\}}(\mathbf{e}_{r'}) Y_{j \{\ell\}}(\mathbf{e}_{r}) \\ \nonumber & \ = \ \frac{2}{(4 \pi)^{\kappa + \tfrac{1}{2}}} \, \frac{j + \kappa}{j + 2\kappa} \big( j + \kappa + \tfrac{1}{2} \big)_{\kappa + \tfrac{1}{2}} P_j^{\big(\kappa - \tfrac{1}{2}, \kappa - \tfrac{1}{2}\big)}(\cos \theta) \; ,\end{aligned}$$ where $\kappa = d/2 - 1$, $\cos \theta = \mathbf{e}_r \cdot \mathbf{e}_{r'}$, $$\label{eq:Pochhammer} (z)_n\ =\ \frac{\Gamma(z+n)}{\Gamma(z)}$$ is the Pochhammer symbol and the $P_j^{(\alpha,\beta)}(z)$ are the Jacobi polynomials. For $d = 1$, $\kappa = - \, 1/2$, $\cos \theta \in \{-1, +1\}$, and we have $$\begin{gathered} P_{j}^{(-1, -1)}(+1) \ = \ 0 \; , \\ P_{j}^{(-1, -1)}(-1) \ = \ \frac{\sin \pi j}{\pi j} \ = \ \begin{cases} 1 \; , \quad j = 0 \\ 0 \; , \quad j \neq 0 \end{cases} \; .\end{gathered}$$ Hence, $G^{(1)}(\varphi; x, x') = G_0(\varphi; r, r')$, as we would expect. For $d = 2$, $\kappa = 0$, and we have $$P_j^{\big(-\tfrac{1}{2}, -\tfrac{1}{2}\big)}(z) \ = \ \frac{T_j(z)}{\sqrt{\pi} \big( j + \tfrac{1}{2} \big)_{\tfrac{1}{2}}} \; ,$$ where $T_j(z)$ is the Chebyshev polynomial of the first kind. We then obtain $$G^{(2)}(\varphi; x, x') \ = \ \frac{1}{\pi} \sum_{j = 0}^{\infty} \cos j \theta \, G_{j}(\varphi; r, r') \; ,$$ where we have used the trigonometric form $T_j(\cos \theta) = \cos j \theta$. For $d = 3$, $\kappa = 1/2$, and $$P_j^{(0, 0)}(z) \ = \ P_j(z) \; ,$$ where $P_j(z)$ are the Legendre polynomials. Thus, we obtain the familiar three-dimensional expansion $$G^{(3)}(\varphi; x, x') \ = \ \frac{1}{4\pi} \sum_{j = 0}^{\infty} (2 j + 1) P_j(\cos \theta) G_j(\varphi; r, r') \; .$$ Finally, for $d = 4$, $\kappa = 1$, and $$P_j^{\big(\tfrac{1}{2}, \tfrac{1}{2}\big)}(z) \ = \ \frac{2}{\sqrt{\pi}} \, \frac{j + 2}{\big( j + \tfrac{3}{2} \big)_{\tfrac{3}{2}} } \, U_j(z) \; ,$$ where $U_j(z)$ are the Chebyshev polynomials of the second kind. Hence, we find $$G^{(4)}(\varphi; x, x') \ = \ \frac{1}{2 \pi^2} \sum_{j = 0}^{\infty} (j + 1) U_j(\cos \theta) G_j(\varphi; r, r') \; ,$$ as appearing in Eq. . Continuum approximation ----------------------- In the coincident limit $x = x'$, $\cos \theta = 1$, and we have $$\begin{gathered} T_j(1) \ = \ 1 \; , \quad P_j(1) \ = \ 1 \; , \quad U_j(1) \ = \ j + 1 \; .\end{gathered}$$ Alternatively, in $d$ dimensions, we may use $$P_j^{(\alpha, \beta)}(1) \ = \ \frac{(\alpha + 1)_j}{\Gamma(j + 1)} \; ,$$ in Eq. , giving $$\begin{aligned} & G^{(d)}(\varphi; x, x') \ = \ \frac{2(4 \pi)^{-\tfrac{d - 1}{2}}}{\Gamma\big(\tfrac{d - 1}{2}\big)} \nonumber \\ & \quad \times \sum_{j = 0}^{\infty}(j + d/2 - 1)\, \frac{\Gamma(j + d - 2)}{\Gamma(j + 1)}\, G_j(\varphi; r, r') \; .\end{aligned}$$ Completing the square in the centrifugal potential in Eq. , we make the following approximation for large $R$: $$\frac{j(j + d - 2)}{R^2} \ = \ \frac{(j + \kappa)^2}{R^2} - \frac{\lambda^2}{4 R^2} \ \approx \ \frac{(j + \kappa)^2}{R^2} \; ,$$ where $\kappa=d/2-1$, as before. We may then promote $(j + \kappa) / R$ to a continuous variable $k$, obtaining $$\begin{aligned} G^{(2)}(\varphi; x, x) \ & = \ \frac{1}{\pi} \int_0^{\infty} \! {\mathrm{d}}k \; G(u, m) \; , \\ G^{(3)}(\varphi; x, x)\ & = \ \frac{1}{2 \pi} \int_0^{\infty} \! {\mathrm{d}}k \; k \, G(u, m) \; , \\ G^{(4)}(\varphi; x, x)\ & = \ \frac{1}{2 \pi^2} \int_0^{\infty} \! {\mathrm{d}}k \; k^2 \, G(u, m) \; , \\ \vdots \nonumber \\ \label{eq:continuousind} G^{(d)}(\varphi; x, x) \ & = \ \frac{2(4 \pi)^{- \tfrac{d - 1}{2}}} {\Gamma \big( \tfrac{d - 1}{2} \big)} \! \int_0^{\infty} \! {\mathrm{d}}k \; k^{d - 2} \, G(u, m) \; , \end{aligned}$$ where we have used the general notation employed in Sec. \[sec:greens\], see Eq. , with $m$ given by Eq. . We note that for $d > 4$, we have dropped terms $\mathcal{O}(k / R)$ and higher within the integrand. Radial function --------------- For large $R$, we neglect the damping term in the radial equation \[Eq. \] and set $r = R$ in the centrifugal potential and discontinuity, giving $$\label{eq:dradialsimple} \bigg[ - \, \frac{{\mathrm{d}}^2}{{\mathrm{d}}r^2} + \frac{j(j + d - 2)}{R^2} + U''(\varphi) \bigg] G_j(\varphi; r, r') \ = \ \frac{\delta(r - r')}{R^{d - 1}} \; .$$ Since the solution depends only on the normalized bounce $u = \tanh[\gamma (r - R)]$, it is convenient to define $$G(u, u', m) \ \equiv \ R^{d - 1} \, G_j(\varphi; r, r') \; ,$$ cf. Sec. \[sec:greens\]. Equation  may then be recast in the form $$\begin{aligned} & \bigg[ \frac{{\mathrm{d}}}{{\mathrm{d}}u} \big( 1 - u^2) \frac{{\mathrm{d}}}{{\mathrm{d}}u} - \frac{m^2}{1 - u^2} + n(n + 1) \bigg] G(u, u', m) \nonumber \\ & \qquad \qquad = \ - \, \gamma^{-1} \delta(u - u') \; ,\end{aligned}$$ where $$n \ = \ 2 \; , \qquad m \ = \ 2 \, \bigg( \! 1 + \frac{j(j + d - 2)} {4 \gamma^2R^2} \bigg)^{\!\!\tfrac{1}{2}} \; .$$ Splitting around the discontinuity at $u = u'$, we decompose $$\begin{aligned} G(u, u', m) \ & = \ \vartheta(u - u') G^>(u, u', m) \nonumber \\ & \qquad \qquad + \vartheta(u' - u) G^<(u, u', m) \; ,\end{aligned}$$ where $G^{\gtrless}(u, u', m)$ are the solutions to the homogeneous equation $$\bigg[ \frac{{\mathrm{d}}}{{\mathrm{d}}u} \big(1 - u^2) \frac{{\mathrm{d}}}{{\mathrm{d}}u} - \frac{m^2}{1 - u^2} + n(n + 1) \bigg] G^{\gtrless}(u, u', m) \ = \ 0 \; .$$ The latter is the associated Legendre differential equation and we obtain the general solutions $$G^{\gtrless}(u, u', m) \ = \ A^{\gtrless} P_2^m(u) + B^{\gtrless} Q_2^m(u) \; ,$$ where $P_n^m(z)$ and $Q_n^m(z)$ are the associated Legendre functions of the first and second kind, respectively. Matching around the delta function in the inhomogeneous equation, we require $$\begin{aligned} & \big( A^> - A^< \big) P_2^m(u') + \big( B^> - B^< \big) Q_2^m(u') \, = \, 0 \; , \\ & \big( A^> - A^< \big) \frac{{\mathrm{d}}}{{\mathrm{d}}u'} P_2^m(u') \nonumber \\ & \quad + \big( B^> - B^< \big) \frac{{\mathrm{d}}}{{\mathrm{d}}u'} Q_2^m(u') = \, - \, \frac{1}{\gamma \big( 1 - u'^2 \big)} \; . \end{aligned}$$ Thus, we find $$\begin{aligned} A^> - A^< \ = \ \frac{1}{\gamma \big( 1 - u'^2 \big)} \, \frac{Q_2^m(u')}{W[P_2^m(u'), Q_2^m(u')]} \; , \\ B^> - B^< \ = \ \frac{1}{\gamma \big( 1 - u'^2 \big)} \, \frac{P_2^m(u')}{W[Q_2^m(u'), P_2^m(u')]} \; , \end{aligned}$$ where $W[P_n^m(z),Q_n^m(z)]$ is the Wronskian, having the explicit form $$W[P_n^m(z), Q_n^m(z)] \ = \ \frac{(n - m + 1)_{2m}}{1 - u'^2} \; ,$$ with the Pochhammer symbol defined in Eq. . We also require the boundary condition that $G(u, u', m)$ go to zero as $u \to \pm \, 1$, giving $$\frac{A_>}{B_>} \ = \ - \, \frac{\pi}{2} \cot m \pi \; , \qquad B_< \ = \ 0 \; .$$ We may now solve for the remaining non-zero coefficients and obtain $$G^>(u, u', m) \ = \ \frac{\pi}{2\gamma} \, \frac{1}{\sin m \pi} \, P_2^{-m}(u) P_2^m(u') \; ,$$ with $G^<(u, u', m) = G^>(u', u, m)$. Here, we have used the identity $$\frac{\pi (n - m + 1)_{2m}}{2\sin m \pi} \, \, P_n^{-m}(z) \ = \ \frac{\pi}{2} \cot m \pi \, P_n^m(z) - Q_n^m(z) \; .$$ Finally, we employ the representation $$P_n^m(z) \ = \ \bigg( \frac{z + 1}{z - 1} \bigg)^{\! \! \tfrac{m}{2}} \! (n - m + 1)_m \, P_n^{(-m, m)}(z)$$ of the associated Legendre function of the first kind in terms of the Jacobi polynomials. For $n = 2$, the polynomial expansion of the latter terminates, and we have $$\begin{aligned} \label{eq:polynomial} & P_2^{(\pm m, \mp m)}(z) \ = \ \frac{1}{2} \Big[(1 \pm m)(2 \pm m) \nonumber \\ & \qquad \qquad - 3(2 \pm m)(1 - u) + 3(1 - u)^2 \Big] \; .\end{aligned}$$ After some algebraic simplification, we then arrive at the final analytic solution, as presented in Eq.  of Sec. \[sec:greens\]. Functional determinant ---------------------- The normalized heat kernel $\widetilde{\mathcal{K}}(\varphi; z, z', \mathbf{k} | \tau)$, see Sec. \[sec:greens\], is given in terms of the inverse Laplace transform $$\widetilde{K}(\varphi; z, z', \mathbf{k} | \tau) \ = \ \mathcal{L}_s^{-1}[\widetilde{G}(u, m)](\tau) \; ,$$ where $$\widetilde{G}(u, m) \ = \ \frac{3}{2 \gamma m} \big( 1 - u^2 \big) \sum_{n = 1}^2 (-1)^n \frac{(n - 1 - u^2)}{m^2 - n^2} \; ,$$ with $$m \ = \ 2 \, \bigg( 1 + \frac{k^2 + s} {4 \gamma^2} \bigg)^{\! \! \tfrac{1}{2}} \; .$$ The inverse Laplace transform may be performed by using the shift, scaling and division properties $$\begin{gathered} \mathcal{L}_{s}^{-1}[F(s + b)](\tau) \ = \ e^{- b\tau} f(\tau) \;, \\ \mathcal{L}_{s}^{-1}[F(a s)](\tau) \ = \ \frac{1}{a} \, f(\tau / a) \; , \\ \mathcal{L}_{s}^{-1}[s^{-1} F(s)](\tau) \ = \ \int_{0}^{\tau} \! {\mathrm{d}}\tau' \; f(\tau') \; , \end{gathered}$$ where $f(\tau) = \mathcal{L}^{-1}_s[F(s)](\tau)$, as well as the elementary transformation $$\mathcal{L}_{s}^{-1}[s^{-z}](\tau) \, = \, \frac{\tau^{z - 1}}{\Gamma(z)} \; , \qquad \mathrm{Re} \, z \ > \ 0 \; .$$ We find $$\mathcal{L}_s^{-1}[m^{-1}(m^2 - n^2)^{-1}](\tau) = \frac{\gamma^2}{n} \, e^{[\gamma^2(n^2 - 4) - k^2]\tau} \mathrm{erf}(n\gamma\sqrt{\tau}) \; ,$$ where $$\mathrm{erf}(z) \ = \ \frac{2}{\sqrt{\pi}} \! \int_0^z \! {\mathrm{d}}t \; e^{-t^2}$$ is the error function. Hence, we have $$\begin{aligned} & \widetilde{K}(\varphi; z, z', \mathbf{k} | \tau) \ = \ - \, \frac{3}{2} \gamma \big( 1 - u^2 \big) e^{-k^2\tau} \nonumber \\ & \times \sum_{n = 1}^2 (-1)^n \bigg( \frac{1 + u^2}{n} - 1 \bigg) e^{\gamma^2(n^2 - 4)\tau} \mathrm{erf}(n \gamma \sqrt{\tau}) \; .\end{aligned}$$ Generalizing to $d$ dimensions, using the continuum limit in Eq. , the correction to the bounce action arising from the functional determinant is therefore $$\begin{aligned} & B^{(1)} \ = \ - \, \frac{3}{2} \frac{\Omega_d (4 \pi)^{- \tfrac{d - 1}{2}}} {\Gamma\big( \tfrac{d - 1}{2} \big)} \! \int_{0}^{\infty} \! {\mathrm{d}}k \; k^{d - 2} \int_{0}^{\infty} \! \frac{{\mathrm{d}}\tau}{\tau} \, e^{-k^2\tau} \nonumber \\ & \times \int_{0}^{\infty} \! {\mathrm{d}}r \; r^{d - 1} \, \gamma \big( 1 - u^2 \big) \nonumber \\ & \times \sum_{n = 1}^2 (-1)^n \bigg( \frac{1 + u^2}{n} - 1 \bigg) e^{\gamma^2(n^2 - 4)\tau} \mathrm{erf}(n \gamma \sqrt{\tau}) \; ,\end{aligned}$$ where $\Omega_d = 2 \pi^{d/2} / \Gamma(d / 2)$ is the solid angle subtended by the $(d - 1)$ dimensional hypersphere. The integral over $r$ is dominated by $r \sim R$, such that (for $n = 1, 2$) $$(-1)^n \! \int_{0}^{\infty} \! {\mathrm{d}}r \; r^{d - 1} \, \gamma \big( 1 - u^2 \big) \bigg( \frac{1 + u^2}{n} - 1 \bigg) \ \approx \ - \, \frac{2}{3} \, R^{d-1} \; .$$ We are then left with $$\begin{aligned} B^{(1)} \ & = \ \frac{\Omega_d (4 \pi)^{-\tfrac{d - 1}{2}}R^{d - 1}} {\Gamma\big(\tfrac{d - 1}{2}\big)} \! \int_0^{\infty} \! {\mathrm{d}}k \; k^{d - 2} \! \int_{0}^{\infty} \! \frac{{\mathrm{d}}\tau}{\tau} \; e^{-k^2\tau} \nonumber \\ & \qquad \times \sum_{n = 1}^2 e^{\gamma^2(n^2 - 4)\tau} \mathrm{erf}(n \gamma \sqrt{\tau})\;,\end{aligned}$$ cf. the form presented in Ref. [@Konoplich:1987yd]. We may now proceed in one of two ways: (i) performing the $\tau$ integration first, we must regularize the $k$ integral, for instance by introducing an ultraviolet cutoff $\Lambda$; or (ii) performing the $k$ integral first, we must instead regularize the $\tau$ integral. The latter is the approach presented in Ref. [@Konoplich:1987yd], which we reproduce in what follows for comparison. \(i) Performing the $\tau$ integral first gives $$\begin{aligned} B^{(1)} \ & = \ - \, \frac{2 \, \Omega_d (4 \pi)^{- \tfrac{d - 1}{2}} R^{d - 1}} {\Gamma \big( \tfrac{d - 1}{2} \big)} \int_{0}^{\Lambda} \! {\mathrm{d}}k \; k^{d - 2} \nonumber \\ & \times \sum_{n = 1}^2 \mathrm{arcsinh} \, \frac{n \gamma}{\sqrt{k^2 - \gamma^2(n^2 - 4)}} \; .\end{aligned}$$ Subsequently, performing the $k$ integral for $d = 4$, we obtain the result in Eq. . \(ii) Instead, performing instead the $k$ integral first, we obtain $$\begin{aligned} B^{(1)} \ & = \ \frac{1}{2} \, \Omega_d R^{d - 1} (4\pi)^{- \tfrac{d - 1}{2}} \! \int_{0}^{\infty} \! \! {\mathrm{d}}\tau \; \tau^{- \tfrac{d + 1}{2}} \nonumber \\ & \qquad \times \sum_{n = 1}^2 e^{\gamma^2(n^2 - 4)\tau} \mathrm{erf}(n \gamma \sqrt{\tau}) \; ,\end{aligned}$$ which is regularized by introducing a large mass $M$ as follows: $$\begin{aligned} B^{(1)} \ & = \ \frac{1}{2} \, \Omega_d R^{d - 1} (4\pi)^{- \tfrac{d - 1}{2}} \lim_{\epsilon \to 0} \frac{{\mathrm{d}}}{{\mathrm{d}}\epsilon} \frac{M^{2\epsilon}}{\Gamma(\epsilon)} \int_{0}^{\infty} \! \! {\mathrm{d}}\tau \nonumber \\ & \qquad \times \tau^{- \tfrac{d + 1}{2} + \epsilon} \sum_{n = 1}^2e^{\gamma^2(n^2 - 4)\tau} \mathrm{erf}(n \gamma \sqrt{\tau}) \; .\end{aligned}$$ We may proceed by using the series representation of the error function $$\mathrm{erf}(z) \ = \ \frac{2}{\sqrt{\pi}} \, e^{-z^2} \sum_{\ell = 0}^{\infty} \frac{2^{\ell}}{(2 \ell + 1)!!} \, z^{2\ell + 1} \; ,$$ where $!!$ denotes the double factorial. The $\tau$ integral may now be performed, and we obtain $$\begin{aligned} & B^{(1)} \ = \ (\gamma R)^{d - 1} \Omega_d \pi^{- \tfrac{d}{2}} \, \lim_{\epsilon \to 0}\frac{{\mathrm{d}}}{{\mathrm{d}}\epsilon} \frac{1}{\Gamma(\epsilon)} \bigg( \frac{M^2}{4 \gamma^2} \bigg)^{\! \! \epsilon} \nonumber \\ & \quad \times \sum_{n = 1}^2 \sum_{\ell = 0}^{\infty} \frac{2^{\ell} (n / 2)^{2 \ell + 1}}{(2 \ell + 1)!!} \, \Gamma(\epsilon + \ell + 1 - d / 2) \; .\end{aligned}$$ Considering the derivative with respect to $\epsilon$, we have $$\begin{aligned} & \frac{{\mathrm{d}}}{{\mathrm{d}}\epsilon} \, \frac{\Gamma(\epsilon + \ell + 1 - d / 2)}{\Gamma(\epsilon)} \bigg( \frac{M^2}{4 \gamma^2} \bigg)^{\! \! \epsilon} \nonumber \\ & \quad = \ \frac{\Gamma( \epsilon + \ell + 1 - d / 2)}{\Gamma(\epsilon)} \bigg(\frac{M^2}{4 \gamma^2}\bigg)^{\! \! \epsilon} \nonumber \\ & \quad \quad \times \bigg[ \ln \frac{M^2}{4 \gamma^2} - \psi(\epsilon) + \psi(\epsilon + \ell + 1 - d / 2) \bigg] \; ,\end{aligned}$$ where $\psi(z)$ is the digamma function. In order to take the limit $\epsilon \to 0$ safely, we must take note of the poles occurring in $\Gamma(z)$ and $\psi(z)$ for non-positive integers. Such poles occur in even dimensions for $\ell = 0,\, 1,\, \dots,\, d - 3$. After treating the limit $\epsilon \to 0$, we find for $d$ odd (including $d = 1$) $$\begin{aligned} B^{(1)} \ & = \ - \, (\gamma R)^{d - 1} \Omega_d \pi^{- \tfrac{d}{2}} \nonumber \\ & \qquad \times \sum_{n = 1}^2 \sum_{\ell = 0}^{\infty} \frac{2^{\ell}(n / 2)^{2 \ell + 1}}{(2 \ell + 1)!!} \, \Gamma(\ell + 1 - d / 2) \; .\end{aligned}$$ On the other hand, for $d$ even, we find $$\begin{aligned} & B^{(1)} \ = \ - \, (\gamma R)^{d - 1} \Omega_d \pi^{- \tfrac{d}{2}} \nonumber \\ & \times \sum_{n = 1}^2 \Bigg[ \sum_{\ell = d - 2}^{\infty} \frac{2^{\ell}(n / 2)^{2 \ell + 1}}{(2 \ell + 1)!!} \, \Gamma(\ell + 1 - d / 2) \nonumber \\ & + \sum_{\ell = 0}^{d - 3} \frac{2^{\ell}(n / 2)^{2 \ell + 1}} {(2 \ell + 1)!!} \, \frac{(-1)^{d / 2 - \ell - 1}}{(d / 2 - \ell - 1)!} \bigg(\ln \frac{M^2}{4 \gamma^2} + H_{d / 2 - \ell - 1} \bigg) \Bigg] \; ,\end{aligned}$$ where $$H_n \ = \ \sum_{k = 1}^n \frac{1}{k}$$ are the harmonic numbers, which we have supplemented with $H_0 \equiv 0$ for notational simplicity. For $d=4$, we then obtain $$\begin{aligned} & B^{(1)} \ = \ - 2 R^3 \gamma^3 \sum_{n = 1}^2 \Bigg[\sum_{\ell = 2}^{\infty} \frac{2^{\ell}(n / 2)^{2 \ell + 1}}{(2 \ell + 1)!!}(\ell - 2)! \nonumber \\ & + \sum_{\ell = 0}^1 \frac{2^{\ell}(n / 2)^{2 \ell + 1}}{(2 \ell + 1)!!} \, \frac{(-1)^{1 - \ell}}{(1 - \ell)!} \bigg(\ln \frac{M^2}{4 \gamma^2} + H_{1 - \ell} \bigg) \Bigg] \; .\end{aligned}$$ Lastly, performing the summations, we arrive at the result $$\label{eq:BK} B^{(1)} \ = \ - \, B \, \bigg( \frac{3 \lambda}{16 \pi^2} \bigg) \bigg( \frac{\pi}{3 \sqrt{3}} - 2 + \ln \frac{4 \gamma^2}{M^2} \bigg) \; .$$ Defining the counterterms in the proper-time representation, see Ref. [@Konoplich:1987yd], and fixing the renormalization conditions as in Eq. , we find the counterterms $$\begin{aligned} \delta m^2 \ & = \ \frac{\lambda \gamma^2}{16 \pi^2} \bigg( \ln \frac{4 \gamma^2}{M^2} + 29 \bigg) \; , \\ \delta \lambda \ & = \ - \, \frac{3 \lambda^2}{32 \pi^2} \bigg( \ln \frac{4 \gamma^2}{M^2} + 3 \bigg) \; , \end{aligned}$$ giving $$\delta B^{(1)} \ = \ B \, \bigg( \frac{3 \lambda}{16 \pi^2} \bigg) \, \bigg( \ln \frac{4 \gamma^2}{M^2} - 23 \bigg) \; .$$ Adding these to Eq. , we obtain agreement with Eq. . Renormalization of the $\bm{N}$-field model {#app:Scalc} =========================================== In this final appendix, we highlight the main technical details of the derivation of the Green’s function and corrections to the bounce from the $\chi$ fields. Proceeding as for the isolated $\varphi$ case, see Sec. \[sec:greens\], the renormalization is fixed using the CW effective potential [@Coleman:1973jx], evaluated in a homogeneous false vacuum. The renormalization conditions are then $$\begin{gathered} \frac{\partial^2 U_{\mathrm{eff}}}{\partial \varphi^2} \bigg|_{\varphi = v , \, \chi_i = 0} \ = \ 4 \gamma^2 \;, \\ \frac{\partial^2 U_{\mathrm{eff}}}{\partial \chi_i^2} \bigg|_{\varphi = v , \, \chi_i = 0} \ = \ 6 \gamma^2 + m_{\chi}^2 \; , \\ \frac{\partial^4 U_{\mathrm{eff}}}{\partial \varphi^4} \bigg|_{\varphi = v , \, \chi_i = 0} \ = \ \lambda \; , \\ \frac{\partial^4 U_{\mathrm{eff}}}{\partial \varphi^2 \partial \chi_i^2} \bigg|_{\varphi = v , \, \chi_i = 0} \ = \ \lambda \; , \\ \frac{\partial^4 U_{\mathrm{eff}}}{\partial \chi_i^2 \partial \chi_j^2} \bigg|_{\varphi = v , \, \chi_i = 0} \ = \ 0 \; , \end{gathered}$$ where the effective potential is $$\begin{aligned} \label{eq:CWU} & U_{\mathrm{eff}} \ = \ U(\varphi, \chi) + \delta U(\varphi, \chi) \nonumber \\ & \ + \frac{N-1}{4 \pi^2} \int_0^{\Lambda} \! {\mathrm{d}}k \; k^2 \bigg( \sqrt{k^2 + M_{\chi}^2} - k \bigg) \nonumber \\ & \ + \frac{1}{4 \pi^2} \int_0^{\Lambda} \! {\mathrm{d}}k \; k^2 \bigg(\sqrt{k^2+M_+^2}+\sqrt{k^2+M_-^2} - 2k \bigg)\;,\end{aligned}$$ with $$\begin{aligned} M_{\chi}^2\ &=\ m_{\chi}^2+\frac{\lambda}{2}\,\varphi^2\;,\\ M_{\varphi}^2\ &=\ -\,2\gamma^2+\frac{\lambda}{2}\,\varphi^2+\frac{\lambda}{2}\,\chi_i^2\;,\\ M_{\pm}^2\ &= \ \frac{M_{\varphi}^2+M_{\chi}^2}{2}\pm\bigg[\bigg(\frac{M_{\varphi}^2-M_{\chi}^2}{2}\bigg)^{\!2}+\lambda^2\varphi^2\chi_i^2\bigg]^{\!1/2}\;,\end{aligned}$$ and \[eq:CWUparts\] $$\begin{aligned} &U(\varphi, \chi) \ = \ - \frac{1}{2!} \, \mu^2 \varphi^2 + \frac{1}{2!} \, m_{\chi}^2 \chi_i^2 \nonumber \\ &\qquad + \frac{1}{4!} \, \lambda \varphi^4 + \frac{1}{4} \, \lambda \varphi^2 \chi_i^2 \; , \\ &\delta U(\varphi, \chi) \ = \ + \frac{1}{2!} \, \delta m_{\varphi}^2 \varphi^2 + \frac{1}{2!}\,\delta m_{\chi}^2 \chi_i^2 \nonumber \\ &\qquad + \frac{1}{4!} \, \delta \lambda_{\varphi} \varphi^4 + \frac{1}{4} \, \delta \lambda_{\varphi\chi} \varphi^2 \chi_i^2 + \frac{1}{4}\,\delta \lambda_{\chi} \chi_i^2\chi_j^2 \; . \end{aligned}$$ In Eqs. –, the summations over $i,j = 1, \dots, N$ have been left implicit for notational convenience. Solving the resulting system, we obtain the set of counterterms $$\begin{aligned} & \delta m_{\chi}^2 \ = \ - \, \frac{\lambda \gamma^2}{16 \pi^2} \bigg(\frac{\Lambda^2}{\gamma^2} - \ln \frac{\gamma^2}{\Lambda^2} - 13\nonumber\\& \quad +\frac{216\gamma^2}{m_{\chi}^2+6\gamma^2}-\frac{360\gamma^2}{m_{\chi}^2+2\gamma^2}\ln\frac{6\gamma^2+m_{\chi}^2}{4\gamma^2}\bigg) \; , \\ & \delta m_{\varphi}^2 \ = \ - \, \frac{\lambda \gamma^2}{16 \pi^2} \bigg[\big( N + 1 \big) \bigg( \frac{\Lambda^2}{\gamma^2} -30 \bigg) - \bigg( \ln \frac{\gamma^2}{\Lambda^2} + 1 \bigg) \nonumber \\ & \ + N \frac{m_{\chi}^2}{2 \gamma^2} \bigg( \ln \frac{6 \gamma^2 + m_{\chi}^2}{4 \Lambda^2} + 1 \bigg) + 27 N \bigg( \frac{m_{\chi}^2 + 2 \gamma^2} {m^2_{\chi} + 6\gamma^2} \bigg)^{\! \! 2} \bigg] \; , \\ & \delta \lambda_{\varphi} \ = \ - \, \frac{3 \lambda^2}{32 \pi^2} \bigg[ \ln \frac{\gamma^2}{\Lambda^2} + 5(N + 1) \nonumber \\ & \ + N \ln \frac{6\gamma^2 + m_{\chi}^2}{4 \Lambda^2} - 3 N \bigg( \frac{m_{\chi}^2 + 2\gamma^2} {m^2_{\chi} + 6\gamma^2} \bigg)^{\! \! 2} \bigg] \; ,\\ & \delta \lambda_{\varphi\chi} \ = \ - \, \frac{\lambda^2}{32 \pi^2} \bigg( \ln \frac{\gamma^2}{\Lambda^2}+4\ln\frac{6\gamma^2+m_{\chi}^2}{4\Lambda^2}\nonumber\\& \quad +\frac{136\gamma^2}{m_{\chi}^2+2\gamma^2}\ln\frac{6\gamma^2+m_{\chi}^2}{4\gamma^2} +9\,\frac{m_{\chi}^2-2\gamma^2}{m_{\chi}^2+6\gamma^2} \bigg) \; , \displaybreak\\ &\delta\lambda_{\chi}\ =\ -\,\frac{\lambda^2}{32\pi^2}\bigg(\frac{1}{2}\ln\frac{\gamma^2}{\Lambda^2}+\frac{\big(m_{\chi}^2-10\gamma^2\big)^2+432\gamma^4}{\big(m_{\chi}^2+2\gamma^2\big)^{2}}\nonumber\\&\quad +24\gamma^2\frac{\big(m_{\chi}^2-2\gamma^2\big)^2-112\gamma^4}{\big(m_{\chi}^2+2\gamma^2\big)^3}\ln\frac{6\gamma^2+m_{\chi}^2}{4\gamma^2}\bigg)\;. \end{aligned}$$ Proceeding as for $\varphi$, we find the unrenormalized tadpole contribution of the $\chi$ fields $$\begin{aligned} & \Sigma(u) \ = \ \frac{\gamma^2\lambda}{16 \pi^2} \Bigg[ \frac{\Lambda^2}{\gamma^2} + \frac{6 \gamma^2 + m_{\chi}^2}{2 \gamma^2} \bigg( \ln \frac{6 \gamma^2 + m_{\chi}^2}{4 \Lambda^2} + 1 \bigg) \nonumber \\ & -3\big(1-u^2\big)\ln \frac{6 \gamma^2 + m_{\chi}^2}{4 \Lambda^2}\nonumber\\ & - 6 \big( 1 - u^2 \big) \sum_{n = 1}^2 (-1)^n \big( n - 1 - u^2 \big) \nonumber \\ & \times \bigg( \frac{6 \gamma^2 + m_{\chi}^2}{n^2 \gamma^2} - 1\bigg)^{\! \! \tfrac{1}{2}} \mathrm{arccot} \bigg( \frac{6 \gamma^2 + m_{\chi}^2} {n^2 \gamma^2} - 1 \bigg)^{\! \! \tfrac{1}{2}} \Bigg] \; .\end{aligned}$$ After adding the counterterms, we obtain $$\begin{aligned} \label{GFren:chi} & \Sigma^R(u) \ = \ \frac{3 \gamma^2\lambda }{16 \pi^2} \Bigg[ 11 - 5 u^2 - 3 \big( 3 - u^2 \big) \bigg( \frac{m_{\chi}^2 + 2\gamma^2}{m_{\chi}^2 + 6\gamma^2}\bigg)^{\! \! 2} \nonumber \\ & - 2 \big( 1 - u^2 \big) \sum_{n = 1}^2 (-1)^n \big( n - 1 - u^2 \big) \nonumber \\ & \times \bigg(\frac{6 \gamma^2 + m_{\chi}^2}{n^2 \gamma^2} - 1 \bigg)^{\! \! \tfrac{1}{2}} \mathrm{arccot} \bigg( \frac{6 \gamma^2 + m_{\chi}^2}{n^2 \gamma^2} - 1 \bigg)^{\! \! \tfrac{1}{2}} \Bigg] \; .\end{aligned}$$ We note that the expression in Eq.  agrees with the renormalized tadpole contribution from $\Phi$ in Eq.  for $m_\chi^2 = - \mu^2$, as we would expect. Assuming $m_{\chi}^2 \gg \gamma^2$, we may expand Eq.  to leading order in $\gamma^2 / m_{\chi}^2$, giving Eq. . The one-loop correction to the bounce action from the determinant over the quadratic fluctuations in the $\chi$ fields is given by $$\begin{aligned} B^{(1)}_{\chi} \ & = \ - \, \frac{N}{2} \int_0^{\Lambda} \! {\mathrm{d}}k \; k^2 \int_0^{\infty} \! \frac{{\mathrm{d}}\tau}{\tau} \int_0^{\infty} \! {\mathrm{d}}r \; r^3 \nonumber \\ & \qquad \times \mathcal{L}^{-1}_s[\widetilde{S}(u, m)](\tau) \; ,\end{aligned}$$ where $\widetilde{S}(u, m)$ is obtained from Eqs.  and  with $$\begin{aligned} m \ = \ \sqrt{6} \bigg( 1 + \frac{k^2 + s + m_{\chi}^2} {6 \gamma^2} \bigg)^{\! \! \tfrac{1}{2}} \; .\end{aligned}$$ Continuing as in Sec. \[sec:greens\], we find $$\begin{aligned} B^{(1)}_{\chi} \ = \ & - N \frac{R^3 \gamma^3}{2} \bigg[ 3 \, \frac{\Lambda^2}{\gamma^2} + 3 \, \frac{m_{\chi}^2 + 4\gamma^2}{2 \gamma^2} \, \ln \frac{6 \gamma^2 + m_{\chi}^2}{4 \Lambda^2} \nonumber \\ & - \frac{m_{\chi}^2 + 2 \gamma^2}{2 \gamma^2} + \frac{2}{3} \sum_{n = 1}^2 n^3 \bigg( \frac{6 \gamma^2 + m_{\chi}^2}{n^2 \gamma^2} - 1\bigg)^{\! \! \tfrac{3}{2}} \nonumber \\ & \times \mathrm{arccot} \bigg( \frac{6 \gamma^2 + m_{\chi}^2}{n^2 \gamma^2} - 1\bigg)^{\! \! \tfrac{1}{2}} \bigg] \; .\end{aligned}$$ Adding the counterterm $$\begin{aligned} \delta B^{(1)}_{\chi} \ = \ & \frac{3}{2} \, N R^3 \gamma^3 \bigg[ \frac{\Lambda^2}{\gamma^2} - 20 + \frac{m_{\chi}^2}{2 \gamma^2} + 21 \bigg( \frac{m_{\chi}^2 + 2 \gamma^2}{m_{\chi}^2 + 6\gamma^2} \bigg)^{\! \! 2} \nonumber \\ & + \frac{m_{\chi}^2 + 4 \gamma^2}{2 \gamma^2} \ln \frac{6 \gamma^2 + m_{\chi}^2}{4 \Lambda^2} \bigg] \; ,\end{aligned}$$ obtained in analogy with Eq. , we find $$\begin{aligned} \label{eq:B1chifull} & B^{(1)}_{\chi} \ = \ - \, N \, \frac{R^3 \gamma^3}{2} \bigg[ 63 - 4 \, \frac{m_{\chi}^2 + 2 \gamma^2}{2 \gamma^2} - 63 \bigg( \frac{m_{\chi}^2 + 2 \gamma^2}{m_{\chi}^2 + 6\gamma^2} \bigg)^{\! \! 2} \nonumber \\ & + \frac{2}{3} \sum_{n = 1}^2 n^3 \bigg(\frac{6 \gamma^2 + m_{\chi}^2}{n^2 \gamma^2} - 1 \bigg)^{\! \! \tfrac{3}{2}} \mathrm{arccot} \bigg( \frac{6 \gamma^2 + m_{\chi}^2}{n^2 \gamma^2} - 1\bigg)^{\! \! \tfrac{1}{2}} \bigg] \; .\end{aligned}$$ The result in Eq.  reduces to that found in Eq.  for $m_{\chi}^2 = - 2 \gamma^2$ and $N = 1$, as we would expect. 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[^2]: It is natural to define the renormalized quantities in the false vacuum, since this is where the physical measurements of these quantities are performed. If it were the case that such measurements were taking place in the true vacuum, or indeed within the wall itself, then the decay rate would be of little concern. [^3]: Using the same renormalization conditions as in Eq. , Ref. [@Konoplich:1987yd] finds (in the notation employed here) $$B^{(1)} \ = \ - \, B \, \bigg( \frac{3 \lambda}{16 \pi^2} \bigg) \bigg( \frac{\pi}{3 \sqrt{3}} + \frac{50}{3} \bigg) \; .$$ Repeating the analysis presented therein, as outlined in Appendix \[app:greens\], we find a result in agreement with Eq.  reported here, suggesting a numerical error in the factor of 50 above.
--- abstract: 'We study the long-time effect of noise on pattern formation for the aggregation model. We consider aggregation kernels that generate patterns consisting of two delta-concentrations. Without noise, there is a one-parameter family of admissible equilibria that consist of two concentrations whose mass is not necessary equal. We show that when a small amount of noise is added, the heavier concentration “leaks” its mass towards the lighter concentration over a very long time scale, eventually resulting in the equilibration of the two masses. We use exponentially small asymptotics to derive the long-time ODE’s that quantify this mass exchange. Our theory is validated using full numerical simulations of the original model – both of the original stochastic particle system and its PDE limit. Our formal computations show that adding noise destroys the degeneracy in the equilibrium solution and leads to a unique symmetric steady state.' author: - 'Joep H.M. Evers$^{\dagger}$ and Theodore Kolokolnikov$^{\dagger}$' bibliography: - '2masses.bib' title: Metastable states for an aggregation model with noise --- Introduction ============ Aggregation is an ubiquitous natural phenomenon that pervades both the animal world and many inanimate physical systems. In the animal kingdom, group formation is observed across all levels from bacterial colonies and insect swarms to complex predator-prey interactions in fish, birds and mammals. Aggregation is also present in physical systems of all scales from the smallest (Bose-Einstein Condensates, DNA buckyball molecules, fluid vortices) to the largest (galaxies). The emergence of group behaviour is often a consequence of individuals (or atoms) following very simple rules, without any external coordination. ![LEFT: Particle simulation of deterministic system (\[deterministic\]) with $f(x)=-x$. Initial conditions consist of $n=200$ particles equally spaced from each other. Particles collapse into a single delta-concentration. MIDDLE: Same ODE system but with noise added (\[noise\]). The delta-concentration is “diffused” into a Gaussian. RIGHT: Particle density distribution from the simulation in the middle panel, averaged over 100 time-steps with $t=10.$ The dashed line is the predicted Gaussian profile given by (\[rhogauss\]). []{data-label="fig:simple"}](simplest.png){width="\textwidth"} One of the simplest models that achieves aggregation is the so-called aggregation model, which has been the subject of intense study in the last two decades; refer to survey papers and references therein. Mathematically, this model may be written as a system of ODE’s for $n$ particles $$\frac{dx_{j}}{dt}=\frac{1}{n}\sum_{k=1}^{n}f(x_{j}-x_{k}) \label{deterministic}$$ where the pairwise interaction force $f(x)$ is assumed to be the gradient of a radial potential function, $f(x)=-\nabla P(\left\vert x\right\vert )=-P^{\prime}(\left\vert x\right\vert )\frac{x}{\left\vert x\right\vert }$. The strength of the force $f(x_{j}-x_{k})$ depends only on the distance between the two particles $x_{j}$ and $x_{k}$, and it acts in the direction between these particles. The system (\[deterministic\]) corresponds to applying the method of steepest descent to determine the minimizer of the pairwise-interaction energy,$$E=\sum_{k,j}P(\left\vert x_{j}-x_{k}\right\vert ).$$ ![ LEFT: Simulation of the deterministic system (\[deterministic\]) with $f(x)=x-x^3$. Initial conditions consist of 80 particles near $x=-0.5$ and 120 particles near $x=0.5$ (corresponding to $M_1=0.4, M_2=0.6$). The long-time dynamics approach two unequal delta-concentrations. The insert in top-left shows the histogram of the final steady state. MIDDLE: Same as the left figure, but with $\sigma=0.075$ noise added. The concentration at the right initially has a larger mass, and it very slowly leaks its mass towards a lighter concentration on the left. RIGHT: The total mass of left and right concentrations of the simulation in the middle is plotted. After a short transient period, a slow mass exchange is apparent, with the two masses gradually equilibrating. The dashed line denotes the asymptotic prediction given by .[]{data-label="fig:cubic"}](2noisy.png){width="\textwidth"} To get confinement, it is further assumed that the particles repel each other at short distances and attract each other at longer distances (see also [@Yao; @Canizo; @Simione]). In many cases this leads to the formation of swarms. The assumption of long-range attraction and short-range repulsion corresponds to $P(r)$ having a minimum at $r=r_{0}$ so that particles at a distance bigger than $r_{0}$ are attracted to each other and those at distance less than $r_{0}$ are repelling. These simple assumptions can give rise to surprisingly complex steady states [@kolokolnikov2011stability; @von2012predicting; @von2012soccer; @fetecau2011swarm; @bertozzi2012aggregation; @kolokolnikov2013singular; @balague2013dimensionality; @fellner2010stable; @von2013swarming] including soccer balls in two and three dimensions [@kolokolnikov2011stability; @von2012predicting; @kolokolnikov2013singular; @balague2013dimensionality] as well as steady states concentrating on points, curves and surfaces [@fellner2010stable; @kolokolnikov2013singular; @balague2013dimensionality; @von2013swarming; @hackett2012aggregation]. Of particular importance for the dimensionality of the steady state is the strength of repulsion near the origin [@kolokolnikov2011stability; @von2012predicting; @balague2013dimensionality]. For this paper, we focus on the simplest case, where the steady state concentrates on a finite number of points (delta-concentrations), which can occur when the repulsion is sufficiently weak at the origin. In this work we are interested in how the noise that is inherently present in most of the physical systems can affect the resulting steady state. That is, we consider the model (\[deterministic\]) with noise, so that (\[deterministic\]) is replaced by stochastic ODE’s $$dx_{j}=\frac{1}{n}\sum_{k=1}^{n}f(x_{j}-x_{k})dt+\sigma\sqrt{dt}\mathcal{N}_{j}; \label{noise}$$ here $\sigma\sqrt{dt}\mathcal{N}_{j}$ is the standard Wiener process with standard deviation $\sigma$; $\mathcal{N}_{j}$ denotes the standard normal distribution of mean zero and variance 1. In the continuum limit as the number of particles $n\rightarrow\infty,$ the average particle density distribution $\rho$ is well approximated by the PDE [@morale2005interacting], $$\rho_{t}+\nabla\cdot\left( v\rho\right) =\varepsilon^{2}\Delta \rho,\ \ \ \ v=\int f\left( x-y\right) \rho\left( y\right) dy. \label{pde}$$ where $\varepsilon^{2}=\sigma^{2}/2.$ Equation (\[pde\]) is the starting point for this paper. The presence of noise can have a profound effect on the steady state, especially if the steady state consists of point concentrations which can be the case when the repulsion is sufficiently weak at the origin [@hughes2013continuum]. As a motivating example, consider the simplest case, $$f(x)=-x.$$ This corresponds to a pontential $P(r)=r^{2}/2$ that is purely attractive, and is only weakly attractive at the origin (i.e. $P^{\prime}(0)=0$). Figure \[fig:simple\] shows the resulting one-dimensional simulations with and without noise for the discrete system (\[noise\]), as well as the associated average density in the presence of noise – computed by averaging the steady state for the last 100 steps of the numerical simulation of (\[noise\]) (for simplicity, we used forward Euler method for these simulations). Without noise, the particle density collapses to a single point (delta function). On the other hand, the noise diffuses the delta function and the resulting average steady state density is a Gaussian, $$\rho(x)=\frac{1}{\sqrt{2\pi\varepsilon^{2}}}\exp\left( -\frac{x^{2}}{2\varepsilon^{2}}\right) . \label{rhogauss}$$ as was already observed in [@hughes2013continuum]. Indeed, the steady state satisfies $\left( v\rho\right) _{x}=\varepsilon^{2}\rho_{xx}$ where $v(x)=\int-\left( x-y\right) \rho(y)dy=-x$, where we assumed that $\rho$ has total mass $M=1$ and its centre of mass is $0$. Integrating once we get the ODE $\varepsilon^{2}\rho_{x}=-x\rho$ whose solution is given by (\[rhogauss\]). In this paper, we are interested in the effect of small amount of diffusion when the steady state consists of more than one delta function. This occurs when repulsion is present (so that the particles do not all collapse into a single point), but the repulsion is sufficiently weak so that far-field attraction causes clumping into two or more delta-concentrations. As was shown in [@fellner2010stable] in one dimension, (and extended in [@kolokolnikov2013singular] to two and three dimensions), the necessary condition for this to happen is that $P^{\prime }(0)=0,$ or equivalently, $f(x)\sim cx$ as $x\rightarrow0$ for some *positive* constant $c>0$. To illustrate this phenomenon as well as the results of the paper, consider the simplest such case, namely the double-well potential, $P(r)=-r^{2}/2+r^{4}/4,$ so that$$f(x)=x-x^{3}. \label{cubic}$$ Figure \[fig:cubic\](left) illustrates the behaviour of the deterministic system (\[deterministic\]) with $n=200$ particles, starting with initial conditions consisting of 80 particles near $x=-0.5$ and 120 particles near $x=0.5$. After some transient time, the system evolves into a steady state consisting of two delta concentrations, with 40% of the mass at the left concentration and 60% at the right. The distance between the two concentrations is $x=1$ corresponding to the root of (\[cubic\]), and this is trivially seen to be a steady state of (\[deterministic\]) since $f(0)=f(1)=0$. Moreover, as was shown in [@fellner2010stable; @kolokolnikov2013singular], such steady state is actually stable. Now suppose there is a small amount of noise present in the system (say $\sigma=0.075$) while keeping all other parameters and initial conditions as in Figure \[fig:cubic\](left). The result is shown in Figure \[fig:cubic\](middle and right). Initially, the system quickly settles to a two-concentration asymmetric steady state with roughly 40% of mass on the left and 60% of mass on the right, except that the noise diffuses the delta concentrations, so that the particles constantly jiggle around, and the average density is approximately a Gaussian. However on a much longer time-scale, there is a very slow exchange of mass that takes place between the left and right concentrations, so that the bigger concentration slowly leaks its mass towards the smaller, until the two concentrations eventually equilibrate. In other words, adding even a small amount of noise eventually symmetrizes the asymmetric steady state over a long time. Quantifying this very slow exchange of mass is the goal of this paper. There are similarities between our work and [@Geigant]. They consider the combination of interactions and diffusion on a one-dimensional interval with periodic boundary conditions. Their main focus is on the $O(1)$ stability of steady states consisting of multiple peaks. By contrast, here we are interested in the slow-time evolution (metastability) of these states. We will illustrate our result now. Our main finding describes the exchange of mass between the asymmetric diffused concentrations. The precise statement, for a general kernel $f(x)$, is given in Proposition \[prop:ode\]. To illustrate the result, consider the cubic kernel (\[cubic\]), refer to Figure \[fig:cubic\]. Starting with arbitrary initial conditions, the system converges, on an $O(1)$ timescale, to a solution consisting of two concentrations. These concentrations will in general have unequal masses (that depend on initial conditions), and their asymptotic profile is a Gaussian spike whose variance depends on their relative masses $M_{1},M_{2}$. Once these spikes form, there is a very slow equilibration process, whereby the mass of a heavier spike leaks towards the lighter one. This process is meta-stable, meaning that it takes an exponentially long time (in $\varepsilon$) for the masses to equilibrate. Specializing Proposition \[prop:ode\] to (\[cubic\]), this slow mass exchange is described asymptotically by an ODE\[odecubic\]$$\frac{d}{dt}M_{1}\sim F(M_{1},M_{2})-F(M_{2},M_{1});\ \ \ M_{1}+M_{2}=M \label{odeM}$$ where $M$ is the total mass, $M_{1}$ and $M_{2}$ are the masses of the two spikes, and$$F(M_{1},M_{2})=\frac{M_{2}}{2\pi}\sqrt{\frac{\left( 2M_{2}-M_{1}\right) }{M}}\left( 2M_{1}-M_{2}\right) \exp\left( \frac{-M_{2}\left( 2M_{1}-M_{2}\right) ^{3}+O(\varepsilon^{2})}{4M^{3}\varepsilon^{2}}\right) . \label{Fm1m2}$$ Note that the $O(\varepsilon^{2})$ term inside the exponent in (\[Fm1m2\]) becomes an $O(1)$ premultiplier when the exponential is expanded, so the constant in front of the exponential is not asymptotically accurate. A higher-order expansion is necessary to obtain that correction, and we do not attempt this here; it is an open problem to do so. However the exponential decay dominates the dynamics and to leading order, the constant in front of the exponential is of lower order when the solution is plotted on a log-time scale. This is explained further in §\[sec: ODE derivation\] – see Figure \[fig:ddp\] there. ![Evolution of the density profile, for $f(x)=x(1-x^{2})$ and $\varepsilon^{2}=0.001$. In the last three plots, the narrow grey lines indicate the heights of the two maxima. Their heights converge at a slow timescale.[]{data-label="fig:profile evol"}](profile_evol_epsSq1e-3.pdf){width="\textwidth"} The summary of the paper is as follows. In §\[sec:ss\] we construct asymptotically the quasi-steady state consisting of two Gaussians of masses $M_{1},M_{2}.$ In §\[sec:ode\] we derive the equations of mass exchange between $M_{1}$ and $M_{2}$, on an exponentially slow timescale, culminating in Proposition \[prop:ode\] which is the main result of this paper. From the equations for mass exchange, we show that the masses equilibrate on a long time scale, that is, $M_{1}=M_{2}$ is the unique global steady state of the long-time dynamics. We compare the full numerical solution of the original aggregation-diffusion PDE for several values of $\varepsilon$, to our asymptotic findings. We conclude with some remarks in §\[sec:discuss\]. Quasi-steady state\[sec:ss\] ============================ In [@fellner2010stable; @kolokolnikov2013singular] the authors constructed a steady state of (\[pde\]) with zero diffusion consisting of discrete number $(N\geq2$) of delta-concentrations. This happens when $f(x)$ is linear near the origin, such as for example (\[cubic\]). More generally, we will assume that:$$\left\{ \begin{tabular} [c]{l}$f(x)$ is odd; \\ $f(x)$ has a positive root at $x=a;$ \\ $f(x)$ is $C^{1}$ at $x=0$ and $x=a$ with $f^{\prime}(0)>0$ and $f^{\prime}(a)<0.$\end{tabular} \ \right. \label{cond}$$ Under these assumptions, a two-delta steady state of (\[pde\]) with $\varepsilon=0$ has the form$$\rho(x,t)\sim M_{1}\delta\left( x-x_{1}\right) +M_{2}\delta(x-x_{2})\ \text{with }x_{2}-x_{1}=a,\ \ M_{1}+M_{2}=M \label{rhodelta}$$ where $M$ is the total mass. Upon substituting (\[rhodelta\]) into $v=f\ast\rho$ we obtain that$$v(x)=M_{1}f(x-x_{1})+M_{2}f(x-x_{2}). \label{v}$$ Turning on $\varepsilon$ in (\[pde\]) diffuses the delta concentrations, so that the $\delta\left( x-x_{1}\right) $ is replaced by a spike that has width of $O(\varepsilon)$. That is, we write$$\rho(x,t)\sim M_{1}\frac{1}{\varepsilon}w_{1}\left( \frac{x-x_{1}}{\varepsilon}\right) +M_{2}\frac{1}{\varepsilon}w_{2}\left( \frac{x-x_{2}}{\varepsilon}\right) \label{rho2}$$ where $w_{i}(y)$ is the spike profile that is to be computed, with $\int_{-\infty}^{\infty}w_{i}(y)dy=1$ and $w_{i}(y)>0$ for all $y.$ Substituting (\[rho2\]) into $v=f\ast\rho$, and expanding the resulting integral in terms of Taylor series, we find that the leading-order expression for the velocity $v$ is then still given by (\[v\]) up to $O(\varepsilon)$ order. To compute the profile of the left spike $w_{1}(y),$ we let $x=x_{1}+\varepsilon y$ and expand (\[rho2\]). We have $v\left( x_{1}\right) =0$ so that$$v\left( x_{1}+\varepsilon y\right) \sim-\varepsilon c_{1}y,\ \ c_{1}=-v^{\prime}(x_{1})\sim-\left( M_{1}f^{\prime}(0)+M_{2}f^{\prime}(a)\right) . \label{vtaylor}$$ We then substitute (\[vtaylor\]) into the steady state equation after discarding $\rho_{t}$ (this is the assumption that $\rho$ is a quasi-steady state). Near $x=x_{1}+\varepsilon y$ we then obtain, up to exponentially small terms,$$\left( yc_{1}w_{1}\right) _{y}\sim w_{1yy}.$$ Assuming decay as $y\rightarrow\pm\infty$ yields$$yc_{1}w_{1}+w_{1y}\sim0$$ so that $$w_{1}\sim\sqrt{\frac{c_{1}}{2\pi}}\exp\left( \frac{-y^{2}}{2}c_{1}\right) .$$ A necessary condition for decay is that $c_{1}>0.$ Performing a similar computation for $w_{2}$ we obtain the following result. \[prop:ss\]Suppose that $f(x)$ satisfies conditions (\[cond\]) and suppose that $M_{1},M_{2}$ satisfy$$-f^{\prime}(0)/f^{\prime}(a)<M_{1}/M_{2}<-f^{\prime}(a)/f^{\prime}(0). \label{mratio}$$ Then (\[pde\]) admits a quasi-equilibrium steady state that has the form $$\rho(x,t)\sim\sum_{j=1}^{2}\frac{M_{j}}{\varepsilon}\sqrt{\frac{c_{j}}{2\pi}}\exp\left( -\left( \frac{x-x_{j}}{\varepsilon}\right) ^{2}\frac{c_{j}}{2}\right) \label{rhoinner}$$ with $x_{2}-x_{1}=a$ and $c_{j}=-v^{\prime}(x_{j})$ with $v$ given by (\[v\]); that is$$c_{1}=-\left( M_{1}f^{\prime}(0)+M_{2}f^{\prime}(a)\right) ;\ \ \ c_{2}=-\left( M_{2}f^{\prime}(0)+M_{1}f^{\prime}(a)\right) . \label{cj}$$ The masses $M_{1},M_{2}$ satisfy $M_{1}+M_{2}=M$ where $M$ is the total mass that is determined by the initial conditions, $M=\int_{-\infty}^{\infty}\rho(x,0)dx.$ As we show in §\[sec:ode\], the masses $M_{1}$ and $M_{2}$ evolve on a timescale much larger than the timescale at which this two-spike profile forms. In Figure \[fig:profile and approx\] we compare the long-time solution of , as shown in the bottom right of Figure \[fig:profile evol\], to the approximation with $M_{1}=M_{2}$. Proposition \[prop:ss\] generalizes naturally to $n$ concentrations. In this case, the sum $\sum_{j=1}^{2}$ in (\[rhoinner\]) is replaced by $\sum _{j=1}^{n}$, with$$c_{j}=-\sum_{k=1}^{n}M_{k}\,f^{\prime}(x_{j}-x_{k}). \label{eqn: def eps j sq N}$$ and the condition $x_{2}-x_{1}=a$ is replaced by a system$$\sum_{k=1}^{n}M_{k}f(x_{j}-x_{k})=0,\ \ j=1\ldots n.$$ Finally, the two conditions (\[mratio\]) are replaced by $n$ conditions$$\sum_{k=1}^{n}M_{k}f^{\prime}(x_{j}-x_{k})<0.$$ ![Density profile at $t=500$, for $f(x)=x(1-x^{2})$ and $\varepsilon^{2}=0.001$; cf. Figure \[fig:profile evol\]. Superimposed is the approximation with $M_{1}=M_{2}$.[]{data-label="fig:profile and approx"}](profile_and_approx_t500_epsSq1e-3.pdf){width="50.00000%" height="30.00000%"} Metastable dynamics\[sec: ODE derivation\] {#sec:ode} ========================================== We now derive the ODE describing the slow-time dynamics for the mass exchange between $M_{1}$ and $M_{2}.$ The starting point is the PDE (\[pde\]) with $v$ as given by (\[v\]). Note from the expansion for $v$ near $x_{1}$ (\[vtaylor\]) that $v(x_{1})=0$ and $v^{\prime}(x_{1})<0$ (which implies that the concentration at $x=x_{1}$ attracts nearby points). Similarly $v(x_{2})=0$ and $v^{\prime}(x_{2})<0.$ By continuity of $v(x),$ there must be a point $\hat{x}$ such that $$\hat{x}:\ \ \hat{x}\in\left( x_{1},x_{2}\right) \text{ with }v(\hat {x})=0\text{ where }v\text{ is given by (\ref{v})} \label{vhat}$$ with $v$ increasing at $\hat{x}.$ We further make the following technical assumption which will be needed for global stability:$$\text{The solution to (\ref{vhat})\ is unique with }v^{\prime}\left( \hat {x}\right) >0\text{.} \label{uniq}$$ This holds for a large class of functions and in particular if $f^{\prime \prime\prime}(x)<0$ for all $x\in\left( 0,a\right) .$ We identify this point $\hat{x}$ as the boundary point between mass belonging to spike $\#1$ and mass belonging to spike $\#2$. Assuming that the density decays away from the $x_{j},$ we have$$M_{1}\sim\int_{-\infty}^{\hat{x}}\rho(x)dx;\ \ \ M_{2}\sim\int_{\hat{x}}^{\infty}\rho(x)dx. \label{eqn: def Mj}$$ Integrating (\[pde\]) from $-\infty$ to $\hat{x}$ we therefore obtain $$\dfrac{d}{dt}M_{1}=\varepsilon^{2}\rho_{x}(\hat{x}),\ \ \dfrac{d}{dt}M_{2}=-\varepsilon^{2}\rho_{x}(\hat{x}) \label{eqn: ODE M}$$ where we used that $v(\hat{x})=0$ and we assumed that $\rho$ decays as $x\rightarrow\pm\infty.$ To derive an expression for $\rho_{x}(\hat{x})$, next we assume that the dynamics are sufficiently slow such that $\rho_{t}$ term can be discarded in (\[pde\]). This assumption will later be seen to be consisent with the final result, which shows exponentially slow (metastable) evolution of mass. Then integrating the resulting ODE we obtain$$\rho v=\varepsilon^{2}\rho_{x}-\varepsilon^{2}\rho_{x}(\hat{x}),\ \ x\in \left( x_{1},x_{2}\right) , \label{apple}$$ where we use that $v(\hat{x})=0$. The solution to (\[apple\]) is given by $$\rho(x)=\left[ \rho(\hat{x})+\rho_{x}(\hat{x})\,\int_{\hat{x}}^{x}\exp(-V(z)/\varepsilon^{2})\,dz\right] \exp(V(x)/\varepsilon^{2}), \label{quake}$$ where $$V(x)=\int_{\hat{x}}^{x}v(x)dx.$$ Note that $-V(x)$ has a global maximum at $x=\hat{x}$ (by (\[vhat\]) and (\[uniq\])), so that we may use Laplace’s method to evalute the integral in (\[quake\]). We obtain:$$\int_{\hat{x}}^{x}\exp(-V(z)/\varepsilon^{2})\,dz\sim\operatorname*{sign}\left( x-\hat{x}\right) \varepsilon\sqrt{\frac{\pi}{2v^{\prime}(\hat{x})}},\ \ \ \left\vert x-\hat{x}\right\vert \gg O(\varepsilon)$$ To determine $\rho_{x}\left( \hat{x}\right) ,$ let $x\rightarrow x_{i},\ i=1,2$, and match (\[quake\])with the inner solution as given by (\[rhoinner\]). Expanding near $x_{j},$ we let $x=x_{j}+\varepsilon y$ and expand$$V(x_{j}+\varepsilon y)=\int_{\hat{x}}^{x_{j}}v(s)ds+v(x_{j})\varepsilon y+v^{\prime}(x_{j})\varepsilon^{2}\frac{y^{2}}{2}+\ldots\sim\int_{\hat{x}}^{x_{j}}v(s)ds-c_{j}\varepsilon^{2}\frac{y^{2}}{2}$$ where $c_{j}=-v^{\prime}(x_{j})$ is as given in (\[cj\]). Therefore the outer region written in inner variables near $x_{j}$ becomes$$\rho(x_{j}+\varepsilon y)\sim\left[ \rho(\hat{x})+\operatorname*{sign}\left( x_{j}-\hat{x}\right) \rho_{x}(\hat{x})\,\varepsilon\sqrt{\frac{\pi }{2v^{\prime}(\hat{x})}}\right] \exp\left( \frac{1}{\varepsilon^{2}}\int_{\hat{x}}^{x_{j}}v(s)ds\right) \exp\left( -c_{j}\frac{y^{2}}{2}\right) . \label{pomme}$$ On the other hand, the inner region near $x=$ $x_{j}+\varepsilon y$ as derived in (\[rhoinner\]) is $$\rho\sim\frac{M_{j}}{\varepsilon}\sqrt{\frac{c_{j}}{2\pi}}\exp\left( -c_{j}\frac{y^{2}}{2}\right) . \label{inner}$$ Matching (\[inner\]) and (\[pomme\]) yields\[oato\]$$\begin{aligned} \frac{M_{1}}{\varepsilon}\sqrt{\frac{c_{1}}{2\pi}}=\left[ \rho(\hat{x})-\rho_{x}(\hat{x})\,\varepsilon\sqrt{\frac{\pi}{2v^{\prime}(\hat{x})}}\right] \exp\left( \frac{1}{\varepsilon^{2}}\int_{\hat{x}}^{x_{1}}v(s)ds\right),\\ \frac{M_{2}}{\varepsilon}\sqrt{\frac{c_{2}}{2\pi}}=\left[ \rho(\hat{x})+\rho_{x}(\hat{x})\,\varepsilon\sqrt{\frac{\pi}{2v^{\prime}(\hat{x})}}\right] \exp\left( \frac{1}{\varepsilon^{2}}\int_{\hat{x}}^{x_{2}}v(s)ds\right) .\end{aligned}$$ Solving for $\rho_{x}\left( \hat{x}\right) $ from (\[oato\]) and then using (\[eqn: ODE M\]) finally yields\[m1m2\]$$\begin{aligned} \frac{dM_{1}}{dt} & \sim\frac{M_{2}}{2}\sqrt{\frac{c_{2}\,v^{\prime}(\hat {x})}{\pi^{2}}}\exp\left( -\frac{1}{\varepsilon^{2}}\int_{\hat{x}}^{x_{2}}v(s)ds\right) -\frac{M_{1}}{2}\sqrt{\frac{c_{1}\,v^{\prime}(\hat{x})}{\pi^{2}}}\exp\left( -\frac{1}{\varepsilon^{2}}\int_{\hat{x}}^{x_{1}}v(s)ds\right) ;\label{M1}\\ \frac{dM_{2}}{dt} & =-\frac{dM_{1}}{dt}.\end{aligned}$$ ![The graph of $v(x)=M_1 f(x)+M_2 f(x-1)$ with $f(x)=x-x^3$ and with $M_1,M_2$ as indicated in the legend. Increasing $M_1$ moves $\hat{x}$ to the left. []{data-label="fig:v"}](v.png){width="50.00000%"} It is clear that the symmetric configuration $M_{1}=M_{2}$ is an equilibrium of the ODE (\[m1m2\]), since in this case, $\hat{x}=x_{1}+a/2$ and $c_{1}=c_{2,}\ \ \int_{\hat{x}}^{x_{1}}v(s)ds=\int_{\hat{x}}^{x_{2}}v(s)ds$. We now show that it is indeed a global attractor, provided (\[uniq\]) holds. From the equation $v(x)\sim M_{1}f(x-x_{1})+M_{2}f(x-x_{2})$ and the fact that $f(x-x_{1})$ is positive for $x\in\left( x_{1},x_{2}\right) ,$ it follows that $v(x)$ is an increasing function of $M_{1}.$ This in turn shows that $\hat{x}$ is a decreasing with $M_{1},$ $\int_{\hat{x}}^{x_{1}}v(s)ds$ is decreasing with $M_{1}$ and $\int_{\hat{x}}^{x_{2}}v(s)ds$ is increasing with $M_{1}$. Refer to Figure \[fig:v\]. It then follows from (\[m1m2\]) that $\frac{dM_{1}}{dt}<0$ whenever $M_{1}>M_{2}$ and $\frac{dM_{1}}{dt}>0$ when $M_{1}<M_{2}.$ This shows that any admissible initial masses satisfying (\[mratio\]) evolve towards the equal-mass $M_{1}=M_{2}$ configuration. We now summarize. \[prop:ode\]Consider the quasi-steady state constructed in Proposition \[prop:ss\]. The spike masses $M_{1}(t),\ M_{2}(t)$ evolve on an exponentially slow time-scale according to (\[m1m2\]), where $v(x)$ is given by (\[v\]) and $\hat{x}$ satisfies (\[vhat\]). Moreover, suppose that in addition to properties (\[cond\]), $f(x)$ also satisfies: $f^{\prime \prime\prime}(x)<0$ for $x\in\left( 0,a\right) .$ Then $M_{1}=M_{2}=M/2$ is the global attactor of (\[m1m2\]) where $M=M_{1}+M_{2}$ is the total mass, so that $M_{1}(t),M_{2}(t)\rightarrow M/2$ as $t\rightarrow\infty.$ Note that the expression for $v(s)$ as given by (\[v\]) is only accurate up to $O(\varepsilon^{2})$ terms. As a result, the constants in front of the exponentials in (\[M1\]) are not asymptotically accurate, since the expansion of the integral in the exponential up to $O(\varepsilon^{2})$ terms will change the these constants by an $O\left( 1\right) $ amount. Nonetheless, the ODE (\[m1m2\]) provides a good estimate of the solution at large time scales as we now show. To illustrate Proposition \[prop:ode\], we take $$f(x)=x(1-x^{2}).$$ Then the right hand side in (\[m1m2\]) can be computed explicitly. Without loss of generality (translation invariance), assume that $x_{1}=0$, hence $x_{2}=1$ so that $v(x)$ given by (\[v\]) becomes $$v(x)\sim x(1-x)\left( M_{1}-2M_{2}+M\,x\right) .$$ The unique $\hat{x}\in(0,1)$ for which $v(\hat{x})=0$ is therefore explicitly given by $$\hat{x}=\dfrac{2M_{2}-M_{1}}{M},$$ and we compute $$v^{\prime}(\hat{x})=\,\dfrac{(2M_{2}-M_{1})(2M_{1}-M_{2})}{M}; \label{1003}$$ and$$\int_{\hat{x}}^{x_{1}}v\left( x\right) dx=\dfrac{M_{1}(2M_{2}-M_{1})^{3}}{4M^{3}};\ \ \ \ \int_{\hat{x}}^{x_{2}}v\left( x\right) dx=\dfrac {M_{2}(2M_{1}-M_{2})^{3}}{4M^{3}}. \label{1004}$$ Substituting (\[1003\]) and (\[1004\]) into (\[m1m2\]) yields the explicit ODE (\[odecubic\]) for the mass exchange dynamics. As mentioned above, the constants in front of exponentials are not asymptotically accurate, since we only computed $O(1)$ contributions of the arguments $\int_{\hat{x}}^{x_{1}}v\left( x\right) dx,\ \int_{\hat{x}}^{x_{2}}v\left( x\right) dx$ inside the exponents, disregarding the $O(\varepsilon^{2})$ corrections which contribute an $O(1)$ amount to the constants in front of the exponential. However the ODE is still asymptotically valid for large time in the following sense. Rewrite the system (\[odecubic\]) in terms of the difference $$d=M_{2}-M_{1}$$ from the equilibrium state. If we assume that $d$ is positive and $d\gg O(\varepsilon^{2})$ then the term $F(M_{2},M_{1})$ in (\[odeM\])$\ $is negligible when compared with $F(M_{1},M_{2})$ and we obtain$$d^{\prime}(t)\sim-C(d)\exp\left\{ -\frac{1}{\varepsilon^{2}}\frac{\left( M+d\right) \left( M-3d\right) ^{3}}{64M^{3}}\right\} \label{dp}$$ where $C(d)$ is some (undetermined) $O(1)$ pre-multiplier. Taking the logarithm on both sides of (\[dp\]) we obtain $$\varepsilon^{2}\log\left( \left\vert d^{\prime}(t)\right\vert \right) =\frac{-\left( M+d\right) \left( M-3d\right) ^{3}}{64M^{3}}+O(\varepsilon ^{2}). \label{logdp}$$ Figure \[fig:ddp\](left) shows direct comparison between the full numerical simulations of the original PDE (\[pde\]) and the asymptotics of the right-hand side given by (\[logdp\]). The dashed asymptotic line there corresponds to the right-hand side of (\[logdp\]). Note how the difference between the dashed line (asymptotics) and solid lines (numerics for several $\varepsilon$) is proportional to $\varepsilon^{2},$ which shows that the error in the exponent in (\[M1\]) is indeed of $O(\varepsilon^{2}).$ The ODE (\[dp\]) is separable, and applying Laplace’s method to its solution (keeping in mind that $\left( M+d\right) \left( M-3d\right) ^{3}$ is a decreasing function of $d$) we obtain the asymptotic solution $$\left( M+d\right) \left( M-3d\right) ^{3}\sim64M^{3}\varepsilon^{2}\left( \log\left( \frac{t}{\varepsilon^{2}64M^{3}}\right) +O(1)\right) \label{logt}$$ valid for large $t.$ Note that to leading order, $d(t)$ is independent of the pre-exponential multiplier $C(d)$ in (\[dp\]) (it appears in the $O(1)$ term in the right hand side), and is also indepedent of the initial conditions as long as $d\gg O(\varepsilon^{2})$. In other words (\[logt\]) is an accurate prediction for $d$ as a function of $\log t$, regardless of the value of $C(d).$ ![LEFT: Validation of the right hand side of the ODE (\[logdp\]). The plot shows $d(t)$ versus ${\varepsilon^2}\log(|d'(t)|)$, where $d(t)$ and $d'(t)$ are computed from numerical simulations of the full PDE for several values of $\varepsilon$ as shown. The dashed line is the leading-order asymptotic prediction (\[logdp\]) ${\varepsilon^2}\log(|d'(t)|)\approx -\frac{(M+d)(M-3d)^3}{64M^3}$. RIGHT: Same simulation/parameters as on the left, except that the mass difference $d(t)$ is plotted versus logarithmic timescale. The theoretical prediction plotted in dashed line is given by (\[logt\]). []{data-label="fig:ddp"}](ddp.pdf "fig:"){width="49.00000%"} ![LEFT: Validation of the right hand side of the ODE (\[logdp\]). The plot shows $d(t)$ versus ${\varepsilon^2}\log(|d'(t)|)$, where $d(t)$ and $d'(t)$ are computed from numerical simulations of the full PDE for several values of $\varepsilon$ as shown. The dashed line is the leading-order asymptotic prediction (\[logdp\]) ${\varepsilon^2}\log(|d'(t)|)\approx -\frac{(M+d)(M-3d)^3}{64M^3}$. RIGHT: Same simulation/parameters as on the left, except that the mass difference $d(t)$ is plotted versus logarithmic timescale. The theoretical prediction plotted in dashed line is given by (\[logt\]). []{data-label="fig:ddp"}](dlogt.pdf "fig:"){width="49.00000%"} In Figure \[fig:ddp\](right) we plot $d$ as a function of $64M^{3}\varepsilon^{2}\left( \log\left( \frac{t}{\varepsilon^{2}64M^{3}}\right) \right) $ for several values of $\varepsilon,$ and then compare with the asymptotic value of $d$ as given by (\[logt\]). Again, the error is observed to be proportional to $\varepsilon^{2}.$ For the full numerical solution to the original PDE (\[pde\]), we took the computational domain to be of size $x\in\left[ 0,3\right]$. Because of the exponential decay outside the spikes, doubling the domain size did not change the results. The initial conditions were taken to be (\[rhoinner\]), with $M_{1}=0.35,\ M_{2}=0.65$ and the centers of the two Gaussian peaks were a distance $1$ apart. We also waited $t=10$ time units to let the transients die out before starting the comparison; at $t=10$ the system already converged to the quasi-steady state. The masses $M_{1}$ and $M_{2}$ in (\[v\]) were computed numerically using (\[eqn: def Mj\]) for each time-step. We used finite differences with semi-implicit time stepping: $v(x)$ is computed explicitly at each time step, while the update for $\rho(x,t+\Delta t)$ is done implicitly. We verified the accuracy by using several stepsizes. Discussion {#sec:discuss} ========== In this paper we focused on the steady states of the aggregation equation with noise (\[pde\]) that consist of two nearly-Dirac concentrations. An important implication of the work presented here, is that not all steady states of the zero-diffusion equation can be recovered as the limit as $\varepsilon\rightarrow0$ of a sequence of steady states of the aggregation-diffusion equation. This concerns in particular the two-Dirac steady states of unequal mass. Nevertheless, on short timescales some reminiscents of these unequal-mass steady states are still present. To be more precise, there is an intermediate timescale in which a state consisting of two Diracs of unequal mass persists as a metastable state. We have shown that the process of equilibration can be described asymptotically by an ODE for the evolution of the mass associated to each of the spikes. This ODE is valid on an exponentially long timescale, after the initial two-spike quasi-steady state profile is formed on an $O(1)$ timescale. An open question is to derive a more precise ODE for the dynamics, which would require the next-order expansion of the quasi-steady state; see figure \[fig:ddp\] and associated discussion in §\[sec: ODE derivation\]. An interesting, but nontrivial complication arises when one wants to derive an ODE for the mass evolution for three or more spikes. In the case of two spikes, these are centred a distance $a$ apart (with $a$ such that $f(a)=0$). For more than two spikes a delicate balance needs to be satisfied: there is an algebraic system of equations involving the masses and mutual distances, such that the velocity at each centre is zero. This algebraic relation, together with an ODE for the evolution of each mass, accounts for the simultaneous evolution of the centres of the spikes and the masses towards equilibrium. As in the case of two spikes, the steady state consisting of three delta-concentrations in the absence of diffusion is degenerate: there is an arbitrariness in how the three masses are distributed among the three holes, and there are two degrees of freedom (three masses subject to constraint $M_{1}+M_{2}+M_{3}=1$). However when the diffusion is turned on, this two-parameter family of steady states should “collapse” into a unique steady state. It would be very interesting to characterize precisely which mass fractions are “selected” by the diffusion. Unlike the two-spike solution where diffusion “chooses” the equal-mass configuration, in the case of the three-spike configuration, the diffusion should in general select unequal mass fractions. It would be interesting to extend these results to two and higher dimensions. In two dimensions, at least three delta-concentrations are required for stability [@kolokolnikov2013singular] (in the case three delta-concentrations, their locations form an equilateral triangle). The construction of the inner solution near the spike is analogous to the derivation in §\[sec:ss\]. On the other hand, the outer region cannot be easily solved, as it requires solving a fully two-dimensional PDE, and performing matching is a nontrivial problem. Nonetheless there is a hope that WKB-type techniques can be used to approximate the solution to the outer region for small diffusion. It would be interesting if similar equilibration results can be obtained in two dimensions. Acknowledgements ================ We thank the anonymous referees for their careful reading of the first draft of the paper and for pointing out that the constant $C(d)$ in the ODE (\[dp\]) was not fully resolved. J.E. is supported by an AARMS Postdoctoral Fellowship. T.K. is supported by NSERC discovery and NSERC accelerator grants.
--- author: - | Saverio Salzo$^1$ and Johan A.K. Suykens$^2$ and Lorenzo Rosasco$^{1,3}$\ $\!^1$LCSL, Istituto Italiano di Tecnologia and Massachusetts Institute of Technology\ Via Morego 30, 16163 Genova, Italy\ `saverio.salzo@iit.it`\ $\!^2$KU Leuven, ESAT-STADIUS\ Kasteelpark Arenberg 10, B-3001 Leuven (Heverlee), Belgium\ `johan.suykens@esat.kuleuven.be`\ $\!^3$DIBRIS, Università degli Studi di Genova\ Via Dodecaneso 35, 16146 Genova, Italy\ `lrosasco@mit.edu` bibliography: - 'publications.bib' title: ' [ **Solving $\ell^p$-norm regularization with tensor kernels** ]{}' --- Introduction ============ Kernel methods are classically formulated as a regularized empirical risk minimization and yields flexible and effective non-parametric models. However, they are restricted to $\ell^2$-regularization. Indeed the so called *kernel trick* crucially rely on a scalar product structure (a Hilbert space). The basic tool of these methods is the kernel function which, evaluated at the training points, allows $(a)$ to formulate a “dual” optimization problem, which is essentially quadratic and finite dimensional, and $(b)$, through the solution of dual problem, to obtain an explicit linear representation of the solution of the original (primal) problem (*the representer theorem*) [@Ste2008; @Vap98]. This dual approach provides a feasible way to deal with non-parametric (infinite dimensional) models, and a possibly easier and more efficient algorithm to tackle the finite dimensional also. It is well known that kernels for other norms can be defined [@Song13; @Zhan09; @Zhan12], but recent results suggest that they are unpractical [@nips11]. In particular, these kernels do not allows to properly express, in closed-form, the dual problem, making the kernel trick inapplicable. In this paper, we question this conclusion. We consider $\ell^p$-regularization for $1<p<2$ and starting from [@sal16b] we illustrate how, for certain values of $p$, a class of tensor kernels make it possible to derive a dual problem that can be efficiently solved. Our main contribution is a dual algorithm, having fast convergence properties, that provides a way to overcome the well-known computational issues related to non-Hilbertian norms, and makes the kernel trick still viable. From the optimization point of view, the challenge is that some standard assumptions are not satisfied. Indeed the dual objective function lacks a global Lipschitz continuous gradient, since it incorporates a convex polynomial of degree strictly greater than 2. Moreover, depending on the choice of the loss, constraints may be present. Considering all these aspects, the proposed algorithm is a dual proximal gradient method with linesearch which in the case of the least square loss and logistic loss we prove to converge linearly. Numerical examples show the effectiveness of the proposed framework and the possible application for variable selection. The rest of the paper is organized as follows. In section \[sec:main\] we explain how tensor kernels arise in $\ell^p$ regularization learning problems and provide an efficient algorithm to solve such problems, which is the main contribution of the paper. In section \[sec:theory\] one finds the main elements of the theoretical analysis. Finally, section \[sec:numerics\] contains the numerical experiments. #### Notation. If $p>1$, $q>1$ is its conjugate exponent, i.e. $1/p + 1/q = 1$. Vectors are denoted by bold fonts and scalars by plain fonts. For every $\bm{x},\bm{x}^\prime \in {\mathbb{R}}^d$, $\bm{x}\odot\bm{x}^\prime \in {\mathbb{R}}^d$ and $\bm{x}\otimes\bm{x}^\prime \in {\mathbb{R}}^{d\times d}$ are their Hadamard and tensor product respectively, and $\mathrm{sum}(\bm{x})\in {\mathbb{R}}$ denotes the sum of the components of $\bm{x}$. If ${\mathbb{K}}$ is a countable set, we denote by $\ell^p({\mathbb{K}})$ the space of $p$-summable sequences indexed in ${\mathbb{K}}$ with $p$-norm ${{\lVert {\bm{w}}\rVert}}_p= \big(\sum_{k \in {\mathbb{K}}} {{\lvert {w_k}\rvert}}^p\big)^{1/p}$. We define the *duality map* of $\ell^q({\mathbb{K}})$ as $J_q \colon \ell^q({\mathbb{K}}) \to \ell^p({\mathbb{K}})$ with $J_q(\bm{u}) = (\operatorname{sign}(u_k) {{\lvert {u_k}\rvert}}^{q-1})_{k \in {\mathbb{K}}}$ [@Sch2012]. Motivation and main contribution {#sec:main} ================================ First, we recall how kernel methods arise for $\ell^2$-regularization. Next, we present the objective of this study, i.e., an effective $\ell^p$-norm regularized learning method. Based on [@sal16b], which showed that this method can be *kernelized* by an appropriate *tensor kernel*, we present a novel dual algorithm which uses the knowledge of the tensor kernel only and converges linearly. Classical kernel methods {#subsec:back} ------------------------ We begin with a look at a simple kernel method, that is, *kernel ridge regression*, and we highlight the role played by duality. Later, this will serve as a guide to generalize the theory to $\ell^p$-regularization. Ridge regression is formulated as the following optimization problem $$\label{eq:ridge} \min_{\bm{w} \in {\mathbb{R}}^d} \frac{\gamma}{2} {{\lVert {\bm{X} \bm{w} -\bm{y}}\rVert}}_2^2 + \frac 1 2 {{\lVert {\bm{w}}\rVert}}_2^2,$$ where $\bm{X} \in {\mathbb{R}}^{n\times d}$ is the data matrix. This problem has a companion dual problem which is $$\label{eq:dualridge} \min_{\bm{\alpha} \in {\mathbb{R}}^n} \frac 1 2 {{\lVert {\bm{X}^* \bm{\alpha}}\rVert}}_2^2 + \frac{1}{2 \gamma} {{\lVert {\bm{\alpha}}\rVert}}_2^2 - {{\left\langle{{\bm{y}},{ \bm{\alpha}}}\right\rangle}}.$$ These two problems are indeed related: writing the optimality conditions for and one obtains $$\bm{X}^*(\bm{X} \bm{w} - \bm{y}) + \gamma^{-1} \bm{w}=0\qquad\text{and} \qquad\bm{X}\bm{X}^*\bm{\alpha} - \bm{y} + \gamma^{-1} \bm{\alpha} = 0$$ respectively; and hence it immediately follows that if $\bar{\bm{\alpha}}$ is the solution of , then $$\label{eq:representer} \bar{\bm{w}} = \bm{X}^* \bar{\bm{\alpha}} = \sum_{i=1}^n \bar{\alpha}_i \bm{x}_i$$ is the unique solution of . Equation is the content of the so called *representer theorem* which ensures that the solution of a regularized regression problem can be written as a linear combination of the data points $\bm{x}_i \in {\mathbb{R}}^d$, $i = 1, \cdots, n$. Moreover, for the linear estimator it holds $$\label{eq:representer2} {{\left\langle{{\bar{\bm{w}}},{ \bm{x}}}\right\rangle}} = \sum_{i=1}^n \bar{\alpha}_i {{\left\langle{{\bm{x}_i},{ \bm{x}}}\right\rangle}} = \sum_{i=1}^n \bar{\alpha}_i K(\bm{x}_i,\bm{x}),$$ where $K\colon {\mathbb{R}}^d \times {\mathbb{R}}^d \to {\mathbb{R}}$ is the *linear kernel function* defined as $K(\bm{x},\bm{x}^\prime) = {{\left\langle{{\bm{x}},{\bm{x}^\prime}}\right\rangle}}$. We note that, since $\bm{X} \bm{X}^* = (K(\bm{x}_i,\bm{x}_j))_{1 \leq i \leq n, 1 \leq j \leq n}$, the dual problem can also be written in terms of the linear kernel function, taking the form of the following quadratic optimization problem $$\label{eq:dualridge2} \min_{\bm{\alpha} \in {\mathbb{R}}^n} \frac 1 2 \sum_{i,j=1}^n K(\bm{x}_i,\bm{x}_j) \alpha_i \alpha_j + \frac{1}{2\gamma} {{\left\langle{{\bm{\alpha}},{\bm{\alpha}}}\right\rangle}} - {{\left\langle{{\bm{y}},{\bm{\alpha}}}\right\rangle}}.$$ So, summarizing, the dual problem and the representation formulas - provide a way to solve the primal problem and to evaluate the optimal linear estimator by relying on the knowledge of the linear kernel function only. This conclusion can then be extended to nonlinear regression models, by introducing general kernel functions defined as $$\label{eq:kernel} K(\bm{x},\bm{x}^\prime) = {{\left\langle{{\Phi(\bm{x})},{\Phi(\bm{x}^\prime)}}\right\rangle}} = \mathrm{sum} (\Phi(\bm{x}) \odot \Phi(\bm{x}^\prime)),$$ for some nonlinear *feature map* $\Phi\colon {\mathbb{R}}^d \to \ell^2$. This is the so called *kernel trick* and it is at the basis of *kernel methods* in machine learning, allowing even to treat infinite dimensional (nonparametric) models. Kernels, defined by , can indeed be characterized as *positive definite functions*, in the sense that for every $n \in {\mathbb{N}}$, $(\bm{x}_i)_{1 \leq i \leq n} \in {({\mathbb{R}}^d)}^n$, and $\bm{\alpha} \in {\mathbb{R}}^n$, $\sum_{i,j=1}^nK(\bm{x}_i,\bm{x}_j)\alpha_i \alpha_j \geq 0$. Moreover, kernels define an associated function space which is a reproducing kernel Hilbert space. There are many significant examples of kernel functions and we cite among the other the Gaussian kernel $K(\bm{x},\bm{x}^\prime) = \exp(- \eta^{-2} {{\lVert {\bm{x} - \bm{x}^\prime}\rVert}}_2^2)$ and the polynomial kernel $K(\bm{x},\bm{x}^\prime) = {{\left\langle{{\bm{x}},{\bm{x}^\prime}}\right\rangle}}^s$, describing the space of homogeneous polynomials of degree $s$. We note that the theory can be further generalized to handle more general loss functions, so to include classification problems too [@Ste2008; @Vap98]. Kernel methods beyond $\ell^2$-regularization --------------------------------------------- In view of the discussion above, a natural question is whether kernel methods can be extended to other regularization terms. In particular $\ell^1$-regularization would be important in view of its properties to provide sparse solutions. Unfortunately, in general $\ell^1$-regularization methods cannot be *kernelized* (although they admit dual) [@Has15; @Osb00] and a useful representer theorem and definition of kernel can be obtained only under severe restrictions [@Song13]. However, it was noted in [@Kol09] that $\ell^p$-regularization can be seen as a proxy to $\ell^1$ for suitable $p$. Moreover, it was recently shown in [@sal16b] that for certain values of $p \in \left]1,2\right[$ (arbitrarily close to $1$), the $\ell^p$-regularization method can indeed be *kernelized*, provided that a suitable definition of tensor kernel is introduced. Here we recall the theory in [@sal16b] for a simple model in order to make it more transparent. Thus, in analogy to section \[subsec:back\], we consider the problem $$\label{eq:ellp} \min_{\bm{w} \in {\mathbb{R}}^d} \frac{\gamma}{2} {{\lVert {\bm{X} \bm{w} - \bm{y}}\rVert}}_2^2 + \frac 1 p {{\lVert {\bm{w}}\rVert}}_p^p:=F(\bm{w}),$$ where $1<p<2$. In this case the dual problem is $$\label{eq:dualellp} \min_{\bm{\alpha} \in {\mathbb{R}}^d}\frac{1}{q} {{\lVert {\bm{X}^* \bm{\alpha}}\rVert}}_q^q + \frac{1}{2 \gamma} {{\lVert {\bm{\alpha}}\rVert}}_2^2 - {{\left\langle{{\bm{y}},{\bm{\alpha}}}\right\rangle}}:=\Lambda(\bm{\alpha}),$$ where $q$ is the conjugate exponent of $p$ (that is $1/p + 1/q = 1$). Now, following the same argument as in section \[subsec:back\], we write the optimality conditions of the two problems. Then we have $$\label{eq:kktellp} \bm{X}^*(\bm{X} \bm{w} - \bm{y}) + \gamma^{-1} J_p(\bm{w}) = 0\quad\text{ and }\quad \bm{X} J_q(\bm{X}^*\bm{\alpha}) - \bm{y} + \gamma^{-1} \bm{\alpha} = 0,$$ where $J_p \colon {\mathbb{R}}^d \to {\mathbb{R}}^d$ and $J_q\colon {\mathbb{R}}^d \to {\mathbb{R}}^d$ are the gradients of $(1/p) {{\lVert {\cdot}\rVert}}_p^p$ and $(1/q) {{\lVert {\cdot}\rVert}}_q^q$ respectively (they are the duality maps). Thus, multiplying by $\bm{X}^*$ the second equation in and taking into account that $J_p \circ J_q = \mathrm{Id}$, it follows that if $\bar{\bm{\alpha}}$ is the solution of , then $\bar{\bm{w}} = J_q(\bm{X}^* \bar{\bm{\alpha}})$ is the solution of . So, in this case the *representer theorem* becomes $$\label{eq:20170511a} \bar{\bm{w}} = J_q(\bm{X}^* \bar{\bm{\alpha}}) = J_q \bigg( \sum_{i=1}^n \bar{\alpha}_i \bm{x}_i\bigg).$$ We remark that, in contrast to the $\ell^2$ case, the above representation is nonlinear in the $\alpha_i$’s, because of the presence of the nonlinear map $J_q$. Indeed this map acts component-wise as the derivative of $(1/q){{\lvert {\cdot}\rvert}}^q$, i.e., $\operatorname{sign}(\cdot) {{\lvert {\cdot}\rvert}}^{q-1}$. Therefore, at first sight it is not clear how to define an appropriate kernel function that can represent the estimator ${{\left\langle{{\bar{\bm{w}}},{\bm{x}}}\right\rangle}}$ in analogy to , and make the kernel trick still successful. So, it comes as a surprise that this is possible if one makes the following assumption [@sal16b] $$\label{q-hypothesis} \boxed{ q \text{ is an even integer and } q \geq 2. }$$ Indeed in that case, for every $\bm{u} \in {\mathbb{R}}^d$, $J_q(\bm{u}) = \big(\operatorname{sign}(u_j) {{\lvert {u_j}\rvert}}^{q-1}\big)_{1 \leq j \leq d} =(u_j^{q-1})_{1 \leq j \leq d}$, and hence, using , we have $$\label{eq:20170514a} {{\left\langle{{\bar{\bm{w}}},{\bm{x}}}\right\rangle}} = \sum_{j=1}^d \bigg(\sum_{i=1}^n \bar{\alpha}_i x_{i,j}\bigg)^{q-1}\!\!\!\! x_j = \sum_{j=1}^d \sum_{i_1,\dots, i_{q-1}=1}^n \!\! x_{i_1,j}\cdots x_{i_{q-1},j} x_j \bar{\alpha}_{i_1} \cdots \bar{\alpha}_{i_{q-1}},$$ where we could expand the power of the summation in a multilinear form since $q$ is an integer. Therefore, we are defining the *linear tensor kernel function* $K$ as $$\label{eq:linearker} K\colon \underbrace{{\mathbb{R}}^d \times \cdots \times {\mathbb{R}}^d}_{q\ \text{times}} \to {\mathbb{R}}, \quad K(\bm{x}^\prime_{1}, \cdots, \bm{x}^\prime_q) = \sum_{j=1}^d x^\prime_{1,j}\cdots x^\prime_{q,j} = \mathrm{sum} (\bm{x}^\prime_1 \odot \cdots \odot \bm{x}^\prime_q),$$ so that, turns to $$\label{eq:20170511b} {{\left\langle{{\bar{\bm{w}}},{\bm{x}}}\right\rangle}} = \hspace{-2ex} \sum_{i_1,\dots, i_{q-1}=1}^n \hspace{-2ex}K(\bm{x}_{i_1}, \cdots, \bm{x}_{i_{q-1}}, \bm{x} ) \bar{\alpha}_{i_1} \cdots \bar{\alpha}_{i_{q-1}}.$$ Comparing and we recognize that we may interpret the tensor kernel as a kind of group-wise similarity measure in the input space. Moreover, since $q$ is even, $${{\lVert {X^* \bm{\alpha}}\rVert}}_q^q = \sum_{j=1}^d \bigg(\sum_{i=1}^n \alpha_i x_{i,j}\bigg)^q =\sum_{j=1}^d \sum_{i_1, \dots, i_q = 1}^n x_{{i_1},j} \cdots x_{{i_q},j} \alpha_{i_1}\cdots \alpha_{i_q}$$ and hence, by exchanging the two summations above, the dual problem becomes $$\label{eq:dualellp2} \min_{\bm{\alpha} \in {\mathbb{R}}^d} \frac 1 q \sum_{i_1, \dots, i_q = 1}^n K(\bm{x}_{i_1}, \dots, \bm{x}_{i_q}) \alpha_{i_1}\cdots \alpha_{i_q} + \frac{1}{2 \gamma} {{\lVert {\bm{\alpha}}\rVert}}^2 - {{\left\langle{{\bm{y}},{\bm{\alpha}}}\right\rangle}}.$$ We see now that, instead of the quadratic problem we have a convex polynomial optimization problem of degree $q$.[^1] The introduction of the tensor kernel allows to parallel the $\ell^2$ case, in the sense that the dual problem and formula provide the solution of the regression problem . Once again, the method can be extended to general feature maps $\Phi\colon {\mathbb{R}}^d \to \ell^q({\mathbb{K}})$, $\Phi(\bm{x}) = (\phi_k(\bm{x}))_{k \in {\mathbb{K}}}$, with ${\mathbb{K}}$ a countable set, provided that, in the definition of $K$, $\bm{x}_i$ is replaced by $\Phi(\bm{x}_i)$. Thus, a general *tensor kernel* is defined as $$\label{def:tensorkernel2} K(\bm{x}^\prime_{1}, \cdots, \bm{x}^\prime_q) = \sum_{k \in {\mathbb{K}}} \phi_k(\bm{x}^\prime_{1})\cdots \phi_k(\bm{x}^\prime_{q}) = \mathrm{sum} (\Phi(\bm{x}^\prime_1) \odot \cdots \odot \Phi(\bm{x}^\prime_q)).$$ It is easy to show that tensor kernels are still symmetric and positive definite, in the sense that - $\forall\, \bm{x}^\prime_1, \dots, \bm{x}^\prime_q \in {\mathbb{R}}^d$, and every permutation $\sigma$ of $\{1,\dots, q\}$, $K({\bm{x}}^\prime_{\sigma(1)} \dots {\bm{x}}^\prime_{\sigma(q)}) = K({\bm{x}}^\prime_{1}, \dots {\bm{x}}^\prime_{q})$; - for every $\bm{x}^\prime_1, \dots, \bm{x}^\prime_n \in {\mathbb{R}}^d$ and every $\bm{\alpha} \in {\mathbb{R}}^n$, $\sum_{i_1,\dots, i_q = 1}^n K({\bm{x}}^\prime_{i_1}, \dots, {\bm{x}}^\prime_{i_q}) \alpha_{i_1} \dots \alpha_{i_q} \geq 0$.[^2] These tensor kernels define an associated function space which is now a reproducing kernel Banach space (See Section A.3 in the supplementary material and [@sal16b; @Zhan09]). Moreover, reasoning as in , the following representation formula can be proved $$\label{eq:20171009b} {{\left\langle{{\bar{\bm{w}}},{\Phi(\bm{x})}}\right\rangle}} =\hspace{-2ex}\sum_{i_1,\dots, i_{q-1}=1}^n \hspace{-3ex} K(\bm{x}_{i_1}, \cdots, \bm{x}_{i_{q-1}}, \bm{x} ) \bar{\alpha}_{i_1} \cdots \bar{\alpha}_{i_{q-1}}.$$ Finally, there do exist cases in which tensor kernel functions can be computed without knowing the feature map $\Phi$ itself. The following *polynomial* and *exponential* tensor kernels are examples of such cases (but, there are others in the class of power series tensor kernels [@sal16b]). #### Polynomial tensor kernel of degree $s \in {\mathbb{N}}$, $s \geq 1:$ $$K(\bm{x}^\prime_1,\dots, \bm{x}^\prime_q) = \Big(\sum_{j=1}^d x^\prime_{1,j} \cdots x^\prime_{q,j} \Big)^{s} = \big(\mathrm{sum}(\bm{x}^\prime_1 \odot \cdots \odot \bm{x}^\prime_q) \big)^s.$$ It describes the space of homogeneous polynomials in $d$ real variables of degree $s$. This corresponds to a finite dimensional model for which ${\mathbb{K}}= \big\{ k \in {\mathbb{N}}^d \,\big\vert\, \sum_{j=1}^d k_j = s\big\}$ and, for every $k \in {\mathbb{N}}^d$, $\phi_k(\bm{x}) = \big( s!/(k_1! \cdots k_d !)\big)^{1/q} \bm{x}^k$, that is $(\phi_k)_{k \in {\mathbb{K}}}$ is the basis of all possible monomials in $d$ variables of degree $s$ and the norm of a polynomial function $f = \sum_{k \in {\mathbb{K}}} w_k \phi_k$ is ${{\lVert {\bm{w}}\rVert}}_p^p = \sum_{k \in {\mathbb{K}}} {{\lvert {w_k}\rvert}}^p$. #### Exponential tensor kernel [:]{} $$K(\bm{x}^\prime_1,\dots, \bm{x}^\prime_q) = \prod_{j=1}^d e^{x^\prime_{1,j}\cdots x^\prime_{q,j}}\\ = e^{\mathrm{sum}(\bm{x}^\prime_1 \odot \cdots \odot \bm{x}^\prime_q)}.$$ This kernel provides an example of an infinite dimensional model, where, ${\mathbb{K}}= {\mathbb{N}}^d$ and, for every $k \in {\mathbb{N}}^d$, the $k$-th component of the feature map is $\phi_k(\bm{x}) = \big(1/\prod_{j=1}^d k_j ! \big)^{1/q} \bm{x}^k$. A dual algorithm {#subsec:dualalgo} ---------------- In this section we present the main contribution of this paper which is an algorithm for solving the problem $$\label{eq:gen1primal} \min_{\bm{w} \in \ell^p({\mathbb{K}})}\! \!\gamma \sum_{i=1}^n \big(y_i - {{\left\langle{{\Phi(\bm{x}_i)},{ \bm{w}}}\right\rangle}}\big)^2 \!+\! \frac 1 p {{\lVert {\bm{w}}\rVert}}_p^p:=F(\bm{w}),$$ where $p=q/(q-1)$ with $q$ an even integer (strictly) grater than $2$, $\gamma>0$, $\displaystyle\Phi\colon {\mathcal{X}}\to \ell^{q}({\mathbb{K}})$ is the feature map, ${\mathbb{K}}$ is a countable set, and $(\bm{x}_i,y_i)_{1 \leq i \leq n} \in ({\mathcal{X}}\times {\mathcal{Y}})^n$ is the training set. Note that reduces to if $\mathcal{X} = {\mathbb{R}}^d$, $\mathbb{K} = \{1,\dots, d\}$ and $\Phi$ is the identity map. The proposed algorithm is based on the minimization of the dual problem , where $K$ is defined as in . This method has two significant characteristics: first, it is entirely formulated in terms of the tensor kernel function, therefore it can also cope with nonparametric (infinite dimensional) tensor kernels, e.g, the exponential-tensor kernel; second, it provides fast convergence. From the optimization viewpoint, we observe that the objective functions in and are smooth. However, none of the two has Lipschitz continuous gradient, since in $1<p<2$ and in the first term is a convex polynomial of degree $q>2$. This poses an issue since most gradient algorithms requires Lipschitz continuous gradient to achieve convergence [@Beck09; @Comb05; @Dau04]. Relaxing this assumption for the more general proximal gradient algorithm has been the objective of a number of recent works [@Cruz15; @Bon15a; @sal17a] that introduce suitable linesearch procedures to determine the gradient stepsizes. In light of these studies, we present a dual gradient descent algorithm with a backtracking linesearch procedure and we prove that, by exploiting the strong convexity of the dual objective function and the dual-primal link, the corresponding primal iterates converge linearly to the solution of . To simplify the exposition we treat here the case $q=4$, that is $p=4/3$. Since the *Gram tensor* $\bm{\mathsf{K}} = (K(\bm{x}_{i_1}, \bm{x}_{i_2}, \bm{x}_{i_3}, \bm{x}_{i_4}))_{ i \in \{1, \dots n\}^4}$ is of order $4$, it can be viewed as a $n^2\times n^2$ symmetric matrix: using a MATLAB-like notation, we define $[\bm{\mathsf{K}}] = \mathrm{reshape}(\bm{\mathsf{K}}, n^2, n^2)$. Likewise, for a $n\times n$ matrix $\bm{B}$, we set $[\bm{B}] = \mathrm{reshape}(\bm{B}, n^2, 1)$ for its vectorization. Then, the dual problem can be equivalently written as $$\label{eq:dualellp3} \min_{\bm{\alpha} \in {\mathbb{R}}^d} \frac 1 q {{\left\langle{{[\bm{\alpha}\otimes \bm{\alpha}]},{[\bm{\mathsf{K}}] [\bm{\alpha} \otimes \bm{\alpha}]}}\right\rangle}} + \frac{1}{2 \gamma} {{\lVert {\bm{\alpha}}\rVert}}^2 - {{\left\langle{{\bm{y}},{\bm{\alpha}}}\right\rangle}}:= \Lambda(\bm{\alpha}).$$ The proposed dual algorithm is detailed below. \[algo:main\] Let $\bm{\alpha}_0 \in {\mathbb{R}}^n$, $\delta, \theta \in \left]0,1\right[$, and initialize the sequence $(\lambda_m)_{m \in {\mathbb{N}}}$ as the constant value $\bar{\lambda} \in \left]0,\gamma/(2(1-\delta))\right[$. Then, for every $m \in {\mathbb{N}}$, $$\label{eq:genaccalgo} \begin{array}{l} \begin{array}{l} \bm{\omega}_m = \mathrm{reshape}([\bm{\mathsf{K}}] [\bm{\alpha}_m \otimes \bm{\alpha}_m], n,n) \bm{\alpha}_m\ \text{(the gradient of the quartic part of $\Lambda$)}\\[1ex] \nabla \Lambda(\bm{\alpha}_m) = \bm{\omega}_m - \bm{y}+ \gamma^{-1} \bm{\alpha}_m\\[1ex] \text{while $\Lambda(\bm{\alpha}_m) - \Lambda(\bm{\alpha}_{m} - \lambda_m \nabla \Lambda(\bm{\alpha}_m))< \lambda_m (1-\delta) {{\lVert {\nabla \Lambda(\bm{\alpha}_m)}\rVert}}^2$ do}\\[1ex] \hspace{-1mm}\vspace{2mm}\left\lfloor \begin{array}{l} \lambda_m := \theta \lambda_m \\ \end{array} \right.\\ \bm{\alpha}_{m+1} = (1 - \lambda_m \gamma^{-1}) \bm{\alpha}_m - \lambda_m (\bm{\omega}_m - \bm{y})\\ \end{array} \end{array}$$ Algorithm \[algo:main\] is given for $q=4$. If $q$ is an even integer greater than $4$, then the leading term of $\Lambda$ is a polynomial of degree $q$ in the variables $\bm{\alpha}=(\alpha_1, \dots, \alpha_m)$, and the formula for its gradient $\bm{\omega}_m$ at $\bm{\alpha}_m$, even if possibly more complicated, can be still expressed in term of the Gram tensor $\bm{\mathsf{K}}$. Our main technical result is the following theorem studying the convergence of the above algorithm. \[thm:main\] Let $(\bm{\alpha}_m)_{m \in {\mathbb{N}}}$ and $(\lambda_m)_{m \in {\mathbb{N}}}$ be generated by Algorithm \[algo:main\]. Then we have $\inf_{m} \lambda_m>0$ and, for every $m \in {\mathbb{N}}$, setting $\bm{w}_m = J_q\big(\sum_{i=1}^n \alpha_{m,i}\Phi(\bm{x}_i) \big)$, it holds $${{\lVert {\bm{w}_m - \bar{\bm{w}}}\rVert}}_p^2 \leq \dfrac{ \big[(2^p q) \big(\Lambda(\bm{\alpha}_0) + (\gamma/2) {{\lVert {\bm{y}}\rVert}}_2^2\big)\big]^{\frac{2-p}{p}} }{C_p} \cdot\bigg(1 - \frac{2}{\gamma} \lambda_m(1 - \delta)\bigg)^m \big( \Lambda(\bm{\alpha}_0) - \min \Lambda\big),$$ for some constant $C_p>0$, depending only on $p$, which tends to zero as $p\to 1$. Therefore, $\bm{w}_m$ converges linearly to the solution $\bar{\bm{w}}$ of problem . An output $\bm{\alpha}_m = (\alpha_{m,1},\dots, \alpha_{m,n})$ of Algorithm \[algo:main\] provides an estimator ${{\left\langle{{\bm{w}_m},{\Phi(\cdot)}}\right\rangle}}$, that can be expressed in terms of the tensor kernel $K$ through the equation $${{\left\langle{{\bm{w}_m},{\Phi(\cdot)}}\right\rangle}} = \sum_{i_1,\dots, i_{q-1}=1}^n K(\bm{x}_{i_1}, \cdots, \bm{x}_{i_{q-1}}, \cdot ) \alpha_{m,i_1} \cdots \alpha_{m,i_{q-1}}.$$ Indeed, this follows from recalling the definition of $\bm{w}_m$ in Theorem \[thm:main\] and by reasoning as in . \[rmk:20171013a\] If $p \in \left]1,2\right[$ is not of the form $p=q/(q-1)$ for some $q$ as in , Theorem \[thm:main\] remains valid provided that in Algorithm \[algo:main\] $\bm{\omega}_m$ is computed directly in terms of the feature map $\Phi$ evaluated at the training points. Clearly, this case is feasible only if the feature map is finite dimensional, that is, if the index set $\mathbb{K}$ is finite. In the following we discuss the most significant aspects of this dual approach. #### Cost per iteration. The complexity of Algorithm \[algo:main\] is mainly related to the computation of the gradient of the quartic form in , which, by exploiting the symmetries of $\bm{\alpha}_m\otimes \bm{\alpha}_m$ and $\bm{\mathsf{K}}$, costs (approximatively) $n^2(n+1)^2/4$ multiplications. We remark that in the infinite dimensional case this algorithm is the only feasible approach to solve problem . However, even in the case $\mathrm{card}({\mathbb{K}})<+\infty$, e.g., for the linear or polynomial tensor kernel, the method may be convenient if $n \ll \mathrm{card}({\mathbb{K}})$. Indeed a standard gradient-type algorithm on costs $2 n \mathrm{card}({\mathbb{K}})$ multiplications ($2 n d$ in case of ). Therefore, Algorithm \[algo:main\] is recommended if $n(n+1)^2/8 \leq \mathrm{card}({\mathbb{K}})$, that is $$\label{eq:compucost} n \leq 2 \big(\mathrm{card}({\mathbb{K}}) \big)^{1/3}.$$ We stress that Algorithm \[algo:main\] has a cost per iteration that depends only on the size $n$ of the data set, while any primal approach will depend on the size of ${\mathbb{K}}$. For instance, in the case of polynomial kernels of degree $s$, we have $\mathrm{card}({\mathbb{K}}) = (d+s-1) \cdots d/s! \geq d^s/s!$, and this implies that the cost of a gradient algorithm on the primal problem grows exponentially with $s$. We also remark that building the Gram tensor $\bm{\mathsf{K}}$ will further require $d\cdot n^2(n+1)^2/4$ multiplications (and $8\cdot n^4/8$ bytes in space). However, the Gram tensor is computed once for all and in a validation procedure for the regularization parameter $\gamma$, it does not need to be recomputed every time. #### Rate of convergence. As mentioned above our dual algorithm has linear convergence rate and can be applied for infinite dimensional kernels. We next discuss the comparison with primal approaches when the kernel is finite dimensional ($\mathrm{card}({\mathbb{K}})<+\infty$). The basic point is that primal approaches will allow only for sublinear rates. Indeed, since the objective function in is the sum of two convex smooth functions, among the various algorithms, appropriate choices are $(a)$ a pure gradient descent algorithm with linesearch (the gradient being that of $F$) and $(b)$ a proximal gradient algorithm (possibly accelerated) with the prox of $(1/p){{\lVert {\cdot}\rVert}}_p^p$. However, concerning $(a)$ and according to [@Bon15a; @sal17a], the algorithm converges, but, since $1<p<2$, the full gradient of $F$ is not even locally Lipschitz continuous, so, the gradient stepsizes may get arbitrarily close to zero, and ultimately the algorithm may exhibit very slow convergence with no explicit rate. Besides, regarding $(b)$, the primal objective function in is only uniformly convex on bounded sets. Therefore, standard convergence results [@Beck09; @Cha15; @Comb05] ensure only convergence of the iterates (without rate) and sublinear convergence rate for the objective values. On the other hand, regarding Algorithm \[algo:main\], we observe that the constant $C_p$, in Theorem \[thm:main\], approaches zero as $p\to 1$, so when $p$ is close to 1 the linear convergence rate for the $\bm{w}_m$’s may degrade. In the numerical experiments, we confirm the above theoretical behaviors: the dual algorithm often converges in a few iterations (of the order of 20), whereas a direct gradient descent method (with linesearch or of proximal-type) on the primal problem may require thousands of iterations to reach the same precision. #### Dealing with general convex loss. Above, we considered, for the sake of simplicity, the least squares loss. However, the proposed dual approach can be generalized to all other convex loss functions commonly used in machine learning: the *logistic loss* and the *hinge loss* for classification and the *$L^1$-loss*, and the *Vapnik-$\varepsilon$-insensitive loss* for regression. In these cases the dual objective function is composed of the same leading polynomial form as in , which has locally Lipschitz continuous gradient, and of a possibly nonsmooth (convex) function, having however a closed-form proximity operator (see Example \[ex:losses\] in the supplementary material). Therefore, according to [@sal17a], for general convex losses, instead of Algorithm \[algo:main\] we use a proximal gradient algorithm with linesearch achieving linear convergence or sublinear convergence depending on the fact that the dual objective function is strongly convex or not. In this respect we note that we have linear convergence for the logistic loss and sublinear convergence for the $\varepsilon$-insensitive loss and the hinge loss. This extension is treated in the next section. Main elements of the theoretical analysis {#sec:theory} ========================================= In this section we further develop the discussion of the previous section and provide the theoretical grounds for the dual approach to $\ell^p$-norm regularized learning problems. The emphasis here is on the duality theory rather than on the tensor kernels. The results are presented for general loss function and any real parameter $p>1$. The most general formulation of our objective is as follows, $$\label{eq:genprimal} \min_{\bm{w} \in \ell^p({\mathbb{K}})} \gamma \sum_{i=1}^n L(y_i, {{\left\langle{{\Phi(\bm{x}_i)},{ \bm{w}}}\right\rangle}})\! +\! \frac 1 p {{\lVert {\bm{w}}\rVert}}_p^p:=F(\bm{w}),$$ where $ p>1,\gamma>0$, $\displaystyle\Phi\colon {\mathcal{X}}\to \ell^{q}({\mathbb{K}})$ is the feature map, $(\bm{x}_i,y_i)_{1 \leq i \leq n} \in ({\mathcal{X}}\times {\mathcal{Y}})^n$ is the training set, and $L\colon {\mathcal{Y}}\times {\mathbb{R}}\to {\mathbb{R}}$ is a loss function which is convex in the second variable. We define the linear *feature operator* $$\label{featmat} \Phi_n \colon \ell^p({\mathbb{K}}) \to {\mathbb{R}}^n,\quad \Phi_n\bm{w}= {\big({{\left\langle{{\Phi(\bm{x}_i)},{\bm{w}}}\right\rangle}}\big)}_{1 \leq i \leq n}.$$ Then its adjoint is $\Phi_n^* \colon {\mathbb{R}}^n \to \ell^q({\mathbb{K}})$, $\Phi_n^* \bm{\alpha}= \sum_{i=1}^n \alpha_i \Phi(\bm{x}_i)$. Duality is based on the following. \[thm:duality\] The dual problem of is $$\label{eq:gendual} \min_{\bm{\alpha} \in {\mathbb{R}}^n} \frac{1}{q}{{\lVert {\Phi_n^* \bm{\alpha}}\rVert}}^{q}_{q} + \gamma \sum_{i=1}^n L^*\Big(y_i, - \frac{\alpha_i}{\gamma}\Big) := \Lambda(\bm{\alpha}),$$ where $L^*(y_i, \cdot)$ is the Fenchel conjugate of $L(y_i, \cdot)$. Moreover, $(i)$ the primal problem has a unique solution, the dual problem has solutions and $\min F = - \min \Lambda$ (strong duality holds); and $(ii)$ the solutions $(\bar{\bm{w}}, \bar{\bm{\alpha}})$ of the primal and dual problems are characterized by the following KKT conditions $$\label{eq:kkt} \begin{cases} \bar{\bm{w}} = J_q( \Phi_n^* \bar{\bm{\alpha}}),\\ \forall\, i \in \{1,\dots, n\}\ \ - \frac{\alpha_i}{\gamma} \in \partial L(y_i, {{\left\langle{{\Phi(\bm{x}_i)},{ \bar{\bm{w}}}}\right\rangle}}), \end{cases}$$ where $\partial L(y_i,\cdot)$ is the subdifferential of $L(y_i, \cdot)$. All the losses commonly used in machine learning admit explicit Fenchel conjugates and we refer to the supplementary material for explicit examples. The connection between the primal and dual problem is further deepened in the following result. \[prop:dual2primal\] Let $\bar{\bm{\alpha}} \in {\mathbb{R}}^n$ be a solution of the dual problem $\eqref{eq:gendual}$ and let $\bar{\bm{w}} = J_q\big( \Phi_n^* \bar{\bm{\alpha}} \big)$ be the solution of the primal problem . Let $\bm{\alpha} \in {\mathbb{R}}^n$ and set $\bm{w} = J_q\big( \Phi_n^* \bm{\alpha} \big)$. Then $$\label{eq:dualineq} \Lambda(\bm{\alpha}) - \min \Lambda \geq \dfrac{C_p}{ \big[(2^{p} q)\big( \Lambda(\bm{\alpha}) + \gamma {{\lVert {\bm{\xi}}\rVert}}_1\big) \big]^{(2-p)/p}} {{\lVert {\bm{w} - \bar{\bm{w}}}\rVert}}^2_p,$$ where, for every $i=1,\dots, n$, $\xi_i = \inf L^*(y_i, \cdot)$ and $C_p>0$ is a constant that depends only on $p$. The above proposition ensures that if an algorithm generates a sequence $(\bm{\alpha}_m)_{m \in {\mathbb{N}}}$ that is minimizing for the dual problem $\eqref{eq:gendual}$, i.e., $\Lambda(\bm{\alpha}_m) \to \min \Lambda$, then the sequence defined by $\bm{w}_m = J_q(\Phi_n^* \bm{\alpha}_m)$, $m \in {\mathbb{N}}$, converges to the solution of the primal problem. Now, for the most significant losses $L$ in machine learning (see Example \[ex:losses\] in the supplementary material), the dual problem has the following form $$\label{eq:201705417a} \min_{\bm{\alpha} \in {\mathbb{R}}^n} \varphi_1(\bm{\alpha}) + \varphi_2(\bm{\alpha})=\Lambda(\bm{\alpha}),$$ where $\varphi_1\colon {\mathbb{R}}^n \to {\mathbb{R}}$ is convex and smooth with locally Lipschitz continuous gradient ($\varphi_1$ will include the term $(1/q) {{\lVert {\Phi_n^* \alpha}\rVert}}_q^q$) and $\varphi_2\colon {\mathbb{R}}^n \to {\mathbb{R}}\cup\{+\infty\}$ is proper, lower semicontinuous, convex, and admitting a closed-form proximity operator. So, the form is amenable by the proximal gradient algorithm with linesearch studied in [@sal17a], which, referring to , takes the following form. \[dualalgo\] Let $\delta \in \left]0,1\right[$, $\bar{\lambda}>0$, and let $\theta \in \left]0,1\right[$. Let $\bm{\alpha}_0 \in {\mathbb{R}}^n$ and define, for every $m \in {\mathbb{N}}$, $$\bm{\alpha}_{m+1} = {\mathrm{prox}}_{\lambda_m \varphi_2} (\bm{\alpha}_m - \lambda_m \nabla \varphi_1(\bm{\alpha}_m)),$$ where $\lambda_m = \bar{\lambda} \theta^{j_m}$ and $j_m$ is taken as the minimum of the indexes $j \in {\mathbb{N}}$ such that $\hat{\bm{\alpha}}_m(j) := {\mathrm{prox}}_{\lambda_m \varphi_2} (\bm{\alpha}_m - \bar{\lambda} \theta^j \nabla \varphi_1(\bm{\alpha}_m))$ satisfies $$\varphi_1\big( \hat{\bm{\alpha}}_m(j)\big) - \varphi_1\big( \bm{\alpha}_m\big) - {{\left\langle{{ \hat{\bm{\alpha}}_m(j) - \bm{\alpha}_m},{\nabla \varphi_1(\bm{\alpha}_m)}}\right\rangle}} \leq \delta/(\bar{\lambda} \theta^j) {{\lVert { \hat{\bm{\alpha}}_m(j) - \bm{\alpha}_m}\rVert}}^2.$$ In contrast to Algorithm \[algo:main\], Algorithm \[dualalgo\] provides rather a general algorithm where $\varphi_1$ and $\varphi_2$ are set depending on the choice of the different losses. If $p=q/(q-1)$ and $q$ satisfies , then the computation of $\nabla \varphi_1(\bm{\alpha})$ in Algorithm \[dualalgo\] can be performed in term of the Gram tensor $\bm{\mathsf{K}}$ (for instance, if $q=4$ the gradient of the quartic part of $\varphi_1$ is as in the first line of Algorithm \[algo:main\]). Moreover, if in addition $L$ is the square loss, then $\Lambda$ is as in and one can take $\varphi_1=\Lambda$ and $\varphi_2=0$; and hence Algorithm \[dualalgo\] reduces to Algorithm \[algo:main\]. The convergence properties of Algorithm \[dualalgo\] are given in the following theorem, which, as opposed to Theorem \[thm:main\], is valid for general loss and any $p \in \left]1,2\right]$. \[thm:dualalgo\] Let $p \in \left]1,2\right]$. Define $(\bm{\alpha}_m)_{m \in {\mathbb{N}}}$ and $(\lambda_m)_{m \in {\mathbb{N}}}$ as in Algorithm \[dualalgo\]. Then, $\inf_{m} \lambda_m>0$ and, for every $m \in {\mathbb{N}}$, setting $\bm{w}_m = J_q(\Phi_n^* \bm{\alpha}_m)$, it holds $${{\lVert {\bm{w}_m - \bar{\bm{w}}}\rVert}}_p \leq o(1/\sqrt{m}).$$ Moreover, if $\Lambda$ is strongly convex (which occurs for the least square loss and the logistic loss), then $\bm{w}_m$ converges linearly to $\bar{\bm{w}}$. Numerical Experiments {#sec:numerics} ===================== We made experiments on simulated data in order to assess the following three points.[^3] ------------------------ --------- --------- --------- ---------- Algorithm $p=4/3$ $p=5/4$ $p=1.1$ $p=1.05$ dual GD + linesearch 12(5) 15(4) 63(22) 258(55) primal GD + linesearch $>5000$ $>5000$ $>5000$ $>5000$ primal FISTA 1158 1542 — — ------------------------ --------- --------- --------- ---------- : Convergence rates[]{data-label="rate"} ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------ ![Left. Convergence rates: dual algorithm vs FISTA on the primal. Right. True and estimated sparse vectors for a linear tensor kernel: $p=4/3$, $n=85$, $d=1500$, and $6$ relevant features.[]{data-label="fig1"}](dualvsprimal.pdf "fig:"){width="52.00000%"} ![Left. Convergence rates: dual algorithm vs FISTA on the primal. Right. True and estimated sparse vectors for a linear tensor kernel: $p=4/3$, $n=85$, $d=1500$, and $6$ relevant features.[]{data-label="fig1"}](ws_vs_wr.pdf "fig:"){width="52.00000%"} ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------ #### Dual vs primal approach (without tensor kernels). We considered problem with different choices of $p$ (not necessarily with $q$ even integer). The purpose is to compare a dual approach against a primal approach per se, thus without considering the tensor kernel function — after all the dual problem is smooth whatever $q$ is. Algorithm \[algo:main\] is therefore modified in such a way that the gradient of the dual term $(1/q){{\lVert {\bm{X}^*\bm{\alpha}}\rVert}}_q^q$ is computed directly as $\bm{X}J_q(\bm{X}^*\bm{\alpha})$.[^4] For the primal approaches we considered two algorithms: $(a)$ the gradient descent method with linesearch and $(b)$ the FISTA algorithm [@Beck09], but with $p \in \{4/3, 5/4\}$, since they are the only cases in which the proximity operator of $(1/p){{\lVert {\cdot}\rVert}}_p^p$ can be computed explicitly [@livre1]. We generated a matrix $\bm{X}$ according to a normal distribution, a sparse vector $\bm{w}_*$, (where the location of the nonzero coefficients was chosen randomly), a normal distributed noise vector $\bm{\varepsilon}$, and we defined $$\bm{y} = \bm{X} \bm{w}_* + \sigma\bm{\varepsilon},\qquad \sigma = 5\cdot 10^{-2}.$$ We chose $n=200$, $d=10^5$ and $10$ relevant features. The regularization parameter was set to $\gamma=10$, so to achieve a reconstruction error of the order of the noise. Table \[rate\] and Figure \[fig1\](Left.) clearly show that the dual approach significantly outperforms the two primal approaches.[^5] #### Tensor kernels in the dual approach. This experiment considered the case treated in section \[subsec:dualalgo\], that is, $q=4$ ($p=4/3$), with the polynomial tensor kernel of degree $2$, i.e., $$\begin{aligned} K(\bm{x}^\prime_1,\bm{x}^\prime_2,\bm{x}^\prime_3, \bm{x}^\prime_4) &= \big(\mathrm{sum}(\bm{x}^\prime_1 \odot \bm{x}^\prime_2\odot \bm{x}^\prime_3 \odot \bm{x}^\prime_4) \big)^2\\ &= \mathrm{sum}(\Phi(\bm{x}^\prime_1) \odot \Phi(\bm{x}^\prime_2) \odot \Phi(\bm{x}^\prime_3) \odot \Phi(\bm{x}^\prime_4)),\end{aligned}$$ where $\Phi(\bm{x}) = \big( x_1^2, \dots, x_d^2, \sqrt[4]{2} x_1 x_2, \dots, \sqrt[4]{2} x_1 x_d, \sqrt[4]{2} x_2 x_3\dots \big)$. The dimension of the feature space is $N = d(d+1)/2$. We generated $\bm{X}$, $\bm{w}_*$, $\bm{\varepsilon}$ as in the previous case and, according to ,[^6] we defined $$\Phi_n = \begin{bmatrix} \Phi(\bm{x}_1)^\top\\ \vdots\\ \Phi(\bm{x}_d)^\top\\ \end{bmatrix} \in {\mathbb{R}}^{n\times N}, \ \bm{y} = \Phi_n \bm{w}_* + \sigma\bm{\varepsilon}, \ \sigma = 5\cdot 10^{-2}.$$ Then we aimed at solving problem with $\bm{X}$ replaced by $\Phi_n$. We examined a situation in which the computational cost per iteration of the dual algorithm is less than the corresponding primal, measuring the gain in CPU time. We set $n=90$, $d=650$ and $6$ relevant features out of the total of $N = 211575$. With these figures, according to the discussion at the end of section \[subsec:dualalgo\], computing the gradient through the tensor kernel, as done in Algorithm \[algo:main\], surely reduces the cost per iteration. Table \[dual+tensor\] shows the CPU time required by the dual algorithm with and without using the tensor kernel. Algorithm CPU time (sec) iterations -------------------------------------------------- ---------------- ------------ build the Gram tensor $\bm{\mathsf{K}}$ 2.73 — dual GD + linesearch (with $\bm{\mathsf{K}}$) 2.49 29 dual GD + linesearch (without $\bm{\mathsf{K}}$) 9.87 28 : The dual algorithm with and without tensor kernels ($p=4/3$).[]{data-label="dual+tensor"} #### Recovering the relevant features. The sparseness properties of an $\ell^p$-regularization method were mentioned in [@Dau04] and later were studied more carefully in [@Kol09], from a statistical viewpoint. In contrast to $\ell^1$-regularization, the $\ell^p$-regularization does not generally provide finite supported vectors, so sparseness here actually means *approximate sparsity* in the sense that the insignificant coefficients are shrunk and the relevant ones are highlighted. Our experiments confirm this property of $\ell^p$ regularization. Indeed in the setting described in the previous scenarios, the solution vector $\bar{\bm{w}}$ always exhibits spikes that corresponds to the non zero coefficients of $\bm{w}_*$. Depending on the value of $p$, on the size $n$ of the data set, and on the feature space dimension $N$, this phenomenon may be more or less notable, but in any case the vector $\bar{\bm{w}}$ either clearly reveals the hidden relevant features (see Figure \[fig1\](Right.)) or can be safely thresholded in order to discard most of the non-relevant features, and reduce the dimensionality of the problem of $1$-$2$ orders of magnitude. Conclusions =========== In this paper we presented a novel and efficient kernel method for $\ell^p$-norm regularized learning problems. The method assumes that $p=q/(q-1)$ with $q$ an even integer grater than $2$. In such case, we provided an algorithm which is based on the minimization of the dual problem and can be formulated in terms of a tensor kernel evaluated at the training points, avoiding the call of the feature map. Therefore, this provides the first viable solution to $\ell^p$-type regularization in infinite dimensional spaces. Moreover, in finite dimension, the proposed approach compares favorably to other solutions in the regime of few sample and large number of variables, and $q$ reasonably low. For example, our experiments show that if $q=4$, the proposed method is practicable and provides an effective variables selection method and/or is able to discard most of the irrelevant features. We remark that, the complexity of the method depends only on the dataset size and does not depend on the dimension of the function space (e.g, the degree of the polynomial kernel). However, there are scenarios and values of $q$ in which using tensor kernels may be cumbersome from the computational point of view, but this difficulty is common to other approaches to nonparametric sparsity and it is certainly a challenge that requires further study. Finally, the experiments are meant to provide a proof of concept for the proposed method and are the starting point for a more systematic empirical study that we defer to a future work. Appendix ======== This section contains proofs and additional details on some of the topics discussed above. Duality in $\ell^p$-regularization ---------------------------------- [*Proof of Theorem \[thm:duality\].*]{} Problem can be written in the form $$\label{FRduality1} \min_{w \in \ell^p({\mathbb{K}})} f(\bm{w}) + g(-\Phi_n \bm{w}),$$ where $f(\bm{w}) = (1/p) {{\lVert {\bm{w}}\rVert}}_p^p$, $g(\bm{\beta}) = \gamma\sum_{i=1}^n L(y_i, -\beta_i)$, and $\Phi_n$ is defined as in . The Fenchel-Rockafellar dual problem of is [@livre1] $$\min_{\bm{\alpha} \in {\mathbb{R}}^n} f^*(\Phi_n^* \bm{\alpha}) + g^*(\bm{\alpha})$$ and the corresponding KKT optimality conditions are $$\bar{\bm{w}} \in \partial f^* (\Phi_n^* \bar{\bm{\alpha}})\qquad\text{and}\qquad \bar{\bm{\alpha}} \in \partial g(-\Phi_n \bar{\bm{w}}).$$ Now it is easy to see that $$(\forall\, \bm{\alpha} \in {\mathbb{R}}^n)\quad g^*(\bm{\alpha}) = \gamma \sum_{i=1}^n L^*\Big(y_i, -\frac{\alpha_i}{\gamma}\Big) \qquad \text{and}\qquad (\forall\, \bm{u} \in \ell^q({\mathbb{K}}))\quad f^*(\bm{u}) = \frac{1}{q}{{\lVert {\bm{u}}\rVert}}_q^q.$$ Therefore, the dual form $\eqref{eq:gendual}$ follows. Statement $(i)$ comes from the fact that $g$ is continuous. Statement $(ii)$ follows from the KKT conditions above by noting that $f^*$ is indeed differentiable and $\nabla f^* = J_q$, and the fact that $g$ is separable. We now specialize Theorem \[thm:duality\] to distance-based and margin-based losses \[3, Definitions 2.24 and 2.32\]. Suppose that $L$ is a convex distance-based loss of the form $L(y,t) = \psi(y - t)$ with ${\mathcal{Y}}={\mathbb{R}}$, for some convex function $\psi\colon {\mathbb{R}}\to {\mathbb{R}}_+$. Then the dual problem $\eqref{eq:gendual}$ becomes $$\label{eq:dual1} \min_{\alpha \in {\mathbb{R}}^n} \frac{1}{q}{{\lVert {\Phi_n^* \bm{\alpha}}\rVert}}^{q}_{q} - \bm{y}^\top \bm{\alpha} + \gamma \sum_{i=1}^n \psi^* \Big(\frac{\alpha_i}{\gamma} \Big)$$ Suppose that $L$ is a convex margin-based loss of the form $L(y,t) = \psi(y t)$ with ${\mathcal{Y}}=\{-1,1\}$, for some convex function $\psi\colon {\mathbb{R}}\to {\mathbb{R}}_+$. Then the dual problem $\eqref{eq:gendual}$ becomes $$\label{eq:dual2} \min_{\alpha \in {\mathbb{R}}^n} \frac{1}{q}{{\lVert {\Phi_n^* \bm{\alpha}}\rVert}}^{q}_{q} + \gamma \sum_{i=1}^n \psi^* \Big(- \frac{y_i \alpha_i}{\gamma} \Big).$$ The following example shows that all the losses commonly used in machine learning admit explicit Fenchel conjugates.   \[ex:losses\] 1. The *least squares loss* is $L(y,t) = \psi(y - t)$ with $\psi = (1/2) {{\lvert {\cdot}\rvert}}^2$. In that case reduces to , which is is strongly convex with modulus $1/\gamma$. 2. The *Vapnik-$\varepsilon$-insensitive loss* for regression is $L(y,t) = \psi(y - t)$ with $\psi = {{\lvert {\cdot}\rvert}}_\varepsilon$. Then, $\psi^* = \varepsilon {{\lvert {\cdot}\rvert}} + \iota_{[-1,1]}$ and the last term in turns out to be $\varepsilon {{\lVert {\bm{\alpha}}\rVert}}_1 + \iota_{\gamma[-1,1]^n}(\bm{\alpha})$. 3. The *Huber loss* is the distance-based loss defined by $$\psi(r) = \begin{cases} r^2/2 &\text{if } {{\lvert {r}\rvert}} \leq \rho\\ \rho {{\lvert {r}\rvert}} - \rho^2/2 &\text{otherwise}. \end{cases}$$ Then $\psi^* = \iota_{[-\rho, \rho]} + (1/2) {{\lvert {\cdot}\rvert}}^2$ \[1, Example 13.7\] and the last term in is $(1/(2\gamma)){{\lVert {\bm{\alpha}}\rVert}}_2^2 + \iota_{\rho\gamma[-1, 1]^n}(\bm{\alpha})$. 4. The *logistic loss* for classification is the margin-based loss with $\psi(r) = \log (1+e^{-r})$. Thus $$\psi^*(s) = \begin{cases} (1+s)\log(1+s)-s\log(-s) &\text{if } s \in \left]-1,0\right[\\ 0&\text{if } s = -1 \text{ or } s = 0\\ +\infty&\text{otherwise}. \end{cases}$$ See [@livre1 Example 13.2(vi)]. It is easy to see that $\psi$ has Lipschitz continuous derivative with constant $1/4$ and hence $\psi^*$ is strongly convex with modulus $4$ [@livre1]. Thus, referring to , we see that in this case $\operatorname{dom}\Lambda = \prod_{i=1}^n (y_i [0,\gamma])$ and $\Lambda$ is differentiable on $\mathrm{int}(\operatorname{dom}\Lambda)$ with locally Lipschitz continuous gradient. Moreover, since $\lim_{s \to1} {{\lvert {(\psi^*)^\prime(s)}\rvert}} = \lim_{s \to0} {{\lvert {(\psi^*)^\prime(s)}\rvert}} = +\infty$, we have that ${{\lVert {\nabla \Lambda (\bm(\alpha))}\rVert}} = +\infty$ on the boundary of $\operatorname{dom}\Lambda$. Finally, it follows from that $0<y_i \bar{\alpha}_i < \gamma$, for $i=1,\dots, n$. Note that in this case we can still apply Algorithm \[dualalgo\] with $\varphi_2=0$ (see [@sal17a Section 4]). 5. The *hinge loss* is the margin-based loss with $\psi(r) = (1-r)_+$. We have $\psi^*(s) = s + \iota_{[-1,0]}(s)$. So the second term in is $-\sum_{i=1}^n y_i \alpha_i + \iota_{\gamma[0,1]}(y_i \alpha_i)$ We also note that in all cases, for every $i \in \{1,\dots, n\}, \inf L^*(y_i,\cdot)>-\infty$, which was a condition considered in Proposition \[prop:dual2primal\]. [*Proof of Proposition \[prop:dual2primal\].*]{} We use the same notation as in the proof of Theorem \[thm:duality\]. It follows from the definitions of $\bm{w}$ and $\bar{\bm{w}}$ and the Young-Fenchel equalities [@livre1] that $f(\bar{\bm{w}}) + f^*(\Phi_n^* \bar{\bm{\alpha}}) = {{\left\langle{{\bar{\bm{w}}},{ \Phi_n^* \bar{\bm{\alpha}}}}\right\rangle}}$ and $f(\bm{w}) + f^*(\Phi_n^* \bm{\alpha}) = {{\left\langle{{\bm{w}},{\Phi^* \bm{\alpha}}}\right\rangle}}$, and hence $$f^*(\Phi_n^* \bm{\alpha}) - f^*(\Phi_n^* \bar{\bm{\alpha}}) = f(\bar{\bm{w}}) - f(\bm{w}) + {{\left\langle{{\Phi_n^* \alpha},{\bm{w}}}\right\rangle}} - {{\left\langle{{\Phi_n^* \bar{\bm{\alpha}}},{\bar{\bm{w}}}}\right\rangle}}.$$ Since $-\Phi \bar{\bm{w}} \in \partial g^*(\bar{\bm{\alpha}})$, we have $$g^*(\bm{\alpha}) - g^*(\bar{\bm{\alpha}}) \geq {{\left\langle{{-\Phi_n \bar{\bm{w}}},{\bm{\alpha} - \bar{\bm{\alpha}}}}\right\rangle}} = {{\left\langle{{\Phi_n^* \bar{\bm{\alpha}} },{\bar{\bm{w}}}}\right\rangle}}- {{\left\langle{{\Phi_n^* \bm{\alpha}},{\bar{\bm{w}}}}\right\rangle}}.$$ Summing the two inequalities above, we get $$\begin{aligned} \Lambda(\bm{\alpha}) - \Lambda(\bar{\bm{\alpha}}) &\geq f(\bar{\bm{w}}) - f(\bm{w}) - {{\left\langle{{ \Phi_n^* \bm{\alpha}},{\bar{\bm{w}} -\bm{ w}}}\right\rangle}}\\ & = f(\bar{\bm{w}}) - f(\bm{w}) - {{\left\langle{{\nabla f(\bm{w})},{\bar{\bm{w}} - \bm{w}}}\right\rangle}},\\ & = \frac 1 p {{\lVert {\bar{\bm{w}}}\rVert}}_p^p - \frac 1 p {{\lVert {\bm{w}}\rVert}}_p^p - {{\left\langle{{J_p(\bm{w})},{\bar{\bm{w}} - \bm{w}}}\right\rangle}}.\end{aligned}$$ Now, since $1<p<2$, it follows from \[2, Corollary 2.6.1\] that $$\frac 1 p {{\lVert {\bar{\bm{w}}}\rVert}}_p^p - \frac 1 p {{\lVert {\bm{w}}\rVert}}_p^p - {{\left\langle{{J_p(\bm{w})},{\bar{\bm{w}} - \bm{w}}}\right\rangle}}\\ \geq \dfrac{C_p}{\big({{\lVert {\bar{\bm{w}}}\rVert}}_p + {{\lVert {\bm{w}}\rVert}}_p \big)^{2 - p}} {{\lVert {\bar{\bm{w}} - \bm{w}}\rVert}}^2_p,$$ for some constant $C_p>0$ that depends only on $p$. Therefore, by the definition of the duality map, $${{\lVert {\bm{w}}\rVert}}_p = {{\lVert {J_q(\Phi_n^* \bm{\alpha})}\rVert}}_p = \big({{\lVert {\Phi_n^* \bm{\alpha}}\rVert}}^q_q\big)^{1/p} \leq q^{1/p} \big( \Lambda(\bm{\alpha}) + \gamma{{\lVert {\bm{\xi}}\rVert}}_1\big)^{1/p},$$ where $\xi_i = \inf L^*(y_i, \cdot)$; and similarly for ${{\lVert {\bar{\bm{w}}}\rVert}}_p$. Then the statement follows. [*Proof of Theorem \[thm:main\].*]{} Since for the least squares loss we have $\xi_i= - (1/2) y_i^2$, it follows from Proposition \[prop:dual2primal\] that for every $m \in {\mathbb{N}}$, $${{\lVert {\bm{w}_m - \bar{\bm{w}}}\rVert}}_p^2 \leq \frac{\big[ (2^p q) \big(\Lambda(\bm{\alpha}_m) + (\gamma/2) {{\lVert {\bm{y}}\rVert}}_2^2 \big)\big]^{(2-p)/p}}{C_p} \big( \Lambda(\bm{\alpha}_m) - \min \Lambda \big).$$ Now it remains to prove that, $\inf_{m} \lambda_m> 0$ and that $$\label{eq:20170517b} (\forall\, m \in {\mathbb{N}})\qquad \Lambda(\bm{\alpha}_{m+1}) - \min \Lambda \leq \big(1 - (2/\gamma) \lambda_m (1-\delta)\big) \big( \Lambda(\bm{\alpha}_{m}) - \min \Lambda \big).$$ First of all, since $q>2$, the gradient of $\Lambda$ is Lipschitz continuous on bounded sets. Therefore, Proposition 3.15 in [@sal17a] yields that $\inf_{m} \lambda_m> 0$. Now, because of the linesearch rule we have that $$\Lambda(\bm{\alpha}_{m+1}) \leq \Lambda(\bm{\alpha}_{m}) - \lambda_m (1 - \delta) {{\lVert {\nabla \Lambda(\bm{\alpha}_m)}\rVert}}_2^2$$ and, since $\Lambda$ is strongly convex with modulus $1/\gamma$, we have $$\Lambda(\bm{\alpha}_m) - \Lambda(\bar{\bm{\alpha}}) \leq \frac{\gamma}{2} {{\lVert {\nabla \Lambda(\bm{\alpha}_m)}\rVert}}_2^2.$$ All together the two inequalities above gives $$\Lambda(\bm{\alpha}_{m+1}) \leq \Lambda(\bm{\alpha}_{m}) - (2/\gamma) \lambda_m (1 - \delta) \big(\Lambda(\bm{\alpha}_m) - \Lambda(\bar{\bm{\alpha}}) \big).$$ Adding $\Lambda(\bar{\bm{\alpha}})$ to both sides, follows and hence the statement. The function Banach space associated to a tensor kernel {#appendixA2} ------------------------------------------------------- In this section we make explicit the space associated to tensor kernels. We assume that $\mathrm{span}(\Phi({\mathbb{R}}^d))$ is dense in $\ell^q({\mathbb{K}})$ – which is equivalent to requiring that the functions $(\phi_k)_{k \in {\mathbb{K}}}$ are $\ell^p$ point-wise independent. Then, we can associate to the feature map $\Phi$ the Banach function space [@Zhan09] $$\label{eq:rkbs} \mathcal{B} = \big\{ {{\left\langle{{\bm{w}},{\Phi(\cdot)}}\right\rangle}} \,\big\vert\, \bm{w} \in \ell^p({\mathbb{K}})\big\}, \qquad {{\lVert {{{\left\langle{{\bm{w}},{\Phi(\cdot)}}\right\rangle}}}\rVert}}_{\mathcal{B}} = {{\lVert {\bm{w}}\rVert}}_p.$$ Note that if $\bm{\alpha} \in {\mathbb{R}}^n$, $\bm{x}_1, \dots, \bm{x}_n \in {\mathbb{R}}^d$, and we set $\bm{w} = J_q \big(\sum_{i=1}^n \alpha_i \Phi(\bm{x}_i) \big)$, then, as in , we have $$\label{eq:20170512a} {{\left\langle{{\bm{w}},{\Phi(\cdot)}}\right\rangle}} = \sum_{i_1,\dots, i_{q-1}=1}^n K(\bm{x}_{i_1}, \cdots, \bm{x}_{i_{q-1}}, \cdot ) \alpha_{i_1} \cdots \alpha_{i_{q-1}},$$ and $$\label{eq:20170519a} {{\lVert {{{\left\langle{{\bm{w}},{\Phi(\cdot)}}\right\rangle}}}\rVert}}_{\mathcal{B}} = \bigg( \sum_{i_1, \dots, i_q = 1}^n K(\bm{x}_1, \dots, \bm{x}_q) \alpha_{i_1}\cdots \alpha_{i_q} \bigg)^{1/p},$$ and the functions are dense in $\mathcal{B}$. Moreover, setting $\Phi^* = J_q \circ \Phi \colon {\mathbb{R}}^d \to \ell^p({\mathbb{K}})$, if $\mathrm{span}(\Phi^*({\mathbb{R}}^d))$ is also dense in $\ell^p({\mathbb{K}})$, then its associated function Banach space $\mathcal{B}^*$ (defined similarly to ) is the topological dual of $\mathcal{B}$ and the following reproducing property holds $$K_{\bm{x}}\colon \bm{x}^\prime \to K(\bm{x}^\prime,\dots, \bm{x}^\prime, \bm{x}) \in \mathcal{B}^*,\quad\text{and} \quad {{\left\langle{{f},{K_{\bm{x}}}}\right\rangle}} = f(\bm{x}).$$ For the case of infinite dimensional power series tensor kernels, which includes the exponential tensor kernels considered here, the density assumptions on $\mathrm{span}(\Phi({\mathbb{R}}^d))$ and $\mathrm{span}(\Phi^*({\mathbb{R}}^d))$ holds, hence the corresponding Banach space can be described through the equations and . The dual algorithm for general loss function and any $p \in \left]1,2\right[$ ----------------------------------------------------------------------------- [*Proof of Theorem \[thm:dualalgo\].*]{} Since $\varphi_1$ is smooth with a locally Lipschitz continuous gradient we can apply Theorem 3.2 and Proposition 3.5 in [@sal17a] and get $\inf_m \lambda_m>0$, $\bm{\alpha}_m \to \bar{\bm{\alpha}}$ and $\Lambda(\bm{\alpha}_m) - \Lambda(\bar{\bm{\alpha}})= o(1/m)$. Then, by Proposition \[prop:dual2primal\], we have ${{\lVert {\bm{w}_m - \bar{\bm{w}}}\rVert}}_p \leq o(1/\sqrt{m})$. Now suppose that $\Lambda$ is $\mu$-strongly convex. We will rely on Proposition 2 in [@Bre08]. Then, strong convexity of $\Lambda$ yields $$\frac \mu 2 {{\lVert {\bm{\alpha}_m - \bar{\bm{\alpha}}}\rVert}}^2 \leq \Lambda(\bm{\alpha}_m) - \Lambda(\bar{\bm{\alpha}})$$ for some constant $\mu>0$. So equation (3.8) in Proposition 2 in [@Bre08] holds. Moreover, defining $$- D_{\lambda_m} (\bm{\alpha}_m) := \varphi_2 (\bm{\alpha}_{m+1}) \big) - \varphi_2 (\bm{\alpha}_{m}) \big) + {{\left\langle{{\bm{\alpha}_{m+1} - \bm{\alpha}_m},{\nabla \varphi_1(\bm{\alpha}_m)}}\right\rangle}},$$ by the definition of $\lambda_m$, and Proposition 3.8 and Proposition 3.9 in [@sal17a], we have $$\label{eq:20170519a} \frac{{{\lVert {\bm{\alpha}_{m+1} - \bm{\alpha}_m}\rVert}}^2}{\lambda_m} \leq D_{\lambda_m} (\bm{\alpha}_m)\quad \text{and}\quad \Lambda(\bm{\alpha}_{m+1}) - \Lambda(\bm{\alpha}_{m}) \leq - (1-\delta) D_{\lambda_m} (\bm{\alpha}_m).$$ Then, since $\inf_{m} \lambda_m>0$ we can proceed as in the proof of Proposition 2 in [@Bre08] and prove that $\Lambda(\bm{\alpha}_m)$ converge linearly to $\Lambda(\bar{\bm{\alpha}})$. Finally, using Proposition \[prop:dual2primal\] the linear convergence of $\bm{w}_m$ follows. Note that Example \[ex:losses\] shows that if $L$ is the least square loss or the logistic loss, then $\Lambda$ is strongly convex. [^1]: The problem is convex since the first term in is equal to $(1/q){{\lVert {X^* \bm{\alpha}}\rVert}}_q^q$. [^2]: However, it is not known whether a function $K\colon \big({\mathbb{R}}^d \big)^q \to {\mathbb{R}}$ satisfying the two properties above can be written as in for some feature map $\Phi\colon {\mathbb{R}}^d \to \ell^q({\mathbb{K}})$. [^3]: All the numerical experiments have been performed in MATLAB^^ environment, on a MacBook laptop with Intel Core 2 Duo, 2 Ghz and 4 GB of RAM. [^4]: Note that in this case the cost per iteration is essentially equal to that of the gradient descent in the primal. [^5]: The optimal values were found by using the dual algorithm and checking that the duality gap was $<10^{-14}$. [^6]: In this case $\mathrm{card}(\mathbb{K}) = N$, so $\ell^p(\mathbb{K})$ can be identified with ${\mathbb{R}}^N$ and the linear map $\Phi_n$ can be thought as a $n\times N$ matrix.
--- author: - 'Yong Chul Ju, Daniel Maurer, Michael Breu[ß]{} and Andr[é]{}s Bruhn' title: Direct Variational Perspective Shape from Shading with Cartesian Depth Parametrisation --- Introduction {#sec:introduction} ============ Shape from Shading (SfS) is a classic task in computer vision. Given information on light reflectance and illumination in a photographed scene, the aim of SfS is to compute based on the brightness variation the 3D structure of a depicted object from a single input image. SfS has a wide variety of applications, ranging from large scale problems such as astronomy [@Rindfleisch1966] or terrain reconstruction [@Bors_TPAMI2003] to small scale tasks such as dentistry [@Abdelrahim_BMVC2011] or endoscopy [@Okatani_CVIU1997; @Wang_MST2009; @Wu_IJCV2010]. [**Classical Methods.**]{} First approaches to SfS go back to 1951 and 1966, respectively, when Van Diggelen [@VanDiggelen1951] and Rindfleisch [@Rindfleisch1966] used SfS techniques to reconstruct the surface of the moon. Later on in the 1970’s, Horn [@Ho70] was the first one to tackle the SfS problem by solving a partial differential equation (PDE) approach. In 1981, he and Ikeuchi were also the first ones to model the SfS problem using a variational framework [@Ikeuchi_AI1981]. The most prominent classical variational approach is given by the work of Horn and Brooks [@HB86]. Assuming a simple [*orthographic*]{} projection model, a light source at [*infinity*]{} as well as a [*Lambertian*]{} reflectance model, they proposed to compute the normals of the unknown surface as minimiser of an energy functional. Those first approaches, however, had several drawbacks. The model assumptions were [*very simple*]{} and mainly suitable in the context of astronomical applications. In fact, the use of an orthographic projection model with a light source located at infinity requires the distances between camera, light source and illuminated object to be huge. Also the [*depth was not estimated directly*]{} such that the SfS process required a postprocessing step that performed a numerical integration of the estimated surface normals. Thereby, [*inconsistent gradient fields*]{} turned out to be a problem, so that extensions of the original model were required that tried to enforce this consistency during or after the estimation [@Frankot_TPAMI1988; @HB89]. Finally, in case of variational methods, the smoothness term was restricted to a quadratic regulariser [@HB86; @Ikeuchi_AI1981]. While such standard smoothness terms simplify the minimisation of the underlying energy, they do not allow to preserve discontinuities in the depth and thus lead to [*oversmoothed solutions*]{} [@Ol91]. For a detailed review of most of the classical methods the reader is referred to [@DFS08; @Ho86; @HB89; @ZTCS99]. [**Perspective Shape from Shading.**]{} At the end of the 1990’s research mainly focused on novel concepts for formulating orthographic SfS such as viscosity solutions [@RouyTourin_SINUM1992] and level set formulations [@Kimmel_IJCV1995]. However, for most applications results were not satisfactory [@ZTCS99]. In the early 2000’s, the situation changed completely. Inspired by the work of Okatani and Deguchi [@Okatani_CVIU1997], independently, Prados and his co-workers [@PF03b; @Prados_CVPR2005] as well as two other research groups [@CCDG04a; @TSY03] proposed to consider a [*perspective*]{} camera model. Evidently, such a model is particularly appropriate for tasks that require the object to be relatively close to the camera such as e.g. in medical endoscopy. In such cases the perspective effects dominate and an orthographic projection model would cause significant systematic errors as shown in [@TSY05]. Secondly, Prados and colleagues proposed to [*shift the light source location from infinity to the camera centre*]{} which can be seen as a good approximation of a camera with photoflash. This made shape from shading attractive for a variety of photo-based applications. Finally, also a physically motivated [*light attenuation term*]{} was introduced that models a quadratic fall-off due to the inverse square law. As discussed in [@BCDFV12], the use of this term largely resolved the convex-concave ambiguity that was inherent to the classical orthographic model although some ambiguities are still present. Even the generalisation of such approaches to advanced reflectance models such as the Oren-Nayar [@Oren_IJCV1995] or the Phong reflectance model [@Phong_CACM1975] have been recently investigated [@AF2006; @VBW08]. However, this evolution of SfS models was accompanied by a different way of formulating the SfS problem. Instead of using variational methods, the perspective SfS problem was formulated in terms of hyperbolic PDEs [@Prados_CVPR2005]. Although such PDE formulations allow for an efficient computation of the solution using fast marching schemes [@Sethian1999], they suffer from two inherent drawbacks: (i) On the one hand, they are prone to [*noise*]{} and [*missing data*]{}, since they do not rely on any form of regularisation or filling-in. This can be particularly problematic in the context of real-world images. (ii) On the other hand, it is [*difficult to extend*]{} the underlying model of such PDE-based schemes by additional constraints such as smoothness terms, multiple views, or additional light sources. While there have been recently some PDE-based approaches to photometric stereo [@MF13], one has to take care of ensuring the uniqueness of the solution if the input data from multiple images is not consistent, cf. the discussion in [@Mecca_SIIMS14]. [**Variational Perspective Shape from Shading.**]{} Given the flexibility and robustness of variational methods, it is not surprising that recently researchers tried to close the evolutionary loop by integrating the perspective SfS model into a suitable variational framework. So far, however, there are only a few works in the literature that deal with this recent idea. On the one hand, there is the work of Ju [*et al.*]{} [@JBB2014] that embeds the PDE of Prados [*et al.*]{} [@Prados_CVPR2005] as data term into a variational model and complements it with a discontinuity-preserving second order smoothness term. However, since the approach penalises deviations from the PDE directly and uses a parametrisation in terms of the radial depth, deviations in both the data and the smoothness term are difficult to interpret geometrically or photometrically. On the other hand, there is the approach of Abdelrahim [*et al.*]{} [@Abdelrahim_ICIP; @2013] that formulates the data term in terms of brightness differences and makes use of a Cartesian depth parametrisation. While the corresponding energy functional is thus more meaningful from a geometric and photometric viewpoint, it defines smoothness based on surface normals and thus needs an additional integrability constraint. Moreover, the corresponding smoothness term is restricted to a simple homogeneous regulariser that does not allow to preserve object edges during the reconstruction. Finally, there are the works of Zhang [*et al.*]{} [@ZYT07] and Wu [*et al.*]{} [@Wu_IJCV2010] that also make use of a Cartesian depth parametrisation but rely on an indirect estimation using auxiliary variables. While the approach of Zhang [*et al.*]{} [@ZYT07] resolves the resulting consistency problem by considering an integrability constraint, the method of Wu [*et al.*]{} [@Wu_IJCV2010] repeatedly integrates the surface normals during computation to ensure valid solutions. Moreover, both approaches use derivations for their surface normals that are based on the orthographic projection model of Horn and Brooks [@HB89]. Unfortunately, the resulting models are thus only valid in case of weak perspective distortions. A final issue that is common to all four of the aforementioned works is the difficulty of minimising the underlying energy. Since this energy is non-convex, two of the methods rely on initialisations provided by closely related PDE-based SfS approaches [@Abdelrahim_ICIP; @2013; @ZYT07]. This, however, contradicts the idea of introducing robustness into the estimation – in particular in the presence of noise or missing data. In contrast, the other two methods estimate the solution from scratch [@JBB2014; @Wu_IJCV2010]. However, those methods do not provide any quantitative assessment of the reconstruction quality. Let us summarise: While from a modelling viewpoint, it would be desirable to design a variational model that directly solves for the Cartesian depth without the need of integrability constraints or repeated integrations steps, it would be helpful from an optimisation viewpoint to develop a minimisation scheme that neither depends on the solution of other SfS techniques as in [@Abdelrahim_ICIP; @2013; @ZYT07] nor requires an accurate initialisation to produce meaningful results. [**Our Contributions.**]{} In this book chapter we contribute to the field of variational SfS in three ways: (i) First, we consider a variational model for perspective SfS that makes use of a Cartesian depth parametrisation and an edge-preserving Cartesian depth regularisation. By penalising deviations from the image brightness in the data term and regularising the Cartesian depth in the smoothness term directly, we obtain an approach that is geometrically and photometrically meaningful. In this context, we also point out a popular mistake in the derivation of the surface normal and show two different ways to derive the normal correctly. (ii) Our method is a direct approach to depth computation, i.e.  it does not yield gradient fields that need to be integrated in a subsequent step, nor do we employ integrability constraints. (iii) Apart from the novel model, we also propose a novel minimisation strategy. By embedding an alternating explicit scheme into a coarse-to-fine scheme, we obtain an optimisation framework that allows to obtain significantly better results than a traditional explicit scheme. Experiments with synthetic and real-world images show the good quality of our reconstructions and the advantages of our numerical scheme. [**Organisation of the Chapter.**]{} In Section 2 we propose a novel PDE-based model for perspective SfS that is based on a Cartesian parametrisation of the depth. In Section 3 we then embed this PDE into a variational framework with appropriate second order smoothness term. Details on the minimisation and the discretisation are provided in Section 4, while Section \[sec:intrinsic\] comments on the integration of intrinsic camera parameters. Finally, a detailed evaluation of our approach is presented in Section \[sec:experimental-results\]. The paper concludes with a summary in Section \[sec:conclusion\]. Perspective SfS with Cartesian Depth Parametrisation {#sec:persp-shape-from} ==================================================== In this section, we introduce a novel PDE-based SfS model that is parametrised in terms of the [*Cartesian*]{} depth. In contrast to most existing SfS models that estimate the radial depth or multiples thereof, such a Cartesian parametrisation expresses the unknown surface directly in terms of the Euclidean distance along the $z$-axis, which is the axis orthogonal to the image plane. ![ Relation between the radial depth factor $u(\mathbf{x})$ (quotient between green and blue distance) that denotes the depth in multiples of the focal length $\mathtt{f}$ and the Cartesian depth $z(\mathbf{x})$ (red distance).[]{data-label="fig:parameterisation"}](figs/pers_proj_Z_bbox.pdf){width="0.6\linewidth"} [**Parametrisation of the Surface.**]{} The starting point for our new model is formed by the classical PDE-approach of Prados [*et al.*]{} [@Prados_CVPR2005] which is originally parametrised in terms of the [*radial*]{} depth. Key assumptions of this SfS model are that a point light source is located at the [*optical centre*]{} of a perspective camera and that the surface reflectance is [*Lambertian*]{} with uniform albedo that is fixed to one. The unknown surface $\mathcal{S}: \overline{\Omega}_{\mathbf{x}} \rightarrow \mathbb{R}^3$ can then be described as $$\label{eq:13} \mathcal{S} \left( \mathbf{x}, u(\mathbf{x}) \right) = \left\{ \dfrac{\mathtt{f}}{\sqrt{\left| \mathbf{x} \right|^2 + \mathtt{f}^2}} \; u(\mathbf{x}) \left[ \left. \begin{array}{c} x \\ y \\ - \mathtt{f} \end{array} \right] \right| \; \mathbf{x} := \left( x, y \right)^\top \in \overline{\Omega}_{\mathbf{x}} \right\} \, ,$$ where $\mathbf{x}=(x, y)^\top \in \overline{\Omega}_{\mathbf{x}}$ is the position in the closure $\overline{\Omega}_{\mathbf{x}}$ of the rectangular image domain $\Omega_{\mathbf{x}} \subset \mathbb{R}^2$, $\mathtt{f}$ denotes the focal length of the camera and $u(\mathbf{x})$ is a multiple of $\mathtt{f}$ that describes the radial distance (depth) of the surface from the camera centre. Since the third component in Eq. corresponds to the negative Cartesian depth $z$, we can derive the following relationship to the radial depth $u \, \mathtt{f}$ $$\label{eq:64} z(\mathbf{x}) = \dfrac{\mathtt{f}}{\sqrt{\left| \mathbf{x} \right|^2 + \mathtt{f}^2}} \, u(\mathbf{x}) \, \mathtt{f} \; \stackrel{\mbox{\tiny \eqref{eq:13}}}{=} \; Q(\mathbf{x}) \, u(\mathbf{x}) \, \mathtt{f} \, ,$$ where $Q(\mathbf{x})$ denotes a spatially variant conversion factor given by $$\label{eq:12} Q(\mathbf{x}) = \dfrac{\mathtt{f}}{\sqrt{|\mathbf{x}|^2 + \mathtt{f}^2}} \, .$$ This relation is illustrated in Figure \[fig:parameterisation\]. Plugging Eq. into Eq. , we then obtain the parametrisation of the original surface $\mathcal{S}$ with respect to the Cartesian depth $z$ $$\label{eq:77} \mathcal{S} \left( \mathbf{x}, z(\mathbf{x}) \right) \stackrel{\mbox{\tiny \eqref{eq:13}}}{=} Q(\mathbf{x}) \, u(\mathbf{x}) \left[ \begin{array}{c} x \vspace{0.5ex}\\ y \vspace{0.5ex}\\ -\mathtt{f} \end{array} \right] \stackrel{\mbox{\tiny \eqref{eq:64}}}{=} Q(\mathbf{x}) \, \left( \dfrac{z(\mathbf{x})}{\mathtt{f} \, Q(\mathbf{x})} \right) \left[ \begin{array}{c} x \vspace{0.5ex}\\ y \vspace{0.5ex}\\ -\mathtt{f} \end{array} \right] = \left[ \begin{array}{c} \dfrac{z(\mathbf{x}) \, x}{\mathtt{f}} \vspace{1ex}\\ \dfrac{z(\mathbf{x}) \, y}{\mathtt{f}} \vspace{1ex}\\ - z(\mathbf{x}) \end{array} \right] \, .$$ [**Brightness Equation.**]{} After we have parametrised the original surface in terms of the Cartesian depth, let us now derive the resulting brightness equation that relates the local orientation of the surface to the image brightness. Assuming a Lambertian reflectance model and a quadratic light attenuation term that follows the inverse square law, we obtain the following general brightness equation [@Prados_CVPR2005]: $$\label{eq:15} I(\mathbf{x}) = \dfrac{1}{r(\mathbf{x})^2} \left( \dfrac{\mathbf{n}( \mathbf{x})}{|\mathbf{n}( \mathbf{x})|} \cdot \mathbf{L}(\mathbf{x}) \right) \, ,$$ where $I$ is the recorded image, ${\mathbf{n}} $ is the surface normal vector, $\mathbf{L}$ stands for the normalised light direction vector, and $r$ is the (radial) distance of the light source to the surface. Knowing that $r=\mathtt{f} \, u$ and using Eq. we can express the quadratic light attenuation term using the Cartesian depth $z$ $$\label{eq:99} r(\mathbf{x}) = \mathtt{f} u(\mathbf{x}) = \dfrac{z(\mathbf{x})}{Q(\mathbf{x})} \; \Rightarrow \;\;\;\; \dfrac{1}{r(\mathbf{x})^2} = \frac{Q(\mathbf{x})^2}{z(\mathbf{x})^2}\, .$$ What remains to be computed in terms of the Cartesian depth are the surface normal $\mathbf{n}$ and the light direction vector $\mathbf{L}$, respectively. [**Surface Normal.**]{} Let us start by deriving the surface normal. Since the surface normal is the normal vector of the tangent plane, we first have to compute the partial derivatives of the surface in Eq. in $x$- and $y$-direction, respectively $$\mathcal{S}_{x}( \mathbf{x}, z ) = \left[ \begin{array}{c} \dfrac{z_{x} \, x + z}{\mathtt{f}} \vspace{1ex}\\ \dfrac{z_{x} \, y}{\mathtt{f}} \vspace{1ex}\\ - z_{x} \end{array} \right] \; , \;\;\;\;\; \mathcal{S}_{y}( \mathbf{x}, z ) = \left[ \begin{array}{c} \dfrac{z_{y} \, x}{\mathtt{f}} \vspace{1ex}\\ \dfrac{z_{y} \, y + z}{\mathtt{f}} \vspace{1ex}\\ - z_{y} \end{array} \right] \, .$$ Here and for the whole paper we dropped the spatial dependency of $z$, $z_x$ and $z_y$ on $\mathbf{x}$ for the sake of clarity. Taking the cross-product then yields the direction of the surface normal $$\label{eq:17} \mathbf{n}( \mathbf{x} ) = \mathcal{S}_{x}( \mathbf{x}, z ) \times \mathcal{S}_{y}( \mathbf{x}, z ) = \left[ \begin{array}{c} \dfrac{z_{x} \, z}{\mathtt{f}} \vspace{1ex}\\ \dfrac{z_{y} \, z}{\mathtt{f}} \vspace{1ex}\\ \dfrac{z \, \left[ \left( \nabla z \cdot \mathbf{x} \right) + z \right]}{\mathtt{f}^2} \end{array} \right] \, .$$ [**Light Direction.**]{} Let us now turn towards the computation of the light direction. Since the light source is assumed to be located in the camera centre which coincides with the origin of the coordinate system, the direction of the light rays and the direction of the optical rays coincide (up to sign). Hence, the light direction can just be read off Eq. as $$\label{eq:18} \mathbf{L}(\mathbf{x}) = \dfrac{1}{\sqrt{|\mathbf{x}|^2 + \mathtt{f}^2}} \left[ \begin{array}{c} -{x} \\ -{y} \\ \mathtt{f} \end{array} \right] \, .$$ [**PDE-Based Model**]{}. By plugging the surface normal from Eq. and the light direction from Eq. into the brightness equation we finally obtain our perspective SfS model with the new Cartesian depth parametrisation $$\label{eq:16} I - \dfrac{Q^3}{z \, \sqrt{ \mathtt{f}^2 \, \left| \nabla z \right|^2 + \left[ \left( \nabla z \cdot \mathbf{x} \right) + z \right]^2 } } = 0 \, .$$ Here and for the whole paper we dropped the spatial dependency of $I$ and $Q$ on $\mathbf{x}$ for the sake of clarity. The main properties of our new model are naturally inherited from the original PDE [@Prados_CVPR2005]: (i) Eq. still belongs to the class of Hamilton-Jacobi equations (HJEs) which have been intensively studied in the SfS literature. (ii) Therefore, well-posedness can be achieved in the viscosity sense [@BCDFV12; @Crandall_TAMS1983; @Prados_CVPR2005]. (iii) Proper numerical discretisations must be considered when solving the HJE. Let us note that the framework of viscosity solutions is a natural setting for HJEs such as Eq. . The basic idea behind the notion of viscosity solutions is to add a (typically, second order) regularisation term to the PDE and study the solution as this term goes to zero. This proceeding yields desirable stability properties and enables to consider even solutions with non-differentiable features like e.g. kinks. We refer the interested reader to [@Barles94; @Crandall_TAMS1983] for studying properties of viscosity solutions and to [@CP14] for their use in computer vision. Furthermore, please note that our model can be seen as a generalisation of the PDE-based approach in [@ZYT07b] that already makes use of the Cartesian depth parametrisation, but does not yet consider the light attenuation term from physics. Variational Model for Perspective SfS with Cartesian Depth Parametrisation {#sec:var-model} ========================================================================== So far we have derived a novel PDE-based model for perspective SfS with Cartesian depth parametrisation. Let us now discuss how this model can be integrated into a variational framework with smoothness term. [**Variational Model.**]{} To this end, we follow the idea from [@JBB2014] and use a quadratic error term based on our novel PDE as data term which is complemented with a suitable second order regulariser. More precisely, we propose to compute the Cartesian depth $z$ as minimiser of the following energy functional $$\label{eq:2} E \left( z \right) = \int_{\overline{\Omega}_{\mathbf{x}}} \, c(\mathbf{x}) \; \underbrace{ D(\mathbf{x} , z, \nabla z) }_{\mbox{Data term}} + \, \; \alpha \; \underbrace{ S(\mbox{Hess}(z)) }_{\!\!\!\!\!\!\!\!\!\!\!\!\!\mbox{Smoothness term}\!\!\!\!\!\!\!\!\!\!\!\!\!} \, \mathrm{d}\mathbf{x} \; ,$$ where $D$ is the data term, $S$ is the smoothness term, $c: \mathbf{x} \in \overline{\Omega}_{\mathbf{x}} \subset \mathbb{R}^2 \rightarrow [0,1]$ is a confidence function and $\alpha \in \mathbb{R}^+$ is a regularisation parameter that steers the degree of smoothness of the solution. As mentioned before our data term is based on a quadratic formulation that penalises deviations from our novel PDE. It is given by $$\label{eq:3} D( \mathbf{x}, z, \nabla z ) = \left(I(\mathbf{x}) - \dfrac{Q(\mathbf{x})^3}{z \,\, W( \mathbf{x}, z, \nabla z ) } \right)^2$$ with $$\label{eq:4} W( \mathbf{x}, z, \nabla z ) = \sqrt{ \mathtt{f}^2 \, \left| \nabla z \right|^2 + \left[ \left( \nabla z \cdot \mathbf{x} \right) + z \right]^2 } \; .$$ As smoothness term, we propose to use the following subquadratic and thus edge-preserving second-order regulariser based on the Frobenius norm of the Hessian $$\label{eq:5} S \left( \mbox{Hess}(z) \right) = \Psi \left( \left\| \mbox{Hess}(z) \right\|^2_{\tiny F} \right) = \Psi \left( z_{xx}^2 + 2 z_{xy}^2 + z_{yy}^2 \right)$$ where $\Psi$ is the Charbonnier function [@Charbonnier_TIP1997] $$\label{eq:11} \Psi(s^2) = 2 \lambda^2 \sqrt{ 1 + \tfrac{s^2}{\lambda^2} }$$ with contrast parameter $\lambda$. Such higher-order smoothness terms have already been successfully applied in the context of perspective SfS parametrised in terms of the radial depth [@JBB2014], orthographic SfS [@Vogel_SSVM2007], image denoising [@Lysaker_TIP2003], optical lithography [@Estellers_TIP2014] and motion estimation [@Demetz_ECCV2014]. Finally, the use of the confidence function $c$ in the data term allows to exclude unreliable image regions which have been identified a priori, e.g.  by a texture detector or by a background segmentation algorithm. Such functions are particularly useful in the context of real-world images that contain texture, noise, or missing data [@CCDG08; @JBB2014]. ---------- -------------------------- -- -- -------------------- -- -- ----------------- -- -- --------------------------- -- -- ----------------- -- -- -------------- -- \[-2mm\] Zhang [*et al.*]{} Wu [*et al.*]{} Abdelrahim [*et al.*]{} Ju [*et al.*]{} [**Our**]{} [@ZYT07] [@Wu_IJCV2010] [@Abdelrahim_ICIP; @2013] [@JBB2014] [**Work**]{} \[-2mm\] \[-2mm\] Cartesian Cartesian Cartesian radial Cartesian \[-2mm\] Parametrisation \[-2mm\] depth depth depth depth depth \[-2mm\] \[-2mm\] Reprojection Error \[-2mm\] $\checkmark$ $\checkmark$ $\checkmark$ $-$ $\checkmark$ \[-2mm\] as Data Term \[-2mm\] \[-2mm\] Light Attenuation Factor $-$ $\checkmark$ $\;\;\;\checkmark^{\; 1}$ $\checkmark$ $\checkmark$ \[-2mm\] \[-2mm\] Correct Surface Normal $\;\;-^{\; 2}$ $\;\;-^{\; 2}$ $\;\;\;\checkmark^{\; 3}$ $\checkmark$ $\checkmark$ \[-2mm\] \[-2mm\] Cartesian Cartesian Cartesian radial Cartesian \[-2mm\] Regularisation \[-2mm\] depth depth surface normal depth depth \[-2mm\] \[-2mm\] Edge Preservation $-$ $-$ $-$ $\checkmark$ $\checkmark$ \[-2mm\] \[-2mm\] No Integrability Term $-$ $\;\;-^{\; 4}$ $-$ $\checkmark$ $\checkmark$ \[-2mm\] \[-2mm\] Direct Estimation $^5$ $-$ $-$ $\checkmark$ $\checkmark$ $\checkmark$ \[-2mm\] ---------- -------------------------- -- -- -------------------- -- -- ----------------- -- -- --------------------------- -- -- ----------------- -- -- -------------- -- : Comparison of the literature on variational models for perspective shape from shading.[]{data-label="tab:methods"} $^1$ factor not expressed in terms of the Cartesian depth\ $^2$ see explanation in appendix\ $^3$ no details given in the paper but derivations shown in [@Ab14]\ $^4$ integrability constraint realised via repeated integration of surface normals\ $^5$ depth is computed without extra variables for surface normals\ [**Properties.**]{} Our variational model from Eq. has the following distinct features: 1. Since the data term in Eq. is inherited from Eq. , the perspective camera projection is already taken into account. Moreover, since the reprojection error is penalised in the data term, deviations have a [*photometric*]{} interpretation. 2. Since the regulariser is applied directly to the Cartesian depth, also deviations from smoothness become now more meaningful than in the case of a radial depth parametrisation. In particular, they can be interpreted [*geometrically*]{}. 3. Moreover, in contrast to most existing approaches, the regulariser is able to [*preserve edges*]{} in the reconstruction despite of the regularisation effect. 4. Unreliable regions can be excluded from the data term via a [*confidence function*]{} such that the smoothness term takes over and fills in information from the neighbourhood. This can be advantageous in the context of texture, noise, or missing data. Please note that in contrast to [@JBB2014], we always guarantee a fixed amount of regularisation by not restricting the smoothness term to unreliable locations. 5. The depth of the surface is [*directly*]{} computed since we minimise for the unknown depth $z$ in Eq. . This is in contrast to most variational methods that estimate the depth in two steps, see e.g. [@BH85; @Frankot_TPAMI1988; @Ikeuchi_AI1981] where first the surface normals are computed by a variational model and then the depth is determined by integration. 6. The solution given by the model fulfils the [*integrability constraint*]{} per construction since we solve for $z$ and use $z_{xy} = z_{yx}$ in the smoothness term. Otherwise, such as in [@Abdelrahim_ICIP; @2013], an additional integrability term would be needed to encourage valid solutions. 7. Another advantage of the new parametrisation is that it allows a straightforward [*combination*]{} with other reconstruction methods such as stereo [@Robert_ECCV1996] or scene flow estimation [@Basha_IJCV2012], since such approaches typically make use of the same Cartesian parametrisation and thus could be easily integrated into a joint framework. 8. A final advantage is the fact that the approach could easily be extended to [*multiple views*]{}, since transformations between the views are simpler if the approach is parametrised in terms of the Cartesian depth instead of the radial depth. To make the difference of our model to other variational approaches from the literature explicit, the features of the different methods are compared in Table \[tab:methods\]. Minimisation {#sec:minimisation} ============ Let us now discuss the minimisation of the proposed energy. To this end, we will first derive the associated Euler-Lagrange equation and then discuss its discretisation. Finally, we will sketch a coarse-to-fine minimisation strategy with an alternating explicit scheme to solve the resulting nonlinear equations. [**Euler-Lagrange Equation**]{}. The calculus of variations [@CourantHilbert1953] tells us that the minimiser $z$ of our energy in Eq. has to fulfil the corresponding Euler-Lagrange equation. Omitting the dependencies on all variables in order to ease the readability, this equation is given by $$\begin{aligned} 0 & \; = \; & \left[ cD + \alpha S \right]_z - \dfrac{\partial}{\partial x} \left[ cD + \alpha S \right]_{z_{x}} - \dfrac{\partial}{\partial y} \left[ cD + \alpha S \right]_{z_{y}} \\[-1mm] & & \; + \dfrac{\partial^2}{\partial x^2} \left[ cD + \alpha S \right]_{z_{xx}} + 2 \dfrac{\partial^2}{\partial x \partial y} \left[ cD + \alpha S \right]_{z_{xy}} + \dfrac{\partial^2}{\partial y^2} \left[ cD + \alpha S \right]_{z_{yy}} \nonumber \\[0mm] & = \; & \left[ cD \right]_z - \dfrac{\partial}{\partial x} \left[ cD \right]_{z_{x}} - \dfrac{\partial}{\partial y} \left[ cD \right]_{z_{y}} + \dfrac{\partial^2}{\partial x^2} \overbrace{ \left[ cD \right]_{z_{xx}} }^{\equiv \, 0} + 2 \dfrac{\partial^2}{\partial x \partial y} \overbrace{ \left[ cD \right]_{z_{xy}} }^{\equiv \, 0} + \dfrac{\partial^2}{\partial y^2} \overbrace{ \left[ cD \right]_{z_{yy}} }^{\equiv \, 0} \nonumber \\[-1mm] & & \; + \underbrace{ \vphantom{\left[ cD \right]_{z_{yy}}} \left[ \alpha S \right]_z}_{\equiv \, 0} - \dfrac{\partial}{\partial x} \underbrace{ \vphantom{\left[ cD \right]_{z_{yy}}} \left[ \alpha S \right]_{z_{x}} }_{\equiv \, 0} - \dfrac{\partial}{\partial y} \underbrace{\left[ \alpha S \right]_{z_{y}} }_{\equiv \, 0} + \dfrac{\partial^2}{\partial x^2} \left[ \alpha S \right]_{z_{xx}} + 2 \dfrac{\partial^2}{\partial x \partial y} \left[ \alpha S \right]_{z_{xy}} + \dfrac{\partial^2}{\partial y^2} \left[ \alpha S \right]_{z_{yy}} \nonumber \\[2mm] & = \; & c \; \left( \left[ D \right]_z - \dfrac{\partial}{\partial x} \left[ D \right]_{z_{x}} - \dfrac{\partial}{\partial y} \left[ D \right]_{z_{y}} \right) + \dfrac{\partial^2}{\partial x^2} \left[ \alpha S \right]_{z_{xx}} + 2 \dfrac{\partial^2}{\partial x \partial y} \left[ \alpha S \right]_{z_{xy}} + \dfrac{\partial^2}{\partial y^2} \left[ \alpha S \right]_{z_{yy}} \nonumber \, , \label{eq:49}\end{aligned}$$ where we exploited the fact that $$\label{eq:28} \dfrac{\partial^2}{\partial x \partial y} \left[ cD + \alpha S \right]_{z_{xy}} = \dfrac{\partial^2}{\partial y \partial x} \left[ cD + \alpha S \right]_{z_{xy}} \, .$$ On a structural level, this Euler-Lagrange equation is somewhat more complicated than its counterparts for indirect methods in [@Wu_IJCV2010; @ZYT07]. Such indirect methods model the surface normal using auxiliary variables $p=z_{x}$ and $q=z_{y}$ and thus do not have the additional data term contributions $\tfrac{\partial}{\partial x} \left[ D \right]_{z_{x}}$ and $\tfrac{\partial}{\partial y} \left[ D \right]_{z_{y}}$. Let us now take a closer look at all the individual terms that occur in Eq. (\[eq:49\]). After some computations we obtain $$\begin{aligned} \label{eq:1} \left[ D \right]_z &=& 2 \, \left( I - \dfrac{Q^3}{z \, W} \right) \, \left( \dfrac{Q^3}{z^2 \, W} + \dfrac{Q^3 }{z \, W^2} \left[ W \right]_z \right) \\ &=& \nonumber 2 \, \left( I - \dfrac{Q^3}{z \, W} \right) \, \dfrac{Q^3}{z \, W} \, \left( \dfrac{1}{z} + \dfrac{ \nabla z \cdot \mathbf{x} + z }{W^2} \right) \end{aligned}$$ $$\begin{aligned} \label{eq:23} \dfrac{\partial}{\partial x} \left[ D \right]_{z_{x}} &=& \left[ 2 \, \left( I - \dfrac{Q^3}{z \, W} \right) \, \dfrac{Q^3}{z \, W^3} \, \left[ W \right]_{x} \right]_{x} \\ &=& \nonumber \left[ 2 \, \left( I - \dfrac{Q^3}{z \, W} \right) \, \dfrac{Q^3}{z \, W^3} \, \left( \mathtt{f}^2 \, z_{x} + \left[ \nabla z \cdot \mathbf{x} + z \right] \, x \right) \right]_{x} \end{aligned}$$ $$\begin{aligned} \label{eq:27} \dfrac{\partial}{\partial y} \left[ D \right]_{z_{y}} &=& \left[ 2 \, \left( I - \dfrac{Q^3}{z \, W} \right) \, \dfrac{Q^3}{z \, W^3} \, \left[ W \right]_{y} \right]_{y} \\ &=& \nonumber \left[ 2 \, \left( I - \dfrac{Q^3}{z \, W} \right) \dfrac{Q^3}{z \, W^3} \, \left( \mathtt{f}^2 \, z_{y} + \left[ \nabla z \cdot \mathbf{x} + z \right] \, y \right) \right]_{y} \end{aligned}$$ as well as $$\label{eq:29} \dfrac{\partial^2}{\partial x^2} \left[ S \right]_{z_{xx}} = 2 \, \dfrac{\partial^2}{\partial x^2} \left[ \Psi'(\left\| \mbox{Hess}(z) \right\|_F^2) \; z_{xx} \right] , \quad$$ $$\label{eq:30} 2 \dfrac{\partial^2}{\partial xy} \left[ S \right]_{z_{xy}} = 4 \, \dfrac{\partial^2}{\partial xy} \left[ \Psi'(\left\| \mbox{Hess}(z) \right\|_F^2) \; z_{xy} \right] , \quad$$ $$\label{eq:31} \dfrac{\partial^2}{\partial y^2} \left[ S \right]_{z_{yy}} = 2 \, \dfrac{\partial^2}{\partial y^2} \left[ \Psi'(\left\| \mbox{Hess}(z) \right\|_F^2) \; z_{yy} \right] \, ,$$ where the derivative of the penaliser function $\Psi(s^2)$ reads $$\label{eq:80} \Psi'(s^2) = \frac{\partial}{\partial (s^2)} \Psi(s^2)= \frac{1}{\sqrt{ 1 + \tfrac{s^2}{\lambda^2}}} \; .$$ While the contributions of the data term are related to the influence of $z$ and $\nabla z$ on the brightness equation, the contributions of the smoothness term define an edge-preserving fourth-order diffusion process. This becomes explicit as follows: Since $\Psi'(s^2)$ becomes small for large values of $s^2$, this reduces the effect of the smoothing at locations with high curvature, i.e. where $\left\| \mbox{Hess}(z) \right\|_F^2$ is large. After we have derived the resulting Euler-Lagrange equation, let us now discuss how this equation can be discretised appropriately. [**Discretisation.**]{} In order to discretise the contributions of the data term given by Eqs. – , we employ the upwind scheme from [@RouyTourin_SINUM1992] in view of the hyperbolic nature of the underlying PDE. In 1D, the corresponding upwind discretisation reads $$\label{eq:7} \widetilde{z}_{x} \approx \max \left( D^- z, - D^+ z, 0 \right) \, ,$$ with $$\label{eq:8} D^- z = \dfrac{z_i - z_{i-1} }{h_{x}} \qquad \mbox{and} \qquad D^+ z = \dfrac{z_{i+1} - z_i }{h_{x}} \, ,$$ where $h_{x}$ denotes the grid size. Please note that in contrast to upwind schemes for eikonal equations [@Sethian1999] that typically approximate only the magnitude of the gradient, the sign matters in our case, such that we have to choose $$\label{eq:z_x} z_x = \left\{ \begin{array}{lcl} D^+ z & & \mbox{if} \quad \widetilde{z}_x = -D^+ z \, , \\ \widetilde{z}_x & & \mbox{otherwise} \, . \end{array} \right.$$ This selects the actual forward difference, if the second argument in (\[eq:7\]) is the maximum [@BCDFV10; @BCDFV12]. This scheme can be extended in a straightforward way to 2D. For discretising the contributions of the smoothness term, a standard central difference scheme is used. Since it is difficult to discretise the Euler-Lagrange equation directly, we followed a [*first discretise then optimise*]{} scheme. To this end, we used the aforementioned finite difference approximations to discretise the energy in applying the upwind scheme for the data term and a central difference approximation for the smoothness term. Then, by computing the derivatives of the discrete energy we obtain a proper discretisation for the Euler-Lagrange equation. Finally, by using the Euler forward time discretisation method $$\label{eq:20} z_t \approx \dfrac{z^{n+1} - z^n}{\tau} \, ,$$ with $\tau$ being a time step size, we can reformulate the solution of Eq. as the steady state of the corresponding evolution equation in artificial time. Thus we obtain the following explicit scheme $$\label{eq:21} \dfrac{z^{n+1} - z^n}{\tau} + EL^n = 0 \qquad \Leftrightarrow \qquad z^{n+1} = z^n - \tau \, EL^n \, ,$$ where $EL^n$ is the discretisation of the Euler-Lagrange equation evaluated at time $n$. Please note that this discretisation may change over time, since we re-discretised the energy in each iteration by adapting the direction of the discretisation of the upwind scheme (forward, backward, no contribution) based on evaluating Eqs. – for the result of the previous time step. In that sense we use a [*lagged*]{} discretisation approach, where the discretisation is updated in each iteration. [**Coarse-to-Fine Approach.**]{} Since the underlying energy functional is highly non-convex, the proposed explicit scheme may get trapped in local minima. To tackle this problem, we propose to embed the estimation into a coarse-to-fine framework. Starting from a very coarse resolution, we successively refine the input image while repeatedly reconstructing the surface. Thereby, solutions from coarser levels serve as initialisation for the finer scales. Similar hierarchical schemes have already been successfully applied to many other problems in computer vision; see e.g. [@BCDFV10; @Broxz_ECCV2004]. Apart from improving the quality of the results by avoiding local minima, coarse-to-fine schemes also render the estimation more robust w.r.t. the choice of the initialisation. In fact, if sufficiently many resolution levels were used, we could hardly observe any impact of the initialisation on the quality of the final results. Since a good initial guess can still be useful to speed up the computation, we propose to initialise the depth by pointwise solving the data term in Eq. for $\nabla z = 0$ $$\label{eq:333} D( \mathbf{x}, z, 0 ) = 0 \;\;\;\; \Rightarrow \;\;\;\; z = \sqrt{\dfrac{Q(\mathbf{x})^3}{I(\mathbf{x})}} \, .$$ This can be seen as an efficient compromise between using the full model which is evidently not feasible and only considering the inverse square law, i.e. $z = 1/\sqrt{I(\mathbf{x})}$, which completely neglects the effect of the surface orientation and thus actually provides a local upper bound for the correct depth. In any case, in contrast to other variational SfS methods from the literature, our technique does not have to rely on initialisations from non-variational SfS approaches [@Abdelrahim_ICIP; @2013; @ZYT07] or surface integration methods [@Wu_IJCV2010] to provide meaningful results. Let us now discuss the details of our coarse-to-fine approach. To this end, we introduce the parameter $\eta$ that specifies the downsampling factor between two consecutive resolution levels and that is typically chosen in the interval $(0.5,1)$. Then the grid size at level $k$ of our coarse-to-fine approach can be computed as $$\begin{aligned} h^k_{x} = h_{x} \cdot \eta^{-k} \; , \;\;\;\;\; h^k_{y} = h_{y} \cdot \eta^{-k} \; .\end{aligned}$$ where $k=0$ is the original resolution and $k=k_{\mathrm{max}}$ is the coarsest level. This tells us that the grid size becomes larger at coarser scales which intuitively makes sense, since the size of the image plane remains constant while the number of pixels decreases. At the same time, however, this increase of the grid size leads to a major problem: Since the contributions of the smoothness term given by Eqs. – involve fourth-order derivatives that scale proportionally to $1/h^4$, the strength of the regularisation actually decreases with $\eta^{4k}$ on coarser scales. In order to compensate for this effect, we thus propose to scale the smoothness weight $\alpha$ according to $$\alpha^k = \eta^{-4k} \cdot \alpha \, .$$ This guarantees a similar amount of regularisation for all resolution levels. [**Alternating Explicit Scheme.**]{} Finally, we observed in our experiments that the terms in Eqs. and that refer to the influence of the depth gradient $\nabla z$ on the brightness equation require to select the time step size $\tau$ rather small. In particular, these terms do not have a weighting parameter such as the smoothness term that can be adjusted appropriately. As a consequence, the minimisation typically needs several thousands or even millions of iterations. To counter this problem, we propose the following alternating estimation strategy at each resolution level: For a fixed number of iterations $n$, instead of performing $n$ iterations using the original explicit scheme, we propose an alternating iterative scheme that first does $n/2$ iterations with a simplified explicit scheme neglecting the two terms in Eqs. and , followed by $n/2$ iterations with the entire explicit scheme. Since the neglected terms are based on second-order derivatives and the remaining terms did not strongly affect the convergence, we empirically found out that we can choose the time step size approximately $\min(h_{x}^{-2}, h_{y}^{-2})$ times larger for the first $n/2$ iterations (given that $h_{x},h_{y}\ll 1$). In our experiments this leads to speed-ups of about one to four orders of magnitude. Moreover, in most cases, even the simplified scheme was sufficient to achieve excellent results. Thereby one should note that, from a numerical viewpoint, the simplified scheme can be understood as an optimisation method for a series of energy functionals of type of Eq. , where the gradient $\nabla z$ is lagging and thus has no direct influence on the optimisation. Intrinsic Parameters {#sec:intrinsic} ==================== So far we have derived a variational model for perspective SfS with Cartesian depth parametrisation that is given in terms of [*image coordinates*]{}. Let us now discuss how the model and the minimisation has to be adapted if we additionally consider the intrinsic camera parameters, i.e. if we express the model in terms of [*pixel coordinates*]{}. [**Coordinate Transformation.**]{} Let the corresponding calibration matrix be given by $$\label{eq:200} K = \left[ \begin{array}{ccc} \mathtt{f} / h_{x} & 0 & c_1\\ 0 & \mathtt{f} / h_{y} & c_2 \\ 0 & 0 & 1\\ \end{array} \right] \, .$$ where $(c_1,c_2)^\top$ denotes the location of the focal point, and $h_{x}$ and $h_{y}$ is the grid size in $x$- and $y$-direction, respectively [@HZ04]. Knowing this matrix allows us to reformulate the image coordinates $\mathbf{x}=(x, y)^\top$ of our original model in terms of pixel coordinates $\mathbf{a}=(a,b)^\top$. The corresponding transformation is given by $$\label{eq:201} \left[ \begin{array}{c} a\\ b\\ - 1\\ \end{array} \right] = K \; \dfrac{1}{\mathtt{f}} \left[ \begin{array}{c} x\\ y\\ - \mathtt{f}\\ \end{array} \right] \; \Rightarrow \; \left[ \begin{array}{c} x\\ y\\ - \mathtt{f}\\ \end{array} \right] = \mathtt{f} \; K^{-1} \left[ \begin{array}{c} a\\ b\\ - 1\\ \end{array} \right] \; ,$$ where one has to take care that the image plane is at distance $\mathtt{f}$ of the camera centre. Plugging Eq. into Eq. then yields $$\begin{aligned} \label{eq:202} \mathbf{x}(\mathbf{a}) = \left[ \begin{array}{c} x (a) \\ y (b) \end{array} \right] = \left[ \begin{array}{cc} h_{x} & 0 \\ 0 & h_{y} \end{array} \right] \left[ \begin{array}{c} a \\ b \end{array} \right] - \left[ \begin{array}{c} h_{x} \, c_1 \\ h_{y} \, c_2 \end{array} \right] \; .\end{aligned}$$ [**Variational Model.**]{} Now we are in the position to reformulate our entire model in terms of pixel coordinates. Substituting Eq. into our original energy and transforming the integration domain $\overline{\Omega}_{\mathbf{a}}=\mathbf{x}^{-1}(\overline{\Omega}_{\mathbf{x}})$ accordingly, we obtain the following variational model expressed in terms of pixel coordinates $$\label{eq:2b} E \left( z(\mathbf{x(a)}) \right) = \int_{\overline{\Omega}_{\mathbf{a}}} \, c(\mathbf{x(a)}) \; \underbrace{ D(\mathbf{x(a)} , z(\mathbf{x(a)}), \nabla z(\mathbf{x(a)})) }_{\mbox{Data term}} + \, \alpha \underbrace{ S(\mbox{Hess}(z)(\mathbf{x(a)})) }_{\!\!\!\!\!\!\!\!\!\!\!\!\!\mbox{Smoothness term}\!\!\!\!\!\!\!\!\!\!\!\!\!} \, \mathrm{d}\mathbf{a} \; . \\[2mm]$$ Please note that we omitted the substitution factor given by $|\mathrm{det}(J(\mathbf{x}(\mathbf{a})))|$, where $J$ is the Jacobian, since this factor is constant and thus does not change the minimiser of our energy. Let us now derive the corresponding Euler-Lagrange equation for our novel model expressed in terms of pixel coordinates. [**Euler-Lagrange Equation.**]{} Analogously to Eq. we drop the dependencies on all variables and obtain the following Euler-Lagrange equation $$\begin{aligned} \label{eq:49b} 0 & \; = \; & c \left( \left[ D \right]_z - \dfrac{\partial}{\partial a} \left[ D \right]_{z_{a}} - \dfrac{\partial}{\partial b} \left[ D \right]_{z_{b}} \right) \nonumber\\ & & \; + \; \alpha \; \left( \dfrac{\partial^2}{\partial a^2} \left[ S \right]_{z_{aa}} + 2 \dfrac{\partial^2}{\partial a \partial b} \left[ S \right]_{z_{ab}} + \dfrac{\partial^2}{\partial b^2} \left[ S \right]_{z_{bb}} \right) \\ & \; = \; & \label{eq:49c} c \left( \left[ D \right]_z - \dfrac{\partial}{\partial x} \left[ D \right]_{z_{x}} - \dfrac{\partial}{\partial y} \left[ D \right]_{z_{y}} \right) \nonumber\\ & & \; + \; \alpha \; \left( \dfrac{\partial^2}{\partial x^2} \left[ S \right]_{z_{xx}} + 2 \dfrac{\partial^2}{\partial x \partial y} \left[ S \right]_{z_{xy}} + \dfrac{\partial^2}{\partial y^2} \left[ S \right]_{z_{yy}} \right) \; , \end{aligned}$$ where we exploited the following relation between derivatives in pixel and image coordinates due to Eq. $$\begin{aligned} \label{eq:203} &&\dfrac{\partial}{\partial a} = h_{x} \; \dfrac{\partial}{\partial x} \; , \;\;\;\;\; \dfrac{\partial}{\partial b} = h_{y} \; \dfrac{\partial}{\partial y} \; , \;\;\;\;\; \dfrac{\partial}{\partial z_{a}} = \dfrac{1}{h_{x}} \; \dfrac{\partial}{\partial z_{x}} \; , \;\;\;\;\; \dfrac{\partial}{\partial z_{b}} = \dfrac{1}{h_{y}} \; \dfrac{\partial}{\partial z_{y}} \; , \;\;\;\;\; \\ && \dfrac{\partial}{\partial z_{aa}} = \dfrac{1}{h_{x} h_{x}} \; \dfrac{\partial}{\partial z_{xx}} \; , \;\;\;\;\; \dfrac{\partial}{\partial z_{ab}} = \dfrac{1}{h_{x} h_{y}} \; \dfrac{\partial}{\partial z_{xy}} \; , \;\;\;\;\; \dfrac{\partial}{\partial z_{bb}} = \dfrac{1}{h_{y} h_{y}} \; \dfrac{\partial}{\partial z_{yy}} \, .\end{aligned}$$ The equality between Eq. and Eq. shows that the Euler-Lagrange equations of our models in pixel and image coordinates are basically identical. One only has to parametrise the terms $\eqref{eq:1}$–$\eqref{eq:31}$ that have been originally derived in image coordinates using the coordinate transform in Eq. . Apart from that, the discretisation can be performed in accordance with our explanations from the previous section. In this context, the grid size is given by the intrinsic parameters $h_{x}$ and $h_{y}$. Moreover, one has to adapt the camera matrix $K$ at each level of the coarse-to-fine scheme. This requires to scale both the grid size and the principal point $(c_1,c_2)^\top$. Evaluation {#sec:experimental-results} ========== [**Test Images and Error Measures.**]{} In order to evaluate our novel approach we make use of four synthetic images with ground truth that fulfil the underlying assumptions regarding reflectance and illumination. This allows us to compute two error measures: one with respect to the reconstructed surface and the other one with respect to reprojected image. The first error measure is the *relative surface error* (RSE) of a point wise computed Euclidean distance between the computed surface $\mathcal{S}$ and the ground truth surface $\mathcal{S}^{\mathrm{gt}}$. It is given by $$\mathrm{RSE} = \frac{ \sum_{\overline{\Omega}_{\mathbf{a}}} \left| \mathcal{S}(\mathbf{x(a)}) - \mathcal{S}^{\mathrm{gt}}(\mathbf{x(a)}) \right| } { \sum_{\overline{\Omega}_{\mathbf{a}}} \left| \mathcal{S}^{\mathrm{gt}}(\mathbf{x(a)}) \right| } \, ,$$ where the normalisation allows to determine the reconstruction error *relative* to the ground truth shape. This in turn makes errors of differently scaled surfaces comparable. The second error measure is the *relative image error* (RIE) between the reprojected image $I$ and the given input image $I^{\mathrm{gt}}$. It is defined as follows $$\mathrm{RIE} = \frac{ \sum_{\overline{\Omega}_{\mathbf{a}}} \left| I(\mathbf{x(a)}) - I^{\mathrm{gt}}(\mathbf{x(a)}) \right| } { \sum_{\overline{\Omega}_{\mathbf{a}}} \left| I^{\mathrm{gt}}(\mathbf{x(a)}) \right| } \, .$$ This time, however, the normalisation is performed with respect to the brightness of the input image to make reprojection results for input images with different brightness scale comparable. Summarising: While the first measure reflects how well the reconstruction matches the ground truth surface, the second measure determines how well the reprojection fits the input data. Let us now discuss the considered test images which are depicted in Fig. \[fig:synthetic\] in detail. The first synthetic test image [*Sombrero*]{} was generated from a known parametric surface, using the following equation $$Z(X, Y) = 0.5 \frac{\sin \left( r(X, Y) \right) }{r(X, Y)} + 1.7 \, , \quad r(X, Y) = \sqrt{ \left( 10 X \right)^2 + \left( 10 Y \right)^2 } \, .$$ The image was rendered using Eq. at a size of $256\!\times\!256$ pixels, where the focal length was set $\mathtt{f} = 1$, the grid size was chosen to be $h_{x} = h_{y} = 1/200$ and the principal point was fixed at $c = (128,128)^\top$. The second test image [*Suzanne*]{} was generated using the open-source software Blender [@BLENDER]. In this context, the Z-buffer of the rendering path and the corresponding intrinsic parameters ($\mathtt{f} = 35$, $h_{x}= 1/16$, $h_{y} = 9/128$, $c = (256,128)^\top$) were extracted and the final image was rendered at a size of $512\!\times\!256$ using Eq. as before. The other two test images [*Stanford Bunny*]{} and [*Dragon*]{} have been computed likewise using 3-D models obtained from the Stanford 3D scanning repository [@STAN3D]. For them a size of $256\!\times\!256$ pixels and the intrinsic parameters ($\mathtt{f} = 35$, $h_{x} = 1/8$, $h_{y} = 9/128$, $c = (128,128)^\top$) were chosen. Finally, all images were saved as $8$-bit grey-value images. Let us finally comment on the selection of the parameters in our experiments. In order to keep the number of parameters low, we choose a preferred standard set of parameters for all the following experiments, unless otherwise stated: A downsampling factor of $\eta = 0.8$ for the coarse-to-fine approach, $n=10^6$ solver iterations on each coarse-to-fine level and a contrast parameter of $\lambda = 10^{-3}$. Moreover, the time step size $\tau$ provided in the different experiments always refers to the simplified explicit scheme. The time step size for the full explicit scheme is $\min(h_{x}^{2}, h_{y}^{2})$ times smaller. \ \ \ \ \ \ \ \ [**Results on Synthetic Test Images.**]{} In our first experiment we evaluate the reconstruction quality of our novel approach. To this end, we applied our perspective SfS algorithm to all four of the previously discussed test images and compared the reprojected image and the reconstruction to the ground truth; see Fig. \[fig:synthetic-sbd\] and Fig. \[fig:synthetic-suzanne\]. Herein, the depth values are colour-coded in such a way that depth increases from red via green to blue. As one can see, both the reprojected image as well as the estimated depth values coincide very well with the ground truth. This is also confirmed by the corresponding surface error maps in Fig. \[fig:errordiff\]. Indeed, only small differences for the *Stanford bunny* (right paw) and the *Dragon* (tail tip) are visible. As a consequence both error measures which are listed in Table \[tab:resultsSyntheticQuadtratic\] are very small. Moreover, one can see that the proposed subquadratic penaliser outperforms a quadratic smoothness term in most cases. Only for the [*Sombrero*]{} which has a rather smooth surface, the reconstruction error is smaller in the quadratic case.\ ---------- ---------------- ----- -- -------------------- ----- -- -------------------- ----- -- -------------------- ----- -- -------------------- --------- -- ---------- -- \[-2mm\] \[-2mm\] \[-2mm\] RSE RIE RSE RIE runtime \[-2mm\] \[-2mm\] Sombrero $\textbf{0.00208}$ $0.00694$ $0.00318$ $\textbf{0.00209}$ $29113$s \[-2mm\] \[-2mm\] Stanford Bunny $0.00546$ $0.00015$ $\textbf{0.00439}$ $\textbf{0.00007}$ $23969$s \[-2mm\] \[-2mm\] Dragon $\textbf{0.01376}$ $\textbf{0.00028}$ $\textbf{0.01376}$ $\textbf{0.00028}$ $25350$s \[-2mm\] \[-2mm\] Suzanne $0.00392$ $0.00011$ $\textbf{0.00251}$ $\textbf{0.00002}$ $48395$s \[-2mm\] ---------- ---------------- ----- -- -------------------- ----- -- -------------------- ----- -- -------------------- ----- -- -------------------- --------- -- ---------- -- : Results for our approach with quadratic and subquadratic penaliser. Error measures are given in terms of the relative surface error (RSE) and the relative image error (RIE). Best results for each test image are highlighted boldface. Same parameters as in Fig. \[fig:synthetic-sbd\] and Fig. \[fig:synthetic-suzanne\].[]{data-label="tab:resultsSyntheticQuadtratic"} [**Influence of the Regularisation.**]{} In our second experiment we investigate the influence of the regularisation on the quality of the reconstruction and its reprojection. To this end, we consider the *Sombrero* test image and vary the regularisation parameter $\alpha$ while the other parameters are kept fixed ($\tau = 0.001$, $n=10^4$). The outcome is visualised in Fig. \[fig:InfluenceRegularization\]. While the reprojection related error measure (RIE) increases for a moderate amount of regularisation but is overall very low, the surface related error measure (RSE) decreases by almost a factor three (from $4.4 \times 10^{-2}$ to $1.7 \times 10^{-2}$). This, however, is not surprising, since the computed surface typically exhibits some form of smoothness and thus benefits from a moderate amount of regularisation. Since the actual purpose of SfS is to find the correct surface, this shows that the regularisation may have an overall positive impact on the quality of the results.\ ![Impact of the amount of regularisation on the reconstruction quality and the reprojection accuracy for the *Sombrero* test image.[]{data-label="fig:InfluenceRegularization"}](figs/sombreroCharRSERIE.pdf){width="0.5\linewidth"} [**Independence of the Initialisation.**]{} In our third experiment we analyse the dependency of our approach on the initialisation. To this end, we use the *Stanford Bunny* ($z \in [1,2]$) and compare our initialisation on the coarsest scale of the proposed coarse-to-fine scheme (cf. Eq. ) with two other initialisations based on plain surfaces ($z=1$, $z=10$). The initial error and the outcome after $n=10^6$ iterations are listed in Table \[tab:resultsInit\]. While the initial error for a good guess ($z=1$) and a poor initialisation ($z=10$) differs significantly, the quality of the reconstruction and the reprojection is identical after sufficiently many iterations. This also holds for our initialisation which can be computed from the input image without requiring a specific knowledge of the depth. That all initialisations converge to the same solution, however, is not surprising since the estimation is embedded in our coarse-to-fine scheme.\ ---------- ------------------ ----- -- ----------- ----- -- ----------- ----- -- ----------- ----- -- ----------- -- \[-2mm\] \[-2mm\] \[-2mm\] RSE RIE RSE RIE \[-2mm\] \[-2mm\] plane ($z = 1$) $0.25804$ $1.63174$ $0.00439$ $0.00007$ \[-2mm\] \[-2mm\] plane ($z = 10$) $6.41960$ $0.97373$ $0.00439$ $0.00007$ \[-2mm\] \[-2mm\] proposed $0.37712$ $0.74363$ $0.00439$ $0.00007$ \[-2mm\] ---------- ------------------ ----- -- ----------- ----- -- ----------- ----- -- ----------- ----- -- ----------- -- : Impact of different initialisations on the reconstruction quality and reprojection accuracy for the *Stanford Bunny* ($\alpha = 7.5 \times 10^{-5}$, $\tau = 10^{-3}$, $n=10^6$).[]{data-label="tab:resultsInit"} [**Comparison of Numerical Schemes.**]{} In our fourth experiment we compare the different numerical schemes proposed in Section \[sec:minimisation\]: the full explicit scheme, the simplified explicit scheme and the alternating explicit scheme. In the first part of the experiment we juxtapose the quality of the different numerical schemes for equal stopping times (iterations $\times$ time step size). As one can see from the results in Table \[tab:resultsDirectionSmallChar\], the full explicit scheme clearly gives the best results in terms of reconstruction quality and reprojection accuracy. However, this comes at the expense of a significantly larger runtime, since more iterations are needed due to the time step restrictions discussed in Section \[sec:minimisation\]. In fact the runtime is up to four orders of magnitude larger making the approach hardly feasible for larger image sizes. In the second part of the experiment we compared the numerical schemes for an equal number of iterations. From the results in Table \[tab:resultsSmallct\] it becomes evident that in this case the simplified explicit scheme and in particular the alternating explicit scheme perform best in most cases in terms of reconstruction quality and reprojection accuracy. This demonstrates that it can be worthwhile to (partly) omit the terms that are added in the full explicit scheme since they slow down the convergence, but doing so does not necessarily compromise the quality. ---------- ---------------------- -- -- -------------------- -- -- ----------- -- -- ----------- -- -- ----------- -- -- -------------------- -- -- -------------------- -- \[-2mm\] \[-2mm\] \[-2mm\] test image RSE RIE RSE RIE RSE RIE \[-2mm\] \[-2mm\] Small Sombrero $0.01823$ $0.01920$ $0.01820$ $0.02048$ $\textbf{0.00785}$ $\textbf{0.00527}$ \[-2mm\] \[-1mm\] ($128 \times 128$) \[-3mm\] \[-2mm\] Small Stanford Bunny $0.00659$ $0.00151$ $0.00667$ $0.00257$ $\textbf{0.00576}$ $\textbf{0.00097}$ \[-2mm\] \[-1mm\] ($128 \times 128$) \[-3mm\] \[-2mm\] Small Dragon $0.01667$ $0.00267$ $0.01673$ $0.00620$ $\textbf{0.01526}$ $\textbf{0.00205}$ \[-2mm\] \[-1mm\] ($128 \times 128$) \[-3mm\] \[-2mm\] Small Suzanne $\textbf{0.00899}$ $0.00514$ $0.01055$ $0.01909$ $0.01022$ $\textbf{0.00203}$ \[-2mm\] \[-1mm\] ($128 \times 96$) \[-3mm\] ---------- ---------------------- -- -- -------------------- -- -- ----------- -- -- ----------- -- -- ----------- -- -- -------------------- -- -- -------------------- -- : Comparison of different numerical schemes for equal stopping time $t = n \times \tau$. Results and runtimes refer to smaller versions of the four test images. Same parameters as in Fig. \[fig:synthetic-sbd\] and Fig. \[fig:synthetic-suzanne\] except for $n$, which is given by $n = t / \tau$. []{data-label="tab:resultsDirectionSmallChar"} ---------- ---------------------- -- -- -------------------- -- -- -------------------- -- -- -------------------- -- -- -------------------- -- -- -------------------- -- -- ----------- -- \[-2mm\] \[-2mm\] \[-2mm\] test image RSE RIE RSE RIE RSE RIE \[-2mm\] \[-2mm\] Small Sombrero $0.02357$ $\textbf{0.00082}$ $0.02392$ $0.00659$ $\textbf{0.00358}$ $0.00319$ \[-2mm\] \[-2mm\] Small Stanford Bunny $0.00390$ $\textbf{0.00001}$ $\textbf{0.00378}$ $0.00004$ $0.00489$ $0.00047$ \[-2mm\] \[-2mm\] Small Dragon $0.00572$ $\textbf{0.00001}$ $\textbf{0.00562}$ $\textbf{0.00001}$ $0.00964$ $0.00170$ \[-2mm\] \[-2mm\] Small Suzanne $\textbf{0.00319}$ $0.00002$ $0.00320$ $\textbf{0.00001}$ $0.00505$ $0.00056$ \[-2mm\] ---------- ---------------------- -- -- -------------------- -- -- -------------------- -- -- -------------------- -- -- -------------------- -- -- -------------------- -- -- ----------- -- : Comparison of different numerical schemes for equal number of iterations. Results refer to the smaller versions of the four test test images, see Tab. \[tab:resultsDirectionSmallChar\]. The same parameters as in Fig. \[fig:synthetic-sbd\] and Fig. \[fig:synthetic-suzanne\] have been used except for $n$, which is given by $n = 10^7$. []{data-label="tab:resultsSmallct"} ---------- ----- ------------------------------------------- -- ----------- ---------------------------------------------- -- ----------- -------------------------- -- ----------- -- \[-2mm\] perforated version sliced version original version \[-2mm\] \[-2mm\] (Fig. \[fig:synthetic-inpaint\], top row) (Fig. \[fig:synthetic-inpaint\], bottom row) (Fig. \[fig:synthetic\]) \[-2mm\] \[-2mm\] RSE $0.00439$ $0.00509$ $0.00439$ \[-2mm\] \[-2mm\] RIE $0.00039$ $0.00249$ $0.00007$ \[-2mm\] ---------- ----- ------------------------------------------- -- ----------- ---------------------------------------------- -- ----------- -------------------------- -- ----------- -- : Evaluation of inpainting properties for degraded versions of the *Stanford Bunny* test image. Same parameters as in Figure \[fig:synthetic-inpaint\].[]{data-label="tab:resultsInpaint"} [**Reconstruction with Inpainting.**]{} In our fifth experiment we demonstrate the inpainting capabilities of the regularisation in combination with the confidence function $c$ embedded in the data term. For this reason we created a pair of degraded *Stanford Bunny* test images together with the corresponding confidence functions, which are both depicted in Fig. \[fig:synthetic-inpaint\]. In addition, the computed depth values and the reprojected images are shown. One can see that in both cases the missing regions in the input image can hardly deteriorate the quality of the results since the smoothness term fills in the information from the neighbourhood. This is also reflected in the error measures given in Table \[tab:resultsInpaint\]. In case of the perforated version the surface error even remains the same compared to the result for the original version.\ \ \ [**Comparison with a PDE-Based Approach.**]{} In our seventh experiment we compare the results of our variational method with the PDE-based approach of Vogel [*et al.*]{} [@Vogel_SSVM2009] with Lambertian reflectance model. This essentially comes down to a comparison to the baseline method of Prados [*et al.*]{} [@Prados_CVPR2005] which is solved by Vogel [*et al.*]{} [@Vogel_SSVM2009] as part of a Phong-based model using an efficient fast marching scheme [@Sethian1999]. In this experiment we consider two scenarios, that nicely demonstrate the advantages and shortcomings of the different types of methods: On the one hand, we use input images without noise, on the other hand, we added Gaussian noise of standard deviation $\sigma=20$ before applying the two methods. The corresponding results are summarised in Tables \[tab:comparison\] and \[tab:comparisonNoise\], respectively. For the test images without noise both approaches give excellent results with errors among or below $1$ percent of the solution. Thereby the approach of Vogel [*et al.*]{} gives slightly better results in terms of the relative surface error (RSE), while the variational approach gives better results in terms of the relative image error (RIE). From the viewpoint of the variational approach this can be explained as follows: While the data term penalises deviations from the photometric reprojection error and thus gives rather small RIE values, the regulariser and the coarse-to-fine scheme yield a moderate smoothing of the surface resulting in slightly higher RSE values. In the case of the noisy input images the findings are completely different. Here, the variational method can take advantage of both the regulariser and the independence of the initialisation. While a higher smoothness weight allows to obtain a smooth surface, the hierarchical initialisation via the coarse-to-fine scheme does not require to rely on noisy solutions at critical points as the PDE-based approach of Vogel [*et al.*]{} As a consequence, the resulting surface errors of $3$ to $6$ percent for our variational approach are significantly lower than those of the PDE-based model ($11$ to $20$ percent). This can also be seen from the depth estimates for the Stanford Bunny depicted in Figure \[fig:bunnyNoise\]. Not surprisingly our findings are in full accordance with the observation in [@JBB2014], in which the robustness of variational methods for perspective SfS has been investigated.\ ------------ ---------------- ----- -- ----------- ----- -- ----------- ----- -- ----------- ----- -- ----------- -- \[-2mm\] \[-2mm\] \[-1.5mm\] \[-2mm\] \[-2mm\] RSE RIE RSE RIE \[-2mm\] \[-2mm\] Sombrero $0.00301$ $0.00495$ $0.00318$ $0.00209$ \[-2mm\] \[-2mm\] Stanford Bunny $0.00266$ $0.00154$ $0.00439$ $0.00007$ \[-2mm\] \[-2mm\] Dragon $0.00422$ $0.00255$ $0.01376$ $0.00028$ \[-2mm\] \[-2mm\] Suzanne $0.00253$ $0.00082$ $0.00251$ $0.00002$ \[-2mm\] ------------ ---------------- ----- -- ----------- ----- -- ----------- ----- -- ----------- ----- -- ----------- -- : Comparison between our variational method and the PDE-based approach of Vogel [*et al.*]{} [@Vogel_SSVM2009] with Lambertian reflectance model ($=$ baseline model of Prados [*et al.*]{} [@Prados_CVPR2005]). Error measures are given in terms of the relative surface error (RSE) and the relative image error (RIE). Same parameters as in Fig. \[fig:synthetic-sbd\] and Fig. \[fig:synthetic-suzanne\].[]{data-label="tab:comparison"} ------------ ---------------------- ----- -- ----------- ----- -- ----------- ----- -- ----------- ----- -- ----------- -- \[-2mm\] \[-2mm\] \[-1.5mm\] \[-2mm\] \[-2mm\] RSE RIE RSE RIE \[-2mm\] \[-2mm\] Noisy Sombrero $0.19530$ $0.27254$ $0.05118$ $0.13239$ \[-2mm\] \[-2mm\] Noisy Stanford Bunny $0.10973$ $0.17347$ $0.03235$ $0.15279$ \[-2mm\] \[-2mm\] Noisy Dragon $0.12240$ $0.19409$ $0.05395$ $0.18767$ \[-2mm\] \[-2mm\] Noisy Suzanne $0.12134$ $0.16783$ $0.01256$ $0.14302$ \[-2mm\] ------------ ---------------------- ----- -- ----------- ----- -- ----------- ----- -- ----------- ----- -- ----------- -- : Performance under noise. Comparison between our variational method and the PDE-based approach of Vogel [*et al.*]{} [@Vogel_SSVM2009] with Lambertian reflectance model ($=$ baseline model of Prados [*et al.*]{} [@Prados_CVPR2005]). Gaussian noise of standard deviation $\sigma=20$. Error measures are given in terms of the relative surface error (RSE) and the relative image error (RIE). The applied parameters are as follows: *Sombrero* ($\alpha = 0.1$, $\tau = 10^{-5}$, $n=10^6$), *Stanford Bunny* ($\alpha = 1.0$, $\tau = 10^{-5}$, $n=10^6$), *Dragon* ($\alpha = 1.0$, $\tau = 10^{-5}$, $n=10^6$), *Suzanne* ($\alpha = 1.0$, $\tau = 5\times10^{-6}$, $n=10^6$).[]{data-label="tab:comparisonNoise"} \ [**Results on Real-World Images.**]{} Finally, in order to evaluate our approach on real-world images, we used two images of faces provided by Prados [@PCF06]. According to Prados, these images have been taken with a cheap digital camera in a dark place, where the scene is illuminated by the flash of the camera. The focal length is $\mathtt{f}=5.8$mm and the grid size is approximately $h_{x} = h_{y} = 0.018$mm. The test images as well as additional images rendered from a new viewpoint using the computed depth are shown in Fig. \[fig:realworld\]. In both cases the results look quite realistic. One can also see how the depth values at the eyes have been inpainted in the reconstruction, since a manually defined confidence function was used to mask out those regions where the assumption of a Lambertian surface is violated.\ \ \ Conclusion {#sec:conclusion} ========== In this paper, we described a novel variational model for perspective shape from shading that not only has many desirable theoretical properties but also yields very convincing reconstruction results for synthetic and real-world input images, even in the presence of noise or other deteriorations in an input image. While the arising optimisation problem has turned out to be challenging, we have proposed an alternating explicit scheme embedded in a coarse-to-fine framework that is robust with respect to the initialisation and that allows reasonable computation times compared to a standard explicit scheme. Besides the results that are documented via extensive experiments in this chapter, let us point out that we see a main contribution of our work in a different context, as we have layed the fundamental building block for a conceptually correct, working variational framework that can combine perspective shape from shading with other techniques from computer vision such as e.g. stereo vision. We aim to explore the arising possibilities in a future work. This work has been partially funded by the Deutsche Forschungsgemeinschaft (DFG) as a joint project (BR 2245/3-1, BR 4372/1-1). Appendix {#sec:appendix} ======== [**Alternative Derivation of the Surface Normal.**]{} Instead of computing the derivatives with respect to the 2-D image coordinates $x$ and $y$, one can also derive the surface normal in an alternative way that is often used in the literature, see e.g. [@Wu_IJCV2010]. The idea is to interpret the original surface in Eq. as a function of the 3-D coordinates $X$, $Y$ and $Z(X, Y)$ $$\mathcal{S} \left( X( \mathbf{x}, z ), Y( \mathbf{x}, z ), Z(X( \mathbf{x}, z ), Y( \mathbf{x}, z )) \right) = \left[ \begin{array}{c} X( \mathbf{x}, z )\\ Y( \mathbf{x}, z )\\ Z(X( \mathbf{x}, z ), Y( \mathbf{x}, z )) \end{array} \right] := \left[ \begin{array}{c} \dfrac{z \, x}{\mathtt{f}} \vspace{1ex}\\ \dfrac{z \, y}{\mathtt{f}} \vspace{1ex}\\ - z \end{array} \right] \, .$$ Dropping the dependency of $X$, $Y$ and $Z(X,Y)$ on $\mathbf{x}$, $z$ and computing the partial derivatives with respect to $X$ and $Y$ via the chain rule $$\frac{\partial X}{\partial Y} = \frac{\partial X}{\partial x} \frac{\partial x}{\partial Y} \, , \qquad \qquad \qquad \frac{\partial Y}{\partial X} = \frac{\partial Y}{\partial y} \frac{\partial y}{\partial X}$$ then gives the tangent vectors to the surface $$\label{eq:s_normal} \mathcal{S}_{X}( \mathbf{x}, z ) = \left[ \begin{array}{c} 1\\[1mm] \dfrac{z_{x} y}{z+z_{x} \, x}\\[3mm] - \dfrac{z_{x} \, \mathtt{f}}{z+z_{x} \, x} \end{array} \right] \; , \;\;\;\;\; \mathcal{S}_{Y}( \mathbf{x}, z ) = \left[ \begin{array}{c} \dfrac{z_{y} \, x}{z+z_{y} \, y}\\[4mm] 1\\[1mm] - \dfrac{z_{y} \, \mathtt{f}}{z+z_{y} \, y} \end{array} \right] \; .$$ After some computations we finally obtain the corresponding normal direction $$\label{eq:normal:alt} \hat{\mathbf{n}}( \mathbf{x} ) = \mathcal{S}_{X}( \mathbf{x}, z ) \times \mathcal{S}_{Y}( \mathbf{x}, z ) = \dfrac{\mathtt{f}^2}{(z+z_{x} \, x)(z+z_{y} \, y)} \,\, \mathbf{n}( \mathbf{x} ) \, .$$ where $\mathbf{n}( \mathbf{x} )$ is the normal direction from Eq. As expected, both vectors only differ by scale, i.e. they have the same direction. Hence, the corresponding normalised vectors $\mathbf{n}/|\mathbf{n}|$ and $\hat{\mathbf{n}}/|\hat{\mathbf{n}}|$ are identical. While this alternative derivation was not used in our paper, it helps to clarify a common mistake in the literature that will be explained in the following. [**Remark.**]{} Please note that, unlike in the orthographic case, the cross derivatives ${\partial X}/{\partial Y}$ and ${\partial Y}/{\partial X}$ do not vanish for the perspective model. Hence, using the orthographic derivation of the normal direction from Horn and Brooks [@HB89] $$\label{eq:normalHB} \mathbf{n_\mathrm{ortho}}(\mathbf{x}) = \frac{\partial}{\partial X}\! \left[ \begin{array}{c} X \\ Y \\ Z \end{array} \right] \times \frac{\partial}{\partial Y}\! \left[ \begin{array}{c} X \\ Y \\ Z \end{array} \right] = \left[ \begin{array}{c} 1 \\ 0 \\ Z_{X} \end{array} \right] \times \left[ \begin{array}{c} 0 \\ 1 \\ Z_{Y} \end{array} \right] = \left[ \begin{array}{c} -Z_{X} \\ -Z_{Y} \\ 1 \end{array} \right]$$ with zero cross derivatives and simply replacing the remaining partial derivatives $Z_{X}$ and $Z_{Y}$ by the corresponding expressions from is [*not completely correct* ]{} for the perspective case. Such an approach has for instance been proposed in [@Wu_IJCV2010; @ZYT07]. It actually mixes the orthographic and the perspective model and thus typically gives worse results in the case of strong perspective distortions. 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--- abstract: 'Distributed intrustion detection systems detect attacks on computer systems by analyzing data aggregated from distributed sources. The distributed nature of the data sources allows patterns in the data to be seen that might not be detectable if each of the sources were examined individually. This paper describes the various approaches that have been developed to share and analyze data in such systems, and discusses some issues that must be addressed before fully decentralized distributed intrusion detection systems can be made viable.' author: - | Michael Treaster\ National Center for Supercomputing Applications (NCSA)\ University of Illinois\ Email: treaster@ncsa.uiuc.edu bibliography: - 'treaster-dids-corr.bib' title: A Survey of Distributed Intrusion Detection Approaches --- Introduction {#introduction .unnumbered} ============ Intrusion detection systems (IDS) have existed since the 1980’s, ever since the rise of the Internet made it possible to attack computer systems from a remote terminal. Although the first such systems operated independently on each machine on which they were installed, eventually the idea was proposed of aggregating IDS data from multiple machines in order to look for patterns across a network. This can improve the system’s ability to detect attacks that might otherwise be undetectable because each single host cannot does not have enough evidence to draw any conclusions. In general, distributed intrusion detection systems leverage some kind of single-node IDS software to monitor security events and collect data. Therefore, research typically focuses more on the sharing, aggregation, and processing of this data from a variety of nodes rather than on the exact nature of the monitoring itself. Existing approaches can be categorized along a variety of axes; here we examine data sharing, the nature of the data analysis, and security and trust features. Data Sharing {#data-sharing .unnumbered} ============ In a distributed IDS system, each agent shares its data with other agents in the system. However, there are a wide variety of sharing schemes that have been developed. These schemes can be viewed as a continuum, with centralized data reporting on one side and completely decentralized sharing on the other. The most extreme centralization is represented by systems in which a commercial vendor collects security information from a wide variety of customers, each running the vendor’s agent software [@deepsight; @dshield]. The vendor typically has multiple machines handling the data collection and analysis load that this widespread deployment incurs. When the vendor detects a possible Internet-scale attack, customers receive alerts and advice from the professional security experts who manage the system. This approach has two primary shortcomings. First, the central management and processing of data represents a single point of failure or vulnerability. Second, it results in a scalability bottleneck, and, due to the volume of incoming data, these systems often have slow response time to new threats. The most common distributed IDS approach is one in which all agents report data to a central server controlled at a domain or enterprise level [@polla98hummingbird; @snapp91dids; @chatzigniannakis04distributed; @huang99large; @jackson91expert]. This is fundamentally the same as in the previous centralization approach, but on a different scale, and this possesses most of the advantages and disadvantages of these larger-scale systems. These are usually oriented towards enterprise security, and are generally unsuitable for use among independent peers on the Internet due to the central control. To address the scalability problem of a centralized system, many techniques use a hierarchical structure [@balasubramaniyan98architecture; @stanifordchen96grids; @porras97emerald]. Data is passed up a hierarchy tree and is processed at each level to search for intrusions and to reduce the amount of information that must be passed higher up the tree. This helps address scalability and allows a system to be deployed across large enterprise-scale networks, but it limits the kinds of intrusions that can be detected at the highest levels. This also helps address the single point of failure problem, since if a higher node in the hierarchy fails the lower tiers can typically continue to function, albeit with reduced detection capabilities. Between the hierarchical approach and the fully distributed approach lie projects such as [@gopalakrishna01framework], which uses a hybrid hierarchical-distributed approach. Each agent publishes “interests” to the network, which are distributed through a hierarchical structure. Agents share data with other nodes who are interested, and all analysis occurs locally at the agent level. Instances of completely distributed solutions are much more rare and are much less well-developed. Gossiping, multicast, or subscription-based data sharing techniques have been proposed [@janakiraman03indra], but none of these have yet been implemented in a distributed IDS system. Other systems [@vlachos04security] ignore the topic entirely or pass it off to the underlying peer-to-peer substrate. Although these examples are still under development, they represent solutions that can be deployed on the Internet at large, independent of any central authority. Nature of Data Analysis {#nature-of-data-analysis .unnumbered} ======================= Although distributed IDS systems are usually independent of the techniques used to detect individual security events, the ways in which these security events are used can vary greatly. Since most systems work in heterogeneous environments, and since the security relationship between, say, a port scan and a buffer overflow attack may not be obvious, how does a system turn event detection into a response? Expert systems are a common approach [@snapp91dids; @jackson91expert], relying on rule sets to process and respond to events. These rules can attempt to define security policies, normal behavior, and/or anomalous behavior, and alerts or actions are generated based on how events match against the rules. [@snapp91dids] attempts to map actions back to a particular human user, such that events can be correlated with the intentions of an individual. Many systems [@barrus98distributed; @stanifordchen96grids] use a threshold scheme. Each security event increases a global alert level. The amount of the increase can be based on any number of factors, such as the particular event that was observed and its relation to other events in time or space. When the alert level exceeds a certain threshold, generic increased security measures are deployed, or an administrator is alerted. Long periods of time without security events can cause the alert level to decrease. Augmented goal trees can be used to model intrusion possibilities [@huang99large]. As more states of the goal tree are fulfilled (based on data from the distributed agents), the system is able to anticipate and counter future stages of the intrusion. An alternative graph-based approach in which connections between machines are logged and constructed into a graph of network activity has also been studied [@stanifordchen96grids]. These graphs are then analyzed by an expert system to detect possible intrusions. Security and Trust {#security-and-trust .unnumbered} ================== Security and trust are crucial aspects of any distributed system. However, in most proposed distributed IDS systems, however, these issues are given a much lower priority than other design considerations. They address the possibility of a rogue agent or a denial of service attack on the system only in passing. In all cases, a complete solution for trust and security is not provided, but sometimes a concrete solution to a limited aspect of the problem is presented. One issue is that of message authentication, allowing agents to ensure that messages come from who they claim to come from. Several systems [@janakiraman03indra; @vlachos04security] use signed messages, relying on a central certificate authority to generate the credentials. This authority does not necessarily participate in the rest of the distributed IDS system. Although this approach validates the source of a message and ensures that the contents have not been tampered with, it cannot protect a legitimate agent sending malicious data. Smaller-scale, centrally controlled systems such as [@polla98hummingbird] can rely upon a login mechanism, such as Kerberos. Agents only acknowledge logged-in systems, providing a measure of trust to the validated agents. This solution is only appropriate for systems with a central login authority, however. This solution, like the signed message approach, is unable to protect against a legitimate agent sending malicious data. The issue of trust can be left to individual agents in the system [@porras97emerald]. Each agent decides whether or not to trust higher level agents (“monitors”) in the system hierarchy. The agent then subscribes to exchange information from those monitors it chooses to trust. By aggregating and forwarding the data they receive from the lower level agents, the monitors are able to distribute data throughout the network. It is not clear how a monitor protects against subscription by rogue agents which then feed it misinformation. Denial of service attacks on agents can be detected using heartbeat signals [@barrus98distributed]. Each agent periodically sends a message to inform the rest of the system that it is functioning properly. If other agents do not receive the heartbeat message on schedule, a denial of service attack is suspected and treated as another security event on the network. Beyond these initial approaches to security and trust, there has been little work in this area with regard to distributed intrusion detection systems, especially in systems with a centralized control component. Most distributed IDS approaches ignore this topic entirely, but some list it among future work. Several projects [@janakiraman03indra; @vlachos04security] suggest the possibility of using a “web of trust” among peers, but this approach has not yet been explored. Future Directions {#future-directions .unnumbered} ================= We believe tolerance of misinformation is a key area in which to focus, due to the lack of attention that it has been given in previous work. In existing systems, a rogue agent might easily corrupt the network by spreading incorrect data. Systems must protect themselves against this type of attack. Centrally managed systems can rely on having complete control over every agent in the network to protect themselves. However, agents in a centrally managed system might be subverted, and fully decentralized systems cannot rely on this at all. One approach that has been suggested is to build a *web of trust* between agents in the network. As an agent reports information that is verified by others, the reputation of the agents is increased and it is trusted more in the future. However, the system must protect against an agent adopting malicious behavior after building up a high level of trust. This approach is closely related to several trust-oriented research endeavors [@abdulrahman97distributed; @chen03poblano; @sniffen00trust]. However, the details of a such a protocol have not yet been carefully specified.
--- abstract: 'We consider large random matrices $X$ with centered, independent entries which have comparable but not necessarily identical variances. Girko’s circular law asserts that the spectrum is supported in a disk and in case of identical variances, the limiting density is uniform. In this special case, the *local circular law* by Bourgade *et. al.* [@Bourgade2014; @BYY_circular2] shows that the empirical density converges even locally on scales slightly above the typical eigenvalue spacing. In the general case, the limiting density is typically inhomogeneous and it is obtained via solving a system of deterministic equations. Our main result is the local *inhomogeneous* circular law in the bulk spectrum on the optimal scale for a general variance profile of the entries of $X$.' bibliography: - 'literature.bib' title: '[Local inhomogeneous circular law]{}' --- *Keywords:* Circular law, local law, general variance profile\ **AMS Subject Classification:** 60B20, 15B52 Introduction ============ The density of eigenvalues of large random matrices typically converges to a deterministic limit as the dimension $n$ of the matrix tends to infinity. In the Hermitian case, the best known examples are the Wigner semicircle law for Wigner ensembles and the Marchenko-Pastur law for sample covariance matrices. In both cases the spectrum is real, and these laws state that the empirical eigenvalue distribution converges to an explicit density on the real line. The spectra of non-Hermitian random matrices concentrate on a domain of the complex plane. The most prominent case is the *circular law*, asserting that for an $n\times n$ matrix $X$ with independent, identically distributed entries, satisfying ${\mathbb{E}}x_{ij} =0$, ${\mathbb{E}}|x_{ij}|^2 =n^{-1}$, the empirical density converges to the uniform distribution on the unit disk $\{ z \, : \, |z| < 1\}\subset{\mathbb{C}}$. Despite the apparent similarity in the statements, it is considerably harder to analyze non-Hermitian random matrices than their Hermitian counterparts since eigenvalues of non-Hermitian matrices may respond very drastically to small perturbations. This instability is one reason why the universality of local eigenvalue statistics in the bulk spectrum, exactly on the scale of the eigenvalue spacing, is not yet established for $X$ with independent (even for i.i.d.) entries, while the corresponding statement for Hermitian Wigner matrices, known as the Wigner-Dyson-Mehta universality conjecture, has been proven recently, see [@ErdoesYau2012] for an overview. The circular law for i.i.d. entries has a long history, we refer to the extensive review [@bordenave2012]. The complex Gaussian case (Ginibre ensemble) has been settled in the sixties by Mehta using explicit computations. Girko in [@Girko1984] found a key formula to relate linear statistics of eigenvalues of $X$ to eigenvalues of the family of Hermitian matrices $(X-z)^*(X-z)$ where $z\in {\mathbb{C}}$ is a complex parameter. Technical difficulties still remained until Bai [@bai1997] presented a complete proof under two additional assumptions requiring higher moments and bounded density for the single entry distribution. After a series of further partial results [@goetze2010; @Pan2010; @tao2008] the circular law for i.i.d. entries under the optimal condition, assuming only the existence of the second moment, was established by Tao and Vu [@tao2010]. Another line of research focused on the local version of the circular law with constant variances, ${\mathbb{E}}|x_{ij}|^2=n^{-1}$, which asserts that the local density of eigenvalues is still uniform on scales $n^{-1/2+\epsilon}$, i.e., slightly above the typical spacing between neighboring eigenvalues. The optimal result was achieved in Bourgade, Yau and Yin [@Bourgade2014; @BYY_circular2] and Yin [@Y_circularlaw] both inside the unit disk (“bulk regime”) and at the edge $|z|=1$. If the first three moments match those of a standard complex Gaussian, then a similar result has also been obtained by Tao and Vu in [@tao2015]. In [@tao2015], this result was used to prove the universality of local eigenvalue statistics under the assumption that the first four moments match those of a complex Gaussian. While there is no proof of universality for general distributions without moment matching conditions yet, similarly to the development in the Hermitian case, the local law is expected to be one of the key ingredients of such a proof in the future. In this paper we study non-Hermitian matrices $X$ with a general matrix of variances $S=(s_{ij})$, i.e., we assume that $x_{ij}$ are centered, independent, but $s_{ij} {\mathrel{\vcenter{\baselineskip0.5ex \lineskiplimit0pt\hbox{\scriptsize.}\hbox{\scriptsize.}}}=}{\mathbb{E}}|x_{ij}|^2$ may depend non-trivially on the indices $i,j$. We show that the eigenvalue density is close to a deterministic density $\sigma$ on the smallest possible scale. As a direct application, our local law implies that the spectral radius $\rho(X)$ of $X$ is arbitrarily close to $\sqrt{\rho(S)}$, where $\rho(S)$ is the spectral radius of $S$. More precisely, we prove that for every ${\varepsilon}>0$ $$\sqrt{\rho(S)} - {\varepsilon}\leq \rho(X) \leq \sqrt{\rho(S)} +{\varepsilon}$$ with a very high probability as $n$ tends to infinity. The fact that the spectral radius of $X$ becomes essentially deterministic is the key mathematical mechanism behind the sharp “transition to chaos” in a commonly studied mean field model of dynamical neural networks [@ChaosInRandomNeuralNetworks]. This transition is described by the stability/instability of the system of ordinary differential equations $$\dot{q_i}(t) = q_i(t) -\lambda \sum_{j=1}^n x_{ij}q_j(t)$$ for $i=1, \ldots, n$ as $\lambda$ varies. Moreover, the number of unstable modes close to the critical value of the parameter $\lambda$ is determined by the behaviour of $\sigma$ at the spectral edge which we also analyze. Such systems have originally been studied under the assumption that the coefficients $x_{ij}$ are independent and identically distributed [@may1972will]. More recently, however, it was argued [@PhysRevLett.114.088101; @Aljadeff2015] that for more realistic applications in neuroscience one should allow $x_{ij}$ to have varying distributions with an arbitrary variance profile $S$. After Girko’s Hermitization, understanding the spectrum of $X$ reduces to analyzing the spectrum of the family of Hermitian matrices $$\label{Hdef} {\boldsymbol H}^z {\mathrel{\vcenter{\baselineskip0.5ex \lineskiplimit0pt\hbox{\scriptsize.}\hbox{\scriptsize.}}}=}\begin{pmatrix} 0 & X- z{\mathds{1}}\\ X^*-\bar z {\mathds{1}}& 0 \end{pmatrix}$$ of double dimension, where $z\in {\mathbb{C}}$. The Stieltjes transform of the spectral density of ${\boldsymbol H}^z$ at any spectral parameter $w$ in the upper half plane ${\mathbb H}{\mathrel{\vcenter{\baselineskip0.5ex \lineskiplimit0pt\hbox{\scriptsize.}\hbox{\scriptsize.}}}=}\{ w\in {\mathbb{C}}\; : \; \operatorname{Im}w >0\}$ is approximated via the solution of a system of $2n$ nonlinear equations, written concisely as $$\begin{split}\label{Seq} -\frac{1}{m_1}= &w + Sm_2 - \frac{|z|^2}{w+ S^t m_1}, \\ -\frac{1}{m_2}= & w + S^t m_1 - \frac{|z|^2}{w+ S m_2}, \end{split}$$ where $m_a= m_a^z(w) \in {\mathbb H}^n$, $a=1,2$ are $n$-vectors with each component in the upper half plane. The normalized trace of the resolvent, $\frac{1}{2n} \mbox{trace} ({\boldsymbol H}^z-w)^{-1}$, is approximately equal to $\frac{1}{n}\sum_j [m_1^{z}(w)]_j =\frac{1}{n}\sum_j [m_2^{z}(w)]_j $ in the $n\to\infty$ limit. The spectral density of ${\boldsymbol H}^z$ at any $E\in {\mathbb{R}}$ is then given by setting $w=E+{\mathrm{i}}\eta$ and taking the limit $\eta\to 0+$ for the imaginary part of these averages. In fact, for Girko’s formula it is sufficient to study the resolvent only along the positive imaginary axis $w\in {\mathrm{i}}{\mathbb{R}}_+$. Heuristically, the equations arise from second order perturbation theory and in physics they are commonly called *Dyson equations*. Their analogues for general Hermitian ensembles with independent or weakly dependent entries play an essential role in random matrix theory. They have been systematically studied by Girko, for example, equation in the current random matrix context appears as the *canonical equation of type* $K_{25}$ in Theorem 25.1 in [@girko2012theory]. In particular, under the condition that all $s_{ij}$ variances are comparable, i.e., $c/n \le s_{ij}\le C/n$ with some positive constants $c, C$, Girko identifies the limiting density. From his formulas it is clear that this density is rotationally symmetric. He also presents a proof for the weak convergence of the empirical eigenvalue distribution but the argument was considered incomplete. This deficiency can be resolved in a similar manner as for the circular law assuming a bounded density of the single entry distribution using the argument from Section 4.4 of [@bordenave2012]. In a recent preprint [@CookNonHermitianRM] Cook *et. al.* substantially relax the condition on the uniform bound $s_{ij}\ge c/n$ by replacing it with a concept of *robust irreducibility*. Moreover, relying on the bound by Cook [@CookSmallestSingularValue] on the smallest singular value of $X$, they also remove any condition on the regularity of the single entry distribution and prove weak convergence on the global scale. The matrix ${\boldsymbol H}^z$ may be viewed as the sum of a *Wigner-type matrix* [@Ajankirandommatrix] with centered, independent (up to Hermitian symmetry) entries and a deterministic matrix whose two off-diagonal blocks are $-z{\mathds{1}}$ and $-\bar z {\mathds{1}}$, respectively. Disregarding these $z$ terms for the moment, has the structure of the *Quadratic Vector Equations* that were extensively studied in [@AjankiQVE; @AjankiCPAM]. Including the $z$-terms, ${\boldsymbol H}^z$ at first sight seems to be a special case of the random matrix ensembles with nonzero expectations analyzed in [@AjankiCorrelated] and is the diagonal part of the corresponding *Matrix Dyson Equation (MDE)*. In [@AjankiCorrelated] an optimal local law was proved for such ensembles. However, the large zero blocks in the diagonal prevent us from applying these results to ${\boldsymbol H}^z$ or even to ${\boldsymbol H}^{z=0}$. In fact, the flatness condition ${{\boldsymbol{\mathrm{A}}}1}$ in [@AjankiCorrelated] (see later) or even its relaxed version ${{\boldsymbol{\mathrm{A}}}1'}$ in [@AjankiCorrelated] prohibit such large zero diagonal blocks. These conditions are essential for the proofs in [@AjankiCorrelated] since they ensure the stability of the corresponding Dyson equation against *any* small perturbation. In this case, there is only one potentially unstable direction, that is associated to a certain Perron-Frobenius eigenvector, and this direction is regularized by the positivity of the density of states at least in the bulk regime of the spectrum. If the flatness condition ${{\boldsymbol{\mathrm{A}}}1}$ is not satisfied, then the MDE can possess further unstable directions. In particular, in our setup, the MDE is not stable in the previously described strong sense; there is at least one additional unstable direction which cannot be regularized by the positivity of the density of states. Owing to the specific structure of ${\boldsymbol H}^z$, the *matrix* Dyson equation decouples and its diagonal parts satisfy a closed system of *vector* equations . Compared to the MDE, the reduced vector equations are rather cubic than quadratic in nature. For this reduced system, however, we can show that there is only one further unstable direction, at least when $S$ is entrywise bounded from below by some $c/n$. The system is not stable against an arbitrary perturbation, but for the perturbation arising in the random matrix problem we reveal a key cancellation in the leading contribution to the unstable direction. Armed with this new insight we will perform a detailed stability analysis of . This delicate stability analysis is the key ingredient for the proof of our main result, the optimal local law for $X$ with an optimal speed of convergence as $n\to\infty$. In this paper we consider the bulk regime, i.e., spectral parameter $z$ inside the disk with boundary $|z|^2=\rho(S)$, where $\rho(S)$ is the spectral radius of $S$. We defer the analysis of the edge of the spectrum of $X$ to later works. In the special case $z=0$, we thoroughly studied the system of equations even for the case when $S$ is a rectangular matrix in [@AltGram]; the main motivation was to prove the local law for random *Gram matrices*, i.e., matrices of the form $XX^*$. Note that in [@AltGram] we needed to tackle a much simpler quadratic system since taking $z=0$ in removes the most complicated nonlinearity. Finally, we list two related recent results. Local circular law on the optimal scale in the bulk has been proven in [@XYY_circularlaw] for ensembles of the form $TX$, where $T$ is a deterministic $N \times M$ matrix and $X$ is a random $M \times N$ matrix with independent, centered entries whose variances are constant and have vanishing third moments. The structure of the product matrix $TX$ is very different from our matrices that could be viewed as the Hadamard (entrywise) product of the matrix $(s_{ij}^{1/2})$ and a random matrix with identical variances. The approach of [@XYY_circularlaw] is also very different from ours: it relies on first assuming that $X$ is Gaussian and using its invariance to reduce the problem to the case when $T^*T$ is diagonal. Then the corresponding Dyson equations are much simpler, in fact they consist of only two scalar equations and they are characterized by a vector of parameters (of the singular values of $T$) instead of an entire matrix of parameters $S$. The vanishing third moment condition in [@XYY_circularlaw] is necessary to compare the general distribution with the Gaussian case via a moment matching argument. We also mention the recent proof of the local *single ring theorem* on optimal scale in the bulk [@ErdosSchnelli2016]. This concerns another prominent non-Hermitian random matrix ensemble that consists of matrices of the form $U\Sigma V$, where $U$, $V$ are two independent Haar distributed unitaries and $\Sigma$ is deterministic (may be assumed to be diagonal). The spectrum lies in a ring about the origin and the limiting density can be computed via free convolution [@guionnet2011single]. #### Acknowledgement We are grateful to David Renfrew for discussing some applications of our results with us and to Dominik Schröder for helping us visualizing our results. #### Notation For vectors $v, w \in {\mathbb{C}}^l$, we write their componentwise product as $vw=(v_i w_i)_{i=1}^l$. If $f\colon U \to {\mathbb{C}}$ is a function on $U \subset {\mathbb{C}}$, then we define $f(v)\in {\mathbb{C}}^l$ for $v\in U^l$ to be the vector with components $f(v)_i = f(v_i)$ for $i = 1, \ldots, l$. We will in particular apply this notation with $f(z) = 1/z$ for $z \in {\mathbb{C}}\setminus \{ 0\}$. We say that a vector $v \in {\mathbb{C}}^l$ is positive, $v >0$, if $v_i>0$ for all $i=1, \ldots, l$. Similarly, the notation $v \leq w$ means $v_i \leq w_i$ for all $i=1, \ldots, l$. For vectors $v, w \in {\mathbb{C}}^l$, we define ${\langle w \rangle} = l^{-1} \sum_{i=1}^l w_i$, ${\langle{v} \mspace{2mu}, {w}\rangle} = l^{-1} \sum_{i=1}^l \overline{v_i} w_i$, ${\lVert w \rVert}_2^2 = l^{-1} \sum_{i=1}^l {\lvert w_i \rvert}^2$ and ${\lVert w \rVert}_\infty = \max_{i=1, \ldots, l} {\lvert w_i \rvert}$, ${\lVert v \rVert}_1 {\mathrel{\vcenter{\baselineskip0.5ex \lineskiplimit0pt\hbox{\scriptsize.}\hbox{\scriptsize.}}}=}{\langle {\lvert v \rvert} \rangle}$. Note that ${\langle w \rangle} = {\langle{1} \mspace{2mu}, {w}\rangle}$, where we used the convention that $1$ also denotes the vector $(1,\ldots, 1) \in {\mathbb{C}}^l$. In general, we use the notation that if a scalar $\alpha$ appears in a vector-valued relation, then it denotes the constant vector $(\alpha, \ldots, \alpha)$. In most cases we will work in $n$ or $2n$ dimensional spaces. Vectors in ${\mathbb{C}}^{2n}$ will usually be denoted by boldface symbols like ${\boldsymbol v}$, ${\boldsymbol u}$ or ${\boldsymbol y}$. Correspondingly, capitalized boldface symbols denote matrices in ${\mathbb{C}}^{2n\times 2n}$, for example ${\boldsymbol R}$. We use the symbol ${\mathds{1}}$ for the identity matrix in ${\mathbb{C}}^{l\times l}$, where the dimension $l=n$ or $l=2n$ is understood from the context. For a matrix $A \in {\mathbb{C}}^{l \times l}$, we use the short notations ${\lVert A \rVert_{\infty}} {\mathrel{\vcenter{\baselineskip0.5ex \lineskiplimit0pt\hbox{\scriptsize.}\hbox{\scriptsize.}}}=}{\lVert A \rVert}_{\infty \to \infty}$ and ${\lVert A \rVert_{2}} {\mathrel{\vcenter{\baselineskip0.5ex \lineskiplimit0pt\hbox{\scriptsize.}\hbox{\scriptsize.}}}=}{\lVert A \rVert}_{2 \to 2}$ if the domain and the target are equipped with the same norm whereas we use ${\lVert A \rVert_{2\to\infty}}$ to denote the matrix norm of $A$ when it is understood as a map $({\mathbb{C}}^l, {\lVert \cdot \rVert}_2) \to ({\mathbb{C}}^l, {\lVert \cdot \rVert}_\infty)$. We define the normalized trace of an $l\times l$ matrix $B = (b_{ij})_{i,j=1}^l\in {\mathbb{C}}^{l\times l}$ as $$\label{eq:def_trace} \operatorname{Tr}B {\mathrel{\vcenter{\baselineskip0.5ex \lineskiplimit0pt\hbox{\scriptsize.}\hbox{\scriptsize.}}}=}\frac{1}{l} \sum_{j=1}^{l} b_{jj}.$$ For a vector $y \in {\mathbb{C}}^{l}$, we write $\operatorname{diag}y$ or $\operatorname{diag}(y)$ for the diagonal $l\times l$ matrix with $y$ on its diagonal, i.e., this matrix acts on any vector $x \in {\mathbb{C}}^l$ as $$\label{eq:def_diag} \operatorname{diag}(y)x = y x.$$ We write ${\mathrm{d}}^2z$ for indicating integration with respect to the Lebesgue measure on ${\mathbb{C}}$. For $a\in {\mathbb{C}}$ and ${\varepsilon}>0$, the open disk in the complex plane centered at $a$ with radius ${\varepsilon}$ is denoted by $D(a,{\varepsilon}) {\mathrel{\vcenter{\baselineskip0.5ex \lineskiplimit0pt\hbox{\scriptsize.}\hbox{\scriptsize.}}}=}\{ b \in {\mathbb{C}}\,\mid\, {\lvert a-b \rvert} < {\varepsilon}\}$. Furthermore, we denote the characteristic function of some event $A$ by ${\ensuremath{\chi}}(A)$, the positive real numbers by ${\mathbb{R}}_+ {\mathrel{\vcenter{\baselineskip0.5ex \lineskiplimit0pt\hbox{\scriptsize.}\hbox{\scriptsize.}}}=}(0,\infty)$ and the nonnegative real numbers by ${\ensuremath{{\mathbb{R}}^{+}_{0}}}{\mathrel{\vcenter{\baselineskip0.5ex \lineskiplimit0pt\hbox{\scriptsize.}\hbox{\scriptsize.}}}=}[0,\infty)$. Main results ============ Let $X$ be a random $n\times n$ matrix with centered entries, ${\mathbb{E}}x_{ij} = 0$, and $s_{ij} {\mathrel{\vcenter{\baselineskip0.5ex \lineskiplimit0pt\hbox{\scriptsize.}\hbox{\scriptsize.}}}=}{\mathbb{E}}{\lvert x_{ij} \rvert}^2$ the corresponding variances. We introduce its variance matrix $S{\mathrel{\vcenter{\baselineskip0.5ex \lineskiplimit0pt\hbox{\scriptsize.}\hbox{\scriptsize.}}}=}(s_{ij})_{i,j=1}^n$. **Assumptions**: (A) The variance matrix $S$ is *flat*, i.e., there are $0<s_*<s^*$ such that $$\label{eq:assumption_A} \frac{s_*}{n} \leq s_{ij} \leq \frac{s^*}{n}$$ for all $i,j=1, \ldots, n$. (B) All entries of $X$ have bounded moments in the sense that there are $\mu_m >0$ for $m \in {\mathbb{N}}$ such that $$\label{eq:assumption_B} {\mathbb{E}}{\lvert x_{ij} \rvert}^m \leq \mu_m n^{-m/2}$$ for all $i,j=1, \ldots, n$. (C) Each entry of $\sqrt{n}\; X$ has a density, i.e., there are probability densities $f_{ij} \colon {\mathbb{C}}\to [0,\infty)$ such that $${\mathbb{P}}\left( \sqrt n \; x_{ij} \in B \right) = \int_B f_{ij} (z) {\mathrm{d}}^2 z$$ for all $i,j=1,\ldots, n$ and $B \subset {\mathbb{C}}$ a Borel set. There are $\alpha,\beta>0$ such that $f_{ij} \in L^{1+\alpha}({\mathbb{C}})$ and $$\label{eq:bounded_density} {\lVert f_{ij} \rVert}_{1+\alpha} \leq n^\beta$$ for all $i,j=1,\ldots, n$. In the following, we will assume that $s_*$, $s^*$, $\alpha$, $\beta$ and the sequence $(\mu_m)_m$ are fixed constants which we will call *model parameters*. The constants in all our estimates will depend on the model parameters without further notice. The Assumption (C) is used in our proof solely for controlling the smallest singular value of $X-z{\mathds{1}}$ with very high probability uniformly for $z \in D(0,{\tau}^*)$ with some fixed ${\tau}^*>0$ in Proposition \[pro:least\_singular\_value\]. All our other results do not make use of Assumption (C). Provided a version of Proposition \[pro:least\_singular\_value\] that tracks the $z$-dependence can effectively be obtained without (C), our main result, the local inhomogeneous circular law in Theorem \[thm:local\_circular\_law\], will hold true solely assuming (A) and (B). For example a very high probability estimate uniform in $z$ in a statement similar to Corollary 1.22 of [@CookSmallestSingularValue] would be sufficient. The density of states of $X$ will be expressed in terms of $v_1^{\tau}$ and $v_2^{\tau}$ which are the positive solutions of the following two coupled vector equations \[eq:v\] $$\begin{aligned} \frac{1}{v^{\tau}_1} & = \eta + S v^{\tau}_2 + \frac{{\tau}}{\eta + S^tv^{\tau}_1}, \label{eq:v_1} \\ \frac{1}{v^{\tau}_2} & = \eta + S^t v^{\tau}_1 + \frac{{\tau}}{\eta + S v^{\tau}_2}. \label{eq:v_2}\end{aligned}$$ for all $\eta\in{\mathbb{R}}_+$ and ${\tau}\in {\ensuremath{{\mathbb{R}}^{+}_{0}}}$. Here, $v_1^{\tau}, v_2^{\tau}\in {\mathbb{R}}_+^n$ and recall that the algebraic operations are understood componentwise, e.g., $(1/v)_i = 1/v_i$ for the $i^{\text{th}}$ component of the vector $v$. The system is a special case of with $w = {\mathrm{i}}\eta$, ${\tau}= {\lvert z \rvert}^2$ and $v_a = {\mathrm{Im}\,}m_a$ for $a=1,2$. The existence and uniqueness of solutions to equations of the type are considered standard knowledge in the literature [@girko2012theory]. The equations can be viewed as a special case of the matrix Dyson equation for which existence and uniqueness was proven in [@Helton01012007]. We explain this connection in more detail in the appendix where we give the proof of Lemma \[lem:existence\_uniqueness\_vf\_equation\] for the convenience of the reader. \[lem:existence\_uniqueness\_vf\_equation\] For every ${\tau}\in {\ensuremath{{\mathbb{R}}^{+}_{0}}}$, there exist two uniquely determined functions $v_1^{\tau}\colon {\mathbb{R}}_+ \to {\mathbb{R}}_+^n$, $v_2^{\tau}\colon {\mathbb{R}}_+ \to {\mathbb{R}}_+^n$ which satisfy . We denote the spectral radius of $S$ by $\rho(S)$, i.e., $$\rho(S) {\mathrel{\vcenter{\baselineskip0.5ex \lineskiplimit0pt\hbox{\scriptsize.}\hbox{\scriptsize.}}}=}\max {\lvert \operatorname*{Spec}(S) \rvert}.$$ Now, we define the density of states of $X$ through the solution to . Let $v_1^{\tau}$ and $v_2^{\tau}$ be the unique positive solutions of . The *density of states* $\sigma\colon {\mathbb{C}}\to {\mathbb{R}}$ of $X$ is defined through $$\label{eq:def_sigma} \sigma(z) {\mathrel{\vcenter{\baselineskip0.5ex \lineskiplimit0pt\hbox{\scriptsize.}\hbox{\scriptsize.}}}=}- \frac{1}{2\pi}\int_0^\infty \Delta_z{\left\langle v_1^{\tau}(\eta)\middle|_{{\tau}={\lvert z \rvert}^2} \right\rangle} {\mathrm{d}}\eta$$ for $ {\lvert z \rvert}^2 <\rho(S)$ and $\sigma(z) {\mathrel{\vcenter{\baselineskip0.5ex \lineskiplimit0pt\hbox{\scriptsize.}\hbox{\scriptsize.}}}=}0$ for ${\lvert z \rvert}^2 \geq\rho(S)$. The right-hand side of is well-defined by part (i) of the following proposition. In the following proposition, we present some key properties of the density of states $\sigma$ of $X$. For an alternative representation of $\sigma$, see later. \[pro:properties\_sigma\] Let $v_1^{\tau}$ and $v_2^{\tau}$ be the unique positive solutions of . Then (i) The function ${\mathbb{R}}_+ \times {\mathbb{C}}\to {\mathbb{R}}_+^{2n}, (\eta, z) \mapsto \left(v_1^{\tau}(\eta),v_2^{{\tau}}(\eta)\right)|_{{\tau}={\lvert z \rvert}^2}$ is infinitely often differentiable and $\eta \mapsto \Delta_z{\left\langle v_1^{\tau}(\eta)\middle|_{{\tau}={\lvert z \rvert}^2} \right\rangle}$ is integrable on ${\mathbb{R}}_+$ for each $z \in D(0,\sqrt{\rho(S)})$. (ii) The function $\sigma$, defined in , is a rotationally symmetric probability density on ${\mathbb{C}}$. (iii) The restriction $\sigma|_{D(0, \sqrt{\rho(S)})}$ is infinitely often differentiable such that for every ${\varepsilon}>0$ each derivative is bounded uniformly in $n$ on $D(0,\sqrt{\rho(S)}-{\varepsilon})$. Moreover, there exist constants $c_1 > c_2 >0$, which depend only on $s_*$ and $s^*$, such that $$\label{eq:uniform_lower_bound_sigma} c_1 \geq \sigma(z) \geq c_2$$ for all $z \in D(0, \sqrt{\rho(S)})$. In particular, the support of $\sigma$ is the closed disk of radius $\sqrt{\rho(S)}$ around zero. The next theorem, the main result of the present article, states that the eigenvalue distribution of $X$, with a very high probability, can be approximated by $\sigma$ on the mesoscopic scales $n^{-a}$ for any $a \in (0,1/2)$. Note that $n^{-1/2}$ is the typical eigenvalue spacing so our result holds down to the optimal local scale. To study the local scale, we shift and rescale the test functions as follows. Let $f \in C_0^2({\mathbb{C}})$. For $z_0 \in {\mathbb{C}}$ and $a >0$, we define $$f_{z_0,a} \colon {\mathbb{C}}\to {\mathbb{C}}, \quad f_{z_0,a}( z) {\mathrel{\vcenter{\baselineskip0.5ex \lineskiplimit0pt\hbox{\scriptsize.}\hbox{\scriptsize.}}}=}n^{2a}f(n^a(z-z_0)).$$ We denote the eigenvalues of $X$ by ${\sigma}_1, \ldots, {\sigma}_n$. \[thm:local\_circular\_law\] Let $X$ be a random matrix which has independent centered entries and satisfies (A), (B) and (C). Furthermore, let $a \in (0,1/2)$, $\varphi>0$, ${\tau}_*>0$ and $\sigma$ defined as in . (i) (Bulk spectrum) For every ${\varepsilon}>0$, $D>0$, there is a positive constant $C_{{\varepsilon},D}$ such that $$\label{eq:Local_law_X} {\mathbb{P}}\left({\left\lvert \frac{1}{n} \sum_{i=1}^n f_{z_0,a}({\sigma}_i) - \int_{{\mathbb{C}}} f_{z_0,a}(z) \sigma(z) {\mathrm{d}}^2 z \right\rvert} \geq n^{-1+2a+{\varepsilon}} {\lVert \Delta f \rVert}_{L^1}\right) \leq \frac{C_{{\varepsilon}, D}}{n^D}$$ holds true for all $n \in {\mathbb{N}}$, for every $z_0 \in {\mathbb{C}}$ satisfying ${\lvert z_0 \rvert}^2 \leq \rho(S)- {\tau}_*$ and for every $f \in C_0^2({\mathbb{C}})$ satisfying $\operatorname{supp}f \subset D(0,\varphi)$. The point $z_0$ and the function $f$ may depend on $n$. (ii) (Away from the spectrum) For every $D>0$, there exists a positive constant $C_D$ such that $$\label{eq:Local_law_X_outside} {\mathbb{P}}\left(\exists i \in\{1, \ldots, n\} \;\middle|\; {\lvert {\sigma}_i \rvert}^2\geq \rho(S) +{\tau}_*\right) \leq \frac{C_D}{n^D}$$ holds true for all $n \in {\mathbb{N}}$. In addition to the model parameters, the constant $C_{{\varepsilon},D}$ in depends only on $a$, $\varphi$ and ${\tau}_*$ (apart from ${\varepsilon}$ and $D$) and the constant $C_D$ in only on ${\tau}_*$ (apart from $D$). The key technical input for the proof of Theorem \[thm:local\_circular\_law\] is the local law for ${\boldsymbol H}^z$ (see Theorem \[thm:local\_law\_H\_z\]). We now state a simple corollary of the local law for ${\boldsymbol H}^z$ on the complete delocalization of the bulk eigenvectors of $X$. \[coro:eigenvector\_delocalization\] Let ${\tau}_*>0$. For all ${\varepsilon}>0$ and $D>0$, there is a positive constant $C_{{\varepsilon},D}$ such that $$\label{eq:eigenvector_delocalization} {\mathbb{P}}\left( {\lVert y \rVert_{\infty}} \geq n^{-1/2 + {\varepsilon}} \right) \leq \frac{C_{{\varepsilon}, D}}{n^D}$$ holds true for all $n \in {\mathbb{N}}$ and for all eigenvectors $y \in {\mathbb{C}}^{n}$ of $X$, normalized as $ \sum_{i=1}^n {\lvert y_i \rvert}^2 =1$, corresponding to an eigenvalue ${\sigma}\in \operatorname*{Spec}X$ with ${\lvert {\sigma}\rvert}^2 \leq \rho(S) - {\tau}_*$. The constant $C_{{\varepsilon},D}$ in depends only on ${\tau}_*$ and the model parameters (in addition to ${\varepsilon}$ and $D$). The proof of Corollary \[coro:eigenvector\_delocalization\] will be given after the statement of Theorem \[thm:local\_law\_H\_z\]. We remark that eigenvector delocalization for random matrices with independent entries was first proved by Rudelson and Vershynin in [@rudelson2015]. [.5]{} [.5]{} Short outline of the proof {#subsec:Idea_Proof_Circular_Law} -------------------------- We start with the Hermitization trick due to Girko which expresses $\sum_{i=1}^n f_{z_0,a}({\sigma}_i)$ in terms of an integral of the log-determinant of $X-z{\mathds{1}}$ for any $z \in{\mathbb{C}}$. Furthermore, the log-determinant of $X-z{\mathds{1}}$ can be rewritten as the log-determinant of a Hermitian matrix ${\boldsymbol H}^z$. Using the log-transform of the empirical spectral measure of $X$, we obtain $$\label{eq:avg_f_lambda_i_as_log_det} \frac{1}{n} \sum_{i=1}^n f_{z_0,a}({\sigma}_i) = \frac{1}{2\pi n}\int_{{\mathbb{C}}} \Delta f_{z_0,a}(z) \log {\lvert \det (X-z{\mathds{1}}) \rvert} {\mathrm{d}}^2 z.$$ To express the log-determinant of $X-z{\mathds{1}}$ in terms of a Hermitian matrix, we introduce the $2n\times 2n$ matrix $$\label{eq:def_H_z} {\boldsymbol H}^z {\mathrel{\vcenter{\baselineskip0.5ex \lineskiplimit0pt\hbox{\scriptsize.}\hbox{\scriptsize.}}}=}\begin{pmatrix} 0 & X- z{\mathds{1}}\\ X^*-\bar z {\mathds{1}}& 0 \end{pmatrix}$$ for all $z \in {\mathbb{C}}$. Note that the eigenvalues of ${\boldsymbol H}^z$ come in opposite pairs and we denote them by ${\lambda}_{2n} \leq \ldots \leq {\lambda}_{n+1} \leq 0 \leq {\lambda}_{n} \leq \ldots \leq {\lambda}_{1}$ with ${\lambda}_i = - {\lambda}_{2n+1-i}$ for $i =1, \ldots, 2n$. We remark that the moduli of these real numbers are the singular values of $X-z{\mathds{1}}$. The Stieltjes transform of its empirical spectral measure is denoted by $m^z$, i.e., $$\label{eq:def_Stieltjes_transform} m^z(w) = \frac{1}{2n} \sum_{i=1}^{2n} \frac{1}{{\lambda}_i(z)-w}$$ for $w \in {\mathbb{C}}$ satisfying ${\mathrm{Im}\,}w>0$. It will turn out that on the imaginary axis ${\mathrm{Im}\,}m^z({\mathrm{i}}\eta)$ is very well approximated by ${\langle v_1^{\tau}(\eta) \rangle} = {\langle v_2^{\tau}(\eta) \rangle}$, where ${\tau}={\lvert z \rvert}^2$ and $(v_1^{\tau},v_2^{\tau})$ is the solution of . This fact is commonly called a *local law* for ${\boldsymbol H}^z$. With this notation, we have the following relation between the determinant of $X-z{\mathds{1}}$ and the determinant of ${\boldsymbol H}^z$ $$\label{eq:log_det_X_H_z} \log {\lvert \det(X-z{\mathds{1}}) \rvert} = \frac{1}{2} \log {\lvert \det {\boldsymbol H}^z \rvert}.$$ We write the log-determinant in terms of the Stieltjes transform (this formula was used by Tao and Vu [@tao2015] in a similar context) $$\label{eq:log_det_Stieltjes} \log{\lvert \det {\boldsymbol H}^z \rvert} = \log{\lvert \det({\boldsymbol H}^z - {\mathrm{i}}T{\mathds{1}}) \rvert} - 2 n \int_0^T {\mathrm{Im}\,}m^z ({\mathrm{i}}\eta){\mathrm{d}}\eta,$$ for any $T>0$. Combining , , and as well as substracting $1/(1+\eta)$ freely and using integration by parts, we obtain $$\begin{aligned} \frac{1}{n} \sum_{i=1}^n f_{z_0,a}({\sigma}_i) - \int_{\mathbb{C}}f_{z_0,a}(z) \sigma(z) {\mathrm{d}}^2 z = & \frac{1}{4\pi n} \int_{{\mathbb{C}}} \Delta f_{z_0,a}(z) \log {\lvert \det({\boldsymbol H}^z-{\mathrm{i}}T{\mathds{1}}) \rvert} {\mathrm{d}}^2 z \nonumber\\ & - \frac{1}{2\pi}\int_{{\mathbb{C}}} \Delta f_{z_0,a} (z) \int_0^T \Big[ {\mathrm{Im}\,}m^z({\mathrm{i}}\eta) - {\left\langle v_1^{\tau}(\eta)\middle|_{{\tau}={\lvert z \rvert}^2} \right\rangle}\Big]\,{\mathrm{d}}\eta\,{\mathrm{d}}^2 z \nonumber\\ & + \frac{1}{2\pi} \int_{\mathbb{C}}\Delta f_{z_0,a}(z) \int_T^\infty \left({\left\langle v_1^{\tau}(\eta)\middle|_{{\tau}={\lvert z \rvert}^2} \right\rangle} -\frac{1}{\eta+1}\right)\,{\mathrm{d}}\eta \,{\mathrm{d}}^2 z. \label{eq:master_formula}\end{aligned}$$ The task is then to prove that each of the terms on the right-hand side of is dominated by $n^{-1+2a}{\lVert \Delta f \rVert}_1$ with very high probability. The parameter $T$ will be chosen to be a large power of $n$, so that the first and the third term will easily satisfy this bound. Estimating the second term on the right-hand side of is much more involved and we focus only on this term in this outline. We split its ${\mathrm{d}}\eta$ - integral into two parts. For $\eta \leq n^{-1+{\varepsilon}}$, ${\varepsilon}\in (0,1/2)$, the integral is controlled by an estimate on the smallest singular value of $X-z{\mathds{1}}$. This is the only step in our proof which uses assumption (C), i.e., that the entries of $X$ have bounded densities . For $\eta\geq n^{-1+{\varepsilon}}$, we use a local law for ${\boldsymbol H}^z$, i.e., an optimal pointwise estimate (up to negligible $n^{\varepsilon}$-factors) on $$\label{eq:local_law_heuristics} {\mathrm{Im}\,}m^z({\mathrm{i}}\eta) - {\left\langle v_1^{\tau}(\eta)\middle|_{{\tau}={\lvert z \rvert}^2} \right\rangle},$$ uniformly in $\eta$ and $z$ (see Theorem \[thm:local\_law\_H\_z\] for the precise formulation). Note that a local law for ${\boldsymbol H}^z$ is needed only at spectral parameters on the imaginary axis. This will simplify the proof of the local law we need in this paper. The proof of the local law is based on a stability estimate of . To write these equations in a more concise form, we introduce the $2n \times 2n$ matrices $$\label{eq:def_Sf} {\boldsymbol S}_o = \begin{pmatrix} 0 & S \\ S^t & 0 \end{pmatrix}, \quad {\boldsymbol S}_d = \begin{pmatrix} S^t & 0 \\ 0 & S \end{pmatrix}.$$ With this notation the system of equations can be written as $$\label{eq:iv_combined} {\mathrm{i}}{\boldsymbol v}+ \left( {\mathrm{i}}\eta + {\boldsymbol S}_o{\mathrm{i}}{\boldsymbol v}- \frac{{\tau}}{{\mathrm{i}}\eta + {\boldsymbol S}_d {\mathrm{i}}{\boldsymbol v}}\right)^{-1} = 0,$$ where we introduced ${\boldsymbol v}{\mathrel{\vcenter{\baselineskip0.5ex \lineskiplimit0pt\hbox{\scriptsize.}\hbox{\scriptsize.}}}=}(v_1, v_2) \in {\mathbb{R}}^{2n}$. Let ${\boldsymbol G}^z(\eta) {\mathrel{\vcenter{\baselineskip0.5ex \lineskiplimit0pt\hbox{\scriptsize.}\hbox{\scriptsize.}}}=}({\boldsymbol H}^z-{\mathrm{i}}\eta{\mathds{1}})^{-1}$, $\eta >0$, be the resolvent of ${\boldsymbol H}^z$ at spectral parameter ${\mathrm{i}}\eta$. We will prove that its diagonal ${\boldsymbol g}(\eta) = ({\langle{\boldsymbol{e}_i} \mspace{2mu}, {{\boldsymbol G}^z(\eta)\boldsymbol{e}_i}\rangle})_{i=1}^{2n}$, where $\boldsymbol{e}_i$ denotes the $i^{\text{th}}$ standard basis vector in ${\mathbb{C}}^{2n}$, satisfies a perturbed version of , $$\label{eq:combined_v_perturbed} {\boldsymbol g}+ \left( {\mathrm{i}}\eta + {\boldsymbol S}_o{\boldsymbol g}- \frac{{\tau}}{{\mathrm{i}}\eta +{\boldsymbol S}_d {\boldsymbol g}} \right)^{-1} = {\boldsymbol d},$$ with ${\tau}= {\lvert z \rvert}^2$ and a small random error term ${\boldsymbol d}$. As $m^z({\mathrm{i}}\eta) = {\langle {\boldsymbol g}(\eta) \rangle}$ (cf. ) obtaining a local law, i.e., an optimal pointwise estimate on , reduces to a stability problem for the *Dyson equation* . Computing the difference of and , we obtain $$\label{eq:first_diff_eq} {\boldsymbol L}\left( {\boldsymbol g}-{\mathrm{i}}{\boldsymbol v}\right) = {\ensuremath{\boldsymbol r}}$$ for some error vector ${\ensuremath{\boldsymbol r}}=O(\|{\boldsymbol d}\|)$ (for the precise definition we refer to below) and with the matrix ${\boldsymbol L}$ defined through its action on ${\boldsymbol y}\in {\mathbb{C}}^{2n}$ via $${\boldsymbol L}{\boldsymbol y}{\mathrel{\vcenter{\baselineskip0.5ex \lineskiplimit0pt\hbox{\scriptsize.}\hbox{\scriptsize.}}}=}{\boldsymbol y}+ {\boldsymbol v}^2 ({\boldsymbol S}_o {\boldsymbol y}) - {\tau}\frac{{\boldsymbol v}^2}{(\eta + {\boldsymbol S}_d{\boldsymbol v})^2} ({\boldsymbol S}_d {\boldsymbol y}). \label{eq:def_Df}$$ Therefore, a bound on ${\boldsymbol g}-{\mathrm{i}}{\boldsymbol v}$ uniformly for $\eta \ge n^{-1+{\varepsilon}}$ requires a uniform bound on the inverse of ${\boldsymbol L}$ down to this local spectral scale. In fact, the mere invertibility of ${\boldsymbol L}$ even for $\eta$ bounded away from zero is a nontrivial fact that is not easily seen from . In Section \[sec:self\_consistent\_equation\] we will factorize ${\boldsymbol L}$ into the form $${\boldsymbol L}= {\boldsymbol V}^{-1}({\mathds{1}}- {\boldsymbol T}{\boldsymbol F}){\boldsymbol V}$$ for some invertible matrix ${\boldsymbol V}$ and self-adjoint matrices ${\boldsymbol T}$ and ${\boldsymbol F}$ with the properties ${\lVert {\boldsymbol T}\rVert_{2}}=1$ and ${\lVert {\boldsymbol F}\rVert_{2}} \le 1-c{\mspace{2 mu}}\eta$ for some $c>0$. In particular, this representation shows the a priori bound ${\lVert {\boldsymbol L}^{-1} \rVert_{2}}\le C\eta^{-1}$ for some $C>0$. The blow-up in the norm of ${\boldsymbol L}^{-1}$ is potentially caused by the two extremal eigendirections ${\boldsymbol f_+}$ and ${\boldsymbol f_-}$ of ${\boldsymbol F}$, which satisfy $${\boldsymbol F}\boldsymbol{f}_{\pm} \,=\, \pm {\lVert {\boldsymbol F}\rVert_{2}} \boldsymbol{f}_\pm\,.$$ However, it turns out that the positivity of the solutions $v_1$, $v_2$ of implies that ${\lVert {\boldsymbol T}{\boldsymbol f_+}\rVert_{2}}$ is strictly smaller than $1$, so that ${\lVert ({\mathds{1}}-{\boldsymbol T}{\boldsymbol F}){\boldsymbol f_+}\rVert_{2}} \geq c {\lVert {\boldsymbol f_+}\rVert_{2}}$ for some constant $c>0$. In this sense the solution of the Dyson equation regularizes the potentially unstable direction ${\boldsymbol f_+}$. In contrast, the other instability caused by ${\boldsymbol f_-}$ persists since we will find that $({\mathds{1}}-{\boldsymbol T}{\boldsymbol F}){\boldsymbol f_-}= O(\eta)$. This problem can only be resolved by exploiting an extra cancellation that originates from the special structure of the random matrix ${\boldsymbol H}^z$. The leading contribution of the random error ${\ensuremath{\boldsymbol r}}=O({\lVert {\boldsymbol d}\rVert})$ from pointing in the unstable direction happens to vanish with a remaining subleading term of order $\eta {\lVert {\boldsymbol d}\rVert}$. The extra $\eta$-factor cancels the $\eta^{-1}$-divergence of ${\lVert {\boldsymbol L}^{-1} \rVert_{2}}$ and allows us to invert the stability operator ${\boldsymbol L}$ in . From this analysis, we conclude ${\lVert {\boldsymbol g}-{\mathrm{i}}{\boldsymbol v}\rVert} \leq C {\lVert {\boldsymbol d}\rVert}$. This result allows us to follow the general arguments developed in [@AjankiCorrelated] for verifying the optimal local law for ${\boldsymbol H}^z$. These steps are presented only briefly in Section \[sec:local\_law\]. Dyson equation for the inhomogeneous circular law {#sec:self_consistent_equation} ================================================= As explained in Section \[subsec:Idea\_Proof\_Circular\_Law\] a main ingredient in the proof of Theorem \[thm:local\_circular\_law\] is the local law for the self-adjoint random matrix ${\boldsymbol H}^z$ with non-centered independent entries above the diagonal. In [@AjankiCorrelated] such a local law was proven for a large class of self-adjoint random matrices with non-centered entries and general short range correlations. For any fixed $z \in {\mathbb{C}}$, the matrix ${\boldsymbol H}^z$ satisfies the assumptions made for the class of random matrices covered in [@AjankiCorrelated] with one crucial exception: ${\boldsymbol H}^z$ is not *flat* (cf. (2.28) in [@AjankiCorrelated]), i.e., for any constant $c>0$, the inequality [ $$\label{MDE:flatness} \begin{split} \frac{1}{n}{\mathbb{E}}{\mspace{2 mu}}{\lvert {\langle{{\boldsymbol a}} \mspace{2mu}, {({\boldsymbol H}-{\mathbb{E}}{\mspace{2 mu}}{\boldsymbol H}){\boldsymbol b}}\rangle} \rvert}^2\,\ge\, c{\mspace{2 mu}}{\lVert {\boldsymbol a} \rVert}_2^2{\lVert {\boldsymbol b} \rVert}_2^2, \end{split}$$ ]{} is not satisfied for ${\boldsymbol H}={\boldsymbol H}^z$ and vectors ${\boldsymbol a},{\boldsymbol b}$ that both have support either in $\{1,\dots,n\}$ or $\{n+1,\dots,2n\}$. Nevertheless we will show that the conclusion from Theorem 2.9 of [@AjankiCorrelated] remains true for spectral parameters ${\mathrm{i}}\eta$ on the imaginary axis, namely that the resolvent ${\boldsymbol G}^z(\eta){\mathrel{\vcenter{\baselineskip0.5ex \lineskiplimit0pt\hbox{\scriptsize.}\hbox{\scriptsize.}}}=}({\boldsymbol H}^z -{\mathrm{i}}\eta{\mathds{1}})^{-1}$ approaches the solution ${\boldsymbol M}^z(\eta)$ of the *Matrix Dyson Equation (MDE)* [ $$\label{MDE on imaginary axis} \begin{split} -{\boldsymbol M}^z(\eta)^{-1}\,=\, {\mathrm{i}}{\mspace{1 mu}}\eta{\mspace{2 mu}}{\mathds{1}}- {\boldsymbol A}^z + {\mathcal}{S}[{\boldsymbol M}^z(\eta)]\,, \qquad \eta > 0\,, \end{split}$$ ]{} as $n\to \infty$. In fact, the solution of is unique under the constraint that the imaginary part ${\mathrm{Im}\,}{\boldsymbol M}{\mathrel{\vcenter{\baselineskip0.5ex \lineskiplimit0pt\hbox{\scriptsize.}\hbox{\scriptsize.}}}=}({\boldsymbol M}- {\boldsymbol M}^*)/(2{\mathrm{i}})$ is positive definite [@Helton01012007]. The data ${\boldsymbol{{A}}}^z \in {\mathbb{C}}^{2n \times 2n}$ and ${\mathcal}{S}\colon{\mathbb{C}}^{2n \times 2n} \to {\mathbb{C}}^{2n \times 2n}$ determining are given in terms of the first and second moments of the entries of ${\boldsymbol H}^z$, [ $$\label{MDE:Data} \begin{split} {\boldsymbol A}^z\,{\mathrel{\vcenter{\baselineskip0.5ex \lineskiplimit0pt\hbox{\scriptsize.}\hbox{\scriptsize.}}}=}\,{\mathbb{E}}{\mspace{2 mu}}{\boldsymbol H}^z\, =\, \begin{pmatrix} 0 & -z \\ -\overline{z} & 0 \end{pmatrix} \,,\qquad {\mathcal}{S}[{\boldsymbol W}]\,{\mathrel{\vcenter{\baselineskip0.5ex \lineskiplimit0pt\hbox{\scriptsize.}\hbox{\scriptsize.}}}=}\, \begin{pmatrix} \operatorname{diag}(S w_2 )&0 \\ 0 & \operatorname{diag}(S^t w_1) \end{pmatrix}\,, \end{split}$$ ]{} for an arbitrary $2n \times 2n$ matrix $$\label{eq:convetion_Wf} {\boldsymbol W}\,=\, (w_{i j})_{i,j=1}^{2n}\,=\, \begin{pmatrix}W_{11}& W_{12} \\ W_{21} & W_{22} \end{pmatrix}\,,\qquad w_1 \,{\mathrel{\vcenter{\baselineskip0.5ex \lineskiplimit0pt\hbox{\scriptsize.}\hbox{\scriptsize.}}}=}\, (w_{ii})_{i=1}^n\,,\qquad w_2 \,{\mathrel{\vcenter{\baselineskip0.5ex \lineskiplimit0pt\hbox{\scriptsize.}\hbox{\scriptsize.}}}=}\, (w_{ii})_{i=n+1}^{2n}\,.$$ In the following, we will not keep the $z$-dependence in our notation and just write ${\boldsymbol M}$, ${\boldsymbol A}$ and ${\boldsymbol G}$ instead of ${\boldsymbol M}^z$, ${\boldsymbol A}^z$ and ${\boldsymbol G}^z$. A simple calculation (cf. the proof of Lemma \[lem:existence\_uniqueness\_vf\_equation\] in the appendix) shows that ${\boldsymbol M}\colon{\mathbb{R}}_+\to {\mathbb{C}}^{2n \times 2n}$ is given by $$\label{eq:def_Mf} {\boldsymbol M}^z( \eta) {\mathrel{\vcenter{\baselineskip0.5ex \lineskiplimit0pt\hbox{\scriptsize.}\hbox{\scriptsize.}}}=}\begin{pmatrix} {\mathrm{i}}\operatorname{diag}\left( v_1^{\tau}(\eta)\right) & - z \operatorname{diag}\left(u^{{\tau}}(\eta)\right) \\ - \bar z \operatorname{diag}\left(u^{\tau}(\eta)\right) & {\mathrm{i}}\operatorname{diag}\left(v_2^{\tau}(\eta)\right) \end{pmatrix},$$ where $z \in {\mathbb{C}}$, ${\tau}={\lvert z \rvert}^2$, $(v_1^{\tau},v_2^{\tau})$ is the solution of and $u^{\tau}{\mathrel{\vcenter{\baselineskip0.5ex \lineskiplimit0pt\hbox{\scriptsize.}\hbox{\scriptsize.}}}=}v_1^{\tau}/(\eta + S^tv_1^{\tau})$. In this section we will therefore analyze the solution and the stability of . Analysis of the Dyson equation ------------------------------- Combining the equations in , recalling ${\boldsymbol v}= (v_1, v_2)$ and the definitions of ${\boldsymbol S}_o$ and ${\boldsymbol S}_d$ in , we obtain $$\label{eq:v_combined} \frac{1}{{\boldsymbol v}} = \eta + {\boldsymbol S}_o {\boldsymbol v}+ \frac{{\tau}}{\eta + {\boldsymbol S}_d {\boldsymbol v}}$$ for $\eta >0$ and ${\tau}\in {\ensuremath{{\mathbb{R}}^{+}_{0}}}$, where ${\boldsymbol v}\colon {\mathbb{R}}_+ \to {\mathbb{R}}_+^{2n}$. This equation is equivalent to . The ${\tau}$-dependence of ${\boldsymbol v}$, $v_1$ and $v_2$ will mostly be suppressed but sometimes we view ${\boldsymbol v}={\boldsymbol v}^{\tau}(\eta)$ as a function of both parameters. The equation has an obvious scaling invariance when $S$ is rescaled to $\lambda S$ for $\lambda>0$. If ${\boldsymbol v}^{\tau}(\eta)$ is the positive solution of , then ${\boldsymbol v}_\lambda^{\tau}(\eta) {\mathrel{\vcenter{\baselineskip0.5ex \lineskiplimit0pt\hbox{\scriptsize.}\hbox{\scriptsize.}}}=}\lambda^{-1/2} {\boldsymbol v}^{{\tau}\lambda^{-1}}(\eta \lambda^{-1/2})$ is the positive solution of $$\label{eq:v_rescaled} \frac{1}{{\boldsymbol v}_\lambda} = \eta + \lambda {\boldsymbol S}_o {\boldsymbol v}_\lambda + \frac{{\tau}}{\eta + \lambda {\boldsymbol S}_d {\boldsymbol v}_\lambda}.$$ Therefore, without loss of generality, we may assume that the spectral radius of $S$ is one, $$\rho(S) = 1,$$ in the remainder of the paper. The following proposition, the first main result of this section, collects some basic estimates on the solution ${\boldsymbol v}$ of . For the whole section, we fix ${\tau}_*>0$ and ${\tau}^*>{\tau}_*+1$ and except for Proposition \[pro:estimates\_v\_small\_z\], we exclude the small interval $[1-{\tau}_*,1+{\tau}_*]$ from our analysis of ${\boldsymbol v}^{\tau}$. Because of the definition of $\sigma$ in – recall ${\tau}= {\lvert z \rvert}^2$ in the definition – we will talk about inside and outside regimes for ${\tau}\in [0,1-{\tau}_*]$ and ${\tau}\in [1+{\tau}_*, {\tau}^*]$, respectively. Recalling $s_*$ and $s^*$ from we make the following convention in order to suppress irrelevant constants from the notation. For nonnegative scalars or vectors $f$ and $g$, we will use the notation $f \lesssim g$ if there is a constant $c>0$, depending only on ${\tau}_*$, ${\tau}^*$, $s_*$ and $s^*$ such that $f \leq cg$ and $f \sim g$ if $f \lesssim g$ and $f \gtrsim g$ both hold true. If $f,g$ and $h$ are scalars or vectors and $h \geq 0$ such that ${\lvert f-g \rvert}\lesssim h$, then we write $f = g + O(h)$. Moreover, we define $${\ensuremath{\mathcal P}}{\mathrel{\vcenter{\baselineskip0.5ex \lineskiplimit0pt\hbox{\scriptsize.}\hbox{\scriptsize.}}}=}\{ {\tau}_*, {\tau}^*, s_*, s^* \}$$ because many constants in the following will depend only on ${\ensuremath{\mathcal P}}$. \[pro:estimates\_v\_small\_z\] The solution ${\boldsymbol v}^{\tau}$ of satisfies $$\label{eq:avg_v1_equal_avg_v2} {\langle v_1^{\tau}(\eta) \rangle} = {\langle v_2^{\tau}(\eta) \rangle}.$$ for all $\eta>0$ and ${\tau}\in {\ensuremath{{\mathbb{R}}^{+}_{0}}}$ as well as the following estimates: (i) (Large $\eta$) Uniformly for $\eta \geq 1$ and ${\tau}\in [0,{\tau}^*]$, we have $$\label{eq:bound_vf_large_eta} {\boldsymbol v}^{\tau}(\eta)\sim \eta^{-1}.$$ (ii) (Inside regime) Uniformly for $\eta\leq 1$ and ${\tau}\in [0,1]$, we have $$\label{eq:bound_vf_small_eta} {\boldsymbol v}^{\tau}(\eta) \sim \eta^{1/3} + (1-{\tau})^{1/2}.$$ (iii) (Outside regime) Uniformly for $\eta\leq 1$ and ${\tau}\in [1,{\tau}^*]$, we have $$\label{eq:bound_vf_small_eta_z_bigger_rho_S} {\boldsymbol v}^{\tau}(\eta) \sim \frac{\eta}{{\tau}-1 + \eta^{2/3} }.$$ We start with proving . By multiplying by $(\eta + S^t v_1)$ and by $(\eta + Sv_2)$ and realizing that both right-hand sides agree, we obtain $$\label{eq:step_to_def_u} \frac{v_1}{\eta + S^tv_1} = \frac{v_2}{\eta + S v_2}.$$ From , we also get $$0 = \eta(v_1 - v_2) + v_1 S v_2 - v_2 S^t v_1.$$ We take the average on both sides, use ${\langle v_1 Sv_2 \rangle} = {\langle{v_1} \mspace{2mu}, {Sv_2}\rangle} = {\langle v_2S^t v_1 \rangle}$ and divide by $\eta >0$ to infer . From , we immediately deduce the following auxiliary bounds $$\label{eq:Sv_sim_avg_v} {\langle v_1 \rangle} \lesssim S^t v_1 \lesssim {\langle v_1 \rangle}, \quad {\langle v_{ 2 } \rangle} \lesssim Sv_2 \lesssim {\langle v_{ 2 } \rangle}.$$ We start with establishing ${\boldsymbol v}\sim {\langle {\boldsymbol v}\rangle}$. Since the entries of $S$ are strictly positive and $\rho(S)=1$ there is a unique vector $p \in {\mathbb{R}}_+^n$ which has strictly positive entries such that $$\label{eq:eigenvector_S_properties} Sp=p, \quad {\langle p \rangle}=1, \quad p \sim 1$$ by the Perron-Frobenius Theorem and . We multiply by $v_1$ as well as $\eta + S^tv_1$ and obtain $\eta+ S^tv_1= v_1 (\eta+ Sv_2)(\eta+S^tv_1) + {\tau}v_1$. Taking the scalar product with $p$ and using ${\langle p \rangle} =1$ and $\rho(S)=1$ yield $$\label{eq:vf_sim_avg_aux1} \eta + {\langle pv_1 \rangle} = {\left\langle pv_1(\eta + S^t v_1)(\eta + Sv_2) \right\rangle} + {\tau}{\langle pv_1 \rangle}.$$ Therefore, , ${\langle v_1 \rangle} = {\langle v_2 \rangle} = {\langle {\boldsymbol v}\rangle}$ by and imply $$\label{eq:vf_sim_avg_aux2} \eta + {\langle {\boldsymbol v}\rangle} \sim \left[ (\eta + {\langle {\boldsymbol v}\rangle})^2 + {\tau}\right] {\langle {\boldsymbol v}\rangle}.$$ We use in and to conclude $$\label{eq:estimates_on_vf_aux_est} {\boldsymbol v}\sim \frac{1}{\eta + \langle {\boldsymbol v}\rangle + \frac{{\tau}}{\eta + \langle {\boldsymbol v}\rangle}} = \frac{\eta + {\langle {\boldsymbol v}\rangle}}{(\eta + {\langle {\boldsymbol v}\rangle})^2 + {\tau}} \sim {\langle {\boldsymbol v}\rangle},$$ where we applied in the last step. Hence, it suffices to prove all estimates , and for ${\boldsymbol v}$ replaced by ${\langle {\boldsymbol v}\rangle}$ only. We start with an auxiliary upper bound on ${\langle {\boldsymbol v}\rangle}$. By multiplying with ${\boldsymbol v}$, we get $1 = \eta {\boldsymbol v}+ {\boldsymbol v}{\boldsymbol S}_o {\boldsymbol v}+ {\tau}{\boldsymbol v}/(\eta + {\boldsymbol S}_d {\boldsymbol v}) \geq {\boldsymbol v}{\boldsymbol S}_o {\boldsymbol v}.$ Hence, $1 \geq {\langle v_1 S v_2 \rangle} \gtrsim {\langle v_1 \rangle} {\langle v_2 \rangle} = {\langle {\boldsymbol v}\rangle}^2, $ where we used in the second step and in the last step. Next, we show . Clearly, implies ${\boldsymbol v}\leq \eta^{-1}$. Moreover, as ${\tau}\leq {\tau}^*$ and $\eta \geq 1 \gtrsim {\langle {\boldsymbol v}\rangle}$ we find $\eta \lesssim \eta^2 {\langle {\boldsymbol v}\rangle}$ from . This gives the lower bound on ${\boldsymbol v}$ in when combined with . We note that immediately implies ${\langle {\boldsymbol v}\rangle} \gtrsim \eta$ for $\eta \leq 1$. Now, we show . For ${\tau}\in [0,1]$, we bring the term ${\tau}{\langle pv_1 \rangle}$ to the left-hand side in and use $v_1 \sim v_2 \sim {\langle {\boldsymbol v}\rangle}$ and as well as ${\langle {\boldsymbol v}\rangle} \gtrsim \eta$ to obtain $$\label{eq:aux_1} \eta + (1-{\tau}) {\langle {\boldsymbol v}\rangle} \sim {\langle {\boldsymbol v}\rangle}^3.$$ From , it is an elementary exercise to conclude for $\eta \leq 1$. Similarly, for $1\leq {\tau}\leq {\tau}^*$, we bring ${\langle pv_1 \rangle}$ to the right-hand side of , use ${\langle {\boldsymbol v}\rangle} \gtrsim \eta$ for $\eta \leq 1$ and conclude $$\label{eq:aux_4} \eta \sim {\langle {\boldsymbol v}\rangle}^3 + ({\tau}-1) {\langle {\boldsymbol v}\rangle}.$$ As before it is easy to conclude from . We leave this to the reader. This finishes the proof of Proposition \[pro:estimates\_v\_small\_z\]. Our next goal is a stability result for in the regime ${\tau}\in {\ensuremath{[0,1-{\tau}_*]}}\cup {\ensuremath{[1+{\tau}_*,{\tau}^*]}}$. In the following proposition, the second main result of this section, we prove that ${\mathrm{i}}{\boldsymbol v}(\eta) $ well approximates ${\boldsymbol g}(\eta)$ for all $\eta >0$ if ${\boldsymbol g}$ satisfies and as long as ${\boldsymbol d}$ is small. However, we will need an additional assumption on ${\boldsymbol g}=(g_1,g_2)$, namely that ${\langle g_1 \rangle}={\langle g_2 \rangle}$ (see below). Note that this is imposed on the solution ${\boldsymbol g}$ of and not directly on the perturbation ${\boldsymbol d}$. Nevertheless, in our applications, the constraint will be automatically satisfied owing to the specific block structure of the matrix ${\boldsymbol H}^z$ from . \[pro:stab\_lemma\] Suppose that some functions ${\boldsymbol d}\colon {\mathbb{R}}_+ \to {\mathbb{C}}^{2n}$ and ${\boldsymbol g}=(g_1, g_2) \colon {\mathbb{R}}_+ \to {\mathbb H}^{2n}$ satisfy and $$\label{eq:assumption_stability} {\langle g_1(\eta) \rangle} = {\langle g_2(\eta) \rangle}$$ for all $\eta >0$. There is a number $\lambda_* \gtrsim 1$, depending only on ${\ensuremath{\mathcal P}}$, such that $$\label{eq:stability_estimate1} {\lVert {\boldsymbol g}(\eta) - {\mathrm{i}}{\boldsymbol v}(\eta) \rVert}_\infty \cdot {\ensuremath{\chi}}\Big({\lVert {\boldsymbol g}(\eta) - {\mathrm{i}}{\boldsymbol v}(\eta) \rVert}_\infty \leq \lambda_*\Big) \lesssim {\lVert {\boldsymbol d}(w) \rVert}_\infty$$ uniformly for $\eta >0$ and ${\tau}\in {\ensuremath{[0,1-{\tau}_*]}}\cup{\ensuremath{[1+{\tau}_*,{\tau}^*]}}$. Moreover, there is a matrix-valued function ${\boldsymbol R}\colon {\mathbb{R}}_+ \to {\mathbb{C}}^{2n\times 2n}$, depending only on ${\tau}$ and $S$ and satisfying ${\lVert {\boldsymbol R}(\eta) \rVert_{\infty}} \lesssim 1$, such that $$\label{eq:stability_estimate_average} {\lvert \langle {\boldsymbol y}, {\boldsymbol g}(\eta)-{\mathrm{i}}{\boldsymbol v}(\eta)\rangle \rvert} \cdot {\ensuremath{\chi}}\Big({\lVert {\boldsymbol g}(\eta) - {\mathrm{i}}{\boldsymbol v}(\eta) \rVert}_\infty \leq \lambda_*\Big) \lesssim {\lVert {\boldsymbol y}\rVert}_\infty{\lVert {\boldsymbol d}(\eta) \rVert}_\infty^2 + {\lvert \langle {\boldsymbol R}(\eta){\boldsymbol y}, {\boldsymbol d}(\eta)\rangle \rvert}$$ uniformly for all ${\boldsymbol y}\in {\mathbb{C}}^{2n}$, $\eta >0$ and ${\tau}\in {\ensuremath{[0,1-{\tau}_*]}}\cup{\ensuremath{[1+{\tau}_*,{\tau}^*]}}$. The proof of this result is based on deriving a quadratic equation for the difference ${\boldsymbol h}{\mathrel{\vcenter{\baselineskip0.5ex \lineskiplimit0pt\hbox{\scriptsize.}\hbox{\scriptsize.}}}=}{\boldsymbol g}- {\mathrm{i}}{\boldsymbol v}$ and establishing a quantitative estimate on ${\boldsymbol h}$ in terms of the perturbation ${\boldsymbol d}$. Computing the difference of and , we obtain an equation for ${\boldsymbol g}- {\mathrm{i}}{\boldsymbol v}$. A straightforward calculation yields $$\label{eq:stability_equation} {\boldsymbol L}{\boldsymbol h}= {\ensuremath{\boldsymbol r}}, \quad \text{for } {\boldsymbol h}= {\boldsymbol g}- {\mathrm{i}}{\boldsymbol v},$$ where we used ${\boldsymbol L}$ defined in and introduced the vector ${\ensuremath{\boldsymbol r}}$ through $${\ensuremath{\boldsymbol r}}{\mathrel{\vcenter{\baselineskip0.5ex \lineskiplimit0pt\hbox{\scriptsize.}\hbox{\scriptsize.}}}=}{\boldsymbol d}+ {\mathrm{i}}{\boldsymbol v}({\boldsymbol h}-{\boldsymbol d}){\boldsymbol S}_o{\boldsymbol h}- {\tau}{\boldsymbol u}\left[ \frac{{\boldsymbol d}-{\boldsymbol g}}{{\mathrm{i}}\eta+{\boldsymbol S}_d{\boldsymbol g}} + {\boldsymbol u}\right] {\boldsymbol S}_d {\boldsymbol h}.\label{eq:def_rf}$$ The vector ${\boldsymbol u}$ in is defined through $$\label{eq:def_u} u {\mathrel{\vcenter{\baselineskip0.5ex \lineskiplimit0pt\hbox{\scriptsize.}\hbox{\scriptsize.}}}=}\frac{v_1}{\eta + S^tv_1} = \frac{v_2}{\eta + S v_2}, \qquad {\boldsymbol u}{\mathrel{\vcenter{\baselineskip0.5ex \lineskiplimit0pt\hbox{\scriptsize.}\hbox{\scriptsize.}}}=}(u, u) = \frac{{\boldsymbol v}}{\eta+ {\boldsymbol S}_d {\boldsymbol v}}$$ which is consistent by . Notice that all terms on the right-hand side of are either second order in ${\boldsymbol h}$ or they are of order ${\boldsymbol d}$, so is the linearization of around . In the following estimates, we need a bound on ${\boldsymbol u}$ as well. Indeed, Proposition \[pro:estimates\_v\_small\_z\] yields $$\label{eq:estimate_uf} {\boldsymbol u}= \frac{{\boldsymbol v}}{\eta + {\boldsymbol S}_d {\boldsymbol v}} \sim \frac{1}{1 +\eta^2}$$ uniformly for $\eta >0$ and ${\tau}\in [0,{\tau}^*]$. To shorten the upcoming relations, we introduce the vector $${\widetilde}{\boldsymbol v}{\mathrel{\vcenter{\baselineskip0.5ex \lineskiplimit0pt\hbox{\scriptsize.}\hbox{\scriptsize.}}}=}(v_2, v_1)$$ and the matrices ${\boldsymbol T}$, ${\boldsymbol F}$ and ${\boldsymbol V}$ defined by their action on a vector ${\boldsymbol y}=(y_1,y_2)$, $y_1, y_2 \in {\mathbb{C}}^n$ as follows \[eq:def\_matrices\] $$\begin{aligned} {\boldsymbol T}{\boldsymbol y}& {\mathrel{\vcenter{\baselineskip0.5ex \lineskiplimit0pt\hbox{\scriptsize.}\hbox{\scriptsize.}}}=}\frac{1}{{\boldsymbol u}} \begin{pmatrix} - v_1 v_2 y_1 + {\tau}u^2 y_2 \\ {\tau}u^2 y_1 -v_1 v_2 y_2 \end{pmatrix}, \label{eq:def_Tf} \\ {\boldsymbol F}{\boldsymbol y}& {\mathrel{\vcenter{\baselineskip0.5ex \lineskiplimit0pt\hbox{\scriptsize.}\hbox{\scriptsize.}}}=}\sqrt \frac{{\boldsymbol v}{\boldsymbol u}}{{\widetilde}{\boldsymbol v}} {\boldsymbol S}_o\left(\sqrt{\frac{{\boldsymbol v}{\boldsymbol u}}{{\widetilde}{\boldsymbol v}}}\, {\boldsymbol y}\right) , \label{FF} \\ {\boldsymbol V}{\boldsymbol y}& {\mathrel{\vcenter{\baselineskip0.5ex \lineskiplimit0pt\hbox{\scriptsize.}\hbox{\scriptsize.}}}=}\sqrt\frac{{\widetilde}{\boldsymbol v}}{{\boldsymbol u}{\boldsymbol v}}\, {\boldsymbol y}.\end{aligned}$$ All these matrices are functions of $\eta$ and ${\tau}$. They provide a crucial factorization of the stability operator ${\boldsymbol L}$; indeed, a simple calculation shows that $$\label{eq:transformation_Lf} {\boldsymbol L}= {\boldsymbol V}^{-1} ( {\mathds{1}}-{\boldsymbol T}{\boldsymbol F}){\boldsymbol V}.$$ This factorization reveals many properties of ${\boldsymbol L}$ which are difficult to observe directly. Owing to , the stability analysis of requires a control on the invertibility of the matrix ${\boldsymbol L}$. The matrices ${\boldsymbol V}$ and ${\boldsymbol V}^{-1}$ are harmless. A good understanding of the spectral decompositions of the simpler matrices ${\boldsymbol F}$ and ${\boldsymbol T}$ will then yield that ${\boldsymbol L}$ has only one direction, in which its inverse is not bounded. We remark that the factorization is the diagonal part of the one used in the stability analysis of the matrix Dyson equation in [@AjankiCorrelated]. Because of , we can study the stability of $$\label{eq:transformed_stability_equation} ({\mathds{1}}-{\boldsymbol T}{\boldsymbol F}) ({\boldsymbol V}{\boldsymbol h}) = {\boldsymbol V}{\ensuremath{\boldsymbol r}}$$ instead of . From Proposition \[pro:estimates\_v\_small\_z\] and , we conclude that $$\label{eq:control_norm_V_V_inverse} {\lVert {\boldsymbol V}\rVert_{\infty}}{\lVert {\boldsymbol V}^{-1} \rVert_{\infty}} \lesssim 1$$ uniformly for all $\eta >0$ and ${\tau}\in {\ensuremath{[0,1-{\tau}_*]}}\cup {\ensuremath{[1+{\tau}_*,{\tau}^*]}}$. Hence, it suffices to control the invertibility of ${\mathds{1}}- {\boldsymbol T}{\boldsymbol F}$. For later usage, we derive two relations for ${\boldsymbol u}$. From , recalling ${\widetilde}{\boldsymbol v}= (v_2, v_1)$, we immediately get $$\label{eq:relation_wt_vf_div_uf} \frac{{\widetilde}{\boldsymbol v}}{{\boldsymbol u}} = \eta + {\boldsymbol S}_o {\boldsymbol v}.$$ We multiply by ${\boldsymbol v}{\boldsymbol u}$ and use to obtain $$\label{eq:defining_eq_v_wt_v_u} {\boldsymbol u}= {\boldsymbol v}{\widetilde}{\boldsymbol v}+ {\tau}{\boldsymbol u}^2, \qquad 1 = \frac{{\boldsymbol v}{\widetilde}{\boldsymbol v}}{{\boldsymbol u}} + {\tau}{\boldsymbol u}.$$ The next lemma collects some properties of ${\boldsymbol F}$. For this formulation, we introduce $${\ensuremath{\boldsymbol{e}_-}}{\mathrel{\vcenter{\baselineskip0.5ex \lineskiplimit0pt\hbox{\scriptsize.}\hbox{\scriptsize.}}}=}(1,-1) \in {\mathbb{C}}^{2n}.$$ \[lem:prop\_Ff\] The eigenspace of ${\boldsymbol F}$ corresponding to its largest eigenvalue ${\lVert {\boldsymbol F}\rVert_{2}}$ is one dimensional. It is spanned by a unique positive normalized eigenvector ${\boldsymbol f_+}$, i.e., ${\boldsymbol F}{\boldsymbol f_+}= {\lVert {\boldsymbol F}\rVert_{2}} {\boldsymbol f_+}$ and ${\lVert {\boldsymbol f_+}\rVert_{2}}=1$. For every $\eta >0$, the norm of ${\boldsymbol F}$ is given by $$\label{eq:norm_F} {\lVert {\boldsymbol F}\rVert_{2}} = 1 - \eta \frac{{\left\langle {\boldsymbol f_+}\sqrt{{\boldsymbol v}/(\eta + {\boldsymbol S}_o {\boldsymbol v})} \right\rangle}}{{\left\langle {\boldsymbol f_+}\sqrt{{\boldsymbol v}(\eta + {\boldsymbol S}_o {\boldsymbol v})} \right\rangle}}.$$ Defining ${\boldsymbol f_-}{\mathrel{\vcenter{\baselineskip0.5ex \lineskiplimit0pt\hbox{\scriptsize.}\hbox{\scriptsize.}}}=}{\boldsymbol f_+}{\ensuremath{\boldsymbol{e}_-}}$, we have $$\label{eq:eigenvalue_relations_ffm} {\boldsymbol F}{\boldsymbol f_-}= - {\lVert {\boldsymbol F}\rVert_{2}} {\boldsymbol f_-}.$$ (i) (Inside regime) The following estimates hold true uniformly for ${\tau}\in{\ensuremath{[0,1-{\tau}_*]}}$. We have $$\label{eq:bound_norm_F_small_eta} 1 -{\lVert {\boldsymbol F}\rVert_{2}} \sim \eta.$$ uniformly for $\eta \in(0,1]$. Furthermore, uniformly for $\eta \geq 1$, we have $$\label{eq:estimate_norm_F_small_z_large_eta} 1-{\lVert {\boldsymbol F}\rVert_{2}} \sim 1.$$ Moreover, uniformly for $\eta \in(0,1]$, ${\boldsymbol f_+}$ satisfies $$\label{eq:ff_+_order_1} {\boldsymbol f_+}\sim 1$$ and there is ${\varepsilon}\sim 1$ such that $$\label{eq:gap_F_order_1} {\lVert {\boldsymbol F}{\boldsymbol x}\rVert_{2}} \leq (1-{\varepsilon}){\lVert {\boldsymbol x}\rVert_{2}}$$ for all ${\boldsymbol x}\in {\mathbb{C}}^{2n}$ satisfying ${\boldsymbol x}\perp {\boldsymbol f_+}$ and ${\boldsymbol x}\perp {\boldsymbol f_-}$. (ii) (Outside regime) Uniformly for all $\eta >0$ and ${\tau}\in {\ensuremath{[1+{\tau}_*,{\tau}^*]}}$, we have $$\label{eq:estimate_norm_F_large_z} 1 -{\lVert {\boldsymbol F}\rVert_{2}} \sim 1.$$ The statements about the eigenspace corresponding to ${\lVert {\boldsymbol F}\rVert_{2}}$ and ${\boldsymbol f_+}$ follow from Lemma 3.3 in [@AltGram]. For the proof of , we multiply by ${\boldsymbol v}$ and take the scalar product of the resulting relation with ${\boldsymbol f_+}\sqrt{{\boldsymbol u}/({\boldsymbol v}{\widetilde}{\boldsymbol v})}$. Using that $${\left\langle{{\boldsymbol f_+}\sqrt{\frac{{\boldsymbol u}}{{\boldsymbol v}{\widetilde}{\boldsymbol v}}} } \,\mspace{2mu},\, {{\boldsymbol v}{\boldsymbol S}_o {\boldsymbol v}}\right\rangle} = {\left\langle{{\boldsymbol f_+}\sqrt{\frac{{\boldsymbol v}{\boldsymbol u}}{{\widetilde}{\boldsymbol v}}} } \,\mspace{2mu},\, {{\boldsymbol S}_o {\boldsymbol v}}\right\rangle} = {\left\langle{ {\boldsymbol S}_o\Bigg( {\boldsymbol f_+}\sqrt{\frac{{\boldsymbol v}{\boldsymbol u}}{{\widetilde}{\boldsymbol v}}}\Bigg) } \,\mspace{2mu},\, {{\boldsymbol v}}\right\rangle} = {\left\langle{ \sqrt{\frac{{\widetilde}{\boldsymbol v}}{{\boldsymbol v}{\boldsymbol u}}} {\boldsymbol F}{\boldsymbol f_+}} \,\mspace{2mu},\, {{\boldsymbol v}}\right\rangle}= {\lVert {\boldsymbol F}\rVert_{2}}{\left\langle{{\boldsymbol f_+}} \,\mspace{2mu},\, {\sqrt{\frac{{\boldsymbol v}{\widetilde}{\boldsymbol v}}{{\boldsymbol u}}}}\right\rangle},$$ this yields $${\lVert {\boldsymbol F}\rVert_{2}} {\left\langle{{\boldsymbol f_+}} \,\mspace{2mu},\, {\sqrt{\frac{{\boldsymbol v}{\widetilde}{\boldsymbol v}}{{\boldsymbol u}}}}\right\rangle} = {\left\langle{{\boldsymbol f_+}\sqrt{\frac{{\boldsymbol u}}{{\boldsymbol v}{\widetilde}{\boldsymbol v}}}} \,\mspace{2mu},\, {1- {\tau}{\boldsymbol u}}\right\rangle} - \eta {\left\langle{{\boldsymbol f_+}\sqrt{\frac{{\boldsymbol u}}{{\boldsymbol v}{\widetilde}{\boldsymbol v}}}} \,\mspace{2mu},\, {{\boldsymbol v}}\right\rangle}.$$ We conclude from applying and to the last relation. Since ${\boldsymbol F}$ from has the form $${\boldsymbol F}= \begin{pmatrix} 0 & F \\ F^t & 0 \end{pmatrix} ,$$ for some $F \in {\mathbb{C}}^{n\times n}$ we have ${\boldsymbol F}({\ensuremath{\boldsymbol{e}_-}}{\boldsymbol y})=-{\ensuremath{\boldsymbol{e}_-}}({\boldsymbol F}{\boldsymbol y})$ for all ${\boldsymbol y}\in {\mathbb{C}}^{2n}$. Thus, we get from ${\boldsymbol F}{\boldsymbol f_+}= {\lVert {\boldsymbol F}\rVert_{2}} {\boldsymbol f_+}$. In the regime ${\tau}\in {\ensuremath{[0,1-{\tau}_*]}}$ and $\eta\in (0,1]$, we have uniform lower and upper bounds on ${\boldsymbol v}$ from Proposition \[pro:estimates\_v\_small\_z\]. Therefore, the estimates in and follow from Lemma 3.3 in [@AltGram]. Combining , and Proposition \[pro:estimates\_v\_small\_z\] yields . In the large $\eta$ regime, i.e., for $\eta\ge 1$, since ${\boldsymbol v}\sim \eta^{-1}$ by Proposition \[pro:estimates\_v\_small\_z\] we obtain $$\label{eq:estimate_F_aux_1} \frac{{\boldsymbol v}}{\eta + {\boldsymbol S}_o {\boldsymbol v}} \sim \eta^{-2}, \quad {\boldsymbol v}(\eta + {\boldsymbol S}_o {\boldsymbol v}) \sim 1.$$ Hence, as ${\boldsymbol f_+}>0$ we conclude $$\label{eq:estimate_F_aux_2} \frac{{\left\langle {\boldsymbol f_+}\sqrt{{\boldsymbol v}/(\eta + {\boldsymbol S}_o {\boldsymbol v})} \right\rangle}}{{\left\langle {\boldsymbol f_+}\sqrt{{\boldsymbol v}(\eta + {\boldsymbol S}_o {\boldsymbol v})} \right\rangle}} \sim \frac{{\langle {\boldsymbol f_+}\rangle}}{ {\langle {\boldsymbol f_+}\rangle}} \frac{1}{\eta} = \frac{1}{\eta},$$ uniformly for all $\eta \geq 1$. This shows that holds true for all $\eta \geq 1$ and ${\tau}\in {\ensuremath{[0,1-{\tau}_*]}}$. We now turn to the proof of (ii). If ${\tau}\in {\ensuremath{[1+{\tau}_*,{\tau}^*]}}$, then ${\boldsymbol v}\sim \eta$ by for $\eta \leq 1$ and therefore $$\frac{{\boldsymbol v}}{\eta + {\boldsymbol S}_o {\boldsymbol v}} \sim 1, \quad {\boldsymbol v}(\eta + {\boldsymbol S}_o {\boldsymbol v}) \sim \eta^{2}.$$ As ${\boldsymbol f_+}>0$, we thus have $$\label{etafraction} \eta \frac{{\left\langle {\boldsymbol f_+}\sqrt{{\boldsymbol v}/(\eta + {\boldsymbol S}_o {\boldsymbol v})} \right\rangle}}{{\left\langle {\boldsymbol f_+}\sqrt{{\boldsymbol v}(\eta + {\boldsymbol S}_o {\boldsymbol v})} \right\rangle}} \sim \frac{{\langle {\boldsymbol f_+}\rangle}}{{\langle {\boldsymbol f_+}\rangle}} = 1.$$ For $\eta \geq 1$, we argue as in and and arrive at the same conclusion . Thus, because of the estimate holds true for all $\eta >0$ and ${\tau}\in {\ensuremath{[1+{\tau}_*,{\tau}^*]}}$. Next, we give an approximation for the eigenvector ${\boldsymbol f_-}$ belonging to the isolated single eigenvalue $-{\lVert {\boldsymbol F}\rVert_{2}}$ of ${\boldsymbol F}$ by constructing an approximate eigenvector. For $\eta \leq 1$ and ${\tau}\in {\ensuremath{[0,1-{\tau}_*]}}$, we define $$\label{eq:def_af} {\boldsymbol a}{\mathrel{\vcenter{\baselineskip0.5ex \lineskiplimit0pt\hbox{\scriptsize.}\hbox{\scriptsize.}}}=}\frac{{\ensuremath{\boldsymbol{e}_-}}({\boldsymbol V}{\boldsymbol v})}{{\lVert {\boldsymbol V}{\boldsymbol v}\rVert_{2}}}$$ which is normalized as ${\lVert {\ensuremath{\boldsymbol{e}_-}}({\boldsymbol V}{\boldsymbol v}) \rVert_{2}}={\lVert {\boldsymbol V}{\boldsymbol v}\rVert_{2}}$. We compute $$\label{eq:ffp} {\boldsymbol F}({\boldsymbol V}{\boldsymbol v}) = \sqrt{\frac{ {\boldsymbol u}}{{\boldsymbol v}{\widetilde}{\boldsymbol v}}} {\boldsymbol v}\left({\boldsymbol S}_o {\boldsymbol v}\right) = \sqrt{\frac{{\boldsymbol u}}{{\boldsymbol v}{\widetilde}{\boldsymbol v}}} \left( 1 - \eta {\boldsymbol v}- {\tau}{\boldsymbol u}\right) = \sqrt{\frac{{\boldsymbol v}{\widetilde}{\boldsymbol v}}{{\boldsymbol u}}} - \eta {\boldsymbol v}\sqrt{\frac{{\boldsymbol u}}{{\boldsymbol v}{\widetilde}{\boldsymbol v}}} = {\lVert {\boldsymbol F}\rVert_{2}} {\boldsymbol V}{\boldsymbol v}+ O(\eta).$$ Here, we used ${\boldsymbol v}{\boldsymbol S}_o {\boldsymbol v}= -\eta {\boldsymbol v}+ {\boldsymbol v}{\widetilde}{\boldsymbol v}/{\boldsymbol u}$ by . For estimating the $O(\eta)$ term we applied , and since ${\tau}\in {\ensuremath{[0,1-{\tau}_*]}}$ and $\eta \leq 1$. Using the block structure of ${\boldsymbol F}$ as in the proof of , we obtain $$\label{eq:af_approximating_eigenvector} {\boldsymbol F}({\ensuremath{\boldsymbol{e}_-}}({\boldsymbol V}{\boldsymbol v})) = - {\lVert {\boldsymbol F}\rVert_{2}} {\ensuremath{\boldsymbol{e}_-}}({\boldsymbol V}{\boldsymbol v}) + O(\eta).$$ The following lemma states that ${\boldsymbol a}$ approximates the nondegenerate eigenvector ${\boldsymbol f_-}$. \[lem:approximating\_eigenvector\] The eigenvector ${\boldsymbol f_-}$ can be approximated by ${\boldsymbol a}$ in the $\ell^\infty$-norm, i.e., $${\lVert {\boldsymbol f_-}- {\boldsymbol a}\rVert_{\infty}} = O(\eta) \label{eq:estimate_ff_-_inf}$$ uniformly for $\eta \leq 1$ and ${\tau}\in {\ensuremath{[0,1-{\tau}_*]}}$. This lemma is proved in Appendix \[proof:approximating\_eigenvalue\]. In the following lemma, we show some properties of ${\boldsymbol T}$. \[lem:pro\_Tf\] The symmetric operator ${\boldsymbol T}$, defined in , satisfies (i) ${\lVert {\boldsymbol T}\rVert_{2}} = 1$, ${\lVert {\boldsymbol T}\rVert_{\infty}} =1$. (ii) The spectrum of ${\boldsymbol T}$ is given by $$\operatorname*{Spec}({\boldsymbol T}) = \{-1\} \cup \left\{ {\tau}{\boldsymbol u}_i - \frac{({\boldsymbol v}{\widetilde}{\boldsymbol v})_i}{{\boldsymbol u}_i} \;\middle| \; i=1, \ldots, n \right\}.$$ (iii) For all $\eta >0$, we have ${\boldsymbol T}({\tau}=0) = -{\mathds{1}}$ and if ${\tau}> 0$, then the eigenspace of ${\boldsymbol T}$ corresponding to the eigenvalue $-1$ is $n$-fold degenerate and given by $$\label{eq:Eigenspace_Tf_minus_one} \mathrm{Eig}(-1, {\boldsymbol T}) = \left\{ (y,-y)\middle| y \in {\mathbb{C}}^n \right\}.$$ (iv) The spectrum of ${\boldsymbol T}$ is strictly away from one, i.e., there is ${\varepsilon}>0$, depending only on ${\ensuremath{\mathcal P}}$, such that $$\label{eq:spec_Tf} \operatorname*{Spec}({\boldsymbol T}) \subset [-1, 1-{\varepsilon}]$$ uniformly for ${\tau}\in {\ensuremath{[0,1-{\tau}_*]}}$ and $\eta\in(0,1]$. The second relation in implies ${\lVert {\boldsymbol T}\rVert_{\infty}} =1$ and ${\boldsymbol T}({\tau}=0) = -{\mathds{1}}$. Moreover, it yields that all vectors of the form $(y,-y)$ for $y \in {\mathbb{C}}^n$ are contained in $\mathrm{Eig}(-1, {\boldsymbol T})$. We define the vector ${\boldsymbol y}^{(j)} \in {\mathbb{C}}^{2n}$ by ${\boldsymbol y}^{(j)} {\mathrel{\vcenter{\baselineskip0.5ex \lineskiplimit0pt\hbox{\scriptsize.}\hbox{\scriptsize.}}}=}(\delta_{i,j} + \delta_{i,j+n})_{i=1}^{2n}$ and observe that $${\boldsymbol T}{\boldsymbol y}^{(j)} = \left({\tau}{\boldsymbol u}_j - \frac{({\boldsymbol v}{\widetilde}{\boldsymbol v})_j}{{\boldsymbol u}_j}\right) {\boldsymbol y}^{(j)}$$ for $j = 1, \ldots, n$. Counting dimensions implies that we have found all eigenvalues, hence (ii) follows. For ${\tau}>0$, we have ${\tau}{\boldsymbol u}_j - ({\boldsymbol v}{\widetilde}{\boldsymbol v})_j/{\boldsymbol u}_j = 2{\tau}{\boldsymbol u}_j -1 >-1$ by and ${\boldsymbol u}_j >0$ for all $j=1, \ldots, n$. This yields the missing inclusion in . Since ${\boldsymbol T}$ is a symmetric operator, ${\lVert {\boldsymbol T}\rVert_{2}}=1$ follows from (ii) and ${\lvert {\tau}{\boldsymbol u}- {\boldsymbol v}{\widetilde}{\boldsymbol v}/{\boldsymbol u}\rvert} \leq 1$ by . For the proof of (iv), we remark that there is ${\varepsilon}>0$, depending only on ${\ensuremath{\mathcal P}}$, such that $2{\boldsymbol v}{\widetilde}{\boldsymbol v}/{\boldsymbol u}\geq {\varepsilon}$ for all $\eta \in (0,1]$ and ${\tau}\in {\ensuremath{[0,1-{\tau}_*]}}$ by and . Thus, $${\tau}{\boldsymbol u}- \frac{{\boldsymbol v}{\widetilde}{\boldsymbol v}}{{\boldsymbol u}} = 1 - 2 \frac{{\boldsymbol v}{\widetilde}{\boldsymbol v}}{{\boldsymbol u}} \leq 1- {\varepsilon}$$ by . This concludes the proof of the lemma. Now we are ready to give a proof of Proposition \[pro:stab\_lemma\] based on inverting ${\mathds{1}}- {\boldsymbol T}{\boldsymbol F}$. We recall that ${\boldsymbol h}= {\boldsymbol g}-{\mathrm{i}}{\boldsymbol v}$. Throughout the proof we will omit arguments, but we keep in mind that ${\boldsymbol g}$, ${\boldsymbol d}$, ${\boldsymbol h}$ and ${\boldsymbol v}$ depend on $\eta$ and ${\tau}$. The proof will be given in three steps. The first step is to control ${\lVert {\ensuremath{\boldsymbol r}}\rVert_{\infty}}$ from in terms of ${\lVert {\boldsymbol h}\rVert_{\infty}}^2$ and ${\lVert {\boldsymbol d}\rVert_{\infty}}$, i.e., to show $$\label{eq:first_step} {\lVert {\ensuremath{\boldsymbol r}}\rVert_{\infty}}{\ensuremath{\chi}}\big({\lVert {\boldsymbol h}\rVert_{\infty}} \leq 1\big)\lesssim {\lVert {\boldsymbol h}\rVert_{\infty}}^2 + {\lVert {\boldsymbol d}\rVert_{\infty}}.$$ Inverting ${\boldsymbol V}^{-1}({\mathds{1}}-{\boldsymbol T}{\boldsymbol F}){\boldsymbol V}$ in , controlling the norm of the inverse and choosing $\lambda_*\leq 1$ small enough, we will conclude Proposition \[pro:stab\_lemma\] from . For any $\eta_*\in(0,1]$, depending only on ${\ensuremath{\mathcal P}}$, this argument will be done in the second step for ${\tau}\in {\ensuremath{[0,1-{\tau}_*]}}\cup{\ensuremath{[1+{\tau}_*,{\tau}^*]}}$ and $\eta \geq \eta_*$ as well as for ${\tau}\in {\ensuremath{[1+{\tau}_*,{\tau}^*]}}$ and $\eta \in (0,\eta_*]$. In the third step, we consider the most interesting regime ${\tau}\in {\ensuremath{[0,1-{\tau}_*]}}$ and $\eta \leq \eta_*$ for a sufficiently small $\eta_*$, depending on ${\ensuremath{\mathcal P}}$ only. In this regime, we will use an extra cancellation for the contribution of ${\ensuremath{\boldsymbol r}}$ in the unstable direction of ${\boldsymbol L}$. ----------- ---------------------------------------------------------------------------------------------------------------------- *Step 1:* For all $\eta >0$ and ${\tau}\in {\ensuremath{[0,1-{\tau}_*]}}\cup{\ensuremath{[1+{\tau}_*,{\tau}^*]}}$, holds true. ----------- ---------------------------------------------------------------------------------------------------------------------- From , we obtain $${\tau}\frac{{\boldsymbol g}-{\boldsymbol d}}{{\mathrm{i}}\eta + {\boldsymbol S}_d {\boldsymbol g}}= 1 + ({\mathrm{i}}\eta +{\boldsymbol S}_o{\boldsymbol g})({\boldsymbol g}-{\boldsymbol d}).$$ We start from , use the previous relation, ${\tau}{\boldsymbol u}= 1+{\mathrm{i}}{\boldsymbol v}({\mathrm{i}}\eta + {\boldsymbol S}_o{\mathrm{i}}{\boldsymbol v})$ by and ${\widetilde}{\boldsymbol v}=(v_2, v_1)={\boldsymbol u}(\eta +{\boldsymbol S}_o{\boldsymbol v})$ by and get $$\begin{aligned} {\ensuremath{\boldsymbol r}}& = {\boldsymbol d}+ {\mathrm{i}}{\boldsymbol v}({\boldsymbol h}-{\boldsymbol d}){\boldsymbol S}_o{\boldsymbol h}- {\boldsymbol u}\left[ {\mathrm{i}}{\boldsymbol v}({\mathrm{i}}\eta +{\boldsymbol S}_o {\mathrm{i}}{\boldsymbol v}) - ({\boldsymbol g}-{\boldsymbol d})({\mathrm{i}}\eta+{\boldsymbol S}_o{\boldsymbol g})\right]{\boldsymbol S}_d {\boldsymbol h}\nonumber\\ & = {\boldsymbol d}+ {\mathrm{i}}{\boldsymbol v}({\boldsymbol h}-{\boldsymbol d}){\boldsymbol S}_o{\boldsymbol h}+ {\boldsymbol u}\left[ {\boldsymbol h}({\mathrm{i}}\eta+{\boldsymbol S}_o{\mathrm{i}}{\boldsymbol v}) + {\boldsymbol g}{\boldsymbol S}_o{\boldsymbol h}\right] {\boldsymbol S}_d {\boldsymbol h}-{\boldsymbol d}{\boldsymbol u}({\mathrm{i}}\eta+{\boldsymbol S}_o{\boldsymbol g}){\boldsymbol S}_d{\boldsymbol h}\nonumber\\ & = {\mathrm{i}}{\boldsymbol v}{\boldsymbol h}{\boldsymbol S}_o{\boldsymbol h}+ {\mathrm{i}}{\widetilde}{\boldsymbol v}{\boldsymbol h}{\boldsymbol S}_d{\boldsymbol h}+{\boldsymbol u}{\boldsymbol g}{\boldsymbol S}_o{\boldsymbol h}{\boldsymbol S}_d{\boldsymbol h}+ {\boldsymbol d}-{\mathrm{i}}{\boldsymbol v}{\boldsymbol d}{\boldsymbol S}_o{\boldsymbol h}-{\boldsymbol d}{\boldsymbol u}({\mathrm{i}}\eta+{\boldsymbol S}_o{\boldsymbol g}){\boldsymbol S}_d{\boldsymbol h}. \label{eq:represenation_rf_5}\end{aligned}$$ Notice that the first three terms are quadratic in ${\boldsymbol h}$ (the linear terms dropped out), while the last three terms are controlled by ${\boldsymbol d}$. Now, we show that all other factors are bounded and hence irrelevant whenever ${\lVert {\boldsymbol g}- {\mathrm{i}}{\boldsymbol v}\rVert_{\infty}} \leq \lambda_*$ for $\eta >0$ and ${\tau}\in {\ensuremath{[0,1-{\tau}_*]}}\cup {\ensuremath{[1+{\tau}_*,{\tau}^*]}}$. In this case, we conclude ${\lVert {\boldsymbol g}\rVert_{\infty}} \lesssim 1$ uniformly for all $\eta >0$ and ${\tau}\in {\ensuremath{[0,1-{\tau}_*]}}\cup {\ensuremath{[1+{\tau}_*,{\tau}^*]}}$ by and from Proposition \[pro:estimates\_v\_small\_z\]. Therefore, starting from and using ${\lVert {\boldsymbol v}\rVert_{\infty}} \lesssim 1$ by and , and ${\lVert {\boldsymbol u}\rVert_{\infty}}\lesssim 1$ by , we obtain . ----------- -------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- *Step 2:* For any $\eta_* \in(0,1]$, there exists $\lambda_* \gtrsim 1$, depending only on ${\ensuremath{\mathcal P}}$ and $\eta_*$, such that holds true for $\eta \geq \eta_*$ and ${\tau}\in {\ensuremath{[0,1-{\tau}_*]}}\cup{\ensuremath{[1+{\tau}_*,{\tau}^*]}}$ as well as for $\eta\in (0,\eta_*]$ and ${\tau}\in {\ensuremath{[1+{\tau}_*,{\tau}^*]}}$. Moreover, with this choice of $\lambda_*$, holds true in these $(\eta, {\tau})$ parameter regimes as well. ----------- -------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- \ Within Step 2, we redefine the comparison relation to depend both on ${\ensuremath{\mathcal P}}$ and $\eta_*$. Later in Step 3 we will choose an appropriate $\eta_*$ depending only on ${\ensuremath{\mathcal P}}$, so eventually the comparison relations for our choice will depend only on ${\ensuremath{\mathcal P}}$. We are now working in the regime, where $\eta \geq \eta_*$ and ${\tau}\in {\ensuremath{[0,1-{\tau}_*]}}\cup{\ensuremath{[1+{\tau}_*,{\tau}^*]}}$ or $\eta \in (0,\eta_*]$ and ${\tau}\in {\ensuremath{[1+{\tau}_*,{\tau}^*]}}$. In this case, to prove , we invert ${\boldsymbol L}= {\boldsymbol V}^{-1}({\mathds{1}}- {\boldsymbol T}{\boldsymbol F}){\boldsymbol V}$ (cf. ) in ${\boldsymbol L}{\boldsymbol h}= {\ensuremath{\boldsymbol r}}$, bound ${\lVert {\boldsymbol L}^{-1} \rVert_{\infty}}\lesssim 1$, which is proved below, and conclude $${\lVert {\boldsymbol h}\rVert_{\infty}} {\ensuremath{\chi}}\big({\lVert {\boldsymbol h}\rVert_{\infty}} \leq 1\big) \lesssim {\lVert {\boldsymbol h}\rVert_{\infty}}^2 + {\lVert {\boldsymbol d}\rVert_{\infty}}$$ from for $\eta \geq \eta_*$ and ${\tau}\in {\ensuremath{[0,1-{\tau}_*]}}\cup{\ensuremath{[1+{\tau}_*,{\tau}^*]}}$ as well as for $\eta \in (0,\eta_*]$ and ${\tau}\in {\ensuremath{[0,1-{\tau}_*]}}$. This means that there are $\Psi_1, \Psi_2 \sim 1$ such that $${\lVert {\boldsymbol h}\rVert_{\infty}} {\ensuremath{\chi}}\big({\lVert {\boldsymbol h}\rVert_{\infty}} \leq 1\big) \leq \Psi_1{\lVert {\boldsymbol h}\rVert_{\infty}}^2 + \Psi_2{\lVert {\boldsymbol d}\rVert_{\infty}}.$$ Choosing $\lambda_* {\mathrel{\vcenter{\baselineskip0.5ex \lineskiplimit0pt\hbox{\scriptsize.}\hbox{\scriptsize.}}}=}\min\{1, (2\Psi_1)^{-1}\}$ this yields $${\lVert {\boldsymbol h}\rVert_{\infty}} {\ensuremath{\chi}}\big({\lVert {\boldsymbol h}\rVert_{\infty}} \leq \lambda_*\big) \leq 2\Psi_2{\lVert {\boldsymbol d}\rVert_{\infty}}.$$ Thus, we are left with controlling ${\lVert {\boldsymbol L}^{-1} \rVert_{\infty}}$, i.e., proving ${\lVert {\boldsymbol L}^{-1} \rVert_{\infty}} \lesssim 1$. In the regime $\eta \geq \eta_*$ and ${\tau}\in {\ensuremath{[0,1-{\tau}_*]}}\cup{\ensuremath{[1+{\tau}_*,{\tau}^*]}}$, we have ${\boldsymbol v}\sim 1/\eta$ by Proposition \[pro:estimates\_v\_small\_z\] and ${\boldsymbol u}\sim 1/\eta^2$ by . Hence, ${\boldsymbol V}\sim \eta$ and ${\boldsymbol V}^{-1} \sim 1/\eta$. Therefore, ${\lVert {\boldsymbol V}\rVert_{\infty}}{\lVert {\boldsymbol V}^{-1} \rVert_{\infty}} \lesssim 1$ and due to ${\lVert {\boldsymbol L}^{-1} \rVert_{\infty}} \lesssim {\lVert {\boldsymbol V}^{-1} \rVert_{\infty}} {\lVert ({\mathds{1}}-{\boldsymbol T}{\boldsymbol F})^{-1} \rVert_{\infty}} {\lVert {\boldsymbol V}\rVert_{\infty}}$, it suffices to show ${\lVert ({\mathds{1}}-{\boldsymbol T}{\boldsymbol F})^{-1} \rVert_{\infty}}\lesssim 1$. Basic facts on the operator ${\mathds{1}}- {\boldsymbol T}{\boldsymbol F}$ are collected in Lemma \[lem:TwoNorms\_to\_InfNorms\] in the appendix. In particular, because of , the $\ell^\infty$ bound follows from ${\lVert ({\mathds{1}}-{\boldsymbol T}{\boldsymbol F})^{-1} \rVert_{2}}\lesssim 1$. Using , and , we get that $1- {\lVert {\boldsymbol F}\rVert_{2}} \sim 1$ for all $\eta \geq \eta_*$ and ${\tau}\in {\ensuremath{[0,1-{\tau}_*]}}\cup{\ensuremath{[1+{\tau}_*,{\tau}^*]}}$. Hence, $1-{\lVert {\boldsymbol T}{\boldsymbol F}\rVert_{2}} \sim 1$ by Lemma \[lem:pro\_Tf\] (i), so the bound ${\lVert ({\mathds{1}}-{\boldsymbol T}{\boldsymbol F})^{-1} \rVert_{2}}\lesssim1 $ immediately follows. This proves for $\eta \geq \eta_*$ and ${\tau}\in {\ensuremath{[0,1-{\tau}_*]}}\cup{\ensuremath{[1+{\tau}_*,{\tau}^*]}}$. For $\eta \leq \eta_*$ and ${\tau}\in {\ensuremath{[1+{\tau}_*,{\tau}^*]}}$, we have ${\boldsymbol v}\sim \eta$ by , ${\boldsymbol u}\sim 1$ by . Thus, ${\boldsymbol V}\sim 1$, ${\boldsymbol V}^{-1} \sim 1$ as well as ${\lVert {\boldsymbol V}\rVert_{\infty}}{\lVert {\boldsymbol V}^{-1} \rVert_{\infty}} \lesssim 1$. As above it is enough to show ${\lVert ({\mathds{1}}-{\boldsymbol T}{\boldsymbol F})^{-1} \rVert_{2}}\lesssim1 $. By Lemma \[lem:pro\_Tf\] (i) and , $1- {\lVert {\boldsymbol T}{\boldsymbol F}\rVert_{2}} \sim 1$ which again leads to ${\lVert ({\mathds{1}}-{\boldsymbol T}{\boldsymbol F})^{-1} \rVert_{2}}\lesssim1 $. We conclude for $\eta \leq \eta_*$ and ${\tau}\in {\ensuremath{[1+{\tau}_*,{\tau}^*]}}$. Next, we verify in these two regimes. Using ${\boldsymbol h}\cdot {\ensuremath{\chi}}({\lVert {\boldsymbol h}\rVert_{\infty}} \leq \lambda_*) = O({\lVert {\boldsymbol d}\rVert_{\infty}})$ by , ${\boldsymbol v}\lesssim 1$ and ${\boldsymbol u}\lesssim 1$, we see that with the exception of ${\boldsymbol d}$, all terms in are second order in ${\boldsymbol d}$. Therefore, $$\label{eq:second_estimate_V_rhs} {\ensuremath{\boldsymbol r}}\cdot {\ensuremath{\chi}}({\lVert {\boldsymbol h}\rVert_{\infty}} \leq \lambda_*) = {\boldsymbol d}\cdot {\ensuremath{\chi}}({\lVert {\boldsymbol h}\rVert_{\infty}} \leq \lambda_*) + O\left({\lVert {\boldsymbol d}\rVert_{\infty}}^2\right)$$ uniformly for $\eta \geq \eta_*$ and ${\tau}\in {\ensuremath{[0,1-{\tau}_*]}}\cup {\ensuremath{[1+{\tau}_*,{\tau}^*]}}$ as well as for $\eta \in (0,\eta_*]$ and ${\tau}\in {\ensuremath{[1+{\tau}_*,{\tau}^*]}}$. We start from ${\boldsymbol L}{\boldsymbol h}= {\ensuremath{\boldsymbol r}}$ and compute $$\label{eq:computation_average_version_stab_estimate} {\langle{{\boldsymbol y}} \mspace{2mu}, {{\boldsymbol h}}\rangle} = {\langle{({\boldsymbol L}^{-1})^*{\boldsymbol y}} \mspace{2mu}, {{\ensuremath{\boldsymbol r}}}\rangle} = {\langle{({\boldsymbol L}^{-1})^*{\boldsymbol y}} \mspace{2mu}, {{\boldsymbol d}}\rangle} + {\langle{({\boldsymbol L}^{-1})^*{\boldsymbol y}} \mspace{2mu}, {{\ensuremath{\boldsymbol r}}- {\boldsymbol d}}\rangle} = {\langle{{\boldsymbol R}{\boldsymbol y}} \mspace{2mu}, {{\boldsymbol d}}\rangle} + {\langle{({\boldsymbol L}^{-1})^*{\boldsymbol y}} \mspace{2mu}, {{\ensuremath{\boldsymbol r}}-{\boldsymbol d}}\rangle}.$$ Here, we defined the operator ${\boldsymbol R}= {\boldsymbol R}(\eta)$ on ${\mathbb{C}}^{2n}$ in the last step through its action on any ${\boldsymbol x}\in {\mathbb{C}}^{2n}$ via $$\label{eq:def_Rf} {\boldsymbol R}{\boldsymbol x}{\mathrel{\vcenter{\baselineskip0.5ex \lineskiplimit0pt\hbox{\scriptsize.}\hbox{\scriptsize.}}}=}\left({\boldsymbol L}^{-1}\right)^* {\boldsymbol x}= {\boldsymbol V}^{-1}( {\mathds{1}}- {\boldsymbol F}{\boldsymbol T})^{-1} {\boldsymbol V}{\boldsymbol x}.$$ Now, we establish that ${\lVert ({\boldsymbol L}^{-1})^* \rVert_{\infty}} \lesssim 1$ in the two regimes considered in Step 2. From this, we conclude that ${\lVert {\boldsymbol R}\rVert_{\infty}} \lesssim 1$ and that the last term in when multiplied by ${\ensuremath{\chi}}({\lVert {\boldsymbol h}\rVert_{\infty}} \leq \lambda_*)$ is bounded by $\lesssim {\lVert {\boldsymbol y}\rVert_{\infty}}{\lVert {\boldsymbol d}\rVert_{\infty}}^{2}$ because of . By Lemma \[lem:pro\_Tf\] (i), , and we have $1-{\lVert {\boldsymbol F}{\boldsymbol T}\rVert_{2}} \sim 1$. Thus, ${\lVert ({\mathds{1}}- {\boldsymbol F}{\boldsymbol T})^{-1} \rVert_{2}} \lesssim 1$ and hence ${\lVert ({\mathds{1}}- {\boldsymbol F}{\boldsymbol T})^{-1} \rVert_{\infty}} \lesssim 1$ by Lemma \[lem:TwoNorms\_to\_InfNorms\] (ii). As ${\lVert {\boldsymbol V}\rVert_{\infty}}{\lVert {\boldsymbol V}^{-1} \rVert_{\infty}} \lesssim 1$ we get ${\lVert ({\boldsymbol L}^{-1})^* \rVert_{\infty}} \lesssim 1$. Therefore, we conclude that holds true uniformly for $\eta \geq \eta_*$ and ${\tau}\in {\ensuremath{[0,1-{\tau}_*]}}\cup{\ensuremath{[1+{\tau}_*,{\tau}^*]}}$ as well as for $\eta \in (0,\eta_*]$ and ${\tau}\in {\ensuremath{[1+{\tau}_*,{\tau}^*]}}$. Thus, we have proved the proposition for these combinations of $\eta$ and ${\tau}$. Finally, we prove the proposition in the most interesting regime, ${\tau}\in {\ensuremath{[0,1-{\tau}_*]}}$ and for small $\eta$: ----------- ------------------------------------------------------------------------------------------------------------------------------------------------- *Step 3:* There exists $\eta_*>0$, depending only on ${\ensuremath{\mathcal P}}$, and $\lambda_* \gtrsim 1$ such that holds true for $\eta\in (0,\eta_*]$ and ${\tau}\in {\ensuremath{[0,1-{\tau}_*]}}$. Moreover, with this choice of $\lambda_*$, holds true for $\eta\in (0,\eta_*]$ and ${\tau}\in {\ensuremath{[0,1-{\tau}_*]}}$. ----------- ------------------------------------------------------------------------------------------------------------------------------------------------- The crucial step for proving and was the order one bound on ${\lVert ({\mathds{1}}-{\boldsymbol T}{\boldsymbol F})^{-1} \rVert_{2}}$. However, in the regime ${\tau}\in {\ensuremath{[0,1-{\tau}_*]}}$ and small $\eta$ such bound is not available since $({\mathds{1}}- {\boldsymbol T}{\boldsymbol F}){\boldsymbol f_-}= O(\eta)$ which can be deduced from below. The simple bound $$\label{eq:trivial_bound_inverse_1-TF_twonorm} {\lVert ({\mathds{1}}-{\boldsymbol T}{\boldsymbol F})^{-1} \rVert_{2}} \lesssim \eta^{-1}$$ which is a consequence of and ${\lVert {\boldsymbol T}\rVert_{2}}=1$ is not strong enough. In order to control ${\lVert ({\mathds{1}}- {\boldsymbol T}{\boldsymbol F})^{-1} {\boldsymbol V}{\ensuremath{\boldsymbol r}}\rVert_{2}}$ we will need to use a special property of the vector ${\boldsymbol V}{\ensuremath{\boldsymbol r}}$, namely that it is almost orthogonal to ${\boldsymbol f_-}$. This mechanism is formulated in the following *Contraction-Inversion Lemma* which is proved in Appendix \[proof:rotation\_inversion\]. It is closely related to the Rotation-Inversion lemmas – Lemma 5.8 in [@AjankiCPAM] and Lemma 3.6 in [@AltGram] – which control the invertibility of ${\mathds{1}}-UF$, where $U$ is a unitary operator and $F$ is symmetric. \[lem:rotation\_inversion\] Let ${\varepsilon}, \eta, c_1, c_2, c_3 >0$ satisfying $\eta \leq {\varepsilon}c_1/(2c_2^2)$ and ${\boldsymbol A}, {\boldsymbol B}\in {\mathbb{C}}^{2n\times 2n}$ be two Hermitian matrices such that $${\lVert {\boldsymbol A}\rVert_{2}} \leq 1, \quad {\lVert {\boldsymbol B}\rVert_{2}} \leq 1 -c_1 \eta. \label{TF}$$ Suppose that there are $\ell^2$-normalized vectors ${\boldsymbol b}_\pm \in {\mathbb{C}}^{2n}$ satisfying $$\label{eq:properties_F} {\boldsymbol B}{\boldsymbol b}_+ = {\lVert {\boldsymbol B}\rVert_{2}} {\boldsymbol b}_+, \quad {\boldsymbol B}{\boldsymbol b}_- = - {\lVert {\boldsymbol B}\rVert_{2}} {\boldsymbol b}_-, \quad {\lVert {\boldsymbol B}{\boldsymbol x}\rVert_{2}} \leq (1 - {\varepsilon}) {\lVert {\boldsymbol x}\rVert_{2}}$$ for all ${\boldsymbol x}\in {\mathbb{C}}^{2n}$ such that ${\boldsymbol x}\perp \operatorname{span}\{{\boldsymbol b}_+, {\boldsymbol b}_-\}$. Furthermore, assume that $$\label{eq:norm_Tf_plus_Tf_minus} {\langle{{\boldsymbol b}_+} \mspace{2mu}, {{\boldsymbol A}{\boldsymbol b}_+}\rangle} \leq 1-{\varepsilon}, \quad {\lVert ({\mathds{1}}+{\boldsymbol A}){\boldsymbol b}_- \rVert_{2}} \leq c_2 \eta.$$ Then there is a constant $C >0$, depending only on $c_1, c_2, c_3$ and ${\varepsilon}$, such that for each ${\boldsymbol p}\in {\mathbb{C}}^{2n}$ satisfying $$\label{eq:assumption_fminus_d} {\lvert {\langle{{\boldsymbol b}_-} \mspace{2mu}, {{\boldsymbol p}}\rangle} \rvert}\leq c_3 \eta {\lVert {\boldsymbol p}\rVert_{2}},$$ it holds true that $${\lVert ({\mathds{1}}-{\boldsymbol A}{\boldsymbol B})^{-1} {\boldsymbol p}\rVert_{2}} \leq C {\lVert {\boldsymbol p}\rVert_{2}}. \label{eq:estimate_stability_lemma}$$ We will apply this lemma with the choices ${\boldsymbol A}= {\boldsymbol T}$, ${\boldsymbol B}= {\boldsymbol F}$, ${\boldsymbol b}_\pm = \boldsymbol f_\pm$ and ${\boldsymbol p}= {\boldsymbol V}{\ensuremath{\boldsymbol r}}$. The resulting bound on ${\lVert ({\mathds{1}}- {\boldsymbol T}{\boldsymbol F})^{-1} {\boldsymbol V}{\ensuremath{\boldsymbol r}}\rVert_{2}}$ will be lifted to a bound on ${\lVert ({\mathds{1}}- {\boldsymbol T}{\boldsymbol F})^{-1} {\boldsymbol V}{\ensuremath{\boldsymbol r}}\rVert_{\infty}}$ by . All estimates in the remainder of this proof will hold true uniformly for ${\tau}\in {\ensuremath{[0,1-{\tau}_*]}}$. However, we will not stress this fact for each estimate. Moreover, the estimates will be uniform for $\eta \in (0,\eta_*]$. The threshold $\eta_*\leq 1$ will be chosen later such that it depends on ${\ensuremath{\mathcal P}}$ only and the assumptions of Lemma \[lem:rotation\_inversion\] are fulfilled. We now start checking the assumptions of Lemma \[lem:rotation\_inversion\]. By Proposition \[pro:estimates\_v\_small\_z\], there is $\Phi_1 \sim 1$ such that $$\label{eq:vf_bounded_above_below} \Phi_1^{-1} \leq {\boldsymbol v}\leq \Phi_1$$ for all $\eta \in(0,1]$. We recall from that there is a constant $c_1\sim 1$ such that ${\lVert {\boldsymbol F}\rVert_{2}} \leq 1 -c_1 \eta$ for all $\eta \in (0,1]$. Recalling the definition of ${\boldsymbol a}$ from , we conclude from the existence of $\Phi_2 \sim 1$ such that $$\label{eq:estimate_b_V_inverse_a} {\lVert {\boldsymbol f_-}- {\boldsymbol a}\rVert_{2}} \leq {\lVert {\boldsymbol f_-}-{\boldsymbol a}\rVert_{\infty}} \leq \Phi_2 \eta$$ for all $\eta \in (0,1]$. Here, we used that ${\lVert {\boldsymbol y}\rVert_{2}} \leq {\lVert {\boldsymbol y}\rVert_{\infty}}$ for all ${\boldsymbol y}\in {\mathbb{C}}^{2n}$ due to the normalization of the $\ell^2$ norm. Since the first and the second $n$-component of the vector ${\boldsymbol V}{\boldsymbol v}$ are the same we have ${\boldsymbol T}{\boldsymbol a}= - {\boldsymbol a}$ by and Lemma \[lem:pro\_Tf\] (iii). Hence, $$\label{eq:Tffm_close_to_minus_ffm} {\lVert {\boldsymbol f_-}+ {\boldsymbol T}{\boldsymbol f_-}\rVert_{2}} \leq {\lVert {\boldsymbol f_-}-{\boldsymbol a}\rVert_{2}} + {\lVert {\boldsymbol T}\rVert_{2}} {\lVert {\boldsymbol f_-}-{\boldsymbol a}\rVert_{2}}\leq 2 \Phi_2 \eta$$ by ${\lVert {\boldsymbol T}\rVert_{2}} = 1$ and . Due to , there exists ${\varepsilon}\sim 1$ such that $${\lVert {\boldsymbol F}{\boldsymbol x}\rVert_{2}} \leq (1- {\varepsilon}){\lVert {\boldsymbol x}\rVert_{2}}$$ for all ${\boldsymbol x}\in {\mathbb{C}}^{2n}$ such that ${\boldsymbol x}\perp {\boldsymbol f_+}$ and ${\boldsymbol x}\perp {\boldsymbol f_-}$ and for all $\eta \in (0,1]$. As ${\boldsymbol T}$ is Hermitian we can also assume by that $${\langle{{\boldsymbol f_+}} \mspace{2mu}, {{\boldsymbol T}{\boldsymbol f_+}}\rangle} \leq 1- {\varepsilon}$$ for all $\eta \in (0,1]$ by possibly reducing ${\varepsilon}$ but keeping ${\varepsilon}\gtrsim 1$. So far we checked the conditions –, it remains to verify with the choice ${\boldsymbol p}= {\boldsymbol V}{\ensuremath{\boldsymbol r}}$. Assuming that ${\langle{{\boldsymbol a}} \mspace{2mu}, {{\boldsymbol V}{\ensuremath{\boldsymbol r}}}\rangle}=0$, we deduce from that $$\label{eq:bound_scalar_b_V_rhs} {\lvert {\langle{{\boldsymbol f_-}} \mspace{2mu}, {{\boldsymbol V}{\ensuremath{\boldsymbol r}}}\rangle} \rvert} \leq {\lvert {\langle{{\boldsymbol a}} \mspace{2mu}, {{\boldsymbol V}{\ensuremath{\boldsymbol r}}}\rangle} \rvert} + {\lVert {\boldsymbol f_-}- {\boldsymbol a}\rVert}_2 {\lVert {\boldsymbol V}{\ensuremath{\boldsymbol r}}\rVert}_2 \leq \Phi_2 \eta {\lVert {\boldsymbol V}{\ensuremath{\boldsymbol r}}\rVert}_2.$$ This is the estimate required in . Hence, it suffices to show that ${\boldsymbol V}{\ensuremath{\boldsymbol r}}$ is perpendicular to ${\boldsymbol a}$, i.e., $$\label{eq:scalar_product_zero} {\langle{{\ensuremath{\boldsymbol{e}_-}}({\boldsymbol V}{\boldsymbol v})} \mspace{2mu}, {{\boldsymbol V}{\ensuremath{\boldsymbol r}}}\rangle} = {\left\langle{{\ensuremath{\boldsymbol{e}_-}}\left({\boldsymbol V}^2 {\boldsymbol v}\right)} \,\mspace{2mu},\, {{\boldsymbol L}{\boldsymbol h}}\right\rangle} = {\left\langle{{\boldsymbol L}^*\left({\ensuremath{\boldsymbol{e}_-}}\frac{{\widetilde}{\boldsymbol v}}{{\boldsymbol u}}\right)} \,\mspace{2mu},\, {{\boldsymbol h}}\right\rangle} = 0,$$ where we used the symmetry of ${\boldsymbol V}$, that ${\boldsymbol V}$ is diagonal and in the first equality, and the notation ${\widetilde}{\boldsymbol v}= (v_2 , v_1)$. We compute $$\label{eq:Lf_adjoint_Vf_af} {\boldsymbol L}^* \left( {\ensuremath{\boldsymbol{e}_-}}\frac{{\widetilde}{\boldsymbol v}}{{\boldsymbol u}}\right) = {\ensuremath{\boldsymbol{e}_-}}\frac{{\widetilde}{\boldsymbol v}}{{\boldsymbol u}} + {\boldsymbol S}_o \left( {\boldsymbol v}^2 {\ensuremath{\boldsymbol{e}_-}}\frac{{\widetilde}{\boldsymbol v}}{{\boldsymbol u}}\right) - {\tau}{\boldsymbol S}_d^t \left({\boldsymbol u}^2 {\ensuremath{\boldsymbol{e}_-}}\frac{{\widetilde}{\boldsymbol v}}{{\boldsymbol u}}\right) = \begin{pmatrix} \eta + Sv_2 - S \left(v_2 \left( \frac{v_1 v_2}{u} + {\tau}u\right)\right) \\ -\eta - S^tv_1 + S^t\left(v_1 \left(\frac{v_1 v_2}{u} + {\tau}u \right) \right) \end{pmatrix} = \eta {\ensuremath{\boldsymbol{e}_-}}.$$ Here, we used in the second step and the $n$-component relations of the second identity in in the last step. Since ${\langle {\ensuremath{\boldsymbol{e}_-}}{\boldsymbol g}\rangle} = {\langle {\ensuremath{\boldsymbol{e}_-}}{\boldsymbol v}\rangle} = 0$ by and , respectively, this proves and therefore as well. Thus, we checked all conditions of Lemma \[lem:rotation\_inversion\]. By possibly reducing $\eta_*$ but keeping $\eta_* \gtrsim 1$, we can assume that $\eta_* \leq {\varepsilon}c_1/(8\Phi_2^2)$. Now, we can apply Lemma \[lem:rotation\_inversion\] with ${\varepsilon}$, $c_1$, $c_2 = 2\Phi_2$, $c_3 = \Phi_2$ for any $\eta \in (0,\eta_*]$. Thus, applying in Lemma \[lem:rotation\_inversion\] to , we obtain ${\lVert {\boldsymbol V}{\boldsymbol h}\rVert_{2}} \lesssim {\lVert {\boldsymbol V}{\ensuremath{\boldsymbol r}}\rVert_{2}}$ and hence ${\lVert {\boldsymbol V}{\boldsymbol h}\rVert_{\infty}} \lesssim {\lVert {\boldsymbol V}{\ensuremath{\boldsymbol r}}\rVert_{\infty}}$ because of . Therefore, for any $\lambda_*>0$, depending only on ${\ensuremath{\mathcal P}}$, we have $${\lVert {\boldsymbol h}\rVert_{\infty}}{\ensuremath{\chi}}\big({\lVert {\boldsymbol h}\rVert_{\infty}}\leq \lambda_*\big) \lesssim {\lVert {\boldsymbol V}^{-1} \rVert_{\infty}} {\lVert {\boldsymbol V}{\ensuremath{\boldsymbol r}}\rVert_{\infty}} {\ensuremath{\chi}}\big({\lVert {\boldsymbol h}\rVert_{\infty}}\leq \lambda_*\big)\lesssim {\lVert {\boldsymbol h}\rVert_{\infty}}^2 + {\lVert {\boldsymbol d}\rVert_{\infty}}$$ uniformly for $\eta \in (0,\eta_*]$ and ${\tau}\in {\ensuremath{[0,1-{\tau}_*]}}$. Here, we used and in the second step. Choosing $\lambda_*>0$ small enough as before, we conclude for $\eta \in (0,\eta_*]$ and ${\tau}\in {\ensuremath{[0,1-{\tau}_*]}}$. Since $\eta_*>0$ depends only on ${\ensuremath{\mathcal P}}$, and $\eta_*$ was arbitrary in the proof of Step 2 we proved for all $\eta >0$ and ${\tau}\in {\ensuremath{[0,1-{\tau}_*]}}\cup {\ensuremath{[1+{\tau}_*,{\tau}^*]}}$. In order to prove , we remark that because of and the estimate holds true for $\eta \in (0,\eta_*]$ and ${\tau}\in {\ensuremath{[0,1-{\tau}_*]}}$ as well. Due to the instability of $({\mathds{1}}- {\boldsymbol T}{\boldsymbol F})^{-1}$ and, correspondingly, of its adjoint, the definition of ${\boldsymbol R}$ in will not yield an operator satisfying ${\lVert {\boldsymbol R}\rVert_{\infty}} \lesssim 1$ in this regime. Therefore, we again employ that the inverse of ${\mathds{1}}- {\boldsymbol T}{\boldsymbol F}$ is bounded on the subspace orthogonal to ${\boldsymbol f_-}$ and the blow-up in the direction of ${\boldsymbol f_-}$ is compensated by the smallness of ${\langle{{\boldsymbol f_-}} \mspace{2mu}, {{\boldsymbol V}{\ensuremath{\boldsymbol r}}}\rangle}$ following from ${\langle{{\boldsymbol a}} \mspace{2mu}, {{\boldsymbol V}{\ensuremath{\boldsymbol r}}}\rangle}=0$ and ${\lVert {\boldsymbol f_-}-{\boldsymbol a}\rVert_{\infty}} =O(\eta)$ by . Let ${\boldsymbol Q}$ be the orthogonal projection onto the subspace ${\boldsymbol f_-}^\perp$, i.e., ${\boldsymbol Q}{\boldsymbol x}{\mathrel{\vcenter{\baselineskip0.5ex \lineskiplimit0pt\hbox{\scriptsize.}\hbox{\scriptsize.}}}=}{\boldsymbol x}- {\langle{{\boldsymbol f_-}} \mspace{2mu}, {{\boldsymbol x}}\rangle} {\boldsymbol f_-}$ for all ${\boldsymbol x}\in {\mathbb{C}}^{2n}$. Recalling the definition of ${\boldsymbol a}$ in , we now define the operator ${\boldsymbol R}={\boldsymbol R}(\eta)$ on ${\mathbb{C}}^{2n}$ as follows: $$\label{eq:def_Rf_2} {\boldsymbol R}{\boldsymbol x}{\mathrel{\vcenter{\baselineskip0.5ex \lineskiplimit0pt\hbox{\scriptsize.}\hbox{\scriptsize.}}}=}{\boldsymbol V}\left(({\mathds{1}}-{\boldsymbol T}{\boldsymbol F})^{-1} {\boldsymbol Q}\right)^*{\boldsymbol V}^{-1} {\boldsymbol x}- {\langle{{\boldsymbol V}^{-1} ({\mathds{1}}-{\boldsymbol T}{\boldsymbol F})^{-1} {\boldsymbol f_-}} \mspace{2mu}, {{\boldsymbol x}}\rangle}{\boldsymbol V}({\boldsymbol f_-}-{\boldsymbol a})$$ for every ${\boldsymbol x}\in {\mathbb{C}}^{2n}$. Note that this ${\boldsymbol R}$ is different from the one given in that is used in the other parameter regimes. Now, we estimate ${\lVert {\boldsymbol R}{\boldsymbol x}\rVert_{\infty}}$. For the first term, we use the bound whose assumptions we check first. The first condition, ${\lVert ({\mathds{1}}-{\boldsymbol T}{\boldsymbol F})^{-1} {\boldsymbol Q}\rVert_{2}} \lesssim 1$, in follows from as with ${\boldsymbol p}= {\boldsymbol Q}{\boldsymbol x}$ is trivially satisfied and hence ${\lVert ({\mathds{1}}-{\boldsymbol T}{\boldsymbol F})^{-1}{\boldsymbol Q}{\boldsymbol x}\rVert_{2}} \lesssim {\lVert {\boldsymbol Q}{\boldsymbol x}\rVert_{2}} \lesssim {\lVert {\boldsymbol x}\rVert_{2}}$. The second condition in is met by and the third condition is exactly . Using ${\lVert {\boldsymbol f_-}\rVert_{\infty}} \lesssim 1$ from , and , we conclude that the first term in is $\lesssim {\lVert {\boldsymbol x}\rVert_{\infty}}$. In the second term, we use the trivial bound $$\label{eq:trivial_bound_inverse_1-TF} {\left\lVert ({\mathds{1}}-{\boldsymbol T}{\boldsymbol F})^{-1} \right\rVert_{\infty}} \lesssim \eta^{-1}$$ which is a consequence of the corresponding bound on ${\lVert (1-{\boldsymbol T}{\boldsymbol F})^{-1} \rVert_{2}}$ in and . The potential blow-up in for small $\eta$ is compensated by the estimate ${\lVert {\boldsymbol f_-}- {\boldsymbol a}\rVert_{\infty}}= O(\eta)$ from . Altogether this yields $ {\lVert {\boldsymbol R}(\eta) \rVert}_\infty \lesssim 1$ for all $\eta \in (0,\eta_*]$. From the definition of ${\boldsymbol R}$, we obtain $$\begin{aligned} \label{yh} {\langle{{\boldsymbol y}} \mspace{2mu}, {{\boldsymbol h}}\rangle} & = {\langle{{\boldsymbol y}} \mspace{2mu}, {{\boldsymbol V}^{-1} ({\mathds{1}}-{\boldsymbol T}{\boldsymbol F})^{-1} {\boldsymbol V}{\ensuremath{\boldsymbol r}}}\rangle} \\ \nonumber & = {\left\langle{{\boldsymbol V}^{-1}{\boldsymbol y}} \,\mspace{2mu},\, {({\mathds{1}}-{\boldsymbol T}{\boldsymbol F})^{-1} {\boldsymbol Q}{\boldsymbol V}({\ensuremath{\boldsymbol r}}-{\boldsymbol d})}\right\rangle}+ {\left\langle{{\boldsymbol y}} \,\mspace{2mu},\, {{\boldsymbol V}^{-1}({\mathds{1}}-{\boldsymbol T}{\boldsymbol F})^{-1} {\boldsymbol f_-}}\right\rangle} {\left\langle{{\boldsymbol f_-}-{\boldsymbol a}} \,\mspace{2mu},\, {{\boldsymbol V}({\ensuremath{\boldsymbol r}}-{\boldsymbol d})}\right\rangle} + {\langle{{\boldsymbol R}{\boldsymbol y}} \mspace{2mu}, {{\boldsymbol d}}\rangle}. \end{aligned}$$ Notice that we first inserted ${\mathds{1}}={\boldsymbol Q}+ |{\boldsymbol f_-}\rangle \langle {\boldsymbol f_-}|$ before ${\boldsymbol V}{\ensuremath{\boldsymbol r}}$, then we inserted the vector ${\boldsymbol a}$ in the second term for free by using ${\langle{{\boldsymbol a}} \mspace{2mu}, {{\boldsymbol V}{\ensuremath{\boldsymbol r}}}\rangle} = 0$ from . This brought in the factor ${\boldsymbol f_-}- {\boldsymbol a}\sim O(\eta)$ that compensates the $({\mathds{1}}- {\boldsymbol T}{\boldsymbol F})^{-1}$ on the unstable subspace parallel to ${\boldsymbol f_-}$. Finally, we subtracted the term ${\boldsymbol d}$ to ${\ensuremath{\boldsymbol r}}$ freely and we defined the operator ${\boldsymbol R}$ exactly to compensate for it. The reason for this counter term ${\boldsymbol d}$ is the formula showing that ${\ensuremath{\boldsymbol r}}-{\boldsymbol d}$ is one order better in ${\boldsymbol d}$ than ${\ensuremath{\boldsymbol r}}$. Thus, the first two terms in the right-hand side of are bounded by ${\lVert {\boldsymbol d}\rVert_{\infty}}^2{\lVert {\boldsymbol y}\rVert_{\infty}}$. The compensating term, ${\langle{{\boldsymbol R}{\boldsymbol y}} \mspace{2mu}, {{\boldsymbol d}}\rangle}$ remains first order in ${\boldsymbol d}$ but only in weak sense, tested against the vector ${\boldsymbol R}{\boldsymbol y}$, and not in norm sense. This is the essential improvement of over . Recalling now ${\boldsymbol h}= {\boldsymbol g}- {\mathrm{i}}{\boldsymbol v}$, the identity together with the bounds we just explained concludes the proof of Proposition \[pro:stab\_lemma\]. Proof of Proposition \[pro:properties\_sigma\] ============================================== As in the previous section, we assume without loss of generality that $\rho(S) =1$. See the remark about . For ${\tau}_* >0$ and ${\tau}^* > {\tau}_* +1$, we define $$\label{eq:def_Dsma_Dbig} {\ensuremath{\mathbb D_{<}}}{\mathrel{\vcenter{\baselineskip0.5ex \lineskiplimit0pt\hbox{\scriptsize.}\hbox{\scriptsize.}}}=}\{ z \in {\mathbb{C}}\;\mid\; {\lvert z \rvert}^2 \leq 1 - {\tau}_* \}, \quad {\ensuremath{\mathbb D_{>}}}{\mathrel{\vcenter{\baselineskip0.5ex \lineskiplimit0pt\hbox{\scriptsize.}\hbox{\scriptsize.}}}=}\{ z \in {\mathbb{C}}\;\mid\; 1 + {\tau}_* \leq {\lvert z \rvert}^2 \leq {\tau}^* \}.$$ Via ${\tau}= {\lvert z \rvert}^2$ these sets correspond to the regimes $[0,1-{\tau}_*]$ and $[1+{\tau}_*,{\tau}^*]$ in the previous section. Since the defining equations in are smooth functions of $\eta$, ${\tau}$ and $({\boldsymbol v}_i)_{i=1,\ldots, 2n}$ and the operator ${\boldsymbol L}$ is invertible for $\eta>0$ the implicit function theorem implies that the function ${\boldsymbol v}\colon {\mathbb{R}}_+\times {\ensuremath{{\mathbb{R}}^{+}_{0}}}\to {\mathbb{R}}_+^{2n}$ is smooth. Therefore, the function ${\mathbb{R}}_+ \times {\mathbb{C}}\to {\mathbb{R}}_+^{2n}, ~~(\eta, z) \mapsto{\boldsymbol v}^{\tau}(\eta)|_{{\tau}= {\lvert z \rvert}^2}$ is also smooth. For $\alpha=(\alpha_1, \alpha_2) \in {\mathbb{N}}^2$, we define $${\partial}^\alpha {\boldsymbol v}{\mathrel{\vcenter{\baselineskip0.5ex \lineskiplimit0pt\hbox{\scriptsize.}\hbox{\scriptsize.}}}=}{\partial}_\eta^{\alpha_1} {\partial}_{\tau}^{\alpha_2} {\boldsymbol v}.$$ For fixed ${\tau}_*>0$ and ${\tau}^*>{\tau}_* +1$, we first prove that for all $\alpha \in {\mathbb{N}}^2$, we have $$\label{eq:vf_bounded_derivates} {\lVert {\partial}^\alpha {\boldsymbol v}\rVert_{\infty}} \lesssim 1$$ uniformly for all $\eta >0$ and ${\tau}\in {\ensuremath{[0,1-{\tau}_*]}}\cup {\ensuremath{[1+{\tau}_*,{\tau}^*]}}$. Differentiating with respect to $\eta$ and ${\tau}$, respectively, yields $$\label{eq:v_eta_and_r_derivative} {\boldsymbol L}({\partial}_\eta {\boldsymbol v}) = - {\boldsymbol v}^2 + {\tau}{\boldsymbol u}^2, \quad {\boldsymbol L}({\partial}_{\tau}{\boldsymbol v}) = - {\boldsymbol u}{\boldsymbol v}.$$ By further differentiating with respect to $\eta$ and ${\tau}$, we iteratively obtain that for any multiindex $\alpha \in {\mathbb{N}}^2$ $$\label{eq:pt_alpha_vf_rf_alpha} {\boldsymbol L}{\partial}^\alpha {\boldsymbol v}= {\ensuremath{\boldsymbol r}}_\alpha,$$ where ${\ensuremath{\boldsymbol r}}_\alpha$ only depends on $\eta$, ${\tau}$ and ${\partial}^\beta {\boldsymbol v}$ for $\beta \in {\mathbb{N}}^2$, ${\lvert \beta \rvert} = \beta_1 + \beta_2 < {\lvert \alpha \rvert}$. In fact, for all $\alpha \in {\mathbb{N}}^2$, we have \[eq:induction\_step\_derivatives\] $$\begin{aligned} {\boldsymbol L}({\partial}^{\alpha + e_1} {\boldsymbol v}) & = {\partial}^\alpha \left(-{\boldsymbol v}^2 + {\tau}{\boldsymbol u}^2\right) - \sum_{\nu\leq \alpha, \nu\neq (0,0)} \begin{pmatrix} \alpha \\ \nu \end{pmatrix} \left({\partial}^{\nu}{\boldsymbol L}\right) \left( {\partial}^{\alpha-\nu + e_1} {\boldsymbol v}\right), \label{eq:induction_step_derivatives1}\\ {\boldsymbol L}({\partial}^{\alpha + e_2} {\boldsymbol v}) & = {\partial}^\alpha \left(-{\boldsymbol v}{\boldsymbol u}\right) - \sum_{\nu\leq \alpha, \nu \neq (0,0)} \begin{pmatrix} \alpha \\ \nu \end{pmatrix} \left({\partial}^{\nu}{\boldsymbol L}\right) \left( {\partial}^{\alpha-\nu + e_2} {\boldsymbol v}\right). \label{eq:induction_step_derivatives2} \end{aligned}$$ As an example, we compute $$\begin{aligned} {\boldsymbol L}{\partial}^2_{\tau}{\boldsymbol v}& = - 2 {\boldsymbol u}{\partial}_{\tau}{\boldsymbol v}+ 2 {\boldsymbol u}^2 {\boldsymbol S}_d {\partial}_{\tau}{\boldsymbol v}-2 {\boldsymbol v}{\partial}_{\tau}{\boldsymbol v}{\boldsymbol S}_o {\partial}_{\tau}{\boldsymbol v}+ \frac{2 {\tau}{\boldsymbol u}^2}{{\boldsymbol v}} {\partial}_{\tau}{\boldsymbol v}{\boldsymbol S}_d {\partial}_{\tau}{\boldsymbol v}-\frac{2{\tau}{\boldsymbol u}^3}{{\boldsymbol v}} \left({\boldsymbol S}_d{\boldsymbol v}\right)^2 \nonumber \\ & = \frac{2}{{\boldsymbol v}} \left( {\partial}_{\tau}{\boldsymbol v}\right)^2 +2{\boldsymbol u}^2 {\boldsymbol S}_d {\partial}_{\tau}{\boldsymbol v}- \frac{2{\tau}{\boldsymbol u}^3}{{\boldsymbol v}} \left({\boldsymbol S}_d {\partial}_{\tau}{\boldsymbol v}\right)^2 ,\label{eq:pt_r^2_vf}\end{aligned}$$ where we used the second relation in in the second step. By induction on ${\lvert \alpha \rvert} = \alpha_1 + \alpha_2$, we prove ${\lVert {\ensuremath{\boldsymbol r}}_\alpha \rVert_{\infty}} \lesssim 1$ and ${\lVert {\partial}^\alpha {\boldsymbol v}\rVert_{\infty}} \lesssim 1$ simultaneously. From , we conclude that ${\ensuremath{\boldsymbol r}}_{\alpha+e_1}$ and ${\ensuremath{\boldsymbol r}}_{\alpha+e_2}$ are bounded in $\ell^\infty$-norm if ${\lVert {\partial}^\nu {\boldsymbol v}\rVert_{\infty}} \lesssim 1$ for all $\nu \leq \alpha$ as the first term on the right-hand side of and , respectively, and ${\partial}^\nu {\boldsymbol L}$ for all $\nu \leq \alpha$ are bounded. In order to conclude that ${\partial}^{\alpha + e_1} {\boldsymbol v}$ and ${\partial}^{\alpha + e_2} {\boldsymbol v}$ are bounded it suffices to prove that ${\lVert {\partial}^\alpha {\boldsymbol v}\rVert_{\infty}} \lesssim {\lVert {\ensuremath{\boldsymbol r}}_\alpha \rVert_{\infty}}$ by controlling ${\boldsymbol L}^{-1}$ in . As in the proof of Proposition \[pro:stab\_lemma\] the norm of ${\boldsymbol L}^{-1}$ is bounded, ${\lVert {\boldsymbol L}^{-1} \rVert_{\infty}} \lesssim 1$, for ${\tau}\in {\ensuremath{[1+{\tau}_*,{\tau}^*]}}$ or ${\tau}\in {\ensuremath{[0,1-{\tau}_*]}}$ and large $\eta$ as well as ${\tau}\in {\ensuremath{[0,1-{\tau}_*]}}$ and small $\eta$ separately. We thus focus on the most interesting regime where ${\tau}\in {\ensuremath{[0,1-{\tau}_*]}}$ and small $\eta$. As for the proof of Proposition \[pro:stab\_lemma\] we apply Lemma \[lem:rotation\_inversion\] in this regime. We only check the condition here since the others are established in the same way as in the proof of Proposition \[pro:stab\_lemma\]. Recall the definition of ${\boldsymbol a}$ in . Using ${\langle {\ensuremath{\boldsymbol{e}_-}}{\partial}^\alpha{\boldsymbol v}\rangle} = 0$ from for all $\alpha \in {\mathbb{N}}^2$, we obtain $${\left\langle{{\boldsymbol a}} \,\mspace{2mu},\, {{\boldsymbol V}{\ensuremath{\boldsymbol r}}_\alpha}\right\rangle} = {\left\langle{{\boldsymbol L}^* ( {\ensuremath{\boldsymbol{e}_-}}{\boldsymbol V}^2 {\boldsymbol v})} \,\mspace{2mu},\, {{\partial}^\alpha{\boldsymbol v}}\right\rangle} = {\langle{\eta{\ensuremath{\boldsymbol{e}_-}}} \mspace{2mu}, {{\partial}^\alpha {\boldsymbol v}}\rangle}= 0$$ for all $\alpha \in {\mathbb{N}}^2$. Here, we used ${\boldsymbol L}^*( {\ensuremath{\boldsymbol{e}_-}}{\boldsymbol V}^2 {\boldsymbol v}) = \eta {\ensuremath{\boldsymbol{e}_-}}$ which is shown in in the proof of Proposition \[pro:stab\_lemma\]. This concludes the proof of . Next, we show the integrability of $\Delta_z {\langle v_1^{\tau}|_{{\tau}= {\lvert z \rvert}^2} \rangle}$ as a function of $\eta$ for $z \in {\ensuremath{\mathbb D_{<}}}$ for fixed ${\tau}_*>0$. Note that ${\langle v_1^{\tau}\rangle} = {\langle {\boldsymbol v}^{\tau}\rangle}$ by . Using $$\Delta_z \left( {\boldsymbol v}^{\tau}|_{{\tau}={\lvert z \rvert}^2} \right) = 4 \left( {\tau}{\partial}_{\tau}^2 {\boldsymbol v}^{\tau}+ {\partial}_{\tau}{\boldsymbol v}^{\tau}\right)|_{{\tau}={\lvert z \rvert}^2}$$ together with and , we obtain $$\label{eq:Lf_Laplace_vf} {\boldsymbol L}\Delta_z \left( {\boldsymbol v}^{\tau}|_{{\tau}={\lvert z \rvert}^2} \right) = 4\left( \frac{2{\tau}}{{\boldsymbol v}} \left({\partial}_{\tau}{\boldsymbol v}\right)^2 + 2 {\tau}{\boldsymbol u}^2 {\boldsymbol S}_d {\partial}_{\tau}{\boldsymbol v}- \frac{2{\tau}^2 {\boldsymbol u}^3}{{\boldsymbol v}}\left({\boldsymbol S}_d {\partial}_{\tau}{\boldsymbol v}\right)^2 - {\boldsymbol u}{\boldsymbol v}\right).$$ From , and , we conclude that ${\boldsymbol u}{\boldsymbol v}\sim (1 +\eta^{3})^{-1}$ and hence $ {\lvert {\partial}_{\tau}{\boldsymbol v}\rvert} \lesssim (1 +\eta^{3})^{-1}$ uniformly for $z \in {\ensuremath{\mathbb D_{<}}}$ since ${\lVert {\partial}^\alpha {\boldsymbol v}\rVert_{\infty}} \lesssim {\lVert {\ensuremath{\boldsymbol r}}_\alpha \rVert_{\infty}}$. Therefore, the right-hand side of is of order $(1+\eta^3)^{-1}$ for $z \in {\ensuremath{\mathbb D_{<}}}$ and hence using the control on ${\boldsymbol L}^{-1}$ as before, we conclude that ${\lvert \Delta_z \left( {\boldsymbol v}^{\tau}|_{{\tau}={\lvert z \rvert}^2} \right) \rvert} \lesssim (1 + \eta^3)^{-1}$ uniformly for $\eta >0$. Thus, $\Delta_z {\langle v_1^{\tau}|_{{\tau}= {\lvert z \rvert}^2} \rangle} = \Delta_z {\langle {\boldsymbol v}^{\tau}|_{{\tau}= {\lvert z \rvert}^2} \rangle}$ as a function of $\eta$ is integrable on ${\mathbb{R}}_+$ and the integral is a continuous function of $z \in {\ensuremath{\mathbb D_{<}}}$. As ${\tau}_*>0$ was arbitrary, this concludes the proof of part (i) of Proposition \[pro:properties\_sigma\] and shows that $\sigma$ is a rotationally invariant function on ${\mathbb{C}}$ which is continuous on $D(0,1)$. Now, we establish that for ${\tau}<1$, the derivative of the average of ${\boldsymbol u}$ with respect to ${\tau}$ gives an alternative representation of the density of states as follows $$\label{eq:sigma_in_terms_of_vf_0} \sigma(z) = \frac{1}{\pi} {\partial}_{\tau}\left({\tau}{\langle {\boldsymbol u}_0 \rangle}\right)\big\rvert_{{\tau}={\lvert z \rvert}^2} = - \frac{2}{\pi} {\langle{{\boldsymbol S}_o {\boldsymbol v}_0} \mspace{2mu}, {{\partial}_{\tau}{\boldsymbol v}_0}\rangle}\big\rvert_{{\tau}={\lvert z \rvert}^2},$$ where ${\boldsymbol u}_0 {\mathrel{\vcenter{\baselineskip0.5ex \lineskiplimit0pt\hbox{\scriptsize.}\hbox{\scriptsize.}}}=}\lim_{\eta \downarrow 0} {\boldsymbol u}(\eta)$ and ${\boldsymbol v}_0 {\mathrel{\vcenter{\baselineskip0.5ex \lineskiplimit0pt\hbox{\scriptsize.}\hbox{\scriptsize.}}}=}\lim_{\eta \downarrow 0} {\boldsymbol v}(\eta)$. The first relation in will be proved below and the second one follows immediately using ${\tau}{\boldsymbol u}_0 = 1 - {\boldsymbol v}_0 {\boldsymbol S}_o {\boldsymbol v}_0 $ by and for $\eta\downarrow 0$, as well as ${\boldsymbol S}_o^t = {\boldsymbol S}_o$. We first give a heuristic derivation of the first equality in . Writing the resolvent ${\boldsymbol G}^z$ of ${\boldsymbol H}^z$ as $${\boldsymbol G}^z = \begin{pmatrix} G_{11} & G_{12} \\ G_{21} & G_{22} \end{pmatrix},$$ we obtain $$\operatorname{Tr}G_{12} = \operatorname{Tr}\left[ \left((X-z)(X^*-\bar z) + \eta^2\right)^{-1} (X-z)\right] = -{\partial}_{\bar z} \operatorname{Tr}\log\left( (X-z)(X^*-\bar z) + \eta^2\right) = -\frac{2}{n}{\partial}_{\bar z } \log{\lvert \det({\boldsymbol H}^z-{\mathrm{i}}\eta) \rvert}$$ for the normalized trace of $G_{12}$ (see ). Since $\Delta_z= 4 {\partial}_z {\partial}_{\bar z}$, taking the ${\partial}_z$-derivative of the previous identity, we obtain $$\label{eq:pro_2_3_aux_1} \frac{1}{2n} \Delta_z \log{\lvert \det({\boldsymbol H}^z-{\mathrm{i}}\eta) \rvert} = - {\partial}_z \operatorname{Tr}G_{12}.$$ Using , and ${\mathrm{Im}\,}m^z \approx {\langle v_1^{\tau}|_{{\tau}={\lvert z \rvert}^2} \rangle}$, the left-hand side of is approximately $\pi\sigma(z)$ after taking the $\eta\downarrow 0$ limit. On the other hand, ${\boldsymbol G}^z$ converges to ${\boldsymbol M}^z$ for $n\to \infty$. Thus, by the right-hand side of can be approximated by ${\partial}_z\left( z {\langle u^{\tau}|_{{\tau}={\lvert z \rvert}^2}(\eta) \rangle}\right)$. Therefore, taking $\eta \downarrow 0$, we conclude $$\pi \sigma(z) \approx {\partial}_z z {\langle u_0^{\tau}|_{{\tau}={\lvert z \rvert}^2} \rangle} = \left({\partial}_{\tau}{\tau}{\langle u_0^{\tau}\rangle}\right)|_{{\tau}= {\lvert z \rvert}^2}.$$ In fact, this approximation holds not only in the $n\to\infty$ limit but it is an identity for any fixed $n$. This completes the heuristic argument for . We now turn to the rigorous proof of the first relation in . In fact, for ${\tau}<1$, we prove the following integrated version $$\label{eq:relation_integral_sigma} \int_{{\lvert z' \rvert}^2 \leq {\tau}} \sigma(z') {\mathrm{d}}^2 z' = {\tau}{\langle {\boldsymbol u}_0^{\tau}\rangle}.$$ Since $\sigma$ is a continuous function on $D(0,1)$ differentiating with respect to ${\tau}$ immediately yields . In order to justify the existence of the limits of ${\boldsymbol v}$ and ${\boldsymbol u}$ for $\eta\downarrow 0$ and the computations in the proof of , we remark that by , $(\eta, z) \mapsto {\boldsymbol v}^{\tau}(\eta)|_{{\tau}={\lvert z \rvert}^2}$ can be uniquely extended to a positive $C^\infty$ function on $[0,\infty) \times D(0,1)$. In the following, ${\boldsymbol v}$ and ${\boldsymbol v}_0^{\tau}{\mathrel{\vcenter{\baselineskip0.5ex \lineskiplimit0pt\hbox{\scriptsize.}\hbox{\scriptsize.}}}=}{\boldsymbol v}^{{\tau}}|_{\eta = 0}$ denote this function and its restriction to $\{0\} \times [0,1)$, respectively. In particular, the restriction ${\boldsymbol v}_0^{\tau}|_{{\tau}= {\lvert z \rvert}^2}$ is a smooth function on $D(0,1)$ which satisfies $$\label{eq:v_eta=0} \frac{1}{{\boldsymbol v}_0^{\tau}} = {\boldsymbol S}_o {\boldsymbol v}_0^{\tau}+ \frac{{\tau}}{{\boldsymbol S}_d {\boldsymbol v}_0^{\tau}}$$ with ${\tau}={\lvert z \rvert}^2$. Moreover, derivatives of ${\boldsymbol v}$ in $\eta$ and ${\tau}$ and limits in $\eta$ and ${\tau}$ for ${\tau}< 1$ can be freely interchanged. For the proof of , we use integration by parts to obtain $$\label{eq:relation_integral_sigma1} \int_{{\lvert z' \rvert}^2 \leq {\tau}} \sigma(z') {\mathrm{d}}^2 z' = - 2{\tau}\int_0^\infty {\partial}_{\tau}{\langle {\boldsymbol v}\rangle} {\mathrm{d}}\eta = - {\tau}\int_0^\infty {\partial}_{\tau}\left( {\langle {\boldsymbol v}\rangle} + {\langle {\widetilde}{\boldsymbol v}\rangle}\right) {\mathrm{d}}\eta.$$ We recall ${\widetilde}{\boldsymbol v}= (v_2, v_1)$ and get $${\boldsymbol v}= \frac{\eta +{\boldsymbol S}_d {\boldsymbol v}}{(\eta + {\boldsymbol S}_d{\boldsymbol v})(\eta + {\boldsymbol S}_o{\boldsymbol v}) + {\tau}}, \quad {\widetilde}{\boldsymbol v}= \frac{\eta +{\boldsymbol S}_o {\boldsymbol v}}{(\eta + {\boldsymbol S}_d{\boldsymbol v})(\eta + {\boldsymbol S}_o{\boldsymbol v}) + {\tau}}$$ from . This implies the identity $${\partial}_\eta \log\left( (\eta + {\boldsymbol S}_d {\boldsymbol v})(\eta + {\boldsymbol S}_o {\boldsymbol v}) + {\tau}\right) = {\boldsymbol v}+ {\widetilde}{\boldsymbol v}+ {\widetilde}{\boldsymbol v}{\boldsymbol S}_d {\partial}_\eta{\boldsymbol v}+ {\boldsymbol v}{\boldsymbol S}_o {\partial}_\eta {\boldsymbol v}.$$ Using $${\langle {\widetilde}{\boldsymbol v}{\boldsymbol S}_d {\partial}_\eta {\boldsymbol v}\rangle} + {\langle {\boldsymbol v}{\boldsymbol S}_o{\partial}_\eta {\boldsymbol v}\rangle} = {\langle {\boldsymbol v}{\boldsymbol S}_o {\partial}_\eta {\boldsymbol v}\rangle} + {\langle {\boldsymbol v}{\boldsymbol S}_o{\partial}_\eta {\boldsymbol v}\rangle} = {\partial}_\eta {\langle {\boldsymbol v}{\boldsymbol S}_o {\boldsymbol v}\rangle}$$ and recalling ${\boldsymbol v}_0 {\mathrel{\vcenter{\baselineskip0.5ex \lineskiplimit0pt\hbox{\scriptsize.}\hbox{\scriptsize.}}}=}\lim_{\eta\downarrow 0} {\boldsymbol v}(\eta)$, we find for the expression $$\label{eq:relation_integral_sigma2} \int_0^\infty {\partial}_{\tau}\left( {\langle {\boldsymbol v}\rangle} + {\langle {\widetilde}{\boldsymbol v}\rangle} \right) {\mathrm{d}}\eta = - {\langle {\partial}_{\tau}\log\left( ({\boldsymbol S}_d {\boldsymbol v}_0)( {\boldsymbol S}_o {\boldsymbol v}_0) + {\tau}\right) \rangle} + {\partial}_{\tau}{\langle {\boldsymbol v}_0 {\boldsymbol S}_o {\boldsymbol v}_0 \rangle}.$$ Hence, due to $${\langle {\partial}_{\tau}\log\left( ({\boldsymbol S}_d {\boldsymbol v}_0)( {\boldsymbol S}_o {\boldsymbol v}_0) + {\tau}\right) \rangle} = {\langle {\boldsymbol u}\rangle} + {\langle {\widetilde}{\boldsymbol v}_0 {\boldsymbol S}_d {\partial}_{\tau}{\boldsymbol v}_0 \rangle} + {\langle {\boldsymbol v}{\boldsymbol S}_o {\partial}_{\tau}{\boldsymbol v}_0 \rangle} = {\langle {\boldsymbol u}\rangle} + {\partial}_{\tau}{\langle {\boldsymbol v}_0 {\boldsymbol S}_o {\boldsymbol v}_0 \rangle}.$$ we obtain from . The formula was also obtained in [@CookNonHermitianRM] with a different method. We prove (iii) before (ii). As ${\boldsymbol v}_0$ is infinitely often differentiable in ${\tau}$ and ${\tau}={\lvert z \rvert}^2$, we conclude from that $\sigma$ is infinitely often differentiable in $z$. The following lemma shows which finishes the proof of part (iii). \[lem:sigma\_strictly\_positive\] Uniformly for $z \in D(0,1)$, we have $$\label{eq:sigma_sim_1} \sigma(z) \sim 1,$$ where $\sim$ only depends on $s_*$ and $s^*$. We will compute the derivative in and prove the estimate first for $z \in {\ensuremath{\mathbb D_{<}}}$ and arbitrary ${\tau}_*>0$ depending only on $s_*$ and $s^*$. Then we show that there is ${\tau}_*>0$ depending only on $s_*$ and $s^*$ such that holds true for $z \in D(0,1)\setminus {\ensuremath{\mathbb D_{<}}}$. In this proof, we write ${\mathscr D}(y) {\mathrel{\vcenter{\baselineskip0.5ex \lineskiplimit0pt\hbox{\scriptsize.}\hbox{\scriptsize.}}}=}\operatorname{diag}(y)$ for $y \in {\mathbb{C}}^l$ for brevity. Furthermore, we introduce the $2n \times 2n$ matrix $${\boldsymbol E}{\mathrel{\vcenter{\baselineskip0.5ex \lineskiplimit0pt\hbox{\scriptsize.}\hbox{\scriptsize.}}}=}\begin{pmatrix} {\mathds{1}}& {\mathds{1}}\\ {\mathds{1}}& {\mathds{1}}\end{pmatrix}.$$ In the following, ${\boldsymbol v}$ and all related quantites will be evaluated at ${\tau}= {\lvert z \rvert}^2$. We start the proof from , recall ${\boldsymbol L}= {\boldsymbol V}^{-1} ( {\mathds{1}}- {\boldsymbol T}{\boldsymbol F}) {\boldsymbol V}$ and use the second relation in as well as to obtain $$\begin{aligned} \sigma(z) = & -\frac{2}{\pi}{\langle{{\boldsymbol S}_o {\boldsymbol v}_0} \mspace{2mu}, {{\partial}_{\tau}{\boldsymbol v}_0}\rangle} = \lim_{\eta\downarrow 0} \frac{2}{\pi}{\left\langle{{\boldsymbol V}^{-1}\frac{{\widetilde}{\boldsymbol v}}{{\boldsymbol u}}} \,\mspace{2mu},\, { ( {\mathds{1}}- {\boldsymbol T}{\boldsymbol F})^{-1} {\boldsymbol V}({\boldsymbol v}{\boldsymbol u}) }\right\rangle} = \lim_{\eta\downarrow 0} \frac{2}{\pi} {\left\langle{\sqrt{{\boldsymbol v}{\widetilde}{\boldsymbol v}}} \,\mspace{2mu},\, { \frac{1}{\sqrt{{\boldsymbol u}}} ( {\mathds{1}}- {\boldsymbol T}{\boldsymbol F})^{-1} \sqrt{{\boldsymbol u}} \sqrt{{\boldsymbol v}{\widetilde}{\boldsymbol v}}}\right\rangle} \nonumber \\ & = \lim_{\eta\downarrow 0} \frac{2}{\pi} {\left\langle{\sqrt{{\boldsymbol v}{\widetilde}{\boldsymbol v}}} \,\mspace{2mu},\, { \left ( {\mathds{1}}- {\mathscr D}({\boldsymbol u}^{-1/2}) {\boldsymbol T}{\boldsymbol F}{\mathscr D}({\boldsymbol u}^{1/2}) \right)^{-1}\sqrt{{\boldsymbol v}{\widetilde}{\boldsymbol v}}}\right\rangle}. \label{eq:pos_sigma_aux2}\end{aligned}$$ Note that the inverses of ${\mathds{1}}- {\boldsymbol T}{\boldsymbol F}$ and ${\mathds{1}}- {\tau}{\mathscr D}({\boldsymbol u}^{-1/2}){\boldsymbol T}{\boldsymbol F}{\mathscr D}({\boldsymbol u}^{1/2})$ exist by Lemma \[lem:pro\_Tf\] and Lemma \[lem:prop\_Ff\] as $\eta >0$ and ${\tau}<1$. Due to and , we have ${\boldsymbol T}= -{\mathds{1}}+ {\tau}{\boldsymbol u}{\boldsymbol E}$ which implies $$\begin{aligned} {\mathds{1}}- {\mathscr D}({\boldsymbol u}^{-1/2}){\boldsymbol T}{\boldsymbol F}{\mathscr D}({\boldsymbol u}^{1/2}) & = {\mathds{1}}+ {\mathscr D}({\boldsymbol u}^{-1/2}){\boldsymbol F}{\mathscr D}({\boldsymbol u}^{1/2}) - {\tau}{\mathscr D}({\boldsymbol u}^{1/2}) {\boldsymbol E}{\boldsymbol F}{\mathscr D}({\boldsymbol u}^{1/2}) \nonumber\\ & = \left( {\mathds{1}}- {\tau}{\mathscr D}({\boldsymbol u}^{1/2}){\boldsymbol E}{\boldsymbol F}({\mathds{1}}+ {\boldsymbol F})^{-1}{\mathscr D}({\boldsymbol u}^{1/2}) \right) \left({\mathds{1}}+ {\mathscr D}({\boldsymbol u}^{-1/2}){\boldsymbol F}{\mathscr D}({\boldsymbol u}^{1/2}) \right). \label{eq:pos_sigma_aux8}\end{aligned}$$ From and , we deduce $ \sqrt{{\boldsymbol u}} {\boldsymbol F}\sqrt{{\boldsymbol v}{\widetilde}{\boldsymbol v}/{\boldsymbol u}} = \sqrt{{\boldsymbol v}{\widetilde}{\boldsymbol v}} + O(\eta)$. Hence, due to , yields $$\label{eq:pos_sigma_aux9} \sigma(z) = \lim_{\eta\downarrow 0} \frac{1}{\pi}{\left\langle{\sqrt{{\boldsymbol v}{\widetilde}{\boldsymbol v}} } \,\mspace{2mu},\, {\left( {\mathds{1}}- {\tau}{\mathscr D}({\boldsymbol u}^{1/2}) {\boldsymbol E}{\boldsymbol F}({\mathds{1}}+ {\boldsymbol F})^{-1} {\mathscr D}({\boldsymbol u}^{1/2}) \right)^{-1} \sqrt{{\boldsymbol v}{\widetilde}{\boldsymbol v}}}\right\rangle}.$$ Defining the matrix $F \in {\mathbb{C}}^{n\times n}$ through $F y = \sqrt{v_1u/v_2} S \sqrt{v_2u/v_1}\, y$ for $y \in {\mathbb{C}}^n$, we obtain $$\label{eq:pos_sigma_aux3} {\boldsymbol F}= \begin{pmatrix} 0 & F \\ F^t & 0 \end{pmatrix}, \qquad \left( {\mathds{1}}+ {\boldsymbol F}\right)^{-1} = \begin{pmatrix} ({\mathds{1}}- FF^t)^{-1} & - ( {\mathds{1}}-FF^t)^{-1} F \\ - F^t ( {\mathds{1}}-F F^t)^{-1} & ( {\mathds{1}}- F^tF)^{-1} \end{pmatrix}.$$ Furthermore, we introduce the $n\times n$ matrix $A$ by $$A {\mathrel{\vcenter{\baselineskip0.5ex \lineskiplimit0pt\hbox{\scriptsize.}\hbox{\scriptsize.}}}=}2\cdot {\mathds{1}}+ (F^t -{\mathds{1}}) ({\mathds{1}}- FF^t)^{-1} + (F-{\mathds{1}}) ( {\mathds{1}}- F^t F)^{-1}.$$ From the computation $$\begin{aligned} {\boldsymbol E}{\boldsymbol F}( {\mathds{1}}+ {\boldsymbol F})^{-1} & = \begin{pmatrix} {\mathds{1}}+ (F^t -{\mathds{1}}) ({\mathds{1}}- FF^t)^{-1} & {\mathds{1}}+ (F-{\mathds{1}}) ( {\mathds{1}}- F^t F)^{-1} \\ {\mathds{1}}+ (F^t -{\mathds{1}}) ({\mathds{1}}- FF^t)^{-1} & {\mathds{1}}+ (F-{\mathds{1}}) ( {\mathds{1}}- F^t F)^{-1} \end{pmatrix},\end{aligned}$$ we conclude that $$\label{eq:pos_sigma_aux10} \left ( {\mathds{1}}- {\tau}{\mathscr D}({\boldsymbol u}^{1/2}){\boldsymbol E}{\boldsymbol F}({\mathds{1}}+ {\boldsymbol F})^{-1} {\mathscr D}({\boldsymbol u}^{1/2}) \right)^{-1} \begin{pmatrix} x \\ x \end{pmatrix} = \begin{pmatrix} ({\mathds{1}}- {\tau}{\mathscr D}(u^{1/2}) A {\mathscr D}(u^{1/2}))^{-1} x \\ ({\mathds{1}}- {\tau}{\mathscr D}(u^{1/2}) A {\mathscr D}(u^{1/2}))^{-1} x \end{pmatrix}$$ for all $x \in {\mathbb{C}}^n$. Before applying this relation to , we show that ${\mathds{1}}- {\tau}{\mathscr D}(u^{1/2}) A {\mathscr D}(u^{1/2})$ is invertible for ${\tau}<1$. The relations in yield $$\label{eq:pos_sigma_aux5} {\langle{x} \mspace{2mu}, {A x}\rangle} = 2{\lVert x \rVert_{2}}^2 - 2{\left\langle{\begin{pmatrix} x \\ x \end{pmatrix}} \,\mspace{2mu},\, {( {\mathds{1}}+ {\boldsymbol F})^{-1} \begin{pmatrix} x \\ x \end{pmatrix}}\right\rangle}$$ for all $x \in {\mathbb{C}}^n$ and $\eta>0$. In particular, since ${\lVert {\boldsymbol F}\rVert_{2}} \leq 1$ by we conclude $A \leq {\mathds{1}}$. Hence, ${\tau}u = 1- v_1 v_2/u <1$ for ${\tau}<1$ by implies that ${\mathds{1}}- {\tau}{\mathscr D}(u^{1/2}) A {\mathscr D}(u^{1/2})$ is invertible for ${\tau}<1$. Thus, we apply to and obtain for $z \in D(0,1)$ $$\label{eq:pos_sigma_aux12} \sigma(z) = \frac{2}{\pi}\lim_{\eta\downarrow 0}{\left\langle{\sqrt{v_1v_2}} \,\mspace{2mu},\, {\left( {\mathds{1}}- {\tau}{\mathscr D}(u^{1/2}) A {\mathscr D}(u^{1/2})\right)^{-1} \sqrt{v_1 v_2}}\right\rangle}.$$ Let ${\tau}_* >0$ depend only on $s_*$ and $s^*$. From and , we conclude that ${\lvert \sigma \rvert} \lesssim 1$ uniformly for $z \in {\ensuremath{\mathbb D_{<}}}$ because of . This proves the upper bound in for $z \in {\ensuremath{\mathbb D_{<}}}$. For the proof of the lower bound, we infer some further properties of $A$ and ${\mathds{1}}- {\tau}{\mathscr D}(u^{1/2}) A {\mathscr D}(u^{1/2})$, respectively, from information about ${\boldsymbol F}$ via . In the following, we use versions of Proposition \[pro:estimates\_v\_small\_z\], and Lemma \[lem:prop\_Ff\] extended to the limiting case $\eta =0+$. Recalling ${\boldsymbol v}_0 = \lim_{\eta\downarrow 0} {\boldsymbol v}$, these results are a simple consequence of the uniform convergence ${\partial}^\alpha {\boldsymbol v}\to {\partial}^\alpha {\boldsymbol v}_0$ for $\eta \downarrow 0$ and all $\alpha \in {\mathbb{N}}^2$ by . Since ${\boldsymbol f_-}= ( \sqrt{v_1v_2/u}, - \sqrt{v_1v_2/u})+O(\eta)$ by there are $\eta_*, {\varepsilon}\sim 1$ by Lemma \[lem:prop\_Ff\] such that $\operatorname*{Spec}({\boldsymbol F}|_W) \subset [-1 +{\varepsilon}, 1]$ on the subspace $W {\mathrel{\vcenter{\baselineskip0.5ex \lineskiplimit0pt\hbox{\scriptsize.}\hbox{\scriptsize.}}}=}\{ (x,x) | x \in {\mathbb{C}}^n\}\subset {\mathbb{C}}^{2n}$ as ${\boldsymbol f_-}\perp W$ uniformly for all $\eta\in[0,\eta_*]$. Therefore, for ${\lVert x \rVert_{2}} = 1$, the right-hand side of is contained in $[2({\varepsilon}-1 )/{\varepsilon}, 1]$. Since $\left( F^t ( {\mathds{1}}- FF^t)^{-1}\right)^t = F ( {\mathds{1}}- F^t F)^{-1}$ the matrix $A$ is real symmetric and hence the spectrum of $A$ is contained in $[2({\varepsilon}-1 )/{\varepsilon}, 1]$ for all $\eta \in [0,\eta_*]$ as well. The real symmetric matrix $A$ has a positive and a negative part, i.e., there are positive matrices $A_+$ and $A_-$ such that $A = A_+ - A_-$. Hence, we have $$\label{eq:pos_sigma_aux11} {\mathds{1}}- {\tau}{\mathscr D}(u^{1/2}) A {\mathscr D}(u^{1/2}) = {\mathds{1}}- {\tau}{\mathscr D}(u^{1/2}) A_+ {\mathscr D}(u^{1/2}) + {\tau}{\mathscr D}(u^{1/2}) A_- {\mathscr D}(u^{1/2}).$$ The above statements about yield $\operatorname*{Spec}A_+ \subset [0,1]$ and $\operatorname*{Spec}A_- \subset [0,2(1-{\varepsilon})/{\varepsilon}]$. As $0 \leq u {\tau}$ we conclude from that the spectrum of ${\mathds{1}}- {\tau}{\mathscr D}(u^{1/2}) A {\mathscr D}(u^{1/2})$ is contained in $(0, 2/{\varepsilon}]$ for all $\eta\in [0,\eta_*]$. Therefore, using , we obtain $$\sigma(z) = \frac{2}{\pi}\lim_{\eta\downarrow 0}{\left\langle{\sqrt{v_1v_2}} \,\mspace{2mu},\, {\left( {\mathds{1}}- {\tau}{\mathscr D}(u^{1/2}) A {\mathscr D}(u^{1/2})\right)^{-1} \sqrt{v_1 v_2}}\right\rangle} \geq \frac{{\varepsilon}}{\pi} {\langle {\boldsymbol v}_0{\widetilde}{\boldsymbol v}_0 \rangle} \gtrsim 1$$ uniformly for all $z \in {\ensuremath{\mathbb D_{<}}}$. Here, we used in the last step. This shows for $z \in {\ensuremath{\mathbb D_{<}}}$ for any ${\tau}_*>0$ depending only on $s_*$ and $s^*$. We now show that there is ${\tau}_*>0$ depending only on $s_*$ and $s^*$ such that holds true for $z \in D(0,1)\setminus {\ensuremath{\mathbb D_{<}}}$. This is proved by tracking the blowup of $({\mathds{1}}-{\tau}{\mathscr D}(u^{1/2})A{\mathscr D}(u^{1/2}))^{-1}$ in $1-{\tau}$ for ${\tau}\uparrow 1$ in and establishing a compensation through $v_1 \sim v_2 \sim (1-{\tau})^{1/2}$ due to . This yields the upper and lower bound in . Since ${\mathds{1}}-{\tau}{\mathscr D}(u^{1/2})A{\mathscr D}(u^{1/2})$ in is also invertible for $\eta =0$ we may directly set $\eta =0$ in the following argument. We multiply the first component of the first relation in by ${\tau}$ and solve for ${\tau}u$ to obtain $${\tau}u = \frac{1}{2}\left(1+ \sqrt{1-4{\tau}v_1 v_2} \right) = 1 - {\tau}v_1 v_2 + O\left((1-{\tau})^2\right).$$ Therefore, using $v_1 \sim v_2 \sim (1-{\tau})^{1/2}$, we have $${\tau}{\mathscr D}(u^{1/2})A{\mathscr D}(u^{1/2}) = A - \frac{{\tau}}{2} \left( {\mathscr D}(v_1v_2)A + A {\mathscr D}(v_1v_2)\right) + O\left((1-{\tau})^2\right).$$ Moreover, from we conclude that $Aa = a$ for $a {\mathrel{\vcenter{\baselineskip0.5ex \lineskiplimit0pt\hbox{\scriptsize.}\hbox{\scriptsize.}}}=}\sqrt{v_1 v_2/u}/{\lVert \sqrt{v_1 v_2/u} \rVert_{2}}$. Here, we also used and with $\eta=0$. Thus, the smallest eigenvalue of the positive operator ${\mathds{1}}- {\tau}{\mathscr D}(u^{1/2})A{\mathscr D}(u^{1/2})$ satisfies $$\lambda_{\text{min}}\left({\mathds{1}}-{\tau}{\mathscr D}(u^{1/2})A{\mathscr D}(u^{1/2})\right) = \lambda_{\text{min}}\left({\mathds{1}}-A\right) + {\tau}{\langle a^2 v_1 v_2 \rangle}+ O\left((1-{\tau})^2\right) = {\tau}{\langle a^2 v_1 v_2 \rangle}+ O\left((1-{\tau})^2\right).$$ Here, we used multiple times that $Aa=a$. Therefore, as $A$ is symmetric we conclude from that $$\sigma(z) =\frac{2}{\pi} {\left\langle{\sqrt{v_1v_2}} \,\mspace{2mu},\, {\left( {\mathds{1}}- {\tau}{\mathscr D}(u^{1/2}) A {\mathscr D}(u^{1/2})\right)^{-1} \sqrt{v_1 v_2}}\right\rangle} \geq \frac{{\langle{a} \mspace{2mu}, {\sqrt{v_1v_2}}\rangle}^2}{{\tau}{\langle a^2v_1v_2 \rangle}} + O\left(1-{\tau}\right).$$ Since $a\sim 1$ and $v_1 \sim v_2 \sim (1-{\tau})^{1/2}$ there is ${\tau}_* \sim 1$ such that the lower bound in holds true for $z \in D(0,1)\setminus {\ensuremath{\mathbb D_{<}}}$. Starting from , we similarly obtain $$\sigma(z) \leq \frac{{\langle v_1v_2 \rangle}}{{\tau}{\langle a^2v_1v_2 \rangle}} + O\left(1-{\tau}\right).$$ Using the positivity of $a$, $v_1 \sim v_2 \sim (1-{\tau})^{1/2}$ and possibly shrinking ${\tau}_*\sim 1$ the upper bound in for $z \in D(0,1)\setminus {\ensuremath{\mathbb D_{<}}}$ follows. This concludes the proof of Lemma \[lem:sigma\_strictly\_positive\]. As $\sigma(z) =0$ for ${\lvert z \rvert}\geq 1$ we conclude from that $\sigma$ is nonnegative on ${\mathbb{C}}$. We use to compute the total mass of the measure on ${\mathbb{C}}$ defined by $\sigma$. Clearly, ${\boldsymbol u}_0 = {\boldsymbol v}_0/{\boldsymbol S}_d {\boldsymbol v}_0$ and using and , we obtain $$\lim_{{\tau}\uparrow 1} \int_{{\lvert z' \rvert}^2 \leq {\tau}} \sigma(z') {\mathrm{d}}^2 z' = 1 - \lim_{{\tau}\uparrow 1} {\langle {\boldsymbol v}_0 {\boldsymbol S}_{o} {\boldsymbol v}_0 \rangle} = 1.$$ Here, we used that $\lim_{{\tau}\uparrow 1} {\boldsymbol v}_0 = 0$ by . Hence, as $\sigma(z) =0$ for ${\lvert z \rvert}\geq 1$ it defines a probability density on ${\mathbb{C}}$ which concludes the proof of Proposition \[pro:properties\_sigma\]. In fact, it is possible to compute the jump height of the density of states $\sigma$ at the edge ${\tau}= {\lvert z \rvert}^2 = 1$. Let $s_1$ and $s_2$ be two eigenvectors of $S^t$ and $S$, respectively, associated to the eigenvalue $1$, i.e., $S^t s_1 =s_1$ and $S s_2 =s_2$. Note that $s_1$ and $s_2$ are unique up to multiplication by a scalar. With this notation, expanding ${\boldsymbol v}^{\tau}$ for ${\tau}\leq 1$ around ${\tau}=1$ yields $$v_1 = \sqrt{1-{\tau}}\left( \frac{{\langle s_1 s_2 \rangle}{\langle s_2 \rangle}}{{\langle s_1^2s_2^2 \rangle}{\langle s_1 \rangle}} \right)^{1/2} s_1 + O\left( (1-{\tau})^{3/2}\right), \quad v_2 = \sqrt{1-{\tau}}\left( \frac{{\langle s_1 s_2 \rangle}{\langle s_1 \rangle}}{{\langle s_1^2s_2^2 \rangle}{\langle s_2 \rangle}} \right)^{1/2} s_2 + O\left( (1-{\tau})^{3/2}\right).$$ Therefore, solving for ${\tau}u$ and expanding in $1-{\tau}$, we obtain that $\sigma$ has a jump of height $$\lim_{{\lvert z \rvert}^2 \uparrow 1} \sigma (z) = \frac{1}{\pi} \lim_{{\tau}\uparrow 1} {\partial}_{\tau}\left( {\tau}{\langle {\boldsymbol u}_0 \rangle} \right) = \frac{1}{\pi} \frac{{\langle s_1s_2 \rangle}^2}{{\langle s_1^2 s_2^2 \rangle}}.$$ Local law {#sec:local_law} ========= We begin this section with a notion for high probability estimates. \[def:stochastic\_domination\] Let $C \colon {\mathbb{R}}_+^2 \to {\mathbb{R}}_+$ be a given function which depends only on $a$, $\varphi$, ${\tau}_*$, ${\tau}^*$ and the model parameters. If $\Phi = (\Phi^{(n)})_{n}$ and $\Psi = (\Psi^{(n)})_{n}$ are two sequences of nonnegative random variables, then we will say that $\Phi$ is **stochastically dominated** by $\Psi$, $\Phi \prec \Psi$, if for all ${\varepsilon}>0$ and $D>0$ we have $${\mathbb{P}}\left( \Phi^{(n)} \geq n^{\varepsilon}\Psi^{(n)} \right) \leq \frac{C({\varepsilon},D)}{n^D}$$ for all $n \in {\mathbb{N}}$. As a trivial consequence of ${\mathbb{E}}x_{ij} = 0$, and we remark that $$\label{eq:trivial_control_x_ij_prec} {\lvert x_{ij} \rvert} \prec n^{-1/2}.$$ Local law for ${\boldsymbol H}^z$ {#sec:local_law_H_z} --------------------------------- Let $(v_1^{\tau},v_2^{\tau})$ be the positive solution of and $u^{\tau}$ defined as in . In the whole section, we will always evaluate $v_1^{\tau}$, $v_2^{\tau}$ and $u^{\tau}$ at ${\tau}= {\lvert z \rvert}^2$ and mostly suppress the dependence on ${\tau}$ and ${\lvert z \rvert}^2$, respectively, in our notation. Recall that ${\boldsymbol M}^z$ is defined in . Note that although $v_1$, $v_2$ and $u$ are rotationally invariant in $z \in {\mathbb{C}}$, the dependence of ${\boldsymbol M}^z$ on $z$ is not rotationally symmetric. For the following theorem, we remark that the sets ${\ensuremath{\mathbb D_{<}}}$ and ${\ensuremath{\mathbb D_{>}}}$ were introduced in . \[thm:local\_law\_H\_z\] Let $X$ satisfy (A) and (B) and let ${\boldsymbol G}={\boldsymbol G}^z$ be the resolvent of ${\boldsymbol H}^z$ as defined in . For fixed ${\varepsilon}\in (0,1/2)$, the entrywise local law $$\label{entrywise matrix local law} {\big\lVert {\boldsymbol G}^z(\eta) - {\boldsymbol M}^z(\eta) \big\rVert}_{\max} \prec \begin{cases} \frac{1}{\sqrt{n\eta}} & \text{ for } z \in \mathbb{D}_<\,,\; \eta \in [n^{-1+{\varepsilon}},1] \,, \\ \frac{1}{\sqrt{n}} +\frac{1}{n \eta}& \text{ for } z \in \mathbb{D}_>\,,\; \eta \in [n^{-1+{\varepsilon}},1] \,, \\ \frac{1}{\sqrt{n}{\mspace{2 mu}}\eta^2}& \text{ for } z \in \mathbb{D}_< \cup \mathbb{D}_> \,,\; \eta \in [1,\infty)\,, \end{cases}$$ holds true. In particular, $$\label{eq:local_law_H_z_version} {\big\lVert {\boldsymbol g}(\eta) - {\mathrm{i}}{\boldsymbol v}(\eta) \big\rVert}_\infty \prec \begin{cases} \frac{1}{\sqrt{n\eta}} & \text{ for } z \in \mathbb{D}_<\,,\; \eta \in [n^{-1+{\varepsilon}},1] \,, \\ \frac{1}{\sqrt{n}} +\frac{1}{n \eta}& \text{ for } z \in \mathbb{D}_>\,,\; \eta \in [n^{-1+{\varepsilon}},1] \,, \\ \frac{1}{\sqrt{n}{\mspace{2 mu}}\eta^2}& \text{ for } z \in \mathbb{D}_< \cup \mathbb{D}_> \,,\; \eta \in [1,\infty)\,, \end{cases}$$ where ${\boldsymbol g}=({\langle{\boldsymbol{e}_i} \mspace{2mu}, {{\boldsymbol G}\boldsymbol{e}_{i}}\rangle})_{i=1}^{2n}$ denotes the vector of diagonal entries of the resolvent ${\boldsymbol G}^z$. For a non-random vector ${\boldsymbol{{y}}} \in {\mathbb{C}}^{2n}$ with ${\lVert {\boldsymbol{{y}}} \rVert}_\infty \le 1$ we have $$\label{eq:local_law_H_z_averaged} \big|{\langle{{\boldsymbol{{y}}}} \mspace{2mu}, {{\boldsymbol g}(\eta) - {\mathrm{i}}{\boldsymbol v}(\eta)}\rangle} \big| \prec \begin{cases} \frac{1}{{n\eta}} & \text{ for } z \in \mathbb{D}_<\,,\; \eta \in [n^{-1+{\varepsilon}},1] \,, \\ \frac{1}{{n}} +\frac{1}{(n \eta)^2}& \text{ for } z \in \mathbb{D}_>\,,\; \eta \in [n^{-1+{\varepsilon}},1] \,, \\ \frac{1}{{n} \eta^2}& \text{ for } z \in \mathbb{D}_< \cup \mathbb{D}_> \,,\; \eta \in [1,\infty)\,. \end{cases}$$ As an easy consequence we can now prove Corollary \[coro:eigenvector\_delocalization\]. Let $y \in {\mathbb{C}}^n$ be an eigenvector of $X$ corresponding to the eigenvalue $\sigma \in \operatorname*{Spec}X$ with ${\lvert \sigma \rvert}^2 \leq \rho(S) - {\tau}_*$. Then the $2n$-vector $(0,y)$ is contained in the kernel of ${\boldsymbol H}^\sigma$. Therefore, is an easy consequence of (Compare with the proof of Corollary 1.14 in [@Ajankirandommatrix]). We recall our normalization of the trace, $\operatorname{Tr}{\mathds{1}}= 1$, from . Recall from the beginning of Section \[sec:self\_consistent\_equation\] how our problem can be cast into the setup of [@AjankiCorrelated]. In the regime $z \in \mathbb{D}_<$ we follow the structure of the proof of Theorem 2.9 in [@AjankiCorrelated] and in the regime $z \in \mathbb{D}_>$ the proof of Proposition 7.1 in [@AjankiCorrelated] until the end of Step 1. In fact, the arguments from these proofs can be taken over directly with three important adjustments. The flatness assumption is used heavily in [@AjankiCorrelated] in order to establish bounds (Theorem 2.5 in [@AjankiCorrelated]) on the deterministic limit of the resolvent and for establishing the stability of the matrix Dyson equation, cf. below, (Theorem 2.6 in [@AjankiCorrelated]). Since this assumption is violated in our setup we present appropriately adjusted versions of these theorems (Proposition \[pro:estimates\_v\_small\_z\] and Proposition \[pro:stab\_lemma\] in [@AjankiCorrelated]). We will also take over the proof of the fluctuation averaging result (Proposition \[prp:Fluctuation averaging\] below) for ${\boldsymbol H}^z$ from [@AjankiCorrelated] since the flatness did not play a role in that proof at all. Note that the $\eta^{-2}$-decay in the spectral parameter regime $\eta\ge1$ was not covered in [@AjankiCorrelated]. But this decay simply follows by using the bounds ${\lVert {\boldsymbol M}^z(\eta) \rVert}_{{\mathrm}{max}}+{\lVert {\boldsymbol G}^z(\eta) \rVert}_{{\mathrm}{max}}\le \frac{2}{\eta}$ instead of just ${\lVert {\boldsymbol M}^z(\eta) \rVert}_{{\mathrm}{max}}+{\lVert {\boldsymbol G}^z(\eta) \rVert}_{{\mathrm}{max}}\le C$ along the proof. As in [@AjankiCorrelated] we choose a pseudo-metric $d$ on $\{1, \dots, 2n\}$. Here this pseudo-metric is particularly simple, $$d(i,j)\,{\mathrel{\vcenter{\baselineskip0.5ex \lineskiplimit0pt\hbox{\scriptsize.}\hbox{\scriptsize.}}}=}\, \begin{cases} 0 & \text{ if } i=j \text{ or } i=j+n \text{ or } j=i+n \,, \\ \infty & \text{otherwise}\,, \end{cases} \qquad i,j =1, \dots, 2n\,.$$ With this choice of $d$ the matrix ${\boldsymbol H}^z$ satisfies all assumptions in [@AjankiCorrelated] apart from the [flatness]{}. We will now show that as in [@AjankiCorrelated] the resolvent ${\boldsymbol G}^z$ satisfies the *perturbed matrix Dyson equation* [ $$\label{Perturbed MDE} \begin{split} -{\mathds{1}}\,=\, ({\mathrm{i}}{\mspace{1 mu}}\eta{\mspace{2 mu}}{\mathds{1}}- {\boldsymbol A}^z + \widetilde{{\mathcal}{S}}[{\boldsymbol G}^z(\eta)]){\boldsymbol G}^z(\eta) + {\boldsymbol D}(\eta)\,. \end{split}$$ ]{} Here, ${\boldsymbol A}^z$ is given by , [ $$\label{definition of error matrix D} \begin{split} {\boldsymbol D^z}(\eta)\,{\mathrel{\vcenter{\baselineskip0.5ex \lineskiplimit0pt\hbox{\scriptsize.}\hbox{\scriptsize.}}}=}\,-(\widetilde{{\mathcal}{S}}[{\boldsymbol G}^z(\eta)]+{\boldsymbol H}^z-{\boldsymbol A}^z){\boldsymbol G}^z(\eta) \,, \end{split}$$ ]{} is a random error matrix and $\widetilde{{\mathcal}{S}}$ is a slight modification of the operator ${\mathcal}{S}$ defined in , [ $$\label{definition of tilde S} \begin{split} \widetilde{{\mathcal}{S}}[{\boldsymbol W}]\,{\mathrel{\vcenter{\baselineskip0.5ex \lineskiplimit0pt\hbox{\scriptsize.}\hbox{\scriptsize.}}}=}\, {\mathbb{E}}({\boldsymbol H}^z-{\boldsymbol A}^z){\boldsymbol W}({\boldsymbol H}^z-{\boldsymbol A}^z)\,=\, \begin{pmatrix} \operatorname{diag}(S{w_2} )& T \odot W_{21}^t \\ T^* \odot W_{12}^t & \operatorname{diag}(S^t {w_1}) \end{pmatrix}\,. \end{split}$$ ]{} Here, $\odot$ denotes the Hadamard product, i.e., for matrices $A=(a_{ij})_{i,j=1}^l$ and $B=(b_{ij})_{i,j=1}^l$, we define their Hadamard product through $(A \odot B)_{ij} {\mathrel{\vcenter{\baselineskip0.5ex \lineskiplimit0pt\hbox{\scriptsize.}\hbox{\scriptsize.}}}=}a_{ij} b_{ij}$ for $i,j=1, \ldots, l$. Moreover, we used the conventions from for ${\boldsymbol W}$ and introduced the matrix $T \in {\mathbb{C}}^{n \times n}$ with entries $$t_{i j}\,{\mathrel{\vcenter{\baselineskip0.5ex \lineskiplimit0pt\hbox{\scriptsize.}\hbox{\scriptsize.}}}=}\, {\mathbb{E}}{\mspace{2 mu}}x_{i j}^2\,.$$ Note that in contrast to [@AjankiCorrelated] the matrix ${\boldsymbol M}$ solves , which is given in terms of the operator ${\mathcal}{S}$ and not $\widetilde{{\mathcal}{S}}$. As we will see below this will not effect the proof, since the entries of the matrix $T$ are of order $N^{-1}$ and thus the off-diagonal terms in of $\widetilde{{\mathcal}{S}}$ are negligible. We will see that ${\boldsymbol D}={\boldsymbol D}^z$ is small in the entrywise maximum norm $${\lVert {\boldsymbol W}\rVert}_{{\mathrm}{max}}\,{\mathrel{\vcenter{\baselineskip0.5ex \lineskiplimit0pt\hbox{\scriptsize.}\hbox{\scriptsize.}}}=}\, \max_{i,j=1}^{2n}{\lvert {w}_{i j} \rvert}\,,$$ ${\boldsymbol W}=(w_{ij})_{i,j=1}^{2n}$, and use the stability of to show that ${\boldsymbol G}(\eta)={\boldsymbol G}^z(\eta)$ approaches ${\boldsymbol M}(\eta)={\boldsymbol M}^z(\eta)$ defined in as $n\to \infty$, i.e., we will show that [ $$\label{definition of Lambda} \begin{split} \Lambda(\eta)\,{\mathrel{\vcenter{\baselineskip0.5ex \lineskiplimit0pt\hbox{\scriptsize.}\hbox{\scriptsize.}}}=}\, {\lVert {\boldsymbol G}(\eta)- {\boldsymbol M}(\eta) \rVert}_{{\mathrm}{max}}\,, \end{split}$$ ]{} converges to zero. For simplicity we will only consider the most difficult regime $z \in \mathbb{D}_<$ and $\eta \le 1$ inside the spectrum. The cases $z \in \mathbb{D}_>$ and $\eta \ge 1$ are similar but simpler and left to the reader. We simply follow the proof in Section 3 of [@AjankiCorrelated] line by line until the flatness assumption is used. This happens for the first time inside the proof of Lemma 3.3. We therefore replace this lemma by the following modification. \[lmm:D norm bounds\]Let $z \in {\ensuremath{\mathbb D_{<}}}$. Then $${\lVert {\boldsymbol D}( \eta) \rVert}_{{\mathrm}{max}}\,\prec\, \frac{1}{\sqrt{n}}\,,\qquad \eta \ge 1\,.$$ Furthermore, we have [ $$\label{D max norm bound} \begin{split} {\lVert {\boldsymbol D}(\eta) \rVert}_{{\mathrm}{max}}\,\chi(\Lambda(\eta)\le n^{-{\varepsilon}})\,\prec\, \frac{1}{\sqrt{n\eta}}\,, \qquad\eta \in [n^{-1+{\varepsilon}},1]. \end{split}$$ ]{} To show Lemma \[lmm:D norm bounds\] we follow the proof of its analog, Lemma 3.3 in [@AjankiCorrelated], where the flatness assumption as well as the assumptions that the spectral parameter is in the bulk of the spectrum (formulated as $\rho(\zeta)\ge \delta$ in [@AjankiCorrelated]) are used only implicitly through the upper bound on ${\boldsymbol{{M}}}$ (Theorem 2.5 in [@AjankiCorrelated]). However, the conclusion of this theorem clearly still holds in our setup because ${\boldsymbol{{M}}}$ has the $2\times 2$-diagonal structure and the vectors $v_1,v_2$ and $u$ are bounded by Proposition \[pro:estimates\_v\_small\_z\] and . We continue following the arguments of Section 3 of [@AjankiCorrelated] using our Lemma \[lmm:D norm bounds\] above instead of Lemma 3.3 there. The next step that uses the flatness assumption is the stability of the MDE (Theorem 2.6 in [@AjankiCorrelated]) which shows that the bound also implies $$\Lambda(\eta)\,\chi(\Lambda(\eta)\le n^{-{\varepsilon}})\prec\frac{1}{\sqrt{n\eta}}\,.$$ In our setup this stability result is replaced by the following lemma whose proof is postponed until the end of the proof of Theorem \[thm:local\_law\_H\_z\]. \[lmm:MDE stability\] Suppose that some functions $D_{ab}, G_{ab}: {\mathbb{R}}_+ \to {\mathbb{C}}^{n \times n}$ for $a,b=1,2$ satisfy with [ $$\label{Block decomposition for D and G} \begin{split} {\boldsymbol{{D}}}\,{\mathrel{\vcenter{\baselineskip0.5ex \lineskiplimit0pt\hbox{\scriptsize.}\hbox{\scriptsize.}}}=}\, \begin{pmatrix} D_{11} & D_{12} \\ D_{21} & D_{22}\end{pmatrix}\,,\qquad {\boldsymbol{{G}}}\,{\mathrel{\vcenter{\baselineskip0.5ex \lineskiplimit0pt\hbox{\scriptsize.}\hbox{\scriptsize.}}}=}\, \begin{pmatrix} G_{11} & G_{12} \\ G_{21} & G_{22}\end{pmatrix}\,, \end{split}$$ ]{} and the additional constraints [ $$\label{symmetry constraints on G} \begin{split} \operatorname{Tr}G_{11} \,=\, \operatorname{Tr}G_{22}\,,\qquad \operatorname{Im}{\boldsymbol G}\,=\, \frac{1}{2{\mathrm{i}}}({\boldsymbol G}-{\boldsymbol G}^*)\text{ is positive definite }. \end{split}$$ ]{} There is a constant $\lambda_* \gtrsim 1$, depending only on $\mathcal{P}$, such that [ $$\label{MDE Stability} \begin{split} {\lVert {\boldsymbol G}-{\boldsymbol M}\rVert}_{\mathrm{max}}\,\chi \,\lesssim\, {\lVert {\boldsymbol D}\rVert}_{\mathrm{max}} +\frac{1}{n}\,, \qquad \chi\,{\mathrel{\vcenter{\baselineskip0.5ex \lineskiplimit0pt\hbox{\scriptsize.}\hbox{\scriptsize.}}}=}\,\chi ({\lVert {\boldsymbol G}-{\boldsymbol M}\rVert}_{\mathrm{max}}\le \lambda_*)\,, \end{split}$$ ]{} uniformly for all $z \in {\mathbb D_{<}}\cup{\mathbb D_{>}}$, where ${\boldsymbol M}(\eta)={\boldsymbol M}^z(\eta)$ is defined in . Furthermore, there exist eight matrix valued functions $R_{ab}^{(k)}: {\mathbb{R}}_+ \to {\mathbb{C}}^{n \times n}$ with $a,b,k=1,2$, depending only on $z$ and $S$, and satisfying ${\lVert R^{(k)}_{ab} \rVert}_\infty\lesssim 1$, such that [ $$\label{averaged matrix stability} \begin{split} \Big|\operatorname{Tr}[\operatorname{diag}({\boldsymbol{{y}}})({\boldsymbol G}-{\boldsymbol M})]\Big|\,\chi\lesssim\max_{a,b,k=1,2}\Big|\operatorname{Tr}[\operatorname{diag}(R_{ab}^{(k)}y_k)D_{ab}]\Big|+{\lVert {\boldsymbol{{y}}} \rVert}_\infty\Big(\frac{1}{n}+{\lVert {\boldsymbol{{D}}} \rVert}_{{\mathrm}{max}}^2\Big), \end{split}$$ ]{} uniformly for all $z \in {\mathbb D_{<}}\cup{\mathbb D_{>}}$ and ${\boldsymbol{{y}}}=(y_1,y_2) \in {\mathbb{C}}^{2n}$. The important difference between Theorem 2.6 in [@AjankiCorrelated] and Lemma \[lmm:MDE stability\] above is the additional assumption imposed on the solution of the perturbed MDE. This assumption is satisfied for the resolvent of the matrix ${\boldsymbol H}^z$ because of the $2\times2$-block structure . In fact with the block decomposition for ${\boldsymbol G}$ as in we have $$G_{11}(\eta)\,=\,\frac{{\mathrm{i}}{\mspace{1 mu}}\eta{\mspace{1 mu}}{\mathds{1}}}{(X-z{\mathds{1}})(X-z{\mathds{1}})^*+\eta^2{\mspace{1 mu}}{\mathds{1}}} \,,\qquad G_{22}(w)\,=\,\frac{{\mathrm{i}}{\mspace{1 mu}}\eta{\mspace{1 mu}}{\mathds{1}}}{(X-z{\mathds{1}})^*(X-z{\mathds{1}})+\eta^2{\mspace{1 mu}}{\mathds{1}}}\,.$$ Using Lemma \[lmm:MDE stability\] in the remainder of the proof of the entrywise local law in Section 3 of [@AjankiCorrelated] finishes the proof of . To see we use the fluctuation averaging mechanism, which was first established for generalized Wigner matrices with Bernoulli entries in [@EYYBern]. The following proposition is stated and proven as Proposition 3.4 in [@AjankiCorrelated]. Since the flatness condition was not used in its proof at all, we simply take it over. \[prp:Fluctuation averaging\] Let $z \in \mathbb{D}_<\cup\mathbb{D}_>$, ${\varepsilon}\in (0,1/2)$, $\eta\geq n^{-1}$ and $\Psi$ a non-random control parameter such that $n^{-1/2} \leq \Psi \leq n^{-{\varepsilon}}$. Suppose the local law holds true in the form $${\lVert {\boldsymbol G}(\eta)-{\boldsymbol M}(\eta) \rVert}_{{\mathrm}{max}}\,\prec\, \Psi\,.$$ Then for any non-random vector $y \in {\mathbb{C}}^{n}$ with ${\lVert {y} \rVert}_\infty\leq 1$ we have $$\max_{a,b=1,2}\Big|\operatorname{Tr}[\operatorname{diag}(y)D_{ab}]\Big|\,\prec\,\Psi^2\,,$$ where $D_{ab} \in {\mathbb{C}}^{n \times n}$, $a,b=1,2$, are the blocks of the error matrix $${\boldsymbol{{D}}}(\eta)\,=\, \begin{pmatrix} D_{11} & D_{12} \\ D_{21} & D_{22}\end{pmatrix}\,,$$ which was defined in . Using this proposition the averaged local law follows from and . This finishes the proof of Theorem \[thm:local\_law\_H\_z\]. We write in the $2\times 2$ - block structure [ $$\label{MDE in block form} \begin{split} &\begin{pmatrix}\operatorname{diag}({\mathrm{i}}{\mspace{1 mu}}\eta + Sg_2) & z{\mspace{1 mu}}{\mathds{1}}\\ \overline{z}{\mspace{1 mu}}{\mathds{1}}& \operatorname{diag}({\mathrm{i}}{\mspace{1 mu}}\eta + S^tg_1)\end{pmatrix} \begin{pmatrix} G_{11} & G_{12} \\ G_{21} & G_{22}\end{pmatrix} \\ &\mspace{100mu}=\, -\begin{pmatrix} {\mathds{1}}& 0 \\ 0 & {\mathds{1}}\end{pmatrix} -\begin{pmatrix} D_{11}+(T \odot G_{21}^t)G_{21} & D_{12}+(T \odot G_{21}^t)G_{22} \\ D_{21}+(T^* \odot G_{12}^t )G_{11}& D_{22}+(T^* \odot G_{12}^t )G_{22}\end{pmatrix} , \end{split}$$ ]{} where we introduced ${\boldsymbol g}=(g_1,g_2) \in {\mathbb{C}}^{2n}$, the vector of the diagonal elements of ${\boldsymbol G}$. We restrict the following calculation to the regime where ${\lVert {\boldsymbol G}(\eta)-{\boldsymbol M}(\eta) \rVert}_{\mathrm{max}}\le \lambda_*$ for some sufficiently small $\lambda_*$ in accordance with the characteristic function on the left hand side of . In particular, [ $$\label{MDE:diagonal a priori bound} \begin{split} {\lVert {\boldsymbol g}(\eta)-{\mathrm{i}}{\mspace{1 mu}}{\boldsymbol v}(\eta) \rVert}_\infty\,\le\, \lambda_*\,. \end{split}$$ ]{} Since by and the identity $$\begin{pmatrix}{\mathrm{i}}\operatorname{diag}(\eta + Sv_2(\eta)) & z{\mspace{1 mu}}{\mathds{1}}\\ \overline{z}{\mspace{1 mu}}{\mathds{1}}& {\mathrm{i}}\operatorname{diag}(\eta + S^tv_1(\eta))\end{pmatrix}^{-1} \,=\,- {\boldsymbol M}(\eta)\,,$$ holds we infer from the smallness of ${\lVert {\boldsymbol g}-{\mathrm{i}}{\boldsymbol v}\rVert}_{\mathrm{max}}$ that the inverse of the first matrix factor on the left hand side of is bounded and satisfies [ $$\label{approximate of lhs inverse matrix} \begin{split} \bigg\|\begin{pmatrix}\operatorname{diag}({\mathrm{i}}{\mspace{1 mu}}\eta + Sg_2) & z{\mspace{1 mu}}{\mathds{1}}\\ \overline{z}{\mspace{1 mu}}{\mathds{1}}& \operatorname{diag}({\mathrm{i}}{\mspace{1 mu}}\eta + S^tg_1)\end{pmatrix}^{-1}+{\boldsymbol M}\bigg\|_{\mathrm{max}}\,\lesssim\, {\lVert {\boldsymbol g}-{\mathrm{i}}{\mspace{1 mu}}{\boldsymbol v}\rVert}_{\mathrm{max}}\,. \end{split}$$ ]{} Using this in yields [ $$\label{MDE: G equation} \begin{split} {\boldsymbol G}+\begin{pmatrix}\operatorname{diag}({\mathrm{i}}{\mspace{1 mu}}\eta + Sg_2) & z{\mspace{1 mu}}{\mathds{1}}\\ \overline{z}{\mspace{1 mu}}{\mathds{1}}& \operatorname{diag}({\mathrm{i}}{\mspace{1 mu}}\eta + S^tg_1)\end{pmatrix}^{-1}\mspace{-10mu}={\boldsymbol M}{\boldsymbol{{D}}} +O\Big( {\lVert {\boldsymbol g}-{\boldsymbol v}\rVert}_{\mathrm{max}}{\lVert {\boldsymbol D}\rVert}_{{\mathrm}{max}}+{\lVert {\boldsymbol G}-{\boldsymbol M}\rVert}_{\mathrm{max}}^2+\frac{1}{n}\Big),\mspace{-3mu} \end{split}$$ ]{} where we applied the simple estimate [ $$\label{estimate on TGG} \begin{split} {\lVert (T \odot G_{ab}^t)G_{cd} \rVert}_{\mathrm{max}}\,\lesssim\, {\lVert {\boldsymbol G}-{\boldsymbol M}\rVert}_{\mathrm{max}}^2+\frac{1}{n}{\lVert {\boldsymbol G}-{\boldsymbol M}\rVert}_{\mathrm{max}}{\lVert {\boldsymbol M}\rVert}_{\mathrm{max}}+\frac{1}{n} {\lVert {\boldsymbol M}\rVert}_{\mathrm{max}}^2 \,\lesssim\, {\lVert {\boldsymbol G}-{\boldsymbol M}\rVert}_{\mathrm{max}}^2+\frac{1}{n}\,, \end{split}$$ ]{} which follows from $${\lVert T \rVert}_{\mathrm{max}}\,\lesssim\, \frac{1}{n}\,.$$ Thus the diagonal elements ${\boldsymbol g}$ of ${\boldsymbol G}$ satisfy with an error term ${\boldsymbol d}$ that is given by [ $$\label{MDE: d estimate} \begin{split} {\boldsymbol d}\,=\, (({\boldsymbol M}{\boldsymbol D})_{ii})_{i=1}^{2n}+ O\Big({\lVert {\boldsymbol G}-{\boldsymbol M}\rVert}_{\mathrm{max}}^2+\frac{1}{n}\Big)\,. \end{split}$$ ]{} Here we used ${\lVert {\boldsymbol D}\rVert}_{\mathrm{max}}\lesssim{\lVert {\boldsymbol G}-{\boldsymbol M}\rVert}_{\mathrm{max}}$, which follows directly from and . With and in Proposition \[pro:stab\_lemma\], the stability result on , we conclude that [ $$\label{conclusion from entrywise stability} \begin{split} {\lVert {\boldsymbol g}-{\mathrm{i}}{\mspace{1 mu}}{\boldsymbol v}\rVert}_\infty\,\lesssim\, {\lVert {\boldsymbol D}\rVert}_{\mathrm{max}}+{\lVert {\boldsymbol G}-{\boldsymbol M}\rVert}_{\mathrm{max}}^2+\frac{1}{n}\,, \end{split}$$ ]{} and that [ $$\label{conclusion from averaged stability} \begin{split} {\lvert {\langle{{\boldsymbol{{y}}}} \mspace{2mu}, {{\boldsymbol g}-{\mathrm{i}}{\mspace{1 mu}}{\boldsymbol v}}\rangle} \rvert}\,\lesssim\,\Big|\operatorname{Tr}[\operatorname{diag}({\boldsymbol{{R}}}{\boldsymbol{{y}}}){\boldsymbol M}{\boldsymbol D}]\Big|+ {\lVert {\boldsymbol D}\rVert}_{\mathrm{max}}^2+{\lVert {\boldsymbol G}-{\boldsymbol M}\rVert}_{\mathrm{max}}^2+ \frac{1}{n}\,, \end{split}$$ ]{} for some bounded ${\boldsymbol{{R}}} \in {\mathbb{C}}^{2n \times 2n}$ and any ${\boldsymbol{{y}}} \in {\mathbb{C}}^{2n}$ with ${\lVert {\boldsymbol{{y}}} \rVert}_\infty\le 1$, respectively. Combining with and yields $${\lVert {\boldsymbol G}-{\boldsymbol M}\rVert}_{\mathrm{max}}\,\lesssim\, {\lVert {\boldsymbol D}\rVert}_{\mathrm{max}}+{\lVert {\boldsymbol G}-{\boldsymbol M}\rVert}_{\mathrm{max}}^2+ \frac{1}{n}\,.$$ By choosing $\lambda_*$ sufficiently small we may absorb the quadratic term of the difference ${\boldsymbol G}-{\boldsymbol M}$ on the right hand side into the left hand side and follows. Using in to estimate the term ${\lVert {\boldsymbol G}-{\boldsymbol M}\rVert}_{\mathrm{max}}^2$ proves . We use a standard argument to conclude from the following statement about the number of eigenvalues ${\lambda}_i(z)$ of ${\boldsymbol H}^z$ in a small interval centered at zero. \[lemma:small\] Let ${\varepsilon}>0$. Then $$\label{eq:estimate_number_of_eigenvalues_weak_local_law} \# \big\{ i\; :\; {\lvert {\lambda}_i(z) \rvert}\le \eta \big\} \prec n {\mspace{1 mu}}\eta\,,$$ uniformly for all $\eta \ge n^{-1+{\varepsilon}}$ and $z \in {\ensuremath{\mathbb D_{<}}}$. Furthermore, we have $$\label{eq:strong} \sup_{z\in {\ensuremath{\mathbb D_{>}}}}\frac{1}{{\lvert {\lambda}_i(z) \rvert}}\,\prec\, n^{1/2}\,.$$ For the proof of we realize that implies a uniform bound on the resolvent elements up to the spectral scale $\eta \ge n^{-1+{\varepsilon}}$. Thus we have $$\frac{\#\Sigma_\eta}{2\eta}\,\le\, \sum_{i \in \Sigma_\eta}\frac{\eta }{\eta^2 + {\lambda}_i(z)^2} \,\le\,2n \operatorname{Im}\operatorname{Tr}{\boldsymbol G}^{z}(\eta)\,\prec\, n \,,$$ where $\Sigma_\eta{\mathrel{\vcenter{\baselineskip0.5ex \lineskiplimit0pt\hbox{\scriptsize.}\hbox{\scriptsize.}}}=}\{i: {\lvert {\lambda}_i(z) \rvert}\le \eta\}$. Here, we used the normalization of the trace . Before proving , we first establish that $$\label{eq:estimate_number_of_eigenvalues_weak_local_law_outside} \frac{1}{{\lvert {\lambda}_i(z) \rvert}}\,\prec\, n^{1/2}\,,$$ uniformly for $z \in {\ensuremath{\mathbb D_{>}}}$. We use and ${\langle {\boldsymbol{{v}}}(\eta) \rangle}\sim \eta$ to estimate $$\label{SG} \frac{\eta}{\eta^2+{\lambda}_i(z)^2}\,\le\, 2n \operatorname{Im}\operatorname{Tr}{\boldsymbol G}^{z} (\eta)\,\prec\, n {\mspace{1 mu}}\eta + \frac{1}{n \eta^2}\,,$$ with the choice $\eta{\mathrel{\vcenter{\baselineskip0.5ex \lineskiplimit0pt\hbox{\scriptsize.}\hbox{\scriptsize.}}}=}n^{-1/2-{\varepsilon}}$ for any ${\varepsilon}>0$. This immediately implies ${\lvert {\lambda}_i(z) \rvert}^{-1}\prec n^{1/2+{\varepsilon}}$, hence . For the stronger bound we use that $z\mapsto \operatorname{Im}\operatorname{Tr}{\boldsymbol G}^{z}(\eta)$ is a Lipschitz continuous function (with a Lipschitz constant $C\eta^{-2}$ uniformly in $z$) and that ${\ensuremath{\mathbb D_{>}}}$ is compact, so the second bound in holds even after taking the supremum over $z\in {\ensuremath{\mathbb D_{>}}}$. Thus $$\sup_{z\in {\ensuremath{\mathbb D_{>}}}} \frac{\eta}{\eta^2+{\lambda}_i(z)^2}\,\le\, 2n \sup_{z\in {\ensuremath{\mathbb D_{>}}}} \operatorname{Im}\operatorname{Tr}{\boldsymbol G}^{z} (\eta)\,\prec\, n {\mspace{1 mu}}\eta + \frac{1}{n \eta^2}\,$$ holds for $\eta{\mathrel{\vcenter{\baselineskip0.5ex \lineskiplimit0pt\hbox{\scriptsize.}\hbox{\scriptsize.}}}=}n^{-1/2-{\varepsilon}}$. From the last inequality we easily conclude . Local inhomogeneous circular law -------------------------------- We start with an estimate on the smallest singular value of $X-z{\mathds{1}}$ which will be used to control the ${\mathrm{d}}\eta$-integral in the second term on the right-hand side of for $\eta \leq n^{-1+{\varepsilon}}$. Notice that Proposition \[pro:least\_singular\_value\] is the only result in our proof of Theorem \[thm:local\_circular\_law\] which requires the entries of $X$ to have a bounded density. Adapting the proof of [@bordenave2012 Lemma 4.12] with the bounded density assumption to our setting, we obtain the following proposition. \[pro:least\_singular\_value\] Under the condition , there is a constant $C$, depending only on $\alpha$, such that $$\label{eq:small_singular_value_estimate} {\mathbb{P}}\left( \min_{i=1}^{2n}{\lvert {\lambda}_i(z) \rvert}\leq \frac{u}{n} \right) \leq C u^{2\alpha/(1+\alpha)} n^{\beta + 1}$$ for all $u>0$ and $z \in {\mathbb{C}}$. We follow the proof in [@bordenave2012] and explain the differences. Let $R_1, \ldots, R_n$ denote the rows of $\sqrt{n}X-z{\mathds{1}}$. Proceeding as in [@bordenave2012] but using our normalization conventions, we are left with estimating $${\mathbb{P}}\left( n {\lvert {\langle{R_i} \mspace{2mu}, {y}\rangle} \rvert} \leq \frac{u}{\sqrt n} \right)$$ uniformly for $u$ and for arbitrary $y \in {\mathbb{C}}^n$ satisfying ${\lVert y \rVert_{2}} = 1/\sqrt{n}$. We choose $j \in \{1, \ldots, n\}$ such that ${\lvert y_j \rvert} \geq 1/\sqrt n$ and compute the conditional probability $${\mathbb{P}}_{ij} {\mathrel{\vcenter{\baselineskip0.5ex \lineskiplimit0pt\hbox{\scriptsize.}\hbox{\scriptsize.}}}=}{\mathbb{P}}\Big( n{\lvert {\langle{R_i} \mspace{2mu}, {y}\rangle} \rvert} \leq \frac{u}{\sqrt n} \Big| x_{i1}, \ldots, {\widehat}{x_{ij}}, \ldots, x_{in} \Big) = \int_{\mathbb{C}}{\ensuremath{\chi}}\left( {\left\lvert \frac{a}{y_{j}} + w \right\rvert} \leq \frac{u}{y_j \sqrt n}\right) f_{ij}(w) {\mathrm{d}}^2 w,$$ where $a$ is independent of $x_{ij}$. Using and ${\lvert y_j \rvert} \geq 1/\sqrt n$, we get $${\lvert {\mathbb{P}}_{ij} \rvert} \leq {\left\lvert \pi\frac{u}{y_j \sqrt n} \right\rvert}^{2\alpha/(1+\alpha)} {\lVert f_{ij} \rVert}_{1+\alpha} \leq (\pi u)^{2\alpha/(1+\alpha)} n^{ \beta}.$$ Thus, ${\mathbb{P}}\left( n{\lvert {\langle{R_i} \mspace{2mu}, {y}\rangle} \rvert} \leq u/\sqrt{n} \right) \leq (\pi u)^{2\alpha/(1+\alpha)} n^{\beta}$ which concludes the proof of as in [@bordenave2012]. For the following proof of Theorem \[thm:local\_circular\_law\] we recall that without loss of generality, we are assuming that $\rho(S) =1$ which can be obtained by a simple rescaling of $X$. Moreover, from , for ${\tau}_*>0$ and ${\tau}^* > 1 + {\tau}_*$, we recall the notations $${\ensuremath{\mathbb D_{<}}}{\mathrel{\vcenter{\baselineskip0.5ex \lineskiplimit0pt\hbox{\scriptsize.}\hbox{\scriptsize.}}}=}\{ z \in {\mathbb{C}}\;\mid\; {\lvert z \rvert}^2 \leq 1 - {\tau}_* \}, \quad {\ensuremath{\mathbb D_{>}}}{\mathrel{\vcenter{\baselineskip0.5ex \lineskiplimit0pt\hbox{\scriptsize.}\hbox{\scriptsize.}}}=}\{ z \in {\mathbb{C}}\;\mid\; 1 + {\tau}_* \leq {\lvert z \rvert}^2 \leq {\tau}^* \}.$$ We start with the proof of part (i) of Theorem \[thm:local\_circular\_law\]. We will estimate each term on the right-hand side of . Let $z_0 \in {\ensuremath{\mathbb D_{<}}}$. We suppress the ${\tau}$ dependence of $v_1$ in this proof but it will always be evaluated at ${\tau}= {\lvert z \rvert}^2$. As $\operatorname{supp}f \subset D_\varphi(0)$, $a>0$ and $z_0 \in {\ensuremath{\mathbb D_{<}}}$ we can assume that the integration domains of the ${\mathrm{d}}^2 z$ integrals in are ${\ensuremath{\mathbb D_{<}}}$ instead of ${\mathbb{C}}$. Hence, it suffices to prove every bound along the proof of (i) uniformly for $z\in {\ensuremath{\mathbb D_{<}}}$. To begin, we estimate the first term in . Since $$\log {\lvert \det({\boldsymbol H}^z-{\mathrm{i}}T{\mathds{1}}) \rvert} = 2 n \log T + \sum_{j=1}^n \log \left( 1 + \frac{{\lambda}_j^2}{T^2}\right)$$ and the integral of $\Delta f_{z_0,a}$ over ${\mathbb{C}}$ vanishes as $f \in C_0^2({\mathbb{C}})$, we obtain $$\label{eq:master_formula_first_term} {\left\lvert \frac{1}{4\pi n} \int_{\mathbb{C}}\Delta f_{z_0,a}(z) \log {\lvert \det({\boldsymbol H}^z -{\mathrm{i}}T{\mathds{1}}) \rvert} {\mathrm{d}}^2 z \right\rvert} \leq \frac{1}{2\pi} \int_{\mathbb{C}}{\lvert \Delta f_{z_0,a}(z) \rvert} \frac{\operatorname{Tr}\left(({\boldsymbol H}^z)^2\right)}{T^2} {\mathrm{d}}^2 z.$$ Here, we used $\log (1 + x) \leq x$ for $x\geq 0$. Furthermore, if ${\lvert z \rvert} \leq 1$, then we have $$\label{eq:computation_trace_H_z} \operatorname{Tr}(({\boldsymbol H}^z)^2) = \frac{1}{n} \sum_{i,j=1}^{n} (x_{ij} - z \delta_{ij})(\overline{x_{ij}} - \bar z \delta_{ij}) \leq \frac{2}{n} \sum_{i,j=1}^n {\lvert x_{ij} \rvert}^2 + 2 {\lvert z \rvert}^2 \prec 1,$$ where we applied in the first and in the last step. Therefore, choosing $T {\mathrel{\vcenter{\baselineskip0.5ex \lineskiplimit0pt\hbox{\scriptsize.}\hbox{\scriptsize.}}}=}n^{100}$, we conclude from and that the first term in is stochastically dominated by $n^{-1+2a} {\lVert \Delta f \rVert}_1$. To control the second term on right-hand side of , we define $$\label{eq:estimate_eta_integral} I(z) {\mathrel{\vcenter{\baselineskip0.5ex \lineskiplimit0pt\hbox{\scriptsize.}\hbox{\scriptsize.}}}=}\int_0^T {\left\lvert {\mathrm{Im}\,}m^z({\mathrm{i}}\eta) - {\langle v_1( \eta ) \rangle} \right\rvert} {\mathrm{d}}\eta$$ for $z \in {\ensuremath{\mathbb D_{<}}}$. We will conclude below the following lemma. \[lem:moment\_bound\_I\_z\] For every $\delta>0$ and $p \in {\mathbb{N}}$, there is a positive constant $C$, depending only on $\delta$ and $p$ in addition to the model parameters and ${\tau}_*$, such that $$\label{eq:moments_I_z} \sup_{z\in{\ensuremath{\mathbb D_{<}}}} {\mathbb{E}}I(z)^p \leq C \frac{n^{\delta p}}{n^p}.$$ We now show that this moment bound on $I(z)$ will yield that the second term in is $\prec n^{-1 + 2a} {\lVert \Delta f \rVert}_1$. Indeed, for every $p\in{\mathbb{N}}$ and $\delta >0$, using Hölder’s inequality, we estimate $$\begin{aligned} {\mathbb{E}}{\left\lvert \int_{\mathbb{C}}\Delta f_{z_0,a}(z) \int_0^T \left[{\mathrm{Im}\,}m^z({\mathrm{i}}\eta) - {\langle v_1(\eta) \rangle} \right] {\mathrm{d}}\eta\; {\mathrm{d}}^2 z \right\rvert}^p & \leq \int_{\mathbb{C}}\ldots \int_{\mathbb{C}}\prod_{i=1}^p {\lvert \Delta f_{z_0,a}(z_i) \rvert} \prod_{i=1}^p \left({\mathbb{E}}I(z_i)^p\right)^{1/p} {\mathrm{d}}^2 z_1 \ldots {\mathrm{d}}^2 z_p \nonumber \\ & \leq C {\lVert \Delta f \rVert}_1^p\frac{n^{\delta p + 2a p}}{n^p} . \label{eq:moment_bound_second_term}\end{aligned}$$ Applying Chebyshev’s inequality to and using that $\delta>0$ and $p$ were arbitrary, we get $${\left\lvert \int_{\mathbb{C}}\Delta f_{z_0,a}(z) \int_0^T {\mathrm{Im}\,}m^z({\mathrm{i}}\eta) - {\langle v_1(\eta) \rangle} {\mathrm{d}}\eta \; {\mathrm{d}}^2 z \right\rvert} \prec n^{-1 + 2a } {\lVert \Delta f \rVert}_1.$$ Hence, the bound on the second term on the right-hand side of follows once we have proven . For the third term in , notice that the integrand is bounded by $C\eta^{-2}$ so it is bounded by $n^{2a} T^{-1}{\lVert \Delta f \rVert}_1$. This concludes the proof of (i) of Theorem \[thm:local\_circular\_law\] up to the proof of Lemma \[lem:moment\_bound\_I\_z\] which is given below. We now turn to the proof of (ii). We will use an interpolation between the random matrix $X$ and an independent Ginibre matrix ${\widehat}X$ together with the well-known result that a Ginibre matrix does not have any eigenvalues ${\lvert \lambda \rvert} \geq 1 + {\tau}_*$ with very high probability. With the help of we will control the number of eigenvalues outside of the disk of radius $1+{\tau}^*$ along the flow. We fix ${\tau}^* > 1 + {\tau}_*$. Let $({\widehat}x_{ij})_{i,j=1}^n$ be independent centered complex Gaussians of variance $n^{-1}$, i.e., ${\mathbb{E}}\,{\widehat}x_{ij} = 0$ and ${\mathbb{E}}{\lvert {\widehat}x_{ij} \rvert}^2 = n^{-1}$. We set ${\widehat}X {\mathrel{\vcenter{\baselineskip0.5ex \lineskiplimit0pt\hbox{\scriptsize.}\hbox{\scriptsize.}}}=}({\widehat}x_{ij})_{i,j=1}^n$, i.e. ${\widehat}X$ is a Ginibre matrix. We denote the eigenvalues of ${\widehat}X$ by ${\widehat}{\sigma}_1, \ldots, {\widehat}{\sigma}_n$. For $t\in [0,1]$, we denote the spectral radius of the matrix $t S + (1-t)E$ by $\rho_t {\mathrel{\vcenter{\baselineskip0.5ex \lineskiplimit0pt\hbox{\scriptsize.}\hbox{\scriptsize.}}}=}\rho(t S + (1-t)E)$, where $E$ is the $n\times n$ matrix with entries $e_{ij} {\mathrel{\vcenter{\baselineskip0.5ex \lineskiplimit0pt\hbox{\scriptsize.}\hbox{\scriptsize.}}}=}1/n$, $E = (e_{ij})_{i,j=1}^n$. Furthermore, we define $$X^t {\mathrel{\vcenter{\baselineskip0.5ex \lineskiplimit0pt\hbox{\scriptsize.}\hbox{\scriptsize.}}}=}\rho_t^{-1/2}\left(t X + (1-t){\widehat}X\right), \quad {\boldsymbol H}^{z,t} {\mathrel{\vcenter{\baselineskip0.5ex \lineskiplimit0pt\hbox{\scriptsize.}\hbox{\scriptsize.}}}=}\begin{pmatrix} 0 & X^t - z{\mathds{1}}\\ \left(X^t - z {\mathds{1}}\right)^* & 0 \end{pmatrix}$$ for $t \in [0,1]$. The eigenvalues of $X^t$ and ${\boldsymbol H}^{z,t}$ are denoted by ${\sigma}^t_i$ and ${\lambda}_k^t(z)$, respectively, for $i=1, \ldots, n$ and $k=1,\ldots, 2n$. The one parameter family $t\mapsto X^t$ interpolates between $X$ and ${\widehat}X$ by keeping the spectral radius of the variance matrix at constant one. Note that ${\lVert (X^t-z)^{-1} \rVert_{2}} = \max_{k=1}^{2n} {\lvert {\lambda}_k^t(z) \rvert}^{-1}$. We can apply Lemma \[lemma:small\] to the matrices $X^t$ for any $t$ to get $$\sup_{z\in {\ensuremath{\mathbb D_{>}}}} {\left\lVert (X^t-z)^{-1} \right\rVert_{2}} \prec n^{1/2}$$ uniformly in $t$ from . In fact, the estimate can be strengthened to $$\label{eq:resolvent_bound} \sup_{t \in [0,1]} \sup_{z\in {\ensuremath{\mathbb D_{>}}}} {\left\lVert (X^t-z)^{-1} \right\rVert_{2}} \prec n^{1/2}$$ exactly in the same way as was strengthened to , we only need to observe that the two parameter family $(z, t) \mapsto \operatorname{Im}\operatorname{Tr}{\boldsymbol G}^{z,t}(\eta)$ is Lipschitz continuous in both variables, where ${\boldsymbol G}^{z,t}$ denotes the resolvent of ${\boldsymbol H}^{z,t}$. Let $\gamma$ be the circle in ${\mathbb{C}}$ centered at zero with radius $1 +{\tau}_*$. For $t\in [0,1]$, we have $$N(t) {\mathrel{\vcenter{\baselineskip0.5ex \lineskiplimit0pt\hbox{\scriptsize.}\hbox{\scriptsize.}}}=}\# \{ i \mid {\lvert {\sigma}_i^t \rvert} \leq 1 +{\tau}_* \} = \frac{n}{2\pi{\mathrm{i}}} \int_{\gamma} \operatorname{Tr}\left((X^t-z)^{-1}\right) {\mathrm{d}}z,$$ where $\operatorname{Tr}\colon {\mathbb{C}}^{n\times n} \to {\mathbb{C}}$ denotes the normalized trace, i.e., $\operatorname{Tr}{\mathds{1}}= 1$. Due to $N(t)$ is a continuous function of $t$. Thus, $N(t)$ is constant as a continuous integer valued function. Using Corollary 2.3 of [@fey2008], we obtain that $\# \{ k \mid {\lvert {\widehat}{\sigma}_k \rvert} \geq {\tau}^*\} = 0$ with very high probability. Furthermore, $\# \{ k \mid {\widehat}{\sigma}_k \in {\ensuremath{\mathbb D_{>}}}\} = 0$ with very high probability by . Thus, $$N(1) = N(0) = n - \# \{ k \mid {\widehat}{\sigma}_k \in {\ensuremath{\mathbb D_{>}}}\} - \# \{ k \mid {\lvert {\widehat}{\sigma}_k \rvert} \geq {\tau}^*\} = n$$ with very high probability which concludes the proof of (ii) and hence of Theorem \[thm:local\_circular\_law\]. In the above proof we showed that $\| {\boldsymbol H}^z \|\le C$ with very high probability via an interpolation argument using the norm-boundedness of a Ginibre matrix and the local law for the entire interpolating family. Robust upper bounds on the norm of random matrices are typically proven by a simple moment method. Such approach also applies here. For example, one may follow the proof of Lemma 7.2 in [@EYYbulk], and estimate every moment ${\mathbb{E}}|x_{ij}|^k$ by its maximum over all $i, j$. The final constant estimating $\|{\boldsymbol H}^z\|$ will not be optimal due to these crude bounds, but it will still only depend on $s^*$ and $\mu_m$ from , . This argument is very robust, in particular it does not use Hermiticity. To show , we use the following estimate which converts a bound in $\prec$ into a moment bound. For every nonnegative random variable satisfying $Y \prec 1/n$ and $Y \leq n^c$ for some $c>0$ the $p^{\text{th}}$ moment is bounded by $$\label{eq:prec_to_moment} {\mathbb{E}}Y^p \leq {\mathbb{E}}Y^p {\ensuremath{\chi}}(Y \leq n^{\delta-1}) + \left({\mathbb{E}}Y^{2p}\right)^{1/2} \left({\mathbb{P}}\left(Y > n^{\delta-1}\right)\right)^{1/2} \leq C\frac{n^{p\delta}}{n^p},$$ for all $p\in{\mathbb{N}}$, $\delta >0$ and for some $C>0$, depending on $c$, $p$ and $\delta$. As a first step in the proof of , we choose ${\varepsilon}\in (0,1/2)$, split the ${\mathrm{d}}\eta$ integral in the definition of $I(z)$, , and consider the regimes $\eta \leq n^{-1+{\varepsilon}}$ and $\eta \geq n^{-1+{\varepsilon}}$, separately. For $\eta\le n^{-1+{\varepsilon}}$, we compute $$\int_0^{n^{-1 + {\varepsilon}}} {\mathrm{Im}\,}m^z({\mathrm{i}}\eta) {\mathrm{d}}\eta = \frac{1}{2n} \sum_{i=1}^n \log \left( 1 + \frac{n^{-2+2{\varepsilon}}}{{\lambda}_i^2} \right).$$ We recall that ${\lambda}_1, \ldots, {\lambda}_{2n}$ are the eigenvalues of ${\boldsymbol H}^z$. Therefore, yields $$\begin{aligned} \int_0^T \left[{\mathrm{Im}\,}m^z({\mathrm{i}}\eta) - {\langle v_1(\eta) \rangle}\right] {\mathrm{d}}\eta & = \frac{1}{n} \sum_{{\lvert {\lambda}_i \rvert} < n^{-l}} \log \left( 1 + \frac{n^{-2+2{\varepsilon}}}{{\lambda}_i^2} \right) + \frac{1}{n} \sum_{{\lvert {\lambda}_i \rvert} \geq n^{-l}} \log \left( 1 + \frac{n^{-2+2{\varepsilon}}}{{\lambda}_i^2} \right) - \int_0^{n^{-1+{\varepsilon}}} {\langle v_1(\eta) \rangle} {\mathrm{d}}\eta \nonumber \\ & ~~ + \int_{n^{-1+{\varepsilon}}}^1 \left[{\mathrm{Im}\,}m^z({\mathrm{i}}\eta) - {\langle v_1(\eta) \rangle}\right] {\mathrm{d}}\eta + \int_1^T\left[ {\mathrm{Im}\,}m^z({\mathrm{i}}\eta) - {\langle v_1(\eta) \rangle}\right] {\mathrm{d}}\eta. \label{eq:decomposition_second_term_master_formula}\end{aligned}$$ Here, $l\in{\mathbb{N}}$ is a large fixed integer to be chosen later. We will estimate each of the terms on the right-hand side of individually. For the first term in , we compute $${\mathbb{E}}\left( \frac{1}{n}\sum_{{\lvert {\lambda}_i \rvert}\leq n^{-l}} \log \left( 1 + \frac{n^{-2 +2{\varepsilon}}}{{\lambda}_i^2} \right) \right)^p \leq {\mathbb{E}}\left[ \log^p \left( 1 + \frac{n^{-2+2{\varepsilon}}}{{\lambda}_n^2}\right) {\ensuremath{\chi}}({\lambda}_n \leq n^{-l}) \right] \leq C {\mathbb{E}}\left[ {\lvert \log {\lambda}_n \rvert}^{p}{\ensuremath{\chi}}({\lambda}_n \leq n^{-l}) \right]$$ for some constant $C>0$ independent of $n$. We compute the expectation directly $${\mathbb{E}}\left[{\lvert \log {\lambda}_n \rvert}^{p} {\ensuremath{\chi}}({\lambda}_n \leq n^{-l})\right] = p\int_{{ l \log n}}^\infty {\mathbb{P}}\left( {\lambda}_n \leq {\mathrm{e}}^{-t}\right)t^{p-1} {\mathrm{d}}t \leq C n^{\beta + 1 + 2\alpha/(1+\alpha)} \int_{{ l \log n}}^\infty t^{p-1} {\mathrm{e}}^{-2\alpha t/(1+\alpha)} {\mathrm{d}}t.$$ Here, we applied in Proposition \[pro:least\_singular\_value\] with $u = {\mathrm{e}}^{-t}{n}$. Choosing $l$ large enough, depending on $\alpha$, $\beta$ and $p$, we obtain that the right-hand side is smaller than $n^{-p}$. This shows the bound for the first term in . We will apply for estimating the absolute value of the second, fourth and fifth term on the right-hand side of . For the first term, we will need a separate argument based on Proposition \[pro:least\_singular\_value\]. To estimate the second term on the right-hand side of , we decompose the sum into three regimes, $n^{-l} \leq {\lvert {\lambda}_i \rvert} < n^{-1+{\varepsilon}}$, $n^{-1+{\varepsilon}} \leq {\lvert {\lambda}_i \rvert} < n^{-1/2}$ and $n^{-1/2} \leq {\lvert {\lambda}_i \rvert}$. For the first regime, we use with $\eta = n^{-1+{\varepsilon}}$ and $\log( 1+ n^{-2 + 2 {\varepsilon}+l}) \leq C \log n$ to get $$\label{eq:smallest_singular_value_term4} \frac{1}{n} \sum_{{\lvert {\lambda}_i \rvert} \in [n^{-l},n^{-1+{\varepsilon}}] } \log\left(1 + \frac{n^{-2 +2 {\varepsilon}}}{{\lambda}_i^2}\right) \leq \frac{C \log n}{n} \# \{ i \colon {\lvert {\lambda}_i \rvert} \leq n^{-1 +{\varepsilon}} \} \prec \frac{n^{\varepsilon}}{n}.$$ As this sum is clearly polynomially bounded in $n$ we can apply to conclude that the first regime of the second term in fulfills the moment bound in . For the intermediate regime, due to the symmetry $\operatorname*{Spec}({\boldsymbol H}^z) = - \operatorname*{Spec}({\boldsymbol H}^z)$, we only consider the positive eigenvalues. We decompose the interval $[n^{-1+{\varepsilon}}, n^{-1/2}]$ into dyadic intervals of the form $[\eta_k, \eta_{k+1}]$, where $\eta_k {\mathrel{\vcenter{\baselineskip0.5ex \lineskiplimit0pt\hbox{\scriptsize.}\hbox{\scriptsize.}}}=}2^k n^{-1 + {\varepsilon}}$. Thus, we obtain $$\label{eq:smallest_singular_value_term2} \frac{1}{n} \sum_{{\lvert {\lambda}_i \rvert} \in [n^{-1+{\varepsilon}},n^{-1/2}] } \log\left(1 + \frac{n^{-2 +2 {\varepsilon}}}{{\lambda}_i^2}\right) \leq \frac{2}{n} \sum_{k=0}^{N} \sum_{{\lambda}_i \in [\eta_k, \eta_{k+1}]} \log\left(1 + \frac{n^{-2 +2 {\varepsilon}}}{{\lambda}_i^2}\right) \prec \frac{n^{\varepsilon}}{n},$$ where we introduced $N = O(\log n)$ in the first step. Moreover, we used the monotonicity of the logarithm, $\log ( 1 + x ) \leq x$ in the last step and the following consequence of : $$\# \{ i \colon {\lambda}_i \in [\eta_k, \eta_{k+1}]\}\leq \# \{ i \colon {\lvert {\lambda}_i \rvert} \leq \eta_{k+1}\} \prec n^{\varepsilon}2^{k+1}.$$ The left-hand side of is trivially bounded by $\log 2$. Therefore, applying to the left-hand side of , we conclude that it satisfies the moment estimate in . For estimating the second term in in the third regime, employing ${\lvert {\lambda}_i \rvert} \geq n^{-1/2}$ and $\log( 1 + x) \leq x$, we obtain $$\label{eq:smallest_singular_value_term3} \frac{1}{n}\sum_{{\lvert {\lambda}_i \rvert} \geq n^{-1/2}} \log\left( 1+ \frac{n^{-2 + 2{\varepsilon}}}{{\lambda}_i^2}\right) \leq \frac{1}{n} \sum_{{\lvert {\lambda}_i \rvert} \geq n^{-1/2}} \log\left( 1+ n^{-1 + 2{\varepsilon}}\right) \leq \frac{2n^{2{\varepsilon}}}{n}.$$ Here, we used that ${\boldsymbol H}^z$ has $2n$ eigenvalues (counted with multiplicities). This deterministic bound and imply that the moments of this sum are bounded by the right-hand side in . Combining the estimates in these three regimes, , and , we conclude that the second term in satisfies the moment bound in . We now estimate the third term on the right-hand side of . Since ${\boldsymbol v}\sim 1$ for $z \in {\ensuremath{\mathbb D_{<}}}$ and $\eta\leq 1$ by , the $p^{\text{th}}$ power of the third term is immediately bounded by the right-hand side of . To bound the fourth and fifth term in , we note that ${\mathrm{Im}\,}m^z({\mathrm{i}}\eta) = {\langle {\boldsymbol g}(\eta) \rangle}$ for $\eta >0$ and recalling the choice $T=n^{100}$, we obtain $$\label{eq:bounds_master_formula_large_eta} \int_{n^{-1+{\varepsilon}}}^1 {\left\lvert {\mathrm{Im}\,}m^z({\mathrm{i}}\eta) - {\langle v_1(\eta) \rangle} \right\rvert} {\mathrm{d}}\eta \prec \frac{n^{\varepsilon}}{n}, \qquad \int_1^T {\left\lvert {\mathrm{Im}\,}m^z({\mathrm{i}}\eta) - {\langle v_1(\eta) \rangle} \right\rvert} {\mathrm{d}}\eta \prec \frac{1}{n}$$ from the first and third regime in with ${\boldsymbol y}=1$. As the integrands are bounded by $n^2$ trivially yields that the moments of the fourth and fifth term in are bounded by the right-hand side in . Since ${\varepsilon}\in (0,1/2)$ was arbitrary this concludes the proof of . Proof of Lemma \[lem:existence\_uniqueness\_vf\_equation\] ========================================================== The existence and uniqueness of the solution of will be a consequence of the existence and uniqueness of the matrix Dyson equation $$\label{eq:matrix_valued} - {\boldsymbol M}^{-1}(\eta) = {\mathrm{i}}\eta {\mathds{1}}-{\boldsymbol A}+ \mathcal S[{\boldsymbol M}(\eta)].$$ Note that ${\boldsymbol A}\in{\mathbb{C}}^{2n\times 2n}$ and ${\mathcal}{S}\colon {\mathbb{C}}^{2n\times 2n} \to {\mathbb{C}}^{2n\times 2n}$ were defined in . The matrix Dyson equation, , has a unique solution under the constraint that the imaginary part $${\mathrm{Im}\,}{\boldsymbol M}{\mathrel{\vcenter{\baselineskip0.5ex \lineskiplimit0pt\hbox{\scriptsize.}\hbox{\scriptsize.}}}=}\frac{1}{2{\mathrm{i}}}({\boldsymbol M}- {\boldsymbol M}^*)$$ is positive definite. This was established in [@Helton01012007]. In the context of random matrices, was studied in [@AjankiCorrelated]. In the following proof, for vectors $a, b, c, d\in {\mathbb{C}}^n$, we will denote the $2n\times 2n$ matrix having diagonal matrices with diagonals $a, b,c,d$ on its top-left, top-right, lower-left and lower-right $n\times n$ blocks, respectively, by $$\begin{pmatrix} a & b \\ c & d \end{pmatrix}{\mathrel{\vcenter{\baselineskip0.5ex \lineskiplimit0pt\hbox{\scriptsize.}\hbox{\scriptsize.}}}=}\begin{pmatrix} \operatorname{diag}a & \operatorname{diag}b \\ \operatorname{diag}c & \operatorname{diag}d \end{pmatrix}\in {\mathbb{C}}^{2n\times 2n}.$$ We show that there is a bijection between the solutions of with positive definite imaginary part ${\mathrm{Im}\,}{\boldsymbol M}$ and the positive solutions of . We remark that implies that there are vector-valued functions $a,b,c,d \colon {\mathbb{R}}_+ \to {\mathbb{C}}^n$ such that for all $\eta >0$ we have $$\label{eq:structure_M} {\boldsymbol M}(\eta) = \begin{pmatrix} a(\eta) & b(\eta) \\ c(\eta) & d(\eta) \end{pmatrix} .$$ First, we show that ${\mathrm{Im}\,}\operatorname{diag}{\boldsymbol M}$ is a solution of satisfying ${\mathrm{Im}\,}\operatorname{diag}{\boldsymbol M}>0$ if ${\boldsymbol M}$ satisfies and ${\mathrm{Im}\,}{\boldsymbol M}$ is positive definite. Due to , multiplying by ${\boldsymbol M}$ yields that is equivalent to $$\label{eq:matrix_valued_as_vector_equations} -1 = {\mathrm{i}}\eta a + a Sd + b \bar z ,\quad 0 = {\mathrm{i}}\eta b + z a + b S^t a ,\quad 0 = {\mathrm{i}}\eta c + \bar z d + c S d ,\quad -1 = {\mathrm{i}}\eta d + d S^ta + z c$$ Solving the second relation in for $b$ and the third relation in for $c$, we obtain $$\label{eq:matrix_valued_aux1} b = -\frac{z a}{{\mathrm{i}}\eta + S^t a}, \qquad c = -\frac{\bar z d}{{\mathrm{i}}\eta + S d}.$$ Plugging the first relation in into the first relation in and the second relation in into the fourth relation in and dividing the results by $a$ and $d$, respectively, imply $$-\frac{1}{a} = {\mathrm{i}}\eta + Sd - \frac{{\lvert z \rvert}^2}{{\mathrm{i}}\eta + S^t a} , \quad -\frac{1}{d} = {\mathrm{i}}\eta + S^ta - \frac{{\lvert z \rvert}^2}{{\mathrm{i}}\eta + S d}.$$ Therefore, if $a$ and $d$ are purely imaginary then $({\mathrm{Im}\,}a, {\mathrm{Im}\,}d) = -{\mathrm{i}}(a,d)$ will fulfill . In order to prove that $a$ and $d$ are purely imaginary, we define $${\widetilde}{\boldsymbol M}{\mathrel{\vcenter{\baselineskip0.5ex \lineskiplimit0pt\hbox{\scriptsize.}\hbox{\scriptsize.}}}=}\begin{pmatrix} {\widetilde}a(\eta) & {\widetilde}b(\eta) \\ {\widetilde}c(\eta) & {\widetilde}d(\eta) \end{pmatrix} {\mathrel{\vcenter{\baselineskip0.5ex \lineskiplimit0pt\hbox{\scriptsize.}\hbox{\scriptsize.}}}=}\begin{pmatrix} - \bar a & \frac{z}{\bar z} \bar b \\ \frac{\bar z }{z} \bar c & - \bar d \end{pmatrix}.$$ The goal is to conclude ${\boldsymbol M}= {\widetilde}{\boldsymbol M}$, and hence $a = - \bar a$ and $d= - \bar d$, from the uniqueness of the solution of with positive definite imaginary part. Since the relations are fulfilled if $a$, $b$, $c$, $d$ are replaced by ${\widetilde}a$, ${\widetilde}b$, ${\widetilde}c$, ${\widetilde}d$, respectively, ${\widetilde}{\boldsymbol M}$ satisfies . For $j = 1, \ldots, n$, we define the $2\times 2$ matrices $$M_j {\mathrel{\vcenter{\baselineskip0.5ex \lineskiplimit0pt\hbox{\scriptsize.}\hbox{\scriptsize.}}}=}\begin{pmatrix} a_j & b_j \\ c_j & d_j \end{pmatrix}, \quad {\widetilde}M_j {\mathrel{\vcenter{\baselineskip0.5ex \lineskiplimit0pt\hbox{\scriptsize.}\hbox{\scriptsize.}}}=}\begin{pmatrix} {\widetilde}a_j & {\widetilde}b_j \\ {\widetilde}c_j & {\widetilde}d_j \end{pmatrix}.$$ Note that ${\mathrm{Im}\,}{\boldsymbol M}$ is positive definite if and only if ${\mathrm{Im}\,}M_j$ is positive definite for all $j=1, \ldots, n$. Similarly, the positive definiteness of ${\mathrm{Im}\,}{\widetilde}{\boldsymbol M}$ is equivalent to the positive definiteness of ${\mathrm{Im}\,}{\widetilde}M_j$ for all $j=1, \ldots, n$. We have $${\mathrm{Im}\,}M_j = \begin{pmatrix} {\mathrm{Im}\,}a_j & \frac{1}{2{\mathrm{i}}}( b_j - \bar c_j) \\ \frac{1}{2{\mathrm{i}}}(c_j - \bar b_j) & {\mathrm{Im}\,}d_j \end{pmatrix}, \quad {\mathrm{Im}\,}{\widetilde}M_j {\mathrel{\vcenter{\baselineskip0.5ex \lineskiplimit0pt\hbox{\scriptsize.}\hbox{\scriptsize.}}}=}\begin{pmatrix} {\mathrm{Im}\,}a_j & \frac{z}{2{\mathrm{i}}\bar z} ( \bar b_j - c_j) \\ \frac{\bar z}{2{\mathrm{i}}z} ( \bar c_j - b_j) & {\mathrm{Im}\,}d_j \end{pmatrix}.$$ As $\operatorname{Tr}{\mathrm{Im}\,}{\widetilde}M_j = \operatorname{Tr}{\mathrm{Im}\,}M_j$ and $\det {\mathrm{Im}\,}{\widetilde}M_j = \det {\mathrm{Im}\,}M_j$ for all $j =1, \ldots,n $ we get that ${\widetilde}{\boldsymbol M}$ is a solution of with positive definite imaginary part ${\mathrm{Im}\,}{\widetilde}{\boldsymbol M}$. Thus, the uniqueness of the solution of implies ${\boldsymbol M}= {\widetilde}{\boldsymbol M}$ as well as $a= - \bar a$ and $d = - \bar d$. Moreover, since $${\mathrm{Im}\,}{\boldsymbol M}= \begin{pmatrix} {\mathrm{Im}\,}a & ( b - \bar c)/(2{\mathrm{i}}) \\ (c - \bar b)/(2{\mathrm{i}}) & {\mathrm{Im}\,}d \end{pmatrix}$$ is positive definite we have that ${\mathrm{Im}\,}a >0$ and ${\mathrm{Im}\,}d >0$. Hence, $({\mathrm{Im}\,}a, {\mathrm{Im}\,}d)$ is a positive solution of . Conversely, let ${\boldsymbol v}= (v_1, v_2) \in {\mathbb{C}}^{2n}$ be a solution of satisfying ${\boldsymbol v}>0$ and $u$ be defined as in . Because of , we obtain that ${\boldsymbol M}= {\boldsymbol M}^z$, defined as in , is a solution of . To conclude that ${\mathrm{Im}\,}{\boldsymbol M}$ is positive definite, it suffices to show that $\det {\mathrm{Im}\,}M_j>0$ for all $j = 1, \ldots,n$ with $$M_j {\mathrel{\vcenter{\baselineskip0.5ex \lineskiplimit0pt\hbox{\scriptsize.}\hbox{\scriptsize.}}}=}\begin{pmatrix}{\mathrm{i}}(v_1)_j & -z u_j \\ - \bar z u_j & {\mathrm{i}}(v_2)_j \end{pmatrix}$$ as $\operatorname{Tr}{\mathrm{Im}\,}M_j = (v_1)_j + (v_2)_j >0$. Since $z u_j - \overline{\bar z u_j} = 0$ for all $j=1, \ldots, n$ by we obtain $$\det {\mathrm{Im}\,}M_j = (v_1)_j (v_2)_j - \frac{1}{4} {\lvert z u_j - \overline{\bar z u_j} \rvert}^2 = (v_1)_j (v_2)_j >0.$$ Therefore, there is a bijection between the solutions of with positive definite imaginary part and the positive solutions of . Appealing to the existence and uniqueness of proved in [@Helton01012007] concludes the proof of Lemma \[lem:existence\_uniqueness\_vf\_equation\]. Contraction-Inversion Lemma {#app:aux_results} =========================== \[proof:rotation\_inversion\] The bounds imply that ${\mathds{1}}-{\boldsymbol A}{\boldsymbol B}$ is invertible and $${\lVert ({\mathds{1}}-{\boldsymbol A}{\boldsymbol B})^{-1} \rVert_{2}}\le \frac{1}{c_1\eta}.$$ The main point of this lemma is to show that $({\mathds{1}}-{\boldsymbol A}{\boldsymbol B})^{-1}{\boldsymbol p}$ can be bounded independently of $\eta$ for ${\boldsymbol p}$ satisfying . We introduce ${\boldsymbol h}{\mathrel{\vcenter{\baselineskip0.5ex \lineskiplimit0pt\hbox{\scriptsize.}\hbox{\scriptsize.}}}=}({\mathds{1}}-{\boldsymbol A}{\boldsymbol B})^{-1}{\boldsymbol p}$. Thus, is equivalent to ${\lVert {\boldsymbol h}\rVert_{2}} \leq C {\lVert {\boldsymbol p}\rVert_{2}}$ for some $C>0$ which depends only on $c_1, c_2, c_3$ and ${\varepsilon}$. Without loss of generality, we may assume that ${\lVert {\boldsymbol h}\rVert_{2}}=1$. We decompose $$\label{eq:stab_lemma_decomposition} {\boldsymbol h}= \alpha {\boldsymbol b}_- + \beta {\boldsymbol b}_+ + \gamma {\boldsymbol x},$$ where $\alpha = {\langle{{\boldsymbol b}_-} \mspace{2mu}, {{\boldsymbol h}}\rangle}$, $\beta = {\langle{{\boldsymbol b}_+} \mspace{2mu}, {{\boldsymbol h}}\rangle}$ and $ {\boldsymbol x}\perp {\boldsymbol b}_\pm$ satisfying ${\lVert {\boldsymbol x}\rVert_{2}} = 1$, thus ${\lvert \alpha \rvert}^2 + {\lvert \beta \rvert}^2 +{\lvert \gamma \rvert}^2=1$. Since ${\boldsymbol B}={\boldsymbol B}^*$, we have ${\boldsymbol b}_+\perp {\boldsymbol b}_-$ and ${\boldsymbol B}{\boldsymbol x}\perp {\boldsymbol b}_\pm$. Hence, we obtain $${\lVert {\boldsymbol A}{\boldsymbol B}{\boldsymbol h}\rVert_{2}}^2 \leq {\lVert {\boldsymbol B}{\boldsymbol h}\rVert_{2}}^2 \leq {\lvert \alpha \rvert}^2 {\lVert {\boldsymbol B}\rVert_{2}} + {\lvert \beta \rvert}^2 {\lVert {\boldsymbol B}\rVert_{2}} + {\lvert \gamma \rvert}^2 {\lVert {\boldsymbol B}{\boldsymbol x}\rVert_{2}}^2 \leq 1 - {\varepsilon}+ {\varepsilon}({\lvert \alpha \rvert}^2 + {\lvert \beta \rvert}^2 ),$$ where we used ${\lVert {\boldsymbol A}\rVert_{2}} \leq 1$, ${\lVert {\boldsymbol B}\rVert_{2}}\le 1$ and ${\lVert {\boldsymbol B}{\boldsymbol x}\rVert_{2}}\le 1-{\varepsilon}$ in the last step. Therefore, if ${\lvert \alpha \rvert}^2 + {\lvert \beta \rvert}^2 \leq 1 - \delta$ for some $\delta>0$ to be determined later, then ${\lVert {\boldsymbol A}{\boldsymbol B}{\boldsymbol h}\rVert_{2}} \leq \sqrt{1 - {\varepsilon}\delta}{\lVert {\boldsymbol h}\rVert_{2}} \leq (1 - {\varepsilon}\delta/2) {\lVert {\boldsymbol h}\rVert_{2}}$ and thus $$\label{eq:stab_lemma_final_estimate_exact} 1={\lVert {\boldsymbol h}\rVert_{2}} \leq \frac{2}{{\varepsilon}\delta} {\lVert {\boldsymbol p}\rVert_{2}}.$$ For the rest of the proof, we assume that ${\lvert \alpha \rvert}^2 + {\lvert \beta \rvert}^2 \geq 1- \delta$. In the regime, where ${\lvert \alpha \rvert}$ is relatively large, we compute ${\langle{{\boldsymbol b}_-} \mspace{2mu}, { ({\mathds{1}}-{\boldsymbol A}{\boldsymbol B}){\boldsymbol h}}\rangle}$, capitalize on the positivity of ${\langle{{\boldsymbol b}_-} \mspace{2mu}, { ({\mathds{1}}-{\boldsymbol A}{\boldsymbol B}){\boldsymbol b}_-}\rangle}$ and treat all other terms as errors. In the opposite regime, where ${\lvert \beta \rvert}$ is relatively large, we use the positivity of ${\langle{{\boldsymbol b}_+} \mspace{2mu}, { ({\mathds{1}}-{\boldsymbol A}{\boldsymbol B}){\boldsymbol b}_+}\rangle}$. Using , we compute $${\langle{{\boldsymbol b}_-} \mspace{2mu}, {{\boldsymbol p}}\rangle} = {\langle{{\boldsymbol b}_-} \mspace{2mu}, {({\mathds{1}}-{\boldsymbol A}{\boldsymbol B}){\boldsymbol h}}\rangle} = \alpha (1 + {\lVert {\boldsymbol B}\rVert_{2}}{\langle{{\boldsymbol b}_-} \mspace{2mu}, {{\boldsymbol A}{\boldsymbol b}_-}\rangle}) - \beta {\lVert {\boldsymbol B}\rVert_{2}} {\langle{{\boldsymbol b}_-} \mspace{2mu}, {{\boldsymbol A}{\boldsymbol b}_+}\rangle} - \gamma {\langle{{\boldsymbol b}_-} \mspace{2mu}, {{\boldsymbol A}{\boldsymbol B}{\boldsymbol x}}\rangle}.$$ From ${\lVert {\boldsymbol A}\rVert_{2}} \leq 1$, the Hermiticity of ${\boldsymbol A}$, ${\langle{{\boldsymbol b}_-} \mspace{2mu}, {{\boldsymbol B}{\boldsymbol x}}\rangle} =0$, and , we deduce $${\lvert {\langle{{\boldsymbol b}_-} \mspace{2mu}, {{\boldsymbol A}{\boldsymbol b}_-}\rangle} \rvert} \leq 1, \quad {\lvert {\langle{{\boldsymbol b}_-} \mspace{2mu}, {{\boldsymbol A}{\boldsymbol b}_+}\rangle} \rvert} = {\lvert {\langle{{\boldsymbol b}_- + {\boldsymbol A}{\boldsymbol b}_-} \mspace{2mu}, {{\boldsymbol b}_+}\rangle} \rvert} \leq c_2 \eta, \quad {\lvert {\langle{{\boldsymbol b}_-} \mspace{2mu}, {{\boldsymbol A}{\boldsymbol B}{\boldsymbol x}}\rangle} \rvert} = {\lvert {\langle{{\boldsymbol b}_- + {\boldsymbol A}{\boldsymbol b}_-} \mspace{2mu}, {{\boldsymbol B}{\boldsymbol x}}\rangle} \rvert} \leq c_2 \eta(1-{\varepsilon}).$$ Employing these estimates, ${\lVert {\boldsymbol B}\rVert_{2}} \leq 1 - c_1 \eta$ and , together with ${\lvert \gamma \rvert}^2\le \delta$, we obtain $$\label{eq:aux_d_estimate2} c_3 {\lVert {\boldsymbol p}\rVert_{2}} \geq {\lvert \alpha \rvert} c_1 - {\lvert \beta \rvert} c_2 - \sqrt{\delta} c_2 (1- {\varepsilon})$$ after dividing through by $\eta>0$. If ${\lvert \alpha \rvert}c_1 \geq c_2 {\lvert \beta \rvert} + \sqrt \delta c_2 (1-{\varepsilon}) + \delta{\varepsilon}c_3/2$ then we obtain . Therefore, it suffices to show in the regime $$\label{eq:stab_lemma_additional_assumptions} {\lvert \gamma \rvert}^2 \leq \delta, \qquad {\lvert \alpha \rvert}c_1 \leq c_2 {\lvert \beta \rvert} + \sqrt \delta c_2 (1-{\varepsilon}) + \delta{\varepsilon}c_3/2.$$ For this regime, we use and obtain $${\langle{{\boldsymbol b}_+} \mspace{2mu}, {{\boldsymbol p}}\rangle}= {\langle{{\boldsymbol b}_+} \mspace{2mu}, {({\mathds{1}}-{\boldsymbol A}{\boldsymbol B}){\boldsymbol h}}\rangle} = \beta (1- {\lVert {\boldsymbol B}\rVert_{2}}{\langle{{\boldsymbol b}_+} \mspace{2mu}, {{\boldsymbol A}{\boldsymbol b}_+}\rangle}) - \alpha{\lVert {\boldsymbol B}\rVert_{2}}{\langle{{\boldsymbol b}_+} \mspace{2mu}, {{\boldsymbol A}{\boldsymbol b}_-}\rangle} - \gamma {\langle{{\boldsymbol b}_+} \mspace{2mu}, {{\boldsymbol A}{\boldsymbol B}{\boldsymbol x}}\rangle}\label{eq:aux_scalar1}.$$ We employ , , the Hermiticity of ${\boldsymbol A}$ and ${\langle{{\boldsymbol b}_-} \mspace{2mu}, {{\boldsymbol b}_+}\rangle} = 0$ to obtain $$\label{eq:equations_estimates_stab_lemma} {\langle{{\boldsymbol b}_+} \mspace{2mu}, {{\boldsymbol A}{\boldsymbol b}_+}\rangle} \leq 1- {\varepsilon}, \quad {\lvert {\langle{{\boldsymbol b}_+} \mspace{2mu}, {{\boldsymbol A}{\boldsymbol b}_-}\rangle} \rvert} = {\lvert {\langle{{\boldsymbol b}_+} \mspace{2mu}, {{\boldsymbol b}_- + {\boldsymbol A}{\boldsymbol b}_-}\rangle} \rvert} \leq c_2 \eta, \quad {\lvert {\langle{{\boldsymbol b}_+} \mspace{2mu}, {{\boldsymbol A}{\boldsymbol B}{\boldsymbol x}}\rangle} \rvert} \leq 1 - {\varepsilon}. \\$$ Applying to , yields $$\label{eq:aux_d_estimate1} {\lVert {\boldsymbol p}\rVert_{2}} \geq {\lvert {\langle{{\boldsymbol b}_+} \mspace{2mu}, {{\boldsymbol p}}\rangle} \rvert} \geq {\lvert \beta \rvert}{\varepsilon}- {\lvert \alpha \rvert} c_2 \eta - {\lvert \gamma \rvert} (1-{\varepsilon}) \geq {\lvert \beta \rvert}{\varepsilon}- {\lvert \alpha \rvert} \frac{{\varepsilon}c_1}{2c_2} - \sqrt{\delta} (1-{\varepsilon}),$$ where we used the assumption $\eta\le {\varepsilon}c_1/2c_2^2$. Since ${\lvert \alpha \rvert} c_1/c_2 \le {\lvert \beta \rvert} + O(\sqrt{\delta})$ from , we obtain that ${\lVert {\boldsymbol p}\rVert_{2}} \ge {\lvert \beta \rvert}{\varepsilon}/3$ for any $\delta\le \delta_0(c_1, c_2, c_3, {\varepsilon})$ sufficiently small. Furthermore, ${\lvert \alpha \rvert}^2 + {\lvert \beta \rvert}^2 \ge 1-\delta$ and the fact that ${\lvert \beta \rvert}$ is large compared with ${\lvert \alpha \rvert}$ in the sense guarantee that ${\lvert \beta \rvert}^2\ge \frac{1}{3}[1+ (c_2/c_1)^2 ]^{-1}$, if $\delta$ is sufficiently small. In particular, ${\lVert {\boldsymbol p}\rVert_{2}} \ge {\varepsilon}\delta/2$ can be achieved with a small $\delta$, i.e., holds true in the regime as well. This concludes the proof of Lemma \[lem:rotation\_inversion\]. \[lem:TwoNorms\_to\_InfNorms\] (i) Uniformly for $z\in {\ensuremath{\mathbb D_{<}}}\cup {\ensuremath{\mathbb D_{>}}}$ and $\eta >0$, we have $$\label{eq:twoinfnorm_Tf_Ff} {\lVert {\boldsymbol F}\rVert_{2\to\infty}} \lesssim 1, \quad {\lVert {\boldsymbol T}{\boldsymbol F}\rVert_{2\to\infty}} \lesssim 1, \quad {\lVert {\boldsymbol F}{\boldsymbol T}\rVert_{2\to\infty}} \lesssim 1.$$ (ii) If $w \notin \operatorname*{Spec}({\boldsymbol T}{\boldsymbol F}) \cup \{0\}$ and ${\lVert (w {\mathds{1}}- {\boldsymbol T}{\boldsymbol F})^{-1} {\boldsymbol y}\rVert_{2}} \lesssim {\lVert {\boldsymbol y}\rVert_{2}}$ for some ${\boldsymbol y}\in {\mathbb{C}}^{2n}$ then $$\label{eq:control_infnorm_with_twonorm_for_TF} {\lVert (w{\mathds{1}}- {\boldsymbol T}{\boldsymbol F})^{-1} {\boldsymbol y}\rVert_{\infty}} \lesssim \frac{1}{{\lvert w \rvert}}{\lVert {\boldsymbol y}\rVert_{\infty}}.$$ A similar statement holds true for $(\bar w {\mathds{1}}- {\boldsymbol F}{\boldsymbol T})^{-1} = \left[ ( w {\mathds{1}}- {\boldsymbol T}{\boldsymbol F})^{-1} \right]^*$. (iii) For every $\eta_*>0$, depending only on ${\tau}_*$ and the model parameters, such that $$\label{eq:inverse_id_minus_TF_Q_conditions} {\lVert ({\mathds{1}}- {\boldsymbol T}{\boldsymbol F})^{-1} {\boldsymbol Q}\rVert_{2}} \lesssim 1, \quad 1- {\lVert {\boldsymbol F}\rVert_{2}} \gtrsim \eta, \quad {\lVert {\boldsymbol f_-}+{\boldsymbol T}{\boldsymbol f_-}\rVert_{2}} \lesssim \eta, \quad {\lVert {\boldsymbol f_-}\rVert_{\infty}}\lesssim 1$$ uniformly for all $\eta \in (0,\eta_*]$ and $z \in {\ensuremath{\mathbb D_{<}}}$, we have $${\lVert \left(({\mathds{1}}-{\boldsymbol T}{\boldsymbol F})^{-1}{\boldsymbol Q}\right)^* \rVert}_\infty \lesssim 1 \label{eq:estimate_stability_inf_projection}$$ uniformly for $\eta \in (0,\eta_*]$ and $z \in {\ensuremath{\mathbb D_{<}}}$. Here, ${\boldsymbol Q}$ denotes the orthogonal projection onto the subspace ${\boldsymbol f_-}^\perp$, i.e., ${\boldsymbol Q}{\boldsymbol y}{\mathrel{\vcenter{\baselineskip0.5ex \lineskiplimit0pt\hbox{\scriptsize.}\hbox{\scriptsize.}}}=}{\boldsymbol y}- {\langle{{\boldsymbol f_-}} \mspace{2mu}, {{\boldsymbol y}}\rangle}{\boldsymbol f_-}$ for every ${\boldsymbol y}\in {\mathbb{C}}^{2n}$. The estimate is proved similarly as (5.28) in [@AjankiQVE]. As ${\lVert {\boldsymbol S}_o \rVert_{2\to\infty}} \lesssim 1$ by , we obtain from Proposition \[pro:estimates\_v\_small\_z\], and $${\lVert {\boldsymbol F}\rVert_{2\to\infty}} \leq {\lVert {\boldsymbol V}^{-1} \rVert_{\infty}} {\lVert {\boldsymbol S}_o \rVert_{2\to\infty}}{\lVert {\boldsymbol V}^{-1} \rVert_{2}} = {\left\lVert \frac{{\boldsymbol u}{\boldsymbol v}}{{\widetilde}{\boldsymbol v}} \right\rVert_{\infty}} {\lVert {\boldsymbol S}_o \rVert_{2\to\infty}}\lesssim 1$$ uniformly for all $\eta >0$ and $z\in {\ensuremath{\mathbb D_{<}}}\cup {\ensuremath{\mathbb D_{>}}}$. This proves the first estimate in . From Lemma \[lem:pro\_Tf\] (i), we conclude the second and the third estimate in . We set ${\boldsymbol x}{\mathrel{\vcenter{\baselineskip0.5ex \lineskiplimit0pt\hbox{\scriptsize.}\hbox{\scriptsize.}}}=}(w {\mathds{1}}-{\boldsymbol T}{\boldsymbol F})^{-1}{\boldsymbol y}$. By assumption there is $C \sim 1$ such that $${\lVert {\boldsymbol x}\rVert_{2}} \leq C {\lVert {\boldsymbol y}\rVert_{2}} \leq C {\lVert {\boldsymbol y}\rVert_{\infty}} .$$ Moreover, since $ w{\boldsymbol x}= {\boldsymbol T}{\boldsymbol F}{\boldsymbol x}+ {\boldsymbol y}$ we obtain from the previous estimate $${\lvert w \rvert} {\lVert {\boldsymbol x}\rVert_{\infty}} \leq {\lVert {\boldsymbol T}{\boldsymbol F}{\boldsymbol x}\rVert_{\infty}} + {\lVert {\boldsymbol y}\rVert_{\infty}} \leq \left( {\lVert {\boldsymbol T}{\boldsymbol F}\rVert_{2\to\infty}} C + 1\right) {\lVert {\boldsymbol y}\rVert_{\infty}}.$$ Using the second estimate in , this concludes the proof of . The statement about $(\bar w{\mathds{1}}- {\boldsymbol F}{\boldsymbol T})^{-1}$ follows in the same way using the third estimate in instead of the second. For the proof of , we remark that the first condition in implies that $$\label{eq:estimate_adjoint} {\left\lVert \left(({\mathds{1}}-{\boldsymbol T}{\boldsymbol F})^{-1} {\boldsymbol Q}\right)^* \right\rVert_{2}} = {\left\lVert ({\mathds{1}}-{\boldsymbol T}{\boldsymbol F})^{-1} {\boldsymbol Q}\right\rVert_{2}} \lesssim 1.$$ The second assumption in yields $$\label{eq:trivial_bound_app} {\left\lVert ({\mathds{1}}- {\boldsymbol T}{\boldsymbol F})^{-1} \right\rVert_{2}} \lesssim \eta^{-1}.$$ Take ${\boldsymbol y}\in {\mathbb{C}}^{2n}$ arbitrary. We get $[{\boldsymbol T}, {\boldsymbol Q}]{\boldsymbol y}= {\langle{{\boldsymbol T}{\boldsymbol f_-}+{\boldsymbol f_-}} \mspace{2mu}, {{\boldsymbol y}}\rangle}{\boldsymbol f_-}- {\langle{{\boldsymbol f_-}} \mspace{2mu}, {{\boldsymbol y}}\rangle}({\boldsymbol T}{\boldsymbol f_-}+{\boldsymbol f_-})$, where $[{\boldsymbol T},{\boldsymbol Q}] = {\boldsymbol T}{\boldsymbol Q}- {\boldsymbol Q}{\boldsymbol T}$ denotes the commutator of ${\boldsymbol T}$ and ${\boldsymbol Q}$. Therefore, $$\label{eq:estimate_comm_TQ} {\lVert [{\boldsymbol T},{\boldsymbol Q}] \rVert_{2}} \leq 2 {\lVert {\boldsymbol f_-}+ {\boldsymbol T}{\boldsymbol f_-}\rVert_{2}} \lesssim \eta$$ by the third condition in . We set ${\boldsymbol x}{\mathrel{\vcenter{\baselineskip0.5ex \lineskiplimit0pt\hbox{\scriptsize.}\hbox{\scriptsize.}}}=}{\boldsymbol Q}({\mathds{1}}-{\boldsymbol F}{\boldsymbol T})^{-1} {\boldsymbol y}= \left(({\mathds{1}}-{\boldsymbol T}{\boldsymbol F})^{-1}{\boldsymbol Q}\right)^*{\boldsymbol y}$ and compute $${\boldsymbol x}= {\boldsymbol F}{\boldsymbol T}{\boldsymbol x}+ {\boldsymbol Q}{\boldsymbol y}- {\boldsymbol F}[{\boldsymbol T},{\boldsymbol Q}]({\mathds{1}}-{\boldsymbol F}{\boldsymbol T})^{-1}{\boldsymbol y},$$ where we commuted ${\mathds{1}}-{\boldsymbol F}{\boldsymbol T}$ and ${\boldsymbol Q}$ and used that ${\boldsymbol F}$ and ${\boldsymbol Q}$ commute. Hence, using ${\lVert {\boldsymbol x}\rVert_{2}} \lesssim{\lVert {\boldsymbol y}\rVert_{2}} \lesssim{\lVert {\boldsymbol y}\rVert_{\infty}}$ by , ${\lVert {\boldsymbol Q}\rVert_{\infty}} \leq 1 + {\lVert {\boldsymbol f_-}\rVert_{\infty}}$, and , we obtain $${\lVert {\boldsymbol x}\rVert_{\infty}} \lesssim \left( {\lVert {\boldsymbol F}{\boldsymbol T}\rVert_{2\to\infty}} + 1 + {\lVert {\boldsymbol f_-}\rVert_{\infty}} + {\lVert {\boldsymbol F}\rVert_{2\to\infty}} \right) {\lVert {\boldsymbol y}\rVert_{\infty}}\lesssim {\lVert {\boldsymbol y}\rVert_{\infty}}.$$ Here, we used the fourth assumption in and . Notice that the $\eta^{-1}$ factor from the trivial estimate was compensated by the smallness of the commutator $[{\boldsymbol T}, {\boldsymbol Q}]$ which was a consequence of the third assumption in . This concludes the proof of . \[proof:approximating\_eigenvalue\] We first prove that $${\lVert {\boldsymbol f_-}- {\boldsymbol a}\rVert_{2}} = O(\eta) \label{eq:estimate_ff_-}.$$ uniformly for $\eta \leq 1$ and ${\tau}\in {\ensuremath{[0,1-{\tau}_*]}}$. To that end, we introduce the auxiliary operator $${\boldsymbol A}{\mathrel{\vcenter{\baselineskip0.5ex \lineskiplimit0pt\hbox{\scriptsize.}\hbox{\scriptsize.}}}=}{\lVert {\boldsymbol F}\rVert_{2}}{\mathds{1}}+ {\boldsymbol F}.$$ Therefore, we obtain from ${\boldsymbol F}{\boldsymbol f_-}= - {\lVert {\boldsymbol F}\rVert_{2}} {\boldsymbol f_-}$ and $${\boldsymbol A}{\boldsymbol f_-}= 0, \quad {\boldsymbol A}{\boldsymbol a}= O(\eta).$$ Let ${\boldsymbol Q}$ be the orthogonal projection onto the subspace ${\boldsymbol f_-}^\perp$ orthogonal to ${\boldsymbol f_-}$, i.e., ${\boldsymbol Q}{\boldsymbol y}{\mathrel{\vcenter{\baselineskip0.5ex \lineskiplimit0pt\hbox{\scriptsize.}\hbox{\scriptsize.}}}=}{\boldsymbol y}- {\langle{{\boldsymbol f_-}} \mspace{2mu}, {{\boldsymbol y}}\rangle} {\boldsymbol f_-}$ for ${\boldsymbol y}\in {\mathbb{C}}^{2n}$. We then obtain ${\boldsymbol A}{\boldsymbol Q}{\boldsymbol a}= O(\eta)$ which implies ${\boldsymbol Q}{\boldsymbol a}= O(\eta)$ as ${\boldsymbol A}$ is invertible on ${\boldsymbol f_-}^\perp$ and ${\lVert ({\boldsymbol A}|_{{\boldsymbol f_-}^\perp})^{-1} \rVert_{2}} \sim 1$ by . We infer . For the proof of , we follow the proof of , replace ${\boldsymbol T}$ by $-{\mathds{1}}$ and use Lemma \[lem:prop\_Ff\] (i) instead of the second and fourth condition in .
--- abstract: 'Current generative frameworks use end-to-end learning and generate images by sampling from uniform noise distribution. However, these approaches ignore the most basic principle of image formation: images are product of: (a) Structure: the underlying 3D model; (b) Style: the texture mapped onto structure. In this paper, we factorize the image generation process and propose Style and Structure Generative Adversarial Network (${\text{S}^2}$-GAN). Our ${\text{S}^2}$-GAN has two components: the Structure-GAN generates a surface normal map; the Style-GAN takes the surface normal map as input and generates the 2D image. Apart from a real vs. generated loss function, we use an additional loss with computed surface normals from generated images. The two GANs are first trained independently, and then merged together via joint learning. We show our ${\text{S}^2}$-GAN model is interpretable, generates more realistic images and can be used to learn unsupervised RGBD representations.' author: - 'Xiaolong Wang, Abhinav Gupta' bibliography: - 'local.bib' title: Generative Image Modeling using Style and Structure Adversarial Networks --- Introduction ============ Unsupervised learning of visual representations is one of the most fundamental problems in computer vision. There are two common approaches for unsupervised learning: (a) using a discriminative framework with auxiliary tasks where supervision comes for free, such as context prediction [@Doersch2014; @CarlUnsup2015] or temporal embedding [@WangUnsup2015; @nyuUnsup15; @ngVideo12; @LiEdge16; @of_iccv2015; @misra2016unsupervised]; (b) using a generative framework where the underlying model is compositional and attempts to generate realistic images [@goodfellow2014generative; @Kingma14; @Gregor15; @Li15]. The underlying hypothesis of the generative framework is that if the model is good enough to generate novel and realistic images, it should be a good representation for vision tasks as well. Most of these generative frameworks use end-to-end learning to generate RGB images from control parameters ($z$ also called noise since it is sampled from a uniform distribution). Recently, some impressive results [@Alec15] have been shown on restrictive domains such as faces and bedrooms. ![image](fig_teaser.pdf){width="99.00000%"} However, these approaches ignore one of the most basic underlying principles of image formation. Images are a product of two separate phenomena: [**Structure:**]{} this encodes the underlying geometry of the scene. It refers to the underlying mesh, voxel representation etc. [**Style:**]{} this encodes the texture on the objects and the illumination. In this paper, we build upon this IM101 principle of image formation and factor the generative adversarial network (GAN) into two generative processes as Fig. \[fig:teaser\]. The first, a structure generative model (namely Structure-GAN), takes $\hat{z}$ and generates the underlying 3D structure ($y_{3D}$) for the scene. The second, a conditional generative network (namely Style-GAN), takes $y_{3D}$ as input and noise $\tilde{z}$ to generate the image $y_I$. We call this factored generative network Style and Structure Generative Adversarial Network (${\text{S}^2}$-GAN). [**Why ${\text{S}^2}$-GAN?**]{} We believe there are fourfold advantages of factoring the style and structure in the image generation process. Firstly, factoring style and structure simplifies the overall generative process and leads to more realistic high-resolution images. It also leads to a highly stable and robust learning procedure. Secondly, due to the factoring process, ${\text{S}^2}$-GAN is more interpretable as compared to its counterparts. One can even factor the errors and understand where the surface normal generation failed as compared to texture generation. Thirdly, as our results indicate, ${\text{S}^2}$-GAN allows us to learn RGBD representation in an unsupervised manner. This can be crucial for many robotics and graphics applications. Finally, our Style-GAN can also be thought of as a learned rendering engine which, given any 3D input, allows us to render a corresponding image. It also allows us to build applications where one can modify the underlying 3D structure of an input image and render a completely new image. However, learning ${\text{S}^2}$-GAN is still not an easy task. To tackle this challenge, we first learn the Style-GAN and Structure-GAN in an independent manner. We use the NYUv2 RGBD dataset [@Silberman12] with more than 200K frames for learning the initial networks. We train a Structure-GAN using the ground truth surface normals from Kinect. Because the perspective distortion of texture is more directly related to normals than to depth, we use surface normal to represent image structure in this paper. We learn in parallel our Style-GAN which is conditional on the ground truth surface normals. While training the Style-GAN, we have two loss functions: the first loss function takes in an image and the surface normals and tries to predict if they correspond to a real scene or not. However, this loss function alone does not enforce explicit pixel based constraints for aligning generated images with input surface normals. To enforce the pixel-wise constraints, we make the following assumption: if the generated image is realistic enough, we should be able to reconstruct or predict the 3D structure based on it. We achieve this by adding another discriminator network. More specifically, the generated image is not only forwarded to the discriminator network in GAN but also a input for the trained surface normal predictor network. Once we have trained an initial Style-GAN and Structure-GAN, we combine them together and perform end-to-end learning jointly where images are generated from $\hat{z}, \tilde{z}$ and fed to discriminators for real/fake task. Related Work ============ Unsupervised learning of visual representation is one of the most challenging problems in computer vision. There are two primary approaches to unsupervised learning. The first is the discriminative approach where we use auxiliary tasks such that ground truth can be generated without labeling. Some examples of these auxiliary tasks include predicting: the relative location of two patches [@CarlUnsup2015], ego-motion in videos [@Agrawal15; @Jayaraman15], physical signals [@Owens16; @pintoicra16; @pintoeccv2016]. A more common approach to unsupervised learning is to use a generative framework. Two types of generative frameworks have been used in the past. Non-parametric approaches perform matching of an image or patch with the database for tasks such as texture synthesis [@efros99] or super-resolution [@Freeman02]. In this paper, we are interested in developing a parametric model of images. One common approach is to learn a low-dimensional representation which can be used to reconstruct an image. Some examples include the deep auto-encoder [@Bengio07; @quocle12] or Restricted Boltzmann machines (RBMs) [@Ranzato10; @Osindero2008; @hinton06; @Honglak09; @Taylor06]. However, in most of the above scenarios it is hard to generate new images since sampling in latent space is not an easy task. The recently proposed Variational auto-encoders (VAE) [@Kingma14; @Gregor15] tackles this problem by generating images with variational sampling approach. However, these approaches are restricted to simple datasets such as MNIST. To generate interpretable images with richer information, the VAE is extended to be conditioned on captions [@Mansimov15] and graphics code [@Kulkarni15]. Besides RBMs and auto-encoders, there are also many novel generative models in recent literature [@Dosovitskiy15; @TatarchenkoDB15; @Theis15; @Oord16]. For example, Dosovitskiy et al. [@Dosovitskiy15] proposed to use CNNs to generate chairs. In this work, we build our model based on the Generative Adversarial Networks (GANs) framework proposed by Goodfellow et al. [@goodfellow2014generative]. This framework was extended by Denton et al. [@Denton15] to generate images. Specifically, they proposed to use a Laplacian pyramid of adversarial networks to generate images in a coarse to fine scheme. However, training these networks is still tricky and unstable. Therefore, an extension DCGAN [@Alec15] proposed good practices for training adversarial networks and demonstrated promising results in generating images. There are more extensions include using conditional variables [@Mirza15; @MathieuCL15; @RGAN16]. For instance, Mathieu et al. [@MathieuCL15] introduced to predict future video frames conditioned on the previous frames. In this paper, we further simplify the image generation process by factoring out the generation of 3D structure and style. In order to train our ${\text{S}^2}$-GAN we combine adversarial loss with 3D surface normal prediction loss [@deep3d15; @eigen15; @Fouhey13a; @Ladicky14b] to provide extra constraints during learning. This is also related to the idea of combining multiple losses for better generative modeling [@MakhzaniSJG15; @LarsenSW15; @DosovitskiyB16]. For example, Makhzani et al. [@MakhzaniSJG15] proposed an adversarial auto-encoder which takes the adversarial loss as an extra constraint for the latent code during training the auto-encoder. Finally, the idea of factorizing image into two separate phenomena has been well studied in [@barrow1978; @josh2000; @fouhey15; @sczhu98], which motivates us to decompose the generative process to structure and style. We use the RGBD data from NYUv2 to factorize and learn a ${\text{S}^2}$-GAN model. Background for Generative Adversarial Networks ============================================== The Generative Adversarial Networks (GAN) [@goodfellow2014generative] contains two models: generator $G$ and discriminator $D$. The generator $G$ takes the input which is a latent random vector $z$ sampled from uniform noise distribution and tries to generate a realistic image. The discriminator $D$ performs binary classification to distinguish whether an image is generated from $G$ or it is a real image. Thus the two models are competing against each other (hence, adversarial): network $G$ will try to generate images which will be hard for $D$ to differentiate from real image, meanwhile network $D$ will learn to avoid getting fooled by $G$. Formally, we optimize the networks using gradient descent with batch size $M$. We are given samples as $\textbf{X} = (X_1, ..., X_M)$ and a set of $z$ sampled from uniform distribution as $\textbf{Z} = (z_1,...,z_M)$. The training of GAN is an iterative procedure with 2 steps: (i) fix the parameters of network $G$ and optimize network $D$; (ii) fix network $D$ and optimize network $G$. The loss for training network $D$ is, [$$\begin{aligned} \label{eq:loss_d} L^{D}(\textbf{X},\textbf{Z}) = \sum_{i=1}^{M/2} L(D(X_i), 1) + \sum_{i=M/2 + 1}^{M} L(D(G(z_i)),0).\end{aligned}$$ ]{} Inside a batch, half of images are real and the rest $G(z_i)$ are images generated by $G$ given $z_i$. $D(X_i) \in [0,1]$ represents the binary classification score given input image $X_i$. $L(y^*, y) = -[y \log(y^*) + (1-y)log(1-y^*)]$ is the binary entropy loss. Thus the loss Eq. \[eq:loss\_d\] for network $D$ is optimized to classify the real image as label $1$ and the generated image as $0$. On the other hand, the generator $G$ is trying to fool $D$ to classify the generated image as a real image via minimizing the loss: [$$\begin{aligned} \label{eq:loss_g} L^{G}(\textbf{Z}) = \sum_{i=M/2 + 1}^{M} L(D(G(z_i)),1).\end{aligned}$$ ]{} Style and Structure GAN ======================= GAN and DCGAN approaches directly generate images from the sampled $z$. Instead, we use the fact that image generation has two components: (a) generating the underlying structure based on the objects in the scene; (b) generating the texture/style on top of this 3D structure. We use this simple observation to decompose the generative process into two procedures: (i) Structure-GAN - this process generates surface normals from sampled $\hat{z}$ and (ii) Style-GAN - this model generates the images taking as input the surface normals and another latent variable $\tilde{z}$ sampled from uniform distribution. We train both models with RGBD data, and the ground truth surface normals are obtained from the depth. Structure-GAN ------------- We can directly apply GAN framework to learn how to generate surface normal maps. The input to the network $G$ will be $\hat{z}$ sampled from uniform distribution and the output is a surface normal map. We use a 100-d vector to represent the $\hat{z}$ and the output is in size of $72\times72\times3$ (Fig. \[fig:example\]). The discriminator $D$ will learn to classify the generated surface normal maps from the real maps obtained from depth. We introduce our network architecture as following. [**[Generator network]{}**]{}. As Table \[tbl:arch\] (top row) illustrates, we apply a 10-layer model for the generator. Given a $100$-d $\hat{z}$ as input, it is first fully connected to a 3D block ($9 \times 9 \times 64$). Then we further perform convolutional operations on top of it and generate the surface normal map in the end. Note that “uconv” represents fractionally-strided convolution [@Alec15], which is also called as deconvolution. We follow the settings in [@Alec15] and use Batch Normalization [@BN15] and ReLU activations after each layer except for the last layer, where a TanH activation is applied. [**[Discriminator network]{}**]{}. We show the 6-layer network architecture in Table \[tbl:arch\] (bottom left). Taking an image as input, the network outputs a single number which predicts the input surface normal is real or generated. We use LeakyReLU [@Maas13; @xu15] for activation functions as in [@Alec15]. However, we do not apply Batch Normalization here. In our case, we find that the discriminator network easily finds trivial solutions with Batch Normalization. ![image](fig_example2.pdf){width="\textwidth"} Style-GAN --------- Given the RGB images and surface normal maps from Kinect, we train another GAN in parallel to generate images conditioned on surface normals. We call this network Style-GAN. First, we modify our generator network to a conditional GAN as proposed in [@Mirza15; @Denton15]. The conditional information, i.e., surface normal maps, are given as additional inputs for both the generator $G$ and the discriminator $D$. Augmenting surface normals as an additional input to $D$ not only forces the generated image to look real, but also implicitly enforces the generated image to match the surface normal map. While training this discriminator, we only consider real RGB images and their corresponding surface normals as the positive examples. Given more cues from surface normals, we generate higher resolution of $128 \times 128 \times 3$ images with the Style-GAN. Formally, we have a batch of RGB images $\textbf{X} = (X_1, ..., X_M)$ and their corresponding surface normal maps $\textbf{C} = (C_1,...,C_M)$, as well as samples from noise distribution ${\tilde{\textbf{Z}}} = (\tilde{z}_1, ..., \tilde{z}_M)$. We reformulate the generative function from $G(\tilde{z}_i)$ to $G( C_i, \tilde{z}_i)$ and discriminative function is changed from $D(X_i)$ to $D( C_i, X_i)$. Then the loss of discriminator network in Eq. \[eq:loss\_d\] can be reformulated as, [$$\begin{aligned} \label{eq:loss_d2} L^{D}_{cond}(\textbf{X},\textbf{C},\tilde{\textbf{Z}}) = \sum_{i=1}^{M/2} L(D(C_i,X_i), 1) + \sum_{i=M/2 + 1}^{M} L(D(C_i, G(C_i,\tilde{z}_i)),0),\end{aligned}$$ ]{} and the loss of generator network in Eq. \[eq:loss\_g\] can be reformulated as, [$$\begin{aligned} \label{eq:loss_g2} L^{G}_{cond}(\textbf{C},\tilde{\textbf{Z}}) = \sum_{i=M/2 + 1}^{M} L(D(C_i, G(C_i,\tilde{z}_i)),1).\end{aligned}$$ ]{} We apply the same scheme of iterative training. By doing this, we can generate the images with network $G$ as visualized in Fig. \[fig:example\] (right). ![image](fig_network2.pdf){width="95.00000%"} [**Network architecture.**]{} We show our generator as Fig. \[fig:G2\_arch\]. Given a $128 \times 128 \times 3$ surface normal map and a 100-d $\tilde{z}$ as input, they are firstly forwarded to convolutional and deconvolutional layers respectively and then concatenated to form $32 \times 32 \times 192$ feature maps. On top of these feature maps, 7 layers of convolutions and deconvolutions are further performed. The output of the network is a $128 \times 128 \times 3$ RGB image. For the discriminator, we apply the similar architecture of the one in Structure-GAN (bottom right in Table. \[tbl:arch\]). The input for the network is the concatenation of surface normals and images ($128 \times 128 \times 6$). Multi-task Learning with Pixel-wise Constraints ----------------------------------------------- The Style-GAN can make the generated image look real and also enforce it to match the provided surface normal maps implicitly. However, as shown Fig. \[fig:example\], the images are noisy and the edges are not well aligned with the edges in the surface normal maps. Thus, we propose to add a pixel-wise constraint to explicitly guide the generator to align the outputs with the input surface normal maps. ![image](fig_model3.pdf){width="90.00000%"} We make the following assumption: If the generated image is real enough, it can be used for reconstructing the surface normal maps. To encode this constraint, we train another network for surface normal estimation. We modify the Fully Convolutional Network (FCN) [@Long15] with the classification loss as mentioned in [@deep3d15] for this task. More specifically, we quantize the surface normals to $40$ classes with k-means clustering as in [@deep3d15; @Ladicky14] and the loss is defined as [$$\begin{aligned} \label{eq:loss_fcn} L^{FCN}(\textbf{X},\textbf{C}) = \frac{1}{K \times K} \sum_{i=1}^{M} \sum_{k=1}^{K \times K} L_s (F_k(X_i), C_{i,k}),\end{aligned}$$ ]{} where $L_s$ means the softmax loss and the output surface normal map is in $K \times K$ dimension, and $K = 128$ is in the same size of input image. $F_k(X_i)$ is the output of $k$th pixel in the $i$th sample. $C_{i,k} (1 \leqslant C_{i,k} \leqslant 40)$ is the label for the $k$th pixel in sample $i$. Thus the loss is designed to enforce each pixel in the image to generate accurate surface normal. Note that when training the FCN, we use the RGBD data which provides indoor scene images and ground truth surface normals. The model is trained from scratch without ImageNet pre-training. [**FCN architecture**]{}. We apply the AlexNet [@Krizhevsky12] following the same training scheme as [@Long15], with modifications on the last 3 layers. Given a generated $128 \times 128$ image, it is first upsampled to $512 \times 512$ before feeding into the FCN. For the two layers before the last layer, we use smaller kernel numbers of $1024$ and $512$. The last layer is a deconvolutional layer with stride 2. In the end, upsampling (4x resolution) is further applied to generate the high quality results. Given the trained FCN model, we can use it as an additional supervision (constraint) in the adversarial learning. Our final model is illustrated in Fig. \[fig:model2\]. During training, not only the gradients from the classification loss of $D$ will be passed down to $G$, but also the surface normal estimation loss from the FCN is passed through the generated image to $G$. This way, the adversarial loss from $D$ will make the generated images look real, and the FCN will give pixel-wise constraints to make the generated images aligned with surface normal maps. Formally, we combine the two losses in Eq. \[eq:loss\_g2\] and Eq. \[eq:loss\_fcn\] for the generator $G$, [$$\begin{aligned} \label{eq:loss_g3} L^{G}_{multi}(\textbf{C},\tilde{\textbf{Z}}) = L^{G}_{cond}(\textbf{C},\tilde{\textbf{Z}}) + L^{FCN}(G(\textbf{C},\tilde{\textbf{Z}}),\textbf{C}),\end{aligned}$$ ]{} where $G(\textbf{C},\tilde{\textbf{Z}})$ represents the generated images given a batch of surface normal maps $\textbf{C}$ and noise $\tilde{\textbf{Z}}$. The training procedure for this model is similar to the original adversarial learning, which includes three steps in each iteration: - Fix the generator $G$, optimize the discriminator $D$ with Eq. \[eq:loss\_d2\]. - Fix the FCN and the discriminator $D$, optimize the generator $G$ with Eq. \[eq:loss\_g3\]. - Fix the generator $G$, fine-tune FCN using generated and real images. Note that the parameters of FCN model are fixed in the beginning of multi-task learning, i.e., we do not fine-tune FCN in the beginning. The reason is the generated images are not good in the beginning, so feeding bad examples to FCN seems to make the surface normal prediction worse. ![image](fig_model_joint2.pdf){width="95.00000%"} Joint Learning for ${\text{S}^2}$-GAN -------------------------------------- After training the Structure-GAN and Style-GAN independently, we merge all networks and train them jointly. As Fig. \[fig:model\_joint\] shows, our full model includes surface normal generation from Structure-GAN, and based on it the Style-GAN generates the image. Note that the generated normal maps are first passed through an upsampling layer with bilinear interpolation before they are forwarded to the Style-GAN. Since we do not use ground truth surface normal maps to generate the images, we remove the FCN constraint from the Style-GAN. The discriminator in Style-GAN takes generated normals and images as negative samples, and ground truth normals and real images as positive samples. For the Structure-GAN, the generator network receives not only the gradients from the discriminator of Structure-GAN, but also the gradients passed through the generator of Style-GAN. In this way, the network is forced to generate surface normals which not only are realistic but also help generate better RGB images. Formally, the loss for the generator network of Structure-GAN can be represented as combining Eq. \[eq:loss\_g\] and Eq. \[eq:loss\_g2\], [$$\begin{aligned} \label{eq:loss_g_joint} L^{G}_{joint}(\hat{\textbf{Z}}, \tilde{\textbf{Z}}) = L^{G}(\hat{\textbf{Z}}) + \lambda \cdot L^{G}_{cond}(G(\hat{\textbf{Z}}),\tilde{\textbf{Z}})\end{aligned}$$ ]{} where $\hat{\textbf{Z}} = (\hat{z}_1, ..., \hat{z}_M)$ and ${\tilde{\textbf{Z}}} = (\tilde{z}_1, ..., \tilde{z}_M)$ represent two sets of samples drawn from uniform distribution for Structure-GAN and Style-GAN respectively. The first term in Eq. \[eq:loss\_g\_joint\] represents the adversarial loss from the discriminator of Structure-GAN and the second term represents that the loss of the Style-GAN is also passed down. We set the coefficient $\lambda = 0.1$ and smaller learning rate for Structure-GAN than Style-GAN in the experiments, so that we can prevent the generated normals from over fitting to the task of generating RGB images via Style-GAN. In our experiments, we find that without constraining $\lambda$ and learning rates, the loss $L^{G}(\hat{\textbf{Z}})$ easily diverges to high values and the Structure-GAN can no longer generate reasonable surface normal maps. ![image](fig_results2.pdf){width="99.00000%"} ![image](fig_cmp.pdf){width="95.00000%"} ![image](fig_full4.pdf){width="\textwidth"} Experiments =========== We perform two types of experiments: (a) We qualitatively and quantitatively evaluate the quality of images generates using our model; (b) We evaluate the quality of unsupervised representation learning by applying the network for different tasks such as image classification and object detection. **Dataset.** We use the NYUv2 dataset [@Silberman12] in our experiment. We use the raw video data during training and extract 200K frames from the 249 training video scenes. We compute the surface normals from the depth as [@Ladicky14b; @deep3d15]. **Parameter Settings.** We follow the parameters in [@Alec15] for training. We trained the models using Adam optimizer [@Kingma14adam] with momentum term $\beta_1 = 0.5, \beta_2 = 0.999$ and batch size $M=128$. The inputs and outputs for all networks are scaled to $[-1,1]$ (including surface normals and RGB images). During training the Style and Structure GANs separately, we set the learning rate to $0.0002$. We train the Structure-GAN for 25 epochs. For Style-GAN, we first fix the FCN model and train it for 25 epochs, then the FCN model are fine-tuned together with 5 more epochs. For joint learning, we set learning rate as $10^{-6}$ for Style-GAN and $10^{-7}$ for Structure-GAN and train them for 5 epochs. **Baselines.** We have 4 baseline models trained on NYUv2 training set: (a) DCGAN [@Alec15]: it takes uniform noise as input and generate $64 \times 64$ images; (b) DCGAN+LAPGAN: we train a LAPGAN [@Denton15] on top of DCGAN, which takes lower resolution images as inputs and generates $128 \times 128$ images. We apply the same architecture as our Style-GAN for LAPGAN (Fig. \[fig:G2\_arch\] and Table. \[tbl:arch\]). (c) DCGANv2: we train a DCGAN with the same architecture as our Structure-GAN (Table. \[tbl:arch\]). (d) DCGANv2+LAPGAN: we train another LAPGAN on top of DCGANv2 as (b) with the same architecture. Note that baseline (d) has the same model complexity as our model. ![image](fig_changez4.pdf){width="99.00000%"} Qualitative Results for Image Generation ----------------------------------------- **Style-GAN Visualization**. Before showing the image generation results of the full ${\text{S}^2}$-GAN model, we first visualize the results of our Style-GAN given the ground truth surface normals on the NYUv2 test set. As illustrated in the first 3 rows of Fig. \[fig:result\], we can generate nice rendering results which are well aligned with the surface normal inputs. By comparing with the original RGB images, we show that our method can generate a different style (illumination, color, texture) of image with the same structure. We also make comparisons on the results of Style-GAN with/without pixel-wise constraints as visualized in Fig. \[fig:cmp\]. We show that if we train the model without the pixel-wise constraint, the output is less smooth and noisier than our approach. **Rendering on Synthetic Scenes**. One application of our Style-GAN is rendering synthetic scenes. We use the 3D models annotated in [@Guo13] to generate the synthetic scenes. We use the scenes corresponding to the NYUv2 test set and make some modifications by rotation, zooming in/out. As the last two rows of Fig. \[fig:result\] show, we can obtain very realistic rendering results on 3D models. **${\text{S}^2}$-GAN Visualization**. We now show the results of our full generative model. Given the noise $\hat{z}, \tilde{z}$, our model generate both surface normal maps ($72 \times 72$) and RGB images ($128 \times 128$) after that, as shown in Fig. \[fig:full\](a). We compare with the baselines including DCGAN(Fig. \[fig:full\](b)) and DCGAN+LAPGAN (Fig. \[fig:full\](c)). We can see that our method can generate more structured indoor scenes, i.e., it is easier to figure out the structure and objects in our image. We also find that using LAPGAN does not help much improving the qualitative results. **Walking the latent space.** One big advantage of our model is that it is interpretable. Recall that we have two random uniform vectors $\hat{z}, \tilde{z}$ as inputs for Structure and Style networks. We conduct two experiments here: (i) Fix $\tilde{z}$ (style) and manipulate the structure of images by changing $\hat{z}$; (ii) Fix $\hat{z}$ (structure) and manipulate the style of images by changing $\tilde{z}$. Specifically, given an initial set of $\hat{z}$ and $\tilde{z}$, we pick up a series of 10 random points in $\hat{z}$ or $\tilde{z}$ and gradually add $0.1$ to these points for $6-7$ times. We show that we can obtain smooth transitions in the outputs by interpolating the inputs as Fig. \[fig:latent\]. For the example in the first two rows of Fig. \[fig:latent\], we show that by interpolating $\hat{z}$, we can gradually “grow” a 3D cube in the room and the style of the RGB images are consistent since we fix the $\tilde{z}$. For the last rows in Fig. \[fig:latent\], we fix the structure of the image and interpolate the $\tilde{z}$ so that the window of the room is gradually shut down. **User study.** We collect 1000 pairs of images randomly generated by our method and DCGAN. We let the AMT workers to judge which one is more realistic in each pair and $71\%$ of the time they think our approach generates better images. **Nearest Neighbors Test.** To estimate the novelness of our generated images, we apply nearest neighbors test on them. We apply the AlexNet pre-trained on the Places dataset [@Zhou14] as feature extractor. We extract the Pool5 feature of the generated images as well as the real images (both training and testing) from the dataset. We show the results as Fig. \[fig:nntest\]. In each row, the first image is generated by our model, which is used as a query. We show the top 7 retrieved real images. We observe that while the images are semantically related, they have different style and structure as compared to nearest neighbors. ![image](fig_nn_big2.pdf){width="95.00000%"} Quantitative Results for Image Generation ----------------------------------------- To evaluate the generated images quantitatively, we apply the AlexNet pre-trained (supervised) on Places [@Zhou14] and ImageNet dataset [@ILSVRC15] to perform classification and detection on them. The motivation is: If the generated images are realistic enough, state of the art classifiers and detectors should fire on them with high scores. We compare our method with the three baselines mentioned in the beginning of experiment: DCGAN, DCGANv2 and DCGANv2+LAPGAN. We generate 10K images for each model and perform evaluation on them. **Classification on generated images.** We apply the Places-AlexNet [@Zhou14] to perform classification on the generated images. If the image is real enough, the Places-AlexNet will give high response in one class during classification. Thus, we can use the maximum norm $||\cdot||_{\infty}$ of the softmax output (i.e., the maximum probability) of Places-AlexNet to represent the image quality. We compute the results for this metric on all generated images and show the mean for different models as Fig. \[fig:test\](a). ${\text{S}^2}$-GAN is around $2\%$ better than the baselines. **Object detection on generated images.** We used Fast-RCNN detector [@girshickICCV15fastrcnn] fine-tuned on the NYUv2 dataset with ImageNet pre-trained AlexNet. We then apply the detector on generated images. If the image is realistic enough, the detector should find objects (door, bed, sofa, table, counter etc). Thus, we want to investigate on which images the detector can find more foreground objects. We plot the curves shown in Fig. \[fig:test\](b) (the x-axis represents the detection threshold, and the y-axis represents average number of detections). We show that the detector can find more foreground objects in the images generated by ${\text{S}^2}$-GAN. At $0.3$ threshold, there are on average $2.2$ detections per image and $1.72$ detections on images generated by DCGAN. ![image](fig_nn3.pdf){width="95.00000%"} Representation Learning for Recognition Tasks --------------------------------------------- We now explore whether the representation learned by the discriminator network in our Style-GAN can be transferred to tasks such as scene classification and object detection. Since the input for the network is RGB image and surface normal map, our model can be applied to recognition tasks in RGBD data. We perform the experiments on scene classification on SUN RGB-D dataset [@Song15; @Janoch11; @Xiao13; @Silberman12] as well as object detection on NYUv2 dataset. **Scene Classification.** We use the standard train/test split for scene classification in SUN RGB-D dataset, which includes 19 classes with 4852 training and 4660 testing images. We use our model, taking RGB images and normals as inputs, to extract the feature of the second-to-last layer and train SVM on top of it. We compare our method with the discriminator network in DCGAN and the baselines reported in [@Song15]: GIST [@GIST11] feature as well as Places-AlexNet [@Zhou14]. For the networks trained with only RGB data, we follow [@Song15; @SGupta14], which directly use them to extract feature on the depth representation. Then the features extracted from both RGB and depth are concatenated together as inputs for SVM classifier. Note that all models are not fine-tuned on the dataset. As Fig. \[fig:test\](c) shows, our model is $8.2\%$ better than DCGAN and $3.7\%$ away from the Places-AlexNet. **Object Detection.** In this task, we perform RGBD object detection on the NYUv2 dataset. We follow the Fast-RCNN pipeline [@girshickICCV15fastrcnn] and use the code and parameter settings provided in [@SGupta16]. In our case, we use surface normal to represent the depth. To apply our model for the detection task, we stacked two fully connected layer ($4096$-d) on top of the last convolutional layer and fine-tune the network end-to-end. We compare against four baselines: network with the same architecture trained from scratch, network pre-trained with DCGAN, DCGANv2, and ImageNet pre-trained AlexNet. For networks pre-trained on only RGB data, we fine-tune them on both the RGB and surface normal inputs separately and average the detection results during testing as [@SGupta16]. We apply Batch Normalization [@BN15] except for ImageNet pre-trained AlexNet. We show the results in Table \[tab:det\]. Our approach has $1.5\%$ improvement compared to the model trained from scratch. Conclusion ========== We present a novel Style and Structure GAN which factorizes the image generation process. We show our model is more interpretable and generates more realistic images compared to the baselines. We also show that our method can learn RGBD representations in an unsupervised manner.\ Supplementary Material: Generated Normals and Images from ${\text{S}^2}$-GAN ============================================================================ ----------------------------------------------- ![image](result_1_24.pdf){width="110.00000%"} ----------------------------------------------- ------------------------------------------------ ![image](result_25_48.pdf){width="110.00000%"} ------------------------------------------------ ------------------------------------------------ ![image](result_49_72.pdf){width="110.00000%"} ------------------------------------------------ ------------------------------------------------ ![image](result_73_96.pdf){width="110.00000%"} ------------------------------------------------
--- abstract: | We fulfill the rough topological analysis of the problem of the motion of the Kovalevskaya top in a double field. This problem is described by a completely integrable system with three degrees of freedom not reducible to a family of systems with two degrees of freedom. The notion of a topological atlas of an irreducible system is introduced. The complete topological analysis of the critical subsystems with two degrees of freedom is given. We calculate the types of all critical points. We present the parametric classification of the equipped iso-energy diagrams of the complete momentum map pointing out all chambers, families of 3-tori, and 4-atoms of their bifurcations. Basing on the ideas of A.T.Fomenko, we introduce the notion of the simplified net iso-energy invariant. All such invariants are constructed. Using them, we establish, for all parametrically stable cases, the number of critical periodic solutions of all types and the loop molecules of all rank 1 singularities. *Keywords*: Integrable Hamiltonian systems, momentum map, iso-energy diagrams, net topological invariants, classification author: - 'M.P. Kharlamov[^1], P.E. Ryabov[^2]' date: 'Submitted to J. Fundam. and Appl. Math. 13.08.2014' title: Topological atlas of the Kovalevskaya top in a double field --- Introduction {#sec1} ============ The investigation of bifurcations in integrable Hamiltonian systems with two degrees of freedom and a non-linear additional integral was started in [@Kh1976; @Kh1979; @KhPMM; @KhDAN; @KhBook]. A complete theory of such systems was built by A.T.Fomenko, H.Zieschang, and A.V.Bolsinov (see [@BolFom]). A theory of topological invariants for integrable systems with many degrees of freedom was created by A.T.Fomenko in the works [@fom91; @fomams91]. At the same time, due to the absence of practical methods to analyze non-trivial examples, up to this moment no investigation has been fulfilled of a three dimensional topology of a system not reducible to a family of systems with two degrees of freedom. The main source of examples of integrable systems with two degrees of freedom is the problem of the motion of a rigid body about a fixed point in an axially symmetric force field and its generalizations to some Lie co-algebras. Non-symmetric force fields lead to the systems with three degrees of freedom without a possibility of the global order reduction. This type of problem is obtained as the generalization of the classical S.V.Kovalevskaya case to the motion of a gyrostat in a double force field. The complete integrability of this system is proved in the works [@Bogo; @Yeh; @ReySem] by consequent generalizations of the classical integrals. Here, we consider the system supposing that the gyrostatic momentum is zero. This case usually is called the Kovalevskaya top in a double field. We give the complete investigation of the three-dimensional phase topology partially announced in [@KhRy2012]. The knowledge of basic definitions and facts on the singularities of momentum maps and bifurcations in the case of many degrees of freedom [@Fom1986; @Fom1989; @BolsOsh2006] is supposed. The corresponding theory, in details, is given in [@BolFom]. The structure of the present paper is based on the idea of a topological atlas of an irreducible system with three degrees of freedom depending on a set of some physical parameters [@KhGDIS; @KhVVMSH13]. In this case, the rough iso-energy invariant is not any more a one-dimensional graph, but can be represented as a so-called equipped iso-energy diagram, which is the bifurcation diagram of the restriction of the momentum map to an energy level; the diagram is stratified by the rank of the momentum map and the types of critical points in the pre-image. Moreover, the notion of an equipped diagram includes its span, i.e., it is supplemented with two-dimensional chambers which are the components of the plane of the constants of additional integrals cut out by the bifurcation diagram together with the number of the regular tori in the chambers and the way in which these tori are united in the so-called families. To this end, one needs to obtain a complete classification of the iso-energy diagrams and analyze their evolution with respect to the parameters of the system. First, we obtain the stratification of the phase space by the critical subsystems. These subsystems are formed by the sets of critical points lying in the pre-images of the smooth surfaces bearing the bifurcation diagram $\Sigma$ of the complete momentum map in three-dimensional space. The critical subsystems have less than three degrees of freedom, thus all up-to-date experience of the topological analysis is applicable to them. In particular, we calculate the bifurcation diagrams of the critical subsystems, the types of points of ranks 0 and 1 which are critical *inside* the subsystems, and also the corresponding atoms of two-dimensional bifurcations. At the same time, the equations of the phase spaces of the critical subsystems provide a way to calculate explicitly the so-called *external* type of each point of any critical subsystem, in particular, for all points of rank 2. Combining this information, we obtain the complete classification of the critical points with respect to their types in the initial system with three degrees of freedom. Let $c$ be the vector of physical parameters, $H$ the Hamiltonian of a system. In the non-resonant case, if there are no exotic bifurcations (having the so-called splitting atoms) the cross sections of the bifurcation diagram $\Sigma$ by the planes of the constant values $H=h$ undergo transformations if either the iso-energy manifold $H_h=\{H=h\}$ has a non-empty intersection with the set of points of rank 0 and degenerate points of rank 1, or the Hamiltonian has an extremum on the set of degenerate points of rank 2. For a fixed value of the vector $c$ these conditions define only a finite set of points. Writing out the dependencies at such points $h=h(c)$, we obtain the set of surfaces in the space of the parameters $(c,h)$ which separate inequivalent parametrically stable iso-energy diagrams. Thus, to obtain the *topological atlas* we need to construct the set of separating surfaces and, for each component of the supplement to the separating set in the space of $(c,h)$, to point out the corresponding equipped diagram. Further, comparing these diagrams we also fulfil the description of the families of regular tori. The experience of investigating the integrable systems, even in the case of two degrees of freedom, shows that the bifurcation diagrams and the chambers generated by them usually have a row of specific singularities connected with the existence of “extra small” regions. One of the solutions of this problem is to pass from the real forms of the diagrams to the net topological invariants of A.T.Fomenko [@fom91]. While applying to the concrete systems, it is reasonable to consider this invariant in some simplified form. Let us represent the span of an iso-energy diagram (which is the image of the corresponding iso-energy manifold depending on $h$ and $c$) as a two-dimensional cell complex. For non-separating values $(c,h)$ we supply all 1-cells with the notation of the corresponding 4-atom and all 2-cells (chambers) with the number of the Liouville tori in the pre-image. The result is also one of the possible forms of the rough topological invariant of the system. Passing to the conjugate complex makes this invariant even more clear. In it, the 0-cells are supplied with natural numbers (the number of 3-tori), 1-cells obtain the notation of the 4-atoms, and the 2-cells represent critical closed orbits (critical motions built of the points of rank 1). Adding one 0-cell with the number 0 that represents the chamber with empty integral manifolds, we turn the conjugate complex to a cellular partition of the two-dimensional sphere. We call this equipped complex a simplified net invariant of the restriction of the initial system to an iso-energy level. Having the full set of net invariants let us analyze the atoms at the boundary of each 2-cell. From this we obtain the complete information on the stability of the corresponding closed orbits and construct the loop molecule (in the rough sense). For non-degenerate orbits of low complexity basing on the corresponding classification [@BolFom] we can also establish the exact topology of loop molecules. The result of such investigation can be called a *complete topological atlas* of an integrable system. In what follows, we construct such an atlas for the Kovalevskaya top in a double field. Equations and integrals. The notion of a critical subsystem {#sec2} =========================================================== The problem of the motion of the Kovalevskaya top in a double field is described by the system of equations [@Bogo] $$\label{eq2_1} \begin{array}{lll} \displaystyle{2 \, \dot \omega_1=\omega_2\omega_3+\beta_3,} &\displaystyle{ \dot \alpha_1=\alpha_2\omega_3-\alpha_3\omega_2,} & \displaystyle{\dot\beta_1=\beta_2\omega_3-\omega_2\beta_3,}\\ \displaystyle{2 \, \dot \omega_2 = - \omega_1 \omega_3-\alpha_3,} & \displaystyle{\dot \alpha_2=\omega_1\alpha_3-\omega_3\alpha_1,}& \displaystyle{\dot\beta_2=\omega_1\beta_3-\omega_3\beta_1,}\\ \displaystyle{\dot\omega_3=\alpha_2-\beta_1,}& \displaystyle{\dot \alpha_3=\alpha_1\omega_2-\alpha_2\omega_1,} & \displaystyle{\dot \beta_3=\beta_1\omega_2-\beta_2\omega_1.} \end{array}$$ Here $\boldsymbol\omega$ is the angular velocity vector. The constant in space vectors $\boldsymbol \alpha, \boldsymbol\beta$ characterize the action of the forces. We choose the notation to have $|\boldsymbol\alpha|\geqslant|\boldsymbol\beta|.$ As shown in [@Kh34], without loss of generality we may consider the force fields mutually orthogonal. Then the geometric integrals of system can be written in the form ($a \geqslant b \geqslant 0$) $$\begin{aligned} \label{eq2_2} |{\boldsymbol\alpha}|^2=a^2,\quad |{\boldsymbol\beta}|^2=b^2,\quad {\boldsymbol\alpha}\cdot {\boldsymbol\beta}=0.\end{aligned}$$ Using the components of the kinetic momentum $M_1 =2 \omega_1$, $M_2 =2 \omega_2$, $M_3 = \omega_3$, we take to the space ${\bR}^9({\boldsymbol \omega,\boldsymbol\alpha, \boldsymbol\beta})$ the Lie–Poisson bracket of the Lie co-algebra $e(3,2)^*=\{(\mathbf{M},\boldsymbol{\alpha}, \boldsymbol{\beta})\}$. This bracket was introduced in [@Bogo]. Then equations take the form $\dot x=\{H,x\}$, where $x$ is any of the coordinates and $$\label{eq2_5} H=\frac{1}{2}(2\omega_1^2+2\omega_2^2+\omega_3^2)-\alpha_1-\beta_2.$$ The Casimir functions of the Lie–Poisson bracket are the left-handed parts of equations . Therefore, the vector field restricted to the six dimensional manifold $\mP^6$ defined in ${\bR}^9(\boldsymbol\omega,\boldsymbol\alpha,\boldsymbol\beta)$ by these equations is a Hamiltonian system with three degrees of freedom. For $b=0$ system describes the case of S.V.Kovalevskaya of the motion of a rigid body in the gravity field. If $a=b$, then we have the case of H.M.Yehia [@Yeh]. These two limit problems admit a symmetry group and are reduced to families of integrable systems with two degrees of freedom having the 2-sphere as the configuration space. The classical Kovalevskaya case is studied in [@KhPMM; @KhDAN; @BolFomRich]. The Yehia case and its generalizations were considered in [@ZotMTT; @JPA]. The global approach to the classification of the integrable systems on the 2-sphere generated by the problems of the rigid body dynamics with axially symmetric potentials is realized in the works [@Fom1991; @BolFom1994; @BolKozFom1995]. In the sequel we consider the non-symmetric case $a>b>0$, which is not reducible to two degrees of freedom. The first integrals of system found in [@Bogo] and [@ReySem] $$\label{eq2_6} \begin{array}{l} K=(\omega_1^2-\omega_2^2+\alpha_1-\beta_2)^2+(2\omega_1\omega_2+\alpha_2+\beta_1)^2,\\ G=\left[\omega_1\alpha_1+\omega_2\alpha_2+\frac{1}{2} \alpha_3 \omega_3\right]^2 + \left[ \omega_1 \beta_1 + \omega_2 \beta_2 + \frac{1}{2} \beta_3 \omega_3 \right]^2+\\ \phantom{G=}+\omega_3\left[(\alpha_2\beta_3-\alpha_3\beta_2)\omega_1+(\alpha_3\beta_1-\alpha_1\beta_3)\omega_2+ \frac{1}{2}(\alpha_1\beta_2-\alpha_2\beta_1)\omega_3\right]-\\ \phantom{K=}-\alpha_1b^2-\beta_2 a^2 \end{array}$$ together with $H$ form on $\mP^6$ a complete involutive set. We define the corresponding momentum map ${\mF }: \mP^{6} \to {\bR}^3$ as ${\mF }(x)=\bigl( G(x), K(x), H(x)\bigr)$. Let $\mK$ denote the set of critical points of the momentum map, i.e., the points at which $\rk d{\mF }(x)<3$. The set of critical values $\Sigma={\mF }(\mK) \subset{\bR}^3$ is called the [*bifurcation diagram*]{}. The set $\mK$ is stratified by the rank of the momentum map $\mK =\mK^0 \cup \mK^1 \cup \mK^2$. Here $\mK^r= \{x\in \mP^6 | \rk d{\mF}(x)=r \}$. Accordingly, the diagram $\Sigma$ becomes a cell complex $\Sigma =\Sigma^0 \cup \Sigma^1 \cup \Sigma^2$. On the other hand, practically bifurcation diagrams are described in terms of some surfaces in the space of the integral constants. It is often possible to get the equations of these surfaces (implicit or parametric) even without calculating the critical points; these surfaces are the discriminant sets of some polynomials connected, for example, with singularities of algebraic curves associated with Lax representations. We denote such surfaces by $\pov _i$ and write down the representation $\Sigma= \bigcup_i \Sigma_i$, where $\Sigma_i = \Sigma \cap \pov _i$. The value of such representation is that the critical set $\mK$ turns out to be the union of the naturally arising invariant sets $\mM_i\subset \mK \,\cap \mF^{-1}(\pov_i)$. If $\pov_i$ is given by the regular equations of the type $$\label{eq2_11} \phi_i(g,k,h) = 0,$$ then $\mM_i$ is defined as the set of critical points of the integral $\phi_i(G,K,H)$ belonging to its zero level, while the components of the gradient of the function $\phi_i$ calculated at the point of $\mM_i$ after substituting the values of the integrals $G,K,H$ provide the coefficients of a zero linear combination of the differentials $dG,dK,dH$. At any point of a transversal intersection of two surfaces $\pov_i$ and $\pov_j$ we get two independent zero combinations, therefore at the points of the corresponding intersection $\mM_i \cap\mM_j$ the rank of $\mF$ is equal to 1. Obviously, the points of a transversal intersection of three surfaces (the angles of the bifurcation diagram) are generated by the points where $\rk \mF =0$. The sets $\mM_i$ with the induced dynamics on them are called *critical subsystems*. The critical subsystems and the equations of the surfaces $\pov_i$ in the problem considered were found in the works [@Bogo; @Kh32; @Kh2005]. The detailed description of the stratification of the critical set by the rank of the momentum map is given in [@Kh36]. Also in [@Kh36], in terms of explicit inequalities for the energy constant, the regions of the motion existence on the surfaces $\pov_i$ are pointed out. These regions are the sets $\Sigma_i$ forming the bifurcation diagram. The obtained inequalities provide the set in the parameters space separating different types of the cross sections of the diagram $\Sigma$ by the planes of constant energy, i.e., the types of the bifurcation diagrams of the map $G{\times}K$ restricted to iso-energy surfaces $\{H=h\} \subset \mP^6$. It appears that the critical subsystems are integrable and almost everywhere Hamiltonian systems with less than three degrees of freedom. For them, in turn, the induced momentum map is defined. The bifurcation diagram $\Sigma_i^*$ for the map $\mF|_{\mM_i}$ is identified, in the obvious way, with a subset in the union of the skeletons of the set $\Sigma_i$ of dimensions 0 and 1. Here it is natural to introduce the stratification of $\Sigma_i$ geometrically, considering the existing intersections of the type $\Sigma_i\cap \Sigma_j$. Then the 0-skeleton $\Sigma^1_i$ may also contain the tangency points of two surfaces. In the pre-image of such points the rank of $\mF|_{\mM_i}$ does not decrease, so formally these points do not belong to $\Sigma_i^*$. Of course, the corresponding points of the set $\mK$ will be *degenerate* critical points of $\mF$, but the system ${\mM_i}$ may not notice this fact. The description of the diagrams $\Sigma_i^*$ and the bifurcations [*inside*]{} the critical subsystems is completed in the works [@Zot; @KhSav; @Kh2006; @Kh2009]. The classification of the points of the set $\mK$ with respect to the complete initial system with three degrees of freedom on $\mP^6$ is obtained in the work [@RyKh2012]. In the next two sections we give a short exposition of the necessary results of the above cited papers. Description of critical subsystems and classes of singularities {#sec3} =============================================================== This section contains the results dealing with finding the critical set and classifying the critical points according to their rank. For a compact description of the critical subsystems we use the change of variables proposed in [@Kh32]: $$\begin{array}{l} \begin{array}{ll} x_1 = (\alpha_1 - \beta_2) + \rmi (\alpha_2 + \beta_1),& x_2 = (\alpha_1 - \beta_2) - \rmi (\alpha_2 + \beta_1 ), \\ y_1 = (\alpha_1 + \beta_2) + \rmi (\alpha_2 - \beta_1), & y_2 = (\alpha_1 + \beta_2) - \rmi (\alpha_2 - \beta_1), \\ z_1 = \alpha_3 + \rmi \beta_3, & z_2 = \alpha_3 - \rmi \beta_3, \end{array}\\ \begin{array}{lll} w_1 = \omega_1 + \rmi \omega_2 , & w_2 = \omega_1 - \rmi \omega_2, & w_3 = \omega_3. \end{array} \end{array}$$ Let us introduce the following functions $$\begin{array}{l} Z_1=w_1^2+x_1, \qquad Z_2=w_2^2+x_2, \\ F_1 = \sqrt{x_1 x_2} w_3 - \ds{\frac{(x_2 z_1 w_1 + x_1 z_2 w_2)}{\sqrt{x_1 x_2}}},\qquad \displaystyle{F_2 =\frac{x_2}{x_1}Z_1-\frac{x_1}{x_2}Z_2}, \\ R_1 =\displaystyle{\frac{w_2 x_1+w_1 y_2+w_3 z_1}{w_1}-\frac{w_1 x_2+w_2 y_1+w_3 z_2}{w_2},} \\ R_2 = \displaystyle{(w_2 z_1+w_1 z_2)w_3^2+\Bigl[\frac{w_2 z_1^2}{w_1}+\frac{w_1 z_2^2}{w_2}+w_1 w_2(y_1+y_2)+}\\ \phantom{R_2 =} \displaystyle{+ x_1 w_2^2+x_2 w_1^2\Bigr]w_3 +\frac{w_2^2 x_1 z_1}{w_1} + \frac{w_1^2 x_2 z_2}{w_2}+}\\ \phantom{R_2 =} + \displaystyle{ x_1 z_2 w_2+ x_2 z_1 w_1 +(w_1 z_2-w_2 z_1)(y_1-y_2)}. \end{array}$$ We define the parameters $p>r>0$, putting $p^2 = a^2 + b^2$ and $r^2 = a^2 - b^2$. Below we use them along with $a$ and $b$ when it is convenient for the sake of brevity. \[theo1\] The critical set of the map $\mF$ consists of the four critical subsystems $\mM_i$ $(i=1,\ldots,4)$ defined in $\mP^6$ by the following systems of equations: $$\begin{array}{l} \mm:\quad Z_1 = 0, \quad Z_2 = 0,\\ \mn:\quad F_1 = 0, \quad F_2 = 0,\\ \mo:\quad R_1 = 0, \quad R_2 = 0,\\ \ml:\quad w_1 =0,\quad w_2=0, \quad z_1=0,\quad z_2=0. \end{array}$$ $\mF$-images of the sets $\mM_i$, further denoted by $\pov_i$, in the space $\bR^3(h,k,g)$ of the integral constants $H,K,G$ are described by the following systems of equations $$\label{eq3_9} \begin{array}{ll} \pov_1: \left\{ \begin{array}{l} k=0, \\ g = \ds{\frac{1}{2}}p^2 h -\ds{\frac{1}{4}} f^2; \end{array}\right. & \pov_2: \left\{ \begin{array}{l} k=r^4 m^2, \\ g = \ds{\frac{1}{2}} (p^2 h - r^4 m); \end{array}\right. \\ \pov_3: \left\{ \begin{array}{l} \displaystyle{k = 3 s^2 - 4 h s + p^2 + h^2 - \frac{a^2 b^2} {s^2}}, \\ \displaystyle{g = -s^3 + h s^2 + \frac{a^2 b^2}{s}}; \end{array}\right. & \pov_4: \left\{ \begin{array}{l} k=(a \mp b)^2,\\ g=\pm a b h. \end{array}\right. \end{array}$$ Here $f,m,s$ stand for the constants of the partial integrals $F,M,S$ in the subsystems $\mm,\mn,\mo$ respectively: $$\label{fms} \begin{array}{l} F = w_1 w_2 w_3+z_2 w_1+z_1 w_2, \\ \displaystyle{M = \frac{1}{2r^2}(\frac{x_2}{x_1}Z_1+\frac{x_1}{x_2}Z_2)}, \\ \displaystyle{S=-\frac{1}{4} \big( \frac {y_2 w_1+x_1 w_2+z_1 w_3}{w_1}+\frac{x_2 w_1+y_1 w_2+z_2 w_3}{w_2} \big).} \end{array}$$ Let us give some comments. Obviously, the systems of equations (invariant relations) describing the sets $\mn$ and $\mo$ have singularities. Here we in fact consider the closure in $\mP^6$ of the corresponding sets of solutions of these systems in their domain of definition. The subsystem $\mm$ and the integral $F$ are found in the work [@Bogo], the subsystem $\mn$ and the integral $M$ are pointed out in the work [@Kh32]. The subsystems $\mo$ and $\ml$ completing the description of the critical set, together with the integral $S$ that is the analog of the Kovalevskaya variable taking a constant value on the critical motions of the 4th Appelrot class, were found in [@Kh2005]. The set $\ml$ given by four equations is a smooth two-dimensional manifold. It is diffeomorphic to the union of two cylinders $S^1{\times}\bR$. The induced Hamiltonian system has one degree of freedom. The sets $\mm$ and $\mn$ are smooth four-dimensional manifolds, though $\mn$ is non-orientable (see [@Zot; @KhGEOPHY]). Using the explicit parametric equations of the set $\mo$ obtained in [@Kh2007] (see also [@RyKh2012]) it can be shown that $\mo$ is a smooth four-dimensional manifold everywhere except for the points common with $\ml$. At these points $\mo$ has a transversal self-intersection which is the two-dimensional manifold $\mo \cap\ml$ with boundary. According to presentation , it is convenient to describe the regions of the existence of critical motions $\Sigma_i$ and the bifurcation diagrams $\Sigma_i^*$ of the critical subsystems in terms of the partial momentum maps. For the first three subsystems these are the maps $\mF_i : \mM_i \to \bR^2$ defined as $$\begin{array}{l} \mF_1 = F^2{\times}H, \quad \mF_2 = M{\times}H, \quad \mF_3 = S{\times}H. \end{array}$$ For the subsystem $\ml$ with one degree of freedom it is natural to put $\mF_4=H: \ml \to \bR$. For the sequel we need to denote different classes of the critical points and the images of these classes under the momentum maps. The rank of the points always is given with respect to the complete momentum map $\mF$. \[rem1\] From now on we agree to use the same notation for the image of some special critical point or of some definite family of critical points no matter what momentum map from the above defined ones is considered. This will not cause any ambiguity. The only exception is the set of points in the self-intersection of the surface $\pov_3$. The points of this set, after unfolding it to the plane $(s,h)$, get two representations with different values of $s$. Two points on the $(s,h)$-plane which give the same point $(h,k,g)$ will be provided by the upper index “plus” or “minus” for the larger and smaller values of $s$ respectively. It is convenient to emphasize first the classes of motions $Q_i$ in the subsystems $\mM_i$ $(i=1,2,3)$ that correspond to the subsets on which the form induced by the symplectic structure degenerates. It is known (see e.g. [@FomSimGeom]) that on a manifold defined as a common level of two independent functions such degeneration takes place on a subset of zeros of the Poisson bracket of these functions. At the points of the subsystems $\mM_i$ the following identities hold [@Zot; @KhSav; @Kh2007] $$\begin{array}{l} \ds \{Z_1,Z_2\} \equiv -2 \rmi F, \quad \{F_1,F_2\} \equiv -2 \rmi r^2 L, \quad \{R_1,R_2\} \equiv \frac{8 \rmi}{S} U. \end{array}$$ Here $F$ and $S$ are defined in , while $L$ and $U$ are the integrals of the subsystems $\mn$ and $\mo$ respectively and have the form $$\label{eqelu} \displaystyle{L = \frac{1} {{\sqrt {x_1 x_2 } }}[w_1 w_2 + {{x_1 x_2 + z_1 z_2}} M]}, \qquad U =2 S^4-2 H S^3+a^2b^2.$$ Thus, $Q_1=\mm\cap\{F=0\}$, $Q_2=\mn\cap\{L=0\}$, and $Q_3=\mo\cap\{U=0\}$. Almost all points of these sets have rank 2, but all of them including the finite set of periodic solutions consisting of the points of rank 1, as shown in [@RyKh2012], are degenerate critical points of the momentum map $\mF$. By definition $Q_i \subset \mM_i$. In addition to that ${Q_1 \subset \mn}$ and ${Q_2 \subset \mo}$, though the restriction of the symplectic structure to $\mn$ is non-degenerate at the points of $Q_1$, and the restriction of the symplectic structure to $\mo$ is non-degenerate at the points of $Q_2$. According to Remark \[rem1\], we denote the images of the sets $Q_1,Q_2,Q_3$ under the complete momentum map $\mF$ and the partial maps $\mF_i$ by $\Delta_1,\Delta_2,\Delta_3$ respectively. We now classify the critical points with respect to their rank and belonging to the critical subsystems. The system has exactly four points of rank 0 (see [@KhZot]). At these points we obviously have $\alpha_1=\pm a$, $\beta_2=\pm b$, all other components of ${\boldsymbol\alpha}$ and ${\boldsymbol\beta}$, the same as the vector ${\boldsymbol\omega}$, equal zero. We denote these points in the increasing order of the value of $H$ by $p_0,p_1,p_2,p_3$. None of these points belong to $\mm$ and all of them belong to $\mn\cap\mo\cap\ml$. The index $i=0,\ldots,3$ is equal to the Morse index of the function $H$ at these points [@KhZot]. We denote by $P_i$ the images of $\imm_i$ under the momentum maps. The coordinates of $P_i$ are easily calculated from , . In what follows, only the $h$-coordinates equal to $\mp a \mp b$ are important. All critical points of rank 1 are organized in nine families of periodic motions denote by $\mD_i$ ($i=1,2,3$) and $\mL_j$ ($j=1,\ldots,6$). The families $\mD_1,\mD_2,\mD_3$ were first described in [@Zot] as the sets of critical points of the pair of integrals $H,F$ on $\mm$. Explicit algebraic expressions of the phase variables in terms of one auxiliary variable connected with the time $t$ by elliptic quadrature are given in the work [@Kh361]. In particular, it is proved that the union of these families coincides with the intersection of the subsystems $\mm$ and $\mo$. In [@Kh361], the parametric expressions are proposed of the values of the first integrals (general and partial) on these families, where the parameter is the constant of the integral $S$. Denoting the one-dimensional images of the families $\mD_i$ by $\delta_i$ ($i=1,2,3$), we get the equations $$\label{eq3_20} \begin{array}{l} \delta_1: \left\{ \begin{array}{l} \displaystyle{h=2s-\frac{1}{s}\sqrt{(a^2-s^2)(b^2-s^2)} }\\[2mm] \displaystyle{f^2={-\frac{2}{s}\sqrt{(a^2-s^2)(b^2-s^2)}}(\sqrt{a^2-s^2}+ \sqrt{b^2-s^2})^2}\\[3mm] \displaystyle{g=\frac{1}{s}(s^4-s^2\sqrt{(a^2-s^2)(b^2-s^2)}+a^2b^2)}, \quad s \in [-b,0) \end{array} \right. ; \\ \delta_2: \left\{ \begin{array}{l} \displaystyle{h=2s+\frac{1}{s}\sqrt{(a^2-s^2)(b^2-s^2)} }\\[2mm] \displaystyle{f^2={\frac{2}{s}\sqrt{(a^2-s^2)(b^2-s^2)}}(\sqrt{a^2-s^2}- \sqrt{b^2-s^2})^2}\\[3mm] \displaystyle{g=\frac{1}{s}(s^4+s^2\sqrt{(a^2-s^2)(b^2-s^2)}+a^2b^2)}, \quad s \in (0,b] \end{array} \right. ; \\ \delta_3: \left\{ \begin{array}{l} \displaystyle{h=2s-\frac{1}{s}\sqrt{(s^2-a^2)(s^2-b^2)} }\\[2mm] \displaystyle{f^2={\frac{2}{s}\sqrt{(s^2-a^2)(s^2-b^2)}}(\sqrt{s^2-b^2}- \sqrt{s^2-a^2})^2}\\[3mm] \displaystyle{g=\frac{1}{s}(s^4-s^2\sqrt{(s^2-a^2)(s^2-b^2)}+a^2b^2)}, \quad s \in [a,+\infty) \end{array} \right. . \end{array}$$ Simultaneously we have in mind that $\mD_i \subset \mM_1$, so here $k=0$. Denote the points having the boundary values of $s$ (respectively, $s=-b, b, a$) by $e_i$ ($i=1,2,3$). On the plane $\bR^2(f^2,h)$ the curves $\delta_i$ have the boundary points $e_i$ on the axis $f^2=0$, the curve $\delta_3$ has a cusp point denoted by $e_4$ and corresponding to the value $s_0$ that is a unique root of the equation $$\label{eq3_21} 3s^8 - 4p^2s^6+6 a^2 b^2 s^4-a^4 b^4 = 0$$ on the half-line $s>a$. On the plane $\bR^2(s,h)$ all three curves $\delta_i$ have no singular points (except for the boundary points), and the value $s_0$ corresponds to the minimum of $h$ on $\delta_3$. The families $\mL_j$ are pendulum type motions (oscillations or rotations) about the principal inertia axes of the body and in the phase space have the form $$\begin{array}{l} {\mL}_{1,2}=\{{\boldsymbol \alpha } \equiv \pm a{\mbe}_1, \; {\boldsymbol \beta } = b({\mbe}_2 \cos \theta - {\mbe}_3 \sin \theta ), \; {\boldsymbol \omega } = \theta ^ {\boldsymbol \cdot} {\mbe}_1 , \; 2\theta ^{ {\boldsymbol \cdot} {\boldsymbol \cdot} } = - b\sin \theta\},\\ {\mL}_{3,4}=\{{\boldsymbol \alpha } = a({\mbe}_1 \cos \theta + {\mbe}_3 \sin \theta ), \; {\boldsymbol \beta } \equiv \pm b{\mbe}_2 ,\; {\boldsymbol \omega } = \theta ^ {\boldsymbol \cdot} {\mbe}_2 , \; 2\theta ^{ {\boldsymbol \cdot} {\boldsymbol \cdot} } = - a\sin \theta\}, \\ {\mL}_{5,6}=\{{\boldsymbol{\alpha }} = a({\mbe}_1 \cos \theta - {\mbe}_2 \sin \theta ),\; {\boldsymbol{\beta }} = \pm b({\mbe}_1 \sin \theta + {\mbe}_2 \cos \theta ), \; {\boldsymbol{\omega }} = \theta ^ {\boldsymbol \cdot} {\mbe}_3 ,\; \\ \qquad\,\,\quad\theta ^{ {\boldsymbol \cdot} {\boldsymbol \cdot} } = - (a \pm b)\sin \theta\}. \end{array}$$ Here ${\mbe}_1 {\mbe}_2 {\mbe}_3$ is the canonical basis in $\bR^3$. The upper sign corresponds to the family with the first number. Bifurcations in the families happen at the singular points $p_i$ ($i=0,\ldots,3$) with the above noted values of $h$, namely, $h=\mp a\mp b$. Let us consider that the manifolds $\mL_i$ include also such special trajectories, i.e., they are in fact the *closure* of the corresponding families of periodic trajectories. It is easily seen that the bifurcations inside the families (the birth of oscillations and the transformation of oscillations into rotations) take place according to the following inclusions: $\imm_0 \in \mL_1\cap\mL_3\cap\mL_5$, $\imm_1 \in \mL_1\cap\mL_4\cap\mL_6$, $\imm_2 \in \mL_2\cap\mL_3\cap\mL_6$, and $\imm_3 \in \mL_2\cap\mL_4\cap\mL_5$. In particular, the minimal values of $H$ on $\mL_i$ are $-a-b$ on $\mL_1,\mL_3$ and $\mL_5$; $a-b$ on $\mL_2$; $-a+b$ on $\mL_4$ and $\mL_6$. As was mentioned above, all the points of the trajectories in the sets of degeneration of the forms induced by the symplectic structure are degenerate critical points of the momentum map $\mF$ for any rank. The formal proof is given in [@RyKh2012]. There are no such points among the points of rank 0. Let us discuss degenerate periodic trajectories. Let $\tau_i=\mD_i \cap Q_1$. It is known (see [@Zot]) that $\tau_1$ and $\tau_2$ consist of one trajectory each, and $\tau_3$ consists of two trajectories. The images of the sets $\tau_i$ are the above introduced points $e_i$ ($i=1,2,3$), so $e_i=\Delta_1 \cap \delta_i$. In the image of $\mF_2$ these points lie on the axis $m=0$. The same trajectories are the intersections $\mD_i \cap \mn$ and the union of them coincides with the intersection of $Q_1$ with $\mo$. The sets $\mD_i$ have no common points with $Q_2$. The only non-empty intersection with $Q_3$ exists for $\mD_3$ and consists of the pair of trajectories $\tau_4$, corresponding to the above mentioned value $s_0$ which is the root of equation . The intersections of $Q_i$ with $\mL_j$ are as follows: - $Q_1$ intersects $\mL_3,\mL_4,\mL_2$ by the above mentioned trajectories $\tau_1,\tau_2,\tau_3$ and does not have any other intersections with $\mL_j$; - $Q_2$ intersects $\mL_4$ by the pair of rotational trajectories $\tau_5$ with $h=\frac{a^2+3b^2}{2b}$ and $\mL_2$ by the pair of rotational trajectories $\tau_6$ with $h=\frac{3a^2+b^2}{2a}$; there are no other intersections of $Q_2$ with $\mL_j$; - the trajectories $\tau_5$ and $\tau_6$ also serve as the intersections of $Q_3$ with $\mL_4$ and $\mL_2$ respectively; moreover $Q_3$ intersects $\mL_5$ by one oscillation type trajectory $\tau_7$ ($h=-2\sqrt{ab}$) and the pair of rotational trajectories $\tau_8$ ($h=2\sqrt{ab}$). To understand the whole picture of the position of the families $\mL_i$ in the critical subsystems, it is useful to mention the following facts. \[prop1\] $1.$ The subsystem $\mm$ does not have common motions with $\mL_1,\mL_5,\mL_6$ and contains the trajectories $\tau_1,\tau_2,\tau_3$ of the families $\mL_3,\mL_4,\mL_2$. These trajectories also lie in $Q_1$ and serve as its intersections with the families $\mD_1,\mD_2,\mD_3$. $2.$ The families $\mL_1,\ldots,\mL_4$ lie completely in the intersection $\mn \cap \mo$. $3.$ The family $\mL_5$ intersects the subsystem $\mn$ by the bifurcational trajectories of the levels $h=\pm(a+b)$; the trajectories of the family $\mL_5$ lie in $\mo$ for all values $h \notin (-2\sqrt{ab},2\sqrt{ab})$. The boundary values of $h$ correspond to the trajectories $\tau_7,\tau_8$. $4.$ The family $\mL_6$ intersects with the subsystem $\mn$ by the bifurcational trajectories of the levels $h=\pm(a-b)$ and completely lies in $\mo$. $5.$ The subsystem $\ml$ is the union of the families $\mL_5$ and $\mL_6$; the part of $\ml$ not belonging to other subsystems consists of the trajectories of $\mL_5$ with the values $h \in (-2\sqrt{ab},2\sqrt{ab})$. Let us denote the images of the families $\mL_i$ under the momentum maps by $\lambda_i$ ($i=1,\ldots,6$). These are one-dimensional objects. The image points of the special trajectories $\tau_i$ will be denoted by $e_i$ ($i=1,\ldots,8$). Let us now collect the information on the images of the above mentioned special points and subsets of the phase space. [|c|c|c|c|c|c|c|]{}\ ${}$ & ${h}$ & ${k}$ & ${g}$ & ${f}$ & ${m}$ & ${s}$\ ${e_1}$ & ${-2b}$ & ${0}$ & ${-b p^2}$ & ${0}$ & ${0}$ & ${-b}$\ ${e_2}$ & ${2b}$ & ${0}$ & ${bp^2}$ & ${0}$ & ${0}$ & ${b}$\ ${e_3}$ & ${2a}$ & ${0}$ & ${ap^2}$ & ${0}$ & ${0}$ & ${a}$\ ${e_4}$ & ${h(s_0)}$ & ${0}$ & ${g(s_0)}$ & ${f(s_0)}$ & ${-}$ & ${s_0}$\ ${e_5}$ & ${\ds \frac{a^2+3b^2}{2b}}$ & ${\ds \frac{r^4}{4 b^2}}$ & ${\ds \frac{p^4-b^2 r^2}{2b}}$ & ${-}$ & ${-\ds \frac{1}{2b}}$ & ${b}$\ ${e_6}$ & ${\ds \frac{3a^2+b^2}{2a}}$ & ${\ds \frac{r^4}{4a^2}}$ & ${\ds \frac{p^4+a^2 r^2}{2a}}$ & ${-}$ & ${-\ds \frac{1}{2a}}$ & ${a}$\ ${e_7}$ & ${-2\sqrt{ab}}$ & ${(a-b)^2}$ & ${-2(ab)^{3/2}}$ & ${-}$ & ${-}$ & ${-\sqrt{ab}}$\ ${e_8}$ & ${2\sqrt{ab}}$ & ${(a-b)^2}$ & ${2(ab)^{3/2}}$ & ${-}$ & ${-}$ & ${\sqrt{ab}}$\ ${e_9}$ & ${p\sqrt{2}}$ & ${\ds \frac{r^4}{2p^2}}$ & ${\ds \frac{2p^4+r^4}{2p\sqrt{2}}}$ & ${-}$ & ${-\ds \frac{1}{p\sqrt{2}}}$ & $\ds{\frac{p}{\sqrt{2}}}$\ In Table \[tabee\], we give the values of all first integrals at the points of the trajectories $\tau_i$, i.e., the coordinates of the points $e_i$ in all image spaces of the momentum maps. Here for the point $e_4$ the coordinates $h,g$ and $f$ are calculated by the formulas for $\delta_3$ with the above defined value $s_0$. \[rem2\] In Table $\ref{tabee}$, we add a new point $e_9$. It corresponds to the minimal value of the energy $H$ on the set of degenerate critical points $Q_2$. Unlike the sets $Q_1$ and $Q_3$, where all extremal energy values provide the appearance of degenerate periodic orbits, i.e., correspond to bifurcations of degenerate two-dimensional tori, on the set $Q_2$ the minimal value of $H$ is reached without any additional bifurcations of the degenerate critical tori of rank 2. Nevertheless this value is essential for the classification of the iso-energy invariants. The images of the sets $Q_i$ of degenerate critical points of rank 1 and 2 under the map $\mF$ have the form $$\begin{array}{l} \Delta_1: \left\{\begin{array}{l} k=0, \\ 2g=p^2h \end{array}\right., \quad h \geqslant -2b. \\ \Delta_2: \left\{\begin{array}{l} k=\ds{\frac{1}{r^4}(2g-p^2h)^2} \\ g=g_{\pm}(h)=\ds{\frac{1}{4p^2}} \left[(2p^4-r^4)h \pm r^4 \sqrt{\mstrut h^2-2p^2} \right] \end{array}\right., \quad h \geqslant \sqrt{\mstrut 2p^2}. \\ \Delta_3: \left\{\begin{array}{l} h=\ds{\frac{a^2b^2+3 s^4}{2 s^3}}\\ k=\ds{-\frac{3s^2}{4}+ a^2+b^2 -\frac{3a^2b^2}{2 s^2}+\frac{a^4b^4}{4 s^6}} \\ g=\ds{\frac{3a^2b^2+s^4}{2s}} \end{array}\right., \quad 0 < s \leqslant s_0. \end{array}$$ The equations for the sets $\Delta_i$ in the image planes of the partial momentum maps are given in Table \[tabdel\]. [|c|c|c|c|]{}\ & --------------------- In ${\bR}^2(f^2,h)$ --------------------- & ------------------- In ${\bR}^2(m,h)$ ------------------- & ------------------- In ${\bR}^2(s,h)$ ------------------- \ $\Delta_1$ & ------------------------- $f=0$, $h\geqslant -2b$ ------------------------- & ------------------------- $m=0$, $h\geqslant -2b$ ------------------------- & $e_1(-b,-2b),e_2(b,2b),e_3(a,2a)$\ $\Delta_2$ & – & --------------------------- $2p^2m^2+2h m+1 =0,$$m<0$ --------------------------- & ------------------------ $2s^2-2hs+p^2=0,$$s>0$ ------------------------ \ $\Delta_3$ & – & – & ------------------------------------- $3s^4-2hs^3+a^2b^2=0,$$s\in(0,s_0]$ ------------------------------------- \ On the families ${\mL}_j$, the values of the first integrals fill the curves $\lambda_j$. In $\bR^3(h,k,g)$, the equations of these curves are $$\begin{array}{l} \lambda_{1,2}=\{g = a^2 h\pm a r^2,k=(h\pm 2a)^2,h \geqslant \mp(a\pm b)\},\\ \lambda_{3,4}=\{g = b^2 h\mp b r^2, k=(h\pm 2b)^2, h \geqslant -(a\pm b)\},\\ \lambda_{5,6}=\{g=\pm abh,k=(a\mp b)^2,h\geqslant -(a\pm b)\}. \end{array}$$ On the plane $\bR^2(m,h)$ the images of $\mL_1,\ldots,\mL_4$ under $\mF_2$ are the half-lines $$\begin{array}{l} \lambda_{1,2}=\{h = r^2 m \mp 2a, h \geqslant \mp(a\pm b)\},\qquad \lambda_{3,4}=\{h =-r^2 m \mp 2b, h \geqslant -(a\pm b)\}. \end{array}$$ On the plane $\bR^2(s,h)$ the images of all six families under $\mF_3$ are $$\begin{array}{ll} \lambda_{1,2}=\{s = \mp a, h \geqslant \mp(a\pm b)\}, & \lambda_{3,4}=\{s = \mp b, h \geqslant -(a\pm b)\},\\[1mm] \lambda_{5}=\{h=s + \ds \frac{ab}{s}, s\in [-a,-b] \cup (0,+\infty)\}, & \lambda_{6}=\{h=s - \ds \frac{ab}{s}, s\in [-a,0) \cup [b,+\infty)\},\\ \end{array}$$ In all these formulas the upper sign corresponds to the family with the first number. Collecting the images on the planes under the partial momentum maps $\mF_1,\mF_2,\mF_3$ of the specific sets of critical points described above, we obtain the bifurcation diagrams $\Sigma_1^*, \Sigma_2^*, \Sigma_3^*$ of the subsystems $\mm,\mn,\mo$ respectively. They are shown in Fig. \[fig\_bifset1\] – \[fig\_bifset3\]. Let us give some comments on these figures. 1. The typical inverse image of the points on the curves $\Delta_1$ and $\Delta_2$ in the critical set, as it was already mentioned, consists of degenerate critical points of rank 2, which normally are not bifurcational inside a critical subsystem with two degrees of freedom. Nevertheless, with this choice of the partial integrals, $\Delta_1$ and $\Delta_2$ become the outer boundaries of the regions of the existence of motions for the subsystems $\mm$ and $\mn$, therefore they are the parts of the bifurcation diagrams. The phenomena taking place in the neighborhood of the pre-images of these curves inside the subsystems $\mm$ and $\mn$ are in details studied in [@Zot1; @KhGEOPHY]. In the third subsystem $\mo$ the curves $\Delta_2,\Delta_3$ do not cause any bifurcations with respect to $\mo$. But they are the images of degenerate points of the complete momentum map and thus separate the points of $\mo$ having different outer type. Thus, these sets are also shown in Fig. \[fig\_bifset3\] (with dashed lines). 2. The diagrams $\Sigma_i^*$ ($i=1,2,3$) split the points of rank 2 regular with respect to the subsystems $\mm,\mn$ and $\mo$ into definite classes (for $\mo$ we add to the separating set also the curves $\Delta_2,\Delta_3$). These classes (to be more exact, the subregions arising in the image of the partial momentum map and corresponding to non-empty integral manifolds) are denoted for the first three subsystems by the symbols, respectively, $a,b,c$ supplied with indices. The connected components of the supplement to bifurcation diagrams are usually called *chambers* [@BolFom]. Here for the partition into chambers we also take into account the images of the 2-tori degenerate in $\mP^6$. 3. The diagram $\Sigma_4^*$ consists of the isolated energy values separating different types of periodic solutions. These values are $h=\pm a\pm b$ at the points of rank 0 and $h=\pm 2\sqrt{ab}$; the latter separate the motions in the family $\mL_5$ belonging to to the subsystem $\mo$ and isolated from another critical subsystems. For the family $\mL_5$ the image segment which is not shown in the figures and correspond to the values $h\in(-2\sqrt{ab},2\sqrt{ab})$ is denoted by $\lambda_{50}$. \[remsegm\] The curves $\Delta_i$, $\lambda_j$ are divided into different segments by the node points $e_1,\ldots,e_9$. A similar division on the curve $\delta_3$ is generated by the point $e_4$. We will mark such segments by putting a second index after the curve number. With the above information, let us give an explicit description of the bifurcation diagram $\Sigma$ of the map $\mF$. Introduce some notation. On the curves , we define the inversions of the dependencies $h(s)$ on monotonous segments: $$\begin{array}{llll} \delta_1:& s=s_1(h), & h\in[-2b,+\infty), & s_1(h)\in [-b,0),\\ \delta_2: & s=s_2(h), & h \geqslant 2b, & s_2(h)\in (0,b],\\ \delta_{31}: & s=s_{31}(h), & h \in [h_0,2a], & s \in [a, s_0],\\ \delta_{32}: & s=s_{32}(h), & h \in [h_0,+\infty), & s \in [s_0+\infty). \end{array}$$ Here $h_0$ is the value $h(s_0)$ on the curve $\delta_3$. We now obtain the equation for the value $s_0$ from the condition for the minimum of $h$ on the curve $\delta_3$ in the following form $$\label{eq3_27} \frac{1}{s}\sqrt{(s^2-a^2)(s^2-b^2)}=\sqrt{s^2-b^2}-\sqrt{s^2-a^2}.$$ Here the fact that it has a unique solution at $s>a$ is obvious. From equations for $\pov_1$ we find the dependency on $\delta_1$: $$g = g_1(h) = s^3+\frac{ab}{s}-s^2 \phi(s)|_{s=s_1(h)}, \quad h \geqslant -2b.$$ Considering the intervals where $h(s)$ is monotonous on the curves $\lambda_5 - \lambda_6$, we denote $$\displaystyle{s_{51}(h)= \frac{h - \sqrt{h^2-4ab}}{2},}\quad \displaystyle{s_{52}(h)= \frac{h + \sqrt{h^2-4ab}}{2},}\quad \displaystyle{s_6(h)= \frac{h + \sqrt{h^2+4ab}}{2}}.$$ Now the bifurcation diagram is completely described by the following theorem [@Kh36], which is formulated in such a way that all the conditions for the parameters on the surfaces are explicit inequalities when the energy value $h$ is fixed. \[theo2\] $1.$ The set $\Sigma_1=\pov_1 \cap \Sigma$ has the form $$\left\{\begin{array}{l} h \geqslant -2b \\ k = 0,\quad g_1(h) \leqslant g \leqslant \frac{1}{2}p^2h \end{array} \right. .$$ $2.$ The set $\Sigma_2=\pov_2 \cap \Sigma$ lies in the half-space $h \geqslant -(a+b)$ and is described by the following collection of the systems of inequalities: $$\left\{ \begin{array}{l} -(a+b)\leqslant h \leqslant p\sqrt{2}\\ b^2 h -b r^2\leqslant g \leqslant a^2 h +a r^2 \end{array} \right.; \quad \left\{ \begin{array}{l} h \geqslant p\sqrt{2}\\ b^2 h -b r^2 \leqslant g \leqslant g_{-}(h) \end{array} \right.; \quad \left\{ \begin{array}{l} h \geqslant p\sqrt{2}\\ g_{+}(h) \leqslant g \leqslant a^2 h +a r^2 \end{array} \right. .$$ $3.$ The set $\Sigma_3=\pov_3 \cap \Sigma$ is completely described by the following collection of conditions in the $(s,h)$-plane. For the negative values of $s$ $$\left\{ \begin{array}{l} -(a+b)\leqslant h \leqslant -2 \sqrt{ab}\\ s \in [-a,s_{51}(h)] \cup [s_{52}(h),-b] \end{array} \right.; \quad \left\{ \begin{array}{l} -2 \sqrt{ab}\leqslant h \leqslant -2b\\ s \in [-a,-b] \end{array} \right.; \quad \left\{ \begin{array}{l} h > -2b\\ s \in [-a,s_1(h)] \end{array} \right..$$ For the positive values of $s$ $$\begin{array}{ll} \left\{ \begin{array}{l} -a+b\leqslant h \leqslant 2b\\ s \in [b,s_6(h)]\end{array} \right.; & \left\{ \begin{array}{l} 2b \leqslant h \leqslant h_0\\ s \in [s_2(h),s_6(h)] \end{array} \right.; \\[4mm] \left\{ \begin{array}{l} h_0 \leqslant h \leqslant 2a\\ s \in [s_2(h),s_{31}(h)]\cup[s_{32}(h),s_6(h)] \end{array} \right.; & \left\{ \begin{array}{l} h > 2a\\ s \in [s_2(h),a]\cup[s_{32}(h),s_6(h)] \end{array} \right.. \end{array}$$ $4.$ The set $\Sigma_4=\pov_4 \cap \Sigma$ consists of two half-lines $$\left\{ \begin{array}{l} h \geqslant - (a + b)\\ g = abh,\quad k = (a - b)^2 \end{array} \right.; \quad \left\{ \begin{array}{l} h \geqslant - a + b\\ g = - abh,\quad k = (a + b)^2 \end{array} \right. .$$ \[remtheo2\] This theorem provides a possibility to present in details any cross section of the diagram $\Sigma$ by the planes of fixed $h$ and to trace with the computer graphics the evolution of these cross section with the energy change. The boundary conditions for the value of $h$ are separating values of the energy. In particular, the conditions for $\Sigma_2$ make it clear why the point $e_4$ was introduced as the extremal value of $h$ on $\Delta_2$ $($see Remark $\ref{rem2})$. Nevertheless, the boundary values of $h$ do not necessarily form the complete separating set. More exact statement will be given below. Classification of critical points by type {#sec4} ========================================= All necessary definitions dealing with the notion of a non-degenerate critical point and the type of a critical point are given in [@BolFom]. Let us give one comment on the terminology used here. In the topological analysis of the systems with two degrees of freedom $\cite{BolFom}$ the following terms are used: a 3-atom to describe bifurcations in the neighborhood of a non-degenerate point of rank $1$ (it is a connected component of the pre-image of a small segment transversal to a smooth segment of the bifurcation diagram; this component is foliated into Liouville 2-tori with one singular fiber) and a 4-atom to describe a saturated neighborhood of a non-degenerate point of rank $0$ (usually in terms of an almost direct product of some atoms from the systems with one degree of freedom). \[remt\] Dealing with a system with three degrees of freedom we fix the term 3-atom to characterize a bifurcation in the neighborhood of a point of rank $1$ *inside* the corresponding critical subsystem with two degrees of freedom, while a 4-atom will always mean a bifurcation in the complete six-dimensional space in the neighborhood of a non-degenerate point of rank $2$. Thus, we define a 4-atom as a foliated into Liouville 3-tori with one singular fiber connected component of the pre-image of a small segment transversal to a smooth leave of the bifurcation diagram of the momentum map $\mF:\mP^6\to \bR^3$. Studying the iso-energy diagrams we may consider such a small segment belonging to the corresponding fixed energy level, since at any non-degenerate point of rank 2 the Hamiltonian is a regular function. At four points of rank 0 (singular points of the initial system) the Hamiltonian $H$ is a Morse function and, as it was already mentioned, ${\rm ind}\,H(p_i)=i$. This fact almost completely determines the character of the system in the neighborhood of these points. Nevertheless, the strict classification demands establishing the types of them as the critical points of the momentum map. \[theo3\] All critical points of rank $0$ are *non-degenerate* in $\mP^6$. Moreover, $\imm_0$ has the type “center-center-center”, $\imm_1$ has the type “center-center-saddle”, $\imm_2$ has the type “center-saddle-saddle”, and $\imm_3$ has the type “saddle-saddle-saddle”. [|c|c|c|c|c|]{}\ $\mK^0$ & ------------------ Image in ${\bR}^3(h,k,g)$ ------------------ & Type in $\mn$ & ---------------- Image in ${\bR}^2(s,h)$ ---------------- & Type in $\mo$\ $\imm_0$ &$P_0$ &center-center & ----------- $P^-_{0}$ $P^+_{0}$ ----------- & --------------- center-center center-center --------------- \ $\imm_1$ &$P_1$ &center-saddle & ----------- $P^-_{1}$ $P^+_{1}$ ----------- & --------------- center-saddle center-center --------------- \ $\imm_2$ &$P_2$ &center-saddle & ----------- $P^-_{2}$ $P^+_{2}$ ----------- & --------------- saddle-saddle center-saddle --------------- \ $\imm_3$ & $P_3$ &saddle-saddle & ----------- $P^-_{3}$ $P^+_{3}$ ----------- & --------------- saddle-saddle saddle-saddle --------------- \ For the proof, in [@RyKh2012] the explicit form of the characteristic polynomials of the symplectic operator $A_H$ are calculated at the points $\imm_i$. Note that at the points $\imm_i$, three local critical subsystems meet, namely, the subsystem $\mn$ and the two parts of the subsystem $\mo$ (recall that this subsystem has at these points a singularity of the self-intersection type). In particular, on the $(s,h)$-plane each point $\imm_i$ is represented by two points. In this sense, each point of the pair $P^{\pm}_{i}$ of the diagram $\Sigma_3^*$ has its own type (the type of the point $\imm_i$ with respect to some chosen smooth part of $\mo$ in the neighborhood of $\imm_i$). The corresponding description of the ctical points of rank 0 in the critical subsystems $\mn$ and $\mo$ is given in Table \[tab41\]. Now let us turn to the points of rank 1 organized into special periodic trajectories. \[theo4\] The points of rank $1$ forming the families of trajectories $\mL_i$ è $\mD_j$ ${(i=1,\ldots,6;}$ $j=1,2,3)$ are non-degenerate as the singularities of the map $\mF$ except for the following values of the energy: on $\mD_1$: $h = -2b$; on $\mD_2$: $h = 2b$; on $\mD_3$: $h = 2a$, $h=h(s_0)$; on $\mL_2$: $h = 2a$, $h={\frac{3a^2+b^2}{2a}}$; on $\mL_3$: $h = -2b$; on $\mL_4$: $h = 2b$, $h={\frac{a^2+3b^2}{2b}}$; on $\mL_5$: $h =\pm 2\sqrt{a b}$. Depending on the family and the energy value, the type of non-degenerate singularities in $\mP^6$ and the corresponding 3-atoms in the critical subsystems are given in Table $\ref{tab42}$. [|c|l|c|c|c|c|c|]{}\ $\mK^1$ & & -------------- Number of trajectories -------------- & [Type in $\mP^6$]{} & ---------- 3-atom in $\mm$ ---------- & ---------- 3-atom in $\mn$ ---------- & ---------- 3-atom in $\mo$ ---------- \ \ $\mK^1$ & & -------------- Number of trajectories -------------- & [Type in $\mP^6$]{} & ---------- 3-atom in $\mm$ ---------- & ---------- 3-atom in $\mn$ ---------- & ---------- 3-atom in $\mo$ ---------- \ & $\lambda_{11}:-(a+b)<h<-(a-b)$ & [1]{} & center-center &&[$A$]{}&[$A$]{}\ & $\lambda_{12}:h>-(a-b)$ & [2]{} & center-center &&[$2A$]{}&[$2A$]{}\ & $\lambda_{21}:a-b<h<a+b$ & [1]{} & saddle-saddle &&[$B$]{}&[$B$]{}\ & $\lambda_{22}:a+b<h<\frac{3a^2+b^2}{2a}$ & [2]{} & saddle-saddle &&[$2B$]{}&[$2B$]{}\ & $\lambda_{23}:\frac{3a^2+b^2}{2a}<h<2a$ & [2]{} & saddle-saddle &&[$2B$]{}&[$2B$]{}\ & $\lambda_{24}: h>2a$ & [2]{} & saddle-center &&[$2B$]{}&[$2A$]{}\ & $\lambda_{31}:-(a+b)<h<-2b$ & [1]{} & center-center &&[$A$]{}&[$A$]{}\ & $\lambda_{32}:-2b<h<a-b$ & [1]{} & center-saddle &&[$A$]{}&[$B$]{}\ & $\lambda_{33}: h>a-b$ & [2]{} & center-saddle &&[$2A$]{}&[$C_2$]{}\ & $\lambda_{41}:-(a-b)<h<2b$ & [1]{} & saddle-center &&[$B$]{}&[$A$]{}\ & $\lambda_{42}:2b<h<a+b$ & [1]{} & saddle-saddle &&[$B$]{}&[$B$]{}\ & $\lambda_{43}: a+b<h <\frac{a^2+3b^2}{2b}$ & [2]{} & saddle-saddle &&[$2B$]{}&[$C_2$]{}\ & $\lambda_{44}: h >\frac{a^2+3b^2}{2b}$ & [2]{} & saddle-saddle &&[$2B$]{}&[$C_2$]{}\ & $\lambda_{50}:-2\sqrt{ab}<h<2\sqrt{ab}$ & [1]{} & focus-focus &&&\ & $\lambda_{51}:-(a+b)<h<-2\sqrt{ab}$ & [1]{} & center-center &&&[$A,A$]{}\ & $\lambda_{52}:2\sqrt{ab}<h<a+b$ & [1]{} & saddle-saddle &&&[$B,B$]{}\ & $\lambda_{53}:h>a+b$ & [2]{} & saddle-saddle &&&[$2A^*,2B$]{}\ & $\lambda_{61}:-(a-b)<h<a-b$ & [1]{} & saddle-center &&&[$B,A$]{}\ & $\lambda_{62}:h>a-b$ & [2]{} & saddle-center &&&[$2A^*,2A$]{}\ ${\mD}_1$ & $\delta_1:s\in(-b,0)$ & [2]{} & center-center &[$2A$]{}&&[$2A$]{}\ ${\mD}_2$ & $\delta_2:s\in(0,b)$ & [2]{} & saddle-center &[$2B$]{}&&[$2A$]{}\ & $\delta_{31}: s\in(a,s_0)$ & [4]{} & saddle-center &[$4B$]{}&&[$4A$]{}\ & $\delta_{32}: s\in(s_0,+\infty)$ & [4]{} & center-center &[$4A$]{}&&[$4A$]{}\ Note that the cases of degeneration correspond to the points $e_1 - e_8$ on the bifurcation diagrams. Let us comment the order of words characterizing the type of points with respect to $\mP^6$ in Table \[tab42\]. For the families $\mL_j$ with $i=1,\ldots,4$ the first type is the type of the point of rank 1 in the subsystem $\mn$, and the second type is the type of this point in the subsystem $\mo$. For the families $\mD_{1,2,3}$ the first type is the type of the point of rank 1 in the subsystem $\mm$, and the second type is the type of this point in the subsystem $\mo$. The families $\mL_{5,6}$ (except for the pre-image of the focus type segment $\lambda_{50}$, which is isolated in the space of the integral constants, and for its boundary points with degenerate rank 1 critical points in the pre-image) form the set of the transversal self-intersection of the phase space of the subsystem $\mo$. This implies that each trajectory corresponds to the pair of values $s$ and to two points $\lambda^{\pm}_{ij}$ ($i=5,6$) in the diagram of the subsystem $\mo$ (see Fig. \[fig\_bifset3\]). The minus sign corresponds to the smaller value of $s$. Therefore, in Table \[tab42\] for these families the first type is the type of the point of rank 1 in the part of the subsystem $\mo$ with smaller $s$, and the second type is the type of this point in the part of the subsystem $\mo$ with larger $s$. For the proof of the theorem, in the work [@RyKh2012], the pairs of the first integrals generating the Cartan subalgebra of dimension 2 in the algebra of the symplectic operators are explicitly written out and it is shown that the following integrals generate regular elements in these subalgebras: $$\begin{array}{ll} \textrm{íà}\quad{\mL}_{1,2}:& \vp_{1,2}=K-2(h\pm 2a)H, \\ \textrm{íà}\quad{\mL}_{3,4}:& \vp_{3,4}=K-2(h\pm 2b)H, \\ \textrm{íà}\quad{\mL}_{5,6}:& \vp_{5,6}=\pm abH-G, \\ \textrm{íà}\quad{\mD}_{i}: & \psi_i=2G-(p^2-\vk_i)H\qquad (i=1,2,3), \end{array}$$ where $$\begin{array}{ll} \displaystyle{\vk_1=(\sqrt{a^2-s^2}+\sqrt{b^2-s^2})^2}, & s \in [-b,0); \\ \displaystyle{\vk_2=(\sqrt{a^2-s^2}-\sqrt{b^2-s^2})^2}, & s \in (0,b]; \\ \displaystyle{\vk_3=-(\sqrt{s^2-b^2}-\sqrt{s^2-a^2})^2}, & s \in [a,+\infty). \end{array}$$ The atoms of all bifurcations taking place inside the critical subsystems $\mm,\mn,\mo$ at the periodic trajectories consisting of the rank 1 points are established as a result of the topological analysis of these subsystems (see [@Zot; @KhSav; @Kh2009]). Let us consider the set $\mK^2$ of the critical points of rank 2. It is the union of three subsystems $\mm$, $\mn$ and $\mo$, from which we remove already investigated points $\mK^0 \cup \mK^1$. According to dimensions the type of a non-degenerate rank 2 singularity can only be *elliptic* (“center”) or *hyperbolic* (“saddle”). Since the property to be non-degenerate is applied to the whole Liouville torus and then all the points of this torus have the same type, we speak about elliptic or hyperbolic tori. First of all note that any integral 2-torus regular in the subsystem $\mm$ and non-degenerate in $\mP^6$ is elliptic. Indeed, the integral $K=Z_1^2+Z_2^2$ is everywhere non-negative function which vanishes exactly on $\mm$. \[theo5\] All critical points of rank 2 on the manifold $\mm$, except for the points of the zero level of the integral $F$, are non-degenerate of the elliptic type. On $\mm$, there are no points of dependence of the integrals $H$ and $G$ except for the sets $\mD_1 - \mD_3$ [@Kh2005] (note that the dependence of the *restrictions* $H|_{\mm}$ and $F|_{\mm}$ or, which is the same, of the *restrictions* $H|_{\mm}$ and $G|_{\mm}$ is investigated in [@Zot; @Zot1]). At the same time everywhere on $\mm$ we have $dK=0$. The characteristic equation of the operator $A_K$ in $\bR^9$ is easily written out with the help of the equations of the manifold $\mm$; it has seven zero roots and the remaining factor $\mu^2+4 f^2$ has two different imaginary roots for $f\ne 0$. The theorem is proved. In the image of the momentum maps the value $F=0$ gives the set $\Delta_1 \subset \bR^3(h,k,g)$. Thus, for this set we have proved that the corresponding critical points are degenerate. On the manifold $\mn$, system has an explicit algebraic solution [@KhSav]. This means that all phase variables are expressed as rational functions of two auxiliary variables $s_1,s_2$ and some radicals of the form $\sqrt{s_i - c_j}$, where $c_j$ are constant values depending on the constants of the integrals $H,M$. In the variables $s_1,s_2$, the equations of motion separate and reduce to elliptic quadratures. The corresponding formulas can also be found in [@RyKh2012]. \[theo6\] All critical points of rank 2 on the manifold $\mn$, except for the points in the pre-image of the curves $\Delta_1$ and $\Delta_2$, are non-degenerate. They have elliptic type for $m>0$ and hyperbolic type for $m<0$. Due to quite simple equations of the leave $\pov_2$, it is convenient to take as a single integral playing the role of and having a singularity at each point of $\mn \cap \mK^2$ the function $$\label{eq4_5} \Phi = \phi_2(H,K,G)=(2G-p^2H)^2-r^4 K.$$ For this function, after the necessary factorization by the zero kernel space and substitution of the explicit expressions for the phase variables via the separation variables, the characteristic equation of the operator $A_\Phi$ takes the form $$\label{eq4_6} \mu^2+4r^{12} \ell^2 m=0,$$ and, except for the cases $m=0$ ($\Delta_1$) and $\ell=0$ ($\Delta_2$), has two different roots, purely imaginary for $m>0$ and real for $m<0$. Here $\ell$ is the constant of the partial integral $L$ defined in . The theorem is proved. As it is shown in [@KhSav], $\mn$ consists of critical points of the zero level of the functions $2G -p^2H \pm r^2 \sqrt{K}$, one of which is the sum of the squares of two regular smooth functions ($m>0$), and the other one is the difference of such squares ($m<0, \ell \ne 0$). This is the reason why the tori are elliptic in the first case and hyperbolic in the second one. Equation strictly proves the non-degeneracy of such points. It is interesting to note the connection between degenerate points and the analytic solution. When $m=0$, the degree of the polynomial standing under the radical in the separated equations of the Abel–Jacobi type jumps from four to three. When $\ell=0$, the variables $s_1,s_2$ in the separated equations have only even powers, which provides an additional symmetry. As a whole, $\mn$ is non-orientable and the main role here plays the neighborhood of the set ${\ell=0}$ [@KhGEOPHY]. On the manifold $\mo$, the explicit algebraic solution of system is given in [@Kh2009; @Kh2007]. All phase variables are expressed as rational functions of two auxiliary variables $t_1,t_2$ and some radicals of the form $\sqrt{t_i - c_j}$, where $c_j$ are constant values depending on the constants of the integrals $H,S$. In the variables $t_1,t_2$, the equations of motion separate and have the form of the Abel–Jacobi equations or the equations of Kovalevskaya with a polynomial of degree six under the radical. This separation is hyperelliptic. The corresponding formulas also are given in [@RyKh2012]. \[theo7\] All regular two-dimensional tori of the subsystem $\mo$ consist of non-degenerate critical points of rank $2$ of the momentum map ${\mF}$, except for the points in the pre-images of the sets $\Delta_2, \Delta_3$. The torus is elliptic if the value $ s(2s^2-2h s+a^2+b^2)[s^4+2(s-h)s^3+a^2b^2] $ is negative and hyperbolic if it is positive. As an integral having singularity on $\mo$ one can take the function of the type obtained, for example, by eliminating $s$ in equations for the surface $\pov_3$. Nevertheless, the result is too huge and this approach is not convenient. It is better to consider the function with Lagrange multipliers introduced in [@Kh2005] for finding the equations of critical subsystems, $\Psi = 2 G + 2s(s-h) H+s K$, where $s$ and $h$ are first considered as undefined multipliers. As shown in [@Kh2005], [*after*]{} writing down the condition that $\Psi$ has a critical point the constants $s,h$ on $\mo$ turn out to be the values of the integrals $S,H$. Thus, calculating the characteristic polynomial of the operator $A_\Psi$ we consider $s$ and $h$ to be some constants, and after that we substitute in the obtained expression the dependencies of the phase variables on the separation variables. Then we get the characteristic equation $$\mu^2-\frac{2(2s^2-2h s+a^2+b^2)}{s}\Bigl[s^4+2(s-h)s^3+a^2b^2\Bigr]=0.$$ This proves the theorem. Finally, let us again turn to Fig. \[fig\_bifset1\] – \[fig\_bifset3\], on which the images of the critical points of rank 2 fill the regions $a_1 - a_3$, $b_1-b_9$, and $c_1 - c_{17}$. For the critical subsystems these regions are in fact the chambers, since we include in $\Sigma_i^*$ also the image of the degenerate 2-tori. According to Theorems \[theo5\]–\[theo7\] we establish the type of all points of rank 2. \[propr2\] The critical points of rank 2 have the elliptic type in the pre-images of the regions $a_1-a_3$, $b_1-b_3$, $c_1 - c_4,c_{10},c_{13},c_{17}$ and the hyperbolic type in the pre-images of the rest of the regions. In the pre-images with elliptic bifurcations the 4-atoms are as follows: $A$ for the regions $c_1,b_1$; $2A$ for the regions $a_1,b_2,c_2,c_3,c_4,c_{10},c_{13}$; $4A$ for the regions $b_4,b_5,b_6,c_{12},c_{16} $; $8A$ for the region $a_3$. In the pre-images of the sets $\Delta_1,\Delta_2$, and $\Delta_3$ the two-dimensional tori consist of degenerate points of rank 2. The information on the number of critical 2-tori in the pre-images of all mentioned regions is obtained during the rough topological analysis of the critical subsystems in the works [@Zot1; @KhSav; @Kh2009] and, for the sake of brevity, it is shown in Table \[tabatom\] of the next section. It is clear that the elliptic 4-atoms by this information are totally defined. For the hyperbolic atoms the necessary foundations will be given later. Thus, we gave a total classification of the critical points of the momentum map with respect to their types and 3-atoms, and this completes the description of the rough phase topology of the critical subsystems. Iso-energy atlas {#sec5} ================ Iso-energy diagram $\sigh$ is the bifurcation diagram of the restriction of the momentum map $\mF$ to the iso-energy level $H_h=\{H=h\}\subset \mP^6$. The natural identification is used of this restriction with the map to the plane of the constants $g,k$: $$\mF|_{\{H=h\}} \cong (G{\times}K)|_{\{H=h\}}: H_h \to \bR^2(g,k).$$ For brevity, the last map is denoted by $\mF(h)$. We now consider the problem of classification of the diagrams $\sigh$ equipped by additional information on the topology of the pre-image, i.e., we also point out the number of the families of regular tori in the supplement to the diagram and the atoms of those bifurcations that take place when crossing the 1-dimensional skeleton of the diagram. The diagram $\sigh$ itself is considered as a stratified manifold. Smooth segments of the one-dimensional skeleton are the images of non-degenerate points of rank 2 and zero-dimensional skeleton is the image of all critical points of rank 1 and degenerate critical points of rank 2. Here we speak, of course, of the points on a fixed regular iso-energy level $H_h$. Having all previous information and the information revealed here, one can easily endow the points of the zero-dimensional skeleton with the description of the saturated neighborhood of the pre-image. Due to the lack of space, we restrict ourselves to pointing out in the next section the rough loop molecules of hyperbolic rank 0 singularities. Generally speaking, the diagram can have a very complicated structure. For this reason until now there is no strict and unique definition of the diagrams equivalence. In our case, $\sigh$ is a plane one-dimensional stratified manifold. Therefore we call two diagrams equivalent if there exists a diffeomorphism of their neighborhoods in the plane preserving taking one diagram to another together with the above introduced equipment. What does it mean to point out the bifurcation? Let $\gamma$ be a smooth segment of the 1-skeleton of $\sigh$ and $z=(g,k)\in \gamma$. Consider a small one-dimensional segment $\eps$ drawn through $z$, transversal to $\gamma$ and having no common points with $\sigh$ other than $z$. The pre-image of $\eps$ in the phase space is a four-dimensional manifold $\mF^{-1}(\eps)\subset H_h$ foliated into Liouville 3-tori with one singular fiber on each connected component. The set of critical points on a singular fiber consists of a finite number of 2-tori, all points of which have rank 2. We call a connected component of $\mF^{-1}(\eps)$ a tom (see Remark \[remt\]). Some examples of 4-atoms are the products of the standard 3-atoms with a circle. For theses atoms we keep the generally accepted notation $A,B,C_2,A^*$ etc. Thus, any smooth edge of the equipped diagram must have the notation of the corresponding 4-atom. On the other hand, any smooth edge of $\sigh$ is a cross section at the level $h$ of the corresponding chamber of one of the critical subsystems and the 4-atom is obtained when we consider the bifurcation taking place while crossing transversally the corresponding smooth leaf that is the image of the critical subsystem. So it is convenient to show on the edges of the iso-energy diagrams the notation of the corresponding chambers of the critical subsystems, and then, collecting the obtained information and analyzing the evolution of the diagrams $\sigh$, to point out all 4-atoms corresponding to those chambers. Obviously $\sigh$ depends not only on $h$ but also on the physical parameters $a,b$. The point $(a,b,h)$ will be called separating if any its neighborhood in $\bR^3$ contains points with non-equivalent diagrams. Since the stratified manifold $\sigh$ is built from the $h$-sections $\Sigma_i(h)$ of the sets $\Sigma_i$ $(i=1,\ldots,4)$, the transformations of $\sigh$ take place if and only if one of the stratified manifolds $\Sigma_i(h)$ is transformed. It follows from Theorem \[theo2\] (see Remark \[remtheo2\]) that the cross sections $\Sigma_i(h)$ are transformed when $h$ crosses the boundary values met in this theorem. This gives eight separating values of $h$ (depending on $a,b$), i.e., eight separating surfaces in $\bR^3(a,b,h)$. In general, the separating values of $h$ are the critical values of $h$ considered as a function on the stratified manifolds $\Sigma_i$ $(i=1,\ldots,4)$ [@KhGDIS; @KhVVMSH13]. One can easily see from Fig. \[fig\_bifset1\] – \[fig\_bifset3\] that the critical values of $h$ on $\Sigma_i$ are exactly the $h$-coordinates of the points $P_0 - P_3$ and $e_1 - e_9$. Looking at the Table \[tabee\] we come to the following theorem [@Kh36]. \[theo8\] In the space of the parameters $\bR^3(a,b,h)$ there exists $13$ separating surfaces for the classification of the iso-energy diagrams $\sigh$. These surfaces are considered under the natural restriction $0\leqslant b \leqslant a$ and have the form $$\label{eq5_1} \begin{array}{l} \Q_1-\Q_4 :\, h = \mp a \mp b; \\ \Q_5:\,h=-2b; \qquad \Q_6:\,h=2b; \qquad \Q_7:\,h=2a; \\ \Q_{8} :\, \left\{ \begin{array}{l} \ds h = s\left(3 - \frac{s^2}{a^2}\right) - \frac{1}{a^2}\sqrt{(s^2 - a^2)^3} \\[3mm] \ds b = \frac{s}{a}\sqrt{s \left[s\left(3 - \frac{2s^2}{a^2}\right) - \frac{2}{a^2}\sqrt{(s^2 - a^2)^3} \right]} \end{array} \right., \quad s \in [a,\frac{2}{\sqrt{3}}a];\\ \ds \Q_9 :\,h = \frac{1}{2b}(a^2 + 3b^2); \qquad \ds \Q_{10}:\,h = \frac{1}{2a}(3a^2 + b^2); \\ \Q_{11}:\, h = - 2\sqrt {ab}; \qquad \Q_{12}:\, h = 2\sqrt {ab}; \qquad \ds \Q_{13} :\,h = \sqrt{2(a^2+b^2)}. \end{array}$$ Non-empty diagrams exist for the region $h \geqslant -a-b$. All cross sections by the planes of constant $a \ne 0$ are taken to the cross section $a=1$ by the homothetic transformation $h'=h/a$, $b'=b/a$. It corresponds to the fact that $a$ can be taken as a measurement unit for the intensities of the force fields. Here the comments are needed only for the surface $\Q_{8}$. It is obtained as a parametric form of representation of the value $h(s_0)$ at the point $e_4$ by using equation and the corresponding equation for $h$ on the curve $\delta_3$. The separating set is shown in Fig. \[fig\_chart\] under the projection onto the cross section $a=1$. Iso-energy diagrams are non-empty in 19 regions in the parameters space. Let us choose one point in each region and connect them with paths to obtain convenient ways to visualize and analyze the diagrams transformations. According to Fig. \[fig\_chart\], let us call these paths “the left circle”, “the right circle”, “the line” and “the block”. The circles have common region 1, the right circle and the line meet at the region 9, and the line and the block have common region 14. The corresponding diagrams are shown in Fig. \[fig\_leftc1\] – \[fig\_block1\]. To obtain a more clear picture, we made some changes by plane diffeomorphisms, i.e., in the same equivalence class as defined above. All elements of the diagrams are equipped with notation of the corresponding elements introduced above for the diagrams of the critical subsystems. Moreover, the roman numbers ${{\textrm{I}}}$ – ${{\textrm{IX}}}$ enumerate the connected components of the supplement to the bifurcation set of the complete momentum map in the admissible domain (i.e., the domain corresponding to non-empty integral manifolds) in the enhanced space $\bR^5(g,k,h,a,b)$. We keep for such components the term *chamber*. In each chamber the structure of the regular integral manifold stays the same for any change of the integral constants and of the physical parameters. The “external” chamber, which is not admissible (it contains unbounded values of the integrals and thus obviously corresponds to empty integral manifolds), will be denoted by $\varnothing$ and called the zero-chamber. We can now establish the number of regular 3-tori in the chambers, describe the families of these tori and find all types of the 4-atoms. \[theon\] Iso-energy diagrams divide the enhanced space of the parameters into $10$ chambers, one of which $($external$)$ corresponds to empty integral manifolds. In the phase space according to all changes of the parameters there exists exactly $23$ families of three-dimensional regular Liouville tori. The chambers are separated by $29$ walls corresponding to the two-dimensional chambers defined by the diagrams of the critical subsystems. While crossing these walls we have the bifurcations defined by the 4-atoms given in Table $\ref{tabatom}$. [|c|c|c|c||c|c|c|c|]{}\ & ----------- Number of 2-tori ----------- & Transfer & 4-atom & ---------- Segment (region) ---------- & ----------- Number of 2-tori ----------- & Transfer & 4-atom\ \ & ----------- Number of 2-tori ----------- & Transfer & 4-atom & ---------- Segment (region) ---------- & ----------- Number of 2-tori ----------- & Transfer & 4-atom\ $a_1$ & 2 & $\varnothing \to {{\textrm{II}}}$ & $2A$ & $c_4$ & 2 & $\varnothing \to {{\textrm{IV}}}$ & $2A$\ $a_2$ & 4 & $\varnothing \to {{\textrm{V}}}$ & $4A$ & $c_5$ & 2 & ${{\textrm{II}}} \to {{\textrm{V}}}$ & $2B$\ $a_3$ & 8 & $\varnothing \to {{\textrm{VIII}}}$ & $8A$ & $c_6$ & 1 & ${{\textrm{I}}} \to {{\textrm{III}}}$ & $B$\ $b_1$ & 1 & $\varnothing \to {{\textrm{I}}}$ & $A$ & $c_7$ & 2 & ${{\textrm{II}}} \to {{\textrm{IV}}}$ & $2A^*$\ $b_2$ & 2 & $\varnothing \to {{\textrm{III}}}$ & $2A$ & $c_8$ & 2 & ${{\textrm{IV}}} \to {{\textrm{VII}}}$ & $2B$\ $b_3$ & 4 & $\varnothing \to {{\textrm{VII}}}$ & $4A$ & $c_9$ & 2 & ${{\textrm{III}}} \to {{\textrm{VI}}}$ & $2B$\ $b_4$ & 4 & ${{\textrm{VII}}} \to {{\textrm{VIII}}}$ & $4B$ & $c_{10}$ & 2 & ${{\textrm{III}}} \to {{\textrm{VI}}}$ & $2A$\ $b_5$ & 4 & ${{\textrm{VI}}} \to {{\textrm{VII}}}$ & $2C_2$ & $c_{11}$ & 2 & ${{\textrm{III}}} \to {{\textrm{VI}}}$ & $2B$\ $b_6$ & 4 & ${{\textrm{VII}}} \to {{\textrm{IX}}}$ & $2C_2$ & $c_{12}$ & 4 & ${{\textrm{V}}} \to {{\textrm{VII}}}$ & $4A^*$\ $b_7$ & 2 & ${{\textrm{III}}} \to {{\textrm{V}}}$ & $2B$ & $c_{13}$ & 2 & ${{\textrm{IV}}} \to {{\textrm{IX}}}$ & $2A$\ $b_8$ & 1 & ${{\textrm{I}}} \to {{\textrm{II}}}$ & $B$ & $c_{14}$ & 2 & ${{\textrm{IV}}} \to {{\textrm{IX}}}$ & $2B$\ $b_9$ & 2 & ${{\textrm{III}}} \to {{\textrm{IV}}}$ & $C_2$ & $c_{15}$ & 2 & ${{\textrm{III}}} \to {{\textrm{VII}}}$ & $2B$\ $c_1$ & 1 & $\varnothing \to {{\textrm{I}}}$ & $A$ & $c_{16}$ & 4 & ${{\textrm{V}}} \to {{\textrm{VIII}}}$ & $4B$\ $c_2$ & 2 & $\varnothing \to {{\textrm{II}}}$ & $2A$ & $c_{17}$ & 4 & ${{\textrm{V}}} \to {{\textrm{VIII}}}$ & $4A$\ $c_3$ & 2 & $\varnothing \to {{\textrm{III}}}$ & $2A$ & & & &\ The proof needs several steps. First, using only known elliptic bifurcation we find the number of regular 3-tori in the chambers and the corresponding families. Recall that a family of regular tori is a foliation to the Liouville tori of a connected component of a set in the phase space which is obtained by removing all singular fibers. The families are fixed inside one chamber, but the same family may be present in several chambers. Walls between chambers (including walls between a regular chamber and the zero-chamber) which correspond to the 4-atoms of the type $kA$ ($k\in \mathbb{N}$) will be called $A$-crossings. Let us put all $A$-crossings on the diagrams using the bifurcations of the type “center” from Proposition \[propr2\]. Then we obtain the information collected in Table \[tab51\]. Thus, we establish the number of regular tori in all chambers and the total number of families of tori in all chambers. As we see, there are 23 families. The common families are present in the pairs of the chambers ${{\textrm{III}}},{{\textrm{VI}}}$ (two families), ${{\textrm{IV}}},{{\textrm{IX}}}$ (two families) and ${{\textrm{V}}},{{\textrm{VIII}}}$ (four families). [|c|c|c|c|c|c|]{}\ Chamber & ---------- Crossing ---------- & Segments & 4-atom & --------- Number of tori --------- & ---------- New families ---------- \ ${{\textrm{I}}}$ & $\varnothing \to {{\textrm{I}}} $ & $c_1,b_1$ & $A$ & [1]{} &[1]{}\ ${{\textrm{II}}}$ & $\varnothing \to {{\textrm{II}}} $ & $a_1,c_2$ & $2A$ & [2]{} &[2]{}\ ${{\textrm{III}}}$ & $\varnothing \to {{\textrm{II}}} $ & $b_2,c_3$ & $2A$ & [2]{} &[2]{}\ ${{\textrm{IV}}}$ & $\varnothing \to {{\textrm{IV}}} $ & $c_4$ & $2A$ & [2]{} &[2]{}\ ${{\textrm{V}}}$ & $\varnothing \to {{\textrm{V}}} $ & $a_2$ & $4A$ & [4]{} &[4]{}\ ${{\textrm{VI}}}$ & ${{\textrm{III}}} \to {{\textrm{VI}}} $ & $c_{10}$ & $2A$ & [4]{} &[2]{}\ ${{\textrm{VII}}}$ & $\varnothing \to {{\textrm{VII}}} $ & $b_3$ & $4A$ & [4]{} &[4]{}\ ${{\textrm{VIII}}}$ & ${{\textrm{V}}} \to {{\textrm{VIII}}} $ & $c_{17}$ & $4A$ & [8]{} &[4]{}\ ${{\textrm{IX}}}$ & ${{\textrm{IV}}} \to {{\textrm{IX}}} $ & $c_{13}$ & $2A$ & [4]{} &[2]{}\ The following steps are dealing with defining all hyperbolic bifurcations. Some of them can be found by the next lemma, the analog of which for two degrees of freedom can be obtained from the results of [@BolFom]. \[lem1\] Let $x_0$ be a non-degenerate critical point of rank $n-2$ of the type “center-saddle” in an integrable system with $n$ degrees of freedom and the momentum map $\mathcal{J}$. Denote by $U_1$ and $U_2$ two local critical subsystems with $n-1$ degrees of freedom transversaly intersecting at $x_0$ and such that in $U_1$ the point $x_0$ has some hyperbolic $n$-atom $V$ and in $U_2$ the $n$-atom of $x_0$ is the elliptic atom $kA$ $(k\in \mathbb{N})$. Let us consider a small two-dimensional section at the point $\mathcal{J}(x_0)$ transversal to the smooth leaves $\mathcal{J}(U_1)$ and $\mathcal{J}(U_2)$. Let $q$ be a point on the edge of this section generated by the leaf $\mathcal{J}(U_2)$. Then the $(n+1)$-atom containing the pre-image of $q$ is $k(V{\times}S^1)$. In Table \[tab52\], we show all the edges of the iso-energy diagrams satisfying the conditions of the lemma, i.e., the edges along which inside the corresponding critical subsystem in the pre-image the boundary bifurcation is elliptic, but the pre-image of the boundary point has the type “center-saddle”. The hyperbolic 3-atom at the boundary point is found from Table \[tab42\], then, on the edges considered, the 4-atoms are defined from the lemma (they are given in the last column of the table). Recall that the 4-atoms which are direct products of 3-atoms and the circle have the same notation as the initial 3-atoms. [|c|c|c|c|c|]{}\ Edge & ---------- Boundary point ---------- & ---------- Boundary crossing ---------- & -------- 3-atom -------- & ------------- 4-atom of the edge ------------- \ $b_8$ & $\lambda_{32}$ & $c_2 \to c_1$ & $B$ â $\mo$& $B$\ $c_6$ & $\lambda_{41}$ & $b_1 \to b_2$ & $B$ â $\mn$ & $B$\ $b_9$ & $\lambda_{33}$ & $c_4 \to c_3$ & $C_2$ â $\mo$& $C_2$\ $c_7$ & $\lambda_{62}^{-}$ & $c_2 \to c_4$ & $2A^*$ â $\mo$& $2A^*$\ $c_5$ & $\delta_{2}$ & $a_1 \to a_2$ & $2B$ â $\mm$ & $2B$\ $c_{16}$ & $\delta_{31}$ & $a_2 \to a_3$ & $4B$ â $\mm$ & $4B$\ Now let us consider the cusps $\Delta_{31} - \Delta_{33}$ of the iso-energy diagrams. Since for theses points we know the 4-atoms on the incoming elliptic edges ($2A$ on the edges $c_{13}$ at $\Delta_{31}$ and $c_{10}$ at $\Delta_{32},\Delta_{33}$ ), we see that the remaining incoming edges $c_{14},c_9$ and $c_{11}$ have the atoms $2B$. The cusp $\Delta_{34}$ does not giva any new information (the atoms on the edges $c_{16}$ and $c_{17}$ are already known). One more block of information is obtained from the following simple reasoning. Let the critical points in the pre-image of some edge form $k$ hyperbolic two-dimensional tori, and crossing the edge between the adjacent chambers changes the number of regular tori from $k$ to $2k$. Then the atom on the edge is $kB$. From here, having the information about 2-tori, we establish the following atoms: $2B$ on the edges $b_7$, $c_8$ and $c_{15}$ (crossings ${{\textrm{III}}}\to {{\textrm{V}}}$, ${{\textrm{IV}}}\to {{\textrm{VI}}}$ and ${{\textrm{III}}}\to {{\textrm{VII}}}$ respectively), $4B$ on the edge $b_4$ (crossing ${{\textrm{VII}}}\to {{\textrm{VIII}}}$). Still we have unknown 4-atoms on the edges $b_5,b_6$ and $c_{12}$. All of them correspond to the crossings between the chambers containing four tori. Let us analyze the information we already have on the non-degenerate periodic trajectories in the pre-images of the following points of the iso-energy diagrams: $\lambda_{43}$ for $b_5$ (region 12), $\lambda_{44}$ for $b_6$ (region 15), and $\lambda_{53}^{\pm}$ for $c_{12}$ (region 13). For all of these points we know the 4-atoms on three incoming edges out of four and also the number of regular tori in the adjacent chambers. From here we find out that the 4-atoms for the edges left are $2C_2$ on the edges $b_5,b_6$ and $4A^*$ on the edge $c_{12}$. The theorem is proved. This completes the description of the equipped iso-energy diagrams and the rough thre-dimensional topology of the regular iso-energy levels. Rough net invariants {#sec6} ==================== The theory of the topological invariants of the integrable systems with arbitrary many degrees of freedom was developed by A.T.Fomenko in the works [@fom91; @fomams91]. At the same time, due to the absence of methods of practical analysis of non-trivial examples up to this moment no investigations of a system not reducible to a family of systems with two degrees of freedom were fulfilled. Such a system is the Kovalevskaya top in a double field studied here. The equipped iso-energy diagrams built above show one of the possible rough topological invariant. Nevertheless, as it can be seen from the above illustrations, the attempt to keep real scales lead to the appearance of extra small regions of the diagrams transformations. Then the clear picture of these transformations is not always possible to obtain and the illustrations themselves in some details are overloaded with information. Here we give the description of the rough topological invariants in the form of the analog of the Fomenko nets on the iso-energy levels for all parametrically stable cases. Let us explain the idea of the Fomenko nets and the simplified analog suggested in this work. Having an integrable system with three degrees of freedom, let us identify to a point each conneted component of integral manifolds. Under some conditions we obtain a three-dimensional stratified manifold (see [@fom91; @Fom1989] for details). The rough topological invariant of the system is a one-dimensional complex built from this manifold. It is a graph with two types of vertices. A vertex of the type “center” replaces each connected component of any three-dimensional stratum. A vertex of the type “atom” stands for each connected component of any two-dimensional wall and is equipped with the notation of the atom of the bifurcation, which takes place in the neighborhood of corresponding critical integral surface. A graph edge is drawn from a center vertex to that atom vertex on which the bifurcation of the corresponding three-dimensional torus takes place when crossing the wall. This invariant is called the net invariant of Fomenko. This invariant may be naturally applied also to reducible systems (i.e., systems having a one-dimensional symmetry group and admitting the reduction to a family of systems with two degrees of freedom). The experience of constructing multidimensional invariants in such systems brought us to the following conclusions. First, it is more rational to build an invariant for the systems induced on nonsingular iso-energy manifolds. Second, in order to describe the rough topology it is convenient to simplify the construction basing on the notion of an iso-energy diagram and generated chambers. Let, as before, $H$ be the Hamilton function of a system, $\mF$ the momentum map, $\Sigma\subset{\bR}^3$ the bifurcation diagram, i.e., the set of critical values of the momentum map. Suppose that all iso-energy manifolds $H_h=\{H=h\}$ are compact and fix a nonsingular $H_h$. The diagram $\sigh$ of the restriction $\mF(h)$ of the map $\mF$ to $H_h$ divides the plane into open connected components (chambers) having regular three-dimensional tori in its pre-image. As it was mentioned above, due to the fact that $H_h$ is compact, the zero-chamber arises containing all sufficiently far points of the plane. Let us add the “infinitely distant” point to the plane to obtain the two-dimensional sphere $S^2$. If there are no focus type singularities, i.e., the diagram does not have isolated points, then the diagram $\sigh$ generates a presentation of $S^2$ as a two-dimensional cell complex. Let us denote this complex by $\csigh$. The equipped diagram adds to the complex the information on the number of regular tori in the pre-images of the 2-cells and on the bifurcations which correspond to 1-cells. Replace each chamber (including the zero-chamber) with a point supplied with the number of regular tori in the pre-image of any point in the chamber. These points are the vertices of the new graph $\Gamma_h$, which is natural to consider lying on the two-dimensional sphere ${S}^2$. The vertices corresponding to two adjacent chambers are joined by the edge, on which the set of atoms is marked taking place in the neighborhood of the preimage of a point on the common wall of the chambers. If the boundary of a chamber contains an isolated point of the diagram $\sigh$ which have a focus type singularity of rank 1 in its pre-image, then at the vertex corresponding to this chamber we draw a loop edge supplied by the symbol F. It is convenient to draw the graph $\Gamma_h$ using the projection onto the plane of the sphere $S^2$ from its “infinitely distant” point. The corresponding zero-vertex is represented by a dashed closed curve surrounding all other vertices and edges of the graph. Note that the $F$-loops does not lie on the sphere, but are only shown on it for simplicity. If there are no such loops (i.e., no focus type singularities), then the sphere together with the graph $\Gamma_h$ is a two-dimensional complex conjugate to the complex $\csigh$. Now we return to the Kovalevskaya top in a double field. According to the above results on the classification of the iso-energy diagrams $\sigh$ with respect to $h$ and the physical parameters $a,b$ (see Fig. \[fig\_chart\]) we come to the following statement. \[theo9\] The parametrically stable diagrams $\sigh$ correspond to $19$ types of the rough topological net invariants $\Gamma_h(a,b)$ of the five-dimensional iso-energy surfaces of the Kovalevskaya top in a double field. Note that the topology of the iso-energy levels is known [@KhZot] and it changes when crossing the values $h=\mp a\mp b$, i.e., the separating surfaces $\Q_1 - \Q_4$. The complete list of the invariants is shown in Fig. \[fig\_leftcnet\] – \[fig\_blocknet\]. Here the circles with numbers inside them stand for the vertices together with the number of regular tori (except for the above described way of presenting the vertex for the outer zero-chamber). On the graph edges we place the notation of the 4-atoms of three-dimensional bifurcations obtained from the corresponding atoms for a system with two degrees of freedom by multiplying by the topological circle. The illustrations are placed in the same way as the corresponding iso-energy diagrams in Fig. \[fig\_leftc1\] – \[fig\_block1\]. The obtained isoenergetic invariants give a possibility to construct the complete net invariant of Fomenko in the sense of [@fom91] on the six-dimensional phase space. To this end, we must watch the dynamics of $\Gamma_h(a,b)$ along the fixed values of $a,b$ and unite these graphs on $h$ in the corresponding enhanced space. Recall that $a$ is not an essential parameter and all objects are preserved by the homothetic transformation $h'=h/a,b'=b/a$. The complete invariant changes when the line $b'={\rm const}$ in the plane $(b',h')$ crosses one of the double points of the set of the separating curves. The corresponding values of $b'$ are straightforwardly defined from . In particular, the point of intersection of the curves $\Q_8$ and $\Q_9$ is defined by the unique positive root $b_0$ of the equation $6591b'^8+612b'^3-262b'^2-28b'-1=0$. \[theo10\] The problem of the motion of the Kovalevskaya top in a double field has four stable with respect to the physical parameter $b/a\in [0,1]$ rough net invariants of Fomenko. The separating values of the parameter are $0$ $($the Kovalevskaya case$)$, $3-2\sqrt{2}$, $\frac{1}{3}$, $b_0$ è $1$ $($the reducible Yehia case$).$ Knowing the simplified net invariant, let us establish the number and the character of periodic solutions and degenerate motions. Since $H_h$ is supposed non-singular the zero-skeleton of $\sigh$ is generated by periodic solutions consisting of critical points of rank $1$. Let us also include in the zero-skeleton the image of degenerate points of rank $2$ filling two-dimensional tori of special type. The graph $\Gamma_h$ (not counting on the $F$-loops) divides the sphere $S^2$ into open connected components; each component is bounded by a cycle of the graph. Thus the 2-cells of the partition of the sphere generated by the graph $\Gamma_h$ correspond to singularities of rank $1$ or to degenerate singularities of rank $2$. Knowing the atoms along the edges bounding a cell corresponding to periodic solutions, one can define the number of these solutions on a given level of the first integrals and the type of these solutions. Ignoring the $F$-loops, we obtain that the solutions of the type “center-center” have on the boundary of the cell only atoms of the type $A$, and the solutions of the type “saddle-saddle” only hyperbolic atoms (here these atoms are $B,A^*$ and $C_2$); for the solutions of the type “center-saddle” on the boundary of the cell we meet atoms of both elliptic and hyperbolic types. The number of periodic solutions in the pre-image of a 2-cell (the node point of the diagram) is defined by the multiplicity of atoms on the edges and the multiplicity of vertices of graph $\Gamma_h$ along the bounding cycle of a 2-cell. To distinguish 2-cells corresponding to degenerate solutions of rank $2$, it is necessary to address the corresponding bifurcation diagram and select the node points which are cusps or tangency points of the bifurcation curves. \[theo11\] The number and the types of singular periodic solutions and the number of degenerate two-dimensional tori on non-singular energy levels of the Kovalevskaya top in a double field in $19$ parametrically stable cases are given in Table $\ref{tab61}$. The regions are enumerated according to Fig. $\ref{fig_chart}$. [|p[40mm]{}|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|]{}\ Region number &**01**&**02**&**03**&**04**&**05**&**06**&**07**&**08**&**09**&**10**&**11**&**12**&**13**&**14**&**15**&**16**&**17**&**18**&**19**\ Type “center-center”&4&3&4&3&2&3&4&4&4&4&4&4&4&4&4&8&8&8&8\ Type “center-saddle”&3&2&2&0&0&1&4&4&6&6&5&6&6&6&6&10&10&8&8\ Type “saddle-saddle”&0&0&0&0&0&0&1&2&3&2&1&6&6&6&6&6&6&4&4\ Type “focus-focus”&1&1&0&0&1&1&1&0&0&1&1&0&0&0&0&0&0&0&0\ Degenerate tori&1&0&0&0&0&1&2&6&6&2&1&6&10&14&12&10&10&12&12\ From the net invariants built above we also get the description of all loop molecules of rank $1$ singularities. If the type of a singularity includes the part “center”, then its saturated neighborhood in $\mP^6$ is presented as the direct product of two 3-atoms and is easily found from Table \[tab42\]. Having a singularities of the type “saddle-saddle”, let us consider its loop molecule on the iso-energy level. In the introduced simplified net invariant such a singularity with necessity generates a 2-cell with four vertices and with the edges marked by the atoms of hyperbolic three-dimensional bifurcations. The complete list of the forms of the saddle type loop molecules met in this problem together with their notation is given in Fig. \[fig\_circle\]. The first three molecules have the analogs in two degrees of freedom; these analogs are the molecules of the critical points of rank $0$ with the saturated neighborhoods $B{\times}B$, $(B{\times}C_2)/\mathbb{Z}_2$ and $B{\times}C_2$ [@Osh1; @Osh2]. Here, of course, we use the standard notations for the 2-atoms. The last molecule does not have such an analog. For all saddle type singularities in our problem we finally get the following iso-energy loop molecules: $\mathrm{NM}_1$ for $\lambda_{42}$ and $\lambda_{52}$; $2\mathrm{NM}_1$ for $\lambda_{23}$; $\mathrm{NM}_2$ for $\lambda_{21}$; $2\mathrm{NM}_2$ for $\lambda_{22}$; $\mathrm{NM}_3$ for $\lambda_{43}$ and $\lambda_{44}$; $2\mathrm{NM}_4$ for $\lambda_{53}$. Áëàãîäàðíîñòè {#áëàãîäàðíîñòè .unnumbered} ============= The authors thank Academician A.T.Fomenko for valuable discussions of the approaches and results. The work is partially supported by the RFBR (grant N 14-01-00119) and the Authority of Volgograd Region (ãðàíò N 13-01-97025). [99]{} Integral manifolds of the reduced system in the problem of inertial motion of a rigid body about a fixed point // Mekh. Tverd. Tela. – 1976. – N 8. – P. 18–23. (In Russian) Phase topology of one integrable case of the rigid body motion // Mekh. Tverd. Tela. – 1979. – N 11. – P. 50–64. 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(In Russian) ![The diagram of the subsystem $\mm$ on the $(f^2,h)$-plane.[]{data-label="fig_bifset1"}](bif_set_1_new.eps){width="25.00000%"} ![The diagram of the subsystem $\mn$ on the $(m,h)$-plane and its fragment.[]{data-label="fig_bifset2"}](bif_set_2_new.eps "fig:"){width="30.00000%"} ![The diagram of the subsystem $\mn$ on the $(m,h)$-plane and its fragment.[]{data-label="fig_bifset2"}](fbif_set_2_new.eps "fig:"){width="22.00000%"} ![The diagram of the subsystem $\mo$ on the $(s,h)$-plane and its fragment.[]{data-label="fig_bifset3"}](bif_set_3_new.eps "fig:"){width="41.00000%"}  ![The diagram of the subsystem $\mo$ on the $(s,h)$-plane and its fragment.[]{data-label="fig_bifset3"}](fbif_set_3_new.eps "fig:"){width="41.00000%"} ![The separation set and the paths of the look-up.[]{data-label="fig_chart"}](chart.eps){width="60.00000%"} ![Equipped diagrams: “left circle”.[]{data-label="fig_leftc1"}](leftc1.eps){width="\cofb\textwidth"} ![Equipped diagrams: “right circle”.[]{data-label="fig_rightc1"}](rightc1.eps){width="\cofb\textwidth"} ![Equipped diagrams: “line”.[]{data-label="fig_line1"}](line1.eps){width="\cofs\textwidth"} ![Equipped diagrams: “block”.[]{data-label="fig_block1"}](block1.eps){width="\cofs\textwidth"} ![Net invariants: “left circle”.[]{data-label="fig_leftcnet"}](leftcnet.eps){width="\cofb\textwidth"} ![Net invariants: “right circle”.[]{data-label="fig_rightcnet"}](rightcnet.eps){width="\cofb\textwidth"} ![Net invariants: “line”.[]{data-label="fig_linenet"}](linenet.eps){width="\cofs\textwidth"} ![Net invariants: “block”.[]{data-label="fig_blocknet"}](blocknet.eps){width="\cofs\textwidth"} ![Loop molecules of the saddle type singularities of rank 1.[]{data-label="fig_circle"}](circle_mol.eps){width="120mm"} [^1]: Russian Academy of National Economy and Public Administration. E-mail: mikeh@inbox.ru [^2]: Financial University under the Government of Russian Federation. E-mail: orelryabov@mail.ru
--- abstract: 'A first measurement of the $e^+e^-$ decay rate of $\phi$ mesons in dAu collisions from the PHENIX detector at RHIC and its comparison to the $K^+K^-$ decay channel is described. The comparison of the two decay channels can be sensitive to chiral symmetry restoration.' address: - ' Physics Department, University of California, Riverside California, 92521 USA' - ' For the full PHENIX Collaboration authors and acknowledgments, see “Collaborations” appendix of this volume.' author: - Richard Seto for the PHENIX Collaboration title: Light Vector Mesons from dAu in PHENIX --- Introduction ============ Deconfinement is the first and most familiar manifestation of the QCD phase transition. Of equal importance is the chiral transition. The spontaneous breakdown of chiral symmetry responsible for the masses of the hadrons. The study of low mass vector mesons is an ideal probe of chiral symmetry restoration in heavy ion collisions. As an example, the lifetime of the $\phi$ is about 50 fm/c in the vacuum and a fraction of $\phi$’s produced in a heavy collision will decay in the fireball. Decays to di-leptons are particularly attractive since the daughter particles are not strongly interacting and can reach the experimental apparatus without rescattering, hence any effects such as mass shifts or broadening of spectral functions due to the onset of chiral symmetry restoration would show themselves in the invariant mass spectra. The E325 experiment at KEK[@E325] and CERES at CERN [@CERES] have seen hints of such effects in both pA and AA collisions. Because of the small Q value of the KK decay channel, a comparison of the $\phi \rightarrow$ KK/$\phi \rightarrow$ ee rate is a particularly sensitive measure of a mass modification of either the $\phi$ or the kaon [@KKEE]. The PHENIX detector is ideally suited for such measurements in that it has good particle identification for kaons with momentum between 300 MeV/c and 2 GeV/c and electrons to 5 GeV/c, as well as good momentum resolution. All data presented in this paper are from deuteron-Gold collisions at $\sqrt{s_{NN}}$=200 GeV measured by PHENIX in 2003. di-electron Analysis ==================== About 31M single-electron triggered events were used for the electron analysis. The trigger required matching hits in the Ring Imaging Cerenkov Counter(RICH) and the Electromagnetic Calorimeter (EMC), where the threshold was set to 600 MeV. An additional data set with a higher threshold was not used for the present analysis. Electrons were identified by requiring two or more phototubes firing in the RICH and an E/p match between the energy measured by the EMC and the momentum measured by the tracking system. Specifically we required $0.5<E/p<1.5$. Cuts are then made to remove electron candidates coming from conversions. The invariant mass spectrum of $e^+e^-$ pairs is then formed as shown in figure 1. In order to form a background sample, opposite sign pairs were taken from different events. Because our sample was triggered, care was taken that the mixed event background had the same characteristics as the events themselves. In particular minimum bias events were used to form the mixed pair, where one of the electrons was required to pass the trigger requirements. Additionally, event used to form the mixed pair were required to have similar centralities and vertex positions. Normalization of the background distribution was done by matching the data to the background in a sideband region between 850 and 950 MeV below the signal and 1100-1200 MeV above the signal. Other methods of normalization were used to estimate the systematic error from this procedure. [cc]{} ![Background subtracted $e^+e^-$ invariant mass spectrum with a fit as described in the text.](eesignalback){width="3in"} ![Background subtracted $e^+e^-$ invariant mass spectrum with a fit as described in the text.](eesignal){width="3in"} The background subtracted signal was then fit to a relativistic Breit-Wigner convoluted with a Gaussian to account for the experimental resolution (figure 2). The width $\Gamma$ was held fixed at the particle data book value. Values obtained from the fit are shown in Table \[tab:tc\]. The value of the mass is consistent with the known value of the mass in the vacuum. In addition the fitted experimental resolution is consistent with simulations of the detector performance. The signal was then divided into 3 bins of $M_T$, and corrections were made for acceptance and efficiencies. A fit was then done to the invariant yield resulting in the yield and inverse slope shown in the table. Major contributors to the systematic error on dN/dy are the normalization of the background and its effect on the inverse slope, and the run-by-run variations from the electron-trigger. $K^+K^-$ Analysis ================= [3.4in]{} ![image](kksignal){width="3.2in"} About 62M minimum bias events were used for the $K^+K^-$ analysis. Kaons were identified in the time-of-flight detector (TOF) which covers $\Delta\phi\approx 40^\circ$. Because of the limited coverage of the TOF, the analysis preferentially accepted higher momentum $\phi's$ as compared with the electron analysis. The invariant mass spectrum was generated by combining unlike sign pairs of kaons. The background shape was formed in a similar manner to the electrons by mixing pairs from events with similar centralities and vertices. The normalization in this case was determined by adjusting the integral of the background to $2\sqrt{N_{++}N_{--}}$ where $N_{++}$ and $N_{--}$ refer to the like sign combinations in each event. Once again the signal was fit to a relativistic Breit-Wigner convoluted with a Gaussian for the experimental resolution. Since the resolution of the $\phi$ in the KK decay channel is rather insensitive to momentum effects, we chose to hold the experimental resolution fixed to the value obtained from the simulation - 1.2 MeV. Results of the fit are shown in the table and figure 3. Both the mass and width are consistent with PDG vacuum values. For both decay channels, $\Gamma$ and $\sigma_{exp}$ are highly correlated, when they are both allowed to vary in the fit. The data are then divided into bins of $M_T$, corrections are made for acceptance and efficiencies and a fit is done to obtain the yield. Because of the good signal to background in the KK channel, systematic errors are considerably smaller than in the electron channel. The systematic errors are dominated by the range in $M_T$ over which the fit was done, the run-by-run changes in efficiency, and the corrections due to the fiducial cuts. Figure 4 shows a comparison of the data points from both the electron and kaon analyses. The fit shown in the figure is to all of the points. When fits were done to extract yields for comparison, the fits were done separately for the electron and kaon channels. The two data samples are consistent in both the yield and the inverse slope as can be seen by looking at the comparisons in figures 5 and 6 of the extracted inverse slopes and yields. .1in Conclusions =========== Within the substantial errors - particularly from the electron analysis - the yields of the $\phi\rightarrow ee$ and $\phi \rightarrow KK$ are consistent with one another. Because of the small fraction which decay inside the relevant volume, the effects from chiral symmetry restoration, particularly in dAu collisions, are expected to be small. PHENIX will make further improvements to this analysis by including the data with a higher trigger threshold and improving systematic errors - particularly from background normalization in the electron channel. PHENIX also sees a clear $\omega \rightarrow ee$ signal, which, with a shorter lifetime than the $\phi$ will increase the number of decays in the fireball. In addition, effects in Au-Au collisions could be considerably stronger since energy densities are higher. Further in the future, upgrades to PHENIX such as the Hadron Blind Detector (HBD) will significantly enhance our capabilities to reject conversion and Dalitz pairs which are the dominant source of background. This will open up the possibility of studying the $\rho \rightarrow ee$ as well as thermal di-electrons. [cc]{} ![A comparison of the inverse slope(right) and yields(left) of the $\phi$ ](pt){width="2.6in"} ![A comparison of the inverse slope(right) and yields(left) of the $\phi$ ](Tdndy){width="3.2in"} $\phi\rightarrow ee$ $\phi\rightarrow KK$ PDG -------------------------- ------------------------- ----------------------------------------- --------- Mass $(GeV/c^2)$ 1.0177$\pm$0.0023 1.0193$\pm$0.0003 1.01946 $\Gamma$ $(MeV/c^2)$ 4.26(fixed) 4.750$\pm$0.67 4.26 $\sigma_{exp} (MeV/c^2)$ 8.1$\pm$2.1 1.2(fixed) N 120 207 dN/dy 0.056$\pm$0.015$\pm$50% 0.0468$\pm$0.0092$^{+0.0095}_{-0.0092}$ T (MeV) 326$\pm$94$\pm$53% 414$\pm$13$\pm$23 : \[tab:tc\]Results of fits to the invariant mass spectra and the yields for both the electron and kaon decay channels of the $\phi$. Statistical errors are listed first, followed by systematic errors. [99]{} K.Ozawa et al, Phys. Rev. Lett. 86-22 5019-5022(2001) CERES Collaboration, G. Agakichiev et al, Phys. Lett. B422 (1998) 405 D. Lissauer and E. V. Shuryak, Phys. Lett. B253, 15 (1991).
--- abstract: 'We study a minimum problem and associated maximum problem for finite, complex, self-adjoint Toeplitz matrices. If $A$ is such a matrix, of size $(N+1)$-by-$(N+1)$, we identify $A$ with the operator it represents on ${{\mathcal P}}_N$, the space of complex polynomials of degrees at most $N$, with the usual Hilbert space structure it inherits as a subspace of $L^2$ of the unit circle. The operator $A$ is the compression to ${{\mathcal P}}_N$ of the multiplication operator on $L^2$ induced by any function in $L^{{\infty}}$ whose Fourier coefficients of indices between $-N$ and $N$ match the matrix entries of $A$. Our minimum problem is to minimize the $L^{{\infty}}$ norm of such inducers. We show there is a unique one of minimum norm, and we describe it. The associated maximum problem asks for the maximum of the ratio of the preceding minimum to the operator norm. That problem remains largely open. We present some suggestive numerical evidence.' address: - 'Department of Mathematics, The University of Iowa, Iowa City, IA  52242-1419, USA' - 'Department of Mathematics, University of California, Berkeley, CA  94720-3840, USA' author: - 'Dennis Courtney${}^*$ and Donald Sarason' title: 'A Mini-Max Problem for Self-Adjoint Toeplitz Matrices' --- [^1] Introduction {#sec1} ============ To begin we consider an $(N+1)$-by-$(N+1)$ complex Toeplitz matrix $A$ ($N$ a positive integer): $$A = (a_{j-k})_{j,k=0}^N.$$ By $L^2$ we shall mean the $L^2$ space of normalized Lebesgue measure on the unit circle, ${{\mathbb T}}$. The subspace of $L^2$ consisting of the complex polynomials of degrees at most $N$ will be denoted by ${{\mathcal P}}_N$. We identify $A$ with the operator it induces on ${{\mathcal P}}_N$, i.e., the operator on ${{\mathcal P}}_N$ whose matrix with respect to the monomial basis is $A$. The operator $A$ is the compression to ${{\mathcal P}}_N$ of the multiplication operator on $L^2$ induced by any function $f$ in $L^{{\infty}}$ of the circle whose Fourier coefficients indexed between $-N$ and $N$ match the matrix elements of $A$, in other words, for which ${\widehat}{f}(n) = a_n$ for $n = -N,-N+1,\dots,0,\dots,N-1,N$. The family of such functions $f$ will be denoted by ${{\mathcal G}}_A$. We pose the problem of minimizing $\|f\|_{{\infty}}$ over the class ${{\mathcal G}}_A$. Standard reasoning shows the minimum is attained; we denote it by $c_A$. It is evident that $\|A\| \le c_A$. In the cases where $A$ is lower triangular or upper triangular, our problem fits into the classical interpolation problem of C. Carathéodory and L. Fejér, dating back 100 years [@CF]. Perhaps surprisingly, the more general problem seems not to have been considered until recently [@BT], [@NF], [@S]. Although we will not do so here, it can be treated as a special case of a general dual extremal problem studied by M. G. Krein and A. A. Nudelman in [@KN] — more details later. In this paper we focus on the case where $A$ is self-adjoint. For this case we show that the minimizing function in ${{\mathcal G}}_A$ is unique, and we describe it: it alternates the values $c_A$ and $-c_A$ on a family of subarcs that partition ${{\mathbb T}}$, are even in number, and number at most $2N$. If the matrix $A$ is self-adjoint but not diagonal, then the minimizing function is not a rational function and so multiplies every nonzero function in ${{\mathcal P}}_N$ outside of ${{\mathcal P}}_N$. It follows that $c_A > \|A\|$. Let ${{\mathcal T}}_{N+1}$ denote the class of $(N+1)$-by-$(N+1)$ self-adjoint Toeplitz matrices. We pose the problem of maximizing the ratio $c_A/\|A\|$ over the nonzero matrices $A$ in ${{\mathcal T}}_{N+1}$. As in the minimum problem, standard reasoning shows that the maximum is attained; we denote it by $c_N$. It is proved in [@S] that $c_1 = \pi/2$. The other cases remain open. We have numerical evidence, presented in Section \[sec7\], that $c_N > \pi/2$ for $N > 1$, and suggesting that $c_N$ increases with $N$. We are able to prove that the preceding inequality holds for infinitely many $N$. It is known, and nontrivial, that the numbers $c_N$ have a common bound. M. Bakonyi and D. Timotin obtain in [@BT] the bound $c_N \le 2$. The interpolation problem they consider is slightly different from ours, but their reasoning applies in our case. L. N. Nikolskaya and Yu. B. Farforovskaya obtain in [@NF] the bound $c_N \le 3$, and their reasoning applies even in the non-self-adjoint case. In both papers [@BT] and [@NF], the inequality obtained is an outcome of a matrix extension problem. In Section \[sec2\] we state what we shall need concerning the Carathéodory–Fejér interpolation problem. Section \[sec3\] recasts our minimum problem in a form that fits into the Carathéodory–Fejér framework. The minimizing functions for our minimum problem are identified in Section \[sec4\] and studied further in Section \[sec5\]. The inequality $c_1 \le \pi/2$ is reproved in Section \[sec6\] in a way that exploits the results from Sections \[sec4\] and \[sec5\]. The concluding Section \[sec8\] contains open questions and conjectures. The maximum part of our mini-max problem remains largely mysterious, while strongly enticing. New ideas are needed. We are indebted to an anonymous referee for alerting us to the relevance to our study of references [@G] and [@KN]. Carathéodory–Fejér Interpolation Problem {#sec2} ======================================== The Carathéodory–Fejér problem asks whether there is a holomorphic function in the unit disk ${{\mathbb D}}$ having prescribed power series coefficients of orders $0,1,\dots,N$ and having a prescribed supremum norm. Carathéodory and Fejér’s solution, recast in our context and in our notation, can be stated as follows: This is actually a mild generalization of what Carathéodory and Fejér prove, because their minimization problem is over $H^{{\infty}}$ functions, not $L^{{\infty}}$ functions. But once one knows their result, one easily generalizes it to the one stated above. Theorem CF forms the basis of the analysis to follow. Reformulation of the Minimum Problem {#sec3} ==================================== Let $A$ be a matrix in the class ${{\mathcal T}}_{N+1}$, and let the function $f$ belong to ${{\mathcal G}}_A$. Then ${\mathop{\mathrm{Re}}\nolimits}f$ is also in ${{\mathcal G}}_A$ and $\|{\mathop{\mathrm{Re}}\nolimits}f\|_{{\infty}} \le \|f\|_{{\infty}}$. Hence $c_A$, the minimum of $\|f\|_{{\infty}}$ over the class ${{\mathcal G}}_A$, equals the minimum of $\|f\|_{{\infty}}$ over the class ${{\mathcal G}}_A^* = \{f \in {{\mathcal G}}_A: f = {\overline}{f}\}$. We let $P_+$ denote the orthogonal projection of $L^2$ onto the Hardy space $H^2 = \{h \in L^2: {\widehat}{h}(n) = 0$ for $n < 0\}$. As is customary, we identify the functions in $H^2$ with their holomorphic extensions in ${{\mathbb D}}$. Suppose $f$ is a function in the class ${{\mathcal G}}_A^*$. Then the function $h = P_+f - a_0/2$ is in $H^2$, and $f = h + {\overline}{h} = 2{\mathop{\mathrm{Re}}\nolimits}h$. We have $$h(z) = a_0/2 + a_1z + \dots + a_Nz^N + O(z^{N+1})\quad (z {\longrightarrow}0).$$ Let ${{\mathcal H}}_A$ denote the class of functions $h$ in $H^2$ that satisfy the preceding condition and for which ${\mathop{\mathrm{Re}}\nolimits}h$ is in $L^{{\infty}}$. Then $2{\mathop{\mathrm{Re}}\nolimits}h$ is in ${{\mathcal G}}_A^*$ for such an $h$. Accordingly, our minimum problem can be restated as the problem of minimizing $2\|{\mathop{\mathrm{Re}}\nolimits}h\|_{{\infty}}$ over the class ${{\mathcal H}}_A$. The Minimum Problem {#sec4} =================== Let $A$ be a matrix in ${{\mathcal T}}_{N+1}$. To eliminate a trivial case, we assume here that $A$ is not diagonal. Suppose $h$ is a minimizing function in ${{\mathcal H}}_A$, i.e., a function in ${{\mathcal H}}_A$ satisfying $2\|{\mathop{\mathrm{Re}}\nolimits}h\|_{{\infty}} = c_A$. To simplify the notation we temporarily write $c$ in place of $c_A$ below. We introduce the domain $${{\mathbb S}}_c = \left\{z \in {{\mathbb C}}: -\frac {c}{2} < {\mathop{\mathrm{Re}}\nolimits}z < \frac {c}{2}\right\},$$ an open vertical strip in the plane bisected by the imaginary axis, of width $c$. The domains ${{\mathbb D}}$ and ${{\mathbb S}}_c$ are conformally equivalent under the map $h_c: {{\mathbb D}}{\longrightarrow}{{\mathbb S}}_c$ given by $$h_c(z) = \frac {c}{\pi i} {\mathop{\mathrm{Log}}\nolimits}\left( \frac {1+z}{1-z}\right);$$ here, ${\mathop{\mathrm{Log}}\nolimits}$ denotes the principal branch of $\log$. The inverse map $h_c^{-1}: {{\mathbb S}}_c {\longrightarrow}{{\mathbb D}}$ is given by $$(h_c^{-1})(z) = i \tan\left( \frac {\pi z}{2c}\right).$$ The composite function $$b = h_c^{-1} \circ h = i \tan\left( \frac {\pi h}{2c}\right)$$ is a self-map of ${{\mathbb D}}$. We write its power series about $0$ as $$b(z) = b_0 + b_1z + \dots + b_Nz^N + O(z^{N+1}).$$ The relation between the coefficients $a_0,\dots,a_N$ of $h$ and $b_0,\dots,b_N$ of $b$ is rather complicated. To illustrate, we exhibit below the expressions for $b_0,b_1,b_2,b_3$ in terms of $a_0,a_1,a_2,a_3$. $$\begin{aligned} b_0 &= i\tan\left( \frac {\pi a_0}{4c} \right),\\ b_1 &= \frac {\pi ia_1}{2c} \sec^2 \left( \frac {\pi a_0}{4c} \right), \\ b_2 &= \frac {\pi ia_2}{2c} \sec^2 \left( \frac {\pi a_0}{4c} \right) + \frac {\pi^2ia_1}{2^2c^2} \sec^2 \left( \frac {\pi a_0}{4c} \right) \tan \left( \frac {\pi a_0}{4c} \right), \\ b_3 &= \frac {\pi ia_3}{2c} \sec^2 \left( \frac {\pi a_0}{4c} \right) + \frac {\pi^2a_1a_2}{2^2c^2} \sec^2 \left( \frac {\pi a_0}{4c} \right) \tan \left( \frac {\pi a_0}{4c} \right) \\ &\quad + \frac {\pi^3ia_1^3}{3 \cdot 2^3 \cdot c^3} \left[ 2\sec^2\left( \frac {\pi a_0}{4c} \right) \tan^2 \left( \frac {\pi a_0}{4c} \right) + \sec^4\left( \frac {\pi a_0}{4c} \right)\right]. \end{aligned}$$ However, for our purpose these complications are not an obstacle. From the rule for composing formal power series one knows that, for $0 \le n \le N$, the coefficient $b_n$ depends only on $c,a_0,\dots,a_n$, and vice versa. That is all we shall need. Let $B$ denote the $(N+1)$-by-$(N+1)$ lower triangular Toeplitz matrix with entries $b_0,b_1,\dots,b_N$ in the first column. By Theorem CF, there is a unique Blaschke product ${\omega}$ of order $\le N$ such that $$\|B\|{\omega}(z) = b_0 + b_1z + \dots + b_Nz^N + O(z^{N+1}).$$ We then have $$((h_c^{-1}) \circ (\|B\|{\omega})) = \frac {a_0}{2} + a_1z + \dots + a_Nz^N + O(z^{N+1}).$$ But in fact $\|B\| = 1$, for the inequality $\|B\| < 1$ would imply that twice the real part of the preceding composite function has $L^{{\infty}}$-norm less than $c = c_A$, contradicting the assumed minimizing property of $h$. Hence $$h = h_c \circ {\omega}= \frac {c_A}{\pi i} {\mathop{\mathrm{Log}}\nolimits}\left( \frac {1+{\omega}}{1-{\omega}}\right).$$ \[def4.1\] For $c > 0$ and $n$ a positive integer, an alternating step function of height $c$ and order $n$ is a function in $L^{{\infty}}$ that assumes alternatively the values $c$ and $-c$ on $2n$ subarcs that form a partition of ${{\mathbb T}}$. For example, the function $$\psi(e^{i{\theta}}) = \begin{cases} 1, &0 < {\theta}< \pi, \\ -1, &-\pi < {\theta}< 0, \end{cases}$$ is an alternating step function of height $1$ and order $1$. For the function $h = h_c \circ {\omega}$ above, we have, on ${{\mathbb T}}$ (with ${\mathop{\mathrm{Arg}}\nolimits}$ denoting the principal value of $\arg$), $$\begin{aligned} 2{\mathop{\mathrm{Re}}\nolimits}h &= \frac {2c_A}{\pi} {\mathop{\mathrm{Arg}}\nolimits}\left( \frac {1+{\omega}}{1-{\omega}} \right) = \frac {2c_A}{\pi} {\mathop{\mathrm{Arg}}\nolimits}((1+{\omega})(1-{\overline}{{\omega}})) \\ &= \frac {2c_A}{\pi} {\mathop{\mathrm{Arg}}\nolimits}({\omega}-{\overline}{{\omega}}) = \frac {2c_A}{\pi} {\mathop{\mathrm{Arg}}\nolimits}(i{\mathop{\mathrm{Im}}\nolimits}{\omega}) \\ &= \frac {2c_A}{\pi} \left( \frac {\pi}{2} \begin{cases} +0 &\text{if $\sin {\omega}> 0$,} \\ -\pi &\text{if $\sin {\omega}< 0$,} \end{cases} \right) \\ &= \begin{cases} c_A &\text{if $\sin {\omega}> 0$,} \\ -c_A &\text{if $\sin {\omega}< 0$.} \end{cases} \end{aligned}$$ Let $n$ be the order of ${\omega}$. The Blaschke product ${\omega}$ wraps the unit circle ${{\mathbb T}}$ around itself $n$ times in the strictly counterclockwise direction. We thus see that the minimizing function $2{\mathop{\mathrm{Re}}\nolimits}h$ is an alternating step function of height $c_A$ and order $n$. The following theorem summarizes what has been proved. To include the diagonal case in the statement, we define an alternating step function of order $0$ to be a constant function. The constant is defined to be the height of the function. \[th4.1\] For $A$ a matrix in ${{\mathcal T}}_{N+1}$, there is a unique function in ${{\mathcal G}}_A$ of minimum $L^{{\infty}}$-norm. It is an alternating step function of height $c_A$ and order at most $N$. Theorem \[th4.1\] as stated excludes much of the information obtained in the reasoning leading up to it— information we shall use later. The theorem as stated can be obtained by fitting it into the general dual extremum problem studied by Krein and Nudelman in Chapter IX of [@KN]. The general problem, which involves a Banach space and its dual, is formulated at the beginning of §1 of Chapter IX, after which their basic theorem on the problem, Theorem 1.1, is established. Taking real $L^1$ of the unit circle as the Banach space in question, it follows fairly simply from this theorem that each of our minimizers has the form $L {\operatorname{sign}}\nu$, where $L$ is a positive number and $\nu$ is a real trigonometric polynomial of order at most $N$; the minimizer is thus an alternating step function. Establishing the uniqueness of the minimizer takes more work but can be done on the basis of Krein-Nudelman’s Theorem 1.3 and its corollaries. See also Theorem 2.1 and 2.2 from Chapter IX of [@KN]. Of course, the approach we use here, which is specific to our setting, is quite distinct from the approach in [@KN]. The reasoning that led to the theorem can be reversed. Start with a nonconstant Blaschke product ${\omega}$ of order $n \le N$. Fix a positive number $c$, and form the function $h = \frac {c}{\pi i} {\mathop{\mathrm{Log}}\nolimits}\left( \frac {1+{\omega}}{1-{\omega}}\right)$. Then $2{\mathop{\mathrm{Re}}\nolimits}h$ is an alternating step function of height $c$ and order $n$. We show that $2{\mathop{\mathrm{Re}}\nolimits}h$ is the minimizing function for the corresponding minimum problem. In view of the analysis that produced Theorem \[th4.1\], it will suffice to show that, among all $H^{{\infty}}$ functions $g$ satisfying ${\widehat}{g}(k) = {\widehat}{{\omega}}(k)$ for $k = 0,\dots,N$, the function ${\omega}$ has the least norm. By Theorem CF, this amounts to showing that the compression $B$ to ${{\mathcal P}}_N$ of the operator on $L^2$ of multiplication by ${\omega}$ has norm $1$. Let $z_1,\dots,z_n$ be the zeros of ${\omega}$. Then ${\omega}$ has the form $${\omega}(z) = {\lambda}^2 \prod_{j=1}^n \left( \frac {z-z_j}{1-{\overline}{z}_jz} \right),$$ where ${\lambda}$ is a constant of modulus $1$. Defining the polynomials $r$ and $s$ by $$r(z) = {\lambda}\prod_{j=1}^n (z-z_j),\quad s(z) = {\overline}{{\lambda}} \prod_{j=1}^n (1-{\overline}{z}_jz),$$ we have ${\omega}= r/s$. The polynomials $r$ and $s$ belong to ${{\mathcal P}}_N$, and we have $\|r\|_2 = \|s\|_2$, plus $Bs = r$. The desired equality, $\|B\| = 1$, follows. (The preceding argument is of course well known.) The question whether every nonconstant alternating step function is obtained by the preceding procedure now arises. That question is addressed in the next section. Inner Functions and Alternating Step Functions {#sec5} ============================================== \[th5.1\] Let $\psi$ be an alternating step function of height $c$ and order $N > 0$. Then there is a Blaschke product ${\omega}$ of order $N$ such that $$\psi = \frac {2c}{\pi} {\mathop{\mathrm{Arg}}\nolimits}\left( \frac {1+{\omega}}{1-{\omega}}\right)$$ on ${{\mathbb T}}$. We assume (without loss of generality) that $c = 1$. Let the real numbers ${\alpha}_1,\dots,{\alpha}_N,{\beta}_1,\dots,{\beta}_N$ satisfy $$0 \le {\alpha}_1 < {\beta}_1 < {\alpha}_2 < {\beta}_2 < \dots < {\alpha}_N < {\beta}_N < {\alpha}_1 + 2\pi := {\alpha}_{N+1}.$$ Let the arcs $\chi_n^-$ and $\chi_n^+$ $(n = 1,\dots,N)$ be defined by $$\chi_n^- = \{e^{i{\theta}}: {\alpha}_n < {\theta}< {\beta}_n\},\quad \chi_n^+ = \{e^{i{\theta}}: {\beta}_n < {\theta}< {\alpha}_{n+1}\}.$$ The function $\psi$ that takes the value $1$ on each arc $\chi_n^-$ and the value $-1$ on each arc $\chi_n^+$ is, to within a sign, the general alternating step function of height $1$ and order $N$. We let $${\alpha}= {\alpha}_1 + \dots + {\alpha}_N,\quad {\beta}= {\beta}_1 + \dots + {\beta}_N.$$ We introduce the polynomials $$p(z) = e^{-i{\alpha}/2} \prod_{n=1}^N (z-e^{i{\alpha}_n}),\quad q(z) = e^{-i{\beta}/2} \prod_{n=1}^N (z-e^{i{\beta}_n}).$$ It is asserted that the polynomial $q - ip$ has degree $N$. In fact, the leading coefficient of $q - ip$ is $e^{-i{\beta}/2} - ie^{-i{\alpha}/2}$. For this to vanish we must have $e^{i({\beta}-{\alpha})/2} = -i = e^{3\pi i/2}$. However $0 < {\beta}- {\alpha}< 2\pi$, so $0 < \frac {{\beta}-{\alpha}}{2} < \pi$, and the assertion follows. We show that the rational function $q/p$ is real valued on ${{\mathbb T}}$ (except at its poles $e^{i{\alpha}_n}$, $n = 1,\dots,N$), and that on each arc $\chi_n := \chi_n^- \cup \chi_n^+ \cup \{e^{i{\beta}_n}\}$ it is an increasing function with range ${{\mathbb R}}$. We have $$\begin{aligned} q(e^{i{\theta}})/p(e^{i{\theta}}) &= e^{i({\beta}-{\alpha})/2} \prod_{n=1}^N (e^{i{\theta}}-e^{i{\beta}_n})\left/ \prod_{n=1}^N (e^{i{\theta}}-e^{i{\alpha}_n}) \right. \\ &= e^{i({\beta}-{\alpha})/2} \prod_{n=1}^N e^{i\left( \frac {{\alpha}_n-{\beta}_n}{2}\right)} \left[ \frac {e^{i\left( \frac {{\theta}-{\beta}_n}{2}\right)} - e^{-i\left( \frac {{\theta}-{\beta}_n}{2} \right)}}{e^{i\left( \frac {{\theta}-{\alpha}_n}{2} \right)} - e^{-i\left( \frac {{\theta}-{\alpha}_n}{2}\right)}} \right] \\ &= \prod_{n=1}^N \frac {\sin\left( \frac {{\theta}-{\beta}_n}{2}\right)}{\sin \left( \frac {{\theta}-{\alpha}_n}{2}\right)}\,. \end{aligned}$$ Thus $q/p$ is real valued on ${{\mathbb T}}$. For ${\alpha}_n < {\theta}< {\alpha}_{n+1}$, the function $\sin\left( \frac {{\theta}-{\alpha}_n}{2}\right)$ is positive, while the function $\sin\left( \frac {{\theta}-{\beta}_n}{2}\right)$ is negative for ${\theta}< {\beta}_n$ and positive for ${\theta}> {\beta}_n$. For $j \ne n$ and ${\alpha}_n < {\theta}< {\alpha}_{n+1}$, the functions $\sin\left( \frac {{\theta}-{\alpha}_j}{2}\right)$ and $\sin\left( \frac {{\theta}-{\beta}_j}{2}\right)$ have the same sign (negative for $j < n$, positive for $j > n$). Together with the expression above for $q/p$, this tells us that $q/p$ is negative on $\chi_n^-$ and positive on $\chi_n^+$. As $e^{i{\alpha}_n}$ and $e^{i{\alpha}_{n+1}}$ are poles of $q/p$, we thus must have $q(e^{i{\theta}})/p(e^{i{\theta}}) {\longrightarrow}-{\infty}$ as ${\theta}\searrow {\alpha}_n$ and $q(e^{i{\theta}})/p(e^{i{\theta}}) {\longrightarrow}+{\infty}$ as ${\theta}\nearrow {\alpha}_{n+1}$. Hence $q/p$ maps the arc $\chi_n$ onto ${{\mathbb R}}$. It must be monotone on each such arc because, being a rational function of degree $N$, it assumes no value with multiplicity greater than $N$. We show that the function ${\omega}= (q-ip)/(q+ip)$ is a Blaschke product of order $N$ satisfying $$\psi = \frac {2}{\pi} {\mathop{\mathrm{Arg}}\nolimits}\left( \frac {1+{\omega}}{1-{\omega}}\right). \leqno(1)$$ In fact, since ${\omega}$ can be rewritten as ${\omega}= \left( \frac {q}{p} - i \right)\left/\left( \frac {q}{p} + i\right)\right.$, it follows by Step 3 that ${\omega}$ is unimodular on ${{\mathbb T}}$. From the definition of ${\omega}$ we see immediately that ${\omega}(e^{i{\alpha}_n}) = 1$ for each $n$ and ${\omega}(e^{i{\beta}_n}) = -1$ for each $n$. We can re-express ${\omega}$ as ${\varphi}\circ \left( \frac {q}{p}\right)$, where ${\varphi}(z) = (z-i)/(z+i)$, a linear-fractional transformation that maps the upper half-plane to ${{\mathbb D}}$ and ${{\mathbb R}}$ to ${{\mathbb T}}\setminus \{1\}$, with ${\varphi}(z)$ moving counterclockwise on ${{\mathbb T}}$ as $z$ moves in the positive direction on ${{\mathbb R}}$. We know from Step 3 that on each arc $\chi_n$ the function $q/p$ is increasing with range ${{\mathbb R}}$. We can conclude that $\arg {\omega}$ undergoes an increment of $2\pi$ on each arc $\chi_n$, hence an increment of $2\pi N$ on ${{\mathbb T}}$. By the argument principle, ${\omega}$ has $N$ zeros in ${{\mathbb D}}$, thus is a Blaschke product of order $N$. Because $\frac {1+{\omega}}{1-{\omega}} = \frac {q}{ip}$, if $q(e^{i{\theta}})/p(e^{i{\theta}}) > 0$, i.e., if (according to Step 3) $e^{i{\theta}}$ is in one of the arcs $\chi_n^+$, we have ${\mathop{\mathrm{Arg}}\nolimits}\left( \frac {1+{\omega}(e^{i{\theta}})}{1-{\omega}(e^{i{\theta}})}\right) = -\frac {\pi}{2} + 0 = -\frac {\pi}{2}$. Similarly, if $e^{i{\theta}}$ is in one of the arcs $\chi_n^-$ then ${\mathop{\mathrm{Arg}}\nolimits}\left( \frac {1+{\omega}(e^{i{\theta}})}{1-{\omega}(e^{i{\theta}})}\right) = -\frac {\pi}{2} + \pi = \frac {\pi}{2}$. The desired equality $(1)$ follows. This concludes the proof of Theorem \[th5.1\]. Theorem \[th5.1\] is a particular case of Theorem 9 in the paper [@G] of P. Gorkin and R.C. Rhoades. Both [@G Theorem 9] and our Theorem \[th5.1\] assert the extistence of certain finite Blaschke products. The proofs of the two theorems are essentially the same, the chief difference being that Gorkin-Rhoades work mainly on the real line while we stick to the unit circle. A conformal isomorphism between the unit disk and the upper half-plane connects the two approaches. We have chosen to include the proof above of Theorem \[th5.1\], rather than refer the reader to [@G] for a proof, for the sake of completeness, and because the aims of [@G] are rather different from ours. (Since Theorem \[th5.1\] is less general than [@G Theorem 9], its proof is also a bit simpler.) To conclude this section we discuss the extent to which the Blaschke product ${\omega}$ produced in the preceding proof fails to be unique. Note that $\left| \frac {1+{\omega}(0)}{1-{\omega}(0)}\right| = 1$. In fact $$\frac {1+{\omega}(0)}{1-{\omega}(0)} = \frac {q(0)}{ip(0)} = -ie^{i({\beta}-{\alpha})/2}.$$ Suppose the Blaschke product ${\varphi}$ of order $n$ satisfies $(1)$ when substituted for ${\omega}$. Then the argument of $\frac {1+{\varphi}}{1-{\varphi}}$ must undergo a jump of $-\pi$ across each point $e^{i{\alpha}_n}$ and a jump of $\pi$ across each point $e^{i{\beta}_n}$. This implies, by a standard argument, that each point $e^{i{\alpha}_n}$ is a simple pole of $\frac {1+{\varphi}}{1-{\varphi}}$ and each point $e^{i{\beta}_n}$ is a simple zero of $\frac {1+{\varphi}}{1-{\varphi}}$. The functions $\frac {1+{\varphi}}{1-{\varphi}}$ and $\frac {1+{\omega}}{1-{\omega}}$ are thus rational functions with the same zeros and poles, so they are constant multiples of each other. It follows that one can obtain ${\varphi}$ by composing ${\omega}$ from the left with a conformal automorphism of ${{\mathbb D}}$. That conformal automorphism must fix the points $1$ and $-1$, so it has the form $z \mapsto \frac {z+r}{1+rz}$, where $-1 < r < 1$. We get ${\varphi}= \frac {{\omega}+r}{1+r{\omega}}$. A calculation produces the equality $$\frac {1+{\varphi}}{1-{\varphi}} = \left( \frac {1+r}{1-r} \right) \left( \frac {1+{\omega}}{1-{\omega}}\right).$$ Thus $\left| \frac {1+{\varphi}(0)}{1-{\varphi}(0)}\right| = 1$ only if $r = 0$, i.e., only if ${\varphi}= {\omega}$. We see that ${\omega}$ becomes unique if one imposes the supplementary condition $\left| \frac {1+{\omega}(0)}{1-{\omega}(0)} \right| = 1$. \[th5.2\] If $\psi$ is an alternating step function of order $N$, then $\psi$ is uniquely determined by the Fourier coefficients ${\widehat}{\psi}(0),{\widehat}{\psi}(1),\dots,{\widehat}{\psi}(N)$, in the sense that if those coefficients are known then the coefficients ${\widehat}{\psi}(N+1),{\widehat}{\psi}(N+2),\dots$ can in principle be found. Let $\psi$ be such a function, and let $c$ be its height. We may assume $N > 0$ (the case $N = 0$ being trivial). By Theorem \[th5.1\], there is a Blaschke product ${\omega}$ of order $N$ such that $\psi = h + {\overline}{h}$, where $$h = \frac {c}{\pi i} {\mathop{\mathrm{Log}}\nolimits}\left( \frac {1+{\omega}}{1-{\omega}} \right).$$ From this we see that, for any $n$, the coefficient ${\widehat}{\psi}(n)$ can be expressed in terms of $c$ and the coefficients ${\widehat}{{\omega}}(0),\dots,{\widehat}{{\omega}}(n)$. As ${\omega}$ is a rational function of degree $N$, the coefficients ${\widehat}{{\omega}}(N+1),{\widehat}{{\omega}}(N+2),\dots$ can be expressed in terms of ${\widehat}{{\omega}}(0),\dots,{\widehat}{{\omega}}(N)$. The desired conclusion follows. \[cor5.3\] Let $E$ be the union of $N$ closed disjoint subarcs of ${{\mathbb T}}$. Then $E$ is uniquely determined by the coefficients ${\widehat}{\chi}_E(0),\dots,{\widehat}{\chi}_E(N)$. It suffices to apply Theorem \[th5.2\] to the alternating step function $2\chi_E-1$. The phenomenon described in the corollary is treated at length in [@C]. Two Dimensions {#sec6} ============== We indicate how the equality $c_1 = \frac {\pi}{2}$ can be deduced from our characterization of minimizing functions. The inequality states that $c_A/\|A\| \le \frac {\pi}{2}$ for all nonzero matrices $A$ in ${{\mathcal T}}_2$. For such a matrix $A$ we know from Theorem \[th4.1\] that the minimizing function in ${{\mathcal G}}_A$ is an alternating step function of order $1$ and height $c_A$. We can thus try to establish the inequality by starting with an alternating step function $\psi$ of order $1$ and a given height, computing the corresponding matrix $A$, finding $\|A\|$, and checking the ratio of interest. And it will clearly suffice to treat the case where $\psi$ has height $1$ and has a discontinuity at the point $1$ with a jump there of $2$. Let ${\alpha}$ be a point in the interval $(0,2\pi)$, and define $\psi$ by $$\psi(e^{i{\theta}}) = \begin{cases} 1, &0 < {\theta}< {\alpha}, \\ -1, &{\alpha}< {\theta}< 2\pi. \end{cases}$$ Then $\psi$ is the general alternating step function of interest. The corresponding matrix $A$ is given by $$A = \begin{pmatrix} {\widehat}{\psi}(0) & {\widehat}{\psi}(-1) \\ {\widehat}{\psi}(1) & {\widehat}{\psi}(0) \end{pmatrix}\,.$$ Calculations give $${\widehat}{\psi}(0) = \frac {{\alpha}-\pi}{\pi},\quad {\widehat}{\psi}(1) = \frac {2e^{-i{\alpha}/2}\sin \frac {{\alpha}}{2}}{\pi}\,.$$ Thus $$A = \frac {1}{\pi} \begin{pmatrix} {\alpha}-\pi & 2e^{i{\alpha}/2}\sin \frac {{\alpha}}{2} \\ 2e^{-i{\alpha}/2}\sin \frac {{\alpha}}{2} & {\alpha}- \pi \end{pmatrix}\,.$$ The eigenvalues of $A$ can be found by the standard procedure. The calculations yield $$\|A\| = \frac {|{\alpha}-\pi| + 2\left| \sin \frac {{\alpha}}{2}\right|}{\pi}\,.$$ As $c_A = 1$, the desired conclusion is $\frac {1}{\|A\|} \le \frac {\pi}{2}$. We have $$\frac {1}{\|A\|} = \frac {\pi}{|{\alpha}-\pi| + 2\left| \sin \frac {{\alpha}}{2}\right|}\,,$$ so we need only verify that $$\pi - {\alpha}+ 2\sin \frac {{\alpha}}{2} \ge 2$$ for $0 \le {\alpha}\le \pi$. Differentiation with respect to ${\alpha}$ shows that the function on the left side is decreasing on $[0,\pi]$. At ${\alpha}= \pi$ it has the value $2$, so the desired inequality holds, and it reduces to an equality only for ${\alpha}= \pi$. The Numbers $c_N$ {#sec7} ================= We present here what we have learned about the maxima $c_N$, including numerical results for small $N > 1$. For $f$ a function in $L^{{\infty}}$ we let $A_{f,N}$ denote the compression to ${{\mathcal P}}_N$ of the multiplication operator on $L^2$ induced by $f$. The inequality $c_N \ge \frac {\pi}{2}$ is easily seen at this point. For example, let $\psi$ be the alternating step function of height $1$ and order $N$ associated with the inner function ${\omega}(z) = z^N$ as in the proof of Theorem \[th4.1\]: $$\psi(z) = 2{\mathop{\mathrm{Re}}\nolimits}\left( \frac {1}{\pi i} {\mathop{\mathrm{Log}}\nolimits}\left( \frac {1+z^N}{1-z^N}\right)\right).$$ Referring to that proof, one sees that $\psi$ alternates the values $1$ and $-1$ on $2N$ consecutive subarcs of ${{\mathbb T}}$, each of length $\frac {\pi}{N}$. For $z {\longrightarrow}0$ we have $$\frac {1+z^N}{1-z^N} = 1 + 2z^N + O(|z|^{2N}),$$ so $$\begin{aligned} \psi(z) &= \frac {2}{\pi} {\mathop{\mathrm{Im}}\nolimits}\left( {\mathop{\mathrm{Log}}\nolimits}\left( \frac {1+z^N}{1-z^N}\right)\right) \\ &= \frac {2}{\pi} {\mathop{\mathrm{Im}}\nolimits}({\mathop{\mathrm{Log}}\nolimits}(1+2z^N)) + O(|z|^{2N}) \\ &= \frac {2}{\pi} {\mathop{\mathrm{Im}}\nolimits}(2z^N) + O(|z|^{2N}) \\ &= \frac {2}{\pi} \left( \frac {2z^N-2{\overline}{z}^N}{2i}\right) + O(|z|^{2N}) \\ &= \frac {2}{\pi i}(z^N - {\overline}{z}^N) + O(|z|^{2N}). \end{aligned}$$ We see that the matrix $A_{\psi,N}$ has only two nonzero entries, $\frac {2}{\pi i}$ in the lower left corner and $-\frac {2}{\pi i}$ in the upper right corner. In particular, $\|A_{\psi}\| = \frac {2}{\pi}$, and the ratio $\frac {\|\psi\|_{{\infty}}}{\|A_{\psi,N}\|} = \frac {1}{\|A_{\psi,N}\|}$ equals $\frac {\pi}{2}$, telling us that $c_N \ge \frac {\pi}{2}$. The following observation produces a generalization of the last inequality. \[prop7.1\] Let the function $f$ be in $L^{{\infty}}$, let $k$ be a positive integer, and let the function $g$ on ${{\mathbb T}}$ be defined by $g(e^{i{\theta}}) = f(e^{ik{\theta}})$. Then $\|A_{f,N}\| = \|A_{g,kN}\|$. To simplify the notation, let $T = A_{g,kN}$. For $p$ in ${{\mathcal P}}_{kN}$, the $m$-th Fourier coefficient of $Tp$ is given by $$(Tp)^{\wedge}(m) = \sum_{\substack{0 \le j \le kN \\ -kN \le m-j \le kN}} {\widehat}{g}(m-j){\widehat}{p}(j).$$ If this coefficient is nonzero, there must be a $j$ such that $k$ divides $m-j$, in other words, such that $j \equiv m\ ({\mathop{\mathrm{mod}}\nolimits}k)$. For $l = 0,\dots,k-1$, let $S_l$ be the subspace of ${{\mathcal P}}_{kN}$ consisting of the polynomials $p$ such that ${\widehat}{p}(j) = 0$ if $j \not\equiv l\ ({\mathop{\mathrm{mod}}\nolimits}k)$. These subspaces are mutually orthogonal, they span ${{\mathcal P}}_{kN}$, and, by the observation above, they are $T$-invariant. Hence $T$ is the direct sum of its restrictions to the subspaces $S_l$. If $p$ is in $S_0$ then there is a $q$ in ${{\mathcal P}}_N$ such that $p(z) = q(z^k)$. For the $km$-th coefficient of $Tp$ we have $$\begin{aligned} (Tp)^{\wedge}(km) &= \sum_{j=0}^N {\widehat}{g}(km-kj){\widehat}{p}(kj) \\ &= \sum_{j=0}^N {\widehat}{f}(m-j){\widehat}{q}(j), \end{aligned}$$ showing that $T\mid S_0$ is unitarily equivalent to $A_{f,N}$. For $p$ in $S_l$ with $l \ne 0$, the $(l+km)$-th coefficient of $Tp$ $(m = 0,\dots,N-1)$ is given by $$(Tp)^{\wedge}(l+km) = \sum_{\substack{j \equiv l\ ({\mathop{\mathrm{mod}}\nolimits}k) \\ l \le j \le l+k(N-1)}} {\widehat}{g}(l+km-j){\widehat}{p}(j).$$ In the summation on the right, $j-l$ runs through the numbers $0,k,\dots,(N-1)k$, and correspondingly, $j$ runs through the numbers $l,l+k,\dots,l+k(N-1)$. The equality can be rewritten as $$(Tp)^{\wedge}(l+km) = \sum_{j=0}^{N-1} {\widehat}{g}((m-j)k){\widehat}{p}(l+jk).$$ A polynomial $p$ in $S_l$ $(l \ne 0)$ can be written as $p(z) = z^lq(z^k)$, where $q$ is in ${{\mathcal P}}_N$ and ${\widehat}{q}(N) = 0$. This gives us $$(Tp)^{\wedge}(l+km) = \sum_{j=0}^{N-1} {\widehat}{f}(m-j){\widehat}{q}(j),$$ showing that the restriction of $T$ to $S_l$ is unitarily equivalent to the $N$-by-$N$ principal minor of $A_{f,N}$, hence is of norm at most that of $A_{f,N}$. The equality $\|A_{f,N}\| = \|A_{g,kN}\|$ now follows. The following corollary generalizes the inequality $c_N \ge \frac {\pi}{2}$. \[cor7.2\] For $k$ a positive integer, $c_{kN} \ge c_N$. It suffices to apply Proposition \[prop7.1\] with $f$ equal to an alternating step function $\psi$ of order at most $N$ for which $\frac {\|\psi\|_{{\infty}}}{\|A_{\psi,N}\|} = c_N$. One way to obtain better lower bounds than $c_N \ge \frac {\pi}{2}$ for $N > 1$ is to find alternating step functions $\psi$ of height $1$ and order $N$ for which $\|A_{\psi, N}\|$ can be shown to be smaller than $\frac {2}{\pi}$. We do this for a number of values of $N$. Our method is to consider a family of step functions involving a number of parameters, and to observe that when the parameters are judiciously chosen, the Fourier coefficients of $\psi$ are related to one another in a way that permits calculation of $\|A_{\psi,N}\|$. A computer was used to assist in the search for relations between Fourier coefficients that would yield nontrivial lower bounds on $c_N$. For any positive $L$ and $M$ satisfying $0 < L + M < \pi$, there is a unique alternating step function $\psi$ of order $2$ taking the value $1$ at $1$ and having discontinuities at $e^{\pm iL}$ and $e^{i(\pi \pm M)}$. Calculations show that $${\widehat}{\psi}(0) = \frac{2L+2M-\pi}{\pi}, \quad {\widehat}{\psi}(1) = \frac{2 (\sin L - \sin M)}{\pi}, \quad {\widehat}{\psi}(2) = \frac{\sin(2M)+\sin(2L)}{\pi}.$$ The values of $L$ and $M$ can additionally be chosen so that the relations ${\widehat}{\psi}(1) = 2 {\widehat}{\psi}(0)$ and ${\widehat}{\psi}(2) = -{\widehat}{\psi}(1)$ are also satisfied, and these choices are unique subject to the conditions $L \in [0,1]$ and $M \in [0,2]$. (To abbreviate the argument: one can show that subject to the inequalities on $L$ and $M$, each equation of Fourier coefficients defines $M$ implicitly as a monotone function of $L$ on $[0,1]$, and use the intermediate value theorem to see that the functions intersect exactly once.) These relations allow one to explicitly compute $\|A_{\psi, 2}\| = 3 |{\widehat}{\psi}(0)|$. If $L$ and $M$ are the unique real numbers satisfying the relations $0 < L + M < \pi$, $L \in [0,1]$, $M \in [0,2]$, $2 L + 2M - \pi = \sin L - \sin M$, and $-2 (\sin L - \sin M) = \sin(2 M) + \sin(2L)$, then $$\label{c2est} c_2 \ge \frac{\pi}{3 |2L+2M-\pi|}.$$ One must find $L \approx .2138$ and $M \approx 1.0263$ to several digits to deduce that this estimate improves on the known $c_2 \ge \frac{\pi}{2}$, but it can be shown that the right hand side of is between $1.6185$ and $1.6186$. For any $L$ in $(0, \frac{\pi}{2})$ there is a unique alternating step function $\psi(e^{i\theta})$ of order $3$ and height $1$ taking the value $1$ at $1$ and having discontinuities at $e^{\pm iL}$, $e^{i(\pi \pm L)}$, and $e^{\pm i\pi/2}$. Calculations show that ${\widehat}{\psi}(0)={\widehat}{\psi}(2)=0$ and $${\widehat}{\psi}(1) = \frac{2}{\pi}(2 \sin L - 1), \quad {\widehat}{\psi}(3) = \frac{2}{3\pi}(2 \sin(3L) + 1).$$ There is a unique $L$ in $(0, \frac{\pi}{2})$ satisfying $\sin(3L)+3\sin L -1=0$. For this value of $L$ we have ${\widehat}{\psi}(1)=-{\widehat}{\psi}(3)$, and therefore $\|A_{\psi,3}\| = \sqrt{2} |{\widehat}{\psi}(1)|$. We deduce $$c_3 \ge \frac{\pi}{2\sqrt{2}(1 - 2 \sin L)}.$$ Trigonometric identities imply that $\sin L$ is the least positive root of the polynomial $1 - 6x+4x^3$. Using this polynomial one can find a similar characterization of $\frac{1}{1-2\sin L}$. If $k$ is the largest root of the polynomial $p(x) = 1-3x-3x^2+3x^3$, then $$\label{c3est} c_3 \ge \frac{\pi}{2 \sqrt{2}} k.$$ It can be checked by hand that $p(\frac{3}{2}) < 0$, and thus $k \ge \frac{3}{2}$, so $c_3 \ge \frac{3\pi}{4 \sqrt{2}} > \frac{\pi}{2}$ can be shown by hand. Working to greater precision one can show that the right hand side of is between $1.6825$ and $1.6826$. Our higher dimensional examples are similar to those just given. The difference is that, with the exception of the case $N=5$, the number of parameters involved makes exact formulas difficult to prove. We can nevertheless give numerical values of parameters that produce a $\psi$ with the property that $\|A_{\psi,N}\|$ is small enough to give new information about $c_N$. For any positive $L,M,N,O$ satisfying $L+M+N+O<\pi$ there is an alterating step function of height $1$ and order $4$ taking the value $1$ at $1$ and having discontinuities at $e^{iL}$, $e^{i(L+M)}$, $e^{i(L+M+N)}$, $e^{i(L+M+N+O)}$ and their complex conjugates. Taking $L \approx .1396$, $M \approx 1.1143$, $N \approx 1.096$, and $O \approx .2724$ one obtains a function $\psi$ for which $\|A_{\psi,4}\|$ can be shown to be between $1.7065$ and $1.7066$. These values of $L,M,N,O$ appear to approximate a tuple with the property that the corresponding list of Fourier coefficients $({\widehat}{\psi}(j))_{j=0}^4$ is a scalar multiple of $(1, -\sqrt{5}-1, 1-\sqrt{5},\sqrt{5}-1,1+\sqrt{5})$. If $L,M,N,O$ can be chosen so that this holds, we would have $c_4 \ge (5 |\psi(0)|)^{-1} = \frac{\pi}{5(\pi-2M-2O)}$. For any positive $L$ and $M$ satisfying $L+M < \frac{\pi}{2}$ there is a unique alternating step function $\psi$ of height $1$ and order $5$ taking the value $1$ at $1$ and having discontinuities at $e^{iL}$, $e^{i(L+M)}$, $e^{i \pi/2}$, $e^{i(\pi - L)}$, $e^{i(\pi - L-M)}$, and their complex conjugates. We have ${\widehat}{\psi}(0) = {\widehat}{\psi}(2) = {\widehat}{\psi}(4) = 0$ for all such $L$ and $M$, and it can be shown that there are unique $L$ and $M$ in the interval $(0,1)$ for which $$\label{fiverel} {\widehat}{\psi}(1) = -2 {\widehat}{\psi}(3) = {\widehat}{\psi}(5).$$ For these values of $L$ and $M$ we deduce that $\|A_{\psi, 5}\| = \frac{3}{2} |{\widehat}{\psi}(1)|$. If $L$ and $M$ are the unique numbers in $(0,1)$ for which the equations hold, then $$\label{c5est} c_5 \ge \frac{\pi}{3 (1 +2 \sin L - 2 \sin(L+M))}.$$ Numerical calculations can show that $L \approx .4304$, $M \approx .2326$, and that the right hand side of is between $1.7353$ and $1.7354$. For any positive $L,M,N,O,P,Q$ whose sum is less than $\pi$, there is a unique alternating step function $\psi$ of height $1$ and order $6$ taking the value $1$ at $1$ and having discontinuities at the points $e^{iL}$, $e^{i(L+M)}$, $e^{i(L+M+N)}$, $e^{i(L+M+N+O)}$, $e^{i(L+M+N+O+P)}$, $e^{i(L+M+N+O+P+Q)}$ and their complex conjugates. Taking $(L,M,N,O,P,Q) \approx (.0989,.7269,.2002,.7702,.7755,.2109)$ one obtains a function $\psi$ for which $\|A_{\psi,6}\|$ can be shown to be between $1.7504$ and $1.7505$. These parameter values appear to approximate a tuple with the property that the list of Fourier coefficients of the corresponding function $\psi$ is proportional to $(1,a,-b,c,-c,b,-a,1)$, where $a<b<c$ are the roots of $8+4x-4x^2-x^3$. For any positive $L,M,N$ satisfying $L+M+N < \frac{\pi}{2}$, there is a unique alternating step function $\psi$ of height $1$ and order $7$ taking the value $1$ at $1$ and having discontinuities at $e^{iL}$, $e^{i(L+M)}$, $e^{i(\pi/2-N)}$, $e^{i\pi/2}$, $e^{i(\pi/2+N)}$, $e^{i(\pi-L-M)}$, $e^{i(\pi-L)}$, and their complex conjugates. Taking $L \approx .0877$, $M \approx .6343$, and $N \approx .6713$ one obtains a function $\psi$ for which it can be shown that $\|A_{\psi,7}\|$ is between $1.7677$ and $1.7678$. These values of $L,M,N$ appear to approximate a tuple with the property that the corresponding list of Fourier coefficients $({\widehat}{\psi}(j))_{j=0}^7$ is a scalar multiple of $(0,-1-\sqrt{2},0,1,0,-1,0,1+\sqrt{2})$. If $L,M,N$ can be chosen so that this holds exactly, we can conclude that $c_7 \ge \frac{(\sqrt{2}+1)\pi}{4 \sqrt{2+\sqrt{2}} (1-2\cos N-2\sin L+2\sin(L+M))}$ for these values of $L,M,N$. In summary, along with the exact value $c_1 = \frac {\pi}{2} \approx 1.5707963$, we have the following lower bounds for $c_2 - c_7$, found numerically: $$\begin{aligned} c_2 &\ge 1.6185 \\ c_3 &\ge 1.6825 \\ c_4 &\ge 1.7065 \\ c_5 &\ge 1.7354 \\ c_6 &\ge 1.7505 \\ c_7 &\ge 1.7677. \end{aligned}$$ Questions and Conjectures {#sec8} ========================= We present some problems for future research suggested by the numerical results of Section \[sec7\], which tell us, in particular, that $c_N > \frac{\pi}{2}$ for $N=2,3,4,5,6,7$. (As noted in Section \[sec7\], for $N = 3$ the inequality can be verified by hand.) By the corollary to Proposition \[prop7.1\], we can conclude that $c_N > \frac{\pi}{2}$ for infinitely many $N$. We conjecture that the following question has an affirmative answer. Our estimates for $c_1$–$c_7$ suggest the following question. Again, we conjecture the answer is affirmative. As noted in Section \[sec1\], the inequality $c_N \le 2$ holds for all $N$, suggesting the next question. We have no conjecture to offer on this one. We have seen that the maximum ratio $c_N$ is attained by an alternating step function of order at most $N$, i.e., there is such a function $\psi$ for which $\|\psi\|_{{\infty}}/\|A_{\psi,N}\| = c_N$. The function $\psi$ here is not unique, because one gets the same ratio if one replaces $\psi$ by a nonzero real scalar multiple of itself, or by a composite (from the right) with a rotation of the circle, or by a composite with complex conjugation on the circle. Two questions are suggested, which we combine into one. Otherwise put, the question asks whether an alternating step function attaining the ratio $c_N$ has $2N$ discontinuities, and whether the set of discontinuities is unique to within rotations and reflections on ${{\mathbb T}}$. We conjecture the answer is affirmative. The questions above are of course part of the underlying problem of “finding,” in some reasonable sense, the maxima $c_N$ and the corresponding maximizers. There must be an interesting underlying structure, one would think, which so far escapes us. Answers to one or more of Questions 1–4 should provide clues to that structure. [NF]{} M. Bakonyi and D. Timotin, [*On an extension problem for polynomials*]{}. Bull. London Math. Soc. [**33**]{} (2001), 599–605. C. Carathéodory and L. Fejér, [*Über den Zusammenhang der Extremen von harmonischen Funktionen mit ihren Koeffizienten und über den Picard–Landau’schen Satz*]{}. Rend. Circ. Mat. Palermo [**32**]{} (1911), 218–239. D. Courtney, [*Unions of arcs from Fourier partial sums*]{}. New York J. Math [**16**]{} (2010), 235-243. P. Gorkin and R. Rhoades, [*Boundary interpolation by finite Blaschke products*]{}. Constr. Approx. [**27**]{} (2008), 75–98. M. G. Krein and A. A. Nudelman, [*The Markov moment problem and extremal moment problems*]{}. Translations of Mathematical Monographs Vol. 50. American Mathematical Society, Providence (1977) L. N. Nikolskaya and Yu. B. Farforovskaya, [*Toeplitz and Hankel matrices as Hadamard–Schur multipliers*]{}. St. Petersburg Math. J. [**15**]{} (2004), 915–928. D. Sarason, [*Algebraic properties of truncated Toeplitz operators*]{}. Algebra and Matrices [**1**]{} (2007), 491–526. [^1]: ${}^*$Partially supported by the University of Iowa Department of Mathematics NSF VIGRE grant DMS-0602242.
--- abstract: 'In the context of the genome rearrangement problem, we analyze two well known models, namely the block transposition and the prefix block transposition models, by exploiting the connection with the notion of permutation pattern. More specifically, for any $k$, we provide a characterization of the set of permutations having distance $\leq k$ from the identity (which is known to be a permutation class) in terms of what we call *generating permutations* and we describe some properties of its basis, which allow to compute such a basis for small values of $k$.' author: - | Giulio Cerbai\ giuliocerbai14@gmail.com - | Luca Ferrari\ luca.ferrari@unifi.it title: Permutation patterns in genome rearrangement problems --- Introduction ============ One of the major trends in bioinformatics and biomathematics is the study of the genome rearrangement problem. Roughly speaking, given a genome, one is interested in understanding how the genome can evolve into another genome. To give a proper formalization, several models for rearranging a genome have been introduced, each of which defines a series of allowed elementary operations to be performed on a genome in order to obtain an adjacent one. For several models, it is possible to define a *distance* between two genomes, by counting the minimum number of elementary operations needed to transform one genome into the other. The investigation of the main properties of such a distance becomes then a key point in understanding the main features of the model under consideration. A common formalization of any such models consists of encoding a genome using a *permutation* (in linear notation) and describing an elementary operation as a *combinatorial* operation on the entries of such a permutation. Many genome rearrangement models have been studied under this general framework. Among them, the following ones are very well known. - The *reversal* model consists of a single operation, defined as follows: a new permutation is obtained from a given one by selecting a cluster of consecutive elements and reversing it. More formally, given $\pi =\pi_1 \pi_2 \cdots \pi_n$, a reversal is performed by choosing $i<j<n$ and then forming the permutation $\sigma =\pi_1 \cdots \pi_{i-1} \boxed{\pi_j \pi_{j-1}\cdots \pi_{i+1} \pi_i}\pi_{j+1}\cdots \pi_n$. This model was introduced in [@WEHM], then studied for instance in [@BaPe1; @HP]. - A variant of the reversal model is the *prefix reversal model*, which is a specialization of the previous one in which the reversal operation can only be performed on a prefix of the given permutation. This is clearly an easier model to investigate, which is also known as *pancake sorting* (see for instance [@GP]). - A very popular and studied model is the *transposition model*, see [@BaPe2]. Given a permutatation $\pi =\pi_1 \cdots \pi_n$, a *transposition operation* consists of taking two adjacent clusters of consecutive elements and interchanging their positions. Formally, one has to choose indices $1\leq i<j<k\leq n+1$, then form the permutation $\sigma =\pi_1 \cdots \pi_{i-1}\boxed{\pi_j \pi_{j+1}\cdots \pi_{k-1}}\boxed{\pi_i \pi_{i+1}\cdots \pi_{j-1}} \pi_k \cdots \pi_n$. - As for the reversal, also for the transposition model there is a “prefix variant". In the *prefix transposition model* the leftmost block of elements to interchange is a prefix of the permutation. Sorting by prefix transposition is studied in [@DM]. Independently from the chosen model, there are some general questions that can be asked in order to gain a better understanding of its combinatorial properties. First of all, the operations of the model often (but not always) allow to define a *distance* $d$ between two permutations $\rho$ and $\sigma$, as the minimum number of elementary operations needed to transform $\rho$ into $\sigma$. Moreover, when the operations are nice enough, the above distance $d$ could even be *left-invariant*, meaning that, given permutations $\pi ,\rho ,\sigma$ (of the same length), $d(\pi ,\rho )=d(\sigma \pi ,\sigma \rho )$. As a consequence, choosing for instance $\sigma =\rho^{-1}$, the problem of evaluating the distance $d(\pi ,\rho )$ reduces to that of sorting $\pi$ with the minimum number of elementary allowed operations. Now, if $d$ is a left-invariant distance on the set $S_n$ of all permutations of the same length, define the *$k$-ball* of $S_n$ to be the set $B_k ^{(d)}(n)=\{ \rho \in S_n \; |\; d(\rho ,id_n )\leq k\}$, where $id_n$ is the identity permutation of length $n$. The following questions are quite natural to ask: - compute the diameter of $B_k ^{(d)}(n)$, i.e. the maximum distance between two permutations of $B_k ^{(d)}(n)$; - compute the diameter of $S_n$, i.e. the maximum distance between two permutations of $S_n$; - characterize the permutations of $\partial B_k ^{(d)}(n)$, i.e. the permutations of $B_k ^{(d)}(n)$ having maximum distance from the identity; - characterize the permutations of $\partial S_n$, i.e. the permutations of $S_n$ having maximum distance from the identity; - characterize and enumerate the permutations of $B_k ^{(d)}(n)$; - design sorting algorithms and study the related complexity issues. In the literature there are several results, concerning several evolution models, which give some insight into the above problems. Our work starts from the observation that, in many cases, the balls $B_k ^{(d)}(n)$ can be characterized in terms of *pattern avoidance*. Recall that, given two permutation $\sigma \in S_k$ and $\tau =\tau_1 \tau_2 \cdots \tau_n \in S_n$, with $k\leq n$, we say that $\sigma$ is a *pattern* of $\tau$ when there exist $1\leq i_1 <i_2 <\cdots <i_k \leq n$ such that $\tau_{i_1}\tau_{i_2}\cdots \tau_{i_k}$ as a permutation is isomorphic to $\sigma$ (which means that $\tau_{i_1}, \tau_{i_2}, \ldots , \tau_{i_k}$ are in the same relative order as the elements of $\sigma$). This notion of pattern in permutation defines an obvious partial order, and the resulting poset is known as the *permutation pattern poset*. When $\sigma$ is not a pattern of $\tau$, we say that $\tau$ *avoids* $\sigma$. A down-set $I$ (also called a permutation class) of the permutation pattern poset can be described in terms of its minimal excluded permutations (or, equivalently, the minimal elements of the complementary up-set): these permutations are called the *basis* of $I$. The idea of studying the balls $B_k ^{(d)}(n)$ in terms of pattern avoidance is not new. As far as we know, the first model which has been investigated from this point of view is the (whole) tandem duplication-random loss model: Bouvel and Rossin [@BR] have in fact shown that, in such a model, the ball $B_k ^{(d)}=\bigcup_{n\geq 0}B_k ^{(d)}(n)$ is a class of pattern avoiding permutations, whose basis is the set of minimal permutations having $d$ descents (here minimal is intended in the permutation pattern order). Subsequent works [@BoPe; @BF; @CGM] have been done concerning the enumeration of the basis permutations of such classes. More recently, Homberger and Vatter [@HV] described an algorithm for the enumeration of any polynomial permutation class, which can be fruitfully used for all the above mentioned distances, since the resulting balls are indeed polynomial classes. However, their results do not allow to find information on the basis of the classes. In the present work we try to enhance what have been obtained in [@HV] in two directions. First, we aim at giving a structural characterization of the balls for some of the above distances, thus complementing the results in [@HV], which is more concerned with computational issues. Second, we provide some insight on the properties of the bases of such balls, hoping to gain a better understanding of them. We will be mainly concerned with the block transposition and the prefix block transposition models, leaving the reversal models to a future paper. Some of the results of the present work are contained in the MSc thesis of the first author [@C]. Block transposition =================== Among the four above mentioned models, the block transposition one is probably the hardest to investigate. Denoting with $td$ the transposition distance, the permutation class $B_k ^{(td)}$ can be described in terms of its *generating permutations*. A *strip* of $\pi =\pi_1 \pi_2 \cdots \pi_n \in S_n$ is a maximal consecutive substring $\pi_i \cdots \pi_{i+k-1}$ such that, for all $j=1,\ldots, i+k-2$, $\pi_{j+1}=\pi_j +1$. A permutation $\pi$ is said to be *plus irreducible* [@AS] when, for all $i=1,\ldots ,n-1$, $\pi_{i+1}\neq \pi_i +1$. In other words, $\pi$ is a plus irreducible permutation when it does not have points that are adjacent both in positions and values, with values increasing. Equivalently, a permutation is plus irreducible if and only if all of its strips have length $1$. Any permutation $\pi$ can be associated with a plus irreducible permutation, denoted $red(\pi )$, which is obtained by replacing each strip of $\pi$ with its minimum element, then suitably rescaling the resulting word. For instance, if $\pi =435612789$, then $red(\pi )=32415$. It is easy to observe that $red(\pi )\leq \pi$ in the permutation pattern order. Moreover, for every permutation $\pi$, we have that $td(\pi )=td(red(\pi ))$. Given $\pi \in S_n$, let $v_1 ,\ldots ,v_n$ be nonnegative integers. The *monotone inflation* of $\pi$ through $v=(v_1 ,\ldots ,v_n )$ is the permutation $\pi [v]=\pi [id_{v_1},\ldots ,id_{v_n}]$ obtained from $\pi$ by replacing each element $\pi_i$ of $\pi$ with the identity permutation $id_{v_i}$ of length $i$ suitably rescaled, so to mantain the relative order of the elements of $\pi$. So, for instance, if $\pi =41352$ and $v=(0,2,1,3,2)$, we have $\pi [v]=\underbrace{\ldots}_4\underbrace{12}_1\underbrace{5}_3\underbrace{678}_5\underbrace{34}_2$. In the following we will denote with $MI(\pi )$ the set of all monotone inflations of a permutation $\pi$ and with $MI(C)$ the set $\bigcup_{\pi \in C}MI(\pi )$, for a given set of permutations $C$. The notion of monotone inflation is clearly related to that of geometric grid class [@AABRV]. More specifically, given a $\{ -1,0,1\}$-matrix $M$ and denoting with $Geom(M)$ the geometric grid class of permutations determined by $M$, if $\pi$ is a permutation and $M_\pi$ is its permutation matrix, then it is not difficult to realize that: 1. $Geom(M_\pi )=Geom(M_{red(\pi )})$; 2. $MI(\pi )=Geom(M_\pi )$; 3. $MI(\pi )=MI(red(\pi ))$. Now define a permutation $\pi$ to be *generating* for $B_k ^{(td)}=\bigcup_{n\geq 0} B_k ^{(td)}(n)$ when it is a maximal plus irreducible permutation of $B_k ^{(td)}$. The set of all generating permutations for $B_k ^{(td)}$ will be called the *generating set* of $B_k ^{(td)}$. We thus have the following fact, whose easy proof is omitted. For every $k$, $B_k ^{(td)}=\bigcup \{ MI(\pi )\, |\, \pi \textnormal{ is generating for $B_k ^{(td)}$} \}$. A very natural description of the balls $B_k ^{(td)}$ is then provided by its generating set. This is our first open problem. **Open problem.***Characterize the generating permutations of $B_k ^{(td)}$, for every $k$*. For instance, it is easy to realize that $B_1 ^{(td)}=MI(1324)$. In the following, we will provide a structural description of the generating permutations of $B_k ^{(td)}$ for a generic $k$. Our approach is based on the observation that a generating permutation for $B_k ^{(td)}$ is a plus irreducible permutation, and so it will be convenient to work inside the poset of plus irreducible permutations, seen as a subposet of the classical permutation pattern poset (notice that this is also a subposet of the poset of *peg permutations*, as defined in [@HV]). In passing, we remark that the enumeration of plus irreducible permutations is well known: denoting with $f_n$ the number of plus irreducible permutations of length $n+1$, we have the recurrence relation $$f_n=nf_{n-1}+(n-1)f_{n-2},$$ for $n\geq 2$. With initial conditions $f_0 =f_1 =1$, we get for $(f_n )_{n\geq 0}$ the exponential generating function $\frac{e^{-x}}{(1-x)^2}$ and the closed form $f_n =\sum_{k=0}^{n}(-1)^k (n+1-k)\frac{n!}{k!}$. This is sequence A000255 in [@S], see also [@AAB]. Suppose $\pi =\pi_1 \pi_2 \cdots \pi_n$ is a plus irreducible permutation of length $n$ in the generating set of $B_k ^{(td)}$. Inflate $\pi$ by choosing three (not necessarily distinct) indices $1\leq i\leq j\leq k\leq n$ and replacing $\pi_i ,\pi_j$ and $\pi_k$ by strips of suitable lengths, as follows: - if the three indices are all distinct, take strips of length 2; - if two of the indices are equal, take the associated strip of length 3; - if all indices are equal, take a strip of length 4. If $I$ is the multiset of the selected indices, the resulting permutation will be denoted $\pi_I$. Now observe that, in all of the above cases, there exists a unique block transposition $\tau_I$ that breaks all the new strips in such a way that, in the resulting permutation, each pair of adjacent elements of a new strip becomes either nonadjacent or adjacent in the reverse way; more specifically, $\tau_I$ is the transposition with indices $i+1,j+2,k+3$. We call $\tilde{\pi}_I$ the permutation obtained from $\tilde{\pi}_I$ by applying $\tau_I$. As an example, consider the permutation $\pi =1324$, and the multiset of indices $I=\{ 2,2,4 \}$; then we get $\pi_I =1345267$ and $\tilde{\pi}_I =1352647$. The following lemma gives some basic properties of the above described construction that will be useful in the sequel. The proof is easy and so left to the reader. \[prel\] Let $\pi =\pi_1 \cdots \pi_n$ be a plus irreducible permutation of length $n$ and $I$ a multiset of indices of $\pi$ of cardinality 3. Then $\tilde{\pi}_I$ is a plus irreducible permutation of length $n+3$; moreover, if $\pi_1 =1$ and $\pi_n =n$, then $(\tilde{\pi}_I )_1 =1$ and $(\tilde{\pi}_I )_{n+3} =n+3$. We are now ready to give an explicit description of the generating set of the ball $B_k ^{(td)}$. Such a result will be preceded by a technical proposition (stated without proof) which gives a recipe to recursively construct the set of permutations which are obtainable by means of a single block transposition. In the proof of the next theorem we also need the definition of breakpoint, which can be found for instance in [@FLRTV]. Given a permutation $\pi =\pi_1 \cdots \pi_n$, a *breakpoint* of $\pi$ is an integer $i\in \{ 0,1,\ldots n\}$ such that $\pi_{i+1}\neq \pi_{i}+1$. By convention, $0$ and $n$ are breakpoints whenever $\pi_1 \neq 1$ and $\pi_n \neq n$, respectively. \[plusone\] Let $\mathcal{I}(n)$ be the set of all multisets of cardinality 3 of $\{ 1,2,\ldots n\}$. For every plus irreducible permutation $\pi \in S_n$, denote with $MI(\pi )^{+1}$ the set of all permutations which can be obtained with a single block transposition from any permutation of $MI(\pi )$. Then $$MI(\pi )^{+1}=\bigcup_{I\in \mathcal{I}(n)}MI(\tilde{\pi}_I ).$$ \[genblock\] For every $k\geq 1$, the generating set of $B_k ^{(td)}$ is the set of all plus irreducible permutations of length $3k+1$ and having distance $k$ from the identity. We start by showing that there exists a finite number $N=N(k)$ of permutations $\alpha^{(1)},\ldots ,\alpha^{(N)}$ which are plus irreducible, of length $3k+1$ and at distance $k$ from the identity, such that $B_k ^{(td)}=\bigcup_{j=1}^{N}MI(\alpha^{(j)})$. We can proceed by induction on $k$. When $k=1$, we have already observed that $B_1 ^{(td)}=MI(1324)$, and 1324 is plus irreducible, has length 3+1=4 and has distance 1 from the identity 1234. Now consider a permutation $\overline{\pi}\in B_{k+1} ^{(td)}\setminus B_k ^{(td)}$; this means, in particular, that there is a permutation $\pi \in B_k ^{(td)}$ such that $\overline{\pi}$ is obtained from $\pi$ by a single block transposition. Thus, using the induction hypothesis, we can say that there exists a plus irreducible permutation $\alpha$ of length $3k+1$ and having distance $k$ from the identity such that $\overline{\pi}\in MI(\alpha )^{+1}$. By Proposition \[plusone\], there exists $I\in \mathcal{I}(3k+1)$ such that $\overline{\pi}\in MI(\tilde{\alpha}_I )$. Notice that $\mathcal{I}(3k+1)$ is finite and that, by Lemma \[prel\], $\tilde{\alpha}_I$ is plus irreducible and has length $3k+4$; so what remains to prove is that $\tilde{\alpha}_I$ has distance $k+1$ from the identity. Clearly $td(\tilde{\alpha}_I )\leq k+1$. On the other hand, since Lemma \[prel\] implies that $\tilde{\alpha}_I$ starts with 1 and ends with $3k+4$, recalling that $\tilde{\alpha}_I$ is plus irreducible, we have that $\tilde{\alpha}_I$ has exactly $3k+3$ breakpoints, since the only indices that are not breakpoints are $0$ and $3k+4$. Therefore, denoting with $Br(\pi )$ the number of breakpoints of $\pi$, since $td(\pi )\geq \left\lceil \frac{Br(\pi )}{3}\right\rceil$ (this follows from an observation in [@BaPe2]), we have that $$td(\tilde{\alpha}_I )\geq \left\lceil \frac{Br(\tilde{\alpha}_I )}{3}\right\rceil =\left\lceil \frac{3k+3}{3}\right\rceil=k+1,$$ as desired. To conclude the proof we now have to show that any plus irreducible permutation $\gamma$ of length $3k+1$ and having distance $k$ from the identity is a generating permutation of $B_k ^{(td)}$. In fact, since $\gamma \in B_k ^{(td)}$, $\gamma$ is the monotone inflation of some generating permutation $\alpha$ of $B_k ^{(td)}$. Therefore $\alpha$ is a plus irreducible permutation of length $3k+1$ at distance $k$ from the identity. So in particular $\gamma$ and $\alpha$ have the same length, which means that necessarily $\gamma =\alpha$. The above theorem allows to design a procedure to list the generating set of $B_k ^{(td)}$: starting from the identity of length $3k+1$, perform repeated monotone inflations as in Proposition \[plusone\] (for $k$ times) so to obtain all generating permutations of $B_k ^{(td)}$. This is similar to the approach used in [@HV]. For instance, when $k=2$, the generating set for $B_2 ^{(td)}$ consists of the eleven permutations 1324657, 1352647, 1354627, 1364257, 1426357, 1436527, 1462537, 1524637, 1536247, 1624357, 1632547. Notice however that, in this way, it is possible to obtain the same generating permutation several times, so in the list of permutations given in output by the above procedure one has to remove duplicates. This is the main reason for which the described approach is not useful for enumerating the generating set. **Open problem.***Enumerate the generating permutations of $B_k ^{(td)}$, for every $k$*. A very interesting information that we can get on $B_k ^{(td)}$ concerns its basis. We start by recalling that monotone inflations are particular geometric grid classes [@AABRV]; as a consequence, the general theory of geometric grid classes allows us to say that $B_k ^{(td)}$ is a permutation class having finite basis (and also that it is *strongly rational*, meaning that its generating function is rational, together with the generating functions of all of its subclasses). What we are able to do is to provide a nontrivial upper bound to the length of the basis elements, which is clearly of great help in effectively computing the basis itself. \[basis\] Every permutation belonging to the basis of $B_k ^{(td)}$ has length at most $3k+1$. We start by observing that, given $\pi$ basis permutation of $B_k ^{(td)}$, if $\pi$ were not plus irreducible, then necessarily $red(\pi )<\pi$ and we have already observed that $td(\pi )=td (red(\pi ))$; so $\pi$ would not be minimal among the permutations at distance $k$ from the identity, which is impossible. Therefore we can assert that all basis permutations of $B_k ^{(td)}$ are plus irreducible. Now it is easy to prove that every basis permutation has length at most $3k+2$. Indeed, it is not difficult to show that a plus irreducible permutation of length $m$ contains as a pattern at least one permutation of length $m-1$ that is plus irreducible as well. Thus, if $\pi$ were a basis permutation having length greater than $3k+2$, then, in the poset of plus irreducible permutations, there would exist at least one plus irreducible permutation $\sigma$ of length greater than $3k+1$ such that $\sigma <\pi$. Since all generating permutations of $B_k ^{(td)}$ have length $3k+1$, $\sigma$ cannot belong to $B_k ^{(td)}$, which is not possible since $\pi$ is a basis permutation. Moreover, suppose that $\pi =\pi_1 \pi_2 \cdots \pi_n$ is a permutation in the basis of $B_k ^{(td)}$. First of all we have that $\pi_1\neq 1$ and $\pi_n \neq n$, since otherwise we could remove $\pi_1$ or $\pi_n$ thus obtaining a smaller permutation having the same distance from the identity, against the minimality of $\pi$. If $\pi$ had length $n=3k+2$, since $\pi$ is plus irreducible, there would exist $\gamma <\pi$ which is plus irreducible as well. Since $\pi$ is minimal in the complement of $B_k ^{(td)}$, necessarily $td(\gamma )=k$. This would imply that $\gamma=\gamma_1 \gamma_2 \cdots \gamma_{3k+1}$ is a generating permutation of $B_k ^{(td)}$. This is however impossible, since it would be $\gamma_1 =1$ and $\gamma_{3k+1}=3k+1$ by Lemma \[prel\] and the construction showed in Theorem \[genblock\], $\pi_1 \neq 1$ and $\pi_{3k+2} \neq 3k+2$ for what we have proved above, and $\gamma$ is obtained from $\pi$ by removing a single entry. The above theorem also suggest a procedure to determine the basis of $B_k ^{(td)}$. In the poset of plus irreducible permutations, consider the set of permutations of length $3k+1$ which are not generating. For each of them (say $\pi$), consider the set of permutation of length $3k$ covered by it: if all of them are also below some generating permutation of $B_k ^{(td)}$, then $\pi$ is in the basis of $B_k ^{(td)}$. Otherwise, just repeat the same procedure starting from the permutations covered by $\pi$ which do not belong to $B_k ^{(td)}$. As an instance, we have the following result. The basis of $B_1 ^{(td)}$ is $\{ 321,2143,2413,3142\}$. Since $B_1 ^{(td)}=MI(1324)$, we perform the above procedure with all permutations of length 4 except $1324$. A direct computation shows that the only permutations which cover only elements of $B_1 ^{(td)}$ are precisely $2143,2413,3142$. Moreover, $321$ is the unique permutation of length 3 which is not in $B_k ^{(td)}$, and all of its coverings are in $B_k ^{(td)}$, so $321$ is in the basis as well. Prefix transposition ==================== If we restrict the block transposition operation to pairs of blocks such that the first one is a prefix of the permutation, we obtain the so-called *prefix transposition model*. It is clearly a special case of the block transposition model and, as such, it is simpler to analyze. Denoting with $ptd$ the prefix transposition distance, our first goal is to characterize the balls $B_k ^{(ptd)}$ in terms of generating permutations. As a first example, it is easy to see that $B_1 ^{(ptd)}=MI(213)$. The approach we use to determine the generating set $B_k ^{(ptd)}$ is slightly different from the one we have used for the block transposition model. In the present case, we are able to give an explicit construction of the generating set of $B_{k+1}^{(ptd)}$ starting from the generating set of $B_k ^{(ptd)}$. \[prefixinfl\] Let $\tau \in S_n$ be a generating permutation of $B_k ^{(ptd)}$. 1. Suppose that $\tau =\pi a \rho b \gamma$, where $a<b\leq n$ and $\pi ,\rho$ and $\gamma$ are the subwords of $\tau$ determined by such a decomposition. Then the permutation $(a+1)\hat{\rho}(b+1)\hat{\pi}a(b+2)\hat{\gamma}$ is a generating permutation of $B_{k+1}^{(ptd)}$, where $\hat{\rho}, \hat{\pi}, \hat{\gamma}$ are obtained from $\rho ,\pi, \gamma$ (respectively) by increasing by 1 all the entries between $a$ and $b$ and by increasing by 2 all the entries that are greater than $b$. 2. Suppose that $\tau =\pi a \rho b \gamma$, where $n\geq a>b$ and $\pi ,\rho$ and $\gamma$ are the subwords of $\tau$ determined by such a decomposition. Then the permutation $(a+2)\hat{\rho}b\hat{\pi}(a+1)(b+1)\hat{\gamma}$ is a generating permutation of $B_{k+1}^{(ptd)}$, where $\hat{\rho}, \hat{\pi}, \hat{\gamma}$ are obtained from $\rho ,\pi, \gamma$ (respectively) by increasing by 1 all the entries between $b$ and $a$ and by increasing by 2 all the entries that are greater than $a$. 3. Suppose that $\tau =\pi a \rho$, where $a\leq n$ and $\pi$ and $\rho$ are the subwords of $\tau$ determined by such a decomposition. Then the permutation $(a+1)\hat{\pi}a(a+2)\hat{\rho}$ is a generating permutation of $B_{k+1}^{(ptd)}$, where $\hat{\pi}, \hat{\rho}$ are obtained from $\pi, \rho$ by increasing by 2 all the entries that are greater than $a$. We will give details only for the first case, the remaining two being analogous. Since the prefix transposition model is a special case of the block transposition one, we have that, if $\tau$ is a generating permutation for $B_k ^{(ptd)}$, we can construct generating permutations for $B_{k+1}^{(ptd)}$ by suitably choosing two elements $a$ and $b$ of $\tau$ (possibly the same one), then suitably inflating them and performing the prefix transposition operation which exchanges the prefix block ending with $a$ with the adjacent block ending with $b$. This is done in analogy with the construction described before Lemma \[prel\]. If $a<b$ and $a$ precedes $b$ in $\tau$, then we can decompose $\tau$ as $\tau =\pi a \rho b \gamma$. After inflating $a$ and $b$, we thus get the permutation $\hat{\pi}a(a+1)\hat{\rho}(b+1)(b+2)\hat{\gamma}$, where the elements of $\pi , \rho$ and $\gamma$ have been renamed, namely all entries greater than $a$ and smaller than $b$ have been increased by 1 and all entries greater than $b$ have been increased by 2. We can now perform the desired prefix transposition, which exchanges the prefix block $\hat{\pi}a$ with the adjacent block $(a+1)\hat{\rho}(b+1)$, thus obtaining the predicted permutation. The above proposition gives a recipe for constructing generating permutations of $B_{k+1} ^{(ptd)}$ starting from those of $B_k ^{(ptd)}$. Notice that, if $\tau$ has length $m$, then the permutations obtained with the previous construction have length $m+2$. Since we have seen that $B_1 ^{(ptd)}=MI(213)$, a simple inductive argument shows that the generating permutations of $B_k ^{(ptd)}$ we have produced all have length $2k+1$. Actually, we have something stronger, which is the analogous of Theorem \[genblock\] in the case of the prefix transposition model. Since the proof is similar, we just give the statement. For every $k\geq 1$, the generating set of $B_k ^{(ptd)}$ is the set of all plus irreducible permutations of length $2k+1$ and having distance $k$ from the identity. However, in this case we can also enumerate the generating sets. The generating set of $B_k ^{(ptd)}$ has cardinality $(2k)!/2^k$. We observe that, if $\sigma =\sigma_1 \sigma_2 \cdots \sigma_{2k+3}$ is a generating permutation for $B_{k+1}^{(ptd)}$, then it has been obtained from a generating permutation of $B_k ^{(ptd)}$ by one of the construction described in Proposition \[prefixinfl\]. However, $\sigma$ cannot be obtained in two different ways. This can be shown by considering the elements $\sigma_1$ and $\sigma_1 -1$ (notice that, in this model, a generating permutation cannot start with 1). 1. If the element on the right of $\sigma_1 -1$ in $\sigma$ is larger than or equal to $\sigma_1 +2$, then $\sigma$ is constructed as in 1 of Proposition \[prefixinfl\]. 2. If the element on the right of $\sigma_1 -1$ in $\sigma$ is smaller than or equal to $\sigma_1 -2$, then $\sigma$ is constructed as in 2 of Proposition \[prefixinfl\]. 3. If the element on the right of $\sigma_1 -1$ in $\sigma$ is equal to $\sigma_1 +1$, then $\sigma$ is constructed as in 3 of Proposition \[prefixinfl\]. Since the three above cases are disjoint, we can conclude that $\sigma$ comes from a unique generating permutation of $B_k ^{(ptd)}$ through the construction of Proposition \[prefixinfl\]. Thus, the total number of generating permutations of $B_{k+1}^{(ptd)}$ is obtained by multiplying the number of generating permutations of $B_k ^{(ptd)}$ by the number of possible inflations of each of them, which is equal to the number of multisets of cardinality 2 of a set of cardinality $2k+1$, i.e. ${2k+2\choose 2}$. Since the generating set of $B_1 ^{(ptd)}$ has cardinality $1={2\choose 2}$, a simple inductive argument shows that the required cardinality is indeed equal to $\prod_{i=1}^{k}{2i\choose i}=(2k)!/2^k$. We have already observed that, for $k=1$, the generating set is $\{ 213\}$. For $k=2$, the generating set is $\{ 32415, 41325, 31425, 24135, 24315, 42135\}$. Concerning the basis of $B_k ^{(ptd)}$, we have been able to prove the analogue of Theorem \[basis\], however the proof is slightly more complicated, so we cannot reproduce it entirely here, due to limited space. Every permutation belonging to the basis of $B_k ^{(ptd)}$ has length at most $2k+1$. [*(sketch).*]{}The fact that the permutations in the basis of $B_k ^{(ptd)}$ must all have length at most $2k+2$ can be proved in a similar way as the first part of Theorem \[basis\]. Now suppose that $\pi =\pi_1 \pi_2 \cdots \pi_{2k+2}$ is a basis permutation for $B_k ^{(ptd)}$ of length $2k+2$, and set $n=2k+2$. Then it can be shown that $\pi$ has to be plus irreducible and that $\pi_{n}\neq n$, i.e. the last element of $\pi$ is not its maximum. From a previous observation, we know that it is possible to remove one element of $\pi$ in such a way that the resulting permutation $\gamma= \gamma_1 \gamma_2 \cdots \gamma_{n-1}$ of length $n-1$ is plus irreducible. However, since $\pi$ belongs to the basis of $B_k ^{(ptd)}$, $\gamma$ has to be a generating permutation of $B_k ^{(ptd)}$. Since it is possible to prove that the last element of any generating permutation of $B_k ^{(ptd)}$ is its maximum, there are only two possibilities: either the last element of $\pi$ is $n-1$ and $\gamma$ is obtained by removing $n$, or the second-to-last element of $\pi$ is $n$ and $\gamma$ is obtained by removing the last element. Since the two cases are symmetric in a well precise sense, we just consider the first one. Our goal is now to show that we can remove another element from $\pi$ (different from $n$) and obtain another plus irreducible permutation, which turns out to be a generating permutation: this is however impossible, since it does not ends with its maximum. In many cases, if we remove the last element $n-1$ of $\pi$, we do obtain a plus irreducible permutation. The only cases in which this does not work are those in which $n-2$ is immediately before $n$ in $\pi$. In such cases, if we remove $n-2$, we indeed get a plus irreducible permutation, unless $n-3$ is immediately before $n-1$ in $\pi$. By repeating this argument, we find that we are always able to remove an element different from $n$ and obtain a plus irreducible permutation, except for the permutation $\sigma= 2468\cdots n1357\cdots (n-1)$ (recall that $n=2k+2$, so $n$ is even). Also in this last case, however, we can remove 1 from $\sigma$ and the permutation thus obtained is easily seen to be plus irreducible. Thanks to the above bound, we are able also in this case to compute the basis for small values of $k$. 1. The basis of $B_1 ^{(ptd)}$ is $\{ 132,321\}$. 2. The basis of $B_2 ^{(ptd)}$ consists of three permutations of length 4, namely 1432, 2143, 4321, and fifteen permutations of length 5, namely 13524, 14253, 24351, 25314, 25413, 35142, 35214, 35241, 41352, 42513, 42531, 43152, 51324, 52413, 53142. ### Acknowledgements {#acknowledgements .unnumbered} Both authors are members of the INdAM Research group GNCS; they are partially supported by INdAM - GNCS 2018 project “Proprietá combinatorie e rilevamento di pattern in strutture discrete lineari e bidimensionali“ and by a grant of the ”Fondazione della Cassa di Risparmio di Firenze“ for the project ”Rilevamento di pattern: applicazioni a memorizzazione basata sul DNA, evoluzione del genoma, scelta sociale". [Com79]{} M. H. Albert, M. D. Atkinson, M. Bouvel, N. Ruskuc and V. Vatter. Geometric grid classes of permutations. , 365:5859–5881, 2013. M. H. Albert, M. D. Atkinson and R. Brignall. Permutation Classes of Polynomial Growth. , 11:249–264, 2007. M. D. Atkinson and T. Stitt. Restricted permutations and the wreath product. , 259:19–36, 2002. V. Bafna and P. A. Pevzner. Genome rearrangements and sorting by reversals. , pp. 148–157, 1993. V. Bafna and P. A. Pevzner. Sorting by transpositions. , 11:224–240, 1998. M. Bouvel and L. Ferrari. On the enumeration of $d$-minimal permutations. , 15:33–48, 2013. M. Bouvel and E. Pergola. Posets and permutations in the duplication-loss model: minimal permutations with $d$ descents. , 411:2487–2501, 2010. M. Bouvel and D. Rossin. A variant of the tandem duplication-random loss model of genome rearrangement. , 410:847–858, 2009. G. Cerbai. Pattern avoiding permutations in genome rearrangement problems. , 2017. K. Chaudhuri, K. Chen, R. Mihaescu and S. Rao. On the tandem duplication-random loss model of genome rearrangement. , pp. 564–-570, 2006. W. Y. C. Chen, C. C. Y. Gu and K. J. Ma. Minimal permutations and 2-regular skew tableaux. , 47:795–812, 2011. Z. Dias and J. Meidanis. Sorting by prefix transposition. , 2476:65–76, 2002. G. Fertin, A. Labarre, I. Rusu, E. Tannier and S. Vialette. . MIT Press, Cambridge, MA, 2009. W. H. Gates and C. H. Papadimitriou. Bounds for sorting by prefix reversal. , 27:47–57, 1979. S. Hannenhalli and P. A. Pevzner. Transforming cabbage into turnip: polynomial algorithm for sorting signed permutations by reversals. , 46:1–27, 1999. C. Homberger and V. Vatter. On the effective and automatic enumeration of polynomial permutation classes. , 76:84–96, 2016. N. J. A. Sloane. . At oeis.org. G. A. Watterson, W. J. Ewens, T. Hall and A. Morgan. The chromosome inversion problem. , 99:1–7, 1982.
--- abstract: 'For every prime number $p$, we show the existence of a solvable number field $\L$ ramified only at $\{p, \infty\}$ whose $p$-Hilbert Class field tower is infinite.' address: - 'Department of Mathematics, University of Massachussetts, Amherst, MA 01003, USA' - 'FEMTO-ST Institute, Université Bourgogne Franche-Comté, CNRS, 15B avenue des Montboucons, 25000 Besançon, FRANCE' - 'Department of Mathematics, Cornell University, Ithaca, USA' author: - 'Farshid Hajir, Christian Maire, Ravi Ramakrishna' title: Infinite class field towers of number fields of prime power discriminant --- [^1] For a number field $\rm L$ of degree $n$ over $\Q$, the root discriminant is defined to be $D_{\rm L}^{1/n}$ where $D_{\rm L}$ is the absolute value of the discriminant of $\rm L$. Given a finite set $S$ of places of $\Q$, it is an old question as to whether there is an infinite sequence of number fields unramified outside $S$ with bounded root discriminant. This question is related to the constants of Martinet [@Martinet] and Odlyzko’s bounds [@Odlyzko]. Since the root discriminant is constant in unramified extensions, an approach to answering the previous question in the positive is to find a number field $\rm L$ (of finite degree) unramified outside $S$ having an infinite class field tower. In the case of $\K/\Q$ quadratic, it is a classical result of Golod and Shafarevich that if $\K/\Q$ is ramified at at least $ 8$ places, then $\K$ has an infinite class field tower. On the other hand, if $p$ is a prime, and $S=\{p,\infty\}$, the question becomes whether there exist number fields with $p$-power discriminant having an infinite unramified extension. Schmitals [@Schmithals] and Schoof [@Schoof] produced a few isolated examples of this type. See also [@Hajir-Maire-JA], [@Leshin], etc. For $p\in \{2,3,5\}$, Hoelscher [@Hoelscher] announced the existence of number fields unramified outside $\{ p, \infty\}$ and having an infinite Hilbert class field tower. Here we prove: \[Theorem:main\] For every prime number $p$, there exists a solvable extension $\rm L/\Q$, ramified only at $\{p,\infty\}$, having an infinite Hilbert $p$-class field tower. Consequently, there exists an infinite nested sequence of number fields of $p$-power discriminant with bounded root discriminant. Our proof is based on the idea of cutting of wild towers introduced in [@HMR]; in particular it does not involve the usual technique of genus theory. For the more refined question where $S$ consists of a single prime number $p$ (i.e. if we focus our attention on totally real fields only), we do not know whether for every prime $p$, there is a totally real number field of $p$-power discriminant having an infinite Hilbert class field tower. In [@Schoof Corollary 4.4] it is shown that $\Q(\sqrt{39345017})$ (which is ramified only at the prime $39345017$) has infinite Hilbert class field tower. In [@Shanks], Shanks studied primes of the form $p=a^2+3a+9$ and the corresponding totally real cubic subfields $\K \subset \Q(\mu_p)$ and showed the minimal polynomials of $\K$ are $x^3-ax^2-(a+3)x-1$. Taking $a=17279$ so $p=298615687$, one can compute that the $2$-part of the class group of $\K$ has rank $6$. It is not hard to see, using the Golod-Shafarevich criterion, that $\K$ has infinite $2$-Hilbert class field tower. Thus some examples exist in the totally real case. The results we need =================== Let $p$ be a prime number. Let $\K/\Q$ be a finite Galois extension. Assume $\mu_p \subset \K$ and moreover that $\K$ is totally imaginary when $p=2$. For a prime $\p$ of $\K$ dividing $p$ denote by $e$ (resp. $f$) the ramification index (resp. the residue degree) of $\p$ in $\K/\Q$. On the group $\G_S$ ------------------- Denote by $S$ the set of places of $\K$ above $p$, and consider $\K_S$ the maximal pro-$p$ extension of $\K$ unramified outside $S$; put $\G_S=\Gal(\K_S/\K)$. Let $g=|S|$ be the number of places of $\K$ above $p$. Let $h_\K'$ be the $S$-class number of $\K$. By class field theory, $h_\K'$ is equal to $[\K':\K]$ where $\K'/\K$ is the maximal abelian of $\K$ unramified everywhere in which all places of $S$ split completely. The Kummer radical of the $p$-elementary subextension $\K'(p)/\K$ of $\K'/\K$ is $$\V_S:=\{x\in \K^\times \mid x{\mathcal O}_\K=\A^p, \ x \in \K_v^{\times p}, \forall v \in S\}.$$ In particular $p\nmid h_\K'$ if and only if $\V_S/\K^{\times p}$ is trivial. By work of Koch and Shafarevich the pro-$p$ group $\G_S$ is finitely presented. More precisely, in our situation one has: \[Theorem:rappel\] Let $\K/\Q$ be a totally imaginary Galois extension containing $\mu_p$. Let $S=\{p,\infty\}$. Then $$\dim H^1(\G_S,\fq_p)=\frac{efg}{2}+1 + \dim H^2(\G_S,\fq_p)$$ and $$\dim H^2(\G_S,\fq_p) = g-1 + \dim \V_S/\K^{\times p}.$$ This is well-known, see for example [@NSW Corollary 8.7.5 and Theorem 10.7.3]. We immediately have: \[coro:main\] If $p\nmid h_\K$ then $\dim H^1(\G_S,\fq_p)= g(\frac{ef}{2}+1)$ and $\dim H^2(\G_S,\fq_p)=g-1 $. The cutting towers strategy --------------------------- ### The Golod-Shafarevich Theorem Let $\G$ be a finitely generated pro-$p$ group. Consider a minimal presentation $1\rightarrow \R \rightarrow \F\stackrel{\varphi}{\rightarrow} \G$ of $\G$, where $\F$ is a free pro-$p$ group. Set $d=d(\G)=d(\F)$, the number of generators of $\G$ and $\F$. Suppose that $\R=\langle \rho_1,\cdots, \rho_r \rangle^{Norm}$ is generated as normal subgroup of $\F$ by a finite set of relations $\rho_i$. We recall the depth function $\omega$ on $\F$. See [@Lazard Appendice] or [@Koch] for more details. The augmentation ideal $I$ of $\fq_p [[\G ]]$ is, by definition, generated by the set of elements $\{g-e\}_{g\in \G}$. Then for $e\neq g\in \F$, define $\omega(g)=\max_k \{ g-e \in I^k\}$; put $\omega(e)= \infty$. It is not difficult to see that $\omega([g,g']) \geq 2$ and that $\omega(g^{p^k}) \geq p^k$ for every $g,g' \in \G$ and $k\in \Z_{>0}$. Observe also that as the presentation $\varphi$ is minimal, $\omega(\rho_i) \geq 2$ for all the relations $\rho_i$. The Golod-Shafarevich polynomial associated to the presentation $\varphi$ of $\G$ is the polynomial $P_\G(t)=1-dt + \sum_i t^{\omega(\rho_i)}$. If $\G$ is finite then $P_\G(t) >0$ for all $t\in ]0,1[$. Of course if we have no information about the $\rho_i$’s we may take $1-dt +rt^2$ (where $r=\dim H^2(G,\fq_p)$) as Golod-Shafarevich polynomial for $\G$: if $1-dt+rt^2$ is negative at $t_0 \in ]0,1[$, then $P_\G(t_0) < 0$ and $\G$ is infinite. We can also define a depth function $\omega_\G$ on $\G$ associated to its augmentation ideal. Then: For every $g \in \G$, one has $$\omega_\G(g)=\max\{ \omega(y), \varphi(y)=g\}.$$ See [@Lazard Appendice 3, Theorem 3.5]. We now study quotients $\Gamma$ of $\G$ such that $d(\G)=d(\Gamma)$. In this case, the initial minimal presentation of $\G$ induces a minimal presentation of $\Gamma$ $$\xymatrix{1 \ar[r] & \R \ar[r] & \F \ar[r]^\varphi \ar@{.>>}[rd] & \G \ar[r] \ar@{->>}[d] & 1 \\ & & & \Gamma &}$$ Suppose that $\Gamma=\G/\langle x_1,\cdots, x_m\rangle^{Norm}$. Here $\langle x_1,\cdots, x_m\rangle^{Norm}$ is the normal subgroup of $\G$ generated by the $x_i$’s. Lift the $x_i$’s to $y_i \in \F$ such that $\omega_\G(x_i)=\omega(y_i)$ for each $i$. Hence, $\Gamma=\F/\R'$, where $\R'=\R\langle y_1, \cdots, y_m\rangle^{Norm}$. In particular, if $\R=\langle \rho_1,\cdots, \rho_r \rangle^{Norm}$, then $\R'=\langle \rho_1, \cdots, \rho_r, y_1, \cdots, y_m\rangle^{Norm}$. If we have no information about the $\rho_i$’s, we can take $P_\Gamma(t)=1-dt+rt^2+\sum_i t^{\omega(y_i)}$ as Golod-Shafarevich polynomial for $\Gamma$. ### Cutting of $\G_S$ {#section:cutting} We want to consider some special quotients $\Gamma$ of $\G_S$, this is what we call “cutting wild towers”. Each place $v\in S$ corresponds to some extension $\K_v/\Q_p$ (in fact these fields are isomorphic as $\K/\Q$ is Galois) of degree $ef$. Then, as $\mu_p \subset \K_v$, the ${\mathbb F}_p$-vector space $\K^\times_v/\K^{\times p}_v$ has dimension $ef+2$, and local class field theory implies the Galois group of the maximal pro-$p$ extension of $\K_v$ is generated by $ef+2$ elements. Thus the decomposition subgroup $\G_v$ of $v$ in $\K_S/\K$ is generated by at most $ef+2$ elements $z_{i,v}$. Consider now the commutators $[z_{i,v},z_{k,v}] $ of all these elements; there are at most $\binom{ef+2}{2}$ such elements. Now we cut $\G_S$ by $\langle [z_{i,v},z_{k,v}], i,k; v\in S\rangle^{Norm}$, and denote by $\Gamma$ the corresponding quotient. As $\omega_{\G_S}([z_{i,v},z_{k,v}]) \geq 2$, one can take $P_\Gamma=1-dt+rt^2+g\binom{ef+2}{2} t^2$ as Golod-Shafarevich polynomial for $\Gamma$; here $d=\dim H^1(\G_S,\fq_p)$ and $r=\dim H^2(\G_S,\fq_p)$. This quotient $\Gamma$ of $\G_S$ corresponds to the maximal subextension $\K_S^{loc-ab}/\K$ of $\K_S/\K$ locally abelian everywhere. Observe that $\K_S^{loc-ab}/\K$ contains the compositum of all $\Z_p$-extensions. Suppose that there exists some $t_0 \in ]0,1[$ such that $P_\Gamma(t_0)<0$. We will then cut the infinite pro-$p$ group $\Gamma$ by all the $z_{v,i}^{p^k}$ for some large $k$. There are $g(ef+2)$ such elements. Denote by $\Gamma_k$ the new quotient and by $\K_S^{[k]}$ the new extension of $\K$ corresponding to $\Gamma_k$. Since $\omega_\Gamma(z_{v,i}^{p^k})\geq p^k$, we may take $P_{\Gamma_k}(t)=P_\Gamma(t) + g(ef+2)t^{p^k}$ as the Golod-Shafarevich polynomial for $\Gamma_k$. When $k$ is sufficiently large, clearly $P_{\Gamma}(t_0) < 0 \implies P_{\Gamma_k}(t_0)<0$, so $\K_S^{[k]}/\K$ is infinite. The main interest of $\K_S^{[k]}/\K$ is: \[prop:basechange\] Suppose $\K_S^{[k]}/\K$ infinite. Then there exists a finite subextension $\rm L/\K$ of $\K_S^{[k]}/\K$ having an infinite Hilbert $p$-class field tower. In $\K_S^{[k]}/\K$ the (wild) ramification is finite: indeed for each $v\in S$, the decomposition groups in $\K_S^{[k]}/\K$ are abelian, finitely generated and of finite exponent. There exists a finite extension $\rm L/\K$ inside $\K_S^{[k]}/\K$ absorbing all the ramification, so $\K_S^{[k]}/\rm L$ is unramified everywhere and infinite. Proof ===== \[prop:0\] Let $\K/\Q$ be finite Galois with $\mu_p \subset \K$. Assume that $g\geq 8$.Then there exists a finite subextension $\rm L/\K$ of $\K_S/\K$ such that the Hilbert $p$-class field tower of $\rm L$ is infinite. Let $\H$ be the “top” of the Hilbert Class Field Tower of $\K$. If $\H/\K$ is infinite, we are done, so suppose $[\H:\K] < \infty$. Note that $\H$ has class number $1$ so by Corollary \[coro:main\], working over $\H$, $\dim H^1(\G_S,\fq_p)=g\left(\frac{ef}{2}+1\right)$ and $\dim H^1(\G_S,\fq_p)=g-1$. As in Section \[section:cutting\], consider the quotient $\Gamma$ of $\G_S$ by the normal subgroup generated by the local commutators at each $v\in S$; one has $\binom{ef+2}{2}$ such commutators. The group $\Gamma$ can be described by $d:=g\left(\frac{ef}{2}+1\right)$ generators and by $r:=g-1+ g\frac{(ef+2)(ef+1)}{2}$ relations. The Golod-Shafarevich polynomial of $\Gamma$ may be written as $P_{\Gamma}(t)=1-d t+r t^2$, when assuming the worst case that all the relations are of depth $2$. Clearly $d/2r <1$, and $P_{\Gamma}(d/2r)=1-\frac{d^2}{4r}$. In particular, if $P_{\Gamma}(d/2r)<0$, then one has room to cut by some large $p$-power of the local generators, in order to obtain at the end some finite local groups. For the result to follow, we thus need $4r < d^2$, or equivalently $$4 \left(g-1+g\frac{(ef+2)(ef+1)}{2}\right) \stackrel{?}{< }\frac{g^2}4 (ef+2)^2$$ which is equivalent to $$16(g-1) +8g(ef+2)(ef+1) \stackrel{?}{<} g^2( ef+2)^2.$$ Replacing the $16(g-1)$ term on the left by $16g$ and dividing by $g$, and setting $x=ef$, we need to verify $$16+8(x+2)(x+1) \stackrel{?}{<}g(x+2)^2.$$ This holds for $g \geq 8$ and $x=ef \geq 1$. Proposition \[prop:basechange\] allows us to conclude $\K_S^{[k]}/\K$ is infinite when $k$ is sufficiently large. [*Proof Theorem \[Theorem:main\]* ]{}: Recall that the principal prime $\p = (1-\zeta_{p^s})$ of $\Q(\zeta_{p^s})$ is the unique prime dividing $p$ and by class field theory $\p$ splits completely in the Hilbert class field $\H$ of $\Q(\zeta_{p^s})$. Thus if the class group has order at least $8$, Proposition \[prop:0\] applied to the solvable number field $\H$ gives the result. In the proof of [@Washington Corollary 11.17], the class number of $\Q(\zeta_{p^r})$ is shown to be at least $10^9$ for $\phi(p^r)=p^{r-1}(p-1) >220$. Choosing $r\geq 9$ for [*any*]{} $p$ completes the proof of the Theorem. A slightly more detailed analysis using Table §3 of [@Washington] shows the fields below suffice: 1em $\begin{array}{cll} p & \K & g=h\\ p>23 & \K=\Q(\zeta_p) & \geq 8\\ 7 \leq p \leq 23 & \K=\Q(\zeta_{p^2}) & \geq 43\\ p=5 & \K=\Q(\zeta_{125}) & 57708445601\\ p=3 & \K=\Q(\zeta_{81}) & 2593 \\ p=2 & \K=\Q(\zeta_{64}) & 17 \end{array}$ $\square$ In [@Hoelscher] a proof of the Theorem for $p=2,3$ and $5$ was given. Our proof is partially modeled on the ideas there, namely considering the Hilbert class field of a cyclotomic field. There are two cases in [@Hoelscher]: Case I, where the Hilbert class field tower is infinite; and Case II, where ramification is allowed at one prime above $p$ in the Hilbert class field $\H$ and a $\Z/p$-extension of $\H$ ramified at exactly this prime is used. Gras has given a criterion for such an extension to exist: see [@gras Chapter V, Corollary 2.4.4]. Gras’ criterion is not verified in [@Hoelscher]. Given the size of the number fields $\H$, it seems very difficult to do so. We therefore we regard the results of [@Hoelscher] as incomplete. [30]{} G. Gras, Class Field Theory, From Theory to practice, corr. 2nd ed., Springer Monographs in Mathematics, Springer (2005), xiii+507 pages. F. Hajir, C. Maire, R. Ramakrishna, [*Cutting towers of number fields*]{}, arXiv:1901.04354, 2019. F. Hajir and C. Maire, [*Unramified subextensions of ray class field towers*]{}, J. Algebra [**249**]{} (2002), no. 2, 528–543. J. L. Hoelscher, [*Infinite class field towers*]{}, Mathematische Annalen [**344**]{} (2009), 923-928. H. Koch, [*Galois Theory of $p$-extensions*]{}, Springer Monographs in Mathematics, Springer-Verlag, Berlin 2002. M. Lazard, [*Groupes analytiques $p$-adiques*]{}, IHES, Publ. Math. [**26**]{} (1965), 389-603. J. Leshin, [*On infinite class field towers ramified at three primes*]{}, New York Journal of Math [**20**]{} (2014), 27-33. J. Martinet, [*Tours de corps de classes et estimations de discriminants*]{}, Inventiones math. [**44**]{} (1978), 65-73. J. Neukirck, A. Schmidt and K. Wingberg, Cohomology of Number Fields, GMW 323, Second Edition, Corrected 2nd printing, Springer-Verlag Berlin Heidelberg, 2013. A. M. Odlyzko, [*Bounds for discriminants and related estimates for class numbers, regulators and zeros of zeta functions: a survey of recent results*]{}, J. Théor. Nombres Bordeaux [**2**]{} (1990), no. 2, 119-141. B. Schmithals, [*Konstruktion imaginärquadratischer Körper mit unendlichem Klassenkörperturm*]{}, (German) Arch. Math. (Basel) [**34**]{} (1980), no. 4, 307-312. R. Schoof, [*Infinite class field towers of quadratic fields*]{}, J. Reine Angew. Math. [**372**]{} (1986), 209-220. D. Shanks [*The simplest cubic fields*]{}, Mathematics of Computation, v.28, no. 128, 1137-1152 (1974). E. B. Vinberg, [*On a theorem concerning on infinite dimensionality of an associative algebra*]{}, Izv. Akad. Nauk SSSR Ser. Mat. [**29**]{} (1965), 208-214; english transl., Amer. Mat. Soc. Transl. (2) [**82**]{} (1969), 237-242. L. C. Washington, Introduction to Cyclotomic Fields, GTM 80, Second Edition, Springer, 1997. [^1]: We all thank Mathematisches Forschungsinstitut Oberwolfach for sponsoring a “Research in Pairs” stay during which this work was done. The second author was partially supported by the ANR project FLAIR (ANR-17-CE40-0012) and by the EIPHI Graduate School (ANR-17-EURE-0002). The third author was supported by Simons collaboration grant 524863.
--- abstract: 'We propose a mechanism to have a smooth transition from a pre-Big Bang phase to a standard cosmological phase. Such transition is driven by gravitational production of heavy massive string states that backreact on the geometry to stop the growth of the curvature. Close to the string scale, particle creation can become effective because the string phase space compensate the exponential suppression of the particle production. Numerical solutions for the evolution of the Universe with this source are presented.' --- BA-TH/11-642\ \ [*Istituto Nazionale di Fisica Nucleare, Sezione di Bari\ Via G. Amendola 173, 70126 Bari, Italy*]{} Introduction {#Intro} ============ The standard cosmological model, even in its inflationary extension, is plagued by an “initial condition problem” [@Borde:1993xh], since it must emerge from an initial singularity. One of the most successful proposal to overcome this problem was the introduction of the so-called pre-Big Bang scenario [@Gasperini:1992em] (for a review, see [@Gasperini:2002bn]), developed in the context of string theory, in which the Universe is supposed to emerge from a string vacuum (so the initial geometry is flat), perturbed by quantum fluctuations of the dilaton (other proposals were put forward in different contexts, such as brane cosmology and Loop Quantum Gravity, see for example [@Khoury:2001wf; @Steinhardt:2001st; @Mukherji:2002ft; @DeRisi:2007dn; @Ashtekar:2006rx; @Ashtekar:2006uz; @DeRisi:2007gp]). This is achieved by implementing purely stringy symmetries on the string cosmology equations. Those symmetries suggest the existence of a phase of growing curvature. However, it is not clear, a priori how to connect smoothly the pre-big bang and the post-big bang phases, since, at the classical level, the two phases are still disconnected by a singularity. Many ideas were proposed to overcome this problem, all involving further assumptions on the dynamics of the Universe, namely higher order correction (either loop or $\a'$ contribution) and negative energy density contribution to obtain the desired “repulsive gravity” (an incomplete list would include [@Gasperini:1996in; @Gasperini:1996fu; @Brustein:1997cv; @Maggiore:1997vw; @Cartier:1999vk; @DeRisi:2001ed]). In this paper we will propose a mechanism to obtain a regular evolution of the curvature that rely on production of heavy massive string states by gravitational backreaction. It is well known [@Birrel] that an evolving Universe will produce particles because of the squeezing of the ground state. It has been speculated in the literature that this effect, though being negligible, could actually have some physical consequences [@Lawrence:1995ct; @Gubser:2003vk; @Chung:1998bt]. The idea is that the exponential suppression of the energy density of particles created during the evolution of the Universe is compensated by the exponential growing of the multiplicity of states above the string scale (the Hagedorn spectrum), thus giving a sensible contribution to the total energy density. Of course, dealing with gravitationally induced particle production brings one to deal with a sort of semiclassical realization of quantum gravity, which is not under control, and in fact the authors themselves of the mentioned papers admit some of the ideas are rather speculative. Here we want to go one step forward with speculations, and assume that the gravitationally produced energy backreacts locally on the geometry of the Universe. We will see that even a rough estimate such as the one we are going to show here is enough to stop the growing of the Ricci curvature and induce a bell-shaped behaviour. Let us stress that all the analysis is performed under the assumtpion that the universe stays in a low-energy (low curvature) regime in which supergravity is a valid approximation, and the only ingredient taken from stringy physics is the behaviour of the high energy spectrum that influences the phase space of low curvature universe. In fact, we are going to treat string massive modes as scalars, assuming however that their multiplicity is controlled by the Hagedorn spectrum. The validity of this assumption is verified by checking that the curvature found vith the modified cosmological equations do not exceed the string scale (which has been set to unity in the numerical evaluationsI. We are aware that some of the passages we will show, though physically well-motivated, can be challenged on a formal basis. Nevertheless, since our goal was only to check if a backreaction mechanism could provide for a graceful exit without invoking any further assumptions and higher order modifications of the string cosmology equations, we just skipped over formal complexities to have an estimate of the order of magnitude of the effect. We plan to come back on all the mathematical issues (among other things we will discuss below) in a forthcoming paper. The paper is organized as follows: Section \[QFT\] is devoted to some formal developments that will lead us to write the enregy density (and pressure) produced by gravitationally excited string states in term of a time dependent Bogoliubov coefficient. In section \[energy\] we will use this expression to find, under certain assumption, a sound expression for the energy density, to be put into the string cosmology equations whcih are solved numerically in section \[cosmology\]. Finally, in section \[conclusions\] we will comment on our results and the (rather strong) approximations assumed in obtaining them, and on how to further develop the present study in several directions. QFT approach to gravitational backreaction {#QFT} ========================================== Let us then consider a massive scalar field living in a curved space. The action is S = d\^4 x (g\^ \_\_- m\^2 \^2 ). \[Action\] Let us assume that the geometry of the Universe is a conformally flat FRW; so, in the conformal time gauge, the metric is ds\^2 = a\^2() (d\^2 - [**dx**]{}\^2 ), \[confmetric\] and the action can be written as S = dd\^3 x a\^2() (’\^2 - ()\^2 - m\^2a\^2()\^2 ), \[confAction\] where $(\nabla \phi)^2$ stands for $\sum_i ({\partial}_i \phi)^2$. Now we introduce the canonical field $\vp (\eta, {\bf x}) = a(\eta) \phi (\eta, {\bf x})$, so that the action can be rewritten as S = dd\^3 x. \[canAction\] This action can be interpreted as describing a scalar field living on a flat space-time, with a potential which controls its gravitational interactions. From the action (\[canAction\]) we can evaluate the (canonical) energy momentum tensor. T\_\^[  ]{} &=& \_-\_\^[  ]{}\ &=& \_\^- \_\^[  ]{}. \[emtensor\] The diagonal elements of this tensor are the energy density and the pressure. The field conjugate to $\vp$ is (, [**x**]{}) = , \[conjfield\] so the Hamiltonian density, to be identified with the energy density, and the pressure along the $i^{th}$ direction are, according to (\[emtensor\]) \_m(,[**x**]{}) &=& ,\ p\_[m,i]{}(,[**x**]{}) &=& , \[enpressdens1\] where, of course, $V(\eta) = m^2 a^2(\eta) -\frac{a''(\eta)}{a(\eta)}$ Now we expand the fields in Fourier modes and, following conventional quantization procedure, we promote ($\eta$-dependent) Fourier coefficients to operators: (,[**x**]{})&=&d\^3k  \_k() e\^[i[**k**]{}]{},\ (,[**x**]{})&=&d\^3k  \_k() e\^[i[**k**]{}]{}, \[modeexp\] where $\pi_k(\eta)=\vp'_k(\eta)$ and $\vp_k(\eta)$ satisfies the (operatorial) evolution equation: ”\_k() + \^2()\_k()=0, \[modeeq\] with $\om_k(\eta)=\sqrt{k^2+V(\eta)}$. Moreover, since the field $\vp$ is real, and thus Hermitean, the modes must satisfy the relation \^\_k()&=&\_[-k]{}(),\ \^\_k()&=&\_[-k]{}(). \[moderel\] We can then write down the energy density and pressure in terms of these modes. By defining the vector \_k()= ( [c]{} \_k()\ \_k()\ ), \[vecfipi\] the densities (\[enpressdens1\]) can be rewritten as \_m(,[**x**]{}) &=& \_k\^()  P\_[k,p]{}() \_p() e\^[i([**k**]{}-[**p**]{})]{},\ p\_[m,i]{}(,[**x**]{}) &=& \_k\^()  Q\^[(i)]{}\_[k,p]{}() \_p() e\^[i([**k**]{}-[**p**]{})]{}, \[enpressdens3\] with P\_[k,p]{}() &=& ( [cc]{} [**k**]{} + V() & 0\ 0 & 1\ ),\ Q\^[(i)]{}\_[k,p]{}() &=& ( [cc]{} [**k**]{} - V() & 0\ 0 & 1\ ). \[P\] This expression is useful, because we know [@Bozza:2003pr; @Bozza:2002fp] that the evolution for $\zeta_k(\eta)$ is given by: \_k()=U\_k(,\_0)\_k(\_0), \[zetaevoleq\] where the propagator can be written as U\_k(,\_0) = ( [cc]{} A\_k(,\_0) & B\_k(,\_0),\ C\_k(,\_0) & D\_k(,\_0)\ ), \[Umatrx\] and the four coefficients of the matrix are A\_k(,\_0) &=& i ,\ B\_k(,\_0) &=& i ,\ C\_k(,\_0) &=& i ,\ D\_k(,\_0) &=& i . \[ABCD\] Here $f_k(\eta)$ is solution of the canonical evolution equation (\[modeeq\]) for modes of the field $\vp$ and $g_k(\eta)$ is solution of the mode equation (which we have not reported) for the conjugate momentum $\pi$ (so, of course, the relation $g_k(\eta) = f'_k(\eta)$ holds), while $\eta_0$ is a suitable time on which we will define initial condition. We will assume that $\eta_0$ is far enough in the past so that it is possible to consider the space to be Minkowski. Using eq. (\[zetaevoleq\]) we can thus write: \_m(,[**x**]{}) = \_k\^(\_0) S\_[k,p]{}(,\_0)  \_p(\_0) e\^[i([**k**]{}-[**p**]{})]{},\ p\_[m,i]{}(,[**x**]{}) = \_k\^(\_0) T\^[(i)]{}\_[k,p]{}(,\_0)  \_p(\_0) e\^[i([**k**]{}-[**p**]{})]{}, \[enpressdens4\] where, S\_[k,p]{}(,\_0)&=&U\_k\^(,\_0)P\_[k,p]{}U\_k(,\_0),\ T\^[(i)]{}\_[k,p]{}(,\_0)&=&U\_k\^(,\_0)Q\^[(i)]{}\_[k,p]{}U\_k(,\_0). \[S\] Now we want to express this in terms of creation-annihilation operators. Notice that on an actual Minkowski space, there are no ambiguities in building a Fock space; so we define: \_k(\_0)=( [c]{}\ -i\ ) a\_k + ( [c]{}\ i\ ) a\^\_[-k]{}, \[creanform\] with $a_k$ and $a^\dag_k$ satisfying the usual canonical commutation rules and $\om_k \equiv \om_k(\eta_0)$. Inserting this expression in (\[enpressdens4\]) we get \_m &=& e\^[i([**k**]{}-[**p**]{})]{},\ p\_[m,i]{} &=& e\^[i([**k**]{}-[**p**]{})]{},\ \[enpressdens5\] where the $S^{(l)}$ and $T^{(i,l)}$ are calculated from the bilinear product of the matrices $S_{k,p}$ and $T^{(i)}_{k,p}$ respectively, with the vectors defined in (\[creanform\]). In taking the VEV of (\[enpressdens5\]), only the term with the correct order survives. Then the $\delta$ function allows us to integrate over one of the two (equal) momenta. Moreover, in order to handle with this expression in a cosmological context (as we are going to do), we assume that particles are created homogeneously throughout the space, so that the integral in (\[enpressdens5\]) depends only on the square momentum $k^2$ of the excitation. We thus get 0|\_m() |0= 2 \_0\^[+]{} dk k\^2 S\^[(4)]{}\_[k,k]{}(,\_0),\ 0|p\_[m,i]{}() |0= 2 \_0\^[+]{} dk k\^2 T\^[(i,4)]{}\_[k,k]{}(,\_0). \[enpressdens6\] The coefficients $S^{(4)}$ and $T^{(i,4)}$ are easily calculated. After a little algebra we get: S\^[(4)]{}\_[k,k]{}(,\_0) &=&\ && + ,\ T\^[(i,4)]{}\_[k,k]{}(,\_0) &=&\ && + . \[S4andT4\] This expression can be further specified by noting that, if the space-time is flat in $\eta = \eta_0$, solutions for the evolution equation (\[modeeq\]) can be written as: f\_k(\_0) &=& e\^[-i\_k(- \_0)]{},\ g\_k(\_0) &=& -i e\^[-i\_k(- \_0)]{}, \[flatfunct\] with $\om_k = \sqrt{k^2+m^2}$ ($m$ is the mass of the excitation). Furthermore, we get a better insight on the expressions of energy density and pressure by writing the mode functions $f_k$ and $g_k$ in terms of time-dependent Bogoliubov operators [@Audretsch:1979uv; @Gubser:2003vk]: f\_k()&=&( -i\_[\_0]{}\^dx \_k(x) ) + ( i\_[\_0]{}\^dx \_k(x) ),\ g\_k()&=&-i \_k()( -i\_[\_0]{}\^dx \_k(x) ) + i\_k()( i\_[\_0]{}\^dx \_k(x) ),\ \[fgtoab\] with $\a_k$ and $\b_k$ satisfying the condition $|\a_k|^2-|\b_k|^2=1$, which follows from the Wronskian condition on $f_k$ and $g_k$. Substituting this in (\[S4andT4\]), and plugging everything into the expressions for the energy density and pressure (\[enpressdens6\]) we get: \_m() &=& 2 \_0\^[+]{} dk k\^2 ( 1+2|\_k()|\^2 )\_k(), \[bendens\]\ p\_m() &=& 2 \_0\^[+]{} dk  ( 1+2|\_k()|\^2 ). \[bpress\] In the next section, we will use this expressions to obtain the energy density as a function of the geometry of the on-shell universe Energy density of the gravitationally excited string states {#Energy} =========================================================== In the previous section, we obtained a formal expression for the energy density and pressure of massive scalar modes during the evolution of the universe. At this stage we can already notice that, since (\[bpress\]) is suppressed by a factor $m$ with respect to (\[bendens\]), the pressure will be negligible with respect to the energy density for large mass (non-relativistic) modes. Therefore, from now on we will assume that the pressure is exactly zero, leaving a more detailed treatments for future works. Asymptotically, the Bogoliubov coefficient $\b_k(\eta)$ can be interpreted as the number density of particles produced via gravitational interaction from an initial vacuum state. Let us stress that, by setting $\b_k(\eta) = 0$, which is true for $\eta \ra \eta_0$, expression (\[bendens\]) reduces just to the unrenormalized zero point energy of the field. Thus we are led to consider a properly renormalized form for (\[bendens\]) as the energy density of massive excitations of the field under consideration, generated by gravitational interaction. We choose to regularize the energy density by subtracting the time-dependent zero-point energy of the oscillations, so to get \_m() = 4 \_0\^[+]{} dk  k\^2 |\_k()|\^2 \_k(). \[finalendens\] Such expression for the energy density has the key feature of vanishing at $\eta = \eta_0$ (as it is expected since we want a vacuum solution to start with), and in general whenever the space-time is flat. In addition, it has the expected form of a sum over energies of each mode[^1], following the interpretation of $\b$ as the number density of gravitationally produced particles, with the spectrum modified by gravitational interaction. We now need to evaluate $\b_k(\eta)$, which is in general a very difficult task. If we assume that the evolution of the Universe is “slow” enough, so that we are in an adiabatic regime [@Birrel], we can use for $\b_k(\eta)$ the approximate expression [@Chung:1998bt] \_k() = d ( -2i \_[\_0]{}\^d ’\_k(’) ). \[beta\_appr\] The integral can be evaluated, to the same level of approximation, with the steepest descendent method [@Gubser:2003vk; @Chung:1998bt] giving: |\_k()|\^2 = , \[beta\_final\] where $\Hef$ and $\Ref$ are respectively the Hubble parameter and the scalar curvature, calculated starting from an effective scale factor () = . \[a\_eff\] With this expression for $\b_k$, the integral in (\[finalendens\]) converges and can be evaluated exactly. After some technical manipulations we get: \_m() = -V\^2() H\^[(1)]{}\_1 ( i ) ( - ), \[int\_en\_dens\] where $H^{(1)}_1$ is the Hankel function of the first kind. For heavy massive states we can approximate the Hankel function with its large argument expansion [@Gradshteyn], thus getting: \_m() = V\^2() ( )\^[3/4]{} ( - ). \[large\_en\_dens\] As expected, the contribution for each massive mode is exponentially suppressed at low (sub-Planckian) energies. Incidentally, being at sub-Planckian energies justifies our quantum field theory approach to production of string modes (which, of course, have masses above that limit), as we stressed in the introduction. Now, as we already mentioned above, the multiplicity of string modes is exponentially growing [@Gubser:2003vk] $N \propto \exp(m/T_H)$ ($T_H$ being the Hagedorn temperature), and this can possibly compensate the exponential suppression of each produced mode. In fact, the total energy density generated by all the stringy massive modes is: \_[tot]{}() = \_[m m\_s]{} \_m() ( ). \[sum\_en\_dens\] Again, this expression can be estimated by noting that $\aef(\eta) \simeq a(\eta)$ for heavy enough states, and approximating the series as an integral, substituting $m \simeq T_H x$, with $x > 0$. We get: \_[tot]{}() && T\_H\^[5/2]{}a\^4 ( H\^2 - )\^[3/4]{} \_0\^[+]{}dx x\^[5/2]{}\ && T\_H\^4 a\^4 ( )\^[5/2]{}. \[tot\_en\_dens\] Since the energy density is a scalar, turning to cosmic time is straightforward. We finally get: \_[tot]{}(t) T\_H\^4 a\^4(t) ( )\^[5/2]{}. \[fin\_en\_dens\] In the next section, we will use this expression as a source in string cosmolgy equations. Cosmology with the backreacting energy density {#cosmology} ============================================== In the last section we have found an anaylitic expression for the energy density massive string states excited by the gravitational interaction with an evolving universe. Now we assume that this energy density backreacts on the geometry of the Universe to favor a smooth transition between a phase of superinflationary accelerated expansion with growing curvature to a (standard) phase of decelerated expansion and decreasing curvature. To see this, we plug the expression (\[fin\_en\_dens\]) into the pre-big bang equations. We assume that the extra dimension are compactified down to the Plank scale, and there is a mechanism for the stabilization of the modula. The string cosmology equations then read [@Gasperini:2002bn] \^2 - 6 H + 6 H\^2 &=& e\^\_[tot]{}, \[Einst\_eq1\]\ 2 + 6 H - \^2 - 6 - 12 H\^2 &=& 0. \[Einst\_eq2\] (we have identified the string mass with $T_H$) where $\r_{tot}$ is given by (\[fin\_en\_dens\]) and we have assumed that matter is minimally coupled to the metric of the string frame, without any direct dilaton coupling, see, e.g. [@Gasperini:2001mr] (we will comment more on this later). Of course, these equations cannot be solved analytically, so we must turn to numeric. Plots for the interesting physical quantities are shown in Fig. \[Pictures\]. Numerical integration has been carried out by fixing a dummy initial timescale so that the maximum value of the curvature occurs at $t = 0$, and by setting suitable (small, in units of string mass) initial condition for the Hubble parameter and the dilaton energy density. As we can see from panel (b), the curvature reaches a maximum and decreases, evolving towards a phase of De Sitter evolution (i.e. with our notations, a constant curvature regime). It is possible to check, in addition, that all the approximation requirements are met. On the other hand, the dilaton energy density, depicted in panel (a) with the squared Hubble parameter is not under control and blows up, after reaching a first local maximum at the same time age of the curvature (see also plot (d)). This is not surprising, since we have not considered the coupling of the generated massive states with the dilaton and the (possible) corresponding dilaton stabilization. Motivated by the successful transition to a phase of decelerated expansion we found in our approach, we seek to extend the present analysis to the dilatonic sector in a forthcoming paper, so to have a full-fledged graceful exit solution. Conclusions and Outlook {#conclusions} ======================= Particle creation generated by gravitational backreaction seems to provide a cogent mechanism to stop superinflationary accelerated expansion and to drive the Universe to a (standard) phase of decelerated expansion. Calculations we have presented in this paper, though being speculative in some of their aspects, show that, as the curvature grows, there is the possibility that gravitational energy excites heavy string states, thus producing some sort of “friction” effect. This can actually slow down and eventually stop the growth of the curvature and prevent the occurrence of the singularity. This can happen before the curvature reaches the string scale, ao before the universe enters in a regime in which quantum field theory and supergravity are no longer valid. However, the analysis we have performed needs to be further developed in many directions: First of all, some of the mathematical derivations need to be posed on a more steady ground (for example, our derivation rely on the assumption that the metric is conformally flat, so that formal results valid in a flat space can be straightforwardly extended to our curved model). In addition, as we already noted, the particle creation mechanism has to include a direct coupling to the dilaton. In fact, even though the geometry is regular everywhere in the toy model we have presented, the dilaton still blows up, indicating a breakdown of the validity of our approximations. Nevertheless, since the dilaton itself is a string mode, it is expected to couple with massive states, so high energetic dilaton modes will somehow excite heavy modes as well. This will probably provide a mechanism to stabilize the dilaton after the transition. Moreover, to actually enter into a phase of standard evolution, one should consider some sort of “reheating process”. The idea is that string states generated via gravitational backreaction decays into radiation and matter, to start the standard phase. Another issue worth to be pursued is to have a complete dynamical system analysis. In fact, our calculations confirm the known result that late-time evolution of the Universe falls toward a de Sitter phase. It is not clear, however, if the de Sitter curvature scale is controlled by the string scale, or can be influenced by backreaction. 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--- abstract: | In this paper, we consider the exact/approximate general joint block diagonalization (GJBD) problem of a matrix set $\{A_i\}_{i=0}^p$ ($p\ge 1$), where a nonsingular matrix $W$ (often referred to as diagonalizer) needs to be found such that the matrices $W^{\operatorname{H}}A_iW$’s are all exactly/approximately block diagonal matrices with as many diagonal blocks as possible. We show that the diagonalizer of the exact GJBD problem can be given by $W=[x_1, x_2, \dots, x_n]\Pi$, where $\Pi$ is a permutation matrix, $x_i$’s are eigenvectors of the matrix polynomial $P(\lambda)=\sum_{i=0}^p\lambda^i A_i$, satisfying that $[x_1, x_2, \dots, x_n]$ is nonsingular, and the geometric multiplicity of each $\lambda_i$ corresponding with $x_i$ equals one. And the equivalence of all solutions to the exact GJBD problem is established. Moreover, theoretical proof is given to show why the approximate GJBD problem can be solved similarly to the exact GJBD problem. Based on the theoretical results, a three-stage method is proposed and numerical results show the merits of the method. [**Key words.**]{} general joint block diagonalization, matrix polynomial, tensor decomposition. [**MSC.**]{} 15A69, 65F15. author: - 'Yunfeng Cai[^1] Guanghui Cheng[^2] Decai Shi[^3]' title: 'Solving General Joint Block Diagonalization Problem via Linearly Independent Eigenvectors of a Matrix Polynomial [^4]' --- Introduction {#sec:intro} ============ The problem of joint block diagonalization of matrices (also called simultaneously block diagonalization problem), is a particular decomposition of a third order tensor in block terms [@de2008decompositions], [@de2008decompositions2], [@de2008decompositions3], [@nion2011tensor]. Over the past two decades, such a decomposition has found many applications in independent component analysis (e.g., [@cardoso1998multidimensional], [@de2000fetal], [@theis2005blind], [@theis2006towards]) and semidefinite programming (e.g., [@bai2009exploiting], [@de2007reduction], [@de2010exploiting], [@gatermann2004symmetry]). For example, in blind source separation (BSS), people aim to recover source signals from the observed mixtures, without knowing either the distribution of the sources or the mixing process [@choi2005blind], [@comon2010handbook], [@hyvarinen2004independent]. Different assumptions on the source signals lead to different models and methods. Typically, there are three cases: first, the source signals are mutually statistically independent, the mixing system can be estimated by joint diagonalization (JD), e.g., JADE [@cardoso1993blind], eJADE [@moreau2001generalization], SOBI [@belouchrani1997blind] and Hessian ICA [@theis2004new], [@yeredor2000blind]; second, there are several groups of signals, in which components from different groups are mutually statistically independent and statistical dependence occurs between components in the same group (known as multidimensional BSS or group BSS), the mixing system can be estimated by joint block diagonalization (JBD) [@cardoso1998multidimensional], [@theis2005blind]; third, the number of groups and the size of each group are unknown, the mixing system can be estimated by general joint block diagonalization (GJBD). To proceed, in what follows we introduce some definitions and notations, then formulate the JBD problem and the GJBD problem mathematically. We call $\tau_n=(n_1,n_2,\dots,n_t)$ a [*partition*]{} of positive integer $n$ if $n_1,n_2,\dots,n_t$ are all positive integers and their sum is $n$, i.e., $\sum_{i=1}^t n_i=n$. The integer $t$ is called the [*cardinality*]{} of the partition $\tau_n$, denoted by $\operatorname{card}(\tau_n)$. The set of all partitions of $n$ is denoted by ${\mathbb{T}_n}$. Given a partition $\tau_n=(n_1,n_2,\dots,n_t)\in{\mathbb{T}_n}$, for any $n$-by-$n$ matrix $A$, define its block diagonal part and off-block diagonal part associated with $\tau_n$ as $$\begin{aligned} \operatorname{Bdiag}_{\tau_n}(A)=\operatorname{diag}(A_{11},A_{22},\dots,A_{tt}),\quad \operatorname{OffBdiag}_{\tau_n}(A)=A-\operatorname{Bdiag}_{\tau_n}(A),\end{aligned}$$ respectively, where $A_{ii}$ is $n_i$-by-$n_i$ for $i=1,2,\dots, t$. A matrix $A$ is referred to as a $\tau_n$-block diagonal matrix if $\operatorname{OffBdiag}_{\tau_n}(A)=0$. The set of all $\tau_n$-block diagonal matrices is denoted by ${\mathbb{D}_{\tau_n}}$. Let ${\mathbb{S}_n}$, ${\mathbb{H}_n}$, ${\mathbb{O}_n}$, ${\mathbb{U}_n}$, ${\mathbb{GL}(n,\mathbb{R})}$ and ${\mathbb{GL}(n,\mathbb{C})}$ denote the $n\times n$ matrix set of real symmetric matrix, complex Hermitian matrix, real orthogonal matrix, complex unitary matrix, real nonsingular matrix and complex nonsingular matrix, respectively. Let ${\mathbb{A}_n}={\mathbb{S}_n}$, ${\mathbb{H}_n}$, ${\mathbb{R}^{n\times n}}$, or ${\mathbb{C}^{n\times n}}$, ${\mathbb{W}_n}={\mathbb{O}_n}$, ${\mathbb{U}_n}$, ${\mathbb{GL}(n,\mathbb{R})}$, or ${\mathbb{GL}(n,\mathbb{C})}$. Then the JBD problem and the GJBD problem can be formulated as follows. **The JBD problem.** Given a matrix set $\mathcal{A}=\{A_i\}_{i=0}^p$ with $A_i\in{\mathbb{A}_n}$, and a partition $\tau_n=(n_1,n_2,\dots,n_t)\in{\mathbb{T}_n}$. Find a matrix $W=W(\tau_n)\in{\mathbb{W}_n}$ such that $W^{\star} A_i W\in{\mathbb{D}_{\tau_n}}$ for $i=0,1,\dots, p$, i.e., $$\label{eq:nujbd} W^{\star} A_i W=\operatorname{diag}(A_{i}^{(11)},A_i^{(22)},\dots, A_{i}^{(tt)}), \quad \mbox{for}\quad i=0,1,\dots, p,$$ where $A_{i}^{(jj)}$ is $n_j$-by-$n_j$ for $j=1,2,\dots,t$, the symbol $(\cdot)^{\star}$ stands for the transpose of a real matrix or the conjugate transpose of a complex matrix. **The GJBD problem.**Given a matrix set $\mathcal{A}=\{A_i\}_{i=0}^p$ with $A_i\in{\mathbb{A}_n}$. Find a partition $\tau_n'=(n_1',n_2',\dots,n_t')\in{\mathbb{T}_n}$ and a matrix $W=W(\tau_n')\in{\mathbb{W}_n}$ such that $$\operatorname{card}(\tau_n')=\max\{\operatorname{card}(\tau_n) \, \big|\, \mbox{there exists a $W=W(\tau_n)$ which solves JBD}\}.$$ The transformation matrix $W$ is often referred to as a [*diagonalizer*]{}. The (G)JBD problem is called symmetric/Hermitian (G)JBD if ${\mathbb{A}_n}={\mathbb{S}_n}/{\mathbb{H}_n}$; exact/approximate (G)JBD if is satisfied exactly/approximately; orthogonal/ non-orthogonal (G)JBD if ${\mathbb{A}_n}={\mathbb{R}^{n\times n}}$ and ${\mathbb{W}_n}={\mathbb{O}_n}/{\mathbb{GL}(n,\mathbb{R})}$; similarly, unitary/non-unitary (G)JBD if ${\mathbb{A}_n}={\mathbb{C}^{n\times n}}$ and ${\mathbb{W}_n}={\mathbb{U}_n}/{\mathbb{GL}(n,\mathbb{C})}$. In practical applications, the matrices $A_i$’s are usually constructed from empirical data, as a result, the exact JBD problem has no solutions and the exact GJBD problem has only trivial solution $((n), I_n)$. Consequently, the approximate (G)JBD problem is considered instead. For the JBD problem, it is natural to formulate it as a constrained optimization problem $C(W)=\min$, where $C(\cdot)$ is a cost function used to measure the off-block diagonal parts of $A_i$’s, $W$ is a diagonalizer in certain feasible set. Different cost functions and feasible sets together with various optimization methods lead to many numerical methods. Since this paper mainly concentrates on algebraic methods, we will not list the detailed literature on the optimization methods, we refer the readers to [@chabriel2014joint], [@de2009survey], [@tichavsky2017non] and references therein. For the GJBD problem, one needs to minimize the off-block diagonal parts of $A_i$’s and maximize the number of diagonal blocks simultaneously, it is difficult to formulate it as a simple optimization problem that can be easily solved. By assuming that the GJBD problem shares the same local minima with the JD problem, the GJBD problem is simply solved with a JD algorithm, followed by a permutation, which is used to reveal the block structure [@abed2004algorithms], [@theis2006towards]. Without good initial guesses, optimization methods may suffer from slow convergence, or converge to degenerate solutions. Algebraic methods, on the other hand, are able to find a solution in finite steps with predictable computational costs. And even if the solutions returned by algebraic methods are “low quality”, they are usually good initial guesses for optimization methods. In current literature, the algebraic methods for the GJBD problem fall into two categories: one is based on matrix $\ast$-algebra (see [@de2011numerical], [@maehara2010numerical], [@maehara2011algorithm], [@murota2010numerical], for the orthogonal GJBD problem and a recent generation [@cai2017algebraic] for the non-orthogonal GJBD problem), the other is based on matrix polynomial (see [@cai2015matrix] for the Hermitian GJBD problem). In the former category, a null space of a linear operator needs to be computed, which requires $\mathcal{O}(n^6)$ flops, thus, for problems with large $n$, such an approach will be quite expensive for both storage and computation. In this paper, we will focus on the latter category, which we will show later that it only requires $\mathcal{O}(n^3)$ flops. As the results in this paper is an extension of those in [@cai2015matrix], in what follows, we summarize some related results therein. For a Hermitian matrix set $\{A_i\}_{i=0}^p$, the corresponding matrix polynomial is constructed as ${P}_{\mathcal{A}}(\lambda)=\sum_{i=0}^p\lambda^i A_i$. Assuming that ${P}_{\mathcal{A}}(\lambda)$ is regular and has only simple eigenvalues, and using the spectral decomposition of the Hermitian matrix polynomial, theoretically, it is shown that the column vectors of the diagonalizer of the exact Hermitian GJBD problem of $\{A_i\}_{i=0}^p$ can be given by $n$ linearly independent eigenvectors (in a certain order) of ${P}_{\mathcal{A}}(\lambda)$ ([@cai2015matrix Corollary 3.5]); all solutions to the Hermitian GJBD problem are equivalent, i.e., all solutions are unique up to block permutations and block diagonal transformations ([@cai2015matrix Theorem 3.8]). Therefore, one can solve the Hermitian GJBD problem by finding $n$ linearly independent eigenvectors $x_1, x_2, \dots, x_n$ of ${P}_{\mathcal{A}}(\lambda)$, followed by determining a permutation $\Pi$ via revealing the block diagonal structure of $\Pi^{\operatorname{T}}[x_1, \dots, x_n]^{\operatorname{H}}A_i [x_1, \dots, x_n]\Pi$ (MPSA-II). Numerically, it is shown that MPSA-II, though designed to solve the exact Hermitian GJBD problem, is able to deal with the approximate Hermitian GJBD problem to some extent. However, the approach in [@cai2015matrix] suffers from the following three disadvantages: first, the proofs are difficult to follow if the readers are unfamiliar with the spectral decomposition of a matrix polynomial; second, there is no theoretical proof to show why the approach is applicable for the approximate Hermitian GJBD problem; third, the approach can not be used to solve general ($A_i$’s are not necessarily Hermitian) GJBD problem directly. [^5] In this paper, we try to give a remedy. For a matrix set $\mathcal{A}=\{A_i\}_{i=0}^p$, we still construct the matrix polynomial as $P_{\mathcal{A}}(\lambda)=\sum_{i=0}^p\lambda^i A_i$. Let $x_1, x_2, \dots, x_n$ be $n$ linearly independent eigenvectors of ${P}_{\mathcal{A}}(\lambda)$. Under the assumption that the geometric multiplicities of the corresponding eigenvalues equal one (much weaker than that “${P}_{\mathcal{A}}(\lambda)$ is regular and all eigenvalues are simple”), we show that the diagonalizer of the exact GJBD problem can be written as $X\Pi=[x_1, x_2, \dots, x_n]\Pi$, where $\Pi$ is a permutation matrix; all solutions to the exact GJBD problem are equivalent. The proofs of these results are easy to follow, without using the spectral decomposition of a matrix polynomial. Furthermore, using perturbation theory, we give a theoretical proof for using $X\Pi$ as the diagonalizer for the approximate GJBD problem. Lastly, a three-stage method, which is modified from MPAS-II in [@cai2015matrix], is proposed. Numerical examples show that the proposed method is effective and efficient. The rest of the paper is organized as follows. In section \[sec:pre\], we give some preliminary results on matrix polynomials and motivations for using a matrix polynomial to solve the GJBD problem. In section \[sec:theory\], the main results are presented. Numerical method and numerical examples are given in sections \[sec:method\] and \[sec:eg\], respectively. Finally, we present some concluding remarks in section \[sec:conclusion\]. [**Notations.**]{}The imaginary unit $\sqrt{-1}$ is denoted by $\imath$. For a matrix $A=[a_{ij}]$, $|A|$, $\|A\|_2$ and $\|A\|_F$ denote $[|a_{ij}|]$, the 2-norm and Frobinius norm, respectively. The eigenvalue sets of a square matrix $A$ and a matrix polynomial $P(\lambda)$ are denoted by $\lambda(A)$ and $\lambda(P)$, respectively. The MATLAB convention is adopted to access the entries of vectors and matrices. The set of integers from $i$ to $j$ inclusive is $i : j$. For a matrix $A$, its submatrices $A{(k:\ell,i:j)}$, $A{(k:\ell,:)}$, $A{(:,i:j)}$ consist of intersections of row $k$ to row $\ell$ and column $i$ to column $j$, row $k$ to row $\ell$ and all columns, all rows and column $i$ to column $j$, respectively. Preliminary and motivation {#sec:pre} ========================== A matrix polynomial of degree $p$ is defined as $$\begin{aligned} \label{mpp} P(\lambda)\triangleq\lambda^p A_p+\lambda^{p-1}A_{p-1}+\dots+A_0,\end{aligned}$$ where the coefficient matrices $A_i$’s are all $n$-by-$n$ matrices and $A_p\ne 0$. A scalar $\lambda$ is called an eigenvalue of $P(\lambda)$ if $\det(P(\lambda))=0$. A nonzero vector is called the corresponding eigenvector if $P(\lambda)x=0$. Such $\lambda$ together with $x$ are called an eigenpair of $P(\lambda)$, denoted by $(\lambda,x)$. The polynomial eigenvalue problem (PEP) $P(\lambda)x=0$ is equivalent to a generalized eigenvalue problem (GEP) $(\lambda M + N)u=0$, where $$\begin{aligned} \label{uxl} u=u(x,\lambda)&\triangleq [x^{\operatorname{T}}, \lambda x^{\operatorname{T}}, \dots, \lambda^{p-1}x^{\operatorname{T}}]^{\operatorname{T}},\end{aligned}$$ $L$ and $M$ are some $np$-by-$np$ matrices. Such a transformation is called [*linearization*]{}. Linearizations are not unique, among which, the commonly used one can be given by $$\begin{aligned} \label{mn} M=\begin{bmatrix} I& 0 &0&\cdots&0\\ 0&I & 0 &\cdots&0\\ \vdots&\ddots&\ddots&\ddots&\vdots\\ 0 & \cdots & 0& I& 0\\ 0 & \cdots & 0 &0 & A_p \end{bmatrix},\quad N=\begin{bmatrix} 0&-I&0&\cdots&0\\ 0&0&-I&\ddots&\vdots\\ \vdots&\vdots&\ddots&\ddots&0\\ 0 & 0& \cdots& 0 & -I\\ A_0& A_1 &\cdots& A_{p-2}& A_{p-1} \end{bmatrix}.\end{aligned}$$ For more linearizations and some structure preserving linearizations for structured matrix polynomials, we refer the readers to [@higham2006symmetric], [@mackey2006vector], [@mehrmann2002polynomial]. The eigenvalues and eigenvectors of PEP can be obtained via those of GEP, vice versa. At first glance, it seems that the matrix polynomial is not related to the GJBD problem at all. But in fact, they are closely related. Let us consider the GJBD problem of three matrices $\{A_0,A_1,A_2\}$ with $$\begin{aligned} A_0=\begin{bmatrix} 7 &8 & 9\\ 4 & -12 & -8\\ 5 & -4 & 7\end{bmatrix},\quad A_1=\begin{bmatrix} -8 & 8 & 8\\ -4 & 4 & 0\\ -4 & 12 & 0 \end{bmatrix},\quad A_2=\begin{bmatrix} 5 & 0 & 3\\ -8 & 4 & -4\\ -5 & 4 & 1 \end{bmatrix}.\end{aligned}$$ The eigenvector matrix $X$ and the eigenvalue matrix $T$ of the PEP $ (\lambda^2 A_2 +\lambda A_1 +A_0)x=0$ can be given by $$\begin{aligned} X&=[x_1, x_2, \dots, x_6]\\ &=\begin{bmatrix} -0.1690 & -0.5774 & 0.3981 + 0.4094\imath & 0.3981 - 0.4094\imath & -0.5774 & 0.4904 \\ -0.9710& -0.5774 & 0.1108 + 0.5792\imath & 0.1108 - 0.5792\imath & -0.5774 & -0.7205\\ -0.1690& 0.5774& 0.3981 + 0.4094\imath &0.3981 - 0.4094\imath & 0.5774 & 0.4904\\ \end{bmatrix},\\ T&=\operatorname{diag}(\lambda_1,\lambda_2,\dots,\lambda_6)\\ &=\operatorname{diag}( -3.5830, 3.0000, -0.5283 + 1.3793\imath, -0.5283 - 1.3793\imath,1.0000, 0.6396).\end{aligned}$$ Let $W=X{(:,[2, 1, 6])}$, then $$\begin{aligned} W^{\operatorname{T}} A_0 W&=\begin{bmatrix} 4.0000 & -0.0000 & 0.0000\\ -0.0000 & -10.5142 & -8.3243\\ 0.0000 & -13.1077 & 0.5034\end{bmatrix},\\ W^{\operatorname{T}} A_1 W&= \begin{bmatrix} -5.3333 & -0.0000 & -0.0000\\ 0.0000 & 6.2833 &7.4702\\ -0.0000 & -6.8800 & -4.5381\end{bmatrix},\\ W^{\operatorname{T}} A_2 W&=\begin{bmatrix} 1.3333 & -0.0000 & 0.0000\\ 0.0000 & 2.5726 & 8.6677\\ 0.0000 & -0.8992 & 5.8646\end{bmatrix}.\end{aligned}$$ By calculations, we can show that the $2$-by-$2$ blocks in $W^{\operatorname{T}}A_0W$, $W^{\operatorname{T}}A_1W$ and $W^{\operatorname{T}}A_2W$ can not be simultaneously diagonalized. Therefore, $((1,2),W)$ is a solution to the GJBD problem of $\{A_0, A_1, A_2\}$. This example shows that the GJBD problem can be indeed solved via linearly independent eigenvectors of a matrix polynomial, in the proceeding sections, we will present the theoretical proofs. Given a matrix set $\mathcal{A}=\{A_i\}_{i=0}^p$, the matrix polynomial $P_{\mathcal{A}}(\lambda)$ is constructed in a particular order of the matrices in $\mathcal{A}$. However, the matrices are not ordered in any way for the GJBD problem of $\mathcal{A}$. Later, we will see that the results in this paper do not depend on such an order. So it suffices to show the results for only one matrix polynomial, say $P_{\mathcal{A}}(\lambda)$. Main results {#sec:theory} ============ In this section, we give our main results for the exact and approximate GJBD problems in subsections \[subsec:exact\] and \[subsec:approx\], respectively. On exact GJBD problem {#subsec:exact} --------------------- In this subsection, we first characterize the diagonalizer $W$, then show the equivalence of the solutions. \[thm1\] Given a matrix set $\mathcal{A}=\{A_i\}_{i=0}^p$. Let $X=[x_1,x_2,\dots, x_n]$, $T=\operatorname{diag}(\lambda_1,\lambda_2,\dots, \lambda_n)$, where $(\lambda_j,x_j)$ for $j=1,2,\dots, n$ are $n$ eigenpairs of $P_{\mathcal{A}}(\lambda)=\sum_{i=0}^{p}\lambda^i A_i$. Assume that $X$ is nonsingular and the geometric multiplicities of $\lambda_1,\lambda_2,\dots, \lambda_n$ all equal one. If $(\tau_n,W)$ solves the GJBD problem, then there exist a permutation matrix $\Pi$ and a nonsingular matrix $D\in{\mathbb{D}_{\tau_n}}$ such that $WD=X\Pi$, i.e., $({\tau}_n,X\Pi)$ also solves the GJBD problem. As $(\tau_n,W)$ is a solution to the GJBD problem, we have $$\begin{aligned} \label{eqB} D_{i}\triangleq W^{\operatorname{H}}A_iW=\operatorname{diag}(A_{i}^{(11)},\dots,A_{i}^{(tt)})\in{\mathbb{D}_{\tau_n}}\mbox{ for } i=0,1,\dots,p.\end{aligned}$$ Since $(\lambda_j, x_j)$ for $j=1,2,\dots,n$ are eigenpairs of $P_{\mathcal{A}}(\lambda)$, we also have $$\begin{aligned} \label{matpolyeq} A_p XT^{p} +A_{p-1}XT^{p-1}+\dots + A_0 X=0.\end{aligned}$$ Pre-multiplying by $W^{\operatorname{H}}$ and using , we get $$\begin{aligned} \label{dyt} D_{p} W^{-1}XT^{p}+D_{p-1}W^{-1}XT^{p-1}+\dots +D_0W^{-1}X=0.\end{aligned}$$ Let $Y=W^{-1}X=[y_1,y_2,\dots, y_n]$, then $(\lambda_j,y_j)$ for $j=1,2,\dots, n$ are eigenpairs of $P_{\mathcal{D}}(\lambda)=\sum_{i=0}^p \lambda^i D_i$, and the geometric multiplicities of $\lambda_j$’s, as eigenvalues of $P(\mathcal{D})$, all equal one. Denote $\mathcal{A}_j=\{A_i^{(jj)}\}_{i=0}^p$, we know that for each $\lambda_j$, it belongs to a unique $\lambda({P}_{\mathcal{A}_k})$ since $\lambda(P_{\mathcal{D}})=\cup_{j=1}^t\lambda({P}_{\mathcal{A}_j})$, and the geometric multiplicities of $\lambda_j$’s all equal one. Let $n_1',n_2',\dots, n_t'$ be the numbers of $\lambda_j$’s in $\lambda({P}_{\mathcal{A}_1}),\lambda({P}_{\mathcal{A}_2}),\dots,\lambda({P}_{\mathcal{A}_t})$, respectively. Then there exists a permutation matrix $\Pi$ such that $$W^{-1}X\Pi={\left[\begin{smallmatrix} }{Y}_{11} & \dots &{Y}_{1t}\\ \vdots& \ddots &\vdots\\ {Y}_{t1} &\dots & {Y}_{tt}{\end{smallmatrix}\right] },\quad \Pi^{\operatorname{T}}T\Pi=\operatorname{diag}({T}_1,\dots,{T}_t),$$ where ${Y}_{jk}\in\mathbb{C}^{n_j\times n_k'}$ for $1\le j,k\le t$, $T_j\in\mathbb{C}^{n_j' \times n_j'}$, $\lambda(T_j)\subset \lambda({P}_{\mathcal{A}_j})$ for $j=1,2,\dots,t$. The assumption that the geometric multiplicities of $\lambda_1,\lambda_2,\dots, \lambda_n$ all equal one implies that $\lambda(T_j)\cap\lambda(\mathcal{P}_{\mathcal{A}_k})=\emptyset$ for $j\ne k$, therefore, by , we have $Y_{jk}=0$ for $j\ne k$. Thus, for any $1\le k\le t$, it holds that $$\begin{aligned} n&=\operatorname{rank}(X\Pi)=\operatorname{rank}(Y_{kk})+\operatorname{rank}(\operatorname{diag}(Y_{11},\dots,Y_{k-1,k-1},Y_{k+1,k+1},\dots,Y_{tt}))\\ &\le\min\{n_k,n_k'\}+\min\{n-n_k,n-n_k'\}\\ &=\min\{n_k,n_k'\}+n-\max\{n_k,n_k'\}\le n.\end{aligned}$$ Then it follows that $\min\{n_k,n_k'\}=\max\{n_k,n_k'\}$, and hence $n_k=n_k'$. Thus, $D\triangleq W^{-1}X\Pi\in{\mathbb{D}_{\tau_n}}$ is a nonsingular $\tau_n$-block diagonal matrix. The conclusion follows. Based on Theorem \[thm1\], we can solve the exact GJBD problem by finding $n$ linearly independent eigenvectors $X$ of $P_{\mathcal{A}}(\lambda)$, then determining a permutation $\Pi$ by revealing the block structure of $\Pi^{\operatorname{T}}X^{\operatorname{H}} A_i X \Pi$. We will discuss the details in section \[sec:method\]. Let $\tau_n=(n_1,n_2,\dots,n_t)\in{\mathbb{T}_n}$, $(\tau_n, W)$ be a solution to the GJBD problem, then $(\tau_n \Pi_t, W D \Pi)$ also solves the GJBD problem, where $\Pi_t\in\mathbb{R}^{t\times t}$ is a permutation matrix, $\Pi\in{\mathbb{R}^{n\times n}}$ is a block permutation matrix corresponding with $\Pi_t$, which can be obtained by replacing the “1” and “0” elements in $i$th row of $\Pi_t$ by a permutation matrix of order $n_i$ and zero matrices of right sizes, respectively, $D\in{\mathbb{D}_{\tau_n}}$ is nonsingular. For two solutions $(\tau_n, W)$, $(\hat{\tau}_n,\widehat{W})$ to the GJBD problem, we say that $(\hat{\tau}_n,\widehat{W})$ is [*equivalent*]{} to $(\tau_n, W)$ if there exist a permutation matrix $\Pi_t$ and a nonsingular matrix $D\in{\mathbb{D}_{\tau_n}}$ such that $(\hat{\tau}_n,\widehat{W})=(\tau_n \Pi_t, W D \Pi)$, where $\Pi$ is the block permutation matrix corresponding with $\Pi_t$. Next, we show that all solutions to the GJBD problem are equivalent, under mild conditions. \[thm2\] Given a matrix set $\mathcal{A}=\{A_i\}_{i=0}^p$. Let $\mathcal{B}=\{B_i\}_{i=0}^q$ be a matrix set such that $\operatorname{span}\{B_0,B_1,\dots,B_q\}=\operatorname{span}\{A_0,A_1,\dots,A_p\}$, [^6] $P_{\mathcal{B}}(\lambda)=\sum_{i=0}^q \lambda^i B_i$. If $P_{\mathcal{B}}(\lambda)$ has $n$ eigenpairs $(\lambda_j,x_j)$ for $j=1,2,\dots, n$ such that $X=[x_1,x_2,\dots, x_n]$ is nonsingular and the geometric multiplicities of $\lambda_1,\lambda_2,\dots, \lambda_n$ all equal one, then all solutions to the GJBD problem of $\mathcal{A}$ are equivalent. First, by Theorem \[thm1\], for any two solutions $(\tau_n,W)$, $(\hat{\tau}_n,\widehat{W})$ to the GJBD problem of $\mathcal{B}$, there exist two permutation matrices $\Pi$ and $\widehat{\Pi}$ and two nonsingular matrices $D\in{\mathbb{D}_{\tau_n}}$ and $\widehat{D}\in\mathbb{D}_{\hat{\tau}_n}$ such that $WD=X\Pi$ and $\widehat{W}\widehat{D}=X\widehat{\Pi}$. Let $H=\sum_{i=0}^q |X^{\operatorname{H}}B_iX|$, $\widetilde{H}=[\tilde{h}_{ij}]$ with $$\begin{aligned} \tilde{h}_{ij}= \begin{cases} 1,\quad \mbox{if $i\ne j$ and $h_{ij}\ne 0$},\\ 0,\quad \mbox{otherwise,} \end{cases}\end{aligned}$$ where $h_{ij}$ is the $(i,j)$ entry of $H$. On one hand, using the fact that $(\tau_n,W)$ and $(\hat{\tau}_n,\widehat{W})$ are both solutions to the GJBD problem, we know that $\Pi^{\operatorname{T}}\widetilde{H}\Pi\in{\mathbb{D}_{\tau_n}}$ and $\widehat{\Pi}^{\operatorname{T}}\widetilde{H}\widehat{\Pi}\in\mathbb{D}_{\hat{\tau}_n}$, and $\operatorname{card}(\tau_n)=\operatorname{card}(\hat{\tau}_n)$. On the other hand, notice that $\widetilde{H}$ is a vertex-adjacency matrix of a graph $G$, and $G$ has $t$ components $G_1,G_2,\dots,G_t$, and $G_i$ is connected with $n_i$ vertices for $i=1,2,\dots,t$, where $n_i$ is the $i$th entry of $\tau_n$. No matter how we relabel the vertices of $G$, the collection of the numbers of vertices of all connected components of $G$ remains invariant. Therefore, there exists a permutation matrix $\Pi_t$ of order $t$ such that $\hat{\tau}_n=\tau_n\Pi_t$ and $\widetilde{\Pi}=\Pi^{\operatorname{T}}\widehat{\Pi}$ is the block permutation matrix corresponding with $\Pi_t$. Then it follows that $$\widehat{W}=X\widehat{\Pi}\widehat{D}^{-1}=(X\Pi D^{-1}) D (\Pi^{\operatorname{T}}\widehat{\Pi})\widehat{D}^{-1} =WD\widetilde{\Pi}\widehat{D}^{-1}=WD(\widetilde{\Pi}\widehat{D}^{-1}\widetilde{\Pi}^{\operatorname{T}})\widetilde{\Pi}.$$ In other words, all solutions to the GJBD problem of $\mathcal{B}$ are equivalent. Second, notice that for a partition $\tau_n$, $W$ is a diagonalizer of the JBD problem of $\mathcal{A}$ if and only if $W$ is a diagonalizer of the JBD problem of $\mathcal{B}$, since $\operatorname{span}\{B_0,B_1,\dots,B_q\}=\operatorname{span}\{A_0,A_1,\dots,A_p\}$. The conclusion follows immediately. Theorem \[thm2\] implies that the solution to the GJBD problem does not depend on the choices of the matrix polynomials, neither the choices of $n$ linearly independent eigenvectors. On approximate GJBD problem {#subsec:approx} --------------------------- In this subsection, we show that the solution to the approximate GJBD problem can also be written in the form $(\tau_n, X\Pi)$, where $\tau_n$ is some partition of $n$, $X$ is a nonsingular matrix whose columns are eigenvectors of $P_{\mathcal{A}}(\lambda)$, $\Pi$ is some permutation matrix. The following two lemmas are needed for the proof. \[lem2\] Let $x$, $y\in\mathbb{C}^n$ be two nonzero vectors, then $$\min_{t\in\mathbb{C}}\|x-t y\|_2=\|x\|_2\sin\angle(x,y),$$ where $\angle(x,y)$ is the angle between the subspaces spanned by $x$ and $y$, respectively. The proof of Lemma \[lem2\] is simple, we will omit it here. The following lemma is rewritten from Theorem 4.1 in [@yuji2016eigenvector]. \[lem:yuji\] Let $(\hat{\lambda}, \hat{x})$ be an approximate eigenpair of $P(\lambda)=\sum_{i=0}^p\lambda^i A_i$ with residual $P(\hat{\lambda})\hat{x}\ne 0$. Suppose $L(\lambda)=\lambda M + N$ is a linearization of $P(\lambda)$ given by , and $L(\lambda)$ has a generalized Schur form $$\begin{aligned} Q^{\operatorname{H}}MZ=\begin{bmatrix}\alpha_1 & \ast \\ 0 & M_1\end{bmatrix}, \quad Q^{\operatorname{H}}NZ=\begin{bmatrix}\beta_1 & \ast \\ 0 & N_1\end{bmatrix}, \quad \lambda_1=-\frac{\beta_1}{\alpha_1},\end{aligned}$$ in which $M_1$, $N_1$ are both upper-triangular. Then the eigenvector $x_1$ of $P(\lambda)$ corresponding with $\lambda_1$ satisfies $$\begin{aligned} \label{sin1} \sin\angle(\hat{x􏰋},x_1) &\le \frac{g\|P(\hat{\lambda})\hat{x}\|_2}{\sqrt{\sum_{i=0}^{p-1}|\hat{\lambda}|^{2i}}\|\hat{x}\|_2}, $$ where $$\begin{aligned} \label{gap} g=g(\lambda_1,\hat{\lambda}; P(\lambda))\triangleq\frac{1}{\|(\hat{\lambda}M_1+N_1)^{-1}\|_2}.\end{aligned}$$ Define a matrix set $\mathcal{W}$ as $$\begin{aligned} \label{setw} \mathcal{W}\triangleq \{W=[w_1,w_2,\dots, w_n]\; \big| \; \det(W)\ne 0, \, \|w_i\|_2=1 \mbox{ for } i=1,2,\dots, n\}.\end{aligned}$$ Now we are ready to present the third main theorem. \[thm3\] Given a matrix set $\mathcal{A}=\{A_i\}_{i=0}^p$. For a partition $\tau_n=(n_1,\dots,n_t)$, assume that there exists a matrix $W\in\mathcal{W}$ such that $$\begin{aligned} \label{offbwaw} \|E_i\|_2\le \mu\|W^{\operatorname{H}}A_i W\|_2, $$ where $E_i=\operatorname{OffBdiag}_{\tau_n}(W^{\operatorname{H}} A_iW)$, $\mu\ge 0$ is a parameter. Let $(\lambda_j,x_j)$ for $j=1,2,\dots, n$ be $n$ eigenpairs of $P_{\mathcal{A}}(\lambda)=\sum_{i=0}^p \lambda^i A_i$ and $X=[x_1,x_2,\dots, x_n]\in\mathcal{W}$. Let $D_i=\operatorname{Bdiag}_{\tau_n}(W^{\operatorname{H}}A_iW)$, $\mathcal{D}=\{D_i\}_{i=0}^p$, $P_\mathcal{D}(\lambda)=\sum_{i=0}^p\lambda^i D_i$, $\mu_j=\mbox{argmin}_{\mu\in\lambda(P_\mathcal{D})} g(\mu, \lambda_j; P_\mathcal{D}(\lambda))$, $y_j$ be the eigenvector of $P_\mathcal{D}(\lambda)$ corresponding with $\mu_j$, for $j=1,2,\dots, n$. Further assume that the geometric multiplicities of $\mu_j$’s all equal one, $Y=[y_1,y_2,\dots, y_n]$ is nonsingular and $(\tau_n, I_n)$ is a solution to the exact GJBD problem of $\mathcal{D}$. Denote $$\begin{aligned} g_j&= g(\mu_j, \lambda_j; P_\mathcal{D}(\lambda)),\quad \mbox{for } j=1,2,\dots, n,\\ \eta &= \max_{1\le j\le n} \frac{ g_j\sum_{i=0}^p |\lambda_j|^i \|W^{\operatorname{H}}A_iW\|_2}{\sqrt{\sum_{i=0}^{p-1} |\lambda_j|^{2i}}}.\end{aligned}$$ Then there exists a permutation matrix $\Pi$ such that $$\begin{aligned} \label{agjbd} \|\operatorname{OffBdiag}_{\tau_n}(\Pi^{\operatorname{T}}X^{\operatorname{H}} A_i X\Pi)\|_2\le C\mu\|X^{\operatorname{H}} A_i X\|_2,\end{aligned}$$ where $$C= \kappa_2(W^{-1}X)\left[\sqrt{n}\eta+\kappa_2(W^{-1}X)(1+\sqrt{n}\mu\eta)(1+\sqrt{n}\mu\eta+\sqrt{n}\eta)\right].$$ First, using the assumption that $(\tau_n,I_n)$ is a solution to the exact GJBD problem of $\mathcal{D}$, together with Theorem \[thm1\], we know that there exists a permutation matrix $\Pi$ such that $Y\Pi\in{\mathbb{D}_{\tau_n}}$. Second, for each $1\le j\le n$, take $(\lambda_j, W^{-1}x_j)$ as an approximate eigenpair of $P_\mathcal{D}(\lambda)$. Using Lemma \[lem:yuji\], we have $$\begin{aligned} \label{sinwyx} \sin\angle(W^{-1}x_j, y_j)\le \frac{g_j\|P_\mathcal{D}(\lambda_j)W^{-1}x_j\|_2}{\sqrt{\sum_{i=0}^{p-1} |\lambda_j|^{2i}}\|W^{-1}x_j\|_2}.\end{aligned}$$ Let $\mathcal{E}=\{E_i\}_{i=0}^p$, $P_{\mathcal{E}}(\lambda)=\sum_{i=0}^p\lambda^i E_i$, it holds that $P_\mathcal{D}(\lambda)+P_{\mathcal{E}}(\lambda)=W^{\operatorname{H}}P_{\mathcal{A}}(\lambda)W$. Then it follows from and that $$\begin{aligned} \sin\angle(W^{-1}x_j, y_j) &\le \frac{g_j(\|W^{\operatorname{H}}P_{\mathcal{A}}(\lambda_j)x_j\|_2+\|P_{\mathcal{E}}(\lambda_j) W^{-1}x_j\|_2)} {\sqrt{\sum_{i=0}^{p-1} |\lambda_j|^{2i}}\|W^{-1}x_j\|_2}\\ &\le \frac{g_j\|P_{\mathcal{E}}(\lambda_j)\|_2 }{\sqrt{\sum_{i=0}^{p-1} |\lambda_j|^{2i}}} \le \frac{g_j\sum_{i=0}^p |\lambda_j|^i \|E_i\|_2 }{\sqrt{\sum_{i=0}^{p-1} |\lambda_j|^{2i}}}\\ &\le \frac{\mu g_j\sum_{i=0}^p |\lambda_j|^i \|W^{\operatorname{H}}A_iW\|_2}{\sqrt{\sum_{i=0}^{p-1} |\lambda_j|^{2i}}}\le \mu\eta.\end{aligned}$$ By Lemma \[lem2\], there exists a $t_j\in\mathbb{C}$ such that $$\|W^{-1}x_j-t_jy_j\|_2=\|W^{-1}x_j\|_2 \sin\angle(W^{-1}x_j,y_j).$$ Let $f_j=W^{-1}x_j-t_jy_j$, then it holds that $\|f_j\|_2\le \mu\eta \|W^{-1}x_j\|_2$. Now denote $\widehat{Y}=[t_1y_1,\dots, t_ny_n]$, $F=[f_1,\dots, f_n]$, we have $X=W(\widehat{Y}+F)$ and $$\label{fbound} \|F\|_2\le\|F\|_F\le \mu\eta\|W^{-1}X\|_F\le \sqrt{n}\mu\eta\|W^{-1}X\|_2.$$ Finally, direct calculations give rise to $$\begin{aligned} &\mbox{}\quad\,\, \|\operatorname{OffBdiag}_{\tau_n}(\Pi^{\operatorname{T}}X^{\operatorname{H}} A_i X\Pi)\|_2\notag\\ &=\|\operatorname{OffBdiag}_{\tau_n}(\Pi^{\operatorname{T}}X^{\operatorname{H}}A_iX\Pi - \Pi^{\operatorname{T}}\widehat{Y}^{\operatorname{H}} D_i\widehat{Y}\Pi)\|_2 \label{xax0}\\ &=\|\operatorname{OffBdiag}_{\tau_n}(\Pi^{\operatorname{T}}X^{\operatorname{H}}A_iX\Pi- \Pi^{\operatorname{T}}\widehat{Y}^{\operatorname{H}}W^{\operatorname{H}}A_iW\widehat{Y}\Pi+\Pi^{\operatorname{T}} \widehat{Y}^{\operatorname{H}}E_i\widehat{Y}\Pi)\|_2 \label{xax1}\\ &\le \|X^{\operatorname{H}}A_iX- \widehat{Y}^{\operatorname{H}}W^{\operatorname{H}}A_iW\widehat{Y}+\widehat{Y}^{\operatorname{H}}E_i\widehat{Y}\|_2\notag\\ &=\| \widehat{Y}^{\operatorname{H}}W^{\operatorname{H}}A_iWF + F^{\operatorname{H}}W^{\operatorname{H}}A_iX+\widehat{Y}^{\operatorname{H}}E_i\widehat{Y}\|_2\label{xax2}\\ &\le \|\widehat{Y}\|_2 \|F\|_2\|W^{\operatorname{H}}A_iW\|_2+ \|X^{-1}W\|_2 \|F\|_2 \|X^{\operatorname{H}}A_iX\|_2+\|\widehat{Y}\|_2^2\|E_i\|_2\notag\\ &\le (\|W^{-1}X-F\|_2\|X^{-1}W\|_2^2 +\|X^{-1}W\|_2)\|F\|_2 \|X^{\operatorname{H}}A_iX\|_2\notag\\ &\mbox{}\hspace{5.7cm}+\mu\|W^{\operatorname{H}}A_iW\|_2\|W^{-1}X-F\|_2^2 \label{xax3}\\ &\le [1+\kappa_2(W^{-1}X)(1+\sqrt{n}\mu\eta)]\kappa_2(W^{-1}X)\sqrt{n}\mu\eta\|X^{\operatorname{H}}A_iX\|_2\notag\\ &\mbox{}\hspace{4.5cm}+\mu(1+\sqrt{n}\mu\eta)^2\kappa_2^2(W^{-1}X)\|X^{\operatorname{H}}A_iX\|_2 \notag\\ &=\mu C\|X^{\operatorname{H}}A_iX\|_2,\notag\end{aligned}$$ where uses $\operatorname{OffBdiag}_{\tau_n}(\Pi^{\operatorname{H}}\widehat{Y}^{\operatorname{H}} D_i\widehat{Y}\Pi)=0$ since $\widehat{Y}\Pi\in{\mathbb{D}_{\tau_n}}$, uses $D_i=W^{\operatorname{H}}A_iW-E_i$, uses $X=W(\widehat{Y}+F)$, uses $\widehat{Y}=W^{-1}X-F$ and . This completes the proof. Several remarks follow. \[rem1\] The assumption that “$(\tau_n, I_n)$ is a solution to the exact GJBD problem of $\mathcal{D}$ ” is equivalent to say that “$\mathcal{D}$ can not be further block diagonalized”. \[rem2\] If $\mu=0$, the approximate GJBD problem becomes the exact GJBD problem. The conclusion in Theorem \[thm3\] agrees with that in Theorem \[thm1\]. \[rem3\] The inequality implies that $X\Pi$ is a “sub-optimal” diagonalizer. The constant $C$ plays a crucial role in bounding the off-block diagonal part of $\Pi^{\operatorname{T}}X^{\operatorname{H}}A_iX\Pi$. Notice that the condition number of $W^{-1}X$ dominates the value of $C$. When $W$ and $X$ are good conditioned, $\kappa_2(W^{-1}X)$ is small. However, if $W$ or $X$ is ill-conditioned, $\kappa_2(W^{-1}X)$ can be quite large, which means that $X\Pi$ can be “low-quality”. In fact, from the perturbation theory of JBD problem [@cai2017perturbation] (see also perturbation theory of JD problem in [@shi2015some], [@afsari2008sensitivity]), the diagonalizer is sensitive to the perturbation when it is ill-conditioned. Therefore, it is not surprising to draw the conclusion that $C$ can be large when $W$ or $X$ is ill-conditioned. Numerical Method {#sec:method} ================ According to Theorems \[thm1\] and \[thm3\], solutions to the exact/approximate GJBD problem can be obtained by the following three-stage procedure: Stage 1 – eigenproblem solving stage. : Compute $n$ linearly independent eigenvectors $x_1,\ldots,x_n$ of the matrix polynomial $P_{\mathcal{A}}(\lambda)$, and let ${X}=[x_1, \dots, x_n]$; Stage 2 – block structure revealing stage. : Determine $\tau_n$ and a permutation matrix $\Pi$ such that $\Pi^{\operatorname{T}}{X}^{\operatorname{H}} A_i{X}\Pi$ for $i=0,1,\ldots,p$ are all approximately $\tau_n$-block diagonal and $\operatorname{card}(\tau_n)$ is maximized. Stage 3 – refinement stage. : Refine $X\Pi$ to improve the quality of the diagonalizer. We call the above three-stage method [*a partial eigenvector approach with refinement (PEAR) method*]{} for the GJBD problem. Next, we will discuss the implement details of each stage of the above procedure. Stage 1. -------- In this stage, we compute $n$ linearly independent eigenvectors of PEP $P_{\mathcal{A}}(\lambda)$. By [@pereira2003solvents Section 3], the number of choices for $n$ linearly independent eigenvectors is no less than $p$. How shall we make such a choice? To answer this question, let us consider the following GJBD problem first. \[eg1\] Let $A_i=VD_iV^{\operatorname{H}}$ for $ i=0, 1, 2$, where $$\begin{aligned} V&=\begin{bmatrix} 0.5377 + 2.7694\imath & 0.8622 + 0.7254\imath & -0.4336 - 0.2050\imath \\ 1.8339 - 1.3499\imath & 0.3188 - 0.0631\imath & 0.3426 - 0.1241\imath \\ -2.2588 + 3.0349\imath & -1.3077 + 0.7147\imath & 3.5784 + 1.4897\imath \\ \end{bmatrix},\\ D_0&= \begin{bmatrix} 0.2939 - 0.7873\imath & 0.0137 + 0.0044\imath & -0.0171 + 0.0038\imath \\ 0.0032 - 0.0086\imath & 0.8884 - 2.9443\imath & -1.0689 + 0.3252\imath \\ 0.0031 - 0.0003\imath & -1.1471 + 1.4384\imath & -0.8095 - 0.7549\imath \\ \end{bmatrix},\\ D_1&= \begin{bmatrix} -0.1649 + 0.6277\imath & -0.0077 + 0.0022\imath & 0.0037 + 0.0075\imath \\ -0.0109 + 0.0055\imath & 1.0933 - 1.2141\imath & -0.8637 - 0.0068\imath \\ 0.0003 + 0.0110\imath & 1.1093 - 1.1135\imath & 0.0774 + 1.5326\imath \\ \end{bmatrix},\\ D_2&= \begin{bmatrix} 1.5442 + 0.0859\imath & -0.0076 + 0.0025\imath & -0.0140 + 0.0062\imath \\ -0.0018 + 0.0142\imath & -1.4916 - 0.6156\imath & -1.0616 - 0.1924\imath \\ -0.0020 + 0.0029\imath & -0.7423 + 0.7481\imath & 2.3505 + 0.8886\imath \\ \end{bmatrix}.\end{aligned}$$ Let $W=V^{-\operatorname{H}}$, then we know that $(\tau_n, W)$ is a solution to the approximate GJBD problem of $\{A_0,A_1,A_2\}$, where $\tau_n=(1,2)$. By calculations, we know that all eigenvalues of the PEP $P_{\mathcal{A}}(\lambda)=\lambda^2 A_2+\lambda A_1 +A_0$ are simple, and the eigenvector matrix $X$, whose columns are all of unit length, can be given by $$\begin{aligned} X(:,1:3)&=\begin{bmatrix} -0.5269 - 0.3384\imath & -0.3566 + 0.4066\imath & 0.0646 - 0.5182\imath \\ -0.1210 - 0.7587\imath & -0.8248 + 0.0209\imath & 0.0578 - 0.7205\imath \\ -0.1247 + 0.0436\imath & -0.0681 + 0.1491\imath & -0.4029 - 0.2062\imath \end{bmatrix}, \\ X(:,4:6)&= \begin{bmatrix} 0.1636 - 0.3165\imath & -0.2841 + 0.2244\imath & 0.4363 - 0.4082\imath \\ 0.7059 + 0.6000\imath & -0.4017 - 0.8314\imath & 0.7443 - 0.1520\imath \\ 0.0065 - 0.1214\imath & -0.0599 + 0.1128\imath & -0.0381 - 0.2541\imath \end{bmatrix}.\end{aligned}$$ Let $\widehat{W}$ be a 3-by-3 matrix whose columns are selected from the columns of $X$. Define $$\begin{aligned} f(\tau_n,\widehat{W})&=\sum_{i=0}^2 \|\operatorname{OffBdiag}_{\tau_n}(\widehat{W}^{\operatorname{H}}A_i\widehat{W})\|_F^2,\\ \theta(W,\widehat{W})&=\max\{ \angle(W(:,1),\widehat{W}(:,1)), \angle(W(:,[2, 3]),\widehat{W}(:,[2, 3])) \}.\end{aligned}$$ For four different choices of $\widehat{W}$, we compute the condition number of $\widehat{W}$, $f(\tau_n,\widehat{W})$ and $\theta(W,\widehat{W})$, and the results are listed in Table \[tab1\]. Case $\widehat{W}$ $\mbox{cond}(\widehat{W})$ $f(\tau_n,\widehat{W})$ $\theta(W,\widehat{W})$ ------ ------------------ ---------------------------- ------------------------- ------------------------- 1 $X(:,[4, 1, 2])$ 2.3e1 0.0048 0.0066 2 $X(:,[4, 2, 3])$ 9.2e0 0.1061 0.0154 3 $X(:,[6, 4, 5])$ 4.9e2 0.0247 0.8855 4 $X(:,[1, 2, 3])$ 1.2e3 9.5550 0.5005 : Condition number, cost function $f(\tau_n,\widehat{W})$ and angle $\theta(W,\widehat{W})$ \[tab1\] From Example \[eg1\] we can see that the qualities of the four approximate diagonalizers $\widehat{W}$ are quite different although they are all consisted of linearly independent eigenvectors: $\widehat{W}$ in Cases 1 and 2 are good in the sense that $f(\tau_n,\widehat{W})$ and $\theta(W,\widehat{W})$ are small, and $\widehat{W}$ in Case 1 is better; $\widehat{W}$ in Case 3 is not good though $f(\tau_n,\widehat{W})$ is small; $\widehat{W}$ in Case 4 is even not an approximate diagonalizer. Based on the above observations and also Remark \[rem3\], it is reasonable to choose $n$ linearly independent eigenvectors $x_1, x_2, \dots, x_n$ such that the condition number of $[x_1, x_2, \dots, x_n]$ is as small as possible. Then the task of Stage 1 is reduced to find such $n$ linearly independent eigenvectors. Classic eigensolvers for PEP concentrate on computing extreme eigenvalues or eigenvalues close to a prescribed number (and their corresponding eigenvectors), which are not suitable for the task. In this paper, we use the following two steps to accomplish the task: [Step 1.]{} Compute $k$ unit length eigenvectors $x_1, \dots, x_{k}$ of the PEP by certain eigensolvers, where $k$ is larger than $n$, say a multiple of $n$, $k=2n, 4n$. In our numerical tests, we first transform the PEP into the GEP $(\lambda M+N)u=0$, where $u$, $M$ and $N$ are given by , , respectively. Then when $p$ is small, we use QZ method to find all eigenvectors of the GEP; When $p$ is large, we use Arnoldi method to compute $k$ largest magnitude eigenvalues and the corresponding eigenvectors of the GEP. Finally, the eigenvectors of the PEP can be obtained via those of the GEP. [Step 2.]{} Compute the QR decomposition of $[x_1, \dots, x_k]$ with column pivoting, i.e., $[x_1,\dots, x_k]P=QR$, where $P$ is a permutation matrix of order $k$, $Q$ is unitary, $R$ is upper triangular with main diagonal entries in a decreasing order. If the $(n,n)$ entry of $R$ is small, [^7] return to Step 1 to find more eigenvectors, else set $X$ as the first $n$ columns of $[x_1,\dots, x_k]P$. The above two-step procedure is perhaps the simplest way to accomplish the task of Stage 1, but it maybe still worth developing some particular eigensolvers for it. Stage 2. -------- In this stage, we need to determine a partition $\tau_n$ and a permutation matrix $\Pi$ such that $\Pi^{\operatorname{T}}X^{\operatorname{H}}A_i X\Pi$’s are all approximately $\tau_n$-block diagonal and $\operatorname{card}(\tau_n)$ is maximized. This stage is of great importance for determining the solutions to the GJBD problem, but without knowing the number of the diagonal blocks, determining a correct $\tau_n$ can be very tricky, especially when the noise is high and the block diagonal structure is fussy. From our numerical experience, the spectral clustering method [@von2007tutorial] is powerful and efficient for finding $\tau_n$ and $\Pi$. Let $$\begin{aligned} H=[h_{ij}]=\sum_{i=0}^p(|X^{\operatorname{H}}A_iX|+|X^{\operatorname{H}}A_i^{\operatorname{H}}X|),\end{aligned}$$ we can define some (normalized) graph Laplacian matrix $L$ from $H$. For example, in this paper, we set $$\begin{aligned} \label{eq:l} L=[l_{ij}] \quad \mbox{with} \quad l_{ij}= \left\{ \begin{array}{cc} -1, & \mbox{if} \ i\ne j, \ h_{ij}> \sum_{k=1}^n h_{kj}/n; \\ 0, & \mbox{if} \ i\ne j, \ h_{ij}\le \sum_{k=1}^n h_{kj}/n; \\ -\sum_{k\ne j} l_{kj}, & \mbox{if}\ i=j. \end{array} \right.\end{aligned}$$ Then the number of diagonal blocks equal the multiplicity of zero as an eigenvalue of $L$, using $k$-means method [@arthur2007k][@macqueen1967some], $\tau_n$ and $\Pi$ can be determined by clustering the eigenvectors corresponding to eigenvalue zero. Stage 3. -------- Theorem \[thm3\] only ensures that $\widehat{W}=X\Pi$ is a “sub-optimal” diagonalizer. In order to improve the quality of the diagonalizer, we propose the following refinement procedure. Let $(\tau_n, \widehat{W})$ be an approximate solution to the GJBD problem produced by the first two stages of PEAR. Suppose $\tau_n=(n_1,\dots,n_t)$, $\widehat{W} =[\widehat{W}_1, \dots, \widehat{W}_t]$ with $\widehat{W}_j\in\mathbb{C}^{n\times n_j}$ for $j=1,\dots,t$. Denote $\widehat{W}_{-j}=[W_1, \dots, W_{j-1}, W_{j+1}, \dots, W_t]$ and $$\begin{aligned} \mathscr{B}_j=\left\{ \begin{array}{cc} \left[A_0 \widehat{W}_{-j}, A_1\widehat{W}_{-j}, \dots, A_p \widehat{W}_{-j}\right]^{\operatorname{H}}, & \mbox{if $A_j$'s are Hermitian;} \\ \left[A_0 \widehat{W}_{-j}, A_0^{\operatorname{H}}\widehat{W}_{-j}, \dots, A_p \widehat{W}_{-j}, A_p^{\operatorname{H}}\widehat{W}_{-j}\right]^{\operatorname{H}} & \mbox{Otherwise.} \end{array} \right.\end{aligned}$$ Fixing $\widehat{W}_{-j}$, we can minimize $f(\tau_n,\widehat{W})=\sum_{i=0}^p \|\operatorname{OffBdiag}_{\tau_n}(\widehat{W}^{\operatorname{H}}A_i\widehat{W})\|_F^2$ [^8] by updating $\widehat{W}_j$ as $V_j$, where the column vectors of $V_j$ are the $n_j$ right singular vectors of $\mathscr{B}_j$ corresponding with the $n_j$ smallest singular values. For $j=1,2,\dots, t$, we update $\widehat{W}_j$ as above, we call it a refinement loop. We can repeat the refinement loop until the diagonalizer is sufficiently good. In our numerical test, three refinement loops are sufficient. Note that the effectiveness of this refinement procedure is built on the assumption that $\tau_n$ obtained in Stage 2 is correct. Without such an assumption, the refinement procedure may make the diagonalizer even more worse. It is also worth mentioning here that the above refinement procedure can be used to update any approximate diagonalizer, not necessarily the one produced by Stage 2. As a matter of fact, the procedure itself can be used to find a diagonalizer, but without a good initial guess, the convergence can be quite slow. In Stage 1, assuming each eigenpair can be found in $\mathcal{O}(1)$ steps, then $k=\mathcal{O}(n)$ eigenpairs can be obtained in $\mathcal{O}(n^3)$ steps. In Stage 2, first a symmetric eigenvalue problem needs to be solved, which can be done in $\mathcal{O}(n^3)$ flops; second, $k$-means needs to be performed, which is so fast in practice that we can ignore its cost. Stage 3 requires $\mathcal{O}(n^3)$ flops, including the matrix-matrix multiplications and singular value decomposition (SVD), etc. So the overall computational cost of PEAR is $\mathcal{O}(n^3)$ flops. When the matrices $A_i$’s are all real, a real diagonalizer is required. But the diagonalizer returned by PEAR is in general complex. Can we make some simple modifications to PEAR to get a real diagonalizer? The answer is positive. In fact, we can add the following “Stage 2.5” to get a real diagonalizer from PEAR. Stage 2.5 --------- Let $\widehat{W}=[\widehat{W}_1, \dots, \widehat{W}_t]$ be the approximate diagonalizer at the beginning of Stage 3, where $\widehat{W}_j\in\mathbb{C}^{n\times n_j}$ for $j=1,\dots,t$. For each $j$, denote $\widehat{W}_j=\widehat{W}_{jR}+\imath \widehat{W}_{jI}$, where $\widehat{W}_{jR}, \widehat{W}_{jI}\in\mathbb{R}^{n\times n_j}$ are the real and imaginary parts of $\widehat{W}_j$, respectively . Let the SVD of $[\widehat{W}_{jR}, \widehat{W}_{jI}]$ be $[\widehat{W}_{jR}, \widehat{W}_{jI}]=U_j\Sigma_j V_j^{\operatorname{T}}$, where $U_j\in\mathbb{R}^{n\times n}, V_j\in\mathbb{R}^{2n_j\times 2n_j}$ are orthogonal, $\Sigma_j\in\mathbb{R}^{n\times 2n_j}$ is diagonal with its main diagonal nonnegative and in a decreasing order. Then we update $\widehat{W}_j$ as $\widehat{W}_j=U_j(:,1:n_j)$. As a consequence, $\widehat{W}=[\widehat{W}_1, \dots, \widehat{W}_t]$ will be a real diagonalizer. The mechanic behind the above procedure is the following critical assumption: A.) [*All solutions to the GJBD problem of $\mathcal{A}$ are equivalent, and the intersection of all diagonalizers and $\mathbb{R}^{n\times n}$ is nonempty.*]{} Based on the above assumption, for the diagonalizer $\widehat{W}\in\mathbb{C}^{n\times n}$ at the beginning of Stage 3, there exists a nonsingular $\tau_n$-block diagonal matrix $D=\operatorname{diag}(D_{11},\dots, D_{tt})$ such that $\widetilde{W}:=\widehat{W}D^{-1}$ is a real diagonalizer. Partition $\widetilde{W}$ as $=[\widetilde{W}_1, \dots, \widetilde{W}_t]$ with $\widetilde{W}_j\in\mathbb{C}^{n\times n_j}$, and let $D_{jj}=D_{jR}+\imath D_{jI}$, where $D_{jR}, D_{jI}\in\mathbb{R}^{n_j\times n_j}$. Then by $\widetilde{W}_j=\widehat{W}_jD_{jj}^{-1}$, we have $$\begin{aligned} \widetilde{W}_j [D_{jR}, D_{jI}] =[\widehat{W}_{jR},\widehat{W}_{jI}].\end{aligned}$$ Then it follows that $\operatorname{span}([\widehat{W}_{jR},\widehat{W}_{jI}])=\operatorname{span}(\widetilde{W}_j [D_{jR}, D_{jI}])=\operatorname{span}(\widetilde{W}_j)$ since $[D_{jR}, D_{jI}]$ is of full row rank. Therefore, updating $\widehat{W}_j$ as $U_j(:,1:n_j)$, we know that $\widehat{W}=[U_1(:,1:n_1),\dots, U_t(:,1:n_t)]$ is a real diagonalizer. The assumption A.) in general holds, but not always. For example, consider the GJBD problem of $\{A_0,A_1\}$, where $$\begin{aligned} A_0=\begin{bmatrix}1 & 1 & 1\\ 1 & 1& -3\\ -3 & 1& 1\end{bmatrix},\quad A_1=\begin{bmatrix}3 & -1 & 1\\ -1 & 3& -3\\ -3 & 1& 3\end{bmatrix}.\end{aligned}$$ Then by calculations, we know that $((1, 2), {\left[\begin{smallmatrix} }1& 1 & 0\\ 1& 0 & 1\\ 0& 1& 1{\end{smallmatrix}\right] })$ and $((1, 1, 1), {\left[\begin{smallmatrix} }1 & 1-\imath & 1+\imath \\ 1 & 1+\imath & 1-\imath \\ 0 & 2 & 2{\end{smallmatrix}\right] })$ are two inequivalent solutions to the GJBD problem with the diagonalizers in $\mathbb{R}^{n\times n}$ and $\mathbb{C}^{n\times n}$, respectively. How to find a real diagonalizer when assumption A.) does not hold is difficult and needs further investigations. In our numerical tests (subsection \[eg2\]), Stage 2.5 works perfectly for finding a real diagonalizer. Numerical Examples {#sec:eg} ================== In this section, we present several examples to illustrate the performance of PEAR. All the numerical examples were carried out on a quad-core $\text{Intel}^{\tiny\textregistered} \text{Xeon}^{\tiny\textregistered}$ Processor E5-2643 running at 3.30GHz with 31.3GB AM, using MATLAB R2014b with machine $\epsilon=2.2\times 10^{-16}$. We compare the performance of PEAR (with and without refinement) with the second GJBD algorithm in [@cai2017algebraic], namely, $\star$-commuting based method with a conservative strategy, SCMC for short, and also two algorithms for the JBD problem, namely, JBD-LM [@cherrak2013non] and JBD-NCG [@nion2011tensor]. For PEAR, three refinement loops are used to improve the quality of the diagonalizer. For SCMC, the tolerance is set as $3n^210^{-\mbox{SNR}/20}$, where $\mbox{SNR}$ is the signal-to-noise ratio defined below. For the JBD-LM method, the stopping criteria are $\|W_{k+1}-W_k\|_F<10^{-12}$, or $|\frac{\phi_k-\phi_{k+1}}{\phi_k}|<10^{-8}$ for successive 5 steps, or the maximum number of iterations, which is set as 200, exceeded. For the JBD-NCG method, the stopping criteria are $|\phi_k-\phi_{k+1}|<10^{-8}$, or $|\frac{\phi_k-\phi_{k+1}}{\phi_k}|<10^{-8}$ for successive 5 steps, or the maximum number of iterations, which is set as 2000, exceeded. Here $W_k$, $\phi_k$ are the $W$ matrix and the value of the cost function in $k$th step, respectively. In JBD-LM and JBD-NCG algorithms, 20 initial values (19 random initial values and an EVD-based initial value [@nion2011tensor]) are used to iterate 20 steps first, and then the iteration which produces the smallest value of the cost function proceeds until one of the stopping criteria is satisfied. Random data {#eg1} ----------- Let $\tau_n=(n_1,n_2,\dots,n_t)$ be a partition of $n$, we will generate the matrix set $\mathcal{A}=\{A_i\}_{i=0}^p$ by the following model: $$\begin{aligned} \label{randmodel} A_i=V^{\operatorname{H}} D_iV, \quad i=0,1,\ldots,p, \end{aligned}$$ where $V$ and $D_i$ are, respectively, the mixing matrix and the approximate $\tau_n$-block diagonal matrices. The elements in V and $\operatorname{Bdiag}_{\tau_n}(D_i)$ are all complex numbers whose real and imaginary parts are drawn from a standard normal distribution, while the elements in $\operatorname{OffBdiag}_{\tau_n}(D_i)$ are all complex numbers whose real and imaginary parts are drawn from a normal distribution with mean zero and variance $\sigma^2$. The signal-to-noise ratio is defined as $\mbox{SNR}=10\log(1/\sigma^2)$. For model , we define the following performance index to measure the quality of the computed diagonalizer $W$, which is used in [@cai2017algebraic] and [@cai2015matrix]: $$\begin{aligned} {PI}(V^{-1},W)=\min_{\pi}\max_{1\leq i\leq t}\operatorname{subspace}(V_i, W_{\pi(i)}), \end{aligned}$$ where $V^{-1}=[V_1, V_2,\dots, V_t]$, $W=[W_1, W_2,\dots, W_t]$, $V_i, W_{\pi(i)}\in\mathbb{C}^{n\times n_i}$ for $i=1,2,\ldots, t$, the vector $(\pi(1),\pi(2),\ldots,\pi(t))$ is a permutation of $\{1,2,\dots t\}$ satisfying $(n_{\pi(1)},n_{\pi(2)},\ldots,n_{\pi(t)})=\tau_n$, the expression $\operatorname{subspace}(E,F)$ denotes the angle between two subspaces specified by the columns of $E$ and $F$, which can be computed by MATLAB function “[subspace]{}”. In what follows, we generate the matrix set by model with the following parameters: P1. : $n=9$, $\tau_n=(3,3,3)$, $p+1=25$. P2. : $n=9$, $\tau_n=(2,3,4)$, $p+1=25$. P3. : $n=9$, $\tau_n=(2,3,4)$, $p+1=20,40,\dots,200$, SNR=80. P4. : $n=9m$, $\tau_n=(2m,3m,4m)$ for $m=1,2,\dots,6$, $p+1=10$, SNR=80. **Experiment 1.** For different SNRs, we generate the data with parameters P1 and P2, respectively. Then for each matrix set generated by those parameters, we perform PEAR for 1000 independent runs. It is known that when the SNR is small, PEAR may fail, specifically, the block diagonal structure in Stage 2 of PEAR is fuzzy, the computed $\hat{\tau}_n$ may not be consistent with the true $\tau_n$, namely, there is no $(0,1)$ matrix $N$ such that $\tau_n=\hat{\tau}_nN$. In Table \[tab:percentage\], we list the percentages of successful runs of [PEAR]{}. From the table we can see that the smaller the SNR is, the more likely PEAR may fail. In our tests, when SNR is no less than 60, PEAR didn’t fail. $30$ $40$ $50$ $60$ $70$ $80$ $90$ $100$ ----- ---------- ---------- ---------- --------- --------- --------- --------- --------- P1. $74.1\%$ $96.1\%$ $99.7\%$ $100\%$ $100\%$ $100\%$ $100\%$ $100\%$ P2. $65.9\%$ $96.4\%$ $99.6\%$ $100\%$ $100\%$ $100\%$ $100\%$ $100\%$ : The percentages of successful runs of [PEAR]{} over $1000$ independent runs[]{data-label="tab:percentage"} **Experiment 2.** For SNR= 40, 60, 80, 100, we generate the data with parameters P1 and P2, respectively. For each matrix set generated by those parameters, we perform JBD-LM, JBD-NCG, SCMC, PEA (PEAR without refinement) and PEAR for 50 independent runs, then compare their performance indices. The box plot (generated by MATLAB function “boxplot”) of the results are displayed in Figures \[Fig\_333\_SNR\] and \[Fig\_234\_SNR\]. ![Performance indices of five methods with different SNRs for P1[]{data-label="Fig_333_SNR"}](./noise333.eps){width="100.00000%"} ![Performance indices of five methods with different SNRs for P2[]{data-label="Fig_234_SNR"}](./noise234.eps){width="100.00000%"} We can see from Figure \[Fig\_333\_SNR\] and Figure \[Fig\_234\_SNR\] that for both cases, when SNR equals 40, 60, 80 or 100, the performance indices produced by JBD-LM, JBD-NCG, SCMC and PEAR are almost the same; the performance indices produced by PEA are larger than those of other four methods, which indicates that the diagonalizers produced by the first two stages of PEAR indeed suffer from low quality, and the refinement stage of PEAR is effective. For all methods, the performance indices decrease as the SNR increases. **Experiment 3.** We generate the data with the parameters in P3 and P4, respectively. 50 independent trials are performed for each matrix set. The average CPU time of two algebraic methods – SCMC and PEAR are displayed in Figures \[Fig\_333\_Num\]. From the left figure we can see that the CPU time of the two methods increase almost linearly with increased matrix number. And from the right figure we can see that the CPU time of SCMC increases dramatically as matrix size increases, meanwhile, the CPU time of PEAR increases much slower. ![(left): CPU time of two methods with different matrix numbers, (right): CPU time of two methods with different matrix sizes[]{data-label="Fig_333_Num"}](./time_matnum.eps "fig:"){width="49.00000%"} ![(left): CPU time of two methods with different matrix numbers, (right): CPU time of two methods with different matrix sizes[]{data-label="Fig_333_Num"}](./time_matsize.eps "fig:"){width="49.00000%"} Separation of convolutive mixtures of source {#eg2} -------------------------------------------- We consider example 4.2 in [@cai2017algebraic], where a real diagonalizer is required. All settings are kept the same. In Figure \[Fig\_cor\], for different SNRs, we plot the correlations between the source signals and the extracted signals obtained from computed solutions by JBD-LM, JBD-NCG, SCMC and PEAR, respectively. All displayed results have been averaged over 50 independent trials. ![[]{data-label="Fig_cor"}](./sig1.eps "fig:"){width="49.00000%"} ![[]{data-label="Fig_cor"}](./sig2.eps "fig:"){width="49.00000%"}\ ![[]{data-label="Fig_cor"}](./sig3.eps "fig:"){width="49.00000%"} We can see from Figure \[Fig\_cor\] that when SNR is larger than 60, the recovered signals obtained from all four methods are all good approximations of the source signals; when SNR is less than 60, SCMC is the best, PEAR is the second best. The reason why SCMC is better than PEAR in this example is that PEAR fails to find the correct partition in Stage 2 when SNR is small, meanwhile, with a large tolerance for SCMC, SCMC is able to find a consistent partition with the correct one. In Stage 2 of PEAR, if we use some normalized Laplacian to find the partition, the numerical results of PEAR can be improved. Conclusion {#sec:conclusion} ========== In this paper, we show how the GJBD problem of a matrix set is related to a matrix polynomial eigenvalue problem. Theoretically, under mild conditions, we show that (a) the GJBD problem of $\{A_i\}_{i=0}^p$ can be solved by $n$ linearly independent eigenvectors of the matrix polynomial $P(\lambda)=\sum_{i=0}^p \lambda^i A_i$; (b) all solutions to the GJBD problem are equivalent; (c) a sub-optimal solution of the approximate GJBD problem can also be given by $n$ linearly independent eigenvectors. Algorithmically, we proposed a three-stage method – PEAR to solve the GJBD problem. Numerical experiments show that PEAR is effective and efficient. 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Signal Process. **80**(5), 897–902 (2000) [^1]: LMAM & School of Mathematical Sciences, Peking University, Beijing, 100871, China, [yfcai@math.pku.edu.cn]{} [^2]: School of Mathematical Sciences, University of Electronic Science and Technology of China, Chengdu, Sichuan 611731, P. R. China, [ghcheng@uestc.edu.cn]{} [^3]: Beijing International Center for Mathematical Research, Peking University, Beijing, 100871, P. R. China, [decaishi@gmail.com]{} [^4]: This research was supported by NSFC under grants 11671023, 11421101 and 11301013. [^5]: By constructing a Hermitian matrix polynomial $\widehat{P}_{\mathcal{A}}(\lambda)\triangleq\lambda^{2p+1}(A_p+A_p^{\operatorname{H}})+ \lambda^{2p} \imath (A_p-A_p^{\operatorname{H}})+\dots+\lambda (A_0+A_0^{\operatorname{H}})+\imath (A_0-A_0^{\operatorname{H}})$, one can still follow the approach in [@cai2015matrix] to solve the general GJBD problem of $\{A_i\}_{i=0}^p$, but the degree of $\widehat{P}_{\mathcal{A}}(\lambda)$ is $2p+1$, almost twice as many as the degree of ${P}_{\mathcal{A}}(\lambda)$. [^6]: The space spanned by several matrices is defined as $\operatorname{span}\{A_0,A_1,\dots,A_p\}=\{\sum_{i=0}^p \alpha_i A_i\; | \; [\alpha_0,\alpha_1,\dots,\alpha_p]^{\operatorname{T}}\in\mathbb{C}^{p+1}\}$. [^7]: This indicates that $x_1,\dots, x_k$ are almost linearly dependent, and it is generally impossible to find a good diagonalizer via those eigenvectors. [^8]: $f(\tau_n,\widehat{W})$ is in fact a commonly used cost function for JBD problem. One can of course use some other cost functions, but this one seems the most simple one for our refinement purpose.
--- abstract: 'In this paper the scattering between a wobbling kink and a wobbling antikink in the standard $\phi^4$ model is numerically investigated. The dependence of the final velocities, wobbling amplitudes and frequencies of the scattered kinks on the collision velocity and on the initial wobbling amplitude is discussed. The fractal structure becomes more intricate due to the emergence of new resonance windows and the splitting of those arising in the non-excited kink scattering. Outside this phase the final wobbling amplitude exhibits a linear dependence of the collision velocity whereas the final frequency is a decreasing function. By contrast these magnitudes are almost independent of the initial wobbling amplitude.' author: - | A. Alonso-Izquierdo$^{(a,b)}$ L.M. Nieto$^{(c,d)}$ and J. Queiroga-Nunes$^{(c)}$,\ $^{(a)}$ Departamento de Matematica Aplicada, Universidad de Salamanca,\ Casas del Parque 2, 37008 - Salamanca, Spain\ $^{(b)}$ IUFFyM, Universidad de Salamanca, Plaza de la Merced 1, 37008 - Salamanca, Spain\ $^{(c)}$ Departamento de Física Teórica, Atómica y Óptica, Universidad de Valladolid,\ 47011 Valladolid, Spain\ $^{(d)}$ Instituto de Matemáticas (IMUVA), Universidad de Valladolid, 47011 Valladolid, Spain title: '**Scattering between wobbling kinks**' --- Introduction ============ Over the last fifty years, topological defects have played an essential role in explaining a wide variety of non-linear phenomena arising in several physical contexts, including Condensed Matter [@Bishop1980; @Eschenfelder1981; @Jona1993; @Strukov], Cosmology [@Vilenkin1994; @Vachaspati2006], Optics [@Mollenauer2006; @Schneider2004; @Agrawall1995], Molecular systems [@Davydov1985; @Bazeia1999], Biochemistry [@Yakushevich2004], etc. This broad range of applications underlies the fact that topological defects are solutions of nonlinear partial differential equations, which behave as extended particles in the physical substrate. Solitons and kinks are paradigmatic examples of this type of solutions, which have been profusely studied both in Physics and Mathematics. They arise, respectively, in the sine-Gordon and $\phi^4$ field theory models, which are endowed with two facing properties: integrability versus non-integrability. Curiously, kink scattering in non-integrable systems exhibits a richer behavior than the one found for integrable systems. The study of the collision between kinks and antikinks in the $\phi^4$ model was initially addressed in the seminal references [@Sugiyama1979; @Campbell1983; @Anninos1991]. The complex relation between the final velocity $v_f$ of the scattered kinks and the initial collision velocity $v_0$ was displayed in these papers. There exist two different scattering channels: *bion formation* and *kink reflection*. In the first case a bound state (called bion) is formed, where kink and antikink collide and bounce back over and over emitting radiation in every impact. In the second case, kink and antikink emerge after the impact and move away with a certain velocity $v_f$. If the initial collision velocity $v_0$ is low enough, a bion is always formed while for large velocities $v_0$ the kinks are reflected. However, the most striking feature in this scheme is that the transition between the two previously described regimes is characterized by a sequence of initial velocity windows with a fractal structure where the kinks collide several times before definitely escaping, see Figure \[fig:velofinalA0000\]. The fractal nature displayed by this final velocity versus initial velocity diagram is twofold: (a) The first two-bounce window arises approximately in the range $v_0\in [0.1920,0.2029]$ and it is infinitely replicated by progressively narrower windows up to the beginning of the one-bounce kink reflection regime at (approximately) $v_0 \approx 0.26$, see Figure \[fig:velofinalA0000\]. (b) Two-bounce windows are surrounded by three-bounce windows, and these ones, in turn, are surrounded by four-bounce windows and so on. Consequently, the previously mentioned diagram displays three clearly differentiated parts: the first one corresponds to zero velocity where the bion state is formed, the second part approximately occurs in the interval \[0.19,0.26\] where the fractal structure emerges and the third one refers to the 1-bounce kink reflection regime characterized by a continuous increasing curve starting at zero final velocity, which we shall call *the one-bounce tail*. Notice that there is no 1-bounce windows in the fractal region. The presence of the $n$-bounce windows can be explained by means of the *resonant energy transfer mechanism*. As it is well known the second order small kink fluctuation operator involves two discrete eigenfunctions: a zero mode (which generates an infinitesimal translational movement of the kink) and a shape mode (an infinitesimal perturbation associated with the internal vibration of the kink). The presence of these modes is the consequence of two different evolutions: (a) the kink travels with constant velocity and (b) the kink vibrates by changing its size. This last behavior defines the so-called *wobbling kink*. The previously mentioned mechanism allows an energy exchange between the zero and shape kink modes. For example, a kink and antikink could approach each other with initial velocity $v_0$, collide and bounce back. The impact could excite the shape mode, which would absorb a part of the kinetic energy. As a consequence, a wobbling kink and a wobbling antikink would emerge after the collision and move away. If the kinetic energy of the resulting kinks was not large enough to make the kinks escape they would end up approaching and colliding again. The new impact would cause a redistribution of the energy among the zero and vibrational modes. It is clear that the wobbling kinks play an important role in the fractal structure of the $n$-bounce windows. In this sense the present study is important to understand the $n$-bounce scattering since after the first collision a wobbling kink and a wobbling antikink emerge with, in general, higher amplitudes than the initial one. A part of the vibrational energy could return to the zero mode making it possible for the kinks to move away and eventually escape. This describes a two-bounce kink scattering event. In general, a $n$-bounce event arises when the resulting kinks need to collide $n$ times before escaping. It is worthwhile to mention that the resonant energy transfer mechanism does not arise for the soliton scattering in the sine-Gordon model. It is assumed that the reason for this is the lack of vibrational (shape) modes in this model. However, this mechanism and other related phenomena are present in a large variety of one-component scalar field theory models, such as in the double sine-Gordon model [@Shiefman1979; @Peyrard1983; @Campbell1986; @Gani1999; @Malomed1989; @Gani2018; @Gani2019], in deformed $\phi^4$ models [@Simas2016; @Gomes2018; @Bazeia2017b; @Bazeia2017a; @Bazeia2019; @Adam2019; @Romanczukiewicz2018; @Adam2020; @Mohammadi2020; @Yan2020], in $\phi^6$-models [@Romanczukiewicz2017; @Weigel2014; @Gani2014; @Bazeia2018b; @Lima2019] and in other more complex models [@Mendoca2019; @Belendryasova2019; @Zhong2020; @Bazeia2020c]. Kink dynamics has also been analyzed in coupled two-component scalar field theory models, see [@Halavanau2012; @Romanczukiewicz2008; @Alonso2018; @Alonso2018b; @Alonso2017; @Alonso2019; @Alonso2020; @Ferreira2019]. The effect of impurities, defects or inhomogeneities on kink dynamics has been discussed in several models, see references [@Fei1992; @Fei1992b; @Goodman2002; @Goodman2004; @Malomed1985; @Malomed1992; @Javidan2006; @Saadatmand2012; @Saadatmand2013; @Saadatmand2015; @Saadatmand2018; @Adam2018; @Adam2019b]. The previous description constitutes an heuristic explanation of the resonant energy transfer mechanism, although this phenomenon has revealed to be more complicated than expected. It has been proved that it can be triggered by the discrete eigenfunctions of combined kink configurations when kink and antikink are close enough and also by quasi-normal modes [@Dorey2011; @Dorey2018; @Campos2020]. This complexity turns the search for an analytical explanation of this phenomenon into a very elusive problem. Indeed, the collective coordinate method was initially introduced in [@Sugiyama1979] to explain the kink dynamics in the $\phi^4$ model and used later to explain the resonant energy transfer mechanism in a satisfactory way. However, the presence of typographical errors in the original paper has been proved [@Takyi2016]. The corrected terms were not sufficient to make the collective coordinate approach fit the data outcome of the scattering process using the harmonic approximation. Recent studies [@Pereira2020] shows that the inclusion of more terms, up to second order, in the effective Lagrangian are necessary to make this analytical method fit with better precision the simulation data. Other important topic in this context is the study of the evolution of the wobbling kink in the $\phi^4$-model. This issue was initially discussed by Getmanov [@Getmanov1976], who interpreted the wobbling kink as a bound state of three non-oscillatory kinks. Some perturbation expansion schemes have been employed to explore the properties of the wobbling kinks, see for example [@Barashenkov2009; @Barashenkov2018; @Segur1983; @Romanczukiewicz2008; @Malomed1992; @Manton1997]. These works show that the amplitude $a(t)$ of the wobbling mode at fourth order in the expansion decays. As a consequence the wobbling kink emits radiation. When $a(0)$ is small, the decay becomes appreciable only after long times $t\sim |a(0)|^{-2}$, see [@Barashenkov2009; @Barashenkov2018]. In this paper we shall investigate the scattering between wobbling kinks. We think that this analysis is interesting for several reasons. The original kink scattering problem where the resonant energy transfer mechanism was initially discovered involves the collisions of wobbling kinks after the first impact. In other words, a $n$-bounce scattering process presumably includes $n-1$ wobbling kink collisions. For example, in a 2-bounce event the kinks collide and a part of the kinetic energy is transferred to the vibrational mode, such that the second collision is a wobbling kink scattering process. Indeed, this last impact causes the more (at first sight) astonishing phenomenon, the two kinks acquire more velocity than that they initially had before colliding. For this reason we think that the study of the wobbling kink scattering can bring new insight in the original problem, particularly those 1-bounce events, where the amplitude and velocity of colliding wobbling kinks can be monitored. This allows us to study the resonant energy transfer mechanism in a more direct way. Obviously, the major part of the results displayed in this paper comes from numerical analysis due to the previously mentioned fact that there are no satisfactory analytical methods to study this problem. The organization of this paper is as follows: in Section 2 the theoretical background of the $\phi^4$ model is introduced. The study of the kink and its linear stability leads us to the description of the wobbling kinks. The kink-antikink scattering is also discussed. Section 3 is devoted to the numerical analysis of the wobbling kink scattering. Here we shall addressed the scattering between weakly wobbling kinks and the scattering between strongly wobbling kinks in two different subsections. The distinction underlies the fact that the amplitude of the wobbling kinks decreases in the course of time. This effect is small for weakly wobbling kinks. So, we can assume in our numerical experiments that the amplitude of these kinks does not significantly change in the interval in which they are initially approaching before the collision. Finally, some conclusions are drawn in Section 4. The $\phi^4$ model and the kink-antikink scattering =================================================== We shall deal with the $\phi^4$ model in (1+1) dimensions, whose dynamics is governed by the action $$\label{action} S=\int d^2 x \,\, {\cal{L}}(\partial_{\mu}\phi, \phi) \hspace{0.5cm},$$ where the Lagrangian density ${\cal{L}}(\partial_{\mu}\phi, \phi)$ is of the form $$\label{lagrangiandensity} {\cal{L}}(\partial_{\mu}\phi, \phi) = \frac{1}{2} \,\partial_\mu \phi \, \partial^\mu \phi - V(\phi) \hspace{0.2cm}, \hspace{1cm} V(\phi) = \frac{1}{2} (\phi^2 -1)^2 \hspace{0.5cm}.$$ The use of dimensionless variables and Einstein summation convention is assumed in (\[action\]) and (\[lagrangiandensity\]). The Minkowski metric $g_{\mu\nu}$ has been chosen as $g_{00}=-g_{11}= 1$ and $g_{12}=g_{21}=0$. The solutions of this model verify the non-linear partial differential equation $$\frac{\partial^2 \phi}{\partial t^2} - \frac{\partial^2 \phi}{\partial x^2} = -\frac{d V}{d \phi} = -2\phi(\phi^2 - 1) \hspace{0.5cm}, \label{pde}$$ which derives from the Euler-Lagrange equations associated with the functional (\[action\]). The energy-momentum conservation laws imply that the total energy and momentum $$E[\phi] = \int dx \Big[ \frac{1}{2} \Big( \frac{\partial \phi}{\partial t} \Big)^2 + \frac{1}{2} \Big( \frac{\partial \phi}{\partial x} \Big)^2 + V(\phi) \Big] \hspace{0.5cm};\hspace{0.5cm} P[\phi] = - \int dx\, \frac{\partial \phi}{\partial t} \, \frac{\partial \phi}{\partial x} \hspace{0.5cm}, \label{invariants}$$ are system invariants. The integrand of the total energy $E[\phi]$ $$\varepsilon[\phi(x)]=\frac{1}{2} \Big( \frac{\partial \phi}{\partial t} \Big)^2 + \frac{1}{2} \Big( \frac{\partial \phi}{\partial x} \Big)^2 + V(\phi)$$ is the energy density of a configuration $\phi(x)$. Time and space independent solutions of are $\phi_V = \pm 1$. Therefore, the set ${\cal{M}}$ of the vacua in this model is ${\cal{M}} = \{-1, 1\}$. Finite energy static solutions of (\[pde\]) are of the form $$\phi_{\rm K}^{(\pm)}(x;x_0) = \pm \tanh (x-x_0) \label{kink} \hspace{0.5cm} ,$$ which are called kink/antikink $(+\,/\,-)$ and connect the two elements of the set ${\cal{M}}$. The kink/antikink energy density $\varepsilon [\phi_{\rm K}^{(\pm)}(x;x_0)] = {\rm sech}^4 (x-x_0)$ is localized around the point $x=x_0$, which is assumed to be the kink center (the value where the field profile vanishes). The Lorentz invariance can be used to construct the traveling kinks/antikinks in the form $$\phi_{\rm K}^{(\pm)}(t,x;x_0,v_0) = \pm \tanh \left[\frac{x-x_0-v_0 t}{\sqrt{1-v_0^2}}\right] \hspace{0.5cm} . \label{travelingkink}$$ Obviously, the kink center $x_C$ for (\[travelingkink\]) moves in the real line following the expression $x_C=x_0+v_0 t$, such that $v_0$ is interpreted as the kink velocity. Now, in order to examine the linear stability of the solution, we consider fluctuations around the static kink/antikink solution (\[kink\]) by adding a small perturbation as $$\widetilde{\phi}_{\rm K}^{(\pm)}(t,x;x_0) = \phi_{\rm K}^{(\pm)}(x;x_0) + \psi(t,x;x_0) \hspace{0.5cm}. \label{perturbation}$$ Expanding it up to first-order in $\psi$ into the equation of motion (\[pde\]) and using the standard separation of variables ansatz $$\psi(t,x;x_0)\, = a \, e^{i\omega t} \psi_{\omega^2}(x;x_0) \hspace{0.5cm},$$ one has the Schrödinger-like equation $$\left[-\frac{d^2}{dx^2} + \textbf{\textit{U}}(x) \right] \psi_{\omega^2}(x;x_0) \, = \, \omega^2\psi_{\omega^2}(x;x_0) \hspace{0.5cm}, \label{schrodingerlike}$$ where $$\textbf{\textit{U}}(x) = \left.\frac{d^2V}{d\phi^2}\right|_{\phi_{\rm K}^{(\pm)}} =4-6\,{\rm sech}^2 (x-x_0) \hspace{0.5cm}.$$ Equation (\[schrodingerlike\]) has one zero mode, an excited mode with eigenvalue $\omega^2=3$ and a continuous spectrum on the threshold value $\omega^2=4$, whose eigenfunctions are given by, $$\begin{aligned} \psi_{\omega^2=0}(x;x_0)&=& \, {\rm sech}^2 (x-x_0) = \frac{\partial \phi_K}{\partial x} \hspace{1.0cm} , \\ \psi_{\omega^2=3}(x;x_0)&=& \, {\rm sinh}\, (x-x_0) \, {\rm sech}^2 (x-x_0) \hspace{0.5cm} , \\ \psi_{\omega^2= 4 + q^2}(x;x_0)&=& \, e^{iq(x-x_0)} [-1-q^2 + 3 \tanh^2 (x-x_0) - 3 i q \tanh (x-x_0)] \hspace{0.5cm}.\end{aligned}$$ This means that perturbations of the static kinks (\[perturbation\]) are solutions of the nonlinear partial differential equation (\[pde\]) at first order in the $a$-expansion. In this sense, the zero mode $\psi_{\omega^2=0}$ describes an infinitesimal translation of the static kink (\[kink\]), or in other words, an infinitesimal evolution of the traveling kink (\[travelingkink\]). The shape mode $\psi_{\omega^2=3}(x;x_0)$ describes a vibrational state of the kink/antikink whose width oscillates with frequency $\omega=\sqrt{3}$. For small amplitudes $a$, a travelling wobbling kink/antikink $\phi_{\rm WK}^{(\pm)}$ is described by the expression $${\phi}_{\rm WK}^{\,(\pm)}(t,x;x_0,v_0,\omega,a) = \pm \tanh \Big[ \frac{x-x_0-v_0 t}{\sqrt{1-v_0^2}} \Big] + a \, e^{i\omega t} \, {\rm sinh} \Big[ \frac{x-x_0-v_0 t}{\sqrt{1-v_0^2}} \Big] \, {\rm sech}^2 \Big[\frac{x-x_0-v_0 t}{\sqrt{1-v_0^2}} \Big] \hspace{0.5cm},\label{wobblingkink}$$ which is a good approximation up to first order. The maximum deviation of the wobbling kink (\[wobblingkink\]) from the kink (\[travelingkink\]) takes places at the points $$x_M^{(\pm)} = x_C \pm \sqrt{1-v_0^2} \,\,{\rm arccosh}\, (2) \hspace{0.5cm} ,$$ where $x_C$ is the kink center. The same result applies to the antikink. The deviation at these points is given by half the wobbling amplitude $$\left|{\phi}_{\rm WK} (x_M^{(\pm)}) - \phi_{\rm K} (x_M^{(\pm)})\right| = \frac{1}{2} \, |a | \hspace{0.5cm }.$$ We shall analyze the evolution of the kink/antikink at these points to study the excitation of the shape modes in the kink scattering processes. When the amplitude $a$ is large enough, this magnitude can depend of the time variable. Indeed, it has been proved that the amplitude $a(t)$ of the wobbling mode at fourth order in the expansion decays following the expression $$|a(t)|^2 = \frac{|a(0)|^2}{1+\omega \,\xi_I\, |a(0)|^2 t} \hspace{0.5cm}, \label{amplitude}$$ where $\xi_I$ is a constant. When $a(0)$ is small, the decay becomes appreciable only after a long time $t\sim |a(0)|^{-2}$, see [@Barashenkov2009; @Barashenkov2018]. The scattering between a kink and an antikink (whose shape eigenfunctions are unexcited, i.e. $a=0$ in (\[wobblingkink\])) has been thoroughly analyzed in the physical and mathematical literature. In this case, a kink and antikink, which are well separated, are pushed together with initial collision velocity $v_0$. Taking into account the spatial reflection symmetry of the system, the kink can be located at the left of the antikink or viceversa. This configuration can be characterized by the concatenation $$\phi_{\rm K}^{(\pm)}(t,x,x_0,v_0) \cup \phi_{\rm K}^{(\mp)}(t,x,-x_0,-v_0) = \left\{ \begin{array}{ll} \phi_{\rm K}^{(\pm)}(t, x, x_0, v_0 ) & \mbox{if} \hspace{0.2cm} x < 0, \\ \phi_{\rm K}^{(\mp)}(t, x, -x_0, -v_0 ) & \mbox{if} \hspace{0.2cm} x \geq 0 \end{array} \right.$$ for $x_0$ large enough. Two different scattering channels have been found in this situation: 1. *Bion formation*: In this case, kink and antikink approach each other, then collide and bounce back. After the impact an exchange of energy from the translational mode to the shape and continuous modes takes place in such a way that the kinetic energy of these two kinks is not big enough to make them escape. Therefore, they approach each other again, collide and bounce back over and over. This is a long living bound kink-antikink state called *bion*. 2. *Kink reflection*: Now, kink and antikink approach each other, collide and bounce back. After the impact a redistribution of the energy among the normal modes occurs. After colliding a finite number $n$ of times, kink and antikink emerge and move away with final velocity $v_f$. These processes will be referred to as $n$-bounce scattering events. If we plot the final velocity of the scattered kinks as a function of the initial collision velocity $v_0$ we find the diagram displayed in Figure \[fig:velofinalA0000\]. Here, it is assumed that the final velocity for a Bion state is zero. It is clear that the bion formation regime arises for low enough values of the collision velocity $v_0$. On the other hand, if $v_0$ is greater than $0.25988$ then kink and antikink reflect each other after colliding once (blue curve in Figure \[fig:velofinalA0000\]). We will referred to this piece of curve as *the one-bounce tail*. Note that a color code has been used in Figure \[fig:velofinalA0000\] to specify the number of collisions that the kinks suffer before escaping. A surprising fractal pattern turns up in the interval $[0.18,0.25988]$, where the bion formation and kink reflection regimes are interlaced, see zoomed area in Figure \[fig:velofinalA0000\]. ![Final velocity $v_f$ of the scattered kinks as a function of the initial collision velocity $v_0$ of the colliding kinks. The final velocity of a Bion is assumed to be zero. The color code is used to specify the number of bounces suffered by the kinks before escaping. The resonance window has been zoomed and inserted in the Figure.[]{data-label="fig:velofinalA0000"}](velofinalamp000){height="3.5cm"} As previously mentioned, after the first collision in a $n$-bounce scattering process the following collisions involve wobbling kinks due to the fact that the first impact usually excites the shape mode of the initially colliding kinks. It is difficult to monitor the velocities and amplitudes of the resulting kinks after the first impact in an $n$-bounce event because the period of time between bounces is usually very short. For this reason it seems reasonable to directly investigate the collision between wobbling kinks. In this situation the velocity and amplitude of the colliding and scattered wobbling kinks can be monitored, at least, in the one-bounce processes. In any case, this type of scattering events can provide us with a lot of information about the resonant energy transfer mechanism. Scattering between wobbling kinks ================================= The goal of this paper is to analyze the scattering between wobbling kinks. In order to accomplish this task we shall employ numerical approaches based on the discretization of the partial differential equation (\[pde\]). The numerical procedure used in this paper corresponds to an energy conservative second-order finite difference algorithm implemented with Mur boundary conditions. The effect of radiation in the simulation is controlled by this algorithm because the linear plane waves are absorbed at the boundaries. As a complement to this numerical method has also been employed the algorithm described in [@Kassam2005] by Kassam and Trefethen. This scheme is spectral in space and fourth order in time and was designed to solve the numerical instabilities of the exponential time-differencing Runge–Kutta method introduced in [@Cox2002]. The initial settings for our scattering experiments are described by two initially well separated wobbling kinks which are pushed together with initial collision velocity $v_0$. This situation is characterized by the concatenation $$\label{configuration} \phi^{(\pm)}_{\rm WK}(t, x, x_0, v_ 0, \omega, a)\cup \phi^{(\mp)}_{\rm WK}(t, x, -x_0, -v_ 0, \omega, a) = \left\{ \begin{array}{ll}\phi_{\rm WK}^{(\pm)}(t, x, x_0, v_0, \omega, a) & \mbox{if }\,\,x < 0 \\ \phi_{\rm WK}^{(\mp)}(t, x, -x_0, -v_0, \omega, a) & \mbox{if }\,\, x\geq 0 \end{array} \right. \hspace{0.5cm},$$ where $x_0$ is large enough and $$\label{wk} \phi_{\rm WK}^{(\pm)}(t,x;x_0,v_0,\omega,a) = \pm \tanh \left( \frac{x-x_0 - v_0 t}{\sqrt{1-v_0^2}}\right) \, \pm a \, \sin(\omega t)\,{\rm{sech}} \left(\frac{x-x_0 - v_0t}{\sqrt{1-v_0^2}}\right) \tanh \left(\frac{x-x_0 - v_0t}{\sqrt{1-v_0^2}}\right) \hspace{0.2cm} ,$$ has been chosen to comply with the initial condition $\phi_{\rm WK}^{(\pm)}(0,x;x_0,v_0,\omega,a) = \phi_{\rm K}^{(\pm)} (t,x,x_0,v_0)$. The configuration (\[configuration\]) consists of a wobbling kink/antikink with center $-x_0$ located at the left side of an wobbling antikink/kink with center $x_0$. It is clear that if $x_0\gg 0$ and $a\ll 1$ the partial differential equation (\[pde\]) is verified by (\[configuration\]) very approximately. The initial conditions for our problem can be derived from (\[configuration\]) by simply taking $t=0$, that is, $\phi(0,x;x_0,v_0,\omega,a)$ and $\frac{\partial \phi}{\partial t}(0,x;x_0,v_0,\omega,a)$ define the starting point of the numerical algorithm. Note that the configuration (\[configuration\]) is invariant under the spatial reflection transformation $x\mapsto -x$, as it is also the evolution equation (\[pde\]), so all the scattering processes will preserve this symmetry. This means that we can extract all the scattering information by analyzing the features of only one of the scattered kinks. In particular, our numerical experiments have been carried out in a spatial interval $x\in [-100,100]$ where the kink and the antikink centers are initially separated by a distance $d=2x_0=60$. These kink centers have been monitored during the evolution, as well as the number of bounces suffered by the topological defects. In the kink reflection regime this information is used to work out the final velocity of the scattered kinks by employing a linear regression when the kinks are far enough apart from each other. This scheme has been performed for a range of initial velocities $v_0$ usually covering the interval $v_0\in [0.1,0.9]$ with initial velocity steps $\Delta v_0=0.001$, which is decreased to $\Delta v_0=0.00001$ in the resonance range. These data allow us to study the dependence of the separation velocity of the scattered kinks as a function of the collision velocity $v_0$, which can be graphically represented by means of diagrams similar to Figure \[fig:velofinalA0000\]. Once the position and the velocity of the kink centers have been determined the wobbling amplitude and frequency are also estimated. To do this, the difference between the numerical profile and a non-excited traveling kink, both of them with the same center $x_C$ and velocities $v_f$ is evaluated at the points $x_M^{\pm}$ for every time step in the simulation. The choice of these points underlies the fact that the shape fluctuation has its maximum/minimum values at the points $x_M^{\pm}$. The time series constructed in this way is analyzed by using a fast Fourier transform algorithm. In order to explore the dependence of the final velocity on the initial wobbling amplitude of the colliding kinks the previously described numerical scheme has been replicated for different values of the amplitude $a$ considered in the initial configuration. In these numerical experiments we shall consider only positive values of $a$. Negative values of $a$ are simply related with the positive ones by adding a phase in the argument of the oscillatory factor $\sin (\omega t)$ in . To get a better understanding of the phenomena associated with this type of scattering processes is convenient to distinguish two different regimes which depend on the magnitude of the amplitude $a$. They are determined as follows: 1. *Scattering between weakly wobbling kinks:* This scenario comprises those scattering processes where the initial amplitude $a$ of the colliding wobbling kinks is $|a| < 0.05$. In these cases the amplitude decay effect is assumed to be negligible such that the wobbling amplitude of the evolving kinks at the time of impact is approximately equal to the initial one. It is clear that this kind of events allows a better control on the variables of the scattering problem. The mechanisms that begin to deform the velocity diagram with respect to the pattern found in Figure \[fig:velofinalA0000\] when $a$ is increased can already be perceived in these cases. These novel behaviors will be discussed in Section 3.1. 2. *Scattering between strongly wobbling kinks:* It is expected that the more intense phenomena take place when the wobbling amplitudes of the colliding kinks are relatively large. We assumed that these cases are determined by the condition $|a| \geq 0.1$. Now, the amplitude decay suffered by the wobbling kinks in the time period lapsed between the beginning of the simulation and the kink collision (approximately $d/(2v_0)$) could be significant. Therefore, it is difficult to estimate the value of the wobbling amplitude immediately before the impact, which is from our point of view the more significant variable. Despite this fact this type of events plays an essential role in the resonance mechanism and for this reason it will be discussed in Section 3.2. We shall analyze the dependence of some scattering parameters on the initial wobbling amplitude of our numerical experiments taking into account that the value of the collision amplitude will be smaller than the initial one. Obviously, the higher the initial magnitude is the higher the collision amplitude is. Scattering between weakly wobbling kinks ---------------------------------------- In this section, we shall consider the scattering of kinks whose initial wobbling amplitudes $a$ are small. As previously mentioned it is assumed that in these cases the decay of the wobbling amplitude is a residual effect. Thus, the magnitude of the wobbling amplitude of the kinks just before colliding must be approximately equal to the initial one. This first regime of kink scattering has been numerically investigated in the initial wobbling amplitude range $a\in [0,0.05]$ taking an amplitude step $\Delta a=0.001$ for $0 \leq a \leq 0.02$ and $\Delta a=0.01$ for $a>0.02$. A characteristic velocity diagram obtained is displayed in Figure \[fig:velofinalA0020\], where the dependence of the final velocity $v_f$ of the scattered kinks on the initial collision velocity $v_0$ is graphically represented when the initial wobbling amplitude is $a=0.02$. Although this value is relatively small the diagram displayed in Figure \[fig:velofinalA0020\] introduces novel features with respect to the classical diagram presented in Figure \[fig:velofinalA0000\]. ![Final velocity $v_f$ of the scattered kinks as a function of the initial collision velocity $v_0$ of the colliding wobbling kinks with initial amplitude $a=0.02$. The final velocity of a Bion is assumed to be zero. The color code is used to specify the number of bounces suffered by the kinks before escaping. The resonance window has been zoomed and inserted in the Figure.[]{data-label="fig:velofinalA0020"}](velofinalamp002){height="3.5cm"} It can be observed that the complexity of the fractal structure grows as the initial wobbling amplitude $a$ increases. A first sign of this fact is that the fractal structure interval is widened as $a$ increases. For example, this interval is approximately $[0.155,0.277]$ for the case $a=0.02$ whereas it is approximately $[0.18,0.26]$ for the case $a=0$. A second indicator is the growth in the number of resonance windows. Indeed, this effect is caused by a resonance window splitting mechanism, which is illustrated in Figure \[fig:windowsplit\]. Before examining this process it is worthwhile to bring our attention to another novel property of the diagram in Figure \[fig:velofinalA0020\]: the presence of isolated 1-bounce windows in the fractal structure, which, in turn, are surrounded by other $n$-bounce windows with $n\geq 2$. This feature does not arise in the classical velocity diagram with zero initial energy on the shape mode displayed in Figure \[fig:velofinalA0000\] and its origin seems to be produced by two different procedures. Figure \[fig:velofinalA0020\] shows the existence of two isolated 1-bounce windows approximately in the intervals $[0.246,0.251]$ and $[0.261,0.272]$, each of them generated by different channels. They are described as follows: 1. *1-bounce reflection tail splitting*: This process is based on the oscillatory behavior of the 1-bounce tail arising for large initial velocities and represented by blue curves in Figures \[fig:velofinalA0000\] and \[fig:velofinalA0020\]. For $a=0$ this 1-bounce tail is a monotonically increasing function, see Figure \[fig:velofinalA0000\]. However, this curve ceases to follow that behavior and begins to oscillate as the amplitude $a$ grows, see Figure \[fig:velofinalA0020\]. The amplitude of these oscillations becomes bigger as the value of $a$ grows, overall at the beginning of the 1-bounce tail. When the amplitude $a$ is large enough the minima of the previously mentioned oscillations can intercept the $v_0$-axis, reaching a zero final velocity. As a consequence an isolated 1-bounce window arises and the gap between this window and the 1-bounce tail is filled with new $n$-bounce windows. This phenomenon can be triggered repeatedly as $a$ increases giving rise to several isolated 1-bounce windows embedded in the resonance regime. The previously described mechanism can be visualized in Figure \[fig:windowsplit\], where the final velocity versus initial velocity diagrams have been plotted for the three close initial amplitudes $a=0.013$, $a=0.014$ and $a=\,0.015$. We can observe the formation of an isolated 1-bounce window approximately in the interval $v_0\in [0.26,0.273]$. ![Velocity diagrams for the wobbling kink scattering with initial amplitudes $a=0.013$, $a= 0.014$ and $a=0.015$ for the initial velocity interval $v_0\in [0.2260,0.2907]$. This sequence of graphics illustrates the formation of isolated 1-bounce windows and the 2-bounce window splitting mechanism.[]{data-label="fig:windowsplit"}](amp0013_vel_0.22-0.29 "fig:"){height="3.2cm"} ![Velocity diagrams for the wobbling kink scattering with initial amplitudes $a=0.013$, $a= 0.014$ and $a=0.015$ for the initial velocity interval $v_0\in [0.2260,0.2907]$. This sequence of graphics illustrates the formation of isolated 1-bounce windows and the 2-bounce window splitting mechanism.[]{data-label="fig:windowsplit"}](amp0014_vel_0.22-0.29 "fig:"){height="3.2cm"} ![Velocity diagrams for the wobbling kink scattering with initial amplitudes $a=0.013$, $a= 0.014$ and $a=0.015$ for the initial velocity interval $v_0\in [0.2260,0.2907]$. This sequence of graphics illustrates the formation of isolated 1-bounce windows and the 2-bounce window splitting mechanism.[]{data-label="fig:windowsplit"}](amp0015_vel_0.22-0.29 "fig:"){height="3.2cm"} 2. *Spontaneous emergence in the resonance phase*: The other process, instead, is characterized by the appearance of windows inside the resonance interval. In these new windows the wobbling kinks collide only once before escaping. The explanation for this fact seems to be that the additional energy carried by the excited kinks in these scattering events due to shape mode allows the kinks to escape in initial velocity windows where this was not possible before. For example, the formation of the first window of this kind happens approximately for $a=0.011$ around the value $v_0=0.249$. As the value of $a$ increases the width of these windows widens. Indeed, this first window can be observed in Figure \[fig:windowsplit\] for the initial amplitudes $a=0.013$, $a=0.014$ and $a=0.015$. The second window of this class arises for $a=0.030$ around $v_0=0.233$. From here the number of the windows grows enormously, see Figure \[fig:1bwindowformation\]. Figure \[fig:1bwindowformation\] illustrates the combined effect of the previously mentioned processes of production of isolated 1-bounce reflection windows. This figure shows the evolution of the velocity diagrams associated to 1-bounce events as the initial wobbling amplitude $a$ increases from $a=0$ (red curve) to $a=0.1$ (dark blue curve). For the sake of clarity, $n$-bounce processes with $n\geq 2$ are not included in this graphics. Together to the reflection tail splitting and the spontaneous emergence processes another curious behavior is displayed in Figure \[fig:1bwindowformation\]. The oscillations of the final versus initial velocity curves for the different values of the amplitude $a$ have common nodes. They intersect each other at the same points (at least in a large degree of approximation). ![Velocity diagram associated with 1-bounce scattering events for initial wobbling amplitudes ranging in the interval $a \in [0,\,0.10]$. This graphics illustrates the formation of isolated 1-bounce windows. For the sake of clarity, $n$-bounce procesees with $n\geq 2$ have not been included in this figure.[]{data-label="fig:1bwindowformation"}](VelFinalA0To01Evo1Bounce){height="3.5cm"} Now, let us return to the previously mentioned resonance window splitting mechanism. If we observe Figure \[fig:windowsplit\] around the initial velocity $v_0=0.24$, we will witness the split of a 2-bounce window into other two narrower 2-bounce windows. As before, the gap between these two new 2-bounce windows is occupied by new $n$-bounce windows with $n>2$. To emphasize the behavior of this newfangled feature the evolution of the first 2-bounce window found in the classical velocity diagram for $a=0$ (see Figure \[fig:velofinalA0000\]) as the value of the wobbling amplitude $a$ increases is shown in Figure \[fig:break2bwindows\]. It can be observed that the initial 2-bounce window $v_0\in [0.1920,0.2028]$ (represented by a red curve) gives rise to three new 2-bounce windows $v_0\in [0.1940, 0.1946]$, $v_0\in [0.1990,0.1998]$ and $v_0\in [0.2039,0.2046]$ (represented by blue curves) for $a=0.02$. This process is repeated for the majority of the resonance windows as the ![Evolution of the first 2-bounce window found in the velocity diagram for $a=0$ (*red curve*) as the value of the initial wobbling amplitude increases up to $a=0.02$ (*dark blue curves*). For the sake of clarity only 2-bounce scattering events have been included in this graphics.[]{data-label="fig:break2bwindows"}](VelFinalA0To01Evo2Bounce){height="3.5cm"} initial wobbling amplitude grows escalating the complexity of the fractal pattern in the resonance phase. Note that the formation of isolated 1-bounce windows leads to an ambiguity in the concept of the *critical velocity* $v_c$. This term was introduced in the context of kink-antikink scattering [@Campbell1983; @Campbell1986] and was defined as the lowest velocity at which the 1-bounce reflection regime takes place, or alternately, the lowest velocity at which the 1-bounce tail starts and no more bion states or more multi-bounces are observed. These two definitions coincide when the two colliding kinks are not wobbling because there is only one (blue) piece of 1-bounce curve, see Figure \[fig:velofinalA0000\], but they can differ in other cases. In order to remove this ambiguity we shall distinguish these two velocities, referring to the first one as the *1-bounce reflection minimum escape velocity* $v_{r}$ whereas the second one will be called *1-bounce tail minimum escape velocity* $v_t$. Since more energy in the vibrational mode means that more energy can be released to the translation mode in the scattering process through the resonant energy transfer mechanism, it is expected that 1-bounce windows become more prevalent as the value of the wobbling amplitude $a$ grows. Consequently, it is presumed that the velocity $v_r$ is a decreasing function of the amplitude $a$. On the other hand, the isolated 1-bounce window formation previously explained implies that the velocity $v_t$ must grow as the amplitude $a$ increases. In the transition from initial amplitude $a_0 = 0$ to $a_0 = 0.014$ the escape velocity $v_t$ is observed to increase logarithmically, however, this pattern is broken by the existence of a discontinuity due to the formation of the first isolated 1-bounce window from the reflection tail. After discussing the features of the velocity diagram as a function of the initial amplitude $a$, we shall illustrate some particular processes. In Figure \[fig:amp002vel028\] (left) a wobbling kink and a wobbling antikink with collision velocity $v_0=0.285$ and initial amplitude $a = 0.02$ approach each other, collide, bounce back and move away with final velocity $v_f=0.144805$ and wobbling amplitude $a_f = 0.175462$ after the collision. These dynamical parameters are the same for the kink and the antikink, in agreement with spatial reflection symmetry. Once they do not collide back after the first bounce the energy stored in the vibrational mode remains there and propagates within the kinks. The process displayed in Figure \[fig:amp002vel028\] (right) describes a 3-bounce event with initial velocity $v_0=0.25737$ and amplitude $a=0.02$. In this case, the scattered kink and antikink travel away with velocity $v_f=0.219$ and wobbling amplitude $a_f=0.003085$. We can see the resonant energy transfer mechanism in action in these cases. In the 1-bounce event the outcome amplitude $a_f$ is bigger than the initial amplitude $a$, which evinces an energy transfer from the translational mode to the shape mode being the final separation velocity $v_f$ less than the initial one $v_0$. On the other hand, in the 3-bounce process this mechanism takes place three times redistributing the energy between the kinetic and vibrational energy pools after every collision. Clearly, in the first impact the shape mode gains energy at the expense of the zero mode, which finally recovers part of that energy in the third collision allowing the kinks to escape. Note that radiation emission is also involved in these processes. ![Scattering processes between two wobbling kinks with initial amplitude $a_0 = 0.02$ and collision velocities $v_0 = 0.285$ (left) and $v_0=0.25737$ (right). The final velocities and wobbling amplitudes for these events are, respectively, $v_f=0.144805$, $a_f = 0.175462$ and $v_f=0.219$ , $a_f = 0.003085$.[]{data-label="fig:amp002vel028"}](amp_0.02_vel_0.285 "fig:"){height="5.0cm"} ![Scattering processes between two wobbling kinks with initial amplitude $a_0 = 0.02$ and collision velocities $v_0 = 0.285$ (left) and $v_0=0.25737$ (right). The final velocities and wobbling amplitudes for these events are, respectively, $v_f=0.144805$, $a_f = 0.175462$ and $v_f=0.219$ , $a_f = 0.003085$.[]{data-label="fig:amp002vel028"}](amp_0.02_vel_0.25737 "fig:"){height="5.0cm"} In Figure \[fig:amp0014vel0\] we have decided to illustrate the behavior of two extreme scattering events, which are near to metastable configurations. In the first process, left plot, kink and antikink approach each other with initial velocity $v_0=0.24691$ and wobbling amplitude $a=0.013$, collide and bounce back. For a long time, they apparently remain motionless at a fixed distance. In this particular simulation this situation takes approximately 400 time units in the dimensionless coordinates introduced in Section 2. Finally, the lumps end up approaching again to form a bion state. In the second simulation Figure \[fig:amp0014vel0\] (right) the kinks initially travel with velocity $v_0=0.27420$ and wobbling amplitude $a=0.015$; after colliding the kink and antikink remain in a similar quasi-metastable estate which was previously described, although in this case after the second collision the kinks are able to escape with final velocity $v_f=0.147041$ and wobbling amplitude $a_f=0.15351$. This type of scattering events are difficult to monitor because there will always be processes whose metastable phase will last more than any simulation time. Indeed, this is the reason for the gap in the velocity diagram introduced in Figure \[fig:windowsplit\] (middle) around $v_0=0.274$. ![Scattering processes between two wobbling kinks with initial velocities and amplitudes $v_0=0.24691$, $a = 0.013$ (left) and $v_0=0.27420$, $a = 0.015$ (right). In the first case a bion state is formed whereas in the second case the scattered kinks have $v_f=0.147041$ and $a_f=0.15351$.[]{data-label="fig:amp0014vel0"}](amp_0.013_vel_0.24691 "fig:"){height="5.0cm"} ![Scattering processes between two wobbling kinks with initial velocities and amplitudes $v_0=0.24691$, $a = 0.013$ (left) and $v_0=0.27420$, $a = 0.015$ (right). In the first case a bion state is formed whereas in the second case the scattered kinks have $v_f=0.147041$ and $a_f=0.15351$.[]{data-label="fig:amp0014vel0"}](amp_0.015_vel_0.27420 "fig:"){height="5.0cm"} Clearly, in the previous simulations the wobbling mode is strongly excited after the first collision. This is a general pattern as we can observe in Figure \[fig:ampxvelamp002\], which exhibits the final wobbling amplitude of the scattered kinks after the last collision as a function of the initial velocity $v_0$ for two different values of the initial wobbling amplitude $a=0.0$ and $a=0.02$. The analysis of these data, specially for the 1-bounce events, can lead to a very valuable information to understand the resonant energy transfer mechanism. 1-bounce events can be considered as elementary processes in the kink scattering because $n$-bounce events can be understood as a reiteration of $n$ 1-bounce events. The most surprising fact is that the final amplitude for the 1-bounce processes is almost independent of the initial wobbling amplitude of the colliding kinks. We can observe that this magnitude follows a linear dependence on $v_0$ very approximately, which can be fitted by the expression $$a(v_0)=0.084 + 0.34 v_0 \hspace{0.5cm}.$$ ![Wobbling amplitude $a_f$ of the scattered kinks after the last impact as a function of the collision velocity for the initial amplitude $a=0$ (top panel) and $a=0.02$ (bottom panel). The same color code employed in the previous figures is used to specify the number of bounces suffered by the kinks before escaping.[]{data-label="fig:ampxvelamp002"}](AmplitudFinalA0000 "fig:"){height="3.5cm"}\ ![Wobbling amplitude $a_f$ of the scattered kinks after the last impact as a function of the collision velocity for the initial amplitude $a=0$ (top panel) and $a=0.02$ (bottom panel). The same color code employed in the previous figures is used to specify the number of bounces suffered by the kinks before escaping.[]{data-label="fig:ampxvelamp002"}](AmplitudFinalA0020 "fig:"){height="3.5cm"} What is clear from Figure \[fig:ampxvelamp002\] is that all the 1-bounce events produce a strong excitation of the wobbling mode, which in all the cases range approximately in the interval $a_f \in [0.15, 0.4]$. Obviously, the more the impact velocity of the colliding kinks is, the more excited the scattered kinks become. For moderate collision impact the final wobbling amplitude is in the range $a_f\in [0.15,0.25]$. A first consequence of this high vibrational excitation in 1-bounce scattering events for initially weakly wobbling kinks is that the $n$-bounce events in this regime necessarily involve the scattering of strongly wobbling kinks in one or several of the intermediate collision processes (at least in the second one). A second consequence is that the direction of energy transfer in these 1-bounce events is always from the kinetic energy to the vibrational energy. Notice that the final velocity $v_f$ of the scattered kinks in 1-bounce events displayed in Figure \[fig:velofinalA0020\] is always less than the initial velocity $v_0$. The isolated 1-bounce windows found in this regime proves that there exist some initial velocity intervals where less energy is transferred to the shape mode, which allows the kinks keep enough kinetic energy to escape. The reverse processes must involve the scattering between strongly wobbling kinks. Another surprising result is given by the dependence of the wobbling frequency $\omega_f$ of the scattered kinks on the collision velocity $v_0$. Note that the shape mode coming from the second order small kink fluctuation operator expressed in (\[schrodingerlike\]) vibrates with the frequency $\omega=\sqrt{3}$. It would seem reasonable to assume that this frequency is kept constant at least by the time the scattered kinks are far away. Figure \[fig:frexvelamp002\] shows this magnitude as a function of the initial velocity $v_0$. It can be observed that the final wobbling frequency $\omega_f$ is a decreasing function on the variable $v_0$, which goes from the value $\sqrt{3}$ for small values of $v_0$ and arrives approximately to $0.865$ at $v_0=0.9$ in the graphics. The previously supposed behavior is confirmed only for values of $v_0< 0.4$. For greater values of $v_0$ the scattered kinks vibrate slower. It is difficult to provide a reason for this behavior. It could be a behavior of the wobbling kinks when they are traveling away at high velocities or it could be caused exclusively by the impact. Recall that the spectral analysis of the wobbling mode requires to study its evolution for a large enough lapse of time. Other remarkable fact is that the two graphics in Figure \[fig:frexvelamp002\] are almost identical, which implies that the behavior of the final frequency is practically independent of the initial wobbling amplitude $a$ for the cases presented in this Section. ![Wobbling frequency of the scattered kinks after the last impact as a function of the collision velocity for the initial amplitude $a=0$ (top panel) and $a=0.02$ (bottom panel). The same color code employed in the previous figures is used to specify the number of bounces suffered by the kinks before escaping.[]{data-label="fig:frexvelamp002"}](FrecuenciaFinalA0000 "fig:"){height="3.5cm"}\ ![Wobbling frequency of the scattered kinks after the last impact as a function of the collision velocity for the initial amplitude $a=0$ (top panel) and $a=0.02$ (bottom panel). The same color code employed in the previous figures is used to specify the number of bounces suffered by the kinks before escaping.[]{data-label="fig:frexvelamp002"}](FrecuenciaFinalA0020 "fig:"){height="3.5cm"} As previously mentioned in this Section, the scattering between strongly wobbling kinks is always present in the resonance phenomenon because independently of the initial velocity $v_0$ the second collision will involve a strongly wobbling kink scattering event. For this reason the next section will be devoted to discuss the properties of this kind of more violent events. We shall emphasize the deviations of this new scenario from that introduced in this section. Scattering between strongly wobbling kinks ------------------------------------------ This class of kink scattering processes are characterized by a relatively large value of the initial wobbling amplitude, which is assumed to be $|a| \geq 0.1$. In this section this regime has been analyzed for events with an initial amplitude $a$ in the interval $a\in [0.1,0.2]$ by taking an amplitude step $\Delta a=0.01$. As usual we shall begin by examining the dependence of the final velocity $v_f$ of the scattered kinks on the initial velocity $v_0$. This function has been plotted in Figure \[fig:velofinalA0200\] for the particular cases $a=0.1$ and $a=0.2$, which exhibit the representative properties of this regime. The global behavior of these velocity diagrams is similar to that described in Section 3.1, see Figure \[fig:velofinalA0020\], although they include important differences. ![Final velocity $v_f$ of the scattered kinks as a function of the initial collision velocity $v_0$ of the colliding wobbling kinks with initial amplitudes $a=0.1$ (top panel) and $a=0.2$ (bottom panel). The final velocity of a Bion is assumed to be zero. The color code is used to specify the number of bounces suffered by the kinks before escaping. The part of the resonance window has been zoomed and inserted in the Figure.[]{data-label="fig:velofinalA0200"}](VeloFinalA0100){height="3.5cm"} ![Final velocity $v_f$ of the scattered kinks as a function of the initial collision velocity $v_0$ of the colliding wobbling kinks with initial amplitudes $a=0.1$ (top panel) and $a=0.2$ (bottom panel). The final velocity of a Bion is assumed to be zero. The color code is used to specify the number of bounces suffered by the kinks before escaping. The part of the resonance window has been zoomed and inserted in the Figure.[]{data-label="fig:velofinalA0200"}](VeloFinalA0200){height="3.5cm"} First, the fractal structure becomes even more intricate than the scenario found in Section 3.1. The interval where the resonance phenomenon takes place keeps widening as $a$ grows. In addition to this, when the value of $a$ is large enough the number of isolated 1-bounce windows explodes and the sequence of these windows forms a fractal structure clustered near the origin of the graphics, see bottom panel in Figure \[fig:velofinalA0200\]. The initial velocities $v_0$ around the peak of these windows can define initial velocity intervals where the scattered kinks move faster than the colliding kinks, $v_f>v_0$. In these cases a part of the vibrational kink energy accumulated in the shape mode is transferred to the kinetic energy, which becomes bigger than its initial value. It can be observed that this phenomenon occurs for low initial velocities and ceases to happen for high values (when the kinetic energy is large). For example, for $a=0.2$ the height of the windows in the resonance phase exceeds the elastic limit approximately when $v_0< 0.34$. Obviously, as the value of the initial amplitude $a$ increases this threshold velocity grows because the vibrational energy becomes bigger. This scenario is a fundamental link in the chain of the resonant energy transfer mechanism because allows relatively slow scattered wobbling kinks to escape in a multiple bounce event in the last collision by transferring vibrational energy to the kinetic energy pool. Figure \[fig:amp02vel028\] illustrates this kind of processes. In this graphics a kink and antikink with initial velocity $v_0=0.1506$ and wobbling amplitude $a=0.2$ approach each other and collide only once before escaping. As we can see, the scattered kink and antikink move away with final velocity $v_f=0.246454$ while its wobbling amplitude is approximately $a=0.0215$. The final outcome in this event is that the kink and antikink are speeded up whereas its wobbling is softened. ![Scattering between two wobbling kinks with initial amplitude $a_0 = 0.2$ and collision velocity $v_0 = 0.1406$. The final velocity $v_f$ of the scattered kinks is $v_f=0.246454$, so the kinks move faster after the collision.[]{data-label="fig:amp02vel028"}](amp_0.2_vel_0.1406){height="5.0cm"} ![Wobbling amplitude of the scattered kinks after the last impact as a function of the collision velocity $v_0$ for the initial amplitude $a=0.1$ (top panel) and $a=0.2$ (bottom panel). The same color code employed in previous Figures is used to specify the number of bounces suffered by the kinks before escaping.[]{data-label="fig:ampxvelamp02"}](AmplitudFinalA0100 "fig:"){height="3.4cm"}\ ![Wobbling amplitude of the scattered kinks after the last impact as a function of the collision velocity $v_0$ for the initial amplitude $a=0.1$ (top panel) and $a=0.2$ (bottom panel). The same color code employed in previous Figures is used to specify the number of bounces suffered by the kinks before escaping.[]{data-label="fig:ampxvelamp02"}](AmplitudFinalA0200 "fig:"){height="3.4cm"} Figure \[fig:amp0014vel0\] (right) represents a 2-bounce kink scattering process, which was introduced in Section 3.1. Here, kink and antikink approach each other with initial velocity $v_0=0.2742$ while vibrating with amplitude $a=0.015$. Notice, thus, that the first collision is a weakly wobbling kink scattering event. As we know, after this first impact an important part of the kinetic energy is devoted to excite the shape mode and emit radiation, such that the resulting kink and antikink move very slowly but vibrates intensely. In these circumstances the attraction force between the kink and the antikink makes them approach again. This evolution is now described by a strongly wobbling kink scattering event. The lumps collide and bounce back, but now the resonant energy transfer mechanism is reversed and the kink and antikink velocities are large enough to let them escape. They travel away with final velocity $v_f=0.147041$ and final wobbling amplitude $a=0.153513$. Figure \[fig:amp002vel028\] (right) represents a 3-bounce kink scattering process, where a similar behavior takes place, although the intermediate stages are much shorter. They finally move away with velocity $v_f=0.219006$ and wobbling amplitude $a=0.00308$. The behavior of the final wobbling amplitude $a_f$ as a function of the initial velocity is plotted in Figure \[fig:ampxvelamp02\]. The amplitude can undergo important fluctuations when the initial velocity varies, which increases as the value of $a$ grows. These oscillations can be observed, for example, in the 1-bounce reflection tail. The range of the wobbling amplitudes found in these cases is similar to that described in Section 3.1 for weakly wobbling kink scattering processes. The minima of these fluctuations can reach very low values. In the resonance phase these oscillations are more accentuated for these 1-bounce events than in the previous regime. Obviously, the detailed behavior of the amplitude in the resonance phase is completely particular for every value of $a$ due to the presence of the fractal structure. Besides, the dependence of the final frequency $\omega_f$ on the initial velocity $v_0$ completely resembles the result found in Section 3.1, see Figure \[fig:frexvelamp02\]. Therefore, the final frequency seems to be a magnitude almost independent of the initial wobbling amplitude $a$ beyond the resonance regime. ![Wobbling frequency of the scattered kinks after the last impact as a function of the collision velocity $v_0$ for the initial amplitude $a=0$ (top panel) and $a=0.02$ (bottom panel). The same color code as in previous figures is used to specify the number of bounces suffered by the kinks before escaping.[]{data-label="fig:frexvelamp02"}](FrecuenciaFinalA0100 "fig:"){height="3.4cm"}\ ![Wobbling frequency of the scattered kinks after the last impact as a function of the collision velocity $v_0$ for the initial amplitude $a=0$ (top panel) and $a=0.02$ (bottom panel). The same color code as in previous figures is used to specify the number of bounces suffered by the kinks before escaping.[]{data-label="fig:frexvelamp02"}](FrecuenciaFinalA0200 "fig:"){height="3.4cm"} Conclusions and further comments ================================ In this paper we have addressed the scattering between wobbling kinks in the $\phi^4$ model. In addition to its intrinsic interest the study of these processes can give us insight into the resonant energy transfer mechanism. We must take into account that two traveling non-excited kinks become wobbling kinks after the first collision due to the energy exchange between the translational mode and the shape mode. In this sense, a $n$-bounce scattering process can be considered as the reiteration of $n$ 1-bounce collisions, most of them between wobbling kinks. In this work the influence of the collision velocity and the initial wobbling amplitude on the scattering processes have been directly investigated. The fractal structure arising in the resonance regime of the final versus initial velocity diagram becomes more intricate as the value of the initial wobbling amplitude of the colliding kinks increases. This growing complexity is caused by two different mechanisms: the 1-bounce reflection tail splitting and the spontaneous emergence of resonance windows. The first case is produced by the oscillations of the 1-bounce reflection tail when the initial wobbling amplitude grows. When the amplitude is large enough this curve can intercept the $v_0$-axis creating an isolated 1-bounce window in the resonance regime. The gap between this new window and the 1-bounce tail is filled with new $n$-bounces windows, with $n > 1$. The same phenomenon is replicated for $n$-bounce windows, which are broken up into narrower new $n$-bounce windows and as before the gap between them is occupied with $N$-bounce windows, with $N > n$. The second mechanism is directly triggered by the extra energy carried by the initially excited shape mode of the wobbling kinks. New bounce windows emerge for ever-smaller initial velocities as the value of the amplitude increases. As a consequence, the fractal structure interval becomes larger and larger as $a$ grows. On the other hand, the final wobbling amplitude of the scattered kinks involve a very approximately linear dependence on the initial velocity outside the resonance phase although some oscillations arise for large enough values of the initial amplitude. 1-bounce events between weakly wobbling kinks always give rise to strongly wobbling kinks moving away. On the other hand, weakly wobbling kinks can emerge from the collision between strongly wobbling kinks only for relatively small values of the initial velocity. A surprising result is that the frequency of the shape mode is a decreasing function of the collision velocity starting from the natural frequency $\omega=\sqrt{3}$ found in the linear analysis. This curve is almost independent of the value of the initial wobbling amplitude. It is worthwhile to mention that for strongly wobbling kink collisions there exist 1-bounce windows where the scattered kinks will travel faster than the colliding kinks. This occurs for relatively low values of the initial velocity. This behavior implies that the last collision in every $n$-bounce scattering event with $n>1$ must involve the presence of strongly wobbling kinks approaching each other at a relatively low speed. In a multiple bounce scattering process the kinks approach each other and bounce back again and again until the next collision velocity and wobbling amplitude fall into one of the previously mentioned 1-bounce windows. In the bion formation regime the successive collisions are not able to excite the shape mode enough to trigger this escape manoeuvre. The research introduced here open some possibilities for future works. The $\phi^6$ model involves a similar resonance regime as the $\phi^4$ model although it does not present vibrational eigenstates in the second order small fluctuation operator. The characteristics of the scattered wobbling kinks can be analyzed to study its influence on the resonant energy transfer mechanism. Alternatively, a twin model to the $\phi^6$ model involving internal modes can be constructed. By doing this, we could compare the scattering processes of the twin model with those of the standard $\phi^6$ model. In this way the role played by the shape modes in the collision process could be examined. Moreover, many other different topological defects (kinks in the double sine-Gordon model, deformed $\phi^4$ models, hybrid and hyperbolic models, etc) could be studied in the new perspective presented here. Acknowledgments {#acknowledgments .unnumbered} =============== A. Alonso-Izquierdo acknowledges the Junta de Castilla y León for financial support under grants BU229P18 and SA067G19. L.M. Nieto acknowledges the Junta de Castilla y León and FEDER Projects for financial support under grants VA057U16, VA137G18 and BU229P18. L J.V. Queiroga-Nunes aknowledges for financial support of Santander Group under scholarship program UVa - Santander Iberoamerica+Asia. 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--- abstract: 'The optimal causal (zero-delay) coding of a partially observed Markov process is studied, where the cost to be minimized is a bounded, non-negative, additive, measurable single-letter function of the source and the receiver output. A structural result is obtained extending Witsenhausen’s and Walrand-Varaiya’s structural results on optimal causal coders to more general state spaces and to a partially observed setting. The decentralized (multi-terminal) setup is also considered. For the case where the source is an i.i.d. process, it is shown that an optimal solution to the decentralized causal coding of correlated observations problem is memoryless. For Markov sources, a counterexample to a natural separation conjecture is presented.' author: - Serdar Yüksel$^1$ title: 'On Optimal Causal Coding of Partially Observed Markov Sources in Single and Multi-Terminal Settings' --- Introduction ============ This paper considers optimal causal encoding/quantization of partially observed Markov processes. We begin with providing a description of the system model. We consider a partially observed Markov process, defined on a probability space $(\Omega,{\cal F},P)$ and described by the following discrete-time equations for $t \geq 0$: $$\begin{aligned} x_{t+1} &=& f(x_t, w_t), \label{sourceModel} \\ y^i_t&=&g^i(x_t,r^i_t), \label{channelModel}\end{aligned}$$ for (Borel) measurable functions $f, g^i,i=1,2$, with $\{w_t, r^i_t, i=1,2\}$ i.i.d., mutually independent noise processes and $x_0$ a random variable with probability measure $\nu_0$. Here, we let $x_t \in \mathbb{X}$, and $y^i_t \in \mathbb{Y}^i$, where $\mathbb{X},\mathbb{Y}^i$ are complete, separable, metric spaces (Polish spaces), and thus, include countable spaces or $\mathbb{R}^n$, $n \in \mathbb{N}_+$. Let an encoder, Encoder $i$, be located at one end of an observation channel characterized by (\[channelModel\]). The encoders transmit their information to a receiver (see Figure \[LLL\]), over a discrete noiseless channel with finite capacity; that is, they quantize their information. The information at the encoders may also contain feedback from the receiver, which we clarify in the following. Let us first define a quantizer. Let ${\cal M} = \{1,2,\dots,M\}$ with $M=|{\cal M}|$. Let $\mathbb{A}$ be a topological space. A quantizer $Q(\mathbb{A};{\cal M})$ is a Borel measurable map from $\mathbb{A}$ to ${\cal M}$. $\diamond$ When the spaces $\mathbb{A}$ and ${\cal M}$ are clear from context, we will drop the notation and denote the quantizer simply by $Q$. We refer by a [**Composite Quantization Policy**]{} $\Pi^{comp,i}$ of Encoder $i$, a sequence of functions $\{Q_t^{comp,i}, t \geq 0 \}$ which are causal such that the quantization output at time $t$, $q^i_t$, under $\Pi^{comp,i}$ is generated by a causally measurable function of its local information, that is, a mapping measurable on the sigma-algebra generated by $$I^i_t=\{y^i_{[0,t]}, q^1_{[0,t-1]}, q^2_{[0,t-1]} \}, \quad t \geq 1,$$ and $I^i_0=\{y^i_0\},$ to a finite set ${\cal M}^i_t$, the quantization output alphabet at time $t$ given by $${\cal M}^i_t := \{1,2,\dots, |{\cal M}^i_t|\},$$ for $0 \leq t \leq T-1$ and $i=1,2$. Here $\{{\cal M}^i_t\}$ are fixed in advance and do not depend on the realizations of the random variables. Here, we have the notation for $t \geq 1$: $$y^i_{[0,t-1]} = \{y^i_s, 0 \leq s \leq t-1 \}.$$ Let $$\mathbb{I}^i_t = \bigg( \prod_{s=0}^{t-1} {\cal M}^1_s \times {\cal M}^2_s \bigg) \times (\mathbb{Y}^i)^{t+1}, \quad t \geq 1, \quad \quad \mathbb{I}^i_0 = \mathbb{Y}^i,$$ be information spaces such that for all $t \geq 0$, $I^i_t \in \mathbb{I}^i_t$. Thus, $$Q_t^{comp,i}: \mathbb{I}^i_t \to {\cal M}^i_t.$$ We may express, equivalently, the policy $\Pi^{comp,i}$ as a composition of a [**Quantization Policy**]{} $\Pi^i$ and a [**Quantizer**]{}. A quantization policy of encoder i, ${\cal T}^i$, is a sequence of functions $\{T^i_t\}$, such that for each $t \geq 0$, $T^i_t$ is a mapping from the information space $\mathbb{I}^i_t$ to a space of quantizers $\mathbb{Q}^i_t$. A quantizer, subsequently is used to generate the quantizer output. That is for every $t$ and $i$, $T^i_t(I_t) \in \mathbb{Q}^i_t$ and for every $I^i_t \in \mathbb{I}^i_t$, we will adopt a following representation $$\begin{aligned} \label{duplicate} Q_t^{comp,i}(I^i_t) = (T^i_t(I^i_t)) (I^i_t), \end{aligned}$$ mapping the information space to ${\cal M}^i_t$ in its most general form. We note that even though there may seem to be duplicated information in (\[duplicate\]) (since a map is used to pick a quantizer, and the quantizer maps the available information to outputs) we will eliminate any informational redundancy: A quantizer action will be generated based on the common information at the encoder and the receiver, and the quantizer will map the relevant private information at the encoder to the quantization output. Such a separation in the design will also allow us to use the machinery of Markov Decision Processes to obtain a structural method to design optimal quantizers, to be clarified further, without any loss in optimality. That is, let the information at the receiver at time $t \geq 0$ be $I^r_t=\{q^1_{[0,t-1]},q^2_{[0,t-1]}\}$. The common information, under feedback information, in the encoders and the receiver is the set $I^r_t \in \bigg( \prod_{s=0}^{t-1} {\cal M}^1_s \times {\cal M}^2_s \bigg)$. Thus, we can express the composite quantization policy as: $$\begin{aligned} \label{form2} Q_t^{comp,i}(I^i_t) = (T^i_t(I^r_t)) (I^i_t \setminus I^r_t),\end{aligned}$$ mapping the information space to ${\cal M}^i_t$. We note that, any composite quantization policy $Q_t^{comp,i}$ can be expressed in the form above; that is there is no loss in the space of possible such policies, since for any $Q_t^{comp,i}$, one could define $$T^i_t(I^r_t) (\cdot) := Q_t^{comp,i}(I^r_t, \cdot).$$ Thus, we let DM$^i$ have policy ${\cal T}^i$ and under this policy generate quantizer actions $\{ Q^i_t, t \geq 0 \}$, $Q^i_t \in \mathbb{Q}^i_t$ ($Q^i_t$ is the quantizer used at time $t$). Under action $Q^i_t$, and given the local information, the encoder generates $q^i_t$, as the [*quantization output*]{} at time $t$. The receiver, upon receiving the information from the encoders, generates its decision at time $t$, also causally: An admissible causal receiver policy is a sequence of measurable functions $\gamma=\{\gamma_t\}$ such that $$\gamma_t : \prod_{s=0}^{t} \bigg( {\cal M}^1_s \times {\cal M}^2_s \bigg) \to \mathbb{U}, \quad \quad t \geq 0$$ where $\mathbb{U}$ denotes the decision space. For a general vector $a$, let ${\bf a}$ denote $\{a^1,a^2\}$ and let ${\bf \Pi} = \{\Pi^1, \Pi^2\}$ denote the ensemble of policies and ${\bf Q}_t=\{Q^1_t,Q^2_t\}$. Hence, ${\bf q}_{[0,t]}$ denotes $\{q^1_{[0,t]},q^2_{[0,t]}\}$. With the above formulation, the objective of the decision makers is the following minimization problem: $$\begin{aligned} \label{CostFunction} \inf_{{\bf \Pi}^{comp}} \inf_{\gamma} E^{{\bf \Pi}^{comp},\gamma}_{\nu_0}[\sum_{t=0}^{T-1} c(x_t,v_t)],\end{aligned}$$ over all policies ${\bf \Pi}^{comp}$, $\gamma$ with the random initial condition $x_0$ having probability measure $\nu_0$. Here $c(\cdot,\cdot),$ is a non-negative, bounded, measurable function and $v_t=\gamma_t({\bf q}_{[0,t]})$ for $t \geq 0$. We also assume that the encoders and the receiver know the apriori distribution $\nu_0$. Before concluding this section, it may be worth emphasizing the operational nature of causality; as different approaches have been adopted in the literature. The encoders at any given time can only use their local information to generate the quantization outputs. The receiver, at any given time, can only use its local information to generate its decision/estimate. These happen with zero delay, that is if there is a common-clock at the encoders and the receiver; the receiver at time $t$ needs to make its decision before the realizations $x_{t+1},y^1_{t+1},y^2_{t+1}$ have taken place. This corresponds to the zero-delay coding schemes of, for example, Witsenhausen and Linder-Lugosi in [@Witsenhausen], [@LinderLugosi]; but is different from the setup of Neuhoff and Gilbert [@NeuhoffGilbert], which allows long delays at the decoder. Our motivation for such zero-delay schemes comes mainly from applications in networked control systems, when sensors need to transmit information to controllers who need to act on a system; such systems cannot tolerate long delays, in particular when the source is open loop unstable and disturbance exists in the evolution of the source. Relevant literature ------------------- Some related studies of the above setup include optimal control with multiple sensors and sequential decentralized hypothesis testing problems and multi-access communications with feedback [@AchilleasAllerton09]. Related papers on real-time coding include the following: [@NeuhoffGilbert] established that the optimal optimal causal encoder minimizing the data rate subject to a distortion for an i.i.d sequence is memoryless. If the source is $k$th-order Markov, then the optimal causal fixed-rate coder minimizing any measurable distortion uses only the last $k$ source symbols, together with the current state at the receiver’s memory [@Witsenhausen]. Reference [@WalrandVaraiya] considered the optimal causal coding problem of finite-state Markov sources over noisy channels with noiseless feedback. [@Teneketzis] and [@MahTen09] considered optimal causal coding of Markov sources over noisy channels without feedback. [@MahajanTeneketzisJSAC] considered the optimal causal coding over a noisy channel with noisy feedback. Reference [@LinderZamir] considered the causal coding of stationary sources under a high-rate assumption. Our paper is particularly related to the following two efforts in the literature: Borkar, Mitter and Tatikonda [@Borkar] studies a related problem of coding of a partially observed Markov source, however, the construction for the encoders is restricted to take a particular form which uses the information at the decoder and the most recent observation at the encoder (not including the observation history). As another point of relevance with our paper, [@Borkar] regarded the actions as the [*quantizer functions*]{}, which we will discuss further. In contrast, the only restriction we have in this paper is causality, in the zero-delay sense. On the other hand, we do not claim the existence results that the authors in [@Borkar] are making. Another work in the literature which is related to ours is by Nayyar and Teneketzis [@NayyarTeneketzis], considering a multi-terminal setup. [@NayyarTeneketzis] considers decentralized coding of correlated sources when the encoders observe conditionally independent messages given a finitely valued random variable and obtain separation results for the optimal encoders. The paper also considers noisy channels. In our setup, there does not exist a finitely valued random variable which makes the observations at the encoders conditionally independent. References [@Weissman] and [@KaspiMerhav] consider optimal causal variable-rate coding under side information and [@YukBasMeynITA08] considers optimal variable-rate causal coding under distortion constraints. The studies in [@KaspiMerhav] and [@YukBasMeynITA08] are in the context of real-time, zero-delay settings; whereas [@Weissman] considers causality in the sense of Neuhoff and Gilbert [@NeuhoffGilbert] as discussed in the previous section. We will also obtain structural results for optimal decentralized coding of i.i.d. sources. There are algorithmic results available in the literature when the encoders satisfy the optimal structure obtained in the paper, important resources in this direction include [@Gray1], [@Gray2] and [@Skoglund]. A parallel line of consideration which has a rate-distortion theoretic nature is on [*sequential-rate distortion*]{} proposed in [@Tatikonda] and the [*feedforward*]{} setup, which has been investigated in [@Feedforward2] and [@Feedforward1]. Our work is also related to Witsenhausen’s [indirect rate distortion]{} problem [@WitsenhausenIndirect] (see also [@Tsybakov]). We will observe that, the separation argument through the [*disconnection principle*]{} of [@WitsenhausenIndirect] applies to our setting in a dynamic context. Further related papers include [@BansalBasarSysCont], [@HuangDey]. In our paper, we also use ideas from team decision theory, see [@WitsenhausenIntrinsic], [@YukTAC09], [@Mahajan], [@MahajanCDC2011] and [@ComoYukConCom] for related discussions and applications. Contributions of the paper -------------------------- - The optimal causal coding of a partially observed Markov source is considered. For the single terminal case, a structural result is obtained extending Witsenhausen’s and Walrand and Varaiya’s structural results on optimal causal (zero-delay) coders to a partially observed setting and to sources which take values in a Polish space. We show that a separation result of a form involving [the decoder’s belief on the encoder’s belief on the state]{} is optimal. - The decentralized (multi-terminal) setup is also considered. For the case where the source is an i.i.d. process, it is shown that the optimal decentralized causal coding of correlated observations problem admits a solution which is memoryless. For Markov sources, a counterexample to a natural separation conjecture is presented. The decentralized control concept of [*signaling*]{} is interpreted in the context of decentralized coding. - The results are applied to a Linear-Quadratic-Gaussian (LQG) estimation/optimization problem. The results above induce an optimality result for [*separation of estimation and quantization*]{}, where the estimation is obtained with a Kalman Filter and the filter output is quantized. We now summarize the rest of the paper. In section II, we present our results on optimal coding of a partially observed Markov process when there is only one encoder. Section III discusses the decentralized setting for a multi-encoder setup and presents a counterexample for a separation conjecture and provides a separation result when the source is memoryless. The paper ends with the concluding remarks of section V, following an application example on linear, Gaussian systems in Section IV. The proofs of the results are presented in the Appendix. Single-Terminal Case: Optimal Causal Coding of a Partially Observed Markov Source ================================================================================= Revisiting the single-terminal, fully observed case --------------------------------------------------- Let us revisit the single-encoder, fully observed case: In this setup, $y_t=x_t$ for all $t \geq 0$. There are two related approaches in the literature as presented explicitly by Teneketzis in [@Teneketzis]; one adopted by Witsenhausen [@Witsenhausen] and one by Walrand and Varaiya [@WalrandVaraiya]. Reference [@Teneketzis] extended the setups to the more general context of non-feedback communication. \[WitsenTheoremFull\]\[Witsenhausen [@Witsenhausen]\] Any [*(causal) composite quantization policy*]{} can be replaced, without any loss in performance, by one which only uses $x_t$ and $q_{[0,t-1]}$ at time $t \geq 1$. $\diamond$ Walrand and Varaiya considered sources living in a finite set, and obtained the following: \[WalVarTheoremFull\]\[Walrand-Varaiya [@WalrandVaraiya]\] Any optimal [*(causal) composite quantization policy*]{} can be replaced, without any loss in performance, by one which only uses the conditional probability measure $P(x_{t}|q_{[0,t-1]})$, the state $x_t$, and the time information $t$, at time $t \geq 1$. $\diamond$ The difference between the structural results above is the following: In the setup suggested by Theorem \[WitsenTheoremFull\], the encoder’s memory space is not fixed and keeps expanding as the decision horizon in the optimization, $T-1$, increases. In Theorem \[WalVarTheoremFull\], the memory space of an optimal encoder is fixed. In general, the space of probability measures is a very large one; however, it may be the case that different quantization outputs may lead to the same conditional probability measure on the state process, leading to a reduction in the required memory. Furthermore, Theorem \[WalVarTheoremFull\] allows one to apply the theory of Markov Decision Processes, for infinite horizon problems. We note that [@Borkar] applied such a machinery to obtain existence results for optimal causal coding of partially observed Markov processes Optimal causal coding of a partially observed Markov source ----------------------------------------------------------- Consider the setup earlier in (\[channelModel\]) with a single encoder. Thus, the system considered is a discrete-time scalar system described by $$\begin{aligned} \label{ProblemModel4} x_{t+1}=f(x_{t},w_t), \quad \quad y_t= g(x_t,r_t),\end{aligned}$$ where $x_t$ is the state at time $t$, and $\{w_t, r_t \}$ is a sequence of zero-mean, mutually independent, identically distributed (i.i.d.) random variables with finite second moments. Let the quantizer, as described earlier, map its information to a finite set ${\cal M}_t$. At any given time, the receiver generates a quantity $v_t$ as a function of its received information, that is as a function of $\{q_0,q_1,\dots,q_t\}$. The goal is to minimize $\sum_{t=0}^{T-1} E[c(x_t,v_t)],$ subject to constraints on the number of quantizer bins in ${\cal M}_t$, and the causality restriction in encoding and decoding. Let for a Polish space $\mathbb{S}$, ${\cal P}(\mathbb{S})$ be the space of probability measures on ${\cal B}(\mathbb{S})$, the Borel $\sigma-$field on $\mathbb{S}$ (generated by open sets in $\mathbb{S}$). At this point pause to provide a brief discussion on the space ${\cal P}(\mathbb{S})$. Let $\Gamma(\mathbb{S})$ be the set of all Borel measurable and bounded functions from $\mathbb{S}$ to $\mathbb{R}$. We first wish to find a topology on ${\cal P}(\mathbb{S})$, under which functions of the form: $$\Theta:= \{ \int_{x\in \mathbb{S}} \pi(dx) f(x), \quad f \in \Gamma(\mathbb{X}) \}$$ are measurable on ${\cal P}(\mathbb{S})$. We will need this to construct the structure of optimal quantizers later in this section. Let $\{\mu_n,\, n\in \mathbb{N}\}$ be a sequence in $\mathcal{P}(\mathbb{S})$. Recall that $\{\mu_n\}$ is said to converge to $\mu\in \mathcal{P}(\mathbb{S})$ *weakly* if $$\int_{\mathbb{S}} c(x) \mu_n(dx) \to \int_{\mathbb{S}}c(x) \mu(dx)$$ for every continuous and bounded $c: \mathbb{S} \to \mathbb{R}$. The sequence $\{\mu_n\}$ is said to converge to $\mu\in \mathcal{P}(\mathbb{S})$ *setwise* if $$\int_{\mathbb{S}} c(x) \mu_n(dx) \to \int_{\mathbb{S}} c(x) \mu(dx)$$ for every measurable and bounded $c: \mathbb{S} \to \mathbb{R}$. For two probability measures $\mu,\nu \in \mathcal{P}(\mathbb{S})$, the *total variation* metric is given by $$\begin{aligned} \|\mu-\nu\|_{TV}&:= & 2 \sup_{B \in {\cal B}(\mathbb{S})} |\mu(B)-\nu(B)| \nonumber \\ &=& \sup_{f: \, \|f\|_{\infty} \leq 1} \bigg| \int f(x)\mu(dx) - \int f(x)\nu(dx) \bigg|, \nonumber\end{aligned}$$ where the infimum is over all measurable real $f$ such that $\|f\|_{\infty} = \sup_{x \in \mathbb{S}} |f(x)|\le 1$. A sequence $\{\mu_n\}$ is said to converge to $\mu\in \mathcal{P}(\mathbb{S})$ in total variation if $\| \mu_n - \mu \|_{TV} \to 0.$ These three convergence notions are in increasing order of strength: convergence in total variation implies setwise convergence, which in turn implies weak convergence. Total variation is a very strong notion for convergence. Furthermore, the space of probability measures under total variation metric is not separable. Setwise convergence also induces an inconvenient topology on the space of probability measures, particularly because this topology is not metrizable ([@Ghosh p. 59]). However, the space of probability measures on a complete, separable, metric (Polish) space endowed with the topology of weak convergence is itself a complete, separable, metric space [@InfiniteDimensionalAnalysis]. The Prohorov metric, for example, can be used to metrize this space, among other metrics. This topology has found many applications in information theory and stochastic control. For these reasons, one would like to work with weak convergence. By the above definitions, it is evident that both setwise convergence and total variation are sufficient for measurability of the function class $\Theta$, since under these topologies $\int \pi(dx) f(x)$ is (sequentially) continuous on ${\cal P}(\mathbb{S})$ for every $f \in \Gamma(\mathbb{S})$. However, as we state in the following, weak convergence is also sufficient (see Theorem 15.13 in [@InfiniteDimensionalAnalysis] or p. 215 in [@Bogachev]). \[MonotoneClassRamon\] Let $\mathbb{S}$ be a Polish space and let $M(\mathbb{S})$ be the set of all measurable and bounded functions $f: \mathbb{S} \to \mathbb{R}$ under which $$\int \pi(dx) f(x)$$ defines a measurable function on ${\cal P}(\mathbb{S})$ under the weak convergence topology. Then, $M(\mathbb{S})$ is the same as $\Gamma(\mathbb{S})$ of all bounded and measurable functions. $\diamond$ Hence, ${\cal P}(\mathbb{S})$ will denote the space of probability measures on $\mathbb{S}$ under weak convergence. Now, define $\pi_t \in {\cal P}(\mathbb{X})$ to be the regular conditional probability measure given by $$\pi_t(A) = P(x_t \in A|y_{[0,t]}), \quad A \in {\cal B}(\mathbb{X}).$$ The existence of this regular conditional probability measure for every realization $y_{[0,t]}$ follows from the fact that both the state process and the observation process are Polish. It is known that the process $\{\pi_t\}$ evolves according to a non-linear filtering equation (see (\[MarkovBelief\])), and is itself a Markov process (see [@BorkarBudhiraja], [@Stettner], [@Borkar]). Let us also define $\Xi_t \in {\cal P}({\cal P}(\mathbb{X}))$ as the regular conditional measure $$\Xi_t(A) = P(\pi_t \in A |q_{[0,t-1]}), \quad A \in {\cal B}({\cal P}(\mathbb{X})).$$ The following are the main results of this section: \[thm1\] Any [*(causal) composite quantization policy*]{} can be replaced, without any loss in performance, by one which only uses $\{\pi_t, q_{[0,t-1]} \}$ as a sufficient statistic for $t \geq 1$. This can be expressed as a quantization policy which only uses $q_{[0,t-1]}$ to generate a quantizer, where the quantizer uses $\pi_t$ to generate the quantization output at time $t$. $\diamond$ \[thm3\] Any optimal [*(causal) composite quantization policy*]{} can be replaced, without any loss in performance, by one which only uses $\{\Xi_t,\pi_t,t\}$ for $t \geq 1$. This can be expressed as an optimal quantization policy which only uses $\{\Xi_t,t\}$ to generate an optimal quantizer, where the quantizer uses $\pi_t$ to generate the quantization output at time $t$. $\diamond$ The proofs of the results above are presented in the Appendix. We present two remarks in the following: Our results above are not surprising. In fact, once one recognizes the fact that $\{\pi_t\}$ forms a Markov source, and the cost function can be expressed as a function $\tilde{c}(\pi,v)$, for some function $\tilde{c}: {\cal P}(\mathbb{X}) \times \mathbb{U} \to \mathbb{R}$, one could almost directly apply Witsenhausen’s [@Witsenhausen] as well as Walrand and Varaiya’s [@WalrandVaraiya] results to recover the structural results above (except the fact that Walrand and Varaiya consider sources living in a finite alphabet). The proofs in the Appendix are presented for completeness and to address the technical intricacies. $\diamond$ Having the actions as the quantizers, and not the quantizer outputs, allows one to define a Markov Decision Problem with well-defined cost functions and state and action spaces. By the proof of Theorem \[thm3\], we will observe that $(\Xi_,Q_t)$ forms a Markov Chain, which is a key observation: Thus, the action space can be constructed as some topological space of quantizers acting on ${\cal P}({\mathbb{X}})$. Borkar, Mitter and Tatikonda [@Borkar] adopted this view while formulating an MDP optimization problem, where the quantizer acts on $\mathbb{Y}$ (As mentioned earlier, our separation result is different from [@Borkar] due to the structure imposed on the quantizers in [@Borkar].). See also [@YukselOptimizationofChannels] and [@YukLinCDC2012] for a topology on quantizers. $\diamond$ Extensions to finite delay decoding and higher order Markov sources ------------------------------------------------------------------- The results presented are also generalizable to settings where, a) the source is Markov of order $m>0$, b) a finite delay $d$ is allowed at the decoder, and c) the observation process depends also on past source outputs in a sense described in (\[ProblemModel114\]) below. For these cases, we consider the following generalization of the source by expanding the state space. Suppose that the partially observed source is such that, the source is Markov of order $m$, or there is a finite delay $d>0$ which is allowed at the decoder. In this case, we can augment the source to obtain $z_t = \{x_{[t-\max(d+1,m)+1,t]}\}$. Note that $\{z_t\}$ is Markov. We can thus consider the following representation: $$\begin{aligned} \label{ProblemModel114} z_{t+1}=f(z_{t},w_t), \quad \quad y_t= g(z_t,r_t),\end{aligned}$$ where $z_t = \{x_{[t-\max(d+1,m)+1,t]}\} \in \mathbb{X}^{\max(d+1,m)}$, and $r_t, w_t$ are mutually independent, i.i.d. processes. Any per-stage cost function of the form $c(x_t,v_t)$ can be written as for some $\tilde{c}$: $\tilde{c}(z_t,v_t)$. For the finite delay case, the cost per-stage can further be specialized as $\tilde{c}(x_{t-d}, v_t)$. For the Markov case with memory, the cost function per-stage writes as $\tilde{c}(x_{[t-m+1,t]}, v_t)$. Now, by replacing $\mathbb{X}$ with $\mathbb{X}^{\max(d+1,m)}$, let $\pi_t \in {\cal P}(\mathbb{X}^{\max(d+1,m)})$ be given by $$\pi_t(A) = P(z_t \in A | y_{[0,t]}), \quad A \in {\cal B}(\mathbb{X}^{\max(d+1,m)})$$ and $\Xi_t \in {\cal P}({\cal P}(\mathbb{X}^{\max(d+1,m)}))$ be the regular conditional measure defined by $$\Xi_t(A) = P(\pi_t \in A |q_{[0,t-1]}), \quad A \in {\cal B}({\cal P}(\mathbb{X}^{\max(d+1,m)})).$$ Hence, we have the following result, which is a direct extension of Theorems \[thm1\] and \[thm3\]. \[thm15\] Suppose that the partially observed source is such that, the source is Markov of order $m$, or there is a finite delay $d>0$ which is allowed at the decoder. With $z_t = \{x_{[t-\max(d+1,m)+1,t]} \}$, $y_t$ satisfies (\[ProblemModel114\]). Then, we have the following extensions:\ i) Any [*causal composite quantization policy*]{} can be replaced, without any loss in performance, by one which only uses $\{\pi_t, q_{[0,t-1]} \}$ as a sufficient statistic for $t \geq 1$. This can be expressed as a quantization policy which only uses $q_{[0,t-1]}$ to generate a quantizer, where the quantizer uses $\pi_t$ to generate the quantization output at time $t$.\ ii) Any optimal [*causal composite quantization policy*]{} can be replaced, without any loss in performance, by one which only uses $\{\Xi_t,\pi_t,t\}$ for $t \geq 1$. This can be expressed as an optimal quantization policy which only uses $\{\Xi_t,t\}$ to generate an optimal quantizer, where the quantizer uses $\pi_t$ to generate the quantization output at time $t$. $\diamond$ For a further case where the decoder’s memory is limited or imperfect, the results apply by replacing the full information at the receiver considered so far in our analysis with the limited memory under additional technical assumptions on the decoder’s update of its memory (in particular, (\[whySeparationHolds\]) in the proof of Theorem \[thm3\] does not apply in general). However, an equivalent result of Theorem \[thm1\] applies also for the limited memory setting. Such memory settings have been considered in [@Witsenhausen], [@WalrandVaraiya] and [@MahajanTeneketzisJSAC]. Multi-Terminal (Decentralized) Setup ==================================== Case with memoryless sources ---------------------------- Let us first consider a special, but important, case when $\{x_t, t \geq 0\}$ is an independent and identically distributed (i.i.d.) sequence. Further, suppose that, the observations are given by $$\begin{aligned} %x_{t+1} &=& f(x_t, w_t), \nonumber \\ y^i_t&=&g^i(x_t,r^i_t), \label{channelModel3}\end{aligned}$$ for measurable functions $g^i,i=1,2$, with $\{r^1_t, r^2_t\}$ (across time) an i.i.d. noise process. We do not require that $r^1_t$ and $r^2_t$ are independent for a given $t$. We note that our result below is also applicable when the process $\{r^1_t, r^2_t\}$ is only independent (across time), but not necessarily identically distributed. One difference with the general setup considered earlier in Section I is that we require the observation spaces $\mathbb{Y}^i, i=1,2$, to be finite spaces ($\mathbb{X}$ can still be Polish). Suppose the goal is again the minimization problem $$\begin{aligned} \label{problemFormulation2} \inf_{{\bf \Pi}^{comp}} \inf_{\gamma} E^{{\bf \Pi}^{comp},\gamma}_{\nu_0}[\sum_{t=0}^{T-1} c(x_t,v_t)],\end{aligned}$$ over all causal coding and receiver decision policies. We now make a definition. In the following $1_E$ denotes the indicator function of an event $E$. We define the class of non-stationary memoryless team policies at $t \geq 0$ as follows: $$\begin{aligned} \label{piNSM} &&\Pi^{NSM}:= \bigg\{{\bf \Pi}^{comp} : P({\bf q}_t | {\bf y}_{[0,t]}) \nonumber \\ && = P(q^1_t | y^1_t,t)P(q^2_t | y^2_t,t) = 1_{\{q^1_t=Q^1_t(y^1_t)\}}1_{\{q^2_t=Q^2_t(y^2_t)\}}, \nonumber \\ && \quad \quad \quad \quad \quad \quad \quad Q^i_t: \mathbb{Y}^i \to {\cal M}^i_t, i=1,2, t \geq 0 \bigg\},\end{aligned}$$ where, in the above, $\{Q^1_t,Q^2_t\}$ are arbitrary measurable functions. \[mainDecentralized\] Consider the minimization of (\[problemFormulation2\]). An optimal composite quantization policy over all causal policies is an element of $\Pi^{NSM}$. Such a policy exists. $\diamond$ The proof is presented in the Appendix. Hence, an optimal composite quantization policy only uses the product form admitted by a non-stationary memoryless team policy. It ignores the past observations and past quantization outputs. We note that, the proof also applies to the case when the source is memoryless, but not necessarily i.i.d.. One may ask why feedback could be useful when the source is i.i.d.. Feedback may be useful for at least two reasons: (i) Feedback can be used as a signaling mechanism for the encoders to communicate with each other (which we discuss further in the next subsections), and (ii) Feedback can provide common randomness to allow a convexification of the space of possibly randomized decentralized encoding strategies. Consider the optimization problem discussed in (\[problemFormulation2\]) $$J({\bf \Pi}^{comp}) = \inf_{\gamma} E^{{\bf \Pi}^{comp},\gamma}_{\nu_0}[\sum_{t=0}^{T-1} c(x_t,v_t)].$$ The function $J({\bf \Pi})$ is concave in the choice of a team policy ${\bf \Pi}$ (see Theorem 4.1 in [@YukLinAllerton2010] for the case with $T=1$, the proof also holds for the current setting). As such, if the space of joint encoding policies is convexified by common randomness, an optimal solution would exist at an extreme point; which in turn does not require a use of common randomness. This explains why common randomness generated by past quantization outputs does not present further benefit in the current setting. We note before ending this subsection that if there is an entropy constraint on the quantizer outputs, then feedback might be useful for finite horizon problems as it provides common randomness, which cannot be achieved by time-sharing in a finite-horizon problem. [@NeuhoffGilbert] observed that randomization of two scalar quantizers (operationally achievable through time-sharing) is optimal in causal coding of an i.i.d. source subject to distortion constraints, which also applies in the side information setting of [@Weissman]. On the other hand, for the zero-delay setting, when one considers the distortion minimization problem subject to an entropy constraint, [@GyorgyLinder] observed that the distortion-entropy curve is non-convex, leading to a benefit of common-randomness for achieving points in the lower convex hull of this curve. Further relevant discussions on randomization and optimal quantizer design are present in [@GaborGyorfi] and [@YukselOptimizationofChannels]. Case with Markov sources: A counterexample with signaling --------------------------------------------------------- We now consider Markov sources and exhibit that it is, in general, not possible to obtain a separation result of the form presented for the single-terminal case. We will consider a two-encoder setup for the following result, where the encoders have access to the feedback from the receiver (Fig. \[LLL\]). We have the following result. \[NegativeResult\] Consider the setup in (\[sourceModel\])-(\[channelModel\]) and let $\pi^i_t=P(x_t|y^i_{[0,t]}), x_t \in \mathbb{X}$, $i=1,2$. An optimal composite quantization policy cannot, in general, be replaced by a policy which only uses $\{{\bf q}_{[0,t-1]},\pi^i_t\}$ to generate $q^i_t$ for $i=1,2$. $\diamond$ **Proof.** It suffices to produce an instance where an optimal policy cannot admit the separated structure. Toward this end, let $z_1,z_2,z_3$ be uniformly distributed, independent, binary numbers; $x_0,x_1$ be defined by: $$x_0= \begin{bmatrix} z_1 \\ z_2 \\ 0 \\ 0 \end{bmatrix}, \quad x_1= \begin{bmatrix} 0 \\ 0 \\ z_2 \\ z_3 \end{bmatrix},$$ such that $x_0(1)=z_1, x_0(2)=z_2, x_0(3)=x_0(4)=0$. Let the observations be given as follows: $$y^1_t= g^1(x_t) = x_t(1) \oplus x_t(3) \oplus x_t(4)$$ $$y^2_t= g^2(x_t) = x_t(1) \oplus x_t(2), \quad t=0,1.$$ where $\oplus$ is the x-or operation. That is, $$y^1_0=\begin{bmatrix} z_1 \end{bmatrix}, \quad y^2_0=\begin{bmatrix} z_1 \oplus z_2\end{bmatrix},$$ $$y^1_1= \begin{bmatrix} z_2 \oplus z_3 \end{bmatrix}, \quad y^2_1= \begin{bmatrix} 0 \end{bmatrix}$$ Let the cost be: $$E\bigg[(x_0(4) - E[x_0(4) | {\bf q}_{[0]}])^2 + (x_1(4) - E[x_1(4) | {\bf q}_{[0,1]}])^2\bigg]$$ That is, the cost is $E[(z_3 - E[z_3 | {\bf q}_{[0,1]}])^2]$, where $q^i_t$ are the information bits sent to the decoder for $t=0$ and $1$. We further restrict the information rates to satisfy: $|{\cal M}^1_0|=|{\cal M}^1_1|=|{\cal M}^2_1|=2$, $|{\cal M}^2_0|=1$. That is, the encoder 2 may only send information at time $t=1$. Under arbitrary causal composite quantization policies, a cost of zero can be achieved as follows: If the encoder 1 sends the value $z_1 $ to the receiver, and at time $1$, encoder 1 transmits $z_2 \oplus z_3$ and encoder 2 transmits $z_2$ (or $z_1 \oplus z_2$), the receiver can uniquely identify the value of $z_3$, for every realization of the random variables. For such a source, an optimal composite policy cannot be written in the separated form, that is, an optimal policy of encoder 2 at time $1$ cannot be written as $h_1({\bf q}_{0},\pi^2_1)$, for some measurable function $h_1$. To see this, note the following: The conditional distribution on $x_1$ at encoder 2 at time $1$ is such that the conditional measure on $(z_2,z_3)$ is uniform and independent, that is $P(z_2=a,z_3=b|z_1 \oplus z_2)=(1/4)$ for all values of $a,b$. If a policy of the structure of $h_1$ is adopted, then it is not possible for encoder 2 to recall its past observation to extract the value of $z_2$. This is because, $\pi^2_1$ will be a distribution only on $z_2$ and $z_3$, which will be uniform and independent, given $z_1 \oplus z_2$. Thus, the information $y^2_0$ will not be available in the memory and the receiver will have access to at most $z_2 \oplus z_3$ and $z_1$ and $P(z_2, z_3 | z_1 \oplus z_2)$ (the last variable containing no useful information). The optimal estimator will be $E[z_3]=1/2$, leading to a cost of $1/4$. $\diamond$ Discussion: Connections with team decision theory ------------------------------------------------- In this subsection, we interpret the results of the previous subsections. We first provide a brief discussion on information structures in a decentralized optimization problem: Consider a collection of decision makers (DMs) where each has access to some local information variable. Such a collection of decision makers who wish to minimize a common cost function and who has an agreement on the system (that is, the probability space on which the system is defined, and the policy and action spaces) is said to be a [*team*]{}. Such a team is [*dynamic*]{} if the information of one DM is affected by the policy of some other DM. If there is a pre-specified order of action for the DMs, the team is said to be [*sequential*]{}. Witsenhausen [@WitsenhausenIntrinsic] provided the following characterization of information structures in a dynamic sequential team: Under a [*centralized information structure*]{}, all DMs have the same information. If a DM’s, say DM$^j$, information is dependent on the policy of another DM, say DM$^k$, and DM$^j$ does not have access to the information available to DM$^k$, this information structure is said to admit a [*non-classical information structure*]{}. A decentralized system admits a [*quasi-classical information structure*]{}, if it is not non-classical. In a decentralized optimization problem, when the information structure is non-classical, DMs might choose to communicate via their control actions: This is known as [*signaling*]{} in decentralized control (see for example [@YukTAC09]). With the characterization of information structures above, every lossy coding problem is non-classical, since a receiver cannot recover the information available at the encoder fully, while its information is clearly affected by the coding policy of the encoder. However, in an encoding problem, the problem itself is the transmission of information. Therefore, we suggest the following: [*Signaling in a coding problem is the policy of an encoder to use the quantizers/encoding functions to transmit a message to other decision makers, or to itself to be used in future stages, through the information sent to the receiver.*]{} In the information theory literature, signaling has been employed in coding for Multiple Access Channels with feedback in [@CoverLeung], [@BrossLapidoth] and [@Sandeep]. In these papers, the authors used active information transmission to allow for coordination between encoders. The reason for the negative conclusion in Proposition \[NegativeResult\] is that in general for an optimal policy, $$\begin{aligned} \label{filtre} P(q^i_{t} | \pi^i_{t}, {\bf q}_{[0,t-1]}, y^i_{[0,t-1]}) \neq P(q^i_{t} | \pi^i_{t}, {\bf q}_{[0,t-1]}),\end{aligned}$$ when the encoders have engaged in signaling (in contrast with what we will have in the proof of the separation results). The encoders may benefit from using the received past observation variables explicitly. Separation results for such dynamic team problems typically require information sharing between the encoders (decision makers), where the shared information is used to establish a sufficient statistic living in a fixed state space and which admits a controlled Markov recursion (hence, such a sufficient statistic can serve as a [*state*]{} for the decentralized system). For the proof of Theorem \[thm3\], we see that $\Xi_t$ forms such a state. For the proof of Theorem \[mainDecentralized\], we see that information sharing is not needed for the encoders to agree on a sufficient statistic, since the source considered is memoryless. Furthermore, for the multi-terminal setting with a Markov source, a careful analysis of the proof of Theorem \[mainDecentralized\] (see (\[longequationDecentralized\]), (\[BeliefOnBelief\]) and the subsequent discussion) reveals that if the encoders agree on $P(dx_t|{\bf y}_{[0,t-1]})$ through sharing their beliefs for $t \geq 1$, then a separation result involving this joint belief can be obtained. See [@YukTAC09] for a related information sharing pattern and discussions. Further results on such a [*dynamic programming approach to dynamic team problems*]{} are present in [@AicardiDavoli], [@Kurtaran], [@NayyarMahajanTeneketzis], [@MahajanCDC2011], among other references. In the context of multi-terminal systems, for the computation of the capacity of Multiple Access Channels with memory and partial state feedback at the encoders, a relevant discussion has been reported in [@ComoYukConCom] (section V). This is in the same spirit as our current paper in that, [@ComoYukConCom] obtains an optimality result when the channel is memoryless; and points out the difficulties arising in the case of channels with memory in view of intractability of the optimization problem: One cannot identify a finite dimensional sufficient statistic for the encoders to use. Along a relevant direction, Section III.D of [@NayyarTeneketzis], in the context of real-time coding, discusses the issue of the growing state space. $\diamond$ Application to Linear Quadratic Gaussian (LQG) Estimation Problems ================================================================== Consider a Linear Quadratic Gaussian setup, where a sensor quantizes its noisy information to an estimator. Let $x_t \in \mathbb{R}^n, y_t \in \mathbb{R}^m$, and the evolution of the system be given by the following: $$\begin{aligned} x_{t+1} &=& Ax_t + w_t, \nonumber \\ y_t&=& Cx_t + r_t, \label{channelModel2}\end{aligned}$$ Here, $\{w_t,r_t\}$ is a mutually independent, zero-mean Gaussian noise sequence with $W=E[w_tw_t'], R=E[r_tr_t']$ (where for a vector $w$, $w'$ denotes its transpose), $x_0$ is a zero mean Gaussian variable, and $A,C$ are matrices of appropriate dimensions. Suppose the goal is the computation of $$\begin{aligned} \label{LQGopt} \inf_{\Pi^{comp}} \inf_{\gamma} E^{\Pi^{comp},\gamma}_{\nu_0}[\sum_{t=0}^{T-1} (x_t-v_t)'Q(x_t-v_t)],\end{aligned}$$ with $\nu_0$ denoting a Gaussian distribution for the initial state, $Q > 0$ a positive definite matrix (See Figure \[KalmanOptimal\]). The conditional measure $\pi_t = P(dx_t|y_{[0,t]})$ is Gaussian for all time stages, which is characterized uniquely by its mean and covariance matrix for all time stages. We have the following. \[LQGApplication\] For the minimization of the cost in (\[LQGopt\]), any [*causal composite quantization policy*]{} can be replaced, without any loss in performance, by one which only uses the output of the Kalman Filter and the information available at the receiver. $\diamond$ **Proof.** The result can be proven by considering a direct approach, rather than as an application of Theorems \[thm1\] and \[thm3\] (which require bounded costs, however, this assumption can be relaxed for this case), exploiting the specific quadratic nature of the problem. Let $||\cdot||_Q$ denote the norm generated by an inner product of the form: $\langle x, y \rangle = x'Qy$ for $x,y \in \mathbb{R}^n$ for positive-definite $Q > 0$. The Projection Theorem for Hilbert Spaces implies that the random variable $x_t - E[x_t|y_{[0,t]}]$ is orthogonal to the random variables $\{y_{[0,t]},q_{[0,t]}\}$, where $q_{[0,t]}$ is included due to the Markov chain condition $$P(dx_t|y_{[0,t]},q_{[0,t]})=P(dx_t|y_{[0,t]}).$$ We thus obtain the following identity. $$\begin{aligned} \label{hilbert} && E[\|x_t - E[x_t|q_{[0,t]}]\|_Q^2] = E[\|x_t - E[x_t|y_{[0,t]}]\|_Q^2] \nonumber \\ && \quad \quad + E\bigg[\bigg\|E[x_t|y_{[0,t]}] - E\bigg[E[x_t|y_{[0,t]}] \bigg| q_{[0,t]} \bigg] \bigg\|_Q^2\bigg]. \nonumber\end{aligned}$$ The second term is to be minimized through the choice of the quantizers. Hence, the term $\tilde{m}_t:=E[x_t|y_{[0,t]}]$, which is computed through a Kalman Filter, is to be quantized (see Figure \[KalmanOptimal\]). Recall that by the Kalman Filter (see [@Kushner]) with $$\Sigma_{0|-1}=E[x_0x_0']$$ and for $t \geq 0$, $$\begin{aligned} && \Sigma_{t+1|t} = A \Sigma_{t|t-1} A' + W \nonumber \\ && \quad \quad - (A \Sigma_{t|t-1}C') (C \Sigma_{t|t-1} C' + R)^{-1} (C \Sigma_{t|t-1} A'), \nonumber\end{aligned}$$ the following recursion holds for $t \geq 0$ and with $\tilde{m}_{-1}=0$: $$\begin{aligned} \label{POMDPtoFOMDPGaussian} && \tilde{m}_t=A\tilde{m}_{t-1} \nonumber \\ && + \Sigma_{t|t-1}C' (C \Sigma_{t|t-1}C' + R)^{-1}(CA (x_{t-1}-\tilde{m}_{t-1}) + r_t).\nonumber\end{aligned}$$ Thus, the pair $(\tilde{m}_t,\Sigma_{t|t-1})$ is a Markov source, where the evolution of $\Sigma_{t|t-1}$ is deterministic. Even though the cost to be minimized is not bounded, since $\tilde{m}_t$ itself is a fully observed process, the proof of Theorem \[thm1\] can be modified to develop the structural result that any causal encoder can be replaced with one which uses $(\tilde{m}_t, \Sigma_{t|t-1})$ and the past quantization outputs (this result can be proven also using Theorem 1 of Witsenhausen [@Witsenhausen], since this source is fully observed by the encoder). Likewise, the proof of Theorem \[thm3\] shows that, for the fully observed Markov source $(\tilde{m}_t,\Sigma_{t|t-1})$, any causal coder can be replaced with one which only uses the conditional probability on $\tilde{m}_t$ and the realization $(\tilde{m}_t,\Sigma_{t|t-1},t)$ at time t. $\diamond$ Thus, the optimality of Kalman Filtering allows the encoder to only use the conditional estimate and the error covariance matrix without any loss of optimality (See Figure \[KalmanOptimal\]), and the optimal quantization problem also has an explicit formulation. The above result is related to findings in [@Tsybakov] (also see [@Ayanoglu] and [@Fine]), and partially improves them in the direction of Markov sources. We note that, the above result also applies to the settings when a controller acts on the system, that is, there exists for $u_t \in \mathbb{R}^m$ and a matrix $B$ such that $x_{t+1} = Ax_t + Bu_t + w_t$. In this case, the well-known principle of separation and control in control theory allows the above results to be applicable. In particular, the conditional estimation error is not affected by the control actions. Conclusion ========== For the optimal causal coding of a partially observed Markov source, the structure of the optimal causal coders is obtained and is shown to admit a separation structure. We observed in particular that [*separation of estimation (conditional probability computation) and quantization (of this probability)*]{} applies, under such a setup. We also observed that the real-time decentralized coding of a partially observed i.i.d. source admits a separation. Such a separation result does not, in general, extend to decentralized coding of partially observed Markov sources. The results and the general program presented in this paper apply also to coding over discrete memoryless noisy channels with noiseless feedback. We note that Walrand and Varaiya [@WalrandVaraiya] considered the noisy channel setting in their analysis in the presence of noiseless feedback. The separation result will likely find many applications in sensor networks and networked control problems where sensors have imperfect observation of a plant to be controlled. One direction is to find explicit results on the optimal policies using computational tools. One promising approach is the expert-based systems, which are very effective once one imposes a structure on the designs, see [@GyorgyLinder2] for details. One further problem is on the existence and design of optimal quantizers. Existence of optimal quantizers, even in the context of vector quantization for $\mathbb{R}^n$-valued random variables, requires stringent conditions. Such proofs typically have the form of Weierstrass theorem: A lower semi-continuous function over a compact set admits a minimum. Existence results for optimal quantizers for a one-stage cost problem has been investigated by Abaya and Wise [@AbayaWise] and Pollard [@Pollard] for continuous cost functions which are non-decreasing in the source-reconstruction distance. [@YukselOptimizationofChannels] obtained existence results for more general cost functions under the restriction that the code bins/cells are convex and the source admits a density function. For dynamic quantizers, [@YukLinCDC2012] established the existence of optimal quantization policies under the assumption that the quantizers admit the structure suggested by Theorem \[thm3\] for fully observed Markov sources and the codecells are convex for a class of Markov sources. Also for dynamic vector quantizers, [@Borkar] investigated the existence results when there is a bound on the quantizer bins. Theorem \[thm3\] motivates the problem of optimal quantization of probability measures. This remains as an interesting problem to be investigated in a real-time coding context, with important practical consequences in control and economics applications. With a separation result paving the way for an MDP formulation, one could proceed with the analysis of [@Borkar] with the evaluation of optimal quantization policies and existence results for infinite horizon problems. Toward this direction, Graf and Luschgy, in [@Graf] and [@Graf2], have studied the optimal quantization of probability measures. One related question to be pursued further is the following: When is there an incentive for signaling in coding problems? When the observations are correlated for sources with memory or when the real-time coding of possibly independent sources over a general MAC type channel is considered, there may be an incentive for signaling. Further results on this will provide some light on some outstanding problems such as the capacity of MAC channels with feedback [@CoverLeung]. Proof of Theorem \[thm1\] ------------------------- We transform the problem into a real-time coding problem involving a fully observed Markov source. At time $t=T-1$, the per-stage cost function can be written as follows, where $\gamma^0_t$ denotes a fixed receiver policy: $$\begin{aligned} \label{costCompactForm} &&E[c(x_{t},\gamma^0_{t}(q_{[0,t]})) | q_{[0,t-1]}] \nonumber \\ && = \sum_{{\cal M}_t} P(q_t=k | q_{[0,t-1]}) \bigg(\int_{\mathbb{X}} P(dx_{t}|q_{[0,t-1]},k) \nonumber \\ && \quad \quad\quad \quad \quad \quad \times c(x_{t},\gamma^0_t(q_{[0,t-1]},k))\bigg) \nonumber \\ && = \int_{\mathbb{X}} \sum_{{\cal M}_t} P(dx_{t}, q_t=k |q_{[0,t-1]}) c(x_{t},\gamma^0_t(q_{[0,t-1]},k))\nonumber \\ && = \int_{{\cal P}(\mathbb{X})} \int_{\mathbb{X}} \sum_{{\cal M}_t} P(dx_{t}, q_t=k , d\pi_t |q_{[0,t-1]}) \nonumber \\ && \quad \quad \quad \quad \times c(x_{t},\gamma^0_t(q_{[0,t-1]},k))\nonumber \\ && = \int_{{\cal P}(\mathbb{X})} \int_{\mathbb{X}} \sum_{{\cal M}_t} P(d\pi_t|q_{[0,t-1]}) \nonumber \\ && \quad \quad \times P(dx_t| \pi_t) P(q_t=k | \pi_t , q_{[0,t-1]}) c(x_{t},\gamma^0_t(q_{[0,t-1]},k)) \nonumber \\ && = \int_{{\cal P}(\mathbb{X})} \sum_{{\cal M}_t} P(d\pi_t|q_{[0,t-1]}) P(q_t=k | \pi_t , q_{[0,t-1]}) \nonumber \\ && \quad \quad \quad \quad \times \int_{\mathbb{X}} P(dx_t| \pi_t) c(x_{t},\gamma^0_t(q_{[0,t-1]},k)) \nonumber \\ && = E[F(\pi_t,q_{[0,t-1]},q_t) | q_{[0,t-1]}],\end{aligned}$$ where, $\pi_t(\cdot)=P(x_t \in \cdot|y_{[0,t]})$ and $$F(\pi_t,q_{[0,t-1]},q_t) = \int_{\mathbb{X}} \pi_t(dx) c(x,\gamma^0_t(q_{[0,t-1]},q_t)).$$ In the above derivation, the fourth equality follows from the property that $$x_t \leftrightarrow P(dx_t|y_{[0,t]}) \leftrightarrow q_{[0,t]}$$ We note that $F(\cdot,\gamma^0_t(q_{[0,t-1]},q_t))$ is measurable on ${\cal P(\mathbb{X})}$ under weak convergence topology by Theorem \[MonotoneClassRamon\] and the fact that the cost is measurable and bounded. It should be noted that for every composite quantization policy, one may define $q_t$ as a random variable on the probability space such that the joint distribution of $(q_t,\pi_t,q_{[0,t-1]})$ matches the characterization that $q_t=Q^{comp}_t(y_{[0,t]},q_{[0,t-1]})$, since $$\begin{aligned} &&P(q_t|\pi_t,q_{[0,t-1]}) \nonumber \\ &&= \int_{\mathbb{Y}^{t+1}} P(q_t|y_{[0,t]},q_{[0,t-1]})) P(dy_{[0,t]} |\pi_t,q_{[0,t-1]}). \nonumber\end{aligned}$$ The final stage cost is thus written as $$E[F(\pi_t,q_{[0,t-1]},q_t) | q_{[0,t-1]}],$$ which is equivalent to, by the smoothing property of conditional expectation, the following: $$E\bigg[E [F(\pi_t,q_{[0,t-1]},q_t) | \pi_t,q_{[0,t-1]}] \bigg| q_{[0,t-1]}\bigg]$$ Now, we will apply Witsenhausen’s two stage lemma [@Witsenhausen], to show that we can obtain a lower bound for the double expectation by picking $q_t$ as a result of a measurable function of $\pi_t,q_{[0,t-1]}$. Thus, we will find a composite quantization policy which only uses $(\pi_t,q_{[0,t-1]})$ which performs as well as one which uses the entire memory available at the encoder. To make this precise, let us fix the decision function $\gamma^0_t$ at the receiver corresponding to a given composite quantization policy at the encoder $Q_t^{comp}$, let $t=T-1$, and define for every $k \in {\cal M}_t$: $$\begin{aligned} && \beta_{k} := \bigg\{\pi_t,q_{[0,t-1]} : F(\pi_t,q_{[0,t-1]},k) \nonumber \\ && \quad \quad \leq F(\pi_t,q_{[0,t-1]},q'), \forall q'\neq k, q' \in {\cal M}_t \bigg\}. \nonumber\end{aligned}$$ These sets are Borel, by the measurability of $F$ on ${\cal P}(\mathbb{X})$. Such a construction covers the domain set consisting of $(\pi_t,q_{[0,t-1]})$ but with overlaps. It covers the elements in ${\cal P}(\mathbb{X}) \times \prod_{t=0}^{T-2} {\cal M}_{t}$, since for every element in this product set, there is a minimizing $k \in {\cal M}_t$ (note that ${\cal M}_t$ is finite). To avoid the overlaps, we adopt the following technique which was introduced in Witsenhausen [@Witsenhausen]. Let there be an ordering of the elements in ${\cal M}_{t}$ as $1, 2, \dots, |{\cal M}_{t}|$, and for $k \geq 1$ in this sequence define a function $Q_t^{comp,*}$ as: $$\begin{aligned} && q_t = Q_t^{comp,*}(\pi_t,q_{[0,t-1]})= k, \nonumber \\ &&\quad \quad \quad \quad \rm{if} \quad (\pi_t,q_{[0,t-1]}) \in \beta_{k} \setminus \cup_{i=1}^{k-1} \beta_{i}, \nonumber\end{aligned}$$ with $\beta_0=\emptyset.$ Thus, for any random variable $q_t$ appropriately defined on the probability space, $$\begin{aligned} &&E\bigg[E [F(\pi_t,q_{[0,t-1]},q_t) | \pi_t,q_{[0,t-1]}] \bigg| q_{[0,t-1]}\bigg] \nonumber \\ && \geq E\bigg[E [F(\pi_t,q_{[0,t-1]},Q_t^{comp,*}(\pi_t,q_{[0,t-1]}) ) | \pi_t,q_{[0,t-1]}] \nonumber \\ && \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \bigg| q_{[0,t-1]}\bigg] \nonumber\end{aligned}$$ Thus, the new composite policy performs at least as well as the original composite coding policy even though it has a restricted structure. As such, any composite policy can be replaced with one which uses only $\{\pi_t,q_{[0,t-1]}\}$ without any loss of performance, while keeping the receiver decision function $\gamma^0_t$ fixed. It should now be noted that $\{\pi_t\}$ is a Markov process: Note that the following filtering equation applies [@BorkarBudhiraja] $$\begin{aligned} && P(dx_t|dy_{[0,t]}) \nonumber \\ &&= {\int_{x_{t-1}} P(dy_t|x_t)P(dx_t|x_{t-1})P(dx_{t-1}|dy_{[0,t-1]}) \over \int_{x_{t-1},x_{t}} P(dy_t|x_t)P(dx_t|x_{t-1})P(dx_{t-1}|dy_{[0,t-1]})}, \nonumber\end{aligned}$$ and $P(dy_t| \pi_{s},s \leq t-1) = \int_{x_t} P(dy_t,dx_t | \pi_{s},s \leq t-1) = P(dy_t| \pi_{t-1})$. These imply that (see [@BorkarBudhiraja], see also the discussion following (\[whySeparationHolds\])) the following defines a Markov kernel (that is, a regular conditional probability measure): $$\begin{aligned} \label{MarkovBelief} P(d\pi_t | \pi_s,s \leq t-1 ) = P(d\pi_t| \pi_{t-1}). %\label{Mark} \\\end{aligned}$$ We have thus obtained the structure of the encoder for the final stage. We iteratively proceed to study the other time stages. In particular, since $\{\pi_t\}$ is Markov, we could proceed as follows (in essence using Witsenhausen’s three-stage lemma [@Witsenhausen]): For a three-stage cost problem, the cost at time $t=2$ can be written as, for measurable functions $c_2, c_3$: $$\begin{aligned} && E\bigg[ c_2(\pi_2,v_2(q_{[1,2]}),q_{[1,2]}) \nonumber \\ && + E[c_3(\pi_3,v_3(q_{[1,2]},Q^{comp,*}_3(\pi_3,q_{[1,2]})))|q_{[1,2]},\pi_{[1,2]}] \nonumber \\ && \quad \quad \quad \quad \quad \quad \quad \quad \bigg| q_{[1,2]},\pi_{[1,2]} \bigg] \nonumber\end{aligned}$$ Since $$P(d\pi_3,q_2,q_1|\pi_2,\pi_1,q_2,q_1)=P(d\pi_3,q_2,q_1|\pi_2,q_2,q_1),$$ and since under $Q^{comp,*}_3$, $q_3$ is a function of $\pi_3$ and $q_1,q_2$, the expectation above is equal to, for some measurable $F_2(.)$, $E[F_2(\pi_2,q_2,q_1)|\pi_2,\pi_1,q_2,q_1]$. Following similar steps as earlier, it can be established now that a composite quantization policy at time $2$ uses $\pi_2$ and $q_1$ and is without any loss. By a similar argument, an encoder at time $t$, $1 \leq t\leq T-1$ only uses $(\pi_t,q_{[0,t-1]})$. The encoder at time $t=0$ uses $\pi_0$, where $\pi_0=\nu_0$ is the prior distribution on the initial state. Now that we have obtained the structure for a composite policy, we can express this as: $$Q_t^{comp}(\pi_t,q_{[0,t-1]}) = Q^{q_{[0,t-1]}}(\pi_t), \quad \forall \pi_t,q_{[0,t-1]}$$ such that the quantizer action $Q^{q_{[0,t-1]}} \in \mathbb{Q}({\cal P}(\mathbb{X});{\cal M}_t)$ is generated using only $q_{[0,t-1]}$, and the quantizer outcome is generated by evaluating $Q^{q_{[0,t-1]}}(\pi_t)$ for every $\pi_t$. $\diamond$ Proof of Theorem \[thm3\] ------------------------- At time $t=T-1$, the per-stage cost function can be written as: $$\begin{aligned} &&E[c(x_{t},\gamma_{t}(q_{[0,t]})) | q_{[0,t]}] \nonumber \\ &&=E[\int_{\mathbb{X}} P(dx_{t}|q_{[0,t-1]},q_t) c(x_{t},\gamma_{t}(q_{[0,t]}))] \nonumber\end{aligned}$$ Thus, at time $t=T-1$, an optimal receiver (which is deterministic without any loss of optimality, see [@Blackwell2]) will use $P(dx_t|q_{[0,t]})$ as a sufficient statistic for an optimal decision (or any receiver can be replaced with one which uses this sufficient statistic without any loss). Let us fix such a receiver policy which only uses the posterior $P(dx_t|q_{[0,t]})$ as its sufficient statistic. Note that $$\begin{aligned} P(dx_t|q_{[0,t]}) &=& \int_{\pi_t} P(dx_t|\pi_t) P(d\pi_t|q_{[0,t]}) \nonumber %&=& \sum_{\mathbb{Y}^{t+1}} P(dx_t|y_{[0,t]}) P(y_{[0,t]}|q_{[0,t]})\end{aligned}$$ and further $$\begin{aligned} && P(d\pi_t|q_{[0,t]}) = { P(q_t,d\pi_t|q_{[0,t-1]}) \over \int_{\pi_t} P(q_t,d\pi_t|q_{[0,t-1]})} \nonumber \\ && \quad \quad = { P(q_t | \pi_t,q_{[0,t-1]}) P(d\pi_t|q_{[0,t-1]}) \over \int_{\pi_t} P(q_t | \pi_t, q_{[0,t-1]}) P(d\pi_t|q_{[0,t-1]}) } \label{whySeparationHolds}\end{aligned}$$ The term $P(q_t | \pi_t , q_{[0,t-1]})$ is determined by the quantizer action $Q_t$ (this follows from Theorem \[thm1\]). Furthermore, given $Q_t$, the relation (\[whySeparationHolds\]) is measurable on ${\cal P}({\cal P}(\mathbb{X}))$ (that is, in $\Xi_t(\cdot) = P(\pi_t \in \cdot|q_{[0,t-1]})$) under weak convergence. To prove this technical argument, consider the numerator in (\[whySeparationHolds\]) and note that the function $\kappa_B: {\cal P}({\cal P}(\mathbb{X})) \to \mathbb{R}$ defined as $\kappa_B(\Xi)=\Xi(B)$ is measurable under weak convergence topology as a consequence of Theorem \[MonotoneClassRamon\], for each $B \in {\cal B}({\cal P}(\mathbb{X}))$. By Theorem 2.1 in Dubins and Freedman [@DubinsFreedman], this implies that the relation in (\[whySeparationHolds\]) is measurable on ${\cal P}({\cal P}(\mathbb{X}))$ (since the topology considered in [@DubinsFreedman] is not stronger than the weak convergence topology, the result in [@DubinsFreedman] holds in this case as well). Let us denote the quantizer applied, given the past realizations of quantizer outputs as $Q_t^{q_{[0,t-1]}}$. Note that $q_t$ is deterministically determined by $(\pi_t,Q_t^{q_{[0,t-1]}})$ and the optimal receiver function can be expressed as $\gamma^0_t(\Xi_t,q_t)$ (as a measurable function), given $Q_t^{q_{[0,t-1]}}$. The cost at time $t=T-1$ can be expressed, given the quantizer $Q_t^{q_{[0,t-1]}}$, for some Borel function $G$, as $G(\Xi_t,Q_t^{q_{[0,t-1]}})$, where $$\begin{aligned} && G(\Xi_t,Q_t^{q_{[0,t-1]}}) \\ && = \int_{{\cal P}(\mathbb{X})} \Xi_t(d\pi_t) \sum_{{\cal M}_t} 1_{\{q_t=Q_t^{q_{[0,t-1]}}(\pi_t)\}} \eta^{Q^{q_{[0,t-1]}}}(\Xi_t,q_t)), \end{aligned}$$ with $$\begin{aligned} \eta^{Q^{q_{[0,t-1]}}}(\Xi_t,q_t)) = \int \pi_t(dx_t) c(x_t,\gamma^0_t(\Xi_t,q_t))\end{aligned}$$ Now, one can construct an equivalence class among the past $q_{[0,t-1]}$ sequences which induce the same $\Xi_t$, and can replace the quantizers $Q_t^{q_{[0,t-1]}}$ for each class with one which induces a lower cost among the finitely many elements in each such class, for the final time stage. Thus, an optimal quantization output may be generated using $\Xi_t(\cdot)= P(\pi_{t} \in \cdot|q_{[0,t-1]})$ and $\pi_t$. Since there are only finitely many past sequences and finitely many $\Xi_t$, this leads to a Borel measurable selection of $\pi_t$ for every $\Xi_t$, leading to a quantizer and a measurable selection in $\Xi_t, \pi_t$. Since such a selection for $Q_t$ only uses $\Xi_t$, an optimal quantization output may be generated using $\Xi_t(\cdot)= P(\pi_{t} \in \cdot|q_{[0,t-1]})$ and $\pi_t$. Hence, $G(\Xi_t,Q_t^{q_{[0,t-1]}})$ can be replaced with $F_t(\Xi_t)$ for some $F_t$, without any performance loss. The same argument applies for all time stages: At time $t=T-2$, the sufficient statistic both for the immediate cost, and the cost-to-go is $P(dx_{t-1}|q_{[0,t-1]})$, and thus for the cost impacting the time stage $t=T-1$, as a result of the optimality result for $Q_{T-1}$. To show that the separation result generalizes to all time stages, it suffices to prove that $\{(\Xi_{t},Q_t)\}$ is a controlled Markov chain, if the encoders use the structure above. Toward this end, we establish that for $t \geq 1$, for all $B \in {\cal B}({\cal P}({\cal P}(\mathbb{X})))$, equation (\[eqnTempHolder\]) (displayed on the next page) holds. $$\begin{aligned} &&P\bigg( P(d\pi_t|q_{[0,t-1]}) \in B \bigg | P(d\pi_s|q_{[0,s-1]}), Q_{s}, s \leq t-1 \bigg) \nonumber \\ &&=P\bigg( \int_{\pi_{t-1}} P(d\pi_t,d\pi_{t-1}|q_{[0,t-1]}) \in B \bigg| P(d\pi_s|q_{[0,s-1]}), Q_{s}, s \leq t-1 \bigg) \nonumber \\ &&=P\bigg( \bigg\{ { \int_{\pi_{t-1}} P(d\pi_t|\pi_{t-1}) P(q_{t-1}|\pi_{t-1}, q_{[0,t-2]}) P(d\pi_{t-1}|q_{[0,t-2]}) \over \int_{\pi_{t-1},\pi_t} P(d\pi_t|\pi_{t-1}) P(q_{t-1}|\pi_{t-1}, q_{[0,t-2]}) P(d\pi_{t-1}|q_{[0,t-2]}) } \bigg\} \in B \bigg| P(d\pi_s|q_{[0,s-1]}), Q_{s}, s \leq t-1 \bigg) \nonumber \\ &&=P\bigg( \bigg\{ { \int_{\pi_{t-1}} P(d\pi_t|\pi_{t-1}) P(q_{t-1}|\pi_{t-1}, q_{[0,t-2]}) P(d\pi_{t-1}|q_{[0,t-2]}) \over \int_{\pi_{t-1},\pi_t} P(d\pi_t|\pi_{t-1}) P(q_{t-1}|\pi_{t-1}, q_{[0,t-2]}) P(d\pi_{t-1}|q_{[0,t-2]}) } \bigg\} \in B \bigg| P(d\pi_{t-1}|q_{[0,{t-2}]}), Q_{t-1}\bigg) \label{SeparationPart22} \\ &&=P\bigg( \int_{\pi_{t-1}} P(d\pi_t,d\pi_{t-1}|q_{[0,t-1]}) \in B \bigg| P(d\pi_{t-1}|q_{[0,t-2]}), Q_{t-1} \bigg) \nonumber \\ &&=P\bigg( P(d\pi_t|q_{[0,t-1]}) \in B \bigg| P(d\pi_{t-1}|q_{[0,t-2]}), Q_{t-1} \bigg)\label{eqnTempHolder}\end{aligned}$$ Here, (\[SeparationPart22\]) uses the fact that $P(q_{t-1}|\pi_{t-1}, q_{[0,t-2]})$ is identified by $\{P(d\pi_{t-1}|q_{[0,{t-2}]}), Q_{t-1}\}$, which in turn is uniquely identified by $q_{[0,t-2]}$ and $Q_{t-1}$. Furthermore, the relation in (\[eqnTempHolder\]) defines a regular conditional probability measure since for all $B \in {\cal B}({\cal P}(\mathbb{X}))$, $$\begin{aligned} &&\Xi_t(B) = P(\pi_t \in B|q_{[0,t-1]}) \nonumber \\ && = \int_{\pi_{t-1}} P(\pi_t \in B,d\pi_{t-1}|q_{[0,t-1]})\nonumber \\ && \quad = \int_{\pi_{t-1}} P(\pi_t \in B|\pi_{t-1}) P(d\pi_{t-1}|q_{[0,t-1]}) \nonumber\end{aligned}$$ is measurable in $\Xi_{t-1}$, given $Q_{t-1}$ (as a consequence of the measurability of (\[whySeparationHolds\]) in $\Xi_{t}$). Hence, by the result of Dubins and Freedman mentioned above (Theorem 2.1 in [@DubinsFreedman]), we conclude that for any measurable function $F_t$ of $\Xi_t$ $$\begin{aligned} E[F_t(\Xi_t)| \Xi_{[0,t-1]}, Q_{[0,t-1]} ] = E[F_t(\Xi_t), Q_{t})| \Xi_{t-1}, Q_{t-1}], \nonumber\end{aligned}$$ for every given $Q_{t-1}$. Now, once again an equivalence relationship between the finitely many past quantizer outputs, based on the equivalence of the conditional measures $\Xi_{t-1}$ they induce, can be constructed. With the controlled Markov structure, we can follow the same argument for earlier time stages. Therefore, it suffices that the encoder uses only $(\Xi_t,t)$ as its sufficient statistic for all time stages, to generate the optimal quantizer. An optimal quantizer uses $\pi_t$ to generate the optimal quantization outputs. $\diamond$ Proof of Theorem \[mainDecentralized\] -------------------------------------- The proof is in three steps, (i), (ii) and (iii) below.\ [**Step (i)**]{} In decentralized dynamic decision problems where the decision makers have the same objective (that is, in team problems), more information provided to the decision makers does not lead to any degradation in performance, since the decision makers can always choose to ignore the additional information. In view of this, let us relax the information structure in such a way that the decision makers now have access to all the previous observations, that is the information available at the encoders $1$ and $2$ are: $$I^i_t=\{y^i_t,{\bf y}_{[0,t-1]}, {\bf q}_{[0,t-1]}\} \quad t \geq 1, \quad i=1,2.$$ $$I^i_0=\{y^i_0\}, \quad i=1,2.$$ The information pattern among the encoders is now the [*one-step delayed observation sharing pattern*]{}. We will show that the past information can be eliminated altogether, to prove the desired result.\ **Step (ii)** The second step uses the following technical Lemma. \[lem2\] Under the relaxed information structure in step $(i)$ above, any decentralized quantization policy at time $t$, $1 \leq t\leq T-1$, can be replaced, without any loss in performance, with one which only uses $(\pi_t, {\bf y}_t,{\bf q}_{[0,t-1]})$, satisfying the following form: $$\begin{aligned} && P({\bf q}_t|{\bf y}_{[0,t]},{\bf q}_{[0,t-1]}) \nonumber \\ && = P(q^1_t | y^1_t,{\bf q}_{[0,t-1]}) P(q^2_t | y^2_t,{\bf q}_{[0,t-1]}) \nonumber \\ && = 1_{\{q^1_t=\bar{Q}^1(y^1_t,{\bf q}_{[0,t-1]})\}}1_{\{q^2_t=\bar{Q}^{2}(y^2_t,{\bf q}_{[0,t-1]})\}},\end{aligned}$$ for measurable functions $\bar{Q}^1$ and $\bar{Q}^2$. $\diamond$\ **Proof.** Let us fix a composite quantization policy ${\bf \Pi}^{comp}$. At time $t=T-1$, the per-stage cost function can be written as: $$\begin{aligned} \label{costCompactForm2} %&&E[c(x_{t},v_{t}({\bf q}_{[0,t]})) | {\bf q}_{[0,t-1]} ] \nonumber \\ && E [\int_{\mathbb{X}} P(dx_{t}|{\bf q}_{[0,t]}) c(x_{t},v_t)| {\bf q}_{[0,t-1]} ]\end{aligned}$$ For this problem, $P(dx_{t}|{\bf q}_{[0,t]})$ is a sufficient statistic for an optimal receiver. Hence, at time $t=T-1$, an optimal receiver will use $P(dx_t|{\bf q}_{[0,t]})$ as a sufficient statistic for an optimal decision as the cost function conditioned on ${\bf q}_{[0,t]}$ is written as: $\int P(dx_t|{\bf q}_{[0,t]}) c(x_t,v_t)$, where $v_t$ is the decision of the receiver. Now, let us fix this decision policy at time $t$. We now note that (\[longequationDecentralized\]), displayed on the next page, follows. $$\begin{aligned} P(dx_t|{\bf q}_{[0,t]}) &=& \sum_{\mathbb{Y}^{t+1}} P(dx_t,{\bf y}_{[0,t]}|{\bf q}_{[0,t]}) = {\sum_{\mathbb{Y}^{t+1}} P(dx_t,{\bf q}_t,{\bf y}_{[0,t]}|{\bf q}_{[0,t-1]}) \over P({\bf q}_t|{\bf q}_{[0,t-1]})} \nonumber \\ &=&{ \sum_{\mathbb{Y}^{t+1}}P({\bf q}_t|{\bf y}_{[0,t]},{\bf q}_{[0,t-1]}) P({\bf y}_t|x_t) P(dx_t|{\bf y}_{[0,t-1]}) P({\bf y}_{[0,t-1]}|{\bf q}_{[0,t-1]}) \over \int_{\mathbb{X}, \mathbb{Y}^{t+1}} P({\bf q}_t|{\bf y}_{[0,t]},{\bf q}_{[0,t-1]}) P({\bf y}_t|x_t) P(dx_t|{\bf y}_{[0,t-1]}) P({\bf y}_{[0,t-1]}|{\bf q}_{[0,t-1]}) } \nonumber \\ &=&{ \sum_{\mathbb{Y}^{t+1}}P({\bf q}_t|{\bf y}_{[0,t]},{\bf q}_{[0,t-1]}) P({\bf y}_t|x_t) \pi(dx_t) P({\bf y}_{[0,t-1]}|{\bf q}_{[0,t-1]}) \over \int_{\mathbb{X}, \mathbb{Y}^{t+1}} P({\bf q}_t|{\bf y}_{[0,t]},{\bf q}_{[0,t-1]}) P({\bf y}_t|x_t) \pi(dx_t) P({\bf y}_{[0,t-1]}|{\bf q}_{[0,t-1]}) } \label{longequationDecentralized}\end{aligned}$$ In (\[longequationDecentralized\]), we use the relation $P(dx_t|{\bf y}_{[0,t-1]})=P(dx_t)=:\pi(dx_t)$, where $\pi(\cdot)$ denotes the marginal probability on $x_t$ (recall that the source is memoryless). The term $P({\bf q}_t|{\bf y}_{[0,t]},{\bf q}_{[0,t-1]})$ in (\[longequationDecentralized\]) is determined by the composite quantization policies: $$\begin{aligned} &&P({\bf q}_t|{\bf y}_{[0,t]},{\bf q}_{[0,t-1]}) \nonumber \\ &&= P(q^1_t | y^1_t,{\bf y}_{[0,t-1]},{\bf q}_{[0,t-1]})P(q^2_t | y^2_t,{\bf y}_{[0,t-1]},{\bf q}_{[0,t-1]}) \nonumber \\ &&= 1_{\{q^1_t=Q_t^{comp,1}(y^1_t,{\bf y}_{[0,t-1]},{\bf q}_{[0,t-1]})\}} \nonumber \\ && \quad \quad \times 1_{\{q^2_t=Q_t^{comp,2}(y^2_t,{\bf y}_{[0,t-1]},{\bf q}_{[0,t-1]})\}} \nonumber\end{aligned}$$ The above is valid since each encoder knows the past observations of both encoders. As such, $P(dx_t|{\bf q}_{[0,t]})$ can be written as, for some function $\Upsilon$: $\Upsilon(\pi,{\bf q}_{[0,t-1]},{\bf Q}_t^{comp}(.))$. Note that ${\bf q}_{[0,t-1]}$ appears due to the term $P({\bf y}_{[0,t-1]}|{\bf q}_{[0,t-1]})$. Now, consider the following space of joint (team) mappings at time $t$, denoted by ${\cal G}_t$: $$\begin{aligned} {\cal G}_t=\{{\bf \Psi}_t: {\bf \Psi}_t =\{\Psi^1_t,\Psi^2_t\}, \Psi^i_t: \mathbb{Y}^i \to {\cal M}^i_t, \quad i=1,2\}. \label{spaceOfTeamActions}\end{aligned}$$ For every composite quantization policy there exists a distribution $P'$ on random variables $({\bf q}_t, \pi, {\bf q}_{[0,t-1]})$ such that $$\begin{aligned} \label{tulsa} && P'({\bf q}_t | \pi, {\bf q}_{[0,t-1]}) = \sum_{(\mathbb{Y}^1 \times \mathbb{Y}^2)^{t+1}} P({\bf q}_t ,{\bf y}_{[0,t]} | \pi, {\bf q}_{[0,t-1]})\nonumber \\ && = \sum_{(\mathbb{Y}^1 \times \mathbb{Y}^2)^{t+1}} \bigg( P(q^1_t |{\bf y}_{[0,t-1]}, y^1_t, {\bf q}_{[0,t-1]},\pi) \nonumber \\ && \quad \quad \quad \quad \quad \times P(q^2_t |{\bf y}_{[0,t-1]}, y^2_t, {\bf q}_{[0,t-1]},\pi) \nonumber \\ && \quad \quad \quad \quad \quad \times P(y^1_t,y^2_t) P({\bf y}_{[0,t-1]} | \pi, {\bf q}_{[0,t-1]}) \bigg)\end{aligned}$$ Furthermore, with every composite quantization policy and every realization of ${\bf y}_{[0,t-1]}, {\bf q}_{[0,t-1]}$, we can associate an element in the space ${\cal G}_t$, ${\bf \Psi}_{{\bf y}_{[0,t-1]},{\bf q}_{[0,t-1]}}$, such that the induced stochastic relationship in (\[tulsa\]) can be obtained: $$\begin{aligned} && P'({\bf q}_t | \pi, {\bf q}_{[0,t-1]}) = \sum_{(\mathbb{Y}^1 \times \mathbb{Y}^2)^{t+1}} P({\bf q}_t, {\bf y}_{[0,t]} | \pi, {\bf q}_{[0,t-1]})\nonumber \\ &=& \sum_{(\mathbb{Y}^1 \times \mathbb{Y}^2)^{t+1}} 1_{\{{\bf \Psi}_{{\bf y}_{[0,t-1]},{\bf q}_{[0,t-1]}}(y^1_t,y^2_t)=(q^1_t,q^2_t) \}} \nonumber \\ && \quad \quad \quad \quad \times P(y^1_t,y^2_t) P({\bf y}_{[0,t-1]} | \pi,{\bf q}_{[0,t-1]}) \nonumber\end{aligned}$$ We can thus, express the cost, for some measurable function $F$ in the following way: $$E[F(\pi,{\bf q}_{[0,t-1]},{\bf \Psi}) | \pi,{\bf q}_{[0,t-1]}],$$ where $$\begin{aligned} &&P({\bf \Psi} | \pi,{\bf q}_{[0,t-1]}) \nonumber \\ &&= \sum_{(\mathbb{Y}^1 \times \mathbb{Y}^2)^{t}} 1_{ \{ {\bf \Psi} = {\bf \Psi}_{{\bf y}_{[0,t-1]},{\bf q}_{[0,t-1]}} \} } P({\bf y}_{[0,t-1]} | \pi,{\bf q}_{[0,t-1]}) \nonumber\end{aligned}$$ Now let $t=T-1$ and define for every realization ${\bf \Psi}_t=(\Psi^1_t,\Psi^2_t) \in {\cal G}_t$ (with the decision policy considered earlier fixed): $$\begin{aligned} &&\beta_{{\bf \Psi}_t} := \bigg\{\pi,{\bf q}_{[0,t-1]} : \nonumber \\ && \quad \quad \quad F(\pi,{\bf q}_{[0,t-1]},{\bf \Psi}_t) \leq F(\pi,{\bf q}_{[0,t-1]},{\bf \Psi}'_t) \nonumber \\ && \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \forall ((\Psi^1_t)',(\Psi^2_t)') \in {\cal G}_t \bigg\}. \nonumber\end{aligned}$$ As we had observed in the proof of Theorem \[thm1\], such a construction covers the domain set consisting of $(\pi,{\bf q}_{[0,t-1]})$ but possibly with overlaps. Note that for every $(\pi,{\bf q}_{[0,t-1]})$, there exists a minimizing function in ${\cal G}_t$, since ${\cal G}_t$ is a finite set. In this sequence, let there be an ordering of the finitely many elements in ${\cal G}_t$ as $\{{\bf \Psi}_t(1),{\bf \Psi}_t(2),\dots, {\bf \Psi}_t(k),\dots\}$, and define a function ${\bf T}^*_t$ as: $$\begin{aligned} &&{\bf \Psi}_t(k)= {\bf T}^*_t(\pi,{\bf q}_{[0,t-1]}), \nonumber \\ && \quad \rm{if} \quad \bigg(\pi,{\bf q}_{[0,t-1]}\bigg) \in \beta_{{\bf \Psi}_t(k)} - \cup_{i=0}^{k-1} \beta_{{\bf \Psi}_t(i)},\nonumber\end{aligned}$$ with $\beta_{{\bf \Psi}_t(0)}=\emptyset$. Thus, we have constructed a policy which performs at least as well as the original composite quantization policy. It has a restricted structure in that it only uses $(\pi,{\bf q}_{[0,t-1]})$ to generate the team action and the local information $y^1_t,y^2_t$ to generate the quantizer outputs. Now that we have obtained the structure of the optimal encoders for the last stage, we can sequentially proceed to study the other time stages. Note that given a fixed $\pi$, $\{(\pi,{\bf y}_t)\}$ is i.i.d. and hence Markov. Now, define $\pi'_t= (\pi,{\bf y}_t)$. For a three-stage cost problem, the cost at time $t=2$ can be written as, for measurable functions $c_2, c_3$: $$\begin{aligned} &&c_2(\pi'_2,v_2({\bf q}_{[1,2]})) \nonumber \\ && \quad + E[c_3(\pi'_3,v_3({\bf q}_{[1,2]},Q_3(\pi'_3,{\bf q}_{[1,2]})))|\pi'_{[1,2]},{\bf q}_{[1,2]}] \nonumber\end{aligned}$$ Since $P(d\pi'_3,{\bf q}_{[1,2]}|\pi'_2,\pi'_1,{\bf q}_{[1,2]})=P(d\pi'_3,{\bf q}_{[1,2]}|\pi'_2,{\bf q}_{[1,2]}),$ the expression above is equal for some $F_2(\pi'_2,{\bf q}_2,{\bf q}_1)$ for some measurable $F_2$. By a similar argument, an optimal composite quantizer at time $t$, $1 \leq t\leq T-1$ only uses $(\pi, {\bf y}_t,{\bf q}_{[0,t-1]})$. An optimal (team) policy generates the quantizers $Q^1_t,Q^2_t$ using ${\bf q}_{[0,t-1]}, \pi$, and the quantizers use $\{y^i_t\}$ to generate the quantizer outputs at time $t$ for $i=1,2$. $\diamond$\ [**Step (iii)**]{} The final step will complete the proof. At time $t=T-1$, an optimal receiver will use $P(dx_t|{\bf q}_{[0,t]})$ as a sufficient statistic for the optimal decision. We now observe that $$\begin{aligned} P(dx_t|{\bf q}_{[0,t]}) &=& \sum_{\mathbb{Y}^{t+1}} P(dx_t|{\bf y}_{[0,t]}) P({\bf y}_{[0,t]}|{\bf q}_{[0,t]}) \label{BeliefOnBelief} \\ &=& \sum_{\mathbb{Y}^{t+1}} P(dx_t|{\bf y}_{t}) P({\bf y}_{[0,t]}|{\bf q}_{[0,t]}) \nonumber \\ &=& \sum_{\mathbb{Y}} P(dx_t|{\bf y}_{t}) \sum_{\mathbb{Y}^{t}} P({\bf y}_{[0,t]}|{\bf q}_{[0,t]}) \nonumber\\ &=& \sum_{\mathbb{Y}} P(dx_t|{\bf y}_{t}) P({\bf y}_{t}|{\bf q}_{[0,t]}) \nonumber\end{aligned}$$ Thus, $P(dx_t|{\bf q}_{[0,t]})$, is a function of $P({\bf y}_t|{\bf q}_{[0,t]})$. Now, let us note that $$\begin{aligned} %&& \nonumber \\ %&& = \int_{\pi_{t-1}} \sum_{{\bf y}_{t-1}} P(d\pi,{\bf y}_t,d\pi_{t-1},{\bf y}_{t-1}|{\bf q}_{[0,t]}) \nonumber \\ && P({\bf y}_t|{\bf q}_{[0,t]}) = { P({\bf q}_t,{\bf y}_t|{\bf q}_{[0,t-1]}) \over \sum_{{\bf y}_t} P({\bf q}_t,{\bf y}_t|{\bf q}_{[0,t-1]})} \nonumber\\ && = { P({\bf q}_t | {\bf y}_t, {\bf q}_{[0,t-1]}) P({\bf y}_t|{\bf q}_{[0,t-1]}) \over \sum_{{\bf y}_t} P({\bf q}_t | {\bf y}_t, {\bf q}_{[0,t-1]}) P({\bf y}_t|{\bf q}_{[0,t-1]}) } \nonumber\\ && = { P({\bf q}_t | {\bf y}_t, {\bf q}_{[0,t-1]}) P({\bf y}_t) \over \sum_{{\bf y}_t} P({\bf q}_t | {\bf y}_t, {\bf q}_{[0,t-1]}) P({\bf y}_t) }, %&& = {\int_{\pi_{t-1}} \sum_{{\bf y}_{t-1}} P({\bf q}_t | \pi,{\bf y}_t , {\bf q}_{[0,t-1]}) P(d\pi,{\bf y}_t|\pi_{t-1},{\bf y}_{t-1}) P(d\pi_{t-1},{\bf y}_{t-1}|{\bf q}_{[0,t-1]}) \over %\int_{\pi_{t-1},\pi} \sum_{{\bf y}_{t-1},{\bf y}_t} P({\bf q}_t | \pi,{\bf y}_t , {\bf q}_{[0,t-1]}) P(d\pi,{\bf y}_t|\pi_{t-1},{\bf y}_{t-1}) P(d\pi_{t-1},{\bf y}_{t-1}|{\bf q}_{[0,t-1]}) } %\nonumber\end{aligned}$$ where the term $P({\bf q}_t | {\bf y}_t , {\bf q}_{[0,t-1]})$ is determined by the quantizer team action ${\bf Q}_t^{comp}$. As such, the cost at time $t=T-1$ can be expressed as a measurable function $G(P({\bf y}_{t}),{\bf Q}_t)$. Thus, it follows that, an optimal quantizer policy at the last stage, $t=T-1$ may only use $P({\bf y}_{t})$ to generate the quantizers, where the quantizers use the local information $y^i_t$ to generate the quantization output. At time $t=T-2$, the sufficient statistic for the cost function is $P(dx_{t-1}|{\bf q}_{[0,t-1]})$ both for the immediate cost, and the cost-to-go, that is the cost impacting the time stage $t=T-1$, as a result of the optimality result for $Q_{T-1}$ and the memoryless nature of the source dynamics. The same argument applies for all time stages. Hence, any policy without loss can be replaced with one in $\Pi^{NSM}$ defined in (\[piNSM\]). 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[Serdar Yüksel]{} received his BSc degree in Electrical and Electronics Engineering from Bilkent University in 2001; MS and PhD degrees in Electrical and Computer Engineering from the University of Illinois at Urbana-Champaign in 2003 and 2006, respectively. He was a post-doctoral researcher at Yale University for a year before joining Queen’s University as an assistant professor of Mathematics and Engineering at the Department of Mathematics and Statistics. His research interests are on stochastic and decentralized control, information theory and applied probability.
--- author: - | [^1]\ European Organization for Nuclear Research (CERN), Switzerland and\ Utrecht University, Netherlands\ E-mail: title: Upgrade of the Inner Tracking System of ALICE --- Upgrade of ALICE ================ The ALICE experiment was designed to study the properties of the quark-gluon plasma created in heavy-ion collisions at the LHC. Since the start of the LHC, ALICE has confirmed the existence of the quark-gluon plasma and extended the precision and kinematic reach of many significant probes measured previously at RHIC and at the SPS. After the second long shut down of the LHC in 2019–2020, ALICE will focus on the measurement of heavy-flavor hadrons, quarkonia and low-mass dileptons at low transverse momenta [@ref:ALICE_LOI]. This requires high-precision measurements at low [$p_\text{T}$]{} resulting in the need of recording large samples of minimum-bias events. In order to achieve this, ALICE is planning to read out all Pb-Pb collisions delivered by the LHC which corresponds to a maximum collision rate of $50$ kHz. The upgraded detector will record an integrated luminosity of $10\text{ nb}^{-1}$ in Pb-Pb collisions which means a factor $100$ compared to the data that will be recorded by 2019. To cope with the physics requirements of ALICE several upgrades will take place during the second long shut down of the LHC. Two new detectors will be installed: a high resolution, low material Inner Tracking System (ITS) and a Muon Forward Telescope (MFT). They will be accompanied by a new beampipe with a smaller radius which will allow the first layer of the ITS to be moved closer to the interaction point resulting in better impact-parameter determination. The readout chambers of the Time Projection Chamber (TPC) will be replaced by Gas Electron Multiplier (GEM) detectors and its readout electronics will be upgraded together with the readout electronics of the Transition Radiation Detector (TRD), the Time Of Flight (TOF) detector and the Muon Spectrometer. The forward trigger detectors, the online system and the offline reconstruction and analysis framework will also be upgraded [@ref:ALICE_LOI]. Upgrade of the Inner Tracking System ==================================== The Inner Tracking System is situated at the center of ALICE closest to the interaction point (Fig. \[fig:alice\]). The data taking by the ITS before the first long shut down of the LHC has been successful and data taking has been already resumed since the restart of the accelerator [@ref:currentITSTalk]. [2012-Aug-02-ALICE\_3D\_v0.jpg]{} (560,670)[ITS]{} (1015,650)[Strip]{} (1010,562)[Drift]{} (960,450)[Pixel]{} (600,660)[(-1,-1.05)[270]{}]{} (610,660)[(1,-0.3)[130]{}]{} (1000,660)[(-1,-0.1)[140]{}]{} (1000,575)[(-1,0.3)[140]{}]{} (970,490)[(-1,0.85)[130]{}]{} However, the current detector is not suited for the physics requirements of ALICE after the second long shut down of the LHC, and an upgrade is necessary to improve the physics performance of ALICE in several ways. The impact parameter resolution at [$p_\text{T}$]{} $=500$ MeV/c will be improved by a factor $5$ in the direction of the beam and a factor $3$ in the transverse direction which will be achieved by different steps. Firstly, the smaller beampipe will allow the first layer of the ITS to be moved as close as $23$ mm to the interaction point instead of the current $39$ mm. Secondly, the material budget of the detector will be reduce from around $1.14\%\text{ X}_0$ to around $0.3\%\text{ X}_0$ for the three innermost layers and finally the pixel size will be reduced to about $30 ~\upmu\text{m} \times 30 ~\upmu\text{m}$ from the current $50~\upmu\text{m} \times 425 ~\upmu\text{m}$. The upgraded detector will also have an improved tracking efficiency and [$p_\text{T}$]{} resolution at low [$p_\text{T}$]{}. To achieve this, the current six layers of hybrid pixel, strip and drift detectors will be replaced by seven layers of monolithic pixel detectors. Right now ALICE is limited by the current ITS in the readout speed to $1$ kHz. After the upgrade, ALICE is planning to read out all Pb-Pb collisions at the maximum rate of the LHC: thanks to the design of the new detector a readout rate of more than $10^5$ Hz in Pb-Pb events and several $10^5$ Hz in pp events will be possible. The requirement to remove and reinsert the detector during the yearly shut down of the LHC places further strong constraints on the design of the mechanical support of the detector. The seven layers of the new ITS cover an area of around $10\text{ m}^2$ and will consists of around $1.25 \times 10^{10}$ . The layers will be grouped according to the following: the three innermost layers make up the Inner Barrel and the four outermost layers the Outer Barrel. The radial arrangement of the layers can be seen in Fig. \[fig:ITS\_layout\]. All the layers will be segmented into elements called staves which extend the full length of the detector in the direction of the beam. A stave is the smallest operable unit of the detector and has a slightly different design in the case of the Inner and Outer Barrel. The first prototypes of both the Inner and Outer Barrel staves can be seen in Fig. \[fig:staves\]. [layout\_small.jpg]{} ( 0,100) ( -170,340) ( 750,400) ( 747,398)[(-2,-1)[95]{}]{} ( 90,340)[(2,-1)[110]{}]{} ( 90,350)[(2,-0.2)[290]{}]{} Since the distance of the layers from the interaction point vary from $23$ mm to $400$ mm, the working environment and the requirements are very different on the innermost and the outermost layers. The requirements for the different layers are summarized in Tab. \[tab:requirements\]. The whole detector is planned to be equipped with the same chip, resulting in very stringent requirements on material budget, spatial resolution, detection efficiency, power density and fake hit rate. On the other hand, the expected radiation environment is relatively modest compared to the radiation levels expected in CMS or ATLAS at the same distance from the collision point. These requirements led to the choice of using Monolithic Active Pixel Sensors for the upgrade of the ITS. [0.49]{} [innerBarrelStave2.png]{} [0.49]{} [outerBarrel7.jpg]{} **Parameter** **Inner barrel** **Outer barrel** ----------------------- ------------------------------------------------------------------- -------------------------------------------------------- Silicon thickness Spatial resolution $5~\upmu$m $10~\upmu$m Power density $<300$ mW$/\text{cm}^2$ $<100$ mW$/\text{cm}^2$ Event resolution Detection efficiency Fake hit rate Average track density $15$–$35$ $\text{cm}^{-2}$ $0.1$–$1$ $\text{cm}^{-2}$ TID radiation $2700$ krad $100$ krad NIEL radiation $1.7 \times 10^{13}$ [1 MeV $\text{n}_{\text{eq}}\text{/cm}^2$]{} $10^{12}$ [1 MeV $\text{n}_{\text{eq}}\text{/cm}^2$]{} : Requirements for the Inner and Outer Barrel for the upgraded ITS. Radiation levels include a safety factor of $10$. Values are taken from [@ref:TDR], but the radiation levels are updated according to the latest estimates.[]{data-label="tab:requirements"} Technology choice ================= In the previous chapter the requirements for the chip for the ITS upgrade was discussed showing that the technology of CMOS Monolithic Active Pixel Sensors (MAPS) suits well these requirements. The first implementation of this technology in a heavy-ion physics experiment was the STAR PXL detector which has been installed in 2014 [@ref:STAR]. The ULTIMATE chip used for the STAR PXL detector however does not meet all the requirements of the ALICE ITS, particularly in terms of readout time. The ULTIMATE chip has a readout time of $190~\upmu$s whereas for the ITS upgrade the design goal is to stay below $30~\upmu$s. The development of a new chip is therefore needed for the ITS upgrade and the chosen technology is the TowerJazz $0.18~\upmu$m CMOS imaging process for which the schematic cross section of a pixel is shown in Fig. \[fig:technology\]. The transistor feature size of $0.18~\upmu$m and the available six metal layers in this technology allow for the implementation of high density and low power digital circuitry. The small transistor feature size and the gate oxide thickness of $3~\upmu$m also make this technology more radiation tolerant than the $0.35~\upmu$m technology used up to now as the baseline in particle-physics applications. The mechanism of the detection of particles in this technology is the following: the crossing particle generates electron-hole pairs in the high resistivity epitaxial layer; these electrons then diffuse until they reach the depleted volume within which they drift towards the collection n-well electrode where they are collected. Being able to produce the chip on a high resistivity epitaxial layer ($\gtrsim~1\text{ k}\Omega$ cm) is important, because it results in larger depletion volumes and thus better charge-collection efficiency. It also results in the electrons being less likely to be trapped, making the technology more radiation hard towards non-ionizing radiation. An important feature of the technology is the implementation of the deep p-well which allows the use of CMOS circuitry within the pixel area by shielding the n-well of the PMOS transistors from the epitaxial layer. Without such a shielding, the n-wells of the CMOS transistors would compete in the collection of the electrons with the collection diode and this would result in a lower charge-collection efficiency. For some prototypes a moderate bias voltage can be applied to the substrate ($< 10$ V) to create a larger depletion volume around the collection diode. This results in a smaller capacitance and consequently in a better charge-collection efficiency. The capacitance can be further reduced by keeping the collection n-well diode small, which in the case of the current prototypes is $2$–$3~\upmu$m. ![Cross section of one pixel in the TowerJazz $0.18~\upmu$m CMOS imaging process. Figure is taken from [@ref:TDR].[]{data-label="fig:technology"}](technology.pdf){width="60.00000%"} Chip architectures ================== ALPIDE and MISTRAL-O -------------------- There are two parallel chip developments ongoing for the upgrade of the ITS, the ALPIDE and the MISTRAL-O. The two chips are designed such that their dimensions, physical and electrical interface are identical, so either of them can be used for the upgrade without any modifications in the mechanical and electrical services. The schematic design of the matrix layout can be seen in Fig. \[fig:schematics\] for the two architectures. [0.49]{} [pALPIDE.png]{} ( 230,640)[ALPIDE]{} ( 650,300)[Continuous or ]{} ( 650,250)[external trigger]{} [0.49]{} [mistral.png]{} ( 300,670)[MISTRAL-O]{} ( 800,300)[Rolling ]{} ( 800,250)[shutter]{} The MISTRAL-O development is a continuation of the ULTIMATE chip development. It has amplification within the pixel, the analog signal is then propagated to the end of column and the digitalization and zero suppression is done at the end of column. In the case of the ALPIDE the amplification, the digitalization and zero suppression is done within the pixel, so that only digital signals are propagated to the end of column. This allows for a data driven readout in the case of the ALPIDE while the MISTRAL-O is read out in the more traditional rolling shutter mode with an event-time resolution of around $20~\upmu$s. The event-time resolution of the ALPIDE is $\lesssim 2~\upmu$s and is determined by the rise time of the pulses. The power consumption, the dead area and the pixel pitch is smaller in the case of the ALPIDE (exact values can be seen in Tab. \[tab:ALPIDE\_MISTRAL\_comparison\]) and both chips show good performance in noise occupancy and detection efficiency measurements which resulted in the choice of the ALPIDE as the baseline for the project. The current document focuses on the ALPIDE architecture, while details on the MISTRAL-O family can be found in [@ref:MistralTalk]. **ALPIDE** **MISTRAL-O** ----------------------- ---------------------------------- ---------------------------------- Pixel pitch $28~\upmu$m $\times$ $28~\upmu$m $36~\upmu$m $\times$ $65~\upmu$m Event time resolution $\lesssim2~\upmu$s $\sim{20}~\upmu$s Power consumption $39\text{ mW}/\text{cm}^2$ $80$–$90\text{ mW}/\text{cm}^2$ Dead area $1.1$ mm $\times$ $30$ mm $1.5$ mm $\times$ $30$ mm : Performance and specification of the ALPIDE and MISTRAL-O architectures.[]{data-label="tab:ALPIDE_MISTRAL_comparison"} pALPIDE-1 --------- The pALPIDE-1 chip is the first prototype of the ALPIDE family with the final chip size of $15$ mm $\times$ $30$ mm. It has $512$ $\times$ $1024$ digital pixels with the size of $28~\upmu$m $\times$ $28~\upmu$m. It is read out using an asynchronous priority encoder network, thanks to which only hit pixels are read out resulting in very fast acquisition of the data from the full matrix. The chip is organized in four sub-matrices consisting of different types of pixels. A picture of the chip can be seen in Fig. \[fig:pALPIDE-1\_photo\] with these sub-matrices indicated. They differ in either the reset mechanism of the pixels or in the spacing between the collection diode and the surrounding p-well. In the following, results will be reported for only one of these pixel variations. [pALPIDEfs2.jpg]{} (475,93) (217,245) (350,300) (470,298) (610,296) (750,294) (430,440)[(-0.05,-1)[11.9]{}]{} (562,440)[(-0.03,-1)[7.2]{}]{} (700,440)[(-0.013,-1)[3.15]{}]{} (465,118)[(-1,0.02)[205]{}]{} (655,112)[(1,-0.02)[205]{}]{} (240,245)[(-0.1,-1)[6]{}]{} (265,418)[(0.1,1)[5.3]{}]{} Characterization of the pALPIDE-1 prototype =========================================== Methods and tools ----------------- In the upgraded ITS the pixel chips will be laser soldered to the flexible printed circuit which will give both the mechanical and the electrical connection, while for testing, the prototypes are wire bonded to a carrier card as can be seen in Fig. \[fig:pALPIDE-1\_photo\]. The carrier card is then connected to the readout board by a PCI Express connector and the readout board is connected via USB to the computer (Fig. \[fig:carrier\]). The first tests of the chips (noise and threshold measurements, noise occupancy determination) are done in this setup while for the detection efficiency and spatial resolution measurements a stack of this setup is used. The stack consists of seven pALPIDE-1 chips where the outer six are the tracking planes and the central one is being tested (Fig. \[fig:testBeamSetup\]). The seven chips are connected to a computer the same way as the individual chips are and the readout is done by the EUDAQ framework [@ref:eudaq]. The analysis of the data collected at beam tests is then done by the EUTelescope software [@ref:eutelescope]. [0.49]{} [DAQ.png]{} (70,560)[DAQ board]{} (700,160)[Carrier card]{} (600,550)[pALPIDE-1]{} ( 690,540)[(0.5,-1)[80]{}]{} [0.49]{} [testBeamSetup.jpg]{} (100,125)[(2,0.77)[700]{}]{} (70,135) (380,160) (380,500) Performance ----------- The pALPIDE-1 prototype has been thoroughly characterized both in laboratory measurements and in several test beam campaigns at different test beam facilities (PS, SPS, DESY, BTF and PAL). The beam energy at these facilities range from $60$ MeV to $120$ GeV and the particles used are electrons, positrons and pions. The following results were obtained at the PS using a $6$ GeV negative pion beam. [0.48]{} ![Results for the pALPIDE-1 from measurements at the PS using a $6$ GeV negative pion beam. The data for both figures were taken applying $-3$ V reverse substrate bias and is shown as function of the threshold current.[]{data-label="fig:results"}](efficiency_noiseOccupancy_irradiationCompare2.pdf){width="100.00000%"} [0.48]{} ![Results for the pALPIDE-1 from measurements at the PS using a $6$ GeV negative pion beam. The data for both figures were taken applying $-3$ V reverse substrate bias and is shown as function of the threshold current.[]{data-label="fig:results"}](resolution_clusterSize_irradiationCompare.pdf){width="100.00000%"} The results are summarized in Fig. \[fig:results\] where the left panel shows the detection efficiency and noise occupancy values of the pALPIDE-1 while the right panel shows the spatial resolution and the average cluster size of the chip. Both figures show the results as a function of the threshold current which is a parameter of the chip used for setting the input charge threshold of the pixels. The requirements for the ITS upgrade is to have a detection efficiency better than $99\%$ and noise occupancy lower than $10^{-5}$ hits/event/pixel. In Fig. \[fig:efficiency\_noiseOccupancy\] it can be seen that both requirements can be fulfilled at a wide operating range of the threshold current. The performance in terms of detection efficiency does not degrade after irradiating the chip by $10^{13}$ [1 MeV $\text{n}_{\text{eq}}\text{/cm}^2$]{}, but the noise occupancy values become slightly higher. In Fig. \[fig:resolution\_clusterSize\] it can be seen that there is also a large operational margin in terms of spatial resolution where the requirement for the chip is to stay below $5~\upmu$m which is fulfilled both before and after irradiation by $10^{13}$ [1 MeV $\text{n}_{\text{eq}}\text{/cm}^2$]{}. [0.46]{} [clusterSize2D.pdf]{} (320,710)[Cluster size vs track]{} (240,650)[impinging point within pixel]{} (100,70)[(0,1)[270]{}]{} (100,340)[(1,0)[440]{}]{} (540,340)[(-1,-0.62)[450]{}]{} (40,20)[1]{} (40,340)[2]{} (500,340) [0.46]{} [clusterSizeCS2.pdf]{} (320,730)[Cross section]{} (100,150)[1]{} (300,150)[2]{} (550,150)[3]{} (850,150)[1]{} In Fig. \[fig:clusterSize2D\_full\] the average cluster size is shown as a function of the impinging point of the track within a pixel. It is smallest if the track goes through the center of the pixel, close to the collection diode and largest if the track goes through the corner of the pixels where it is more likely to have charge-sharing with the neighboring pixels. Summary ======= The Inner Tracking System of ALICE will be upgraded during the second long shut down of the LHC in 2019–2020 to fulfill the physics requirements of ALICE after the shut down. The new detector will have seven layers of Monolithic Active Pixel Sensors and will improve significantly the tracking and [$p_\text{T}$]{} resolution at low [$p_\text{T}$]{}, as well as the impact-parameter resolution. The first large-scale prototypes of both ALPIDE and MISTRAL-O have been thoroughly characterized and show very good performance with a wide operational margin. Tests have also been carried out with irradiated sensors and the chips show good performance after an irradiation level of $10^{13}$ [1 MeV $\text{n}_{\text{eq}}\text{/cm}^2$]{}. Overall the development of the new detector is progressing according to schedule with all aspects of the R&D close to completion. Acknowledgement =============== We would like to thank the host institutes (DESY, BTF Frascati, Pohang and CERN) for their support during our test beam measurements. We would like to thank also the developers of the EUDAQ and EUTelescope software framework for their support. [99]{} ALICE collaboration, *Letter of Intent for the Upgrade of the ALICE Experiment*, CERN-LHCC-2012-012, LHCC-I-022 (2012) D. Colella, *ALICE ITS: the Run 1 to Run 2 transition and recent operational experience*,\ in proceedings of the current conference. ALICE Collaboration, *Technical Design Report for the Upgrade of the ALICE Inner Tracking System*, *J. Phys.* [**G41**]{} (2014) 087002 J. Schambach et al., *A MAPS Based Micro-Vertex Detector for the STAR Experiment*, *\ Phys. Procedia* [**66**]{} 514–519 M. Deveaux, *Monolithic Active Pixel Sensors*, in proceedings of the current conference. *EUTelescope*, http://eudaq.github.io *EUDAQ*, https://eutelescope.web.cern.ch [^1]: on behalf of the ALICE collaboration.
--- abstract: 'We theoretically investigate cross-talk in hyperfine gate control of donor-qubit quantum computer architectures, in particular the Kane proposal. By numerically solving the Poisson and Schrödinger equations for the gated donor system, we calculate the change in hyperfine coupling and thus the error in spin-rotation for the donor nuclear-electron spin system, as the gate-donor distance is varied. We thus determine the effect of cross-talk – the inadvertent effect on non-target neighbouring qubits – which occurs due to closeness of the control gates (20-30nm). The use of compensation protocols is investigated, whereby the extent of crosstalk is limited by the application of compensation bias to a series of gates. In light of these factors the architectural implications are then considered.' address: 'Centre for Quantum Computer Technology, School of Physics, University of Melbourne, Parkville, Victoria 3010, Australia' author: - 'G Kandasamy, C J Wellard and L C L Hollenberg' title: 'Cross-talk compensation of hyperfine control in donor qubit architectures' --- Introduction ============ There is a growing interest in solid state proposals for quantum computation, owing to their promise of a scalable technology given the connection with existing nano-fabrication techniques. An important scheme in silicon was proposed by Kane [@Kane1], which although difficult to realise, has a number of advantages, including long decoherence times of both $^{31}$P nuclear and electron spins [@Tyrishkin]. Dopant based proposals, in general propose qubits to be encoded via the spin state of the donor-nucleus [@Kane1] or electron [@Vrijen] or donor-charge state [@Hollenberg3] of a phosphorus atom in a silicon substrate. Single qubit control in the Kane case is achieved by the application of gate potentials (derived from surface electrodes), which shift the hyperfine interaction of a given target qubit in or out of resonance with a global transverse AC magnetic field. Two-qubit operations are performed by drawing together the two donor-electron wave functions, such that the exchange interaction is modified [@Fang; @Kettle1; @Wellard1]. This requires the donors to be separated at distances of 20-30nm. The standard paradigm for quantum computing requires the ability to individually control single qubits in a register of many qubits, as well as to implement controlled inter-qubit interactions. The proximity of neighbouring qubits results in poor localisation of gate potential, giving rise to a cross-talk effect on non-target qubits. This cross-talk in quantum control renders selective addressability for single qubit control problematic. A similar problem arises in other dopant based proposals [@Vrijen; @Skinner; @Hill], superconducting qubits and in quantum dots[@Tanamoto; @Snider]. Analogous to this control problem, measurement cross-talk in such architectures may also be a problem (similar to the SQUID equivalent [@McDermott]). Although both cross-talk and decoherence processes cause errors in qubit operations, we wish to distinguish the two. Decoherence is a fundamentally non-unitary process caused by interaction with the environment, whereas cross-talk leads to unwanted unitary evolution of non-target qubits during an operation. Given the stringent requirements of control precision that fault-tolerant quantum computation imposes at the 10$^{-4}$ level[@Preskill], determination and elimination of cross-talk is a key issue for scalable quantum computation. This paper is organised as follows: In Sec. 2 the physical architecture and considerations for modelling is described. In Sec. 3 the calculation of donor-electron wave functions and hyperfine coupling is presented with an emphasis to illustrate the cross-talk problem and the fidelity of single qubit addressing. In Sec. 4 the gate compensation to improve selectivity is described and considers alternative configuration of. Modelling the Kane architecture =============================== A schematic of the system we consider is shown in Fig. \[kane\_arch\]. This architecture consists of $^{31}$P (spin $I=\frac{1}{2}$) atoms doped in purified $^{28}$Si ($I=0$). The dopants are separated by 30nm and embedded at a depth of 20nm below the 5nm oxide layer. A series of surface gates referred to as $A$ (aligned above the donor) and $J$ (in between the donors) are placed on a 5nm thick oxide layer. The gates are 10nm wide and separated by 5nm. A backgate is connected to the substrate by a graded n-doped layer such that it forms an ohmic contact with0 the substrate. ![a) A Kane architecture based model, b) schematic configuration of gates and donors.[]{data-label="kane_arch"}](Fig1a_mykane.eps){height="2.5in"} ![a) A Kane architecture based model, b) schematic configuration of gates and donors.[]{data-label="kane_arch"}](Fig1b_don_arr.eps){height="1.5in"} Each $^{31}$P atom forms four covalent bonds with neighbouring $^{28}$Si atoms, with the fifth valence electron loosely bound to the P$^+$ ion, with a Bohr radius of $a \approx$ 2nm. Operations to be performed on the system are governed by the effective spin Hamiltonian for the 2-donor system, given by: $$H^{\rm spin}= H_{Z} + H_{A} + H_J +H_{\rm ac}, \label{hamiltonian}$$ where $H_{Z}$ is the Zeeman term due to a constant magnetic field applied to the entire system $$H_{Z} = \mu_BB({\bf \sigma}^{1e}_z + {\bf \sigma}^{2e}_z ) - g_n\mu_nB({\bf \sigma}^{1n}_z +{\bf \sigma}^{2n}_z ),$$ $H_{A}$ is the contribution of the hyperfine interaction that couples nuclear and electron spins $$H_{A} = A_1{\bf \sigma}^{1e}\cdot{\bf \sigma}^{1n} + A_2{\bf \sigma}^{2e}\cdot{\bf \sigma}^{2n},$$ $H_J$ is the contribution of the exchange interaction coupling neighbouring electrons $$H_J = J_{12}{\bf \sigma}^{1e}\cdot{\bf \sigma}^{2e},$$ and $H_{\rm ac}$ is the contribution of the transverse rotating magnetic field to the Hamiltonian $$\begin{aligned} H_{\rm ac}= -g_n\mu_n&B_{\rm ac}[(\sigma_x^{1n} +\sigma_x^{2n}) \cos(wt) +(\sigma_y^{1n} +\sigma_y^{2n}) \sin(wt)] \\ \nonumber &+ \mu_BB_{\rm ac}[(\sigma_x^{1e}+\sigma_x^{2e}) \cos(wt) +(\sigma_y^{1e} +\sigma_y^{2e}) \sin(wt)].\end{aligned}$$ In the above expressions ${\mu_B}$ is the Bohr magneton, ${\mu_n}$ is the nuclear magneton, ${g_n}$ is the Lande factor for $^31$P, $B$ is the magnitude of the constant magnetic field, $B_{\rm ac}$ is the magnitude of the global transverse AC magnetic field, $w$ the frequency of the transverse field and $\sigma_x$,$\sigma_y$,$\sigma_z$ the Pauli matrices operating on electron and nuclear spins. $A$ and $J$ stand for hyperfine and exchange couplings respectively. For a detailed description of the Kane architecture refer to [@Goan; @Hollenberg2].\ The coupling parameters $A$ and $J$ are controlled via A and J gates, in order to perform qubit operations. The problem of cross talk can be summarised as follows; rather than being dependent on a single control gate, the Hamiltonian parameters $A,J$ are functions of biases applied to all the control gates in the vicinity: $$A_0= A_0({\bf V}_{\rm bias}).$$ where ${\bf V}_{\rm bias}=\{V_{A_0}, V_{J_{01}}, V_{A_1},...\}$ is the set of voltage biases applied on the gates (Fig. \[kane\_arch\] b)). This dependence and the inherent crosstalk has not received investigative attention so far. Gate control of donor-electron wave functions ============================================= The couplings $A$ and $J$ depend on the donor electron wave function, which is determined by the solution of the electronic wave function Hamiltonian. There have been a number of approaches to the calculation of the gate induced shift in the hyperfine coupling strength at various levels of sophistication of both electron wave functions and description of control field, including: perturbation theory with envelope hydrogenic wave function for circular and strip electrodes [@Larionov], envelope hydrogenic basis diagonalisation using numerically determined nano-gate potentials[@Wellard2; @Kettle2], variational effective mass theory in uniform fields[@Friesen], tight-binding calculations with uniform field [@Martins], self-consistent calculation of 3D Poisson and DFT taking into account the six valleys of Si band structure [@Lu] and numerical expansion in a basis of silicon Bloch functions in a uniform field[@Wellard3]. Despite differences in detail, there is good agreement between these various methods on the hyperfine response to external electric fields. A hydrogenic approach has proven to be sufficient for the calculation of the hyperfine shift, whereas the details of the potential field generated by the control gate are vital to the description of the cross-talk problem. Therefore, in this first determination of the extent of, and possible correction procedures for cross-talk in the quantum control problem, we adopt a hydrogenic envelope approximation for the donor electron wave function, but determine the electrostatic fields numerically to get a reliable description. In this analysis we ignore the effects of the image charges of donor nuclei and electrons in the silicon-oxide barrier, as well as the effect of the magnetic fields on the electron wave functions. Both of these issues are discussed in [@Calderon], and will alter the energies of the bound-states of the inerface well that is formed beneath the control gate. This will change the voltage at which it becomes energetically favourable to occupy these surface states for all donors, and will not produce a relative energy shift between the donors. We also ignore any inter-donor coupling, to include this would complicate the calculation considerably as it would be necessary to calculate the ground-state of a many electron system. We expect that inter-donor coupling will only play a role when the electron wavefunctions have been significantly deformed, and we are not interested in any regime in wich more than one donor electron wavefunction is distorted. The ground-state single-donor wave functions are calculated by expanding in the basis of scaled hydrogen-like orbitals. The single-donor electronic Hamiltonian in the presence of an external potential generated by a bias profile ${\bf V}_{\rm bias}$ applied to a surface gates, is given by $$H^{\rm elect}=-\frac{\hbar}{2m}\nabla^2 - \frac{e^2}{4\pi\epsilon_0\epsilon_{si}r} + U(\vec{r},{\bf V}_{\rm bias}) + U_{\rm barrier}(\vec{r}).$$ In the basis of scaled Hydrogen-like orbitals we compute the matrix elements $$\begin{aligned} \langle n,l,m \vert H^{\rm elect} \vert n',l',m' \rangle =\int \Psi^*_{n,l,m}({\vec r}) [ U({\vec r}, {\bf V}_{\rm bias})& + U_{\rm barrier}(\vec{r})] \Psi_{n',l',m'}({\vec r}) d{\vec r} + \nonumber \\ & +\delta_{n,n'}\delta_{l,l'}\delta_{m,m'}\frac{E_n}{n^2}~,\end{aligned}$$ where $n,l,m$ are the electronic, orbital and magnetic quantum numbers and $E_n = E_1/n^2$ is the energy of the orbital. In the scaled donor units, $a_B=2{\rm nm}$ and $E_1=-45{\rm meV}$. The potential $U(\vec{r},{\bf V}_{\rm bias})$ is composed of the externally applied electrostatic potential, and $U_{\rm barrier}(\vec{r})$ is the potential barrier ($3.25\rm{eV}$) across the semiconductor-oxide interface, which arises due to the relative energies of the conduction bands of the two materials. The potentials in the device are calculated by solving the Poisson equation using ISE-TCAD [@tcad]. The calculations are performed for a device temperature of 1K, where convergence is obtained by bootstrapping room temperature data carefully down to required temperature. The matrix elements are calculated numerically using the Monte Carlo integration, and the Hamiltonian is diagonalised to find the donor-electron ground-state as a function of the applied gate biases. Fidelity of single-qubit addressing ----------------------------------- Single qubit operations are performed by controlling the donor nuclear-electron hyperfine coupling $A$, via gate potentials. Both $A$, and the exchange coupling $J$, depend on the wave functions of the donor-electrons. Application of a surface gate bias deforms the donor electron wave functions, thus altering the exchange and hyperfine coupling parameters. The hyperfine coupling between the nucleus and the electron depends on the overlap of their respective wave functions. The probability density of the nucleus is treated as a delta function, so the coupling strength $A$ is given by $$A({\bf V}_{\rm bias})= \frac{2\pi}{3}\mu_{\rm B}g_{\rm n}\mu_{\rm n}\mu_{\rm si} \vert\Psi(0,{\bf V}_{\rm bias})\vert^2\;,$$ where ${\mu_{\rm si}}$ is the permeability of silicon and ${\Psi(0,{\bf V}_{\rm bias})}$ is the donor electron wave function evaluated at the nucleus [@Larionov]. Due to complications arising from the core electrons of the donor, it is difficult to calculate $A$ directly, rather we calculate the relative change in coupling, as a function of applied field [@Wellard3]: $$A({\bf V}_{\rm bias}) = \frac{\vert \Psi(0,{\bf V}_{\rm bias}) \vert ^2}{\vert \Psi(0,0) \vert ^2}A(0),$$ where $A(0)$ is determined from experimental data [@Fletcher]. Single qubit operations are implemented by bringing the target qubit into resonance with an oscillating transverse magnetic field. Changing $A$ changes the resonance frequency of the donor nucleus. To do this selectively requires the ability to sufficiently alter the hyperfine coupling strength of an individual donor, while keeping the other donors sufficiently far off resonance such that their spin state remains unaltered.[ As discussed, the close spacing of the qubits makes unique addressing problematic; individual hyperfine control is hindered by the gate potential cross-talk as shown in Fig. \[basic\_A\] a), where we have plotted the hyperfine coupling coefficient as a function of gate bias for several bits.]{} ![a)Response of the hyperfine coupling of an array of donors (as in Fig. \[kane\_arch\] b)), b) Comparison of the Hyperfine coupling constant ’A’ of target donor $Q_0$ at various depths.[]{data-label="basic_A"}](Fig2a_A_basic.eps "fig:"){height="2in"} ![a)Response of the hyperfine coupling of an array of donors (as in Fig. \[kane\_arch\] b)), b) Comparison of the Hyperfine coupling constant ’A’ of target donor $Q_0$ at various depths.[]{data-label="basic_A"}](Fig2b_A_basic_0_vd.eps "fig:"){height="2in"} The results in figure \[basic\_A\](b) are in good agreement with previous results [@Wellard2; @Kettle2; @Martins; @Smit] and show the donor-electron response when placed extremely close to the oxide layer. The strength of the hyperfine coupling is also affected by donor depth, due to effects of both gate potential and oxide barrier. While the hyperfine coupling at donor depths $\ge$ 12nm from the oxide follow a regular pattern influenced by gate potential, very shallow depths show pronounced effects due to the oxide layer. At very shallow depths, we find that the hyperfine coupling initially increases with the applied positive bias before decreasing. This is due to the fact that when the donor is close to the oxide, the donor-electron at zero bias is pushed away from oxide, a positive bias restores $A$ to the normal value and then $A$ decreases as the wave function is pulled further towards the oxide. As the gate voltage is increased, we find that the electrons from all donors are perturbed, as shown in Fig. \[e\_response\]. It is found that for positive applied biases the electron wave functions of neighbouring donors are perturbed at lower voltages than the target qubit, while for negative biases the behaviour is more intuitive, with the target electron perturbed at a lower bias. This can be explained as follows; the positive gate potential pulls the electron wave functions towards the oxide, while the oxide restricts the wave functions from entering it, forming a potential well at the silicon-oxide interface, underneath the gate. The depth of this potential well is the same for all donors, however donors nearer to the gate have the potential well of their ionic core lowered by the application of the field, more than for donors further from the gate. Therefore, it becomes energetically favourable for the electrons of these distant donors donors to occupy this interface well at a lower potential than is the case for donors closer to the gate (Fig. \[pot\_wells\]). It should be noted however, that the time scales on which these electrons tunnel into this interface well increases as the donors become further from the gate, as discussed in ref [@Calderon]. Therefore, it may be possible to use a kind of temporal selection whereby the gate pulses are timed such that they allow adiabatic evolution of the target donor electron into the interface well, but are too fast for the neighbouring donor electrons to tunnel. The design of such pulses with the requisite fidelity for quantum computation would be a formidable task, involving solution of he time-dependent Schrödinger equation in a very large basis of states. Here we restrict our attention to designing “ground-state” protocols, in which we assume that the pulses are applied in such a fashion that each donor electron is allowed to evolve adiabatically in its ground electronic state. In the case of a negative applied potential, the potential wells of the donor cores are raised in energy by the gate potential, the donors close to the gate are affected more by this than those further away, and so are perturbed at a lower gate bias. However, this results in a very weakly bound donor electron state, which may be subject to significantly increased spin-orbit coupling, leading to increased dephasing rates. Additionally, such weakly bound states are more susceptible to non-reversible ionisation. We are, therefore, hesitant to propose this as a practical method for implementing gate operations. ![Donor-electron probability density under perturbation when +1.0V is applied on A$_0$ gate (all other gates are grounded) for a) target donor $Q_0$ and c) neighbouring donor $Q_1$. Negative gate voltages produces different behaviour as shown when A$_0$ gate is biased at -0.8V b) for $Q_0$ and d) for $Q_1$.[]{data-label="e_response"}](Fig3a_wf_pos_0_new.eps "fig:"){height="1.9in"} ![Donor-electron probability density under perturbation when +1.0V is applied on A$_0$ gate (all other gates are grounded) for a) target donor $Q_0$ and c) neighbouring donor $Q_1$. Negative gate voltages produces different behaviour as shown when A$_0$ gate is biased at -0.8V b) for $Q_0$ and d) for $Q_1$.[]{data-label="e_response"}](Fig3b_wf_pos_1_new.eps "fig:"){height="1.9in"} ![Donor-electron probability density under perturbation when +1.0V is applied on A$_0$ gate (all other gates are grounded) for a) target donor $Q_0$ and c) neighbouring donor $Q_1$. Negative gate voltages produces different behaviour as shown when A$_0$ gate is biased at -0.8V b) for $Q_0$ and d) for $Q_1$.[]{data-label="e_response"}](Fig3c_wf_neg_0_new.eps "fig:"){height="2in"} ![Donor-electron probability density under perturbation when +1.0V is applied on A$_0$ gate (all other gates are grounded) for a) target donor $Q_0$ and c) neighbouring donor $Q_1$. Negative gate voltages produces different behaviour as shown when A$_0$ gate is biased at -0.8V b) for $Q_0$ and d) for $Q_1$.[]{data-label="e_response"}](Fig3d_wf_neg_1_new.eps "fig:"){height="2in"} ![a) The effect of gate potential increases with gate-donor distances, b) Schematic (not to scale) illustration of gate potential wells: i) potential well created by donor-electron in the absence of gate potential, ii) lowering depth of donor-electron potential well due to gate voltage, iii) lowering of depth is less at increased distance from gate, hence $\delta_1 > \delta_0$.[]{data-label="pot_wells"}](Fig4a_pot_wells_exp.eps "fig:"){height="2in"}![a) The effect of gate potential increases with gate-donor distances, b) Schematic (not to scale) illustration of gate potential wells: i) potential well created by donor-electron in the absence of gate potential, ii) lowering depth of donor-electron potential well due to gate voltage, iii) lowering of depth is less at increased distance from gate, hence $\delta_1 > \delta_0$.[]{data-label="pot_wells"}](Fig4b_pot_wells_new.eps "fig:"){height="2.5in"} Effective gate errors --------------------- Single-qubit operations are implemented by bringing the energy difference between spin states of the nucleus into resonance with a globally applied, oscillating transverse field, for a time $t$, which determines the angle of the rotation. The resonance frequency ${\nu}_{\rm res}$, of the nucleus is given by[@Kane1]: $$h\nu_{\rm res} = g_{\rm n}\mu_{\rm n}B - \mu_{\rm B}B+\sqrt{4A^2-(g_{\rm n}\mu_{\rm n}B+\mu_{\rm B}B)^2} + 2A .$$ At time $t$, the probability of measuring the $i^{\rm th}$ nucleus in its spin flipped state is given by Rabi’s formula [@Sakurai]: $$P_i(t)=\frac{\gamma^2}{\gamma^2 + (\nu-\nu_{\rm res}^i)^2/4}\rm{sin}^2\left[t\sqrt{\left( \gamma^2 + \frac{\nu-\nu_{\rm res}^i}{4}\right)}\right],$$ where $\nu$ is the frequency of the field, $\nu_{\rm res}^i$ is the resonant frequency of the nucleus and ${\gamma = 2g_n\mu_nB_{ac}/h}$. For the target qubit $Q_0$, we define the error as $\epsilon_0 = 1 - {\rm max}[P_0(t)]$, which gives the error due to under-rotation of the qubit. In the case of the non-target qubit (off-resonant with the RF field in general), we define the error as $\epsilon_{i \neq 0} = {\rm max}[P_i(t)]$, which quantifies any unwanted rotation of the qubit. In what follows we consider the error in implementing spin flip on a particular target qubit, we assume that there is no error associated with the timing of the pulse and so set the sine function in the above expression to unity. Studies in fault tolerant quantum computation provide a rough estimate of the allowable error in any gate operation, which is no more than one part in $10^4$ [@Preskill]. In the case of single qubit rotations, the error is composed both of contributions associated with the target qubit being out of resonance, as well as neighbouring qubits being insufficiently off resonance so that they undergo a partial rotation. The hyperfine coupling is least sensitive to gate voltage fluctuations when $\partial{A}/\partial{V}$=0, which is the case when either $A=A(0)$, or, with the exception of very shallow depths, for $A=0$ [@Kane2], as seen in (Fig. \[basic\_A\] b)). This suggests that the frequency of the global field should be chosen to be the nuclear resonance frequency of the target qubit in one of those cases. We set the AC field to be resonant with the $A=0$ case, in the first instance. Plotted in Fig. \[error\_f\] a) is the gate error for each qubit as we change the target gate bias, at this frequency. We find that the required error rates are not achieved for positive gate biases. ![Evaluating performance of single qubit operation on a qubit array : a) when resonance frequency is chosen to be nuclear frequency of target when $A_0$=0, b) when resonance frequency is chosen to be the nuclear frequency of the target donor ($Q_0$ does not have any error and hence not plotted). []{data-label="error_f"}](Fig5a_fidelity_basic.eps "fig:"){height="2in"} ![Evaluating performance of single qubit operation on a qubit array : a) when resonance frequency is chosen to be nuclear frequency of target when $A_0$=0, b) when resonance frequency is chosen to be the nuclear frequency of the target donor ($Q_0$ does not have any error and hence not plotted). []{data-label="error_f"}](Fig5b_fidelity_voltres_basic_fullmodel_log.eps "fig:"){height="2in"} To determine optimum frequency for the global field, for the implementation of single qubit operations, we allow the global field to be the nuclear frequency of the target qubit ($\nu=\nu_{res}$) and sweep the gate bias for the relevant range. These results are plotted in (Fig. \[error\_f\] b)), and show that positive biases do not give a practical window of operation. The spike at around 1V is due to the crossover of the hyperfine coupling between target and neighbouring donor-electrons (Fig. \[basic\_A\] a)). It is clear that some kind of compensation is required to reduce the cross-talk. Compensation for cross-talk induced gate errors =============================================== The diffuse nature of the electric fields produced inside the device by the application of a bias voltage to any particular control gate makes selective addressability of qubits problematic. In this section we investigate a simple linear compensation scheme to address this issue. Through the application of compensation biases to neighbouring gates we minimise the electric potential in the vicinity of non-target donors whilst maintaining as much as possible in the vicinity of the target qubit. Therefore we design a bias profile such that both the potential, and x-component of the electric field, at the positions of the non-target qubits be zero. The required compensation gate profile is achieved by the following procedure. Assuming linearity of potentials in the device, the potential and it’s x-gradient, at the donor sites can be described as: $${\bf \Pi} = M {\bf V}_{\rm bias} \label{compmain}$$ where ${\bf V}_{\rm bias}$ refer to the gate biases, and ${\bf \Pi}$ refer to potentials and their gradients at the respective qubits: $${\bf \Pi} = (U_{Q_0},U'_{Q_0},U_{Q_1},U'_{Q_1},...)^T \quad.\\ \nonumber$$ Here $$U_{Q_0} \equiv U(\vec{r}_{Q_0}; {\bf V}_{\rm bias}) = u_{A_0}(\vec{r}_{Q_0})V_{A_0} + u_{J_{01}}(\vec{r}_{Q_0})V_{J_{01}} + ... \quad , \\ \nonumber$$ and $$u_{A_0}(\vec{r}_{Q_0}) = U(\vec{r};\{V_{A_0}=1,0,0,...\}) \quad ,\nonumber$$ with $u_{A_0}(\vec{r}_{Q_0})$, the potential at qubit $Q_0$ due a bias $V_{A_0}=1V$. The matrix $M$ is constructed by solution of the Poisson equation using a commercial software package TCAD, such that Eqn. \[compmain\] can be written as follows $$\left(\begin{array}{c} U_{Q_0}\\ U_{Q_1}\\ U'_{Q_1}\\ \vdots\\ U_{Q_n}\\ U'_{Q_n} \end{array} \right) = \left(\begin{array}{ccccc} u_{A_0}(\vec{r}_{Q_0})&u_{J_{01}}(\vec{r}_{Q_0})&u_{A_1}(\vec{r}_{Q_0})&\cdots&u_{A_n}(\vec{r}_{Q_0})\\ u_{A_0}(\vec{r}_{Q_1})&u_{J_{01}}(\vec{r}_{Q_1})& & & \\ u'_{A_0}(\vec{r}_{Q_1})& & & &\vdots\\ \vdots& &\ddots& &\vdots\\ u_{A_0}(\vec{r}_{Q_n})& & &\ddots &\vdots\\ u'_{A_0}(\vec{r}_{Q_n})&\cdots &\cdots &\cdots &u'_{A_n}(\vec{r}_{Q_n}) \end{array} \right) \left(\begin{array}{c} V_{A_0}\\ V_{J_{01}}\\ V_{A_1}\\ \vdots\\ V_{J_{(n-1)n}}\\ V_{A_n} \end{array} \right) , \label{Mmatrix}$$ where $U'$ is the derivative of potential with respect to x-direction. We require a value of ${\bf V}_{\rm bias}$, such that gives a ${\bf \Pi}$ with $U_{Q_0} =1$, and all other elements zero. The gradient of the potential at the target donor does not need compensation due to symmetry, and hence is ignored. This is obtained by matrix inversion; $${\bf V}_{\rm bias}= M^{-1}{\bf \Pi}.$$ Applying the above compensated bias profile produces a potential that is more localised around the target donor (Fig. \[comp\_top\] b)). The compensation assumed linearity of the medium, which would be the case in the absence of charge carriers. However non-linearities in potentials introduced by the presence of P-dopants ($10^{11}/{\rm cm}^3$) in the semiconductor, result in slight deviation from ideal compensation at all donor-sites, as can be seen in (Fig. \[comp\_A\] a)). ![a) Potential profile before and after application of compensation biases to other gates, b)Hyperfine coupling of donor array after compensation with respect to $V_{A_0}$.[]{data-label="comp_A"}](Fig7a_comp_math_TCAD.eps "fig:"){height="2in"} ![a) Potential profile before and after application of compensation biases to other gates, b)Hyperfine coupling of donor array after compensation with respect to $V_{A_0}$.[]{data-label="comp_A"}](Fig7b_A_comp_v+e.eps "fig:"){height="2in"} ![ Wave function response of a) $Q_0$, b) $Q_1$ after compensation.[]{data-label="dens_comp"}](Fig8a_wf_pos_comp_0_new.eps "fig:"){height="2in"} ![ Wave function response of a) $Q_0$, b) $Q_1$ after compensation.[]{data-label="dens_comp"}](Fig8b_wf_pos_comp_1_new.eps "fig:"){height="2in"} The response of electron-nuclear hyperfine coupling to compensation biases is plotted in Fig. \[comp\_A\] b). One of the consequences of compensation is that larger voltages are required to affect the donors, with a 2V bias required to produce the same effect as 1V in the uncompensated case. This is due to the application of counter-acting voltages on the neigbouring gates, which have the effect of reducing the overall potential within the device. In (Fig. \[dens\_comp\]) we show the probability density for target and neighbouring donor electrons with a compensated bias of 2.6V applied to the target gate $V_{\rm A_0}$. This shows that, in contrast to the uncompensated case, the target qubit feels a greater effect than the neighbour, as required. ![Error probability : When resonance frequency is chosen to be nuclear frequency of target when $A_0$=0 , b)When resonance frequency is chosen to be the nuclear frequency of the target donor ($Q_0$ does not have any error by definition in this case and hence not plotted).[]{data-label="error_ac"}](Fig9a_fidelity_basic_comp_v+e.eps "fig:"){height="2in"} ![Error probability : When resonance frequency is chosen to be nuclear frequency of target when $A_0$=0 , b)When resonance frequency is chosen to be the nuclear frequency of the target donor ($Q_0$ does not have any error by definition in this case and hence not plotted).[]{data-label="error_ac"}](Fig9b_fidelity_voltres_comp_fullmodel_log.eps "fig:"){height="2in"} Evaluating the fidelity of a single qubit rotation on a compensated qubit array, shows a marked difference to the uncompensated case (Fig. \[error\_ac\]). In the positive voltage regime, there exists a small window, at about 2V - 2.5V, in which the required fidelity is achieved (Fig. \[error\_ac\] b)). However, this may still be too constrained. In light of this we consider an architecture which consists of a series of well separated two-qubit cells, where quantum information can be transported between cells via coherent electron transport [@Skinner; @Greentree], as sketched in (Fig. \[2qbit\_cell\] a)). We have calculated the fidelity of single qubit operations for both the uncompensated, and a compensated gate profile, which we show in (Fig. \[2qbit\_cell\] b)). This gives a more robust compensated voltage regime in which to perform single qubit operations. ![a) Donors organised as two-qubit cells interacting via quantum transport mechanisms, b) Error probabilities of rotating non-target qubit when the transverse RF is fixed at the resonant frequency of target nucleus. []{data-label="2qbit_cell"}](Fig10a_2qbit_cell.eps "fig:"){height="1.8in"}![a) Donors organised as two-qubit cells interacting via quantum transport mechanisms, b) Error probabilities of rotating non-target qubit when the transverse RF is fixed at the resonant frequency of target nucleus. []{data-label="2qbit_cell"}](Fig10b_fidelity_voltres_comp_2qbit_log.eps "fig:"){height="2in"} Conclusion ========== Practical quantum devices require scalable architectures where crosstalk would have to be minimised. The feasibility of the conventional Kane architecture has been analysed in light of such problems that arise due to close proximity of qubits. The concept of cross-talk in dopant-qubit architectures has been introduced and the physics behind this effect is explained. We have shown that cross-talk can be reduced using gate voltages based on linear compensation techniques, however, the use of more sophisticated quantum control techniques may be also be beneficial. In response to the various issues for scale-up of the original Kane proposal, which includes the problem of cross-talk analysed here, an alternative quasi 2D donor based architecture has been proposed [@Hollenberg4]. The initial analysis of cross-talk for the two donor interaction cells shows that the problem can be more effectively compensated in such an architecture. Acknowledgment ============== G.K. would like to thank J. H. Cole and T. R. Starling for useful discussions. This work was supported by the Australian Research Council, the Australian Government and by the US National Security Agency (NSA), Advanced Research and Development Activity (ARDA), and the Army Research Office (ARO) under contract number W911NF-04-1-0290. 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=-2truecm =-1truecm =16truecm =24truecm [**Ising Model on A Fibonacci Lattice with Magnetic Field**]{} [**Susanta Bhattacharya**]{}$^a$ and [**Samir K. Paul**]{}$^b$\ $^a$ Ramsaday College\ Amta,Howrah,West Bengal,India\ $^b$ S. N. Bose National Centre For Basic Sciences\ Block-JD, Sector-III, Salt Lake\ Calcutta-700091, India [**Abstract**]{} We present a general procedure for calculating the partition function of an Ising Model on a one dimensional Fibonacci lattice in presence of magnetic field .This partition function can be written as a sum of partition functions of usual open Ising chains in presence of magnetic field with coeffecients having Fibonacci symmetries.An exact expression for the partition function of the usual open open Ising Model is found.We observe that “Mirror Symmetry” is a characteristic property of all Fibonacci generations . Further $nth$ and $(n+6)th$ generations have the same topology.We have also established a recurrence relation among partition functions of different Fibonacci generations from $nth$ to $(n+6)th$. PACS number:05.50+q a\) susanta1005@rediffmail.com\ b) smr@boson.bose.res.in [**Introduction**]{} Studying Ising model on a Fibonacci chain in presence of magnetic field is interesting in its own right since no proper analytical procedure exists for evaluating the partition function.However scaling forms of thermodynamic functions for such system have been studied using renormalization group technique \[1\] through one step decimation.Here we reformulate the expression for the partition function by breaking the transfer matrix into two particular noncommuting matrices . This formulation enables us to calculate the partition function for the usual open Ising chain. The result is quite nontrivial in contrast to the expression for the partition function of the closed one\[2\]. The same formulation helps us to express the partition function of Ising model on Fibonacci chain in presence of magnetic field as a sum of partition functions of usual Ising open chains with coefficients containing Fibonacci symmetry.We have also studied some symmetry properties of the Fibonacci chain.Using a special symmetry property (“Mirror Symmetry”) and the usual trace map relation we have established a recurrence relation among the partition functions of different Fibonacci generations.This includes all the partition functions starting from $nth$ up to $(n+6)th$ generations.We observe that mirror symmetry is a characteristic property of each Fibonacci generation with $nth$ and $(n+6)th$ generations having same topology . [**[Exact partition function for open Ising chain with magnetic field]{}**]{} The one dimensional Ising model consists of a chain of N spins ${S_i}={\pm 1}$ ;$i =1,2,.....,N$ with nearest neighbour interactions ${\epsilon_{i,i+1}}$. The Hamiltonian is given by: $${\mathcal{H}} = - {\sum_{i=1}^{N-1}}{\epsilon_{i,i+1}} {S_i} {S_{i+1}} - H{\sum_{i=1}^{N}}{S_i}$$ For a uniform lattice ${\epsilon_{i,i+1}} = {\epsilon}$,the partition function is given by: $${Z_N^o}(T,H) = {\sum_{{S_1},{S_2},....,{S_N}= - 1}^{+1}} f({S_1},{S_2}) f({S_2},{S_3})......f({S_{N-1}},{S_N}) {f_0}({S_N},{S_1})$$ with $f({S_i},{S_{i+1}}) =exp[{\beta}{\epsilon}{S_i}{S_{i+1}} + {\frac{1}{2}} \noindent {\beta}H({S_i}+{S_{i+1}})]$; ${f_0}({S_N},{S_1}) = {{[f({S_N},{S_1})]}_{\epsilon =0}}$. Here the superscript $o$ stands for the chain with open boundary condition.Therefore the partition function (2) can be written in terms of transfer matrix as: $${Z_N^o}(T,H) = Tr {P^{N-1}}{P_0}$$ where $$P = {\sqrt{r}}(1+{\frac{\lambda}{r}}){{\sigma}_1} = {\sqrt{r}}{{\sigma}_1}(1+ {\frac{{\lambda}^T}{r}})$$ $${P_0}={[P]_{\epsilon =0}}=(1+{\lambda}){\sigma_1}={\sigma_1}(1+{\lambda^T})$$ with $r = exp(-2{\beta}{\epsilon})$ , $\lambda = \left(\matrix{0 & e^{\beta H}\cr e^{-\beta H} & 0}\right),$ ${\lambda^T} = \left(\matrix{0 & e^{-\beta H}\cr e^{\beta H} & 0}\right)$ and ${\sigma_1} = \left(\matrix{0 & 1 \cr 1 & 0}\right).$ The general formula of the partition function for even and odd number of spins (i.e.,odd and even number of bonds) can be derived by using equations \(3) and (4) as : $${Z_{2N}^o}(T,H) = {r^{N-{\frac{1}{2}}}}Tr(1+{x_1})(1+{x_2})....(1+{x_{2N-1}}) (1+{\lambda^T})$$ and $${Z_{2N+1}^o}(T,H) = {r^N}Tr(1+{x_1})(1+{x_2})......(1+{x_{2N}})(1+\lambda ) {\sigma_1}$$ where $${x_{2i+1}} = {\frac{\lambda}{r}}, {x_{2i}} = {\frac{\lambda^T}{r}}; i=integer$$ The above equations show that $\lambda , {\lambda^T}$ are the signatures for the transfer matrices corresponding to bonds in odd and even positions.In the case of a chain with closed boundary condition the last factor in eqn.(2) is $f({S_N},{S_1})$ and consequently the partition function takes the form: $${Z_{2N}^c}(T,H) = {r^N} Tr(1+{x_1})(1+{x_2})......(1+{x_{2N}})$$ $${Z_{2N+1}^c} = {r^{N+{\frac{1}{2}}}}Tr(1+{x_1})(1+{x_2}).....(1+{x_{2N+1}}) {\sigma_1}$$ Here the superscript $c$ indicates closed chain.One can show by elementary calculation that eqns.(9) and (10) reduce to the well known form\[3\] $${Z_N^c}(T,H) = {\lambda_{+}^N} + {\lambda_{-}^N}$$ where $${\lambda_{\pm}} = {r^{-\frac{1}{2}}}[{\cosh{(\beta H)}} \pm {\sqrt{({{\sinh}^2} {(\beta H)} + {r^2})}}]$$ are the eigenvalues of the transfer matrix $P$.The expression for the partition function in the case of an open chain with even number of spins can be derived from eqn.(6) as follows: $$\begin{aligned} {Z_{2N}^o}(T,H) = {r^{N-{\frac{1}{2}}}}Tr(1+{x_1})(1+{x_2}).....(1+{x_{2N-1}})(1+\lambda^T)\nonumber\\ = {\sqrt{r}}{Z_{2N}^c}(T,H) + {r^{N-{\frac{1}{2}}}}(1-r)Tr(1+{x_1})(1+{x_2}) .......(1+{x_{2N-1}})\nonumber\\ = {\sqrt{r}}{Z_{2N}^c}(T,H) + {\sqrt{r}}(1-r){Z_{2(N-1)}^c}(T,H) + {r^{N-{\frac{1}{2}}}}(1-r)\nonumber\\ \times Tr (1+{x_1})(1+{x_2}).......(1+{x_{2N-2}}){x_{2N-1}}\end{aligned}$$ The last term in the above expression can be written in terms of the eigenvalues of the transfer matrix $P$ viz.${\lambda_\pm}$. By following the method of induction : $$\begin{aligned} {r^{N-{\frac{1}{2}}}}(1-r)Tr(1+{x_1})(1+{x_2})......(1+{x_{2N-2}}){x_{2N-1}} \nonumber\\ = (1-r){r^{N-{\frac{1}{2}}}}{\frac{4}{r^2}}{\cosh^2}{(\beta H)} {\sum_{i=0}^{N-2}}{\bigg( {\frac{{\lambda_+}^2}{r}} \bigg)}^{N-2-i} {\bigg( {\frac{{\lambda_-}^2}{r}} \bigg)}^{i}\nonumber\\ =4(1-r){r^{-\frac{1}{2}}}{\cosh^2}(\beta H)\frac{{{\lambda_+}^{2(N-1)}}-{{\lambda_-}^{2(N-1)}}}{{{\lambda_+}^2}- {{\lambda_-}^2}}\end{aligned}$$ So eqn.(13) becomes : $$\begin{aligned} {Z_{2N}^o}(T,H)& = &{\sqrt{r}}{Z_{2N}^c}(T,H)+{\sqrt{r}}(1-r){Z^c_{2(N-1)}}(T,H)\nonumber\\ & &+ 4(1-r){r^{-\frac{1}{2}}}{\cosh^2} {(\beta H)} \times {\frac{{{\lambda_+}^{2(N-1)}}-{{\lambda_-}^{2(N-1)}}}{{{\lambda_+}^2} - {{\lambda_-}^2}}}\end{aligned}$$ Similarly the expression (7) for the open chain partition function with odd number of spins takes the form: $$\begin{aligned} {Z_{2N+1}^o}(T,H) = {\sqrt{r}}{Z_{2N+1}^c}+2(1-r){\cosh{(\beta H)}}\nonumber\\ \times \frac{{{\lambda_+}^{2N}}-{{\lambda_-}^{2N}}}{{{\lambda_+}^2}-{{\lambda_+}^2}}\end{aligned}$$ Exact expressions for the thermodynamic functions can be calculated by well known methods \[3\]. [**[Ising model on Fibonacci chain]{}**]{} A Fibonacci chain can be inflated by two bonds $L(large)$ and $S(small)$ by the inflation rule$L \longrightarrow LS,S \longrightarrow L$.The chain can be represented by the sequence: $$L \longrightarrow LS \longrightarrow LSL \longrightarrow LSLLS \longrightarrow LSLLSLSL \longrightarrow .....$$ In this case the interaction strengths in the Hamiltonian (1) ${\epsilon_{i,i+1}}=\epsilon$ for long bonds and ${\epsilon_{i,i+1}}={\bar \epsilon}$ for the short ones where the bonds are arranged according to the Fibonacci sequence (17).The corresponding partition function of the $nth$ generation Fibonacci chain having $N$ spins with $N-1$ bonds is given by: $${Z_N^o}(F)=Tr P{\bar P}PP{\bar P}.....{P_0}$$ where for long bonds the transfer matrix $P$ is given by eqn.(4) and for short bonds the transfer matrix ${\bar P}$ is given by eqn.(4) with $r$ replaced by ${\bar r}= {r|}_{\epsilon = {\bar \epsilon}}$.Henceforth ${Z_N^o}(F)$ and ${Z_N^o}(I)$ will represent partition functions for Ising models on an open Fibonacci chain and on an open regular lattice respectively.The expressions for the partition functions with odd and even number of bonds take the same forms as shown in eqns.(6) and (7) with $x_i$’s given in eqn.(8) for long bonds whereas for short bonds we replace $r$ by ${\bar r}$ in eqn.(8).The explicit expressions for the partition functions for open and closed chains are: $${Z_{2N}^o}(F)={r^{\frac{N_L}{2}}}{{\bar r}^{\frac{N_S}{2}}}Tr (1+{x_1})(1+{x_2})......(1+{x_{2N-1}})({1+\lambda^T})$$ $${Z_{2N+1}^o}(F)={r^{\frac{N_L}{2}}}{{\bar r}^{\frac{N_S}{2}}}Tr (1+{x_1})(1+{x_2})......(1+x_{2N})({1+\lambda}){\sigma_1}$$ Similarly for closed chain eqs. (18),(19) and (20) take the following forms: $${Z_N^c}(F)= Tr P{\bar P}PP{\bar P}.......P$$ $${Z_{2N}^c}(F)={r^{\frac{N_L}{2}}}{{\bar r}^{\frac{N_S}{2}}}Tr (1+{x_1})(1+{x_2})......(1+{x_{2N-1}})(1+{x_{2N}})$$ $${Z_{2N+1}^c}(F)={r^{\frac{N_L}{2}}}{{\bar r}^{\frac{N_S}{2}}}Tr(1+{x_1})(1+{x_2}).......(1+{x_{2N}})(1+{x_{2N+1}}){\sigma_1}$$ where $N_L$,$N_S$ are number of long and short bonds in a particular sequence. Now ${\bar P}$ is related to P through the following equation: $${\bar P}={\sqrt{\bar r}}(1-{\frac{r}{\bar r}}){\sigma_1}+{\sqrt{{\frac{r}{\bar r}}}} P$$ Using eqn.(24) in eqns.(19) and (20) the Fibonacci partition function for any generation can be written in terms of open Ising partition functions as follows: $${Z_{2N}^o}(F)={h_0}({\epsilon,{\bar \epsilon}})+{\sum_{i=1}^N}{h_{2i}}({\epsilon, {\bar \epsilon}}){Z_{2i}^o}(I)$$ $${Z_{2N-1}^o}(F)={l_0}({\epsilon,{\bar \epsilon}})+{\sum_{i=1}^N}{l_{2i-1}}({\epsilon,{\bar \epsilon}}){Z_{2i-1}^o}(I)$$ where ${Z_{2i}^o}(I)$ and ${Z_{2i-1}^o}(I)$ are given by eqns.(15) and (16) respectively.We observe that the quasiperiodic nature of the Fibonacci chain is encoded in the functions $h(\epsilon,{\bar \epsilon})$ and $l(\epsilon,{\bar \epsilon})$.Though for small generations these functions can be derived exactly still we could not find out their general forms. To circumvent this difficulty we study the recurrence relations among the partition functions of different Fibonacci generations.A servey of different Fibonacci generations depicted by eqn.(17) shows a symmetric pattern in terms of the number of bonds ,viz., $$\begin{aligned} {P_1} : P (odd)\nonumber\\ {P_2} : P{\bar P}(even)\nonumber\\ {P_3} : P{\bar P}P(odd)\nonumber\\ {P_4} : P{\bar P}PP{\bar P}(odd)\nonumber\\ {P_5} : P{\bar P}PP{\bar P}P{\bar P}P(even)\nonumber\\ {P_6} : P{\bar P}PP{\bar P}P{\bar P}PP{\bar P}PP{\bar P}(odd)\end{aligned}$$ and so on. [**[Recurrence relation among partition functions]{}**]{} Let $P_{n-2}$ be the $(n-2)th$ Fibonacci generation with even number of bonds.This automatically ensures that the previous as well as the next two consecutive generations will have odd number of bonds . The recurrence relation for Fibonacci generations is given by: $${P_n}={P_{n-1}}{P_{n-2}}$$ Now adding a term ${D_{n-2}}{P_{n-3}}$ in the above equation gives $${P_n} + {D_{n-2}}{P_{n-3}} = {P_{n-1}}{P_{n-2}} + {D_{n-2}}{P_{n-3}}$$ where ${D_{n-2}}=Det({P_{n-2}})$.The following operations are applied sequentially on eqn.(29): Substitute ${{P_{n-2}}^{-1}}{P_{n-1}}$ in place of ${P_{n-3}}$ on the right hand side and finally use Cayley-Hamilton theorem to get the usual trace map relation on the Fibonacci lattice: $$Tr {P_n} = Tr {P_{n-1}} Tr {P_{n-2}} - {D_{n-2}}Tr {P_{n-3}}$$ The above equation will be necessary for calculating recurrence relation among different Fibonacci generations. For this purpose we must understand symmetry properties of Fibonacci chain.Inspecting different generations of the Fibonacci chain it reveals that if the total number of bonds $N$ of a particular generation is odd then there is a mirror reflection symmetry arround the ${\bigg( {\frac{N-1}{2}} \bigg) }th$ bond ;except the last two bonds.If the special bond arround which mirror symmetry occurs is a short(long) one the Fibonacci generation will have equal number of odd and even short(long) bonds . However if the total number of bonds $N$ is even,the mirror reflection symmetry is arround a cluster of two successive long bonds at the ${\bigg( {\frac{N}{2}} \bigg)}th$ and ${\bigg( {\frac{N-2}{2}} \bigg)}th$ positions of the chain . So “Mirror reflection symmetry” is a characteristic property of a Fibonacci chain. The $nth$ and $(n\pm 3)th$ generations have the mirror reflection symmetry property arround the same kind of bond with last two bonds interchanged , while the $(n\pm 6)th$ generations are topologically same as the $nth$ one. Using recurrence relation (28) we can write $${D_{n-2}}{P_{n-3}}={D_{n-2}}{{P_{n-2}}^{-1}}{P_{n-1}}$$ Using Cayley-Hamilton theorem on the right hand side of eq.(31) we get: $${D_{n-2}}{P_{n-3}}=(Tr{P_{n-2}}){P_{n-1}}-{P_{n-2}}{P_{n-1}}$$ Multiplying eq.(32) by P from the right and taking trace we obtain: $${D_{n-2}}{Z^c_{n-3}}=(Tr{P_{n-2}}){Z^c_{n-1}}-Tr({P_{n-2}}{P_{n-1}}P)$$ In a similar fashion we obtain : $${D_{n-2}}{Z^o_{n-3}}=(Tr{P_{n-2}}){Z^o_{n-1}}-Tr({P_{n-2}}{P_{n-1}}{P_0})$$ The expression ${P_{n-2}}{P_{n-1}}$ in eqns.(33) and (34) is similar to ${P_n}={P_{n-1}}{P_{n-2}}$ with last two bonds interchanged,i.e.,both of them have the same mirror symmetric part $\Omega_n$.Therefore eqns.(33) and (34) can be written as: $${D_{n-2}}{Z^c_{n-3}}={Z^c_{n-1}}(Tr{P_{n-2}})-Tr({\Omega_n}P{\bar P}P)$$ $${D_{n-2}}{Z^o_{n-3}}={Z^o_{n-1}}(Tr{P_{n-2}})-Tr({\Omega_n}P{\bar P}{P_0})$$ The transfer matrix has the property that $P={P^T}$ and ${\bar P}={{\bar P}^T}$ .If such transfer matrices are arranged in a mirror symmetric fashion then the resulting matrix $({\Omega_n})$ will have the following properties: i\) Off diagonal elements are same i.e.,${({\omega_n})_{12}}={({\omega_n})_{21}}$ ii)Diagonal elements are not same but satisfy the condition: ${({\omega_n})_{11}}(p,q)={({\omega_n})_{22}}(q,p)$; where $p={e^{{\beta}H}}$,$q={e^{-{\beta}H}}$. Thus the matrix ${\Omega_n}$ in eqns. (34) and (35) is of the form: ${\Omega_n}=\left(\matrix {{({\omega_n})_{11}} & {({\omega_n})_{12}}\cr {({\omega_n})_{21}} & {({\omega_n})_{22}}}\right)$ Eqns.(35) and (36) can be written explictly in the following way: $$\begin{aligned} {D_{n-2}}{Z^c_{n-3}}& = & {Z^c_{n-1}}(Tr{P_{n-2}}) -r{\sqrt{\bar r}}[{({\omega_n})_{11}}(y+{\frac{pu}{r}})+{({\omega_n})_{12}}(u+v+{\frac{px+qy}{r}})\nonumber\\ +{({\omega_n})_{22}}(x+{\frac{qv}{r}})]\end{aligned}$$ $$\begin{aligned} {D_{n-2}}{Z^o_{n-3}}& = & {Z^o_{n-1}}(Tr{P_{n-2}}) -{\sqrt{r{\bar r}}}[{({\omega_n})_{11}}(y+pu)+{({\omega_n})_{12}}(u+v+px+qy)\nonumber\\ +{({\omega_n})_{22}}(x+qv)]\end{aligned}$$ where we have used $P{\bar P}={\sqrt{r{\bar r}}}\left(\matrix {u & y\cr x & v}\right)$ with $x={\frac{p}{\bar r}}+{\frac{q}{r}}$, $y={\frac{p}{r}}+{\frac{q}{\bar r}}$,$u=1+{\frac{p^2}{r{\bar r}}}$ and $v=1+{\frac{q^2}{r{\bar r}}}$. Eliminating $Tr{P_{n-2}}$ from eqns.(37) and (38) we have: $$\begin{aligned} {\frac{y{V_{n-1}}+pu{{V^\prime}_{n-1}}}{(u+v){V_{n-1}}+(px+qy){{V^\prime}_{n-1}}}} {({\omega_n})_{11}} +{\frac{x{V_{n-1}}+qv{{V^\prime}_{n-1}}}{(u+v){V_{n-1}}+(px+qy){{V^\prime}_{n-1}}}} {({\omega_n})_{22}} +{({\omega_n})_{12}}\nonumber\\ ={\frac{D_{n-2}}{r{\sqrt{\bar r}}}}\times {\frac{{Z_{n-1}^o}{Z_{n-3}^c}-{Z_{n-1}^c}{Z_{n-3}^o}}{(u+v){V_{n-1}}+(px+qy){{V^\prime}_{n-1}}}}\end{aligned}$$ where ${V_{n-1}}={\frac{Z_{n-1}^c}{\sqrt{r}}}-{Z_{n-1}^o}$ and ${{V^\prime}_{n-1}}={\frac{Z_{n-1}^c}{\sqrt{r}}}-{\frac{Z_{n-1}^o}{r}}$. To solve for ${({\omega_n})_{11}}$,${({\omega_n})_{12}}$ and ${({\omega_n})_{22}}$ we need another two equations.These equations are obtained from the usual formulae: ${Z_n^c}=Tr({\Omega_n}{\bar P}PP)$, ${Z_n^o}=Tr({\Omega_n}{\bar P}P{P_0})$. The explicit forms of these two relations are: $$\begin{aligned} {\frac{x+{\frac{pu}{r}}}{u+v+{\frac{qx+py}{r}}}} {({\omega_n})_{11}} +{\frac{y+{\frac{qv}{r}}}{u+v+{\frac{qx+py}{r}}}} {({\omega_n})_{22}} +{({\omega_n})_{12}}\nonumber\\ = {\frac{Z_n^c}{r{\sqrt{\bar r}}}} {\frac{1}{u+v+{\frac{qx+py}{r}}}}\end{aligned}$$ $$\begin{aligned} {\frac{x+pu}{u+v+qx+py}} {({\omega_n})_{11}} +{\frac{y+qv}{u+v+qx+py}} {({\omega_n})_{22}}+{({\omega_n})_{12}}\nonumber\\ = {\frac{Z_n^o}{{\sqrt{r{\bar r}}}}} {\frac{1}{u+v+qx+py}}\end{aligned}$$ By elementary calculation one obtains: $$\begin{aligned} {({\omega_n})_{11}}(x,y)= {\frac{{\Gamma_{n-1}}-{{{\Gamma}^\prime}_{n-1}}}{{\Gamma_{n-1}}{{{\Lambda}^\prime}_{n-1}}-{{{\Gamma}^\prime}_{n-1}}{\Lambda_{n-1}}}}\times {\frac{D_{n-2}}{r\sqrt{\bar r}}}\times {\Delta_{n-1}}\nonumber\\ -{\frac{1}{\sqrt{r{\bar r}}}}{\frac{1}{{\Gamma_{n-1}}{{{\Lambda}^\prime}_{n-1}}-{{{\Gamma}^\prime}_{n-1}}{\Lambda_{n-1}}}}\times ({\Gamma_{n-1}}{Z_n^o}{K^\prime}-{{\Gamma}^\prime}{\frac{Z_n^c}{\sqrt{r}}}K)\end{aligned}$$ $$\begin{aligned} {({\omega_n})_{22}}(x,y)=-{\frac{{\Lambda_{n-1}}-{{{\Lambda}^\prime}_{n-1}}}{{\Gamma_{n-1}}{{{\Lambda}^\prime}_{n-1}}-{{{\Gamma}^\prime}_{n-1}}{\Lambda_{n-1}}}}\times {\frac{D_{n-2}}{r\sqrt{\bar r}}}\times {\Delta_{n-1}}\nonumber\\ +{\frac{1}{\sqrt{r{\bar r}}}}{\frac{1}{{\Gamma_{n-1}}{{{\Lambda}^\prime}_{n-1}}-{{{\Gamma}^\prime}_{n-1}}{\Lambda_{n-1}}}}\times ({\Gamma_{n-1}}{Z_n^o}{K^\prime}-{{\Gamma}^\prime}{\frac{Z_n^c}{\sqrt{r}}}K)\end{aligned}$$ and $${({\omega_n})_{12}}(x,y)={\frac{1}{(qx+py)(1-{\frac{1}{r}})}}\bigg[ -{\frac{1}{\sqrt{r{\bar r}}}}\times {V_n}-{(1-{\frac{1}{r}})}\bigg( pu{({\omega_n})_{11}}+qv{({\omega_n})_{22}} \bigg) \bigg]$$ where $${\Lambda_{n-1}}(x,y)={\frac{y{V_{n-1}}+pu{{V^\prime}_{n-1}}}{(u+v){V_{n-1}}+(px+qy){{V^\prime}_{n-1}}}}-{\frac{x+{\frac{pu}{r}}}{u+v+{\frac{qx+py}{r}}}}$$ $${{{\Lambda}^\prime}_{n-1}}(x,y)={\frac{y{V_{n-1}}+pu{{V^\prime}_{n-1}}}{(u+v){V_{n-1}}+(px+qy){{V^\prime}_{n-1}}}}-{\frac{x+pu}{u+v+qx+py}}$$ $${\Gamma_{n-1}}(x,y)=\frac{x{V_{n-1}}+qv{{V^\prime}_{n-1}}}{(u+v){V_{n-1}}+(px+qy){{V^\prime}_{n-1}}}-\frac{y+{\frac{qv}{r}}}{u+v+{\frac{qx+py}{r}}}$$ $${{{\Gamma}^\prime}_{n-1}}(x,y)= \frac{x{V_{n-1}}+qv{{V^\prime}_{n-1}}}{(u+v){V_{n-1}}+(px+qy){{V^\prime}_{n-1}}}-\frac{y+qv}{u+v+qx+py}$$ $${\Delta_{n-1}}(x,y)={\frac{{Z_{n-1}^o}{Z_{n-3}^c}-{Z_{n-1}^c}{Z_{n-3}^o}}{(u+v){V_{n-1}}+(px+qy){{V^\prime}_{n-1}}}}$$ $$K(x,y)={\frac{1}{u+v+{\frac{qx+py}{r}}}}$$ $${K^\prime}(x,y)={\frac{1}{u+v+qx+py}}$$ Eliminating $D_{n-2}$ from eqns.(37) and (38) we obtain: $$\begin{aligned} Tr{P_{n-2}}={\frac{r{\sqrt {\bar r}}\times {V_{n-3}}}{{Z^o_{n-1}}{Z^c_{n-3}}-{Z^c_{n-1}}{Z^o_{n-3}}}} \bigg[ \bigg( {\alpha_{xy}}+{\beta_{xy}}{\frac{{V^\prime}_{n-3}}{V_{n-3}}}\bigg){({\omega_n})_{11}}(x,y)\nonumber\\ +\bigg( {{\alpha^\prime}_{xy}}+{{\beta^\prime}_{xy}} {\frac{{V^\prime}_{n-3}}{V_{n-3}}} \bigg){({\omega_n})_{22}}(x,y)\nonumber\\ -{\frac{1}{{\sqrt{r\bar r}}(1-{\frac{1}{r}})(qx+py)}}\bigg( (u+v)+(px+qy){\frac{{V^\prime}_{n-3}}{V_{n-3}}} \bigg) {V_n} \bigg]\end{aligned}$$ where ${\alpha_{xy}}=y-{\frac{u+v}{py+qx}}\times pu$, ${\beta_{xy}}={\frac{(x-y)(q-p)}{py+qx}}\times pu$ ${{\alpha^\prime}_{xy}}=x-{\frac{u+v}{py+qx}}\times qv$, ${{\beta^\prime}_{xy}}={\frac{(x-y)(q-p)}{py+qx}}\times qv$ In general $n$th and $(n\pm 2)$th generations have same arrangement of the last two bonds appart from their respective mirror symmetric parts.That is why $Tr{P_n}$ and $Tr{P_{n\pm 2}}$ will have similar expressions.Since we have assumed ${P_n}={\Omega_n}{\bar P}P$ it follows from the expression (27) that ${P_{n-3}}={\Omega_{n-3}}P{\bar P}$.Therefore proceeding in a similar way as above we get: $$\begin{aligned} Tr{P_{n-3}}={\frac{r{\sqrt{\bar r}}\times {V_{n-4}}}{{Z^o_{n-2}}{Z^c_{n-4}}-{Z^c_{n-2}}{Z^o_{n-4}}}}\bigg[ \bigg( {\alpha_{yx}}+{\beta_{yx}}{\frac{{V^\prime}_{n-4}}{V_{n-4}}}\bigg){({{\omega}_{n-1}})_{11}}(y,x)\nonumber\\ +\bigg( {{\alpha^\prime}_{yx}}+{{\beta^\prime}_{yx}}{\frac{{V^\prime}_{n-4}}{V_{n-4}}} \bigg) {({{\omega}_{n-1}})_{11}}(y,x)\nonumber\\ -{\frac{1}{{\sqrt{r\bar r}}(1-{\frac{1}{r}})(px+qy)}}\bigg( (u+v)+(qx+py){\frac{{V^\prime}_{n-4}}{V_{n-4}}} \bigg) {V_{n-1}} \bigg]\end{aligned}$$ Using eqns. (52) and (53) and similar expressions for $Tr{P_n}$ , $Tr{P_{n-1}}$ in the trace map relation (30) we obtain the following recurrence relation among partition functions of different Fibonacci generations as: $$\begin{aligned} {\frac{V_{n-1}}{{Z^o_{n+1}}{Z^c_{n-1}}-{Z^c_{n+1}}{Z^o_{n-1}}}} \bigg[ \bigg( {\alpha_{xy}}+{\beta_{xy}} {\frac{{V^\prime}_{n-1}}{V_{n-1}}} \bigg) {({\omega_{n+2}})_{11}}(x,y)+\bigg( {{\alpha^\prime}_{xy}}+{{\beta^\prime}_{xy}}{\frac{{V^\prime}_{n-1}}{V_{n-1}}} \bigg) {({\omega_{n+2}})_{22}}(x,y)\nonumber\\ -{\frac{1}{{\sqrt{r\bar r}}(1-{\frac{1}{r}})(qx+py)}}\bigg( u+v+(px+qy){\frac{{V^\prime}_{n-1}}{V_{n-1}}} \bigg){V_{n+2}} \bigg] ={\frac{r{\sqrt{\bar r}}\times {V_{n-2}}}{{Z^o_n}{Z^c_{n-2}}-{Z^c_n}{Z^o_{n-2}}}}\nonumber\\ \times {\frac{V_{n-3}}{{Z^o_{n-1}}{Z^c_{n-3}}-{Z^c_{n-1}}{Z^o_{n-3}}}} \bigg[ \bigg( {\alpha_{yx}}+{\beta_{yx}}{\frac{{V^\prime}_{n-2}}{V_{n-2}}} \bigg){({\omega_{n+1}})_{11}}(y,x)+ \bigg( {{\alpha^\prime}_{yx}}+{{\beta^\prime}_{yx}} {\frac{{V^\prime}_{n-2}}{V_{n-2}}} \bigg) {({\omega_{n+1}})_{22}}(y,x)\nonumber\\ -{\frac{1}{{\sqrt{r\bar r}}(1-{\frac{1}{r}})(px+qy)}}\bigg( u+v+(qx+py) {\frac{{V^\prime}_{n-2}}{V_{n-2}}} \bigg){V_{n+1}} \bigg] \times \bigg[ \bigg( {\alpha_{xy}}+{\beta_{xy}} {\frac{{V^\prime}_{n-3}}{V_{n-3}}} \bigg) {({\omega_n})_{11}}(x,y)\nonumber\\ + \bigg( {{\alpha^\prime}_{xy}}+{{\beta^\prime}_{xy}}{\frac{{V^\prime}_{n-3}}{V_{n-3}}} \bigg) {({\omega_n})_{22}}(x,y)- {\frac{1}{{\sqrt{r\bar r}}(1-{\frac{1}{r}})(qx+py)}}\bigg( u+v+(px+qy){\frac{{V^\prime}_{n-3}}{V_{n-3}}} \bigg) {V_n} \bigg]\nonumber\\ -{\frac{{D_{n-2}}{V_{n-4}}}{{Z^o_{n-2}}{Z^c_{n-4}}-{Z^c_{n-2}}{Z^o_{n-4}}}} \bigg[ \bigg( {\alpha_{yx}}+{\beta_{yx}} {\frac{{V^\prime}_{n-4}}{V_{n-4}}} \bigg) {({\omega_{n-1}})_{11}}(y,x) +\bigg( {{\alpha^\prime}_{yx}}+{{\beta^\prime}_{yx}} {\frac{{V^\prime}_{n-4}}{V_{n-4}}} \bigg) {({\omega_{n-1}})_{22}}(y,x)\nonumber\\ -{\frac{1}{{\sqrt{r\bar r}}(1-{\frac{1}{r}})(px+qy)}}\bigg( u+v+(qx+py) {\frac{{V^\prime}_{n-4}}{V_{n-4}}} \bigg) {V_{n-1}} \bigg] \end{aligned}$$ The above equation reveals the reccurence relation among the partition functions of different Fibonacci generations from $(n-4)th$ to $(n+2)th$.The partition functions have entered in the above equation through the quantities given by equations from (42) to (51) .This is in conformity with the symmetry properties of the Fibonacci chain. [**Conclusion**]{} We have found an exact expression for the partition function of an Ising model on a reguler lattice in presence of magnetic field with open boundary conditions.This always includes closed partition functions because of the fact that out of four different spin configurations at the end points of the chain two configurations $\uparrow \uparrow$ and $\downarrow \downarrow$ satisfy closed boundary conditions.We have also shown that “Mirror Symmetry” is a characteristic property of all Fibonacci generations . This is the property which leads to a recurrence relation among the partition functions of different Fibonacci generations from $nth$ to $(n\pm 6)th$ . Since $nth$ and $(n\pm 6)th$ generations are topologically same , one must go through six times decimation renormalization group procedure to find scaling forms of thermodynamic functions. Acknowledgement: We are thankful to S.N.Karmakar of SINP,Calcutta and to S.M. Bhattacharjee of IOP,Bhubaneswar for illuminating discussions.One ofthe authors (SKP) thanks J.P.Chakraborty of IACS,Calcutta for lending him necessary books from his library. \[1\] Yaakov Achian,T.C.Lubensky and E.W.Marshall; Phys.Rev.B33,6460(1986) \[2\] Phase Transition and Critical Phenomena , article by Colin J. Thomson p177 V1 , Edited by C. Domb and M.S.Green; Introduction to Phase Transition and Critical Phenomena by H. Eugene Stanley (Oxford University Press) \[3\] Statistical Mechanics (Second Edition) by Kerson Huang(John Wiley Sons)
--- author: - | Aleksander Czechowski$^1$ and Piotr Zgliczyński$^1$\ [$^1$Institute of Computer Science and Computational Mathematics\ Jagiellonian University\ Łojasiewicza 6, Kraków, 30-348 Poland\ e-mail: [*czechows@ii.uj.edu.pl, zgliczyn@ii.uj.edu.pl*]{} ]{} bibliography: - 'bsq\_bib.bib' title: 'Rigorous numerics for PDEs with indefinite tail: existence of a periodic solution of the Boussinesq equation with time-dependent forcing[^1]' --- [^1]: AC was supported by the Foundation for Polish Science under the MPD Programme ‘‘Geometry and Topology in Physical Models’’, co-financed by the EU European Regional Development Fund, Operational Program Innovative Economy 2007-2013. PZ was supported by Polish National Science Centre grant 2011/03B/ST1/04780.
Introduction ============ Among other metal-silicon systems, the growth of silver (Ag) onto Si(111) surfaces is one of the most extensively studied.[@AgSiH1] Typically, Si(111) surfaces exhibit a wide variety of reconstructions, originated by the tendency to saturate the dangling bonds resulting from the abrupt surface termination. Because of this, deposition of Ag adatoms onto these surface reconstructions results in Ag/Si(111) interfaces with strongly different electronic properties of the interface. By far, the most commonly analyzed Ag/Si(111) systems are those of Ag films onto Si(111)-(7x7) and the Ag-derived Si(111)-($\sqrt{3}$x$\sqrt{3}$)R30$^{o}$ surface reconstruction. The interest in such systems lies on the preparation of well ordered structures which show new applications in optoelectronics as magnetic or optical devices. In general, it is highly desirable to achieve sharp interfaces, in which the conductivity and transport properties depend on their size and spatial distributions. Nevertheless, because of stress effects, the most common growth mode at room temperature (RT) for nonreactive Ag/Si(111) interfaces is the formation of three-dimensional islands with widely varying heights.[@AgSiH2; @AgSiH3; @AgSiH4; @AgSiH5] In order to avoid these effects and to improve the conductivity and transport properties of the Ag films, recent studies have focused their efforts on alternative growth modes and regimes.[@AgSiH6; @AgSiH7; @AgSiH8] In this context, much effort has been devoted to find mechanisms that neutralize dangling bonds at the Si(111) surface.[@AgSiH9; @AgSiH10] With this purpose, interest has been recently revived by the obtention of artificially produced H-terminated surfaces with a high degree of homogeneity. The hydrogenation of the Si(111)-(7x7) restores the (1x1) symmetry of the bulk and seems to destroy the Fermi level (E$_F$) pinning of the non-hydrogenated surface.[@Aminabis; @Amina] The development of both the wet chemical treatment and the atomic hydrogen-based method has allowed the preparation of a stable and unreconstructed flat Si(111) surface,[@AgSiH11; @AgSiH12] with a nearly-defect-free termination. As a consequence of the high-quality of this hydrogenated surface, four remarkably sharp features have been recently resolved by angle-resolved photoelectron spectroscopy (ARPES), which appear to be mostly located around the $\overline{K}$-point.[@AgSiH12; @AgSiH13; @AgSiH14] The origin of these features is attributed to: (i) A surface resonance, with a p$_{x}$-p$_{y}$ symmetry,[@AgSiH14] (ii) A surface state identified as a Si-Si backbond state,[@AgSiH14] and (iii) two higher binding-energy H-Si surface states.[@AgSiH14; @AgSiH15] In spite of the extensive investigations of the electronic properties of Ag films deposited onto clean Si(111) substrates, there remains a lack of studies on the electronic properties of Ag films deposited onto H/Si(111)-(1x1) substrates. The study of the Ag films deposited onto these surfaces has been mostly approached through structural methods,[@AgSiH16; @Naitoh3; @Sakurai] in order to elucidate their growth mode on these passivated surfaces. Growth mode and structure of these Ag films is drastically changed by the hydrogen termination of the Si(111)-(7x7) surface.[@AgSiH16; @Naitoh3] It is well-known that Ag films deposited onto clean Si(111) surfaces grow in quasi-layer-by-layer fashion up to few monolayers (MLs) at RT, with a two-domains Ag-islands distribution.[@dosdominios] At high substrate temperatures, growth of Ag films onto clean Si substrates proceeds in the Stranski-Krastanov mode.[@AgSiH2] Opposite to this behavior, impact-collision ion-scattering spectroscopy showed that growth of Ag films deposited onto H/Si(111)-(1x1) surfaces follows a quasi-layer-by-layer mode even at temperatures of 300 $^{o}$C.[@AgSiH16] In this case, the forming Ag islands appeared to be thinner than those deposited onto the high-temperature clean substrates.[@AgSiH16; @Naitoh3] Moreover, it was observed that Ag films deposited onto H/Si(111)-(1x1) substrates at 300 $^{o}$C tend to follow a mono-domain island distribution.[@AgSiH16] With regard to the presence of H at the interface, it has been shown that its presence gradually decreases with the Ag deposition even at RT,[@AgSiH17; @AgSiH18] indicating a partial replacement of H by Ag atoms. This effect has been recently confirmed by resonant nuclear analysis of the Ag-H/Si(111) interface in 1 ML passivated Si substrates prepared by dosing atomic H at 820 K.[@AgSiH19] These results show that low-temperature deposition of Ag causes part of the H-monolayer migrates to the surface and the rest remaining at the interface. In contrast, deposition at 360 K prevents the presence of H at the metal surface. In this work we report on the electronic properties of Ag films deposited onto H/Si(111)-(1x1) surfaces at RT, determined by ARPES. In Sect. II we describe the experimental setup. The results obtained by ARPES are showed and discussed in Sect. III. In this section we illustrate and discuss the evolution of the Ag valence-band structure as the film thickness increases and the influence of H on it. We finish this section with the analysis of the Fermi surface (FS) of the metallic films prepared. Experiment ========== The experiments were performed at LURE (Orsay, France) using the French-Spanish (PES2) experimental station of the Super-Aco storage ring, described elsewhere.[@AgSiH20] The measurements were carried out in a purpose-built ultra-high vacuum system, with a base pressure of 5x10$^{-11}$ mbar, equipped with an angle resolving 50 mm hemispherical VSW analyzer coupled on a goniometer inside the chamber. The manipulator was mounted in a two-axes goniometer that allows rotation of the sample in: (i) The whole 360$^{o}$ azimuthal angle ($\phi$) and (ii) In the 180$^{o}$ polar emission angle relative to surface normal ($\theta_{off}$), with an overall angular resolution of 0.5$^{o}$. The energy range of light was of 18-150 eV. Photoelectrons were excited with p-polarized synchrotron radiation, that is, with the polarization vector of incident light, the surface normal, and the emitted electrons lying in the same plane. The current incident angle of the light ($\Theta_{i}$) was $\Theta_{i}$=45$^{o}$. Nevertheless, some experiments were carried out as a function of $\Theta_{i}$. With this set-up, the procedure to determine the FS using ARPES is direct. For a given experiment, the photon energy ($h\upsilon$) was fixed and the intensity at the E$_F$ was recorded, along a series of azimuthal scans, for each step of the crystal rotation about its surface normal. This procedure was repeated at different polar angle positions of the analyzer, which allows us to scan a sheet of the Brillouin zone (BZ) for each h$\upsilon $. In order to measure the FS contour at different perpendicular wave vectors, the above procedure can be carried out at different photon energies. In our measurements, the typical polar intervals were 1.5$^{o}$ and the azimuthal angle range was fixed at 180$^{o}$. In these conditions, the typical measuring time was between 4 and 6 hours, depending on statistics and optimal signal-to-noise ratio. The substrates were n-doped Si(111) single crystal, with a nominal resistivity of 100 $\Omega$cm. They were prepared *ex situ* using a wet chemical treatment that results in a passivated H/Si(111)-(1x1) surface.[@AgSiH11] After introducing the substrates in the chamber, the quality of the surfaces was checked out through the sharpness of the features appearing in the valence-band photoemission-spectra at $\overline{K}$-point, which are attributed to intrinsic H-Si surface states.[@AgSiH13; @AgSiH14] Ag was evaporated onto the surface at RT. The rate of evaporation, 0.06 ML/min, was determined by using a quartz microbalance. In these conditions, Ag films of thickness ranging from 0.03 to 8 ML were deposited onto H/Si(111)-(1x1) surfaces at RT. Results and Discussion ====================== Metallization of Ag films deposited on H/Si(111)-(1x1) surfaces --------------------------------------------------------------- Ag films were deposited at RT onto H/Si(111)-(1x1) surfaces with a nominal Ag coverage ($\theta_{Ag}$) lower than 1 ML. Valence-band photoemission measurements were carried out in these films. Figure \[fig1\] shows the energy distribution curves (EDCs) measured in these films with h$\upsilon$=32 eV at $\overline{K}$ and $\overline{\Gamma}$-points of the Si(111) surface Brillouin zone (SBZ). All the spectra are referred, in binding energy, to the E$_F$. With regard to the spectra corresponding to clean substrate, the most remarkable features at $\overline{K}$-point are those appearing with binding energies between 4 and 10 eV (Fig. \[fig1\](a)). These features have been related with emission from surface states ((a), (a’), and (b‘) features) and a surface resonance ((b)-feature).[@AgSiH13; @AgSiH14] Opposite to this, main emission from the substrate at $\overline{\Gamma}$-point is related to bulk Si transitions. As the $\theta_{Ag}$ increases, the substrate features at both $\overline{K}$ and $\overline{\Gamma}$-points tend to be attenuated by the film signal. This effect is expected to be stronger for surface states than for bulk-derived states or surface resonances, due to the intrinsic localization of the surface states.[@AgSiH14] This would explain the faster decrease of intensity of (a) and (a‘) features compared to that of the (b)-feature with $\theta_{Ag}$ (Fig. \[fig1\](a)), which would support the assignment of the (b)-feature as a surface resonance state.[@AgSiH14] In spite of Ag deposition, non-appreciable changes in binding energy and line width of the substrate-derived features are detected as the $\theta_{Ag}$ increases. On the other side, no trace from Ag-induced surface states at these points of the SBZ were detected. These facts suggest that this metal-semiconductor interface shows a nearly abrupt and nonreactive behavior. The insets of Fig. \[fig1\] show in detail the evolution with $\theta_{Ag}$ of the density of states (DOS) at the E$_F$ in these submonolayer Ag films. At $\overline{\Gamma}$-point (Fig. \[fig1\](b)), the DOS at the E$_F$ is negligible and the small emission observed can be attributed to secondary electrons. The only emission expected to appear close to the E$_F$ at $\overline{\Gamma}$-point in metallic Ag(111) films would come from the Shockley Ag(111) surface state. In Ag films deposited onto clean Si substrates, emission from this state appears in films thicker than 3 MLs.[@AgSiH21] Therefore, the lack of emission at $\overline{\Gamma}$-point with h$\upsilon$=32 eV is not really surprising in such nonreactive interfaces. At $\overline{K}$-point, the situation is completely different from that observed at $\overline{\Gamma}$-point. This symmetry point has a parallel wave vector (k$_\|$) of k$_\|$=1.09 Å$^{-1}$, which appears to be slightly smaller than the Fermi vector (k$_F$) of the bulk Ag(111) FS cut with h$\upsilon$=32 eV along the $\overline{\Gamma}$$\overline{K}$ direction (k$_F$=1.13 Å$^{-1}$).[@AgSiH22] Therefore, the evolution of the DOS at the E$_F$ at this symmetry point is directly related with the metallization process in these films due to the *sp*-band crossing the E$_F$. The inset of Fig. \[fig1\](a) elucidates this point. It can be observed that the DOS at the E$_F$ increases with the $\theta_{Ag}$ with an incipient *sp*-derived state close to the E$_F$ at $\theta_{Ag}$=0.56 ML. This can be considered the onset of metallization of Ag films deposited onto H/Si(111)-(1x1) surfaces. These results would confirm the nonreactive behavior of Ag overlayers and suggest that metallization process of Ag films prepared onto H/Si(111)-(1x1) surfaces at RT follows the tendency through bulk-like metallic films in a similar way to that observed in Ag films prepared onto clean Si surfaces at RT,[@AgSiH21] without additional metal-semiconductor surface states induced at the E$_F$. As it is expected for a quasi-layer-by-layer growth of Ag films onto H/Si(111)-(1x1) substrates at RT,[@AgSiH16] the whole metallization process can be monotonously followed in films thicker than 1 ML. Figure \[fig2\] shows, in a similar way, the EDCs measured in films thicker than 1 ML. The main features observed in these spectra at both symmetry points correspond to the Ag *4d* valence band, appearing at binding energies higher than 4 eV. The Ag *4d* features are better resolved than those measured in Ag films with similar thickness as-deposited onto clean Si substrates. This fact can be related to the different growth mode of these films, which is an indicative of the structural improvement obtained in films prepared onto H/Si(111)-(1x1) surfaces at RT. The insets of Fig. \[fig2\] show the evolution of the DOS at the E$_F$ with $\theta_{Ag}$. At $\overline{K}$-point, the gradual increasing of emission at the E$_F$ observed in the submonolayer films defines in a clear peak. At $\theta_{Ag}$=2.4 ML, the bulk-like long lifetime *sp*-band is formed and, consequently, the metallic behavior of these Ag films is completely established at this Ag coverage. At $\overline{\Gamma}$-point, the evolution of the DOS at the E$_F$ is completely different from that observed in the submonolayer films. In addition, Ag films of round to 2 ML show a new incipient emission in the vicinity of the E$_F$ that becomes resolved in thicker films as two features, labelled as SS1 and SS2. These features correspond to occupied states SS1 and SS2 with binding energies of 0.10 eV and 0.35 eV, respectively. The origin of these two features will be discussed in the next section. Surface states of metallic Ag films ----------------------------------- In the previous section, it was pointed out the existence of SS1 and SS2 features in the valence band spectra at $\overline{\Gamma}$-point of Ag films deposited at RT onto H/Si(111)-(1x1) with $\theta_{Ag}>$2 ML. Now, we discuss the origin of the states that give rise to these features. Figure \[fig3\] shows the normal-emission valence-band spectra measured with h$\upsilon $=32 eV in films prepared with different conditions. In this Figure, curves *a*-*c* correspond, respectively, to 4, 5, and 6 ML-thick as-deposited Ag films. Curve *d* corresponds to the latter 6 ML-thick Ag film annealed to 300 $^{o}$C for 5 min. The latter sample was annealed again to 300 $^{o}$C for 15 min and then 1 ML Ag film was deposited (Curve *e*). Curve *f* corresponds to a further Ag deposition of 1 ML and then annealed to 300 $^{o}$C for 5 min. The binding energies of the SS1 and SS2 states are marked by solid lines in these spectra. From these results, we can remark that: (i) These states appear with Ag deposition onto H/Si(111)-(1x1) surfaces in films thicker than 2-3 MLs and their intensities simultaneously increase with $\theta_{Ag}$. (ii) Annealing effects make emission from the SS2 feature to decrease, whereas the SS1 feature is enhanced. Moreover, it appears that this annealing effect is irreversible with additional Ag deposition. (iii) The binding energy of the SS1 state is nearly constant in the whole Ag film-thickness range studied. Nevertheless, the SS2 feature appears slightly shifted to lower binding energy after annealing. Figure \[fig4\] shows the valence-band spectra measured by ARPES along the \[$\overline{1}$10\] direction with h$\upsilon$=32 eV in a 7 ML-thick Ag film deposited at RT onto H/Si(111)-(1x1) and annealed to 300 $^{o}$C for 20 min. Similar results are obtained for the spectra measured along the \[$\overline{2}$11\] symmetry direction (not shown). Small solid bars indicate the position of the SS1 and SS2 states as they disperse. The inset shows the band diagram as extracted from the dispersion of these features. The SS1 and SS2 states appear to cross the E$_F$ at k$_{\Vert }$=0.16 and 0.22 Å$^{-1}$, respectively. Solid lines are parabolic fitting curves of the obtained dispersions, which correspond to an in-plane effective mass (m$^{*}_{\Vert }$) of m$^{*}_{\Vert }$=0.88m$_o$ and 0.56m$_o$ for the SS1 and SS2 states, respectively, being m$_o$ the free electron mass. Dispersion of SS1 and SS2 states along the \[111\] symmetry direction has been also determined. Figure \[fig5\] shows the normal-emission valence-band spectra measured in a 6 ML-thick Ag film deposited at RT onto H/Si(111)-(1x1) and then annealed to 300 $^{o}$C for 5 min, as a function of the h$\upsilon$. Solid lines indicate the binding energies of the SS1 and SS2 states in these spectra. From these results, it can be observed that these states do not disperse in the perpendicular wave vector range studied, which extends over the whole $\Gamma$L symmetry direction of the Ag(111) BZ. The facts observed are: (i) SS1 and SS2 show a parabolic in-plane dispersion (Fig. \[fig4\]) and (ii) they do not disperse with h$\upsilon$ (Fig. \[fig5\]), indicating clearly that SS1 and SS2 states can be considered as two-dimensional states. The first state (SS1) can be identified as the well-known Shockley surface state of bulk Ag(111) (or *sp*-surface state), whose existence is due to the break of crystalline periodicity at the surface.[@Echenique] This state is located in the [*sp*]{} band gap at the L-point of the bulk BZ. ARPES measurements carried out in Ag(111) single crystal show that this state disperses as a parabolic surface state centered at $\overline{\Gamma}$-point, with a maximum binding energy of $\sim$0.12 eV and crossing the E$_F$ at k$_{\Vert }$=0.14 Å$^{-1}$.[@McDougall] On the other side, This state has been observed in Ag films deposited onto clean Si(111) and Cu(001), when these films are thicker than 2-3 MLs.[@AgSiH21; @ss1] All these facts are in concordance with our observations and support the assignment of SS1 as the Ag(111) *sp*-surface state. With regard to the two-dimensional SS2 state, it appears at 0.25 eV below the SS1 surface state. This state is rather confined in the surface and spreads along the surface. At first sight, this state may be tentatively assigned to be a Ag(111) *sp*-derived quantum-well state. Shallow quantum states with m$^{*}_{\Vert }\sim$0.4m$_o$ have been observed at binding energies of 0.5 eV,[@qw; @chiangGL] which is, in fact, quite close the top of the Ag *sp*-band in the $\Gamma$L-direction.[@chiangGL; @bandstruct] Nevertheless, this assignment of the SS2 state seems not to be correct, since it is not expected quantum states at binding energies of 0.35 eV in 2ML-Ag films and this state appears slightly outside the *sp*-band range in the Ag $\Gamma$L direction. In addition, its binding energy appears to be constant with $\theta_{Ag}$ in the thickness range of the Ag films studied. In fact, a shallow Ag(111) *sp*-quantum state should show a shift of $\sim$0.2 eV to lower binding energies in the Ag thickness range of 4-8 MLs, following the phase-accumulation-model.[@phaseaccmod] After these considerations, it appears that the SS2 state is related to a Ag-derived surface state or surface resonance, since it appears quite close, in binding energy, to the bottom of the *sp*-gap of the Ag valence-band projected onto the (111)-face. From our results (Fig. \[fig2\] and Fig. \[fig4\]), we have observed that SS2 emission increases in a similar way to that of SS1 as a function of the Ag deposition. In addition to this, this state appears to be a parabolic state with a m$^{*}_{\Vert }$ quite close to that obtained for the *sp*-surface state of bulk Ag(111).[@masaeffsurfstate] These results suggest that SS2 is a surface state related to a bulk-like termination of the Ag(111) films, as the SS1 state, shifted to higher binding energy. In order to check these assignments for both SS1 and SS2 states, we have analyzed the behavior of their wave functions as a function of $\Theta_{i}$. By means of $\Theta_{i}$, the weight of the components of the polarization vector of light can be tuned, either along the normal to the surface (large $\Theta_{i}$) or contained in the surface plane (small $\Theta_{i}$). In this way, emission from perpendicular-to-surface components of the wave function is enhanced at large $\Theta_{i}$ and decreased at small $\Theta_{i}$. In the case that the emission plane is a mirror plane, p-polarized excitation selects even states with respect to this plane. As we are interested only in *sp*-states, even states refer to *p*-states lying in the emission plane and *s*-states. Therefore, at large $\Theta_{i}$, it would be expected an enhancement of their *p$_z$*-character. Figure \[fig6\] shows the normal-emission valence-band spectra measured with h$\upsilon$=50 eV in a 6 ML-thick Ag film deposited onto H/Si(111)-(1x1) and then annealed to 300 $^{o}$C for 5 min, as a function of the $\Theta_{i}$. The emission plane is that determined by the $\overline{\Gamma}$$\overline{K}$-direction, which can be considered as a mirror plane if we neglect multiple scattering effects.[@AgSiH14] Solid lines indicate the binding energies of the SS1 and SS2 states. As one can observe, intensity of both states increases with $\Theta_{i}$. Nevertheless, the intensity from SS1 increases slower than that from the SS2. In fact, the SS1 increase is the 70% of that of SS2, as calculated after background subtraction. These results can be explained if we consider that SS1 and SS2 are bulk-induced surface states. The Ag(111) *sp*-gap is a Shockley inverted gap. In this case, *sp*-surface states inside this gap are, basically, *s*-like at the top and *p*-like at the bottom. If we assume SS1 and SS2 to be *sp*-surface states, they are expected to show a rather *p*-like behavior since they appear quite close to the bottom of the gap. Therefore, the observed increase of the SS1 and SS2 intensity with $\Theta_{i}$ can be attributed to their *p$_z$*-character. In any case, if they were *only* *p*-like states, they should show the same $\Theta_{i}$-dependence of the intensity. Nevertheless, SS1 shows a smoother $\Theta_{i}$-dependence than that of SS2 which can be attributed to a non-negligible *s*-component. This contribution would be expected to be more important for *sp*-surface states at the top of the gap, which is coherent with our results. The behavior of the two different SS1 and SS2 features as coming from the *sp*-surface state seems to be established. The question now, is to determine the origin of the splitting of the Ag *sp*-surface state into the SS1 and SS2 features. On one side, our results show that annealing processes make an irreversible decrease of the emission from SS2, whereas that from SS1 increases (Fig. \[fig3\]). On the other side, it has been observed that Ag deposited onto H/Si(111)-(1x1) surfaces at RT partially removes H from the metal-semiconductor interface and this desorption process is enhanced at higher deposition temperatures.[@AgSiH17; @AgSiH18; @AgSiH19] This behavior has been also observed in Cu and Au films deposited onto these substrates.[@Amina; @AgSiH19] A similar H-desorption mechanism has also been observed when the studied films RT-deposited were annealed.[@AgSiH18; @Nishiyama] These facts suggest that the SS2 state is related to the presence of H at the metal-semiconductor interface, which somehow makes the bulk Ag(111) surface state appear closer to the *sp*-band gap edge. Several effects can cause a shift of the *sp*-surface state. In the following, we will discuss the different possibilities for H to induce such effect. Let us now discuss the possibility of the existence of different Ag relaxed regions in films deposited onto H/Si(111) surfaces. It has been observed that the binding energy of the *sp*-surface state can differ from one substrate to other, depending on strain.[@karsten] Nevertheless, this seems not to be the case of films as-deposited onto H/Si(111)-(1x1), where only two splitted surface states are observed. If this splitting was induced by strain, the Ag films would be mostly composed by two differently strained domains, which would imply that different points of the Ag BZ are simultaneously probed by ARPES. This fact would make that the measured bulk Ag *4d* features appeared anomalously broadened, which is not observed. The study of the nature of surface states can be analyzed in the workframe of the phase-accumulation model,[@phaseaccmod] based on the idea that surface states can be considered as electrons trapped between the surface gap and the surface barrier.[@Echenique] In this model, differences of the binding energy of surface states are expected via the surface-barrier and crystal phases. In principle, there are two possibilities for H to modify the binding energy of the bulk Ag(111) surface state. The first one is related to the fact that Ag deposition at RT removes H from the interface, which seems to migrate to the metal surface.[@AgSiH19] This H on the surface appears to be mostly removed when films are annealed at high temperatures.[@AgSiH19] Under these conditions, the presence of H on the metallic surface could induce a shift of the surface state to higher binding energies ($\Delta$SS) as far as H produced a decrease of the work-function of the metal at the surface and, consequently, a decrease of the surface-barrier phase.[@phaseaccmod] In fact, these effects have been observed after adsorption of submonolayer alkali-metals on Cu(111) and on Ag(111) films deposited onto graphite and Si(111)-(7x7) at low temperature.[@karsten; @alkali] In these cases, the magnitude of the observed $\Delta$SS was a $\sim$10-20% of the work-function decrease. Nevertheless, in contrast to the observed role of alkali metals on metallic surfaces, it has been observed that the presence of H at the Ag(111) and Ag(110) surfaces increases the metal work-function at the surface by 0.2-0.3 eV,[@plummer1; @plummer2] slightly shifting the Ag *sp*-surface state to lower binding energies. These facts suggest that the splitting of the *sp*-surface state can not be attributed to the presence of H at the metal surface. The second possibility for H to modify the binding energy of the Ag(111) *sp*-surface state comes from the presence of absorbed H at the metal-semiconductor interface. After Ag deposition, and mostly after annealing processes, H can be progressively desorbed from the interface. Nevertheless, large H-covered areas of Si can stay underneath the Ag film, since, when one initial Ag cluster is formed, Ag adatoms appear not to be spontaneously substituted for H atoms around Ag adsorption sites.[@Sakurai] This situation would induce two different limiting conditions as far as the surface-state wave-function could be influenced by surface conditions. It has been early observed that Ag *sp*-surface state shifts to lower binding energy as thicker Ag films are deposited onto 1ML-Au/Ag substrates,[@decaySS] recovering a bulk-like situation in $\sim$10ML-thick Ag films. This fact was explained by the presence of the interlayer Au film, which downshifts the bulk-like Ag surface state. It those systems, the decay length of the surface state was determined to spread along several MLs (28 Å).[@decaySS] In our case, a similar behavior may be expected in Ag films deposited onto H/Si(111)-(1x1) surfaces, where H appears to introduce a larger shift of the surface state than that observed in the Ag/1ML-Au/Ag system. This effect could be also understood in the frame of the phase-accumulation model.[@phaseaccmod] The presence of H at the interface would be expected to increase the reflectivity of the *sp*-surface state, due to its long decay length compared to the film thickness.[@decaySS] This fact would induce an increase of the effective crystal phase and, therefore, the *sp*-surface state would downshift. In order to estimate the magnitude of this effect, an increase of the crystal phase of round to 30-40% would downshift the sp-surface state in $\sim$0.2 eV, which would reproduce quite well the behavior observed in our data for Ag films deposited onto H/Si(111)-(1x1) substrates. Fermi surface ------------- In this section we analyze the electronic properties of the Ag films beyond the metallization onset, centering this study on the determination of the FS of these films. The image in Fig. \[fig7\] shows the photoelectron angular distribution measured at the E$_F$ with h$\upsilon $=32 eV in a 6 ML Ag film deposited onto H/Si(111)-(1x1) and then annealed to 300 $^{o}$C for 5 min. Photoelectron angular distribution measurements were carried out by selecting x-axis corresponding to the $\left[ \overline{1}10\right] $ direction of k$_{\Vert }$. The image has been scaled in such a way that it is linear in photoemission intensity and in k$_{\Vert }$. The photoemission intensity is maximum for the brightest feature and minimum for the darkest one. In this image, well-defined features are labelled, indicating the momentum distribution, as a function of k$_{\Vert }$$_y$ and k$_{\Vert }$$_x$, of initial states lying at the E$_F$. The Ag SBZ contours are also plotted on the image, following the extended zone scheme. In the center of the image it can be observed two different features, labelled as SS1 and SS2. The SS1 feature appears as an intense spot at the center of the image of $\sim$0.10 Å$^{-1}$ in size, whereas the SS2 feature is a homogeneous ring-like feature centered at $\overline{\Gamma}$-point with a radius of 0.23 Å$^{-1}$. These features appear at the FS cut as a consequence of the crossing of the E$_F$ of the SS1 and SS2 surface states, as these states disperse. The fact that these states show such a circular symmetry in the k$_{\Vert }$-plane indicates that these states are isotropic parabolic states, as expected for *sp*-derived surface states in bulk metallic systems.[@Hufner] Besides to the surface-state-related features, additional ones appear in the image, at higher k$_{\Vert }$ values, which have been labelled as FS1, FS2, and FS2’. The FS1 feature appears as a distorted ringlike inside the first SBZ, which corresponds to the cut of the bulk Ag(111) FS in the first BZ.[@juan2] From this feature, the k$_F$ values of the FS of these Ag films can be extracted. In order to do this, one should take into account experimental energetic-window effects, which move the E$_F$ band-crossing-point away from the maximum intensity of emitted photoelectrons at the E$_F$ towards the unoccupied-states side.[@AgSiH22; @straub] In our experiments, the energetic window was 0.2 eV with h$\upsilon $=32 eV. From this analysis, it was obtained that this Ag FS cut shows a maximum and minimum k$_F$ values of k$_F$=1.29 and 1.16 Å$^{-1}$, which occur along the \[$\overline{1}$10\] and \[$\overline{2}$11\] (or \[$\overline{1}\overline{1}$2\]) directions, respectively. These values are in concordance with those obtained in metallic Ag films prepared onto Si(111)-(7x7) surfaces.[@juan2] The resemblance between the FS of both Ag films appears not only in the k$_F$ values as well as in the lifetime of *sp*-states at the E$_F$.[@AgSiH22] Intrinsic photoemission line width of states at the E$_F$ is dominated by photoelectron lifetime.[@smith] In our experiments, the line width of the FS1 feature, where it can be isolated from other contributions, is of 0.28$\pm$0.04 Å$^{-1}$ in average, which implies that the inverse lifetime of photoelectrons is of 4.0$\pm$0.5 eV,[@AgSiH22] coherently with that expected in the bulk material for photoelectrons excited with h$\upsilon $=32 eV.[@AgSiH22; @Goldmann] The rest of features observed in Fig. \[fig7\] (FS2 and FS2’) appear outside of the first SBZ, as a semi-circular feature (FS2) below a wide spot-like feature (FS2‘). The presence of these features is expected from neighboring BZs, which would be measured with h$\upsilon $=32 eV.[@juan2] However, the most remarkable electronic information to be extracted from the FS of these metallic films is that it shows a sixfold symmetry, instead of the typical threefold symmetry found in the three-dimensional Ag single crystal. The loss of symmetry of the FS of these films can be related to the fact that Ag films deposited onto H/Si(111)-(1x1) at RT are composed by two domains rotated 60$^o$. In order to illustrate this point, we have analyzed the influence of the two-domains character of the films on the resulting Ag(111) FS. Figure \[fig8\] shows (on the left) the band dispersion of the *sp*-band of Ag(111) single crystal along the high-symmetry directions of the SBZ: \[$\overline{1}$10\] (k$_{\|x}$), \[$\overline{2}$11\] (k$_{\|y}$), and \[$\overline{1}\overline{1}$2\], as would be obtained by ARPES with h$\upsilon $=32 eV.[@juan2] In this plot, the crossing-points of the *sp*-band with the E$_F$ are indicated, which are labelled as e$_1$-e$_5$. Each one of these points defines one point of the Ag(111) FS cut (middle plot) and, extending this to the whole BZ, a two-dimensional image of the Ag FS can be obtained. This FS cut of Ag single crystal shows the intrinsic threefold symmetry of the Ag BZ. The obtention of such a FS by ARPES assumes that each direction of the hexagonal Ag SBZ is scanned along one experimental azimuth (upper plot). Nevertheless, this seems not to be the case of the Ag films prepared onto H/Si(111)-(1x1) at RT. A film composed of two domains rotated 60$^o$ would artificially produce, by ARPES, a 60$^o$-overlapping of the band structure, of the SBZ, and, therefore, of the FS (right plots in Fig. \[fig8\]). Since one experimental azimuth simultaneously scans two 60$^o$-differing directions of the Ag SBZ, belonging each one to two different domains. The fact that Ag films deposited onto H/Si(111)-(1x1) at RT and then annealed to 300 $^o$C for 5 min are composed by two domains rotated 60$^o$ has been also observed in Ag films of similar thicknesses deposited onto Si(111)-(7x7).[@juan2] Nevertheless, in contrast to the results observed in films deposited onto H/Si(111)-(1x1) at RT, it seems that high-temperature Ag deposition favors a mono-domain growth.[@AgSiH17] These facts suggest that there exists an activation energy for the Ag nucleation beyond that mono-domain Ag growth is favored. In addition to this, the fact that annealing process carried out after deposition appears not to favor this structural transition seems to confirm the fact observed that Ag deposition rate may also play a role in the obtention of mono-domain films.[@Nishiyama] Nevertheless, much work should be done to elucidate this point. In any case, these results are promising and suggest that a good knowledge and control of the growth conditions of Ag films onto H/Si(111)-(1x1) substrates may improve the conductivity and transport properties of these films, since the influence of electronic barriers and hopping processes would be reduced. Summary ======= Silver films of different thickness (0-8 MLs) were deposited at RT onto H-passivated Si(111) surfaces. Their electronic properties have been analyzed by ARPES, probing $\overline{K}$ and $\overline{\Gamma}$-points of the Si(111) SBZ. The valence-band spectra of submonolayer Ag films appeared to follow the evolution expected for an abrupt and nonreactive metal-semiconductor interface, with no-trace observed from Ag-induced surface states at these points of the SBZ. Bulk-like metallization process was followed by analyzing the evolution of the DOS at the E$_F$ at Si(111) $\overline{K}$-point, which coincides, at h$\upsilon $=32 eV, with the crossing-point of the bulk Ag *sp*-band with the E$_F$. At initial Ag growth-stage, films appeared to be semiconducting and the onset of metallization was established at a coverage of $\sim$0.6 MLs. The bulk-like metallic behavior was completely defined in Ag films of 2-3 MLs and the small line width of the Ag *4d*-derived features indicates that high-quality (111)-oriented films can be prepared onto H-passivated Si surfaces. Two well-defined Ag-derived features appeared in the valence-band spectra measured at $\overline{\Gamma}$-point, in films thicker than 2 MLs. These features, labelled as SS1 and SS2, were assigned to occupied states with binding energies of 0.1 and 0.35 eV, respectively. These states were both identified, by means of their perpendicular and parallel dispersion, as two surface states appearing at the *sp*-gap of the Ag SBZ and related to the *sp*-surface state of bulk Ag(111). As expected for such surface state, the wave function of both SS1 and SS2 states shows a pronounced *p*-like behavior, as revealed by photoemission measurements tuning the components of the polarization vector of light. The existence of two different surface states with the same origin was attributed to the non-homogeneous presence of H at the interface, due to the fact that Ag deposition and annealing process partially desorb H from the interface and, therefore, a downshift of the *sp*-surface state is produced in regions where H is still present. The electronic properties of metallic Ag films deposited at RT onto H-passivated substrates were analyzed by measurements of photoelectron angular distribution at the E$_F$ with h$\upsilon $=32 eV. With this technique, a cross-sectional cut of the FS of these films has been obtained with this h$\upsilon $. The k$_F$ values of this FS cut obtained in different symmetry directions appeared to be in good agreement with those expected for bulk-like Ag(111) single crystal. A similar agreement was also found for the photoelectron lifetime of states at the E$_F$. In spite of the fact that the FS cut resembles a two-dimensional cut of the three-dimensional bulk-like Ag FS, it reflects a sixfold symmetry rather than the threefold symmetry expected for Ag single crystal. This indicates that these Ag films are composed by two domains rotated 60$^o$. This work was financed by DGICYT (Spain) (Grant No. PB-97-1199) and the Large Scale Facilities program of the EU to LURE. 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--- abstract: 'A Dirichlet problem is considered for the eikonal equation in an anisotropic medium. The nonlinear boundary value problem (BVP) formulated in the present work is the limit of the diffusion–reaction problem with a diffusion parameter tending to zero. To solve numerically the singularly perturbed diffusion–reaction problem, monotone approximations are employed. Numerical examples are presented for a two-dimensional BVP for the eikonal equation in an anisotropic medium. The standard piecewise-linear finite-element approximation in space is used in computations.' address: - 'Nuclear Safety Institute, Russian Academy of Sciences, 52, B. Tulskaya, Moscow, Russia' - 'North-Eastern Federal University, 58, Belinskogo, Yakutsk, Russia' author: - 'Alexander G. Churbanov' - 'Petr N. Vabishchevich' title: Numerical solution of boundary value problems for the eikonal equation in an anisotropic medium --- The eikonal equation, finite-element method, diffusion–reaction equation, singularly perturbed BVP, monotone approximation Introduction ============ Many applied problems lead to the need of solving a BVP for the eikonal equation. First of all, this nonlinear partial differential equation is used to simulate wave propagation in the approximation of geometric optics [@Born; @Kravtsov]. In computational fluid dynamics, image processing and computer graphics (see, for example, [@sethian1999level; @Gilles]), the solution of BVPs for the eikonal equation is associated with calculating the nearest distance to boundaries of a computational domain. The eikonal equation is a typical example of steady-state Hamilton–Jacobi equations. The issues of the existence and uniqueness of the solution for boundary value problems for such equations are considered, e.g., in [@kruvzkov1975generalized; @lions1982generalized]. To solve numerically BVPs for the eikonal equation, the standard approaches are used, which are based on using difference methods on rectangular grids or finite-element/finite-volume approximations on general irregular grids. In this approach, the main attention is paid to problems of nonlinearity. A boundary value problem is formulated in the following way. The function $u(\bm x)$ is defined as the solution of the equation $$\label{1} |\nabla u|^2 = 1, \quad \bm x \in \Omega$$ in a domain $\Omega$ with the specified boundary conditions $$\label{2} u(\bm x) = 0, \quad \bm x \in \partial \Omega .$$ Computational algorithms for solving BVPs for the eikonal equation can be divided into two classes. Marching methods (the first class of algorithms) are the most widely used. They are based on the hyperbolic nature of the eikonal equation. In this case, the desired solution of the problem (\[1\]), (\[2\]) is obtained by successive moving into the interior of the domain from its boundary, using, for instance, first-order upwind finite differences [@Tsitsiklis; @sethian1999fast]. Among other popular methods, we should mention, first of all, the fast sweeping method [@tsai2003fast; @Zhao], which uses a Gauss–Seidel-style update strategy to progress across the domain. Recently (see, for example, [@jeong2008fast]), a fast iterative method for eikonal equations is actively developed using triangular [@fu2011fast] and tetrahedral [@fu2013fast] grids. Other modern variants of the fast marching method, which are adapted, in particular, to modern computing systems of parallel architecture, have been studied and compared, e.g., in [@gomez2015fast]. The second class of algorithms is associated with a transition from (\[1\]), (\[2\]) to a linear or nonlinear BVP for an elliptic equation [@belyaev2015variational]. Instead of equation (\[1\]), we (see [@li2005level]) minimize the functional $$J(u) = \frac{1}{2} \int_{\Omega} (|\nabla u| - 1)^2 d \bm x .$$ It is possible to solve the BVP for the Euler–Lagrange equation for this functional, which has the form $$\triangle u - \nabla \cdot \left (\frac{\nabla u}{|\nabla u|} \right ) = 0 , \quad \bm x \in \Omega .$$ In [@fares2002differential], the computational algorithm is based on solving the nonlinear boundary value problem for $v = 1/u$. The solution of the problem (\[1\]), (\[2\]) can be related to the solution of the homogeneous Dirichlet problem for $p$-Laplacian: $$\nabla \cdot (|\nabla u_p|^{p-2} \nabla u_p ) = - 1 , \quad \bm x \in \Omega .$$ In this case (see, e.g., [@bhattacharya1989limits; @kawohl1990family]), we have $$u_p(\bm x) \rightarrow u(\bm x) \ \mathrm{as} \ p \rightarrow \infty , \quad \bm x \in \Omega .$$ Thus, to find the solution of the problem (\[1\]), (\[2\]), we need to solve the nonlinear BVPs. In our study, we focus on solving auxiliary boundary value problems for linear equations. This approach (see [@gurumoorthy2009schrodinger; @sethi2012schrodinger])is based on a connection between the nonlinear Hamilton–Jacobi equation and the linear Schrodinger equation. Let $v_\alpha (\bm x)$ be the solution of the boundary value problem $$\label{3} - \alpha^2 \triangle v_\alpha + v_\alpha = 0, \quad \bm x \in \Omega ,$$ $$\label{4} v_\alpha (\bm x) = 1, \quad \bm x \in \partial \Omega .$$ Then, for $u_\alpha (\bm x) = - \alpha \ln (v_\alpha(\bm x)$, we have $u_\alpha (\bm x) \rightarrow u(\bm x)$ as $\alpha \rightarrow 0$. A similar approach, where the auxiliary functions $v_\alpha (\bm x)$ are associated with the solution of the unsteady heat equation, is considered in the paper [@crane2013geodesics]. In the present paper, we consider the eikonal equation in an anisotropic medium that is a more general variant in comparison with (\[1\]). Using the transformation $u(\bm x) = - \alpha \ln (v(\bm x))$, the corresponding BVP of type (\[3\]), (\[4\]) is formulated for the new unknown quantity. In our case, $\alpha \rightarrow 0$ and so, we have a singularly perturbed BVP for the diffusion–reaction equation [@roos2008robust; @miller2012fitted]. The numerical solution is based on using standard Lagrangian finite elements [@brenner2008mathematical; @larson2013finite]. The main attention is paid to the monotonicity of the approximate solution for the auxiliary problem. The paper is organized as follows. A boundary value problem for the eikonal equation in an anisotropic medium is formulated in Section 2. Its approximate solution is based on a transition to a singularly perturbed diffusion–convection equation. In Section 3, an approximation in space is constructed using Lagrangian finite elements and the main features of the problem solution are discussed. Numerical experiments on the accuracy of the approximate solution are presented in Section 5 for model two-dimensional problems. The results of the work are summarized in Section 5. Transformation of BVP for the eikonal equation in an anisotropic medium ======================================================================= In a bounded polygonal domain $\Omega \subset R^m$, $m=1,2,3$ with the Lipschitz continuous boundary $\partial\Omega$, we search the solution of the BVP for the eikonal equation $$\label{5} E u = 1, \quad \bm x \in \Omega .$$ Define the operator $E$ as $$\label{6} E u = \sum_{i=1}^{m} a_i^2 \left ( \frac{\partial u}{\partial x_i} \right )^2$$ with the coefficients $a_i({\bm x}) > 0$. The equation (\[5\]) is supplemented with the homogeneous Dirichlet boundary condition $$\label{7} u({\bm x}) = 0, \quad {\bm x} \in \partial \Omega .$$ The basic problems of numerical solving the boundary value problem (\[1\])–(\[3\]) result from the nonlinearity of the equation (see the operator $E$). Similarly to [@gurumoorthy2009schrodinger; @sethi2012schrodinger], we introduce the transformation $$\label{8} v_\alpha (\bm x) = \exp \left (- \frac{u_\alpha (\bm x) }{\alpha} \right )$$ with a numerical parameter $\alpha > 0$. This type of transformation is widely used in studying differential equations with quadratic nonlinearity $Eu$ (see, e.g., [@bicadze]). Define the elliptic second-order operator $L$ by the relation $$\label{9} L u = \sum_{i=1}^{m} \frac{\partial }{\partial x_i} \left ( a_i^2 \frac{\partial u}{\partial x_i} \right ) .$$ For (\[8\]), we have $$\begin{split} a_i^2 \frac{\partial v_\alpha}{\partial x_i} = &- \frac{1}{\alpha} \exp \left (- \frac{u_\alpha(\bm x) }{\alpha} \right ) a_i^2 \frac{\partial u_\alpha}{\partial x_i} , \\ \frac{\partial }{\partial x_i} \left ( a_i^2 \frac{\partial v_\alpha}{\partial x_i} \right ) = & - \frac{1}{\alpha} \exp \left (- \frac{u_\alpha(\bm x) }{\alpha} \right ) \frac{\partial }{\partial x_i} \left ( a_i^2 \frac{\partial u_\alpha}{\partial x_i} \right ) \\ & + \frac{1}{\alpha^2} \exp \left (- \frac{u_\alpha(\bm x) }{\alpha} \right ) a_i^2 \left (\frac{\partial u_\alpha}{\partial x_i} \right )^2 . \end{split}$$ By virtue of this, we obtain $$\alpha^2 L v_\alpha - v_\alpha = \exp \left (- \frac{u_\alpha(\bm x) }{\alpha} \right ) (E u_\alpha - 1 - \alpha L u_\alpha ) .$$ Let $u_\alpha(\bm x)$ satisfies the equation $$\label{10} \alpha L u_\alpha - E u_\alpha = - 1, \quad \bm x \in \Omega ,$$ and the boundary conditions $$\label{11} u_\alpha ({\bm x}) = 0, \quad {\bm x} \in \partial \Omega .$$ Under these conditions, for $v_\alpha(\bm x)$, we have the equation $$\label{12} \alpha^2 L v_\alpha - v_\alpha = 0, \quad \bm x \in \Omega .$$ In view of (\[8\]), from (\[11\]), we obtain the following boundary condition: $$\label{13} v_\alpha ({\bm x}) = 1, \quad {\bm x} \in \partial \Omega .$$ The equation (\[10\]) can be treated as a regularization of the Hamilton–Jacobi equation via the method of vanishing viscosity [@bookEvans]. The boundary value problem (\[10\]), (\[11\]) produces an approximate solution of the problem (\[5\]), (\[6\]) for small values of $\alpha$: $$u_\alpha (\bm x) \rightarrow u(\bm x) \ \mathrm{as} \ \alpha \rightarrow 0 , \quad \bm x \in \Omega .$$ In this case, $u_\alpha (\bm x)$ is defined according to (\[8\]) from the solution of the linear boundary value problem (\[12\]), (\[13\]). Numerical implementation ======================== An approximate solution of the BVP (\[5\])–(\[7\]) is represented (see (\[8\])) in the form $$\label{14} u_\alpha (\bm x) = - \alpha \ln (v_\alpha (\bm x)) ,$$ at a sufficiently low value of $\alpha$. In this case, $v_\alpha (\bm x)$ is defined as the solution of the BVP (\[12\]), (\[13\]). In the present work, the numerical implementation of this approach is carried out on the basis of standard finite-element approximations [@brenner2008mathematical; @larson2013finite]. The main features of the computational algorithm result from the fact that the BVP of diffusion–reaction (\[12\]), (\[13\]) at small $\alpha$ is singularly perturbed, i.e., we have a small parameter at higher derivatives [@holmes2012introduction; @verhulst2005methods]. Let us consider a standard quasi-uniform triangulation of the domain $\Omega$ into triangles in the 2D case or tetrahedra for 3D case. Let $$V_0 = \{ v \in H^1(\Omega) \ | \ v(\bm x) = 0, \ \bm x \in \partial \Omega \} ,$$ $$V_1 = \{ v \in H^1(\Omega) \ | \ v(\bm x) = 1, \ \bm x \in \partial \Omega \} .$$ Denote by $V^h_0 \subset V_0$ and $V^h_1 \subset V_1$ the linear finite-element spaces. For the BVP (\[12\]), (\[13\]), we put into the correspondence the variational problem of finding the numerical solution $y \in V^h_1$ from the conditions $$\label{15} a(y, v) = 0, \quad \forall v \in V^h_0 .$$ By (\[9\]), for the bilinear form, we have $$a(y,v) = \int_{\Omega} \left ( \sum_{i=1}^{m} \alpha^2 a_i^2 \frac{\partial y}{\partial x_i} \frac{\partial v}{\partial x_i} + y v\right ) d \bm x .$$ The differential problem (\[12\]), (\[13\]) satisfies the maximum principle. In particular, this guarantees the positiveness of the solution. More precisely (see, e.g., [@protter2012maximum; @gilbarg2015elliptic]), for points inside the domain $\Omega$, we have $$0 < v_\alpha (\bm x) < 1, \quad \bm x \in \Omega .$$ This the most important property must be also fulfilled for the solution of the discrete problem (\[15\]): $$\label{16} 0 < y(\bm x) < 1, \quad \bm x \in \Omega .$$ If (\[16\]) holds, we speak of monotone approximations for the solution of the diffusion–reaction problem. Even for regular boundary value problems, where the parameter $\alpha$ in (\[12\]) is not small, monotone approximations can be constructed using linear finite elements with restrictions on the computational grid (Delaunay-type mesh, see, for instance, [@letniowski1992three; @huang2011discrete]). Additional restrictions appear (see, e.g., [@ciarlet1973maximum; @brandts2008discrete]) on the magnitude of the reaction coefficient. With respect to our problem (\[12\]), (\[13\]), for the grid step size, we have $h \leq \mathcal{O}(\alpha)$. Restrictions on the grid due to the reaction coefficient can be removed. The standard approach is related to the correction of approximations for the reaction coefficient based on the lumping procedure (see, e.g., [@Thomee2006]). The standard approach to the solution of singularly perturbed diffusion–reaction problems (see [@roos2008robust; @miller2012fitted]) is based on using computational grids with refinements in the vicinity of boundaries. A refinement of the grid is directly related to the value of the small parameter $\alpha$. Another possibility to monotonize the solution of the problem (\[12\]), (\[13\]) at small values of $\alpha$ is the following approach. As noted in the paper [@cai2014natural], for singularly perturbed problems for the diffusion–convection equation, the use of finite-element approximations of higher order not only increases the accuracy of the approximate solution, but improves the monotonicity property as well. It is interesting to check whether there is the same effect in the numerical solution of singularly perturbed problems for the diffusion–reaction equations. Numerical experiments ===================== The 2D BVP (\[5\])–(\[7\]) in the L-shaped region depicted in Fig. \[f-1\] is considered as a model problem. We start with the simplest case, when $a_i = 1, \ i = 1, 2$. The calculations have been performed on various grids. The basic (medium) computational grid, which contains 10,465 nodes and 20,480 triangles, is shown in Fig. \[f-2\]. (0,0) rectangle +(4,6); (4,0) rectangle +(4,4); (0,0) – (0,6) – (4,6) – (4,4) – (8,4) – (8,0) – (0,0); (0,0) – (0,7); (0,0) – (9,0); (0,4) – (4,4); (4,0) – (4,4); (0,0) – (4,4); (-0.35,-0.35) node [$0$]{}; (4,-0.35) node [$1$]{}; (8,-0.35) node [$2$]{}; (-0.35,4) node [$1$]{}; (-0.35,6) node [$1.5$]{}; (-0.5,6.75) node [$x_2$]{}; (8.75,-0.5) node [$x_1$]{}; In solving this problem, the key point is the dependence of the solution on the small parameter $\alpha$. The numerical solution obtained on a very fine grid with $\alpha = 2^{-8}$ is treated as the exact one. The solution $v_\alpha (\bm x)$ of the auxiliary problem (\[12\]), (\[13\]) under these conditions is presented in Fig. \[f-3\], and the corresponding function $u_\alpha (\bm x)$, determined according to (\[14\]), is shown in Fig. \[f-4\]. The influence of the parameter $\alpha$ can be observed in Fig. \[f-5\], where the solution in the cross section $x_1 = x_2$ is plotted (the red line in Fig. \[f-5\]). In our model problem, a good accuracy is achieved for $\alpha \approx 2^{-7}$. The increase in accuracy can be achieved, first of all, by using finer grids. The solution for various $\alpha$ on the coarse grid (2,673 nodes and 5,120 triangles) is given in Fig. \[f-6\]. In this case, for $\alpha = 2^{-k}, \ k \geq 6$, the solution is non-monotone, i.e., at some nodes of the computational grid we have $y(\bm x) < 0$. Similar data for the basic grid are presented in Fig. \[f-7\]. Here, the non-monotonicity appears at $\alpha = 2^{-k}, \ k \geq 7$. Figure \[f-8\] demonstrates the numerical results obtained on the fine grid (41,409 nodes and 81,920 triangles). The non-monotonicity of the approximate solution occurs at $\alpha = 2^{-k}, \ k \geq 8$. In the practical use of the approach (\[12\])–(\[14\]), it seems reasonable to follow the next strategy. We solve a number of auxiliary problems (\[12\]), (\[13\]) with a step-by-step decrease of the parameter $\alpha$ as long as the maximum principle holds. The solution obtained with the smallest $\alpha$ is taken as the approximate solution of the problem (\[5\])–(\[7\]). Our computational grids consist of rectangular isosceles triangles. Because of this, the non-monotonicity is due to the reaction coefficient only. To monotonize discrete solutions, it is sufficient to apply the standard procedure of the reaction coefficient lumping [@Thomee2006]. The effect of diagonalization of the reactive term in the finite-element approximation in predictions on different computational grids can be observed in Figures \[f-9\]–\[f-11\]. In this case, the maximum principle holds for all $\alpha$. The accuracy of the approximate solution decreases from some value of $\alpha$ as the parameter $\alpha$ decreases. Moreover, the value of this optimal value is close to the value that we had without the lumping procedure. Therefore, we can use the diagonalization procedure for selecting the parameter $\alpha$ using the monotonicity condition for the discrete solution of the standard finite-element approximation. In our case (see Figures \[f-6\]–\[f-8\]), we select $\alpha=2^{-5}$ for the coarse grid, $\alpha=2^{-6}$ — for the basic grid and $\alpha=2^{-7}$ — for the fine grid. Above, we have used linear finite elements. Below, we will present numerical results obtained on the basic grid for Lagrangian finite elements of degree $m$, i.e., for approximations $P_m, \ m > 1$. Figure \[f-12\] demonstrates the approximate solution obtained using finite elements of degree 3. A comparison with the case $m=1$ (see Fig. \[f-7\]) indicates that the solution is more accurate and, in addition, it is possible to carry out calculations with a smaller value of the parameter $\alpha$. These effects become more pronounced when using finite elements of degree 5 (see Fig. \[f-13\]) and degree 7 (see Fig. \[f-14\]). Thus, the computational algorithm for solving the eikonal equation (BVP (\[5\])–(\[7\])) can be based on the solution of the auxiliary problem (\[12\]), (\[13\]). In doing so, we employ the minimum value of the parameter $\alpha$ that provides the monotone solution on sufficiently fine computational grids using higher degree Lagrangian finite elements. Special attention should be paid to the problem (\[5\])–(\[7\]) in the anisotropic case. We have considered a variant with constant coefficients, where in (\[6\]), we had $$a_1^2 = 1, \quad a_2^2 = 4 .$$ The convergence of the approximate solution with decreasing $\alpha$ is given in Fig. \[f-15\]. Calculations have been performed on the basic grid using finite-element approximation with $P_7$. The numerical solution of the problem (\[5\])–(\[7\]) for $\alpha = 2^{-8}$ is shown in Fig. \[f-16\]. Similar data obtained for a more pronounced anisotropy: $$a_1^2 = 1, \quad a_2^2 = 10,$$ are depicted in Fig. \[f-17\], \[f-18\]. Conclusions =========== 1. A Dirichlet problem is considered for the multidimensional eikonal equation in a bounded domain with an anisotropic medium. The main peculiarities of such problems results from the fact that the eikonal equation is nonlinear. 2. An approximate solution is constructed using a transformation of the original nonlinear boundary value problem to a linear boundary value problem for the diffusion–reaction equation for an auxiliary function. The transformed equation belongs to the class of singularly perturbed problems, i.e., there is a small parameter at higher derivatives. 3. Computational algorithms are constructed using standard finite-element approximations on triangular (2D problems) or tetrahedral (3D problems) grids. Monotonization of a discrete solution is achieved not only by using finer grids, but also via a correction of approximations for the reaction coefficient using the lumping procedure. The use of finite elements of high degree is studied, too. 4. Numerical experiments have been performed for 2D problems in order to demonstrate the robustness of the approach proposed in the work for solving boundary value problems for the eikonal equation in an anisotropic medium. In particular, a good accuracy is observed when using sufficiently fine grids and Lagrangian finite elements of higher degree. Acknowledgments {#acknowledgments .unnumbered} =============== Petr Vabishchevich gratefully acknowledges support from the the Russian Federation Government (\# 14.Y26.31.0013). 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--- abstract: 'In this paper we present a novel [algorithm for improved]{} block-online supervised acoustic system identification in adverse noise scenarios by exploiting prior knowledge about the space of Room Impulse Responses (RIRs). The method is based on the assumption that the [variability of the unknown RIRs is]{} controlled by only few physical parameters, describing, e.g., source position movements, and thus [is]{} confined to a low-dimensional manifold which is modelled by a union of affine subspaces. The offsets and bases of the affine subspaces are learned in advance from training data by unsupervised clustering followed by Principal Component Analysis. We suggest to denoise the parameter update of any supervised adaptive filter by projecting it onto an optimal affine subspace which is selected based on a novel computationally efficient approximation of the associated evidence. The proposed method significantly improves the system identification performance of state-of-the-art algorithms in adverse noise scenarios.' address: | Multimedia Communications and Signal Processing, Friedrich-Alexander-University Erlangen-Nürnberg,\ Cauerstr. 7, D-91058 Erlangen, Germany, thomas.haubner@fau.de bibliography: - 'refs.bib' title: '**[Online]{} Supervised [Acoustic System Identification]{} Exploiting Prelearned Local Affine Subspace Models**' --- Online Supervised System Identification, Acoustic Echo Cancellation, Model Learning, Local Affine Subspace, Model Selection Introduction {#sec:intro} ============ is one of the classical tasks in acoustic signal processing with a multitude of applications [@haykin_2002; @diniz_adaptive_filtering]. In this paper we consider linear convolutive applications with high-level interfering noise sources which are prone to non-robust performance. Such situations are typically encountered in hands-free acoustic human-machine interfaces which operate in, e.g., driving cars with open windows, or factories, and often involve negative . is usually tackled by frequency-domain adaptive filter algorithms which take for its optimization the statistical properties of the excitation signals, e.g., non-stationarity, temporal and spatial correlation, into account [@buchner_generalized_2005; @malik_recursive_2011]. Noise and interference in the observations is often addressed by methods which use either binary or smooth adaptation control. Binary adaptation control, which in the context of is applied to cope with double-talk, stipulates halting the adaptation during periods of high interference levels [@gansler_double-talk_1996; @Benesty_new_2000]. In contrast, smooth adaptation control continuously adjusts the step size in dependence of a noise estimate. A powerful model-based approach for smooth adaptation control, based on an online algorithm, was introduced in [@malik_online_2010]. However, -based algorithms still result in limited system identification performance for applications with persistent low . Besides adaptation control, the exploitation of prior knowledge about the unknown system has proven to be beneficial for with high-level interfering noise [@fozunbal_multi-channel_2008; @koren_supervised_2012; @talmon_relative_2013]. This prior knowledge is usually extracted in advance from a training data set of samples. The main assumption behind these approaches is the existence of a low-dimensional manifold that is embedded in the high-dimensional space of adaptive filter parameters for a given scenario. This can be motivated by the assumption that the variability of the unknown is controlled by only few physical parameters, describing, e.g., source position movements, temperature changes or movement of furniture [@talmon_diffusion_2013; @laufer-goldshtein_study_2015]. There is a variety of different approaches to model this manifold with the most prominent one assuming that the are confined to a single affine subspace which can be estimated, e.g., by . In [@fozunbal_multi-channel_2008] this model has been employed by regularizing a cost function with the Mahalanobis distance based on the estimated covariance matrix. The strong assumption of globally-correlated is however only rarely valid in practice, e.g., see [@laufer-goldshtein_study_2015]. Thus, [@koren_supervised_2012] modifies it to a local model, which can be motivated by the assumption of manifolds being locally Euclidean [@tu2010introduction]. By the increased model flexibility, which results from employing several instead of a single one, [@koren_supervised_2012] shows a performance improvement in an offline -based system identification task. Hereby, each is associated with a specific source position and estimated from samples which correspond to local source position movements. By employing several mutually exclusive local models, a model selection is required. As selection criterion [@koren_supervised_2012] suggests the Frobenius norm of the difference of the a-priori-learned model covariance matrices and an estimated covariance matrix. The latter one is estimated from the solutions of several system identification problems with local source position variations. In [@talmon_relative_2013] another offline approach for noise-robust system identification is introduced which represents the training data by a globally[-]{}nonlinear manifold model. As [@koren_supervised_2012] and [@talmon_relative_2013] rely on an affinity measure between a statistic of the adaptive filter estimate and the model parameters, they are susceptible to nonunique solutions to the system identification problems which result, e.g., from cross-correlated input signals [@sondhi_stereophonic_1995; @sondhi_benesty_a_better_understanding]. In this paper we introduce a general method which allows to include prior knowledge about the into any algorithm to enhance its performance in adverse noise scenarios. The method relies on the assumption that the can be modelled by a set of affine subspaces [whose parameters are estimated by unsupervised clustering and ]{}. We suggest to denoise the estimated coefficient updates of any algorithm by projecting it onto an optimally selected affine subspace. Furthermore, we introduce a probabilistic approach for computationally-efficient online model selection by evidence maximization which is independent of the current estimate of the algorithm. Supervised Adaptive [MIMO]{} Filtering {#sec:sup_ad_filt} ====================================== In this section we will define a signal model for . Hereby, it is assumed that there exists a linear functional relationship between the $n$th sample of the $Q$ estimated output signals $$\hat{\boldsymbol{y}}(n) = {{\hat{{\boldsymbol{H}}}}^{\text{T}}}(n) \boldsymbol{x}(n) \in \mathbb{R}^{Q} \label{eq:datMod}$$ and [the most recent $L$]{} samples of the $P$ input signals $${\boldsymbol{x}}(n) = {{\begin{pmatrix} {{\boldsymbol{x}}^{\text{T}}}_1(n), & \dots, & {{\boldsymbol{x}}^{\text{T}}}_P(n) \end{pmatrix}}^{\text{T}}} \in \mathbb{R}^{PL}, \label{eq:inSig_def}$$ with $${\boldsymbol{x}}_p(n) = {{\begin{pmatrix} x_p(n), & \dots, & x_p(n-L+1) \end{pmatrix}}^{\text{T}}} \in \mathbb{R}^{L}.$$ The estimated transmission matrix at time instant $n$ $${\hat{{\boldsymbol{H}}}}(n) = \begin{pmatrix} \hat{\boldsymbol{h}}_{11}(n) & \dots & \hat{\boldsymbol{h}}_{1Q}(n) \\ \vdots & \ddots & \vdots \\ \hat{\boldsymbol{h}}_{P1}(n) & \dots & \hat{\boldsymbol{h}}_{PQ}(n) \\ \end{pmatrix}\label{eq:firMIMODef} \in \mathbb{R}^{PL \times Q}$$ models of length $L$ between each input and each output signal. As most algorithms directly process blocks of observations, we introduce the block output matrix $$\hat{\boldsymbol{Y}}(m) = \begin{pmatrix} \hat{\boldsymbol{y}}(mL), & \dots &, \hat{\boldsymbol{y}}(mL-L+1)\\ \end{pmatrix}\in \mathbb{R}^{Q \times L}$$ which captures $L$ samples into one block indexed by $m$. The estimation of the transmission matrix Eq.  an optimization problem in the high-dimensional parameter space $\mathbb{R}^R$ of dimension $R = PLQ$ with elements ${\tilde{\boldsymbol{h}}}(n) = {\text{vec}({{\hat{{\boldsymbol{H}}}}^{\text{T}}}(n))}$ and $\text{vec}(\cdot)$ being the vectorization operator [@Dhrymes2000]. Then, the generic parameter update for iterative algorithms reads: $$\tilde{\boldsymbol{h}}(m) = \tilde{\boldsymbol{h}}(m-1) + \Delta \tilde{\boldsymbol{h}} (m) \label{eq:parUp}$$ with $\Delta \tilde{\boldsymbol{h}} (m)$ denoting the update term. Note that in the following the block-dependency $m$ of the parameters $\tilde{\boldsymbol{h}}(m)$ is omitted if possible for notational convenience. Local Affine Subspace Models {#sec:parModels} ============================ As discussed in Sec. \[sec:intro\], the latent coefficient vectors often populate only a structured subset of the high-dimensional space $\mathbb{R}^R$ of adaptive filter parameters [@talmon_diffusion_2013], which leads to the assumption of a low-dimensional manifold that can be learned in advance from a set of $G$ training data samples $\tilde{\boldsymbol{h}}_g$ with $g=1,\dots,G$. With the assumption of manifolds being locally Euclidean [@tu2010introduction], the coefficient vector manifold can be approximated by patches of locally tangential hyperplanes $\mathcal{M}_i$ as illustrated exemplarily in Fig. \[fig:manApproxVis\] for $R=3$. Each tangential hyperplane $\mathcal{M}_i$ describes a local approximation of the manifold. This motivates the idea of confining the coefficient vectors ${\tilde{\boldsymbol{h}}}$ to a union $$\mathcal{M}_{\text{loc}} = \mathcal{M}_1 \cup \dots \cup \mathcal{M}_I \label{eq:unAffSub}$$ of $I$ affine subspaces $\mathcal{M}_i := \{ \bar{\boldsymbol{h}}_i + {\boldsymbol{V}}_i \boldsymbol{\beta}_i \vert~ \boldsymbol{\beta}_i \in \mathbb{R}^{D_i} \}$ of dimension $D_i$. Each subspace $\mathcal{M}_i$ is defined by its offset $\bar{\boldsymbol{h}}_i$ and its basis matrix $\boldsymbol{V}_i \in \mathbb{R}^{R \times D_i}$. While estimating the offset and the basis of a single global affine subspace, i.e., $I=1$, by, e.g., , is straightforward, it is not obvious how to learn the parameters of the local models. However, as each affine subspace $\mathcal{M}_i$ denotes a local approximation of the manifold, its parameters can be estimated from the surrounding training data samples. Therefore we first assign each training data sample $\tilde{\boldsymbol{h}}_g$ to a specific cluster $\mathcal{U}_i$ by introducing the indicator variable $z_{gi}$ $$z_{gi} := \begin{cases} 1 &\text{if } \tilde{\boldsymbol{h}}_g \in \mathcal{U}_i, \\ 0 &\text{if } \tilde{\boldsymbol{h}}_g \notin \mathcal{U}_i \end{cases}$$ and then use the clustered data for estimating the model parameters. The mean and covariance matrix of the respective cluster $\mathcal{U}_i$ can be estimated by $$\begin{aligned} \bar{\boldsymbol{h}}_i &= \frac{1}{G_i} \sum_{g=1}^{G} z_{gi}{\tilde{\boldsymbol{h}}}_g \\ \boldsymbol{C}_i &= \frac{1}{G_i - 1} \sum_{g=1}^{G} z_{gi} \left[({\tilde{\boldsymbol{h}}}_g - \bar{\boldsymbol{h}}_i) {{({\tilde{\boldsymbol{h}}}_g - \bar{\boldsymbol{h}}_i)}^{\text{T}}} \right] \end{aligned}$$ with $G_i = \sum_{g=1}^{G} z_{gi}$. A local basis matrix $\boldsymbol{V}_i$ can be computed by, e.g., the eigenvectors $\boldsymbol{u}_i$ corresponding to the largest eigenvalues $d_i$ of the parameter covariance matrix $\boldsymbol{C}_i$. Note that one is by no means limited to for extracting the model parameters and can resort to any other algorithm for estimating a linear representation [@SparseAndRedRepr]. Due to the broadband definition of the filter parameters in Eq. , the covariance matrix $\boldsymbol{C}_i$ describes, in addition to the correlation of different taps of one filter $\hat{\boldsymbol{h}}_{pq}$, also the correlation between different filters. Note that $I=1$ denotes the special case of dimension reduction by [a single]{} which assumes globally-correlated coefficient vectors, i.e., strong correlation between all samples used as training data. The local affine subspace model relaxes this assumption by requiring only a local correlation, i.e., only subsets of the training data are assumed to be correlated. In [@koren_supervised_2012] it was assumed that the clusters represent local source position variations and the assignment of the samples was given by oracle knowledge. As this oracle knowledge cannot be assumed in general and the resulting assignment is by no means guaranteed to be optimum, we suggest to learn the assignment blindly from the data by unsupervised K-Means clustering [@lloyd_least_1982] which employs a Euclidean affinity measure which can only be assumed to be meaningful in a local neighbourhood of the samples. Local Projection-based [Update]{} Denoising {#sec:prop_alg_intr} =========================================== In the previous section we have introduced the union of $I$ affine subspace models as a low-dimensional approximation of the parameter space of coefficient vectors. Now we will describe how to exploit this knowledge for the general update of the form to become more robust against noise. The proposed algorithm is inspired by the theory of manifold optimization, e.g., [@AbsMahSep2008], in which the main idea is to exploit prior knowledge about the structure of the parameter space, e.g., matrix properties, by computing the steepest descent direction with respect to the metric defined by the manifold. Model Selection {#sec:evidence_maximization} --------------- A powerful method for model selection is given by the evidence maximization framework [@bishop2007; @modSelMagazine]. It suggests to employ the likelihood of each model $$p(\boldsymbol{Y}(m)|\mathcal{M}_i) = \int p(\boldsymbol{Y}(m)|{\tilde{\boldsymbol{h}}}, \mathcal{M}_i) p({\tilde{\boldsymbol{h}}}| \mathcal{M}_i) d{\tilde{\boldsymbol{h}}}, \label{eq:ev_def}$$ given by the evidence of the observations, as selection criterion. By assuming i.i.d. observations $\boldsymbol{y}(n)$, the evidence of block $m$ is defined by $$p(\boldsymbol{Y}(m)| \mathcal{M}_i) := \prod_{n=mL-L+1}^{mL} p(\boldsymbol{y}(n)|\mathcal{M}_i). \label{eq:iidAssEv}$$ Note that the assumption of i.i.d. observations is only a simplifying modelling assumption and its validity depends on the statistical properties of the excitation signal and the system. If we assume a linear Gaussian model for the likelihood [@roweis1999unifying] $$p(\boldsymbol{y}(n)|{\tilde{\boldsymbol{h}}}, \mathcal{M}_i) = p(\boldsymbol{y}(n)|{\tilde{\boldsymbol{h}}})= \mathcal{N}\left( \boldsymbol{y}(n) | {{\tilde{\boldsymbol{X}}}^{\text{T}}}(n) {\tilde{\boldsymbol{h}}}, \boldsymbol{L} \right) \label{eq:likelihood}$$ which is independent of the model $\mathcal{M}_i$ and [further assume]{} a Gaussian prior for each model $\mathcal{M}_i$ $$p({\tilde{\boldsymbol{h}}}| \mathcal{M}_i) = \mathcal{N}\left( {\tilde{\boldsymbol{h}}}| \bar{\boldsymbol{h}}_i, \boldsymbol{C}_i \right), \label{eq:prior}$$ the sample-wise evidence is [given]{} by [@bishop2007] $$p(\boldsymbol{y}(n)| \mathcal{M}_i) = \mathcal{N}\left( \boldsymbol{y}(n)|{{\tilde{\boldsymbol{X}}}^{\text{T}}}(n) \bar{\boldsymbol{h}}_i, \boldsymbol{R}_i(n)\right) \label{eq:evidence}$$ with covariance matrix $$\boldsymbol{R}_i(n) = \boldsymbol{L} + {{\tilde{\boldsymbol{X}}}^{\text{T}}}(n) \boldsymbol{C}_i \tilde{\boldsymbol{X}}(n). \label{eq:covDevEv}$$ We introduced here the input signal matrix ${{\tilde{\boldsymbol{X}}}^{\text{T}}}(n) = {{\boldsymbol{x}}^{\text{T}}}(n) \otimes \boldsymbol{I}_Q\in \mathbb{R}^{Q \times R}$ with $\otimes$ denoting the Kronecker product and $\boldsymbol{I}_Q \in \mathbb{R}^{Q \times Q}$ being the identity matrix, and the observation noise covariance matrix $\boldsymbol{L} \in \mathbb{R}^{Q \times Q}$. Instead of employing the logarithmic evidence $\log p(\boldsymbol{Y}(m)| \mathcal{M}_i)$ of block $m$ as objective function for model selection, we suggest to use the recursive average evidence estimator $$\hat{\mathcal{E}}_i(m) = \lambda~\hat{\mathcal{E}}_i(m-1) + (1-\lambda) ~\log p(\boldsymbol{Y}(m)| \mathcal{M}_i) \label{eq:recEstEv}$$ to reflect the smooth trajectories on the manifolds caused by changes. The recursive averaging factor $\lambda \in [0,1]$ in Eq.  models an exponential weighting of temporally preceding observations and needs to be chosen according the time-variance of the . Finally, the optimum model index ${i^*(m)}$ at block index $m$ is computed by $${i^*(m)} = \operatorname*{argmax}_{ i=1,\dots,I} ~ \hat{\mathcal{E}}_i(m). \label{eq:evMax}$$ We will now aim at interpreting the logarithmic evidence $$\log p(\boldsymbol{y}(n)| \mathcal{M}_i) \stackrel{\text{c}}{=} -\frac{1}{2} \left( \log \det \boldsymbol{R}_i(n) + {{\bar{\boldsymbol{e}}}^{\text{T}}}_i(n) \boldsymbol{R}_i^{-1}(n) {\bar{\boldsymbol{e}}_i(n)}\right) \label{eq:logEv1}$$ of the observed sample $\boldsymbol{y}(n)$ given the model $\mathcal{M}_i$ with the estimated average observation error $$\bar{\boldsymbol{e}}_i(n) = \boldsymbol{y}(n) - {{\tilde{\boldsymbol{X}}}^{\text{T}}}(n) \bar{\boldsymbol{h}}_i$$ and $\stackrel{\text{c}}{=}$ denoting equality up to a constant term. As expected for evidence-based model selection [@bishop2007; @modSelMagazine], Eq.  consists of two terms which trade model complexity, described by $\log \det \boldsymbol{R}_i(n)$, against data fitting, described by ${{\bar{\boldsymbol{e}}}^{\text{T}}}_i(n) \boldsymbol{R}_i^{-1}(n) {\bar{\boldsymbol{e}}_i(n)}$. By additionally assuming uncorrelated observations $\boldsymbol{y}(n)$, the logarithmic evidence reduces to a sum of channel-wise measures $$\log p(\boldsymbol{y}(n)| \mathcal{M}_i) \stackrel{\text{c}}{=} -\frac{1}{2} \sum_{q=1}^{Q} \left( \log \det {r}_{iq}(n) + \frac{|\bar{{e}}_i(n) |^2}{r_{iq}(n)}\right). \label{eq:logEv2}$$ The data-fitting term is given by the weighted sum of the squared average observation errors $\bar{{e}}_i(n)$ of model $\mathcal{M}_i$. As the diagonal terms of the covariance matrix (see Eq. ) $r_{iq}(n)$ denote an estimate of the observation power, we can interpret the data-fitting term as a sum of the channel-dependent instantaneous inverse performance measures [@enzner_acoustic_2014] which are well-known in . Thus, the logarithmic evidence can be seen as an extension of the data-fitting performance measure which additionally penalizes complex models. Efficient Evidence Approximation {#sec:eff_approximation} -------------------------------- As the direct evaluation of the logarithmic evidence by Eq.  is computationally demanding, we will now introduce an efficient approximation based on the low-dimensionality assumption of the subspaces. Therefore, we insert the of the prior covariance matrix $\boldsymbol{C}_i= \boldsymbol{U}_i \boldsymbol{D}_i {{\boldsymbol{U}}^{\text{T}}}_i$ of model $\mathcal{M}_i$ into the second term of the evidence covariance matrix computation $$\begin{aligned} {{\tilde{\boldsymbol{X}}}^{\text{T}}}(n) \boldsymbol{C}_i \tilde{\boldsymbol{X}}(n) &= {{\tilde{\boldsymbol{X}}}^{\text{T}}}(n) \boldsymbol{U}_i \boldsymbol{D}_i^{\frac{1}{2}} \boldsymbol{D}_i^{\frac{1}{2}} {{\boldsymbol{U}}^{\text{T}}}_i \tilde{\boldsymbol{X}}(n) \\ &= {{\tilde{\boldsymbol{X}}}^{\text{T}}}(n) \check{\boldsymbol{U}}_i {{\check{\boldsymbol{U}}}^{\text{T}}}_i \tilde{\boldsymbol{X}}(n) \\ & = \sum_{r=1}^{R} \check{{\boldsymbol{y}}}_{ir}(n) {{\check{{\boldsymbol{y}}}}^{\text{T}}}_{ir}(n), \label{eq:outProdCompCov} \end{aligned}$$ which shows that it can be computed by a sum of outer products. The existence of the matrix square root is guaranteed, due to the symmetry and positive semi-definiteness of [the]{} covariance matrix. Each vector $\check{{\boldsymbol{y}}}_{ir}(n)$ of the sum is computed by a multiplication of the input signal matrix with a scaled eigenvector $\check{\boldsymbol{u}}_{ir} = {\boldsymbol{u}}_{ir} \sqrt{d_{ir}}$ of the prior covariance matrix $\boldsymbol{C}_i$. As each [matrix-vector]{} product $ \check{{\boldsymbol{y}}}_{ir}(n) = {{\tilde{\boldsymbol{X}}}^{\text{T}}}(n) \check{\boldsymbol{u}}_{ir}$ corresponds to a linear convolution of the input signals with a scaled eigenvector, i.e., eigenfilter, it can be efficiently computed by an overlap-save block processing structure. The latter also holds for the computation of the estimated average observation ${{\tilde{\boldsymbol{X}}}^{\text{T}}}(n) \bar{\boldsymbol{h}}_{i}$ (see Eq. ). Furthermore, as we originally assumed the existence of a lower-dimensional subspace (see Sec. \[sec:parModels\]), the ordered eigenvalues $d_{ir}$ with $r=1,\dots,R$ of the covariance matrix $\boldsymbol{C}_i$ are assumed to exhibit a pronounced decay of magnitude. Hence, it is reasonable to approximate Eq.  by the $K_i = D_i$ largest terms corresponding to the dominant eigenvalues. Note that often $K_i$ can be chosen much smaller compared to $D_i$, i.e., $K_i \ll D_i$, as the first $K_i$ eigenfilters provide sufficient discrimination for model selection. This allows for computationally efficient low-rank evidence approximations. Projection {#sec:proj} ---------- As each sub model $\mathcal{M}_i$ denotes an affine subspace of $\mathbb{R}^R$, the parameter vector ${\tilde{\boldsymbol{h}}}^{{p}_i}$ resulting from orthogonal projection onto $\mathcal{M}_i$ reads (see, e.g., [@strang2006linear]) $${\tilde{\boldsymbol{h}}}^{{p}_i} = \bar{\boldsymbol{h}}_i + \boldsymbol{P}_i \left({\tilde{\boldsymbol{h}}}- \bar{\boldsymbol{h}}_i \right) \label{eq:projEstimate}$$ with the rank-deficient projection matrix $$\boldsymbol{P}_i = {\boldsymbol{V}_i} ({{{\boldsymbol{V}}_i}^{\text{T}}} {\boldsymbol{V}_i})^{-1} {{{\boldsymbol{V}}}^{\text{T}}}_i. \label{eq:defProj}$$ Note that the projection matrix $\boldsymbol{P}_i$ depends only on the training data and can thus be computed a priori. Algorithmic Description {#sec:probAlg} ----------------------- Alg. \[alg:prop\_alg\_descr\] gives a detailed description of the proposed for . For each block of observations, indexed by $m$, the evidence estimates of all models $\mathcal{M}_i$ are updated by Eq. . Hereby, the evidence $p(\boldsymbol{Y}(m)| \mathcal{M}_i)$ of block $m$, given model $\mathcal{M}_i$, is efficiently computed by an overlap-save processing and the low-rank evidence approximation derived in Sec. \[sec:eff\_approximation\]. If the optimum model index $i^*(m)$ has changed relative to the previous block, the previous parameter estimate ${\tilde{\boldsymbol{h}}}(m-1)$ is projected onto the optimum affine subspace $\mathcal{M}_{i^*(m)}$ by Eq. . This ensures that the updated estimate will be confined to $\mathcal{M}_{\text{loc}}$. Subsequently, the parameter update $\Delta {\tilde{\boldsymbol{h}}}(m)$ is computed by a suitable algorithm and projected onto the optimum affine subspace by multiplication with the projection matrix $\boldsymbol{P}_{i^*(m)}$ (see Eq. ). Finally, the projected update is used for optimizing the adaptive filter coefficient vector (see Eq. ). Update evidences of all $I$ models by Eq.  Compute optimum model $\mathcal{M}_{i^*(m)}$ by Eq.  Project ${{\tilde{\boldsymbol{h}}}}(m-1)$ onto opt. aff. subspace by Eq.  Compute parameter update $\Delta {\tilde{\boldsymbol{h}}}(m)$ Project parameter update: $\Delta {\tilde{\boldsymbol{h}}}(m) \gets \boldsymbol{P}_{i^*(m)} \Delta {\tilde{\boldsymbol{h}}}(m)$ Experiments {#sec:experiments} =========== In this section we will evaluate the proposed algorithm in a simulated environment with respect to its performance in noisy scenarios. Therefore, we consider an acoustic system identification scenario with $Q=2$ microphones of $10$ cm spacing and a single source, i.e., $P=1$, located on a sector of a sphere with a radius of $1.3$ m, an azimuth angle range $\theta \in [30\degree, 150\degree]$ and an elevation angle range $\phi \in [-5\degree, 50\degree]$. All $PQ$ $\boldsymbol{h}_{pq}$ have been simulated according to the image method [@allen1979image; @habets2010room] with maximum reflection order for a room of dimension $[6,~5,~3.5]$ m with a reverberation time of $T_{60}=0.3$ s, a sampling frequency of $f_s=8$ kHz and an length of $W=4096$ samples. The observed microphone signals have been sampled from the Gaussian density $\boldsymbol{y}(n) \sim \mathcal{N}(\boldsymbol{d}(n), \boldsymbol{L})$ with $\boldsymbol{d}(n) = {{\boldsymbol{H}}^{\text{T}}} \boldsymbol{x}(n) \in \mathbb{R}^Q$ denoting the true source image at the microphones and $\boldsymbol{H}$ being the acoustic transmission matrix which includes the true $\boldsymbol{h}_{pq}$ analogously to Eq. . The noise covariance matrix $\boldsymbol{L}$ is a scaled identity matrix with the scale factor determined by the . For assessing the performance of the proposed algorithm, we introduce the signal-dependent average measure $$\text{ERLE} = \frac{1}{ (N_2 - N_1 + 1)Q} \sum_{n=N_1}^{N_2} \sum_{q=1}^Q \left(\frac{d_q(n)^2}{(d_q(n) - \hat{y}_q(n))^2} \right) \label{eq:erleDef}$$ and the signal-independent average system mismatch $$\Upsilon = \frac{1}{(M_2 - M_1 + 1)} \sum_{m=M_1}^{M_2} \Upsilon(m) \label{eq:systMisDef}$$ which is computed by the temporal average of the block-dependent system mismatch $$\Upsilon(m) = \frac{1}{PQ} \sum_{p,q=1}^{P,Q} \left( \frac{||{\boldsymbol{h}_{pq} - \hat{\boldsymbol{h}}_{pq}(m)||_2^2}}{||{\boldsymbol{h}}_{pq}||_2^2} \right). \label{eq:systMisDef1}$$ Note that, as the adaptive filter length $L$ is usually much smaller than the true filter length $W$ of the physical system to be modelled, we only use the first $L$ taps of $\boldsymbol{h}_{pq}$ to obtain an estimate of the attainable system mismatch. The observed signal that is caused by the remaining $W-L$ taps of the true acts as an error in the introduced signal model Eq.  and results in an upper bound for the signal-dependent ERLE measure. It corresponds to the excess error in statistically optimum filtering [@haykin_2002]. As pointed out in Sec. \[sec:prop\_alg\_intr\], the presented method is not tied to any specific algorithm. In this paper we employ, as a fast-converging state-of-the-art algorithm, the [@buchner_generalized_2005] which represents a computationally efficient optimization of the well-known block-recursive least-squares cost function in the frequency domain. For applications the is equivalent to the popular [@haykin_2002] with a recursive estimation and an additional data-dependent dynamical regularization. We use a filter length of $L=1024$ and no block overlap, a constant step size of $\mu=1$, a recursive averaging factor of $\nu=0.9$ and the dynamical regularization parameters $\delta_{\text{max}} = \delta_0=1$. Note that for stationary noise and non-stationary excitation signals, e.g., speech, is still beneficial due to the time-varying . In the following we will evaluate the proposed algorithm against two baselines, i.e., the raw and a . The algorithm is a special case of the with $I=1$. The training data for learning the model consisted of $G=5000$ which were simulated according to randomly drawn source positions. The global affine subspace dimension is set to $D_1=550$ which showed good overall performance. The algorithm consists of $I=40$ clusters of identical local dimension . The cluster assignment was learned by the K-Means algorithm [@lloyd_least_1982; @Arthur07]. Furthermore, the evidence of each model $\mathcal{M}_i$ was approximated by the $K_i=5$ most dominant eigenfilters (see Sec. \[sec:eff\_approximation\]). Fig. \[fig:resResultsTemporal\] shows the block-dependent system mismatch $\Upsilon(m)$ of all algorithms for different types of input signals, i.e., stationary and speech signals, and a of $-5$ dB. For each type of input signal we have averaged $\Upsilon(m)$ over $50$ independent Monte Carlo experiments which are defined by randomly drawing the source position and the source signals from the respective models. This limits the influence of a specific input signal and source position. As speech source signals we employed $20$ different talkers reading out random concatenations of IEEE Harvard sentences [@uwnu_corpus]. As can be concluded from Fig. \[fig:resResultsTemporal\] all algorithms reach their steady-state estimate after approximately $3$ s. While the steady-state performance of the improves only slightly in comparison to the , the results in a significant improvement for both types of excitation signals. By comparing to speech excitation, we observe that shows consistently approximately $10$ dB smaller system mismatch than speech for all algorithms. This reflects the well-known difference in convergence behaviour of adaptive filters caused by the nonstationarity and nonwhiteness of speech signals [@haykin_2002; @buchner_generalized_2005; @breiningAEC]. While for this demanding scenario the state-of-the-art algorithm is not capable of achieving a sufficient system identification performance anymore, the proposed achieves an average system mismatch of after convergence. Additionally, by comparing the initial convergence phases of the algorithms, we observe an almost instantaneous gain of the which is caused by the projection on the estimated affine subspace. This results in superior system identification performance even during the early convergence phase, i.e., the first second. In Fig. \[fig:resResults\] we compare the respective algorithms for different levels in terms of average ERLE and system mismatch. The results are averaged over $10$ s of excitation and $15$ s of speech excitation and $50$ independent Monte Carlo experiments. The respective limits of the sums in Eqs.  and , i.e., $N_1, N_2, M_1, M_2$, are chosen to divide the signals into two parts of equal length. This allows to assess the [Convergence Phase (*CP*)]{}, i.e., the first part, and the Steady-State (*SS*), i.e., the second part, independently. As can be concluded from Fig. \[fig:resResults\] the proposed method significantly outperforms the for all levels in terms of steady-state performance for both and system mismatch $\Upsilon$. This suggests an efficient denoising of the update in low- applications while still preserving a sufficient model flexibility for precise system identification in high- scenarios. Additionally, by comparing the to the algorithm, one can observe the advantage of assuming only local linearity compared to the global linear approach which lacks the aforementioned trade-off opportunity. Finally, we observed that the optimum subspace dimensions $D_i$ are strongly related to the respective which would [allow]{} even higher performance improvements by choosing the signal-dependent optimum for each scenario. Summary and Outlook {#sec:summary_and_outlook} =================== In this paper we presented a novel method for improved in noisy environments by exploiting prior knowledge about the space of for a given acoustic scenario. The proposed method is based on the projection of the parameter update onto an affine subspace which is selected by a novel computationally efficient computation of the associated evidence. The benefit of the proposed update denoising for a state-of-the-art algorithm was corroborated by simulated experiments. Future research aims at evaluating the benefit of various dictionary learning algorithms in comparison to for estimating the model parameters. Furthermore, probabilistic mixtures of subspace models, e.g., [@tipping_mixtures_1999], are of interest to improve the unsupervised clustering of the training data in Sec. \[sec:parModels\]. Finally, an adaptive estimation of the noise variances by, e.g., an framework, and an adaptive computation of the optimum subspace dimension appears to be promising for non-stationary noise signals.
--- abstract: 'The theory of random graphs is being applied in recent years to model neural interactions in the brain. While the probabilistic properties of random graphs has been extensively studied in the literature, the development of statistical inference methods for this class of objects has received less attention. In this work we propose a non-parametric test of hypotheses to test if two samples of random graphs were originated from the same probability distribution. We show how to compute efficiently the test statistic and we study its performance on simulated data. We apply the test to compare graphs of brain functional network interactions built from eletroencephalographic (EEG) data collected during the visualization of point light displays depicting human locomotion.' author: - | Andressa Cerqueira, Daniel Fraiman,\ Claudia D. Vargas and Florencia Leonardi bibliography: - './references.bib' title: A test of hypotheses for random graph distributions built from EEG data --- Introduction ============ The brain consists in a complex network of interconnected regions whose functional interplay is thought to play a major role in cognitive processes [@aertsen; @friston93; @vander10]. Based on an elegant representation of nodes (vertices) and links (edges) between pairs of nodes where nodes usually represent anatomically defined brain regions while links represent functional or effective connectivity [@bullmore2009], random graph theory is progressively allowing to explore properties of this sophisticated network [@calmels; @fraiman]. Such properties have been used so far to infer, for instance, about effects of brain lesion [@wang], ageing [@achard; @wu; @meunier] and neuropsychiatric diseases (for a recent review, see [@bassett09]). From a theoretical point of view, the most famous model of random graphs is the Erdös-Reny model [@erdos-renyi-1960] (introduced by Gilbert in [@gilbert1959]), where the edges of the graphs are independent and identically distributed Bernoulli random variables. Besides its simplicity, this model continues to be actively studied and new properties are being discovered (see for example [@chatterjee2011] and references therein). From the applied point of view the most popular model is the Exponential Random Graph Model (ERGM) that has emerged mainly in the Social Sciences community (see [@rinaldo2009] and references therein). Notwithstanding the crescent interest of the scientific community in the graph theory applications, the development of statistical techniques to compare sets of graphs or network data is still quite limited. Some recent works have addressed the problem of maximum likelihood estimation in ERGM ([@rinaldo2009; @snijders2010; @chatterjee2013]), but the testing problem has been even less developed. As far as we know, the testing problem is restricted only to the identification of differences in some one dimensional graph property ([@fraiman; @stam; @ibanez]). At this point it is important to remark that the number of different graphs with $v$ nodes grows as fast as $2^{v(v-1)/2}$ which in practice is far much larger than a typical sample size analyzed. This is the reason why the testing problem is difficult and relevant given that the graph space has no total order. In this paper we propose a goodness-of-fit test of hypothesis for random graph distributions. The statistic is inspired in a recent work [@busch2009] where a test of hypothesis for random trees is developed. We show how to compute the test statistic efficiently and we prove a Central Limit Theorem. The test makes no assumption on the specific form of the distributions and it is consistent against any alternative hypothesis that differs form the sample distribution on at least one edge-marginal. In a simulation study we show the efficiency of the test and we compare its performance with the simultaneous testing of the edge-marginals. We also apply the test to compare graphs built from electroencephalographic (EEG) signals collected during the observation of videos depicting human locomotion. Definition of the test ====================== Let $V$ denote a finite set of vertices, with cardinal $|V|=v$, and let ${\mathbb G}(V)$ denote the set of all simple undirected graphs over $V$. We identify a graph $g=(V,E)$ with the indicator function $g_{ij}={\mathbf 1}\{(i,j)\in E\}$. Given a graph $g\in{\mathbb G}(V)$, we denote by ${\mathbf 1}-g$ the graph defined by $({\mathbf 1}-g)_{ij} = 1-g_{ij}$. In order to measure a discrepancy between two graphs $g,g'\in{\mathbb G}(V)$ we introduce a distance given by $$D(g,g') = \sum_{ij} (g_{ij} - g'_{ij})^2 \,.$$ Here and throughout the rest of the paper summations will refer to the set of vertices $(i,j)\in V^2$ such that $i<j$ (because $g_{ij}=g_{ji}$). Given a set of graphs $\mathbf{g} = (g^1,\dotsc, g^n)$ and a graph $g\in {\mathbb G}(V)$, we denote by $\bar D_{\mathbf{g}}(g)$ the mean distance of graph $g$ to the set $\mathbf{g}$; that is $$\bar D_{\mathbf{g}}(g) = \frac1n \sum_{k=1}^n D(g,g^k)\,.$$ We also define the function $\overline{{\mathbf{g}}}\colon V^2\to[0,1]$, the *mean* of ${\mathbf{g}}$, by $$\overline{\mathbf{g}}_{ij}\; =\; \frac1n\sum_{k=1}^n g^k_{ij}\,.$$ Assume $g$ is a random graph with distribution $\pi$. Denote by $\pi_{ij}=\pi(g_{ij}=1)$ and let $\Sigma$ denote the covariance matrix of $\pi$. Given another probability distribution $\pi'$ defined on ${\mathbb G}(V)$, we are interested in testing the hypothesis $$\label{hip} H_0\colon \pi=\pi'\quad\text{ versus }\quad H_A\colon \pi\neq\pi'\,.$$ Given an i.i.d sample of graphs $\mathbf{g} = (g^1,\dotsc, g^n)$ with distribution $\pi$, we define the one-sample test statistic $$\label{onesample} W(\mathbf{g}) = \max_{g\in {\mathbb G}(V)} \,| \bar D_{\mathbf{g}}(g) - \pi' D(g,\cdot)\,|\,,$$ where $\pi'D(g,\cdot)$ denotes the mean distance of graph $g$ to a random graph with distribution $\pi'$ and is given by $$\pi'D(g,\cdot) = \sum_{g'\in{\mathbb G}(V)} D(g,g')\pi'(g')\,.$$ In the same way, given two samples $\mathbf{g} = (g^1,\dotsc, g^n)$ and $\mathbf{g'} = (g'^1,\dotsc, g'^m)$ with distributions $\pi$ and $\pi'$ respectively, we define the two-sample test statistic $$\label{twosample} W(\mathbf{g},\mathbf{g'}) = \max_{g\in {\mathbb G}(V)} \,| \bar D_{\mathbf{g}}(g) - \bar D_{\mathbf{g'}}(g)\,|\,.$$ At first sight the computation of (\[onesample\]) or (\[twosample\]) is prohibited for even a small number of vertices. But as we show in the following proposition, it is possible to compute the test statistic in $O(v^2(n+m))$ time. \[main-teo\] For the one-sample test statistic we have that $$\label{wg} W(\mathbf{g}) \;=\; \sum_{ij} \;\bigl|\, \overline{\mathbf{g}}_{ij}- \pi'_{ij} \,\bigr|\,.$$ Analogously, for the two-sample test statistic we have that $$\label{wgg} W(\mathbf{g},\mathbf{g'}) \;=\; \sum_{ij} \;\bigl|\, \overline{\mathbf{g}}_{ij}- \overline{\mathbf{g}}'_{ij} \,\bigr|\,.$$ As a corollary of this proposition we prove the following result about the asymptotic distribution of the test statistic. Let $\Pi=(\pi_{ij})$, $\hat\Pi=(\overline{\mathbf{g}}_{ij})$ and $\hat\Pi'=(\overline{\mathbf{g}}'_{ij})$. Then we can write $W(\mathbf{g})=\|\hat\Pi -\Pi\|$ and $W(\mathbf{g},\mathbf{g'})=\|\hat\Pi -\hat\Pi'\|$, where $\|\cdot\|$ denotes the vectorized 1-norm. \[main-cor\] Under $H_0$, for the one-sample test statistic we have that $$\sqrt{n}\bigl(\hat\Pi-\Pi\bigr) \xrightarrow[n \rightarrow \infty]{D} N(0,\Sigma)\,.$$ Analogously, for the two-sample test statistic we have that $$\sqrt{\frac{nm}{n+m}}\bigl(\hat\Pi-\hat\Pi'\bigr) \xrightarrow[n,m \rightarrow \infty]{D} N(0,\Sigma)\,.$$ The proofs of these results are postponed to the Appendix. Assuming the distribution of $W$ under the null hypothesis is known, the result of the test at the significance level $\alpha$ is $$\text{Reject }H_0\quad\text{ if }\quad W(\mathbf{g}) > q_{1-\alpha}\,,$$ where $q_{1-\alpha}$ is the $(1-\alpha)$-quantile of the distribution of $W$ under the null hypothesis. The result of the test for the two-sample case is obtained in the same way replacing $W(\mathbf{g})$ by $W(\mathbf{g},\mathbf{g'})$. By the form of the resulting test statistic, given in and , and Corollary \[main-cor\] we can deduce that the test is consistent against any alternative hypothesis $\pi'$ with $\pi'_{ij}\neq\pi_{ij}$ for at least one pair $ij$. Performance of the test on simulated data ========================================= In this section we present the results of a simulation study in order to evaluate the performance of the test . In the first simulation example we compute the power function of the one-sample test of a (modified) Erdös-Renyi model of parameter $p\in(0,1)$ with $v=10$ vertices, taking as null hypothesis the classical Erdös-Renyi model with $p_0=0.5$. In the modified model, a percentage $q$ of the edges of the graph (previously chosen) are independent Bernoulli variables with parameter $p$, and the remaining edges are taken with parameter $p_0$ as in the null model. The power function of the test as a function of $p$ and for different values of $q$ is presented in Fig. \[fig:1a\]. The sample size was $n=20$ and the quantile of the distribution of $W$ (under $H_0$) was computed as the empirical $0.95$ quantile of a simulation with $p_0=0.5$ and $10.000$ replications. Even for a somehow small proportion of $25\%$ of different edge probabilities and a small sample size the test performs well and the power function approaches 1 when the absolute difference $|p-p_0|$ grows. In order to compare our results with a classical method, we performed simultaneous hypothesis tests on the edge occurrences by using Bonferroni correction (BC). In this case exact critical regions were obtained from the Binomial distribution. For all values of $q$, the $W$ test performs better than the BC procedure, as shown in Fig. \[fig:1a\]. Alternatively, in Fig. \[fig:1b\] we show a comparison of the power function of a pure Erdös-Renyi model ($q=100\%$) but with different graph sizes (given by the number of nodes $v$). As in the first example we took as null model the Erdös-Renyi model of parameter $p_0=0.5$. In all the simulations the sample size was $n=20$. In this case, as it can be expected the power function is closer to 1 as the number of vertices increases because there is more evidence against the null hypothesis. We emphasize the good performance of the $W$ statistic taking into account the small sample size and the number of possible graphs (that in the case of $v=10$ is $|{\mathbb G}(V)| = 2^{55}$ graphs). In the same figure we show the comparison with the power functions for the BC procedure. For all values of $v$ considered the $W$ statistic outperforms the simultaneous tests with Bonferroni correction. Moreover, the test proposed here gains power as the number of vertices grows meanwhile the BC procedure decreases its power. Finally, we focus our attention in the power function for the Exponential Random Graph Model (ERGM). In this model, the probability of a graph $g$ is given by $$\label{prob} \pi(g|\theta)=\frac{\exp(\theta\cdot S(g))}{z(\theta)}\,,$$ where $\theta=(\theta_1,\dots,\theta_k)$ is the parameter vector, $S(g)$ is a vector of $k$ statistics computed from $g$ (e.g. the number of edges, degree statistics, triangles, etc.) and $z(\theta)$ is the normalizing constant. Depending on the particular $S(g)$ function and parameter vector $\theta$, the model favors graphs with distinct small structures. For example, the so called edge-triangle model, where $S(g)= (n_e(g),n_t(g))$ is the number of edges and the number of triangles present in $g$, penalizes (when $\theta_2<0$) or favors (when $\theta_2>0$) the appearance of triangle structures on the graph. The statistic $S(g)$ can be simply computed from $g$ by the formulas $n_e(g)=\sum_{ij}g_{ij}$ and $n_t(g)=\sum_{ijk}g_{ij}g_{jk}g_{ik}$. In the model with $\theta_1\geq0$ and $\theta_2>0$ ($\theta_1\leq0$, $\theta_2<0$) it happens that the graph with highest probability (eq. \[prob\]) is the complete (null) graph where all edges are present (absent). When $\theta_1<0$ and $\theta_2>0$, the model favors the appearance of triangles in the graphs but penalizing graphs with too many edges. This model is very sensitive to the values of the parameter vector and is not uniquely determined, meaning that different values of the parameter vector can give rise to the same probability distribution. Even more, for $\theta_1\in \mathbb R$ and $\theta_2>0$, when the number of nodes increases, the ERGM model is closed in distribution to a Erdös-Renyi model (cf. [@chatterjee2013] and references therein). Another well studied model is the edge-2star model, defined by $S(g)=(n_e(g),n_s(g))$, with $n_s(g)=\sum_{ijk}g_{ij}g_{jk}$, which generates small graphs with nodes of degree 2 or more (less) for $\theta_2>0$ ($\theta_2<0$). In this last model there is no “incentive” for the two nodes that join one of degree 2 to be linked creating a triangle structure. In our first evaluation of the $W$ statistic for the ERGM we consider the edge-triangle model with parameter vector $(\theta_1,\theta_2)=(-1,\theta)$ and two different values of $v$. To understand the behavior of this model as a function of the parameter vector, we first compute the density of edges $p$ for each value of $\theta$ ranging from $-0.5$ to $1$, the results are summarized in Fig. \[fig:theta-p\]. We can observe that the density of edges grows very fast in the interval $(0.5,0.8)$ for $v=8$ and $(0.3,0.5)$ for $v=12$ and as we will see, this fact is relevant for the behavior of the power function. To consider moderate null models (far from being the full graph or the null graph) we take for each value of $v$ the corresponding value of $\theta$ given a density of edges approximately equal to $p=0.6$, this corresponds to $\theta_2=0.63$ for $v=8$ and $\theta_2=0.38$ for $v=12$. For each one of these null models, we computed the $(1-\alpha)$-quantile of the distribution of $W$ under $H_0$ for the one sample test statistic , using 2000 replications of $W$ with sample size $n=20$. Then we computed the power function of the test against any hypothesis with $\theta_2=\theta$, with $\theta$ ranging from $-0.5$ to $1$. We can see that the power functions grow very fast to 1 and this is a consequence of $\theta_2$ in the null model being in the interval where $p$ grows very fast. The behavior of the power function is very anomalous if we take as null model a value of $\theta_2$ belonging to a flat region of $p$, because in these cases a big difference in $\theta$ does not imply a big difference in $p$, and this is determinant for the value of the power function. In our second example we took the same null models for each one of the two cases $v=8$ and $v=12$ and computed the power function for the edge-2star model with parameter vector $(\theta_1,\theta_2)=(-1,\theta)$, with $\theta$ varying between $-0.6$ and $0.6$. As in our previous example, we computed the density of edges $p$ for each value of $\theta$ (Fig. \[fig:theta-p-kstar\]). In these cases the power function also converges very fast to 1 because the value $p=0.6$ of the null hypothesis lies in a region where the density of edges in the edge-2star model also grows very fast as a function of $\theta$. Discrimination of EEG brain networks ==================================== ![First two phases of the stimulus used in the experiment and the steps to obtain the samples of graphs from the EEG signal. []{data-label="fig:estimulo"}](estimulus3) The data analyzed in this section where first presented in [@ghislain]. A total of sixteen healthy subjects (29.25$\pm$6.3 years) with normal or corrected to normal vision and with no known neurological abnormalities participated in this study. The study was conducted in accordance with the declaration of Helsinki (1964) and approved by the local ethics committee (Comité de Ética em pesquisa do Hospital Universitário Clementino Fraga Filho, Universidade Federal do Rio de Janeiro, 303.416). The stimulus used in the experiment is composed by 10 white luminous points with black background, that represent 10 markers of the human body (head, shoulder, elbow, hand, hip, knee and ankle). The animation of these points permitted a vivid perception of a walker’s movement (which we call biological movement). The stimulus has a total length of 5200ms and is composed by 3 different phases: the visible phase (0 - 1600ms) represents the individual walking, the occlusion phase (1600ms - 3900ms) where the luminous points disappear behind a black wall and the phase of reappearance (3900ms - 5200ms) where the individual is again visible and continues walking. A second stimulus employed in that study consisted on a permuted version of the point lights, thus destroying the gestalt of the human walker motion. This stimulus is called non-biological movement. The results presented in this paper only consider the visible and the occlusion phases of the experiment (0 - 3900ms), a representation of the stimuli can be observed in the top of Fig. \[fig:estimulo\]. The EEG activity was registered using a BrainNet BNT 36 (EMSA) that consists of twenty Ag/AgCl electrodes distributed in the scalp of the individual. To study the brain response to the stimulus, the animations were shown in two blocks with a five-minute inter-block interval. Each block consisted of 25 biological movement and 25 non-biological movement stimuli presented randomly. Each stimulus was displayed for 1.3 seconds, followed by an inter-stimulus interval (ISI) of 5 seconds. In each trial, a fixation cross appeared in the last second of the ISI. A total of 100 point light animations were displayed (2 blocks, 2 conditions \[biological and non-biological movement\], 25 repetitions). To construct the brain functional networks, for each subject, phase and repetition of the experiment we first computed a Spearman correlation between each pair of electrodes for each temporal window $[t, t+333ms]$, for values of $t$ varying every 16.66ms (this corresponds to the *interaction criterion* in Fig. \[fig:estimulo\]). The series of correlations for each pair of electrodes $ij$ (and specific for each subject, phase and repetition) will be denoted by $\{\rho_{t}^{ij}\colon t=t_1,\dots,t_n\}$. For the construction of the graphs we computed a threshold for each pair of electrodes $ij$ based on this series of correlations and we put an edge between these electrodes if the absolute value of the correlation for a given time $t$ was above this threshold (this step corresponds to the *network criterion* in Fig. \[fig:estimulo\]). That means to say that for each pair of electrodes we selected a different threshold value, and the selection of this threshold was done in the following way. Let $c$ be a constant, $0<c<1$, and let $q^{ij}_1$ and $q^{ij}_3$ denote the first and third quartiles of the series of correlations $\{\rho_{t}^{ij}\colon t=t_1,\dots,t_n\}$. For a given time $t$ define $$g_{ij}^{t}=\left \{ \begin{array}{cl} 1, & \mbox{if } \rho^{ij}_{t}\geq \mathrm{max}(c,q^{ij}_3) \mbox{ or } \rho^{ij}_{t} \leq \mathrm{min}(-c,q^{ij}_1)\,; \\ 0, & c.c. \end{array} \right.$$ In this way, the graph of interactions for time $t$ will be given by $g^t=(g^t_{ij})_{1 \leq i < j \leq 20}$. The rationality of the criterion proposed here is that the graphs constructed in this way select the edges between electrodes that behaves similarly from a statistical point of view, and this is done by imposing the first and third quartile condition. Each correlation between two electrodes fluctuates in time, then for a given time $t$ we select the ones that are too small (less than $q^{ij}_1$) or large (greater than $q^{ij}_3$). It is interpreted as follows, a given interaction grows if the two brain regions (principally responsible of the signal) are interacting in an excitatory way feeding back the process, or the interaction can decrease if there exist an inhibitory interaction between them. Both changes are captured by our criterion. The extra condition greater (or less) to the value $c$ ($-c$) is just for obtaining statistical significant correlations. The value chosen for $c$ in this study is 0.5. The samples of graphs constructed with our method consist of 132 graphs for the visible phase and 142 for the occlusion phase of biological movement, for each temporal window (after deletion of spurious repetitions, and considering all subjects). In the same way, we obtained 132 graphs for the visible phase and 137 for the occlusion phase of the non-biological movement, for each temporal window. To perform the tests we selected four non-overlapping windows on each phase, $V_1-V_4$ in the visible phase and $O_1-O_4$ in the occlusion phase (see the top of Fig. \[fig:estimulo\]). We first tested the samples corresponding to visible vs. occlusion windows; that is we tested $V_1$ vs. $O_1$, $V_2$ vs. $O_2$ and so on, for biological and non-biological movement. To compute the $p$-values, we used a permutation procedure [@manly2007]; that is, for each pair of samples we extract two random subsamples from the pooled sample, with the same samples sizes of the original ones. Then we compute the test statistic for the subsamples extracted in this way, and we replicate this procedure 1000 times. The estimated $p$-value is therefore the empirical proportion of values in the vector of size 1000 built up in this way that are greater than the observed $W$ statistic. The $p$-values obtained for the four tests are reported in Fig. \[tab:pvalor\_phase\]. We notice that in both types of movement the $p$-values corresponding to the first windows of visible and occlusion phases are significantly smaller than the other $p$-values. The stimulus onset evokes an event related response [@picton2000] in the first window of the visible phase. This response, also known as visual evoked potential, is absent in the occlusion phase where there is no stimulus presentation. As can be observed the $W$ statistic is able to retrieve this difference from the graphs distributions. It is important to remark that the test of hypotheses proposed here does not discriminate which edges in the graphs contribute more significantly to distinguish the two conditions under analysis. Therefore, to compile the results obtained with the test of hypotheses we plotted a summary graph representing each sample by selecting the 30 more frequent edges. Fig. \[fig:graph\_test.phase\](b)-(c) illustrate the graphs corresponding to $V_1$ and $O_1$ in the biological and non-biological movement conditions for which the smallest $p$-values were found, as illustrated in Fig. \[tab:pvalor\_phase\]. Although the plots of the 30 most frequent edges in the first window of the visible and occlusion phases are quite similar for the biological movement condition, comparatively less edges seem present in the occipital electrodes (O1, Oz and O2) and there is a shift towards the right parietofrontal region in the occlusion period. These results could be taken as an evidence of the hypothesis raised in [@ghislain] that the brain would implicitly “reenact” the observed biological movement during the occlusion period (see for more details). For the non-biological condition, the 30 most frequent edges in the first window of the visual phase clearly connect electrodes in the frontal region whereas the 30 most frequent edges in the first window of the occlusion phase connect electrodes in the central-occipital region. Comparing the biological and non biological conditions during the visible phase, Saunier et al. (2013) [@ghislain] found differences both in the right temporo-parietal and in centro-frontal regions. Using functional connectivity, Fraiman et al. (2014) [@fraiman] confirmed that the left frontal regions may play a major role when it comes to discriminating biological x non biological movements. To confirm these findings we proceeded to test the corresponding windows of the biological and non-biological movement conditions. For the visible phase the smallest $p$-value ($<0.03$) was obtained for the third temporal window (time between 668.3ms and 1001.7ms). The occlusion phase does not report significant results in any of the tested windows, see Fig. \[tab:pvalor\_nonbio\]. We emphasize the fact that this is a more sensible problem compared to the comparison of visible and occlusion windows, in the sense that the differences in the stimuli are very subtle. For that reason it is not surprising that with the actual sample sizes we do not obtain very significant results in this case. ----------- -------- -------- ---------------- -------- 1 2 3 4 Visible 0.1014 0.6621 [**0.0295**]{} 0.5910 Occlusion 0.8227 0.8816 0.3764 0.1292 ----------- -------- -------- ---------------- -------- Discussion ========== In this paper we presented a goodness-of-fit non-parametric test inspired in the recent work [@busch2009] for probability distributions over graphs. To our knowledge this is the first nonparametric goodness-of-fit test of hypothesis for random graphs distributions. We derived a closed and efficient formula for the test statistic that implies that the test is consistent against any alternative hypothesis having at least one different marginal distribution over the set of edges. In this case, the simulations show that our test outperforms the simultaneous testing of the marginal means with Bonferroni correction. As in practice the sample sizes are very small compared with the sample space (in our simulations we took $n=20$ for the sample size versus $2^{28}$ of the sample space for graphs with 8 nodes), our test performs very well even for small differences in the marginal distributions. In the real EEG dataset, we showed the potentiality of the $W$ statistic to detect differences in graphs of interaction built from EEG data. Although the main focus of this paper is on simple non-directed graphs, the generalization of the test statistic to other graph structures could be possible. This could be done for example by taking a more general distance function between graphs or by modifying directly the test statistic formula given in Proposition \[main-teo\]. This would generalize the test to include other graph structures or would enable the test statistic to be consistent even for different graph distributions having the same marginals over the edges. Appendix ======== We will prove the proposition only for the one-sample test. The result for the two-sample test can be derived analogously. Denote by $w_{\mathbf{g}}(g)=\bar D_{\mathbf{g}}(g) -\pi'D(g,\cdot)$. Observe that in order to maximize $|w_{\mathbf{g}}(g)|$ in ${\mathbb G}(V)$ it is sufficient to maximize $w_{\mathbf{g}}(g)$ and $-w_{\mathbf{g}}(g)$. We have that $$\begin{aligned} w_{\mathbf{g}}(g)&=& \dfrac{1}{n}\sum_{k=1}^n D(g,g^{k}) - \sum\limits_{g' \in {\mathbb G}(V)}D(g,g')\pi'(g')\,. \nonumber\\\end{aligned}$$ The first sum equals $$\begin{aligned} \frac{1}{n}\sum_{k=1}^n D(g,g^{k}) &=& \frac{1}{n}\sum_{k=1}^n \; \sum_{ij}(g_{ij} - g^k_{ij})^2 \\ &=& \sum_{ij} ( g_{ij} - 2g_{ij}\overline{\mathbf{g}}_{ij} + \overline{\mathbf{g}}_{ij} )\,. $$ The second sum is $$\begin{aligned} \sum\limits_{g' \in {\mathbb G}(V)}D(g,g')\pi'(g') &=& \sum\limits_{g' \in {\mathbb G}(V)}\pi'(g') \sum_{ij}(g_{ij} - g'_{ij})^2 \\ &=& \sum_{ij} ( g_{ij} - 2g_{ij}\pi'_{ij} + \pi'_{ij} )\,. \end{aligned}$$ Therefore we have that $$\begin{aligned} \label{swg} w_{\mathbf{g}}(g)&=& \sum_{ij} (2g_{ij}-1)(\pi'_{ij} -\overline{\mathbf{g}}_{ij} )\,. \end{aligned}$$ As this is a weighted sum, the graph $g^*\in{\mathbb G}(V)$ that maximizes $w_{\mathbf{g}}(g)$ is given by $$\label{gstar} g^*_{ij}= \begin{cases} 1, & \text{ if }\,\overline{{\mathbf{g}}}_{ij} \leq \pi'_{ij} \\ 0, & \text{c.c.} \end{cases}$$ Similarly, the graph $g^{**}\in{\mathbb G}(V)$ that maximizes $-w_{\mathbf{g}}(g)$ is given by $$\label{gstar2} g^{**}_{ij}= \begin{cases} 1, & \text{ if }\,\overline{{\mathbf{g}}}_{ij} \geq \pi'_{ij} \\ 0, & \text{c.c.} \end{cases}$$ Note also that by a direct calculation from (\[swg\]) and the definitions (\[gstar\]) and (\[gstar2\]) we have that $|w_{\mathbf{g}}(g)|=|-w_{\mathbf{g}}(g)|$. Finally, from (\[wg\]) and (\[gstar\]) we obtain $$W({\mathbf{g}})\,=\,\max_{g\in{\mathbb G}(V)} |w_{\mathbf{g}}(g)| \,=\, w_{{\mathbf{g}}}(g^{*})\,=\,\sum_{ij} |\overline{{\mathbf{g}}}_{ij} - \pi'_{ij}|\,.\qedhere$$ This is a direct consequence of the multidimensional Central Limit Theorem (cf. Theorem 11.10 in [@breiman1992]). Acknowledgments {#acknowledgments .unnumbered} =============== F.L. is partially supported by a CNPq-Brazil fellowship (304836/2012-5) and FAPESP’s fellowship (2014/00947-0). This article was produced as part of the activities of FAPESP Research, Innovation and Dissemination Center for Neuromathematics, grant 2013/07699-0, São Paulo Research Foundation. She also thanks L’Oréal Foundation for a “Women in Science” grant.
--- abstract: | We present a re-reduction and expansion of the Bolocam Galactic Plane Survey, first presented by @Aguirre2011 and @Rosolowsky2010. The BGPS is a 1.1 mm survey of dust emission in the Northern galactic plane, covering longitudes $-10{\ensuremath{^{\circ}}}< \ell < 90{\ensuremath{^{\circ}}}$ and latitudes $|b| < 0.5{\ensuremath{^{\circ}}}$ with a typical $1-\sigma$ RMS sensitivity of 30-100 mJy in a $\sim33{\mbox{$^{\prime\prime}$}\xspace}$ beam. Version 2 of the survey includes an additional $\sim20$ square degrees of coverage in the 3rd and 4th quadrants and $\sim2$ square degrees in the 1st quadrant. The new data release has improved angular recovery, with complete recovery out to $\sim80{\mbox{$^{\prime\prime}$}\xspace}$ and partial recovery to $\sim300{\mbox{$^{\prime\prime}$}\xspace}$, and reduced negative bowls around bright sources resulting from the atmospheric subtraction process. We resolve the factor of 1.5 flux calibration offset between the [[v]{}1.0]{} data release and other data sets and determine that there is no offset between [[v]{}2.0]{}and other data sets. The [[v]{}2.0]{}pointing accuracy is tested against other surveys and demonstrated to be accurate and an improvement over [[v]{}1.0]{}. We present simulations and tests of the pipeline and its properties, including measurements of the pipeline’s angular transfer function. The Bolocat cataloging tool was used to extract a new catalog, which includes [8594]{} sources, with [591]{}in the expanded regions. We have demonstrated that the Bolocat 40and 80apertures are accurate even in the presence of strong extended background emission. The number of sources is lower than in [[v]{}1.0]{}, but the amount of flux and area included in identified sources is larger. author: - 'Adam Ginsburg, Jason Glenn, Erik Rosolowsky, Timothy P. Ellsworth-Bowers, Cara Battersby, Miranda Dunham, Manuel Merello, Yancy Shirley, John Bally, Neal J. Evans II, Guy Stringfellow, James Aguirre' title: 'The Bolocam Galactic Plane Survey IX: Data Release 2 and Outer Galaxy Extension' --- Introduction ============ Observations in the millimeter continuum provide the best method to identify a Galaxy-wide sample of star-forming clumps. The emission is optically thin, minimally affected by temperature and can be surveyed over large areas. Unlike targeted observations, blind surveys allow for a complete and systematic study of dense gas clumps. In the past decade, there have been many Galactic plane surveys at millimeter/submillimeter wavelengths, of which the Bolocam Galactic Plane Survey [@Aguirre2011; @Rosolowsky2010a] was the first to be completed and publicly released. ATLASGAL [@Schuller2009a] surveyed the southern Galactic plane at 870 [$\mu \textrm{m}$]{}. The JCMT Galactic Plane Survey, or JPS, will survey the Northern plane at 850 [$\mu \textrm{m}$]{}. In the past 3 years, the Hi-Gal Galactic Plane survey observed the Galaxy from 70 to 500 [$\mu \textrm{m}$]{}with the Herschel Space Observatory, sensing the peak of the dust SED with minimal spatial filtering [@Molinari2010]. The survey has provided access to the peak of the dust spectral energy distribution at modest ($\lesssim40{\mbox{$^{\prime\prime}$}\xspace}$) resolution [@Traficante2011a]. Together, these surveys provide a complete map of long-wavelength dust emission across the Galactic plane. Long-wavelength data are essential for constraining the dust emissivity, one of the free parameters in greybody spectral energy distribution (SED) fits. @Shetty2009b [@Shetty2009a] demonstrated the need for long-wavelength data to accurately determine both $\beta$, the dust emissivity spectral index, and temperature. @Juvela2012d also showed that adding additional wavelengths to an SED fit, even with lower signal-to-noise, significantly reduces the degeneracy in the fit. Millimeter-wave dust emission also has the advantage of being relatively insensitive to temperature. When looking at cold gas, $T\lesssim20$ K, all of the Herschel bands deviate from a Rayleigh-Jeans temperature approximation. Longer wavelength observations are less affected by temperature assumptions. The 1.1 mm band is in many cases the longest wavelength unaffected by free-free emission, providing the least environmentally-biased view of optically thin dust emission and therefore total dust mass. Similarly, at millimeter wavelengths, the dust opacity is low enough that all clumps detected in the BGPS are expected to be optically thin [with the possible exception of Sgr B2; @Bally2010a]. In combination with the weak temperature dependence, this feature of 1.1 millimeter emission allows for the most direct and straightforward estimates of total dust mass. Millimeter-bright dust clumps are generally associated with high-density, star-forming gas. Previous surveys have found cold, massive molecular clouds via the [$^{12}\textrm{CO}$]{}and [$^{13}\textrm{CO}$]{}1-0 lines [@Dame2001a; @Jackson2006a]. However, these clouds are only moderate density, $n({\ensuremath{\textrm{H}_{2}}})\sim10^2-10^3$ [$\textrm{cm}^{-3}$]{}, while dust-detected clumps have typical densities $n({\ensuremath{\textrm{H}_{2}}})\gtrsim10^4$ [$\textrm{cm}^{-3}$]{}[@Dunham2010a]. The dense gas in these clumps is more directly associated with star and cluster formation [@Dunham2011b; @Battersby2010a], allowing for systematic studies of pre-star-forming and star-forming gas. The BGPS [[v]{}1.0]{}data has been public since 2009, and has been used extensively as both a finder chart and a tool to probe Galactic properites. It was used to examine the properties of maser sources [@Pandian2012; @Chen2012], outflow sources [@Ioannidis2012], and high-mass star-forming regions [@Reiter2011; @Battersby2011; @Dunham2011]. It has served as the basis for studies of forming clusters [@Alexander2012a; @Ginsburg2012a] and intermediate-mass stars [@Arvidsson2010a]. The BGPS and other surveys have served as finder charts for large-scale millimeter line studies of the Galactic plane [@Schenck2011; @Schlingman2011; @Ginsburg2011a; @Shirley2013a]. BGPS clumps have been used as the target sample for distance determinations to large cloud populations [@Ellsworth-Bowers2013a]. @Dunham2011b used the BGPS to measure properties of star forming regions as a function of Galactocentric radius. These and many other ongoing and planned studies demonstrate the need for, and benefits of, publicly available blind legacy surveys. This paper presents [[v]{}2.0]{}of the Bolocam Galactic Plane Survey (BGPS), with a complete data release available at [irsa.ipac.caltech.edu/data/BOLOCAM\_GPS/](irsa.ipac.caltech.edu/data/BOLOCAM_GPS/). In Paper I [@Aguirre2011], the initial processing of the BGPS [[v]{}1.0]{}was described in detail. It was noted in Section 5 of @Aguirre2011 that there was a discrepancy between our survey and previously published results. This discrepancy raised the possibility of a flux calibration error in the Version 1 (hereafter [[v]{}1.0]{}) results: we confirm and correct the error in this paper. In addition, we have made significant improvements to the data pipeline, measured important features of the pipeline including its angular transfer function, improved the pointing accuracy, and added new observations. The paper is as follows: We resolve the flux calibration discrepancy in [[. In , we discuss new observations included in the [[v]{}2.0]{}data. describes changes to the data reduction process and new data products. and measure the angular transfer function of the BGPS [[v]{}2.0]{}pipeline and properties of extracted sources respectively.]{}]{} The paper concludes with a brief discussion of the results and a summary. Calibration {#sec:calibration} =========== The original calibration, along with tables of color correction and a detailed treatment of the filter response, are described in @Aguirre2011a. We discuss important changes in [[v]{}2.0]{}in this section. Why was there a multiplicative offset in the [[v]{}1.0]{}data release? {#sec:caloffset} ---------------------------------------------------------------------- In @Aguirre2011, we reported that a ‘correction factor’ of about 1.5 on average was needed to bring our data into agreement with other 1 mm data sets. We discovered that the published [[v]{}1.0]{}BGPS images have a different calibration reported in their FITS headers than was used in processing the data. The calibration used in the released data was borrowed from a previous observing run, during which a different bias voltage was used, and differed from the pipeline-derived calibration by a factor $\approx 1.5$, completely explaining the discrepancy. Comparing [[v]{}1.0]{}and [[v]{}2.0]{}calibration {#sec:v1v2compare} ------------------------------------------------- We checked the data for consistency with the measured calibration offset. In order to compare flux densities in identical sources, we performed aperture photometry on the [[v]{}2.0]{}data based on the locations of [[v]{}1.0]{}sources using both the ‘source masks’ from Bolocat [[v]{}1.0]{}[@Rosolowsky2010] and circular apertures centered on the Bolocat [[v]{}1.0]{}peaks. Source masks, also known as label masks, are images in which the value of a pixel is either 0 for no source or the catalog number if there is a source associated with that pixel. ="2D We measured the multiplicative offset between [[v]{}1.0]{} and [[v]{}2.0]{}by comparing these aperture-extracted fluxes. For each aperture size, we measured the best-fit line between the [[v]{}1.0]{}and [[v]{}2.0]{}data using a total least squares (TLS[^1]) method weighted by the flux measurement errors as reported in the catalogs. The agreement with $S_{{{\rm v}2.0\xspace}}=1.5~S_{{{\rm v}1.0\xspace}}$, as expected based on [[Section \[sec:caloffset\]]{}]{}, is within 10%, although the larger apertures show a slight excess with $S_{{{\rm v}2.0\xspace}}\approx(1.6\mhyphen1.7)~S_{{{\rm v}1.0\xspace}}$. This excess is expected given the improved extended flux recovery in [[v]{}2.0]{}(see Section \[sec:stf\]). The [[v]{}2.0]{}/[[v]{}1.0]{} flux ratio is weakly dependent on the source flux density, with higher [[v]{}2.0]{}/[[v]{}1.0]{} ratios for brighter sources. Comparison to Other Surveys --------------------------- In Section 5.5 of @Aguirre2011, we compared the BGPS [[v]{}1.0]{} data to other data sets from similar-wavelength observations. We repeat those comparisons here using the [[v]{}2.0]{} data and demonstrate that [[v]{}2.0]{}achieves better agreement with other data sets than [[v]{}1.0]{}. Full details of the comparison were given in @Aguirre2011a. We compare to 3 data sets in the same $\sim 1$ mm atmospheric window. Two data sets from MAMBO II, the @Motte2007 [M07] Cygnus X survey and the @Rathborne2006 [R06] IRDC survey, overlap with the BGPS. The SIMBA 1.3 mm survey of the $\ell=44{\ensuremath{^{\circ}}}$ region is the largest survey in the 1 mm band that overlaps with ours [@Matthews2009 M09]. The comparison data sets have angular transfer functions that differ from the BGPS. In order to account for the difference, we allow for a large angular scale offset between the observations. We fit a line of the form $y=m x + b$ to the data, where $x$ and $y$ represent the pixel values gridded to 7.2pixels. The $b$ value allows for a local offset, i.e. a non-zero $b$ value indicates a substantial difference in the angular transfer function. Since such an additive offset is unlikely to apply across the entire observed region, we also fit the offset for small sub-regions in the M07 and M09 data, focusing on DR21 and a region centered on G45.5+0.1 respectively. The results of that comparison are displayed in Table \[tab:FluxComparison\], which includes the original comparison from @Aguirre2011.[^2] BGPS [[v]{}2.0]{}is in much better agreement with the other data sets than [[v]{}1.0]{}, but it retains a significant additive offset, particularly with respect to MAMBO. The additive offset is explained by a difference in the angular transfer function; the MAMBO observing strategy of fast position switching allows structures on the scale of the array to be preserved, while Bolocam’s fast-scan strategy does not. The differing observing strategy explains why there is an additive offset between Bolocam and MAMBO, but no such offset for SIMBA, which was used in a fast-scan mode similar to Bolocam. The varying backgrounds in separate regions account for some of the remaining multiplicative offset. When individual sub-regions are compared, the additive and multiplicative offsets more clearly separate into independent components, i.e. a line with an additive offset is a better fit to the data. To enable a comparison of flux density between the surveys, we must account for the different spectral bandpasses of the instruments. The relative flux density measured between the instruments depends on the spectral index $\alpha_{\nu}$ of the observed source; $\alpha_{\nu}=2$ corresponds to a perfect black body on the Rayleigh-Jeans tail.[^3] In Table \[tab:relflux\] we show the relative flux densities expected for Bolocam, MAMBO and SIMBA; they differ by at most 19% for spectral indices $\alpha_{\nu}<5$. Bolocam flux densities are expected to be higher because Bolocam has a higher effective central frequency than either of the other instruments. In @Aguirre2011, we measured Bolocam [[v]{}1.0]{}/MAMBO and Bolocam [[v]{}1.0]{}/SIMBA ratios in the range $0.66 < R < 0.83$, indicating a clear disagreement between the surveys. With the [[v]{}2.0]{}  data, we measure ratios $0.97 < R_{SIMBA} < 1.08$ and $0.89 < R_{MAMBO} < 0.99$. These numbers still indicate that the BGPS is too faint by $\lesssim20\%$ relative to the expectations laid out in Table \[tab:relflux\], but with a systematic calibration error no better than 20% in each survey, this level of agreement is reasonable. Expansion of the BGPS and Observations {#sec:observations} ====================================== Thirteen nights of additional data were acquired from December 15th, 2009 to January 1st, 2010. The target fields and areas covered are listed in Table \[tab:observations\] as boxes in Galactic latitude and longitude, with position angles to the Galactic plane indicated. The original observations are described in Section 2 of @Aguirre2011. The new target fields were selected from visual inspection of FCRAO OGS [$^{12}\textrm{CO}$]{} integrated maps, @Dame2001 [$^{12}\textrm{CO}$]{} maps, and IRAS 100 [$\mu \textrm{m}$]{}maps. The fields were selected primarily to provide even spacing in RA in order to maximize observing efficiency, and were therefore not blindly selected. Additionally, the Orion A and B and Mon R2 clouds were observed in observing campaigns by collaborators. These complexes are not directly part of the BGPS, but are included in this data release reduced in the same manner as the Galactic plane data. They are much closer than typical BGPS sources and their selection for mapping is very biased, but we include them in the archival data. Parts of the Orion A nebula remain proprietary as of this release, but are expected to be released upon publication of Kauffmann et al (in prep). The California nebula has also been observed and the data published in @Harvey2013a. Finally, some archival CSO data was recovered and added to the BGPS. These data include maps of M16 and M17. M17 is an extraordinarily bright 1.1 mm source that was poorly covered in the BGPS because it is below $b = -0.5$. The Bolocat cataloging tool was run on these new fields and they have been included in the [[v]{}2.0]{}catalog. Some of their properties are displayed in Section [[\[sec:sourceextraction\]]{}]{}. A total of [591]{}new sources not covered in the [[v]{}1.0]{}survey were extracted.[^4] Data Reduction and Data Products {#sec:datareduction} ================================ A brief review of ground-based millimeter observational techniques ------------------------------------------------------------------ Observations at wavelengths longer than 2 [$\mu \textrm{m}$]{}and shorter than 2 cm from the ground are strongly affected by emission and absorption from our own atmosphere. Optical and radio observations from the ground see through a transparent atmosphere, but millimeter observations are dominated by bright foreground emission that dominates the astrophysical signal. This foreground must be removed in order to create maps of astrophysical emission. @Chapin2013a presented a summary of the techniques used to separate astrophysical and atmospheric signals in (sub)millimeter bolometric observations. The Bolocam observations reported in this paper were conducted with a fast-scanning strategy that places some of the ‘fixed’ astrophysical emission at a different sampling frequency than the varying foreground atmosphere. This approach is one of the most efficient and flexible and has been used predominantly over alternatives, such as a nodding secondary, in most recent large-scale observing campaigns [@Aguirre2011; @Schuller2012a]. A variety of different atmospheric removal algorithms have been successfully utilized, but in addition to removing the atmospheric foreground, these approaches remove some of the astrophysical signal. In order to recover signal on angular scales up to the array size, the most commonly used approach for bright Galactic signals is an iterative reconstruction process. This process assembles a model of the astrophysical emission, subtracts it from the observed timestream, and repeats, each time reducing the amount of astrophysical signal that is removed by the algorithm. This general approach was first used on Bolocam data by @Enoch2006b and refined in @Aguirre2011. We directly examine the effects of the data reduction methods below. Sky Subtraction --------------- We compared a few different methods for atmospheric subtraction and astrophysical image reconstruction, but settled on an approach very similar to that used in [[v]{}1.0]{}. This subsection recounts the minor changes from [[v]{}1.0]{}and includes discussion of alternative approaches. The PCA method [@Enoch2006b; @enoch07] with iterative flux density restoration was used for [[v]{}2.0]{}as for [[v]{}1.0]{}. In the PCA atmosphere removal method, the $n$ eigenvectors corresponding to the highest values along the diagonal of the covariance matrix (the most correlated components) are nulled. We nulled 13 PCA components in both [[v]{}1.0]{}and [[v]{}2.0]{}. The selection of 13 components produced the best compromise between uniform background noise and fully restored peak signal. Simulations show that the point source recovery is a very weak function of number of PCA components nulled (nPCA), while extended flux recovery is a strong function of nPCA. However, residual atmospheric signal was substantially reduced with higher nPCA. In [[v]{}2.0]{}, 20 iterations were used instead of the 50 used in [[v]{}1.0]{}; in both surveys, convergence was clearly achieved by 20 iterations, and generally individual iterations are indistinguishable by $\sim5$ iterations. The iterative process adopts the non-negative flux density above some cutoff as a model of the astrophysical sky and subtracts that flux density from the timestream before repeating the atmospheric subtraction. This approach allows large angular scale structures to be recovered by removing them from the timestreams before they can contribute to the correlated signal. The [[v]{}2.0]{}pipeline was more succesful than in [[v]{}1.0]{}at removing negative bowls (see Section \[sec:catalogmatching\] for visual examples). Negative bowls are introduced because the atmospheric subtraction process assumes that the mean level of any timestream, and therefore any map, is zero. The iterative process allows this assumption to be violated, creating maps with net positive signal. The reduced impact of negative bowls is attributed to a few small changes to the pipeline that each slightly mitigate the bowls. 1. The astrophysical model is created by deconvolving the positive emission rather using positive pixels directly. The deconvolution process, which removes sub-beam-scale noise, was made more stable in [[v]{}2.0]{}by performing a local signal-to-noise cut using the noise maps described in Section \[sec:noisemaps\]; in [[v]{}1.0]{}there was no reliable noise map available during the iterative map making process. 2. Better image co-alignment reduced inter-observation spatial offsets. The astrophysical models therefore better reproduced the timestream data. 3. Improvement in the bolometer gain calibration, which is done on a per-observation basis in [[v]{}2.0]{}, improved the convergence of the iterative map maker. These changes are individually minor, but together resulted in significant improvements to the map quality. The quadratic planar fit sky subtraction method discussed in @Sayers2010 was implemented and tested for 1.1mm Galactic plane data in the v2.0 pipeline, but was not used for the final data products. In principle, this method should do a substantially better job at removing smooth atmospheric signal from timestreams than PCA cleaning because it is based on physically expected atmospheric variation. The spatial recovery was better than the aggressive 13-PCA approach, but as with a simpler median subtraction approach (subtracting the median timestream from all bolometers), a great deal of spurious signal from the atmosphere remained in the maps, and the noise properties were highly non-uniform, rendering source extraction difficult. It was also more computationally expensive and did not remove correlated electronic readout noise, which PCA subtraction did. The @Sayers2010 approach is likely more effective at 143 GHz because the atmosphere is better-behaved at lower frequencies. We speculate that it is also more effective for deep extragalactic fields in which more repeat observations of the same field are able to distinguish atmospheric from real signal on the angular scales of the array. Data products ------------- The BGPS data are available from the Infrared Processing and Analysis Center (IPAC) at [irsa.ipac.caltech.edu/data/BOLOCAM\_GPS/](irsa.ipac.caltech.edu/data/BOLOCAM_GPS/). The [[v]{}2.0]{}data products include the pipeline-processed maps and Bolocat label masks as in [[v]{}1.0]{}. In the [[v]{}2.0]{}data release, there are two new map types released: noise maps and median maps. A variant of the noise maps was produced in [[v]{}1.0]{}, while the median maps are an entirely new data product. ### Noise Maps {#sec:noisemaps} Residual bolometer timestreams are automatically generated as part of the iterative map-making process. The residual is the result of subtracting the astrophysical model (which is smooth, noiseless, and non-negative) from the atmosphere-subtracted data timestream. The resulting timestream should only contain the remaining astrophysical noise. However, maps of the residual timestream contain sharp edge features because the astrophysical model is sharp-edged (i.e., transitions from 0 to a non-zero value from one pixel to the next). These sharp transitions are mitigated in the presence of noise. We therefore created noise maps by taking the local standard deviation of the residual map. Pixels in the original map that were not sampled (i.e., represented by `NaN` in the FITS data file) are ignored when computing this local standard deviation and their values are set to be an arbitrarily high number ($100$ Jy/beam) such that pixels near the map edge are assumed to have extremely high noise (which is reasonable, since these pixels are affected by a variety of artifacts rendering them unreliable measurements of the true astrophysical flux). The local noise is computed within a $FWHM=10$-pixel gaussian, which enforces a high noise level within $\sim2{\mbox{$^\prime$}\xspace}$ of the map edge. This method produces good noise maps (i.e., in agreement with the standard deviation calculated from blank regions of the signal map) and was used both within the iterative process and for cataloging. We show the noise per pixel for each half square degree in the inner galaxy in Figure \[fig:noiseperdeg\]. The noise level in each outer-galaxy field is summarized in Figure \[fig:noiseperfield\]. Because the outer galaxy coverage is irregular, we show the noise per observed region rather than dividing the regions into degree-scale sub-regions. Median Maps ----------- Some artifacts (cosmic ray hits, instrumental artifacts) inevitably remained at the end of the process. In order to mitigate these effects, “median maps” were created. The value of each spatial pixel was set to the median value of the timestream points intersecting that pixel; pixels with fewer than 3 data points were set to `NaN`. The noise in the median maps was in some cases lower than that in the weighted mean maps, particularly for fields with fewer total observations. They uniformly have mitigated instrument-related artifacts such as streaking. These maps are released in addition to the weighted-mean maps, which often have higher signal-to-noise. Pointing {#sec:pointing} -------- In order to get the best possible pointing accuracy in each field, all observations of a given area were median-combined using the [montage]{} package, which performs image reprojections, to create a pointing master map [@Berriman2004a]. Each individual observation was then aligned to the master using a cross-correlation technique [@Welsch2004a]: 1. The master and target image were projected to the same pixel space 2. A cross-correlation image was generated, and the peak pixel in the cross-correlation map was identified 3. Sub-pixel alignment was measured by performing a 2nd-order Taylor expansion around the peak pixel This method is similar to the version 1.0 method, but the new peak-finding method proved more robust than the previous Gaussian fitting approach. The [[v]{}1.0]{}Gaussian fitting approach is often used in astronomy (e.g., <http://www.astro.ucla.edu/~mperrin/IDL/sources/subreg.pro>), but it is biased when images are dominated by extended structure. This bias occurs because the least-squares fitting approach will identify the broader peak that represents auto-correlation of astrophysical structure rather than the cross-correlation between the two images. In [[v]{}1.0]{}, we attempted to mitigate this issue by subtracting off a ‘background’ component before fitting the Gaussian peak, but this method was not robust. The improved approach to pointing resulted in typical RMS offsets between the individual frames and the master $\sigma_{\mathbf{x}} \sim 2$ . The improvement in the point spread function is readily observed (see Section \[sec:catalogmatching\]). Pointing Comparison ------------------- We carefully re-examined the pointing throughout the BGPS using a degree-by-degree cross-correlation analysis between the [[v]{}1.0]{}, [[v]{}2.0]{}, and Herschel Hi-Gal 350 [$\mu \textrm{m}$]{}data. The Herschel data were unsharp-masked (high-pass filtered) by subtracting a version of the data smoothed with a $\sigma=120{\mbox{$^{\prime\prime}$}\xspace}$ Gaussian. The result was then convolved with a $\sigma=8.9{\mbox{$^{\prime\prime}$}\xspace}$ Gaussian to match the Herschel to the Bolocam beam sizes. Errors on the offsets were measured utilizing the Fourier scaling theorem to achieve sub-pixel resolution [inspired by @Guizar2008]. The errors on the best-fit shift were determined using errors estimated from the BGPS data and treating the filtered Hi-Gal data as an ideal (noiseless) model. The tools for this process, along with a test suite demonstrating their applicability to extended structures in images, are publicly available at <http://image-registration.readthedocs.org/>. The cross-correlation technique calculated the $\chi^2$ statistic as a function of the offset. For a reference image $Y$ and observed image $X$ with error per pixel $\sigma_{xy}$, $$\chi^2 = \sum \frac{(X-Y(\Delta x,\Delta y))^2}{\sigma_{xy}^2}$$ where $\Delta x$ and $\Delta y$ are the pixel shifts. Because Y is not actually an ideal model but instead is a noisy image, we increase $\sigma_{xy}$ by the rms of the difference between the aligned images, using a corrected $\sigma_c^2 = \sigma_{BGPS}^2 + RMS(X-Y(\Delta x_b,\Delta y_b))$, where $\Delta x_b,\Delta y_b$ are the best-fit shifts. For the majority of the examined 1-square-degree fields, the signal dominated the noise and we were able to measure the offsets to sub-pixel accuracy. A plot of the longitude / latitude offsets between [[v]{}2.0]{} and [[v]{}1.0]{} and Herschel Hi-Gal is shown in Figure \[fig:offsets\]. Table \[tab:ccoffsets\] lists the measured offsets in arcseconds between images for all 1 degree fields from $\ell=351{\mbox{$^\circ$}\xspace}$ to $\ell=65{\mbox{$^\circ$}\xspace}$. The offsets represent the Galactic longitude and latitude shifts in arcseconds from the reference (left) to the ‘measured’ field (right). Table \[tab:ccoffsetsmeans\] shows the means of the columns in Table \[tab:ccoffsets\], weighted by the error in the measurements and by the number of sources. Weighting by the number of sources is used for comparison with other works that attempt to measure the pointing offset on the basis of catalog source position offsets. None of the measured offsets are significant; in all cases the scatter exceeds the measured offset. Addressing the ATLASGAL offset ------------------------------ @Contreras2012 performed a comparison of the Bolocam and ATLASGAL catalogs, identifying a systematic offset between the catalogs of $\Delta\ell=-4.7{\mbox{$^{\prime\prime}$}\xspace}$, $\Delta b = 1.2 {\mbox{$^{\prime\prime}$}\xspace}$. Because the offset is measured between catalog points, the meaning of this measured offset is not immediately clear. In the BGPS maps in the ATLASGAL-BGPS overlap region, there were 12 individual sub-regions ($3{\mbox{$^\circ$}\xspace}\times1{\mbox{$^\circ$}\xspace}$, with $1{\mbox{$^\circ$}\xspace}\times1{\mbox{$^\circ$}\xspace}$ regions in the CMZ) that could have independendent pointing. Because we did not have direct access to the ATLASGAL maps or catalog at the time of writing, we compared the Bolocam [[v]{}1.0]{}and [[v]{}2.0]{}catalogs to each other determine whether the pointing changes in [[v]{}2.0]{}might account for the observed ATLASGAL offset, assuming the [[v]{}2.0]{}pointing is more accurate than the [[v]{}1.0]{}pointing. We performed an inter-catalog match between [[v]{}1.0]{}and [[v]{}2.0]{}, considering sources between the two catalogs to be a match if the distance between the centroid positions of the two sources is $<40{\mbox{$^{\prime\prime}$}\xspace}$ (this distance is more conservative than that used in Section [[\[sec:catalogmatching\]]{}]{}). We then compared the pointing offset as measured by the mean offset between the catalogs to the offset measured via cross-correlation analysis of the maps on a per-square-degree basis. The catalog and image offsets agree well, with no clear systematic offsets between the two estimators. The scatter in the catalog-based measurements is much greater, which is expected since the source positions are subject to spatial scale recovery differences between the versions and because the sources include less signal than the complete maps. There is no clear net offset between either version of the BGPS and the Herschel Hi-Gal survey, or between the two versions of the BGPS. However, the scatter in the pointing offsets between [[v]{}1.0]{}and Herschel is substantially greater than the [[v]{}2.0]{}-Herschel offsets. The offset measured in @Contreras2012 is likely a result of particularly large offsets in a few fields with more identified sources. As shown in Table \[tab:ccoffsetsmeans\], the mean offset, *weighted by number of sources*, is greater for the ATLASGAL overlap region than overall. We reproduce a number similar to the ATLASGAL-measured longitude offset of $\Delta\ell=-4.7{\mbox{$^{\prime\prime}$}\xspace}$ (our source-count-weighted $\Delta\ell=-3.7{\mbox{$^{\prime\prime}$}\xspace}$), despite a much larger standard deviation and despite no significant offset being measured directly in the images. These measurements imply that the pointing offset measured by @Contreras2012 was localized to a few fields *and* that the offset is corrected in the [[v]{}2.0]{}data. The Angular Transfer Function of the BGPS {#sec:stf} ========================================= Simulations with synthetic sky and atmosphere {#sec:atmotests} --------------------------------------------- In order to determine the angular response of the Bolocam array and BGPS pipeline in realistic observing conditions, we performed simulations of a plausible synthetic astrophysical sky with synthetic atmospheric signal added to the bolometer timestream. To generate the simulated atmosphere, we fit a piecewise power law to a power spectrum of a raw observed timestream (Figure \[fig:powerspecfit\]). The power spectrum varies in amplitude depending on weather conditions and observation length, but the shape is generally well-represented by $1/f$ “pink” noise ($P_\nu\propto \nu^{-1.5}$) for $\nu < 2 $ Hz and flat “white” noise ($P_\nu \sim const$) for $\nu \geq 2$ Hz, where $\nu$ is the frequency. We show a fitted timestream power spectrum in Figure \[fig:powerspecfit\]. The deviations from $1/f$ and white noise have little effect on the reduction process. The Fourier transform of the atmosphere timestream is generated by applying noise to the fitted power spectrum. The power at each frequency is multiplied by a random number sampled from a Gaussian distribution[^5] with width 1.2, determined to be a reasonable match to the data, and mean 1.0. The resulting Fourier-transformed timestream $d(t)$ is $FT(d(t)) = (r_{\nu1} P_f)^{1/2} + i (r_{\nu2} P_f)^{1/2}$, where $r_1$ and $r_2$ are the normally distributed random variables and $P_f$ is the fitted power-law power spectrum. The atmosphere timestream is then created by inverse Fourier transforming this signal. Gaussian noise is added to the atmospheric timestream of each bolometer independently, which renders the correlation between timestreams imperfect. This decorrelation is important for the PCA cleaning, which would remove all of the atmosphere with just one nulled component if the correlation was exact. The noise level set in the individual timestreams determines the noise level in the output map. ### Simulated Map Parameters We simulated the astrophysical sky by randomly sampling signal from an azimuthally symmetric 2D power-law distribution in Fourier space. The power distribution as a function of angular frequency is given by $$P(1/r) \propto (1/r)^{-\alpha_{ps}}$$ where $r$ is the angular size-scale and $\alpha_{ps}$ is the power-law spectral index for power spectra. We modeled this signal using power spectrum power-law indices ranging from -3 to +0.5; in the HiGal $\ell=30{\mbox{$^\circ$}\xspace}$ Science Demonstration Phase (SDP) field, the power-law index measured from the 500 [$\mu \textrm{m}$]{} map is $\alpha_{ps}\sim2$ over the scales of interest for comparison with Bolocam.[^6] The data were smoothed with a model of the instrument PSF to simulate the telescope’s aperture and illumination pattern. For each power-law index, three realizations of the map using different random seeds were created. The signal map was then sampled into timestreams with the Bolocam array using a standard pair of perpendicular boustrophedonic scan patterns. Examples of one of these realizations with identical random seeds and different power laws are shown in Figure \[fig:exp10gridin\]. Simulations performed with $\alpha_{ps}=3$ yielded no recovered astrophysical emission for normalizations in which the astrophysical sky was fainter than the atmosphere. Such a steep power spectrum is inconsistent with both BGPS and other observations: as noted above, Herschel sees structure with $\alpha_{ps}\sim2$. The fact that the BGPS detected a great deal of astrophysical signal, none of which was brighter than the atmosphere, confirms that $\alpha_{ps}=3$ is unrealistic. The Angular Transfer Function {#sec:STF} ----------------------------- We used a subset of these power-law simulations to measure the amount of recovered signal at each angular (spatial) scale. For each power-law in the range $1<\alpha_{ps}<2$, we used three different realizations of the map to measure the angular transfer function, defined as $STF(f) = F_{out}(f)/F_{in}(f)$ where $f$ is the angular frequency, $F_{out}$ is the azimuthally averaged power-spectrum of the pipeline-processed map, and $F_{in}$ is the azimuthally averaged power-spectrum of the simulated input map. The angular transfer function shows only weak dependence on the ratio of astrophysical to atmospheric power, and is approximately constant at $\sim95\%$ recovery over the range of angular scales between the beam size and $\sim1.5{\mbox{$^\prime$}\xspace}$. The angular transfer function is shown in Figure \[fig:stf\]. At larger angular scales, in the range $2{\mbox{$^\prime$}\xspace}-8{\mbox{$^\prime$}\xspace}$, the recovery is generally low ($<80\%$). Our simulations included the full range of observed astrophysical to atmospheric flux density ratios, from $\sim10^{-2}$ for the Central Molecular Zone (CMZ) down to $\sim10^{-4}$ for sparsely populated regions in the $\ell\sim70{\mbox{$^\circ$}\xspace}$ region. @Chapin2013a perform a similar analysis for the SCUBA-2 pipeline. Our transfer function (Figure \[fig:stf\]) cuts off at a scale $\sim 1/6$ the SCUBA-2 scale. While the angular extent of the Bolocam footprint is only slightly smaller than SCUBA-2’s, some feature of the instrument or pipeline allows SCUBA-2 to recover larger angular scales. We speculate that the much larger number of bolometers in the SCUBA-2 array allows the atmosphere to be more reliably separated from astrophysical and internal electrical signals (bias and readout noise), so the SCUBA-2 pipeline is able to run with an atmosphere subtraction algorithm less aggressive than the 13-PCA approach we adopted. Comparison to other data sets: Aperture Photometry {#sec:otherdata} -------------------------------------------------- Given an understanding of the angular transfer function, it is possible to compare the BGPS to other surveys, e.g. Hi-Gal, ATLASGAL, and when it is complete, the JCMT Galactic Plane Survey (JPS), for temperature and $\beta$[^7] measurements. Because of the severe degeneracies in temperature/$\beta$ derivation from dust SEDs [e.g. @Shetty2009a; @Shetty2009b; @Kelly2012a], we recommend a conservative approach when comparing BGPS data with other data sets. For compact sources, aperture extraction with background subtraction in both the BGPS and other data set should be effective. Section [[\[sec:apextract\]]{}]{} discusses aperture extraction in the presence of typical power-law distributed backgrounds. Comparison to other data sets: Fourier-space treatment {#sec:otherdata_fourier} ------------------------------------------------------ In order to compare extended structures, which includes any sources larger than the beam, a different approach is required. The safest approach is to “unsharp mask” (high-pass-filter) both the BGPS and the other data set with a Gaussian kernel with FWHM $\lesssim120{\mbox{$^{\prime\prime}$}\xspace}$ ($\sigma \lesssim 51 {\mbox{$^{\prime\prime}$}\xspace}$). The filtering will limit the angular dynamic range, but will provide accurate results over the angular scales sampled. Direct comparison of power spectra over the reproduced range is also possible. A demonstration of this approach is given in Figure \[fig:higalpowerlaw\], which shows the structure-rich $\ell=30{\ensuremath{^{\circ}}}$ field. The BGPS power spectrum has a shape very similar to that of HiGal. The spectral index is a commonly used measure of the ratio between flux densities at two different wavelengths in the radio, $$\frac{F_2}{F_1} = \left(\frac{\lambda_2}{\lambda_1}\right)^{-\alpha_{\nu}} = \left(\frac{\nu_2}{\nu_1}\right)^{\alpha_{\nu}}$$ The spectral index between the BGPS at 1.1 mm and Herschel at 500 [$\mu \textrm{m}$]{}is $\alpha_{\nu}\sim 3.7$ over the range $33{\mbox{$^{\prime\prime}$}\xspace}< dx < 300{\mbox{$^{\prime\prime}$}\xspace}$, although because the BGPS angular transfer function is low at the large end of this range, this is only an ‘eyeball’ estimate. On the Rayleigh-Jeans tail, $\alpha_{\nu}=\beta+2$, so this spectral index is consistent with typical dust emissivity index $\beta$ measurements in the range $1.5 < \beta < 2$. Source Extraction {#sec:sourceextraction} ================= @Rosolowsky2010 presented the Bolocat catalog of sources extracted from the [[v]{}1.0]{}data with a watershed decomposition algorithm. We have used the same algorithm to create a catalog from the [[v]{}2.0]{}catalog. We have also performed comparisons between the [[v]{}1.0]{}and [[v]{}2.0]{}data based on the extracted sources. The new catalog was derived using the same Bolocat parameters as in [[v]{}1.0]{}. This catalog includes regions that were not part of the [[v]{}1.0]{}survey area, but we restrict our comparison between [[v]{}1.0]{}and [[v]{}2.0]{}to the area covered by both surveys. Aperture Extraction {#sec:apextract} ------------------- One major change from the [[v]{}1.0]{}catalog is that the fluxes in the [[v]{}2.0]{}catalog are reported with background subtracted. The backgrounds are calculated from the mode of the pixels in the range $[2R,4R]$, where R represents the aperture radius (20, 40, or 60). The mode is computed using the IDL astrolib routine `skymod.pro`, which returns the mean of the selected data if the mean $\bar{\mu}$ is less than the median $\mu_{1/2}$ (indicating low “contamination” from source flux) or $3 \mu_{1/2} - 2 \bar{\mu} $ otherwise, then performs iterative rejection of bad pixels [@Landsman1995a; @Stetson1987a]. We performed aperture extraction on simulated data sets to determine what size apertures are appropriate when comparing to other data sets. In Figure \[fig:bgsub\], we show the results of aperture extraction with and without background subtraction on a simulated power-law generated image with $\alpha_{ps}=2$ before and after pipeline processing. The map has had point sources added to it randomly distributed throughout the field with flux densities randomly sampled from the range $[0.1,1]$ Jy, and the power-law extended flux has an amplitude $\approx1.8$ Jy. Sources are extracted from the pipeline-processed map using Bolocat, then the same source locations and masks are used to extract flux measurements from the input map. Figure \[fig:simptsrc\] shows the input, pipeline-processed, and point-source-only maps along with the Bolocat apertures to give the reader a visual reference for an $\alpha_{ps}=2$ background with point sources. The scatter between the flux density measurements derived from the input simulated sky map and the iteratively produced map is small when background subtraction is used (the blue points), but large and unpredictable otherwise (the red points). The agreement between the flux densities extracted from the iterative map and the input synthetic map is excellent for 40 diameter background-subtracted apertures. For these apertures, the RMS of the difference between the iterative map and the input map fluxes is $\sigma=0.03$ Jy when background subtraction is used, indicating the utility and necessity of this approach. The agreement is similarly good for 80apertures ($\sigma=0.10$ Jy), but the 120apertures exhibit a source- and background-brightness dependent bias, so we recommend against apertures that large when comparing to other data sets. There are caveats to this analysis. If the “background” power-law map has a peak flux density $\gtrsim10\times$ the peak point-source flux density, the point sources will not be recovered: the cataloging algorithm will pick out peaks in the power law flux distribution. These cannot be analyzed with simple aperture extraction for an $\alpha_{ps}=2$ flux density distribution. However, for shallower power-law distributions, i.e. $\alpha_{ps}\lesssim1$, aperture extraction effectively recovered accurate flux-densities in the processed maps - shallow power-law distributions more strongly resemble point-source-filled maps. Catalog Matching between [[v]{}1.0]{}and [[v]{}2.0]{} {#sec:catalogmatching} ----------------------------------------------------- We matched the [[v]{}1.0]{} and [[v]{}2.0]{} catalogs based on source proximity. For each source in [[v]{}1.0]{}, we identified the nearest neighbor from [[v]{}2.0]{}, and found that 5741 [[v]{}2.0]{}sources are the nearest neighbor for a [[v]{}1.0]{}source out of 8004 [[v]{}2.0]{}sources in the [[v]{}1.0]{}-[[v]{}2.0]{}overlap region. Similarly, we identified the nearest neighbor in [[v]{}1.0]{}for each [[v]{}2.0]{}source, finding 5745 [[v]{}1.0]{}sources are the nearest neighbor for a [[v]{}2.0]{}source out of 8358 [[v]{}1.0]{}sources. There are 5538 [[v]{}1.0]{}-[[v]{}2.0]{}source pairs for which each member of the pair has the other as its nearest-neighbor. These sources are clearly reliable and stable source identifications and constitute about 70% of the [[v]{}2.0]{}sample. Most of the unmatched sources have low flux density (Figure \[fig:v1vsv2matchhist\]), but some were significantly higher - these generally represent sources that were split or merged going from [[v]{}1.0]{}to [[v]{}2.0]{}. A few examples of how mismatches can happen are shown in Figures \[fig:contourmatch\] and \[fig:contourmatch2\]. The low-flux-density sources were most commonly unmatched in regions where the noise in [[v]{}1.0]{}and [[v]{}2.0]{}disagreed significantly. The high-flux-density mismatches tend to be different decompositions of bright sources and are preferentially found near very bright objects, e.g. in the Galactic center region. Source flux density, size, shape, and location distributions ------------------------------------------------------------ We reproduce parts of @Rosolowsky2010 Figures 17 and 19 as our own Figures \[fig:bolocathistograms\] and \[fig:sizehistograms\]. These figures show the distributions of extracted source properties (flux density, size, and aspect ratio) for the [[v]{}1.0]{} and [[v]{}2.0]{} data. The source flux density distributions above the completeness cutoff are consistent between [[v]{}1.0]{}and [[v]{}2.0]{}, both exhibiting power-law flux density distributions $$\frac{dN}{dS_{\nu}} \propto S_{\nu}^{-\alpha_{src}}$$ with values in the range $\alpha_{src}=2.3-2.5$ for sources with $S_\nu\gtrsim0.5$ Jy. In the left panel of Figure \[fig:bolocathistograms\], we have included the [[v]{}2.0]{}aperture-extracted data both with and without annular background subtraction. The [[v]{}1.0]{}catalog had no background subtraction performed because the backgrounds were thought to be negligible, but the [[v]{}2.0]{}catalog has had background subtraction performed so that the flux densities reported more accurately represent the sky. The [[v]{}2.0]{}data include more large sources. The longitude and latitude source flux density distribution plots, Figure 15 of @Rosolowsky2010, are reproduced in Figure \[fig:latlonhist\]. The properties are generally well-matched, although even with the 1.5$\times$ correction factor to the [[v]{}1.0]{} data, there is more flux density per square degree in [[v]{}2.0]{}sources. The gain in flux density recovery is both because of an increased flux density recovery on large angular scales and because of improved noise estimation, which results in a greater number of pixels being included in sources (see Section \[sec:catalogmatching\] for more details and Figures \[fig:contourmatch\] and \[fig:contourmatch2\] for examples). A two-dimensional histogram providing a broad overview of the survey contents is shown in Figure \[fig:hist2d\]. The ratio of source counts per half square degree is included in panel 3. This figure illustrates that the two catalog versions are broadly consistent, and the regions in which they differ significantly tend to have fewer sources. The most extreme ratios of [[v]{}2.0]{}to [[v]{}1.0]{}source counts tend to occur along field edges both because of preferentially low source counts and because the [[v]{}2.0]{}images have slightly greater extents in latitude than the [[v]{}1.0]{}. Conclusions =========== We presented Version 2 of the Bolocam Galactic Plane Survey, which is a significant improvement over [[v]{}1.0]{}in pointing and flux calibration accuracy. The [[v]{}2.0]{}data show an improvement in large angular scale recovery. The [[v]{}2.0]{}release includes new observations of regions in the outer galaxy. - We have characterized the angular transfer function of the Bolocam pipeline. Flux recovery is $>95\%$ for scales between $33{\mbox{$^{\prime\prime}$}\xspace}< dx \lesssim 80{\mbox{$^{\prime\prime}$}\xspace}$. The angular transfer function shows a sharp drop in recovered power above $\gtrsim100{\mbox{$^{\prime\prime}$}\xspace}$ scales. - We compared the pointing of the BGPS to that in Hi-Gal, and found that the surveys are consistent to within the measurement error $\sigma\approx3.5{\mbox{$^{\prime\prime}$}\xspace}$. - We measured the power spectral density in some regions and compared it to that in Hi-Gal, concluding that the power spectra are consistent with the normally used dust emissivity values in the range $\beta \sim1.5-2$. - A new version of the catalog has been released. The improved quality of the [[v]{}2.0]{}images has some effects on the BGPS catalog but the basic statistical properties of the catalog have not significantly changed. Because of changing noise properties within the images, only 70% of the individual sources in [[v]{}2.0]{}have an obvious [[v]{}1.0]{}counterpart and vice versa. The remaining 30% of sources do not have obvious counterparts because of two effects: 1. At low significance, changing noise levels recover different features at marginal significance. It is likely that low significance sources in [[v]{}1.0]{}and [[v]{}2.0]{}are [*both*]{} real features but have been rejected in the other catalog because of the relatively conservative limits placed on catalog membership. 2. At high significance, the catalog algorithm is dividing up complex structure using the underlying watershed algorithm. In this case, the precise boundaries between objects are sensitive to the shape of the emission. All of the high significance features appear in both catalogs, but the objects to which a piece of bright emission is assigned can vary. Despite these changes in the catalogs, the overall statistical properties of the population show little variation except that the largest sources appear brighter and larger owing to better recovery of the large scale flux density. **Acknowledgements:** We thank the referee for a timely and constructive report. This work was supported by the National Science Foundation through NSF grant AST-1008577. The BGPS project was supported in part by NSF grant AST-0708403, and was performed at the Caltech Submillimeter Observatory (CSO), supported by NSF grants AST-0540882 and AST-0838261. The CSO was operated by Caltech under contract from the NSF. Support for the development of Bolocam was provided by NSF grants AST-9980846 and AST-0206158. ER is supported by a Discovery Grant from NSERC of Canada. NJE and MM were supported by NSF Grant AST-1109116. [48]{} natexlab\#1[\#1]{} , J. E. 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[^3]: If the underyling spectral indices of the emission regions are uncorrelated with the flux, e.g. if they are constant, the slopes in Table \[tab:FluxComparison\] will be unaffected. The assumption of constant spectral index with flux is reasonable since observed spectral index-flux correlations are shallow [@Kelly2012a]. [^4]: In the v2.0 catalog on IPAC, 35 sources in the $\ell=195$ and Orion B fields were inadvertently excluded; these are now included in a v2.1 release. [^5]: We experimented with different noise distributions that reasonably matched the data, including a lognormal distribution, and found that the angular transfer function was highly insensitive to the noise applied to the atmosphere time series power spectrum. [^6]: The Herschel data used were those presented in @Molinari2010a, and the measured power-law was consistent in more recent reductions [@Traficante2011a]. [^7]: $\beta$ is the dust emissivity index, i.e., a modification to a blackbody to create a greybody such that $G(\nu) = B(\nu) (1-e^{-\tau(\nu)})$, and $\tau(\nu) = (\nu/\nu_0)^\beta$.
--- abstract: 'We devise a technique for defining and computing $n$-point functions in the context of a background-independent gravitational quantum field theory. We construct a tentative implementation of this technique in a perturbatively-finite loop/spinfoam model.' author: - | Leonardo Modesto and Carlo Rovelli\ *Centre de Physique Théorique de Luminy, Université de la Méditerranée, F-13288 Marseille, EU* title: | **[Particle scattering in loop quantum gravity]{}\ ** --- The lack of a general technique for computing particle scattering amplitudes is a seriously missing ingredient in nonperturbative quantum gravity [@book; @thomas]. Various problems can be traced to this absence: the difficulty of deriving the low energy limit of a theory; of comparing alternative theories, such as alternative versions of the hamiltonian operator in loop quantum gravity (LQG) or different spinfoam models; or comparing the predictions of a theory with those of perturbative approaches to quantum gravity, such as perturbative string theory. Here we explore one possibility for defining a general formalism aimed at computing scattering amplitudes. We outline a calculation strategy, which can certainly be improved. Our interest is not in a particular theory, but rather in a general technology to be used for analyzing different models. For concreteness, we implement this strategy in the context of a specific model, and present a well-defined and perturbatively finite expression, which, under substantial assumptions and approximations, might be used as a general covariant $n$-point function. In conventional QFT, we can derive all scattering amplitudes from the $n$-point functions $$W(x_{1}, \ldots, x_{n})=Z^{-1}\int D\phi\ \phi(x_{1})\ldots \phi(x_{n}) \ e^{-iS[\phi]} \label{Wn}$$ where the $x_{i},\ i= 1, \ldots, n$ are points of the background spacetime, $\phi$ is the quantum field, $S[\phi]$ its action and $Z$ is the integral of the sole exponential of the action. Alternatively, the $n$-point functions can be derived from their Euclidean continuations, defined by dropping the $i$ factor in the above expression. The integral (\[Wn\]) is well-defined in perturbation theory or as a limit of a lattice regularization, under appropriate renormalization. A well-known difficulty of background independent quantum field theory is given by the fact that if we assume (\[Wn\]) to be well-defined with general-covariant measure and action, then the $n$-point function is easily shown to be constant in spacetime (see for instance [@TW]). This is the difficulty we address here. Consider a spacetime region $R$ such that the points $x_{i}$ lie on its 3d boundary $\Sigma$. Call $\varphi$ the restriction of the field $\phi$ to $\Sigma$. Then (\[Wn\]) can be written in the form $$W(x_{1}, \ldots, x_{n})=Z^{-1}\int D\varphi\ \varphi(x_{1})\ldots \varphi(x_{n})\ W[\varphi,\overline \Sigma] \ W[\varphi,\Sigma] \label{WW}$$ where (in the euclidean case) $$W[\varphi, \Sigma]=\int_{\phi|_{{}_{\Sigma}}=\varphi} D\phi_{R}\ e^{-S_{R}[\phi_{R}]} \label{W}$$ is the integral over the fields in $R$ bounded by the 3d field $\varphi$ on $\Sigma$, and $S_{R}$ is the restriction of the action to $R$; while $W[\varphi, \overline\Sigma]$ is the analogous quantity defined on the complement $\overline R$ of $R$ in spacetime. The field boundary propagator (\[W\]) has been considered in [@oeckl] and [@book] and studied in [@cr]. Assume that we are dealing with an interacting theory, approximated by a free (gaussian) theory $S^{(0)}[\phi]$ in some regime, and that, within a certain approximation of the amplitude (\[Wn\]), the interaction term in the action can be restricted to $R$. Then we can replace $W[\varphi,\overline R]$ with its free theory equivalent $$W_{0}[\varphi, \overline \Sigma] \ =\ \int_{\phi|_{{}_{\Sigma}}=\varphi} D\phi_{\bar R}\ e^{-S^{(0)}_{\bar R}[\phi]}\ \equiv\ \Psi_{\Sigma}[\varphi]. \label{W2}$$ This integral is gaussian and can be performed, giving a gaussian “boundary state" $\Psi_{\Sigma}[\varphi]$, determined by appropriate boundary conditions for the field at infinity. For instance, if we take $R$ to be defined by $t>0$, then $\Psi_{t=0}[\varphi]$ is the vacuum state in the functional Schrödinger representation. In general, we expect the boundary state to be given by some gaussian functional of the boundary field $\varphi$ on $\Sigma$. Consider a diffeomorphism invariant theory including the gravitational field. Assume that the equations above hold, in some appropriate sense. The field $\phi$ represents the gravitational field, as well as any eventual matter field, and we assume action and measure to be diffeomorphism invariant. Two important facts follow [@cdort]. First, because of diffeomorphism invariance the boundary propagator $W[\varphi,\Sigma]$ is independent from (local deformations of) the surface $\Sigma$. Thus in gravity the left hand side of (\[W\]) reads $W[\varphi]$. Second, the geometry of the boundary surface $\Sigma$ is not determined by a background geometry (there isn’t any), but rather by the boundary gravitational field $\varphi$ itself. We can obtain an indication on the possible forms of the boundary state in gravity from the free quantum theory of non-interacting gravitons on Minkowski space. If we take $R$ to be $t>0$, for instance, then $\Psi_{t=0}[\varphi]$ must be approximated by the well-known Schrödinger vacuum wave functional of linearized gravity. This is a gaussian state picked around a classical geometry: the flat geometry of the $t=0$ surface in Minkowski space. In the case of a compact $R$, it is then reasonable to consider a gaussian boundary state $\Psi_{q}[\varphi]$ picked around *some* 3-geometry $q$ of the boundary surface $\Sigma$. Thus, we may expect an expression of the form $$W(x_{1}, \ldots, x_{n};\ q)=Z^{-1}\int D\varphi\ \varphi(x_{1})\ldots \varphi(x_{n})\ \Psi_{q}[\varphi]\ W[\varphi] \label{Wn4}$$ to approximate (\[Wn\]) when the interaction term can be neglected outside $R$. For this equation to be significant, we have to fix the meaning of the coordinates $x_{i}$, since the rest of the expression is generally covariant. There is an obvious choice: the points $x_{i}$ can be defined with respect to the geometry $q$. For instance, if $n=4$, $t^0_{1}=t^0_{2}=0$ and $t^0_{3}=t^0_{4}=T$, (we use $x=(t,\vec x)$) we can take $q$ to be the geometry of a rectangular box of height $T$ and side $L$ and interpret $\vec x_{i}$ as proper distances from the boundaries of the box. In other words, *the localization of the arguments of the $n$-point function can be defined with respect to geometry over which the boundary state is picked*. Notice that $x_{i}$ in (\[Wn4\]) are then *metric* coordinates: they refer to gravitational field values. They are not anymore general-covariant coordinates as in (\[Wn\]). In this manner, we can give meaning to $n$-point functions in a background independent context. Physically, we can interpret $R$ as a finite spacetime region where a scattering experiment is performed. The quantities $x_{i}$ are then relative distances and relative proper time separations, measured along the boundary of this region, and determined by (the mean value of) the gravitational field (hence the geometry) on this boundary. This is precisely the correct general-relativistic description of the position measurements in a realistic scattering experiment. In order to give (\[Wn4\]) a fully well-defined meaning, and compute $n$-point functions concretely, we need four ingredients: (i) A proper definition of the space of the 3d fields $\varphi$ integrated over, and a well-posed definition of the integration measure. (ii) An explicit expression for the boundary propagator $W[\varphi]$. (iii) An explicit expression for the boundary state $\Psi_{q}[\varphi]$. (iv) A definition of the field operator $\varphi(x)$. In the following we analyze the status of these four ingredients in the loop and spinfoam approach to quantum gravity. We consider for simplicity pure gravity without matter. \(i) In quantum theory, the boundary values of Feynman integrals can be taken to be the classical dynamical variables only if the corresponding operators have continuum spectrum. If the spectrum is discrete, the boundary values are the quantum numbers that label a basis of eigenstates (see [@book]). In our case, the boundary field $\varphi$ represents the metric of a 3d surface. Let us assume here the results of LQG that the 3d metric is quantized [@book]. Therefore we must replace the continuum gravitational field variable $\varphi$ with the quantum numbers labelling a basis that diagonalizes some metric degrees of freedom. These can be taken to be (abstract) spin networks $s$, or $s$-knots. An $s$-knot is here an equivalence class under (extended [@winston]) diffeomorphisms of embedded spin networks $S$. An embedded spin network is a graph immersed in space, labeled with spins and intertwiners. The $s$-knots are discrete [@winston]. Thus, we rewrite (\[Wn4\]) in the form $$W(x_{1}, \ldots x_{n};\ g)=Z^{-1}\! \sum_{s} c(s)\ \varphi(x_{1})\ldots \varphi(x_{n}) \Psi_{q}[s]\ W[s]. \label{W4s}$$ where the meaning of $\varphi(x)$ will be specified later on. The discrete measure $c(s)$ on the space of the $s$-knots is defined by the projection of the scalar product of the space of the embedded spin networks ($c(s)$=1) except for discrete symmetries of $s$.) For simplicity, and in order to match with the spinfoam formalism that we use below, we restrict here the space of the spin networks to the four-valent ones and we identify spin networks with the same graph, spins and intertwiners (i.e., we ignore knotting and linking). \(ii) The boundary propagator $W[s]$ is a now a function of a boundary spin network. A natural possibility is to identify it with the boundary propagator $W[s]$ defined by the spin foam models [@Perez]. For concreteness, let us choose here the model defined by the $SO(4)/SO(3)$ group field theory [@PR], which gives a perturbation expansion finite at all orders [@finiteness]. This is the model denoted $GFT/C$ in [@book]. The amplitude of a spin network $s$ is given in this model by $$W[s]= \int D\Phi\ f_{s}[\Phi]\ e^{-\int \Phi^2-\lambda \int \Phi^5}. \label{GFT}$$ Here $\Phi$ is a function on $[SO(4)]^4$ and the precise meaning of the (symbolic) integrals in the exponent is detailed in [@Perez] and [@book]. The quantity $f_{s}[\Phi]$ is a polynomial in the field $\Phi$, determined by $s$. It is defined by picking one factor $$\Phi^i_{\alpha_{1}\ldots\alpha_{4}} = \int dg_1\ldots dg_4\ \Phi(g_{1},\ldots,g_{4})\ R^{(j_1)}_{\alpha_{1}}{}^{\beta_1}(g_{{1}}) \ldots R^{(j_4)}_{\alpha_4}{}^{\beta_{{4}}}(g_{{4}}) \ v^{i}_{\beta_{{1}}\ldots\beta_4}$$ per node of $s$, where $v^i$ is the intertwiner of the node and $j_{1},\ldots, j_{4}$ the colors of the adjacent links, and contracting the indices $\alpha_{i}$ according to the connectivity of the graph of $s$. The expression (\[GFT\]) is well-defined and finite order by order in $\lambda$. (The rigorous proof of this statement is complete up to certain degenerate graphs [@Perez].) The explicit computation of $W[s]$ is entirely combinatorial and can be performed in terms of combinations of $nJ$ Wigner symbols [@book]. For completeness, recall that the reason for the definition (\[GFT\]) is that the expansion of $W[s]$ in $\lambda$ can be written as a sum over spinfoams bounded by the spin network $s$ $$W[s]=\sum_{\partial\sigma=s} A(\sigma), \label{sf}$$ where the spinfoam amplitude $A(\sigma)$ is the Barrett-Crane discretization of the exponential of the Einstein-Hilbert action of the discrete four-geometry defined by the spinfoam $\sigma$. Therefore the definition (\[GFT\]) of $W[s]$ can be interpreted as a (background independent) discretization of the functional integral (\[W\]). \(iii) An expression for the boundary state $\Psi_{q}[s]$ can be obtained from the analysis of the coherent states in LQG [@coherent; @coherent2; @florian]. For concreteness, let us pick here Conrady’s definition of a coherent state [@florian]. Other more refined expression could be used instead. Conrady has defined a state $\Psi_0[S]$ that describes the Minkoski vacuum as a function of embedded spin networks $S$, under certain approximations and assumptions. This function has the property of being picked on spin networks that are “weaves", namely that approximate a flat metric $q$ when averaged over regions large compared to the Planck scale [@weave]. This vacuum state can be written as follows. Pick cartesian coordinates $x^a, a=1,2,3$, on a 3d surface equipped with a flat metric $q$ and with total volume $V$. Fix a triangulation $\mathcal{T}$ of lattice spacing $a$, small compared to the Planck length $l_{p}$ in the metric $q$. Restrict the attention to embedded spin networks $S$ living on $\mathcal{T}$. Define the form factor of a spin network as $$\begin{aligned} F^{ab}_{S}(\vec{x}) & = &\frac{\pi l_P^4}{96a^3} \, \sum_{v\in S}\sum_{e,e'\in v}\int_0^1dt \int_0^1dt^{\prime} \, \dot{e}^a(t) \, \dot{e}^b(t^{\prime}) \, \delta(\vec{x} - \vec{x}_v),\end{aligned}$$ where $v$ are the vertices of the spin network $S$; $\vec x_{v}$ their position; $e:t\mapsto e^a(t)$ the edges; $e \in v$ indicates the edges $e$ adjacent to the vertex $v$; and $\dot e^a=de^a/dt$. Its Fourier transform is $ F^{ab}_{S}(\vec{k}) = V^{-1} \int d^3 x \ \mbox{e}^{- \mbox{i} \vec{k} \cdot \vec{x}} \ F^{ab}_{S}(\vec{x}).$ Then $$\begin{aligned} \Psi_{0}[S] = {\mathcal{N}} \, \mbox{exp}\Big[-\frac{1}{4 \, l_p^2} \, \sum_{\vec{k}} |\vec{k}|\ \big| F^{ab}_{S}(\vec{k}) \, j_e(j_e +1) - \sqrt{V} \delta^{ab} \, \delta_{\vec{k},0} \big|^2 \Big], \label{vacuum2}\end{aligned}$$ where the momenta summed over are the discrete modes on the triangulation; $j_{e}$ is the spin associated to the edge $e$; $ \mathcal{N}$ is a normalization factor. To understand this construction, notice that if we consider the gravitational field associated to the spin network, (in the sense of the weaves) $q_{S}^{ab}(\vec{x}) = F^{ab}_{S}(\vec{x}) \, j_e(j_e +1)$, then $\Psi_0[S]=\Psi_0[q_S]$ where $$\begin{aligned} \Psi_0[q] & = & \mbox{e}^{-\frac{1}{4 \hbar \kappa} \int d^3x\int d^3y \, [(\tilde{q}^{ab}(\vec{x}) - \delta^{ab}) \, W_{\Lambda}(\vec{x} - \vec{y}) \, (\tilde{q}^{ab}(\vec{y}) - \delta^{ab})]} \label{vacuum1}\end{aligned}$$ is the Schödinger functional representation of the linearized vacuum state. $W_{\Lambda}(\vec{x} - \vec{y})$ is a lattice regularization of the vacuum covariance. We can extend this construction to a 3d (euclidean) rectangular boundary $\Sigma$ simply taking the product of the Conrady states associated to each of the eight faces forming $\Sigma$. We need to carry this result over to the diffeomorphism invariant $s$-knot states. Given an abstract spin network $s$ there will be in general one embedded spin network $S(s)$ that maximizes the state $\Psi_{0}[S]$. We can then tentatively define $\Psi_{0}[s]=\Psi_{0}[S(s)]$. Notice that if $\Psi_{0}[S]$ is picked on weaves, then the diffeomorphism invariant state $\Psi_{0}[s]$ defined is picked on the corresponding (“weavy") discrete 3-geometries. The maximization condition can be interpreted as a gauge choice, picking the coordinate system in which the 3-geometry is closest to the euclidean metric. The gauge invariant state is then chosen to be the restriction of the state to this gauge surface. In the spirit of [@florian], we restrict to embedded spin networks $S$ living on $\mathcal{T}$. Given an $s$-knot $s$, there is only a discrete number of such spin networks that are in the class $s$: we choose $S(s)$ that maximizes (\[vacuum2\]) among these. We expect this definition (possibly with an appropriate correction of the Conrady vacuum state) to converge for fine triangulations, making the background structure chosen effectively irrelevant for a triangulation sufficiently finer than the Planck length. This construction provides a finite definition of $\Psi_{0}[s]$, diffeomorphism invariant by definition. \(iv) Finally, we need to define the field $\phi(x)$ appearing in (\[W4s\]). Following [@florian] we write $ h_{s}^{ab}(x) = ({q}_{S(s)}^{ab}(x) - \delta^{ab})$, where the point $x$ is defined in terms of the boundary metric $q$ and ${q}_{S(s)}^{ab}$ is defined above (\[vacuum1\]). An alternative, which we do not pursue here, is to derive $h^{ab}(x)(S)$ from the action of two $SU(2)$ generators [@simone]. We can now bring together the various pieces discussed. To start with, consider a parallelepiped in 4d euclidean space with hight $T$ and cubic base of side $L$. Let $\Sigma$ be its boundary, equipped with the induced metric $q$. Fix a triangulation of $\Sigma$. The simplest choice is to start from a cubic triangulation of $\Sigma$, and to obtain a four-valent lattice, by splitting each (six-valent) vertex of the cubic lattice into two vertices. Replacing the various items discussed into the formal expression (\[Wn4\]) we obtain $$W^{a_{1}b_{1} \ldots a_{n}b_{n}}(x_{1}, \ldots x_{n}; L, T)=Z_{LT}^{-1} \sum_{s} c(s)\ {h}_{s}^{a_{1}b_{1}}({x_{1}})\ldots\, {h}_{s}^{a_{n}b_{n}}({x_{n}})\ \Psi_{q}[s]\ W[s] \label{W4D}$$ where the sum is over all the $s$-knots that can be embedded in the triangulation. The normalization factor is the “vacuum to vacuum" amplitude $ Z_{LT}= \sum_{s} c(s)\ \Psi_{q}[s] \, W[s]$. (\[W4D\]) can be expanded in powers $\lambda^n$. $n$ is the number of vertices of the spinfoam, which is the number of 4-simplices in a simplicial complex dual to the spinfoam, if this exists. As a rough estimate, we can imagine each 4-simplex to have Planck size: if classical configurations dominate, the main contribution should come from $n$ of the order of the 4-volume of the interaction region in Planck units. Hypotheses and approximations used to get to (\[W4D\]) are severe. But all quantities in (\[W4D\]) are well defined. The expression is probably finite at any order in $\lambda$. We can thus take (\[W4D\]) as a tentative definition of an $n$-point function within the formalism of non-perturbative quantum gravity. More precisely, we can consider (\[W4D\]) as a tentative concrete definition of the quantity formally given by $$\begin{aligned} W^{a_{1}b_{1} \ldots a_{n}b_{n}}(x_{1}, \ldots, x_{n})= Z^{-1}\int Dg\ g^{a_{1}b_{1}}(x_1) \ldots\, g^{a_{n}b_{n}}(x_{n})\ e^{-S_{EH}[g]} \label{form}\end{aligned}$$ where $S_{EH}$ is the Einstein-Hilbert action, computed at relative spacetime distances $x_{1}, \ldots, x_{n}$ evaluated in terms of the mean value of the quantum gravitational field itself, on a box encircling the interaction region. The construction can probably be ameliorated and varied in a number of ways, and many issues remain open. There are important missing steps to get to the definition of quantities that can be interpreted as particle transition amplitudes. (On the physical interpretation of “particle" states defined on finite spacial regions, see [@daniele].) The key questions are whether the expression (\[W4D\]) is indeed finite, convergent, and independent from the auxiliary structures uses to define it, when the triangulation is sufficiently finer than the Planck scale, and whether this construction leads, in a first approximation, to the general relativity scattering tree amplitudes. [9]{} C Rovelli, *Quantum Gravity* (Cambridge University Press, Cambridge, 2004) T Thiemann, “Introduction to Modern Canonical Quantum General Relativity", gr-qc/0110034 NC Tsamis, RP Woodard, *Annals of Phys* [**215**]{} (1992) 96-155 R Oeckl, *Phys Lett* [**B575**]{} (2003) 318-324; *Class and Quantum Grav* [**20**]{} (2003) 5371-5380 F Conrady, C Rovelli, *Int J Mod Phys* [**A 19**]{}, (2004) 1-32. 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--- abstract: | We present a Kleene realizability semantics for the intensional level of the Minimalist Foundation, for short , extended with inductively generated formal topologies, Church’s thesis and axiom of choice. This semantics is an extension of the one used to show consistency of the intensional level of the Minimalist Foundation with the axiom of choice and formal Church’s thesis in previous work. A main novelty here is that such a semantics is formalized in a constructive theory represented by Aczel’s constructive set theory [**CZF**]{} extended with the regular extension axiom. address: - 'Dipartimento di Matematica “Tullio Levi Civita”, Università di Padova, Italy' - 'Dipartimento di Matematica “Tullio Levi Civita”, Università di Padova, Italy' - 'School of Mathematics, University of Leeds, UK' author: - Maria Emilia Maietti - Samuele Maschio - Michael Rathjen bibliography: - 'bibliopsp.bib' title: 'A realizability semantics for inductive formal topologies, Church’s Thesis and Axiom of Choice' --- [^1] Introduction ============ A main motivation for introducing the Minimalist Foundation, for short [[**MF**]{}]{}, in [@mtt; @m09] was the desire to provide a foundation where to formalize constructive point-free topology in a way compatible with most relevant constructive foundations. In particular, [[**MF**]{}]{} was designed with the purpose of formalizing the topological results developed by adopting the approach of Formal Topology by P. Martin-L[ö]{}f and G. Sambin introduced in [@S87]. This approach was further enriched with the introduction of Positive Topology by Sambin in [@somepoint]. A remarkable novelty of this approach to constructive topology was the advent of inductive topological methods (see [@CSSV03; @cms13]) to represent the point-free topologies of the real number line, of Baire space and of Cantor space. However, while the basic notions of Formal Topology can be formalized in the Minimalist Foundation in [@m09], the construction of inductively generated topologies cannot. This is indeed done on purpose since the Minimalist Foundation, for short [[**MF**]{}]{}, was introduced to be a minimalist foundation compatible with (or interpretable in) the most relevant constructive and classical foundations for mathematics in the literature (see [@m09]). Observe indeed that the intensional level of [**MF**]{} is quite weak in proof-theoretic strength being interpretable in the fragment of Martin-L[ö]{}f’s type theory with one universe, or directly in Feferman’s theory of non-iterative fixpoints [$\widehat{ID}_1$]{} as first shown in [@ms16]. Moreover, [**MF**]{} is presented in [@m09] as a two level system in accordance with the notion of constructive foundation in [@mtt]. Indeed [[**MF**]{}]{} consists of an intensional level based on an intensional type theory à la Martin-L[ö]{}f, aimed at exhibiting the computational contents of mathematical proofs, and an extensional level formulated in a language as close as possible to that of present day mathematics which is interpreted in the intensional level by means of a quotient model (see [@m09]). Here we present an extension [[**MF$_{ind}$**]{}]{} of [[**MF**]{}]{} with the inductive definitions sufficient to define inductively generated formal topologies and necessary to define inductive suplattices. This is due to the fact that in [@CSSV03] the problem of generating formal topologies inductively is reduced to that of generating inductive suplattices. The rules added to the intensional level  of [[**MF**]{}]{}to form the intensional level of [[**MF$_{ind}$**]{}]{}, called [[**mTT**]{}$_{ind}$]{}, are driven by those of well-founded sets in Martin-L[ö]{}f’s type theory in [@PMTT] without assuming generic well-founded sets as in the representations given in [@CSSV03; @silviobar]. The main purpose of our paper is then to show that the intensional level [[**mTT**]{}$_{ind}$]{} of [[**MF$_{ind}$**]{}]{}  is consistent with the axiom of choice (${\mbox{\bf AC}}$) and the formal Church’s thesis (${\mbox{\bf CT}}$). More in detail ${\mbox{\bf AC}}$ states that from any total relation we can extract a type-theoretic function as follows: $$({\bf AC})\;(\forall x\in A)\,(\exists y\in B)\,R(x,y)\rightarrow (\exists f\in (\Pi x\in A)\,B)\,(\forall x\in A)\,R(x,\mathsf{Ap}(f,x))$$ with $A$ and $B$ generic collections and $R(x,y)$ any relation, while ${\mbox{\bf CT}}$ (see also  [@DT88]) states that from any total relation on natural numbers we can extract a (code of a) recursive function by using the Kleene predicate $T$ and the extracting function $U$ $$({\bf CT})\;(\forall x\in\mathsf{N})\,(\exists y\in \mathsf{N})\,R(x,y)\rightarrow (\exists e\in \mathsf{N})\,(\forall x\in \mathsf{N})\,(\exists z\in \mathsf{N})\,(T(e,x,z)\wedge R(x,U(z))).$$ Such a consistency property is essential to fulfill the requirement of the intensional level of a constructive foundation proposed in [@mtt]. In order to meet our purpose, we produce a realizability semantics for [[**mTT**]{}$_{ind}$]{} by extending the one used to show the consistency of the intensional level of [[**MF**]{}]{}  with +  in [@IMMS], which in turn extends Kleene realizability interpretation of intuitionistic arithmetic. A main novelty of our semantics is that it is formalized in a constructive theory as the (generalized) predicative set theory [[**CZF+REA**]{}]{}, namely Aczel’s constructive Zermelo-Fraenkel set theory extended with the regular extension axiom [**REA**]{}. To this purpose it is crucial to modify the realizability interpretation in [@IMMS] in the line of the realizability interpretations of Martin-Löf type theories in extensions of Kripke-Platek set theory introduced in [@R93] (published as [@RG94]). Therefore, contrary to the semantics in [@IMMS], which was formalized in a classical theory as Feferman’s theory of non-iterative fixpoints [$\widehat{ID}_1$]{}, here we produce a proof that [[**mTT**]{}$_{ind}$]{}, and hence ${\mbox{{\bf mTT}}}$, is [*constructively*]{} consistent with +. As in [@IMMS], we actually build a realizability model for a fragment of Martin-L[ö]{}f’s type theory [@PMTT], called , where [[**mTT**]{}$_{ind}$]{} extended with the axiom of choice can be easily interpreted. As it turns out, $\mathbf{CZF}+\mathbf{REA}$ and  possess the same proof-theoretic strength. In the future we intend to further extend our realizability to model [[**mTT**]{}$_{ind}$]{} enriched with coinductive definitions to represent Sambin’s generated Positive Topologies. Another possible line of investigation would be to employ our realizability semantics to establish the consistency strength of [[**mTT**]{}$_{ind}$]{} or the extension of  with particular inductively generated topologies, like that of the real line. The extension [[**MF$_{ind}$**]{}]{}  with inductively generated formal topologies {#mfind} ================================================================================== Here we describe the extension [[**MF$_{ind}$**]{}]{} of [[**MF**]{}]{}capable of formalizing most relevant examples of formal topologies defined by inductive methods introduced in [@CSSV03]. In that paper, the problem of generating the minimal formal topology which satisfies some given axioms is reduced to show how to generate a complete suplattice in terms of an infinitary relation called [**basic cover relation**]{} $$a\cov V$$ between elements $a$ of a set $A$, thought of as [*basic opens*]{}, and subsets $V$ of $A$, meaning that [*the basic open $a$ is covered by the union of basic opens in the subset $V$*]{}. Then the elements of the generated suplattice would be fixpoints of the associated [*closure operator*]{} $$\cov(-) : \mathcal{ P}(A)\ \longrightarrow \ \mathcal{ P}(A)$$ defined by putting $$\cov(V)\, \equiv\, \{ \ x\in A\ \mid\ x\covV\ \}$$ which are [*complete*]{} with respect to families of subsets [*indexed over a set*]{}. Furthermore, a formal topology is defined as a basic cover relation satisfying a convergence property and a positivity predicate (see [@CSSV03; @mv04; @cms13]). Indeed in this case the resulting complete suplattice of $\cov$-fixpoints actually forms a [*predicative locale*]{} which is [*overt*]{} (or [*open*]{} in the original terminology by Joyal and Tierney) for the presence of the positivity predicate. The tool of basic covers appears to be the only one available in the literature to represent complete suplattices in most-relevant predicative constructive foundations including Aczel’s [**CZF**]{}, Martin-L[ö]{}f’s type theory and also [[**MF**]{}]{}. The reason is that [*there exist no non-trivial examples of complete suplattices that form a set*]{} in such predicative foundations (see [@Cu10]). As a consequence, there exist no non-trivial examples of locales which form a set and the approach of formal topology based on a cover relation seems to be compulsory (see also [@whyp]) when developing topology in a constructive predicative foundation, especially in [[**MF**]{}]{}. In [@CSSV03] it was introduced a method for generating basic covers inductively starting from [*an indexed set of axioms*]{}, called [*axiom set*]{}. Such a method allows to generate a formal topology inductively when the basic cover relation $\cov$ is defined on [*a preordered set $(A, \leq)$* ]{}and it is generated by an axiom set satisfying a so called [*localization condition*]{} which refers to the preorder defined on $A$. An algebraic study of the relation between basic covers and formal covers including their inductive generation is given in [@cms13].\ In the following we describe a suitable extension of [[**MF**]{}]{} capable of representing inductively generated basic covers, and hence also formal topologies. We start by describing how to enrich the extensional level  of [[**MF**]{}]{} in [@m09] with such inductive basic covers. The reason is that the language of   is more apt to represent the topological axioms given that it is very close to that of everyday mathematical practice (with proof-irrelevance of propositions and an encoding of the usual language of first order arithmetic and of subsets of a set, see [@m09]). We recall that in  we have four kinds of types, namely [**collections**]{}, [**sets**]{}, [**propositions**]{} and [**small propositions**]{} according to the following subtyping relations: $$\xymatrix@C=4em@R=1em{ {\mbox{\bf small propositions}}\ar@{^{(}->}[dd]\ar@{^{(}->}[rr]& & {\mbox{\bf sets}}\ar@{^{(}->}[dd] \\ &&\\ {\mbox{\bf propositions}}\ar@{^{(}->}[rr] && {\mbox{\bf collections}} }$$ where collections include the power-collection ${\mathcal P}(A)$ (which is not a set!) of any set $A$ and small propositions are defined as those propositions closed under intuitionistic connectives and quantifiers restricted to sets. We first extend  with new primitive [*small propositions*]{} $$a\triangleleft_{I,C} V\ prop_s$$ expressing that [*the basic open $a$ is covered by the union of basic opens in $V$*]{} for any $a$ element of a [*set*]{} $A$, $V$ subset of $A$, assuming that the basic cover is generated by a family of (open) subsets of $A$ indexed on a family of sets $I(x)\ set \ [x\in A]$ and representing by $$C(x,j) \in {\mathcal P}( A)\ [x\in A, j\in I(x)].$$ The precise rules extending  to form a new type system  are the following: [**Rules of inductively generated basic covers in** ]{} $$\begin{array}{l} \mbox{\rm F-$\triangleleft$} \ \displaystyle{\frac{\begin{array}{l} A \ set \ \ \ \ I(x)\ set \ [x\in A] \ \ \ C(x,j) \in {\mathcal P}( A)\ \ [x\in A, j\in I(x)]\\ V\in {\mathcal P}( A)\qquad a\in A\ \end{array} } {\displaystyle a\triangleleft_{I,C} V\ prop_s }}\\[20pt] \mbox{\rm rf-$\triangleleft$} \displaystyle{\frac{\begin{array}{l} A \ set \ \ \ \ I(x)\ set \ [x\in A] \ \ \ C(x,j)\in {\mathcal P}( A) \ \ [x\in A, j\in I(x)]\\ V\in {\mathcal P}( A) \qquad\qquad a\,\epsilon\, V\ true \end{array} } {\displaystyle a\triangleleft_{I,C} V\ true}}\\ \\ \mbox{\rm tr-$\triangleleft$} \ \displaystyle{ \frac{\begin{array}{l} A\ set \ \ \ \ I(x)\ set \ [x\in A] \ \ \ C(x,j)\in {\mathcal P}( A) \ \ [x\in A, j\in I(x)]\\ a\in A\qquad i\in I(a)\qquad \qquad V\in {\mathcal P}( A)\\ \forall_{y\epsilon C(a,i)}\ \ y\triangleleft_{I,C} V\ true\end{array} } {\displaystyle a\triangleleft_{I,C}V \ true }} \\[20pt] \mbox{ind-$\triangleleft$} \displaystyle{\frac{ \begin{array}{l} A\ set \ \ \ \ I(x)\ set \ [x\in A] \ \ \ C(x,j)\ \in {\mathcal P}( A)\ \ [x\in A, j\in I(x)]\\ P(x)\ prop\ [x\in A]\qquad V\in {\mathcal P}( A) \qquad \mathsf{cont}(V,P)\ true\ \\ a\in A\qquad a \cov_{I,C} V\ true \end{array} }{ P(a)\ true\ }} \end{array}$$ where $$\begin{array}{rl} \mathsf{cont}(V,P)\ \equiv\ & \forall_{x\in A}\ (\ x\,\epsilon\, V\ \rightarrow \ P(x) \ )\ \\ & \qquad \&\ \forall_{x\in A}\ \ (\ \forall_{j \in I(x)}\ \forall_{y\in A}\ (y\,\epsilon \, C(x,j)\ \rightarrow \ P(y) \ ) \ \rightarrow\ P(x)\ ) \end{array}$$\ where above we adopted the convention of writing $\phi\ true$ for a proposition $\phi$ instead of $\mathsf{true}\in \phi$ as in [@m09]. A main example of formal topology that can be formalized in with the rules above is that of [*real line*]{}, represented by Joyal’s inductive formal cover $\cov_{r}$ of Dedekind real numbers defined on the set $ \mathbb Q\, \times\, \mathbb Q$ which acts as $A$ in the rules above and where $\mathbb Q$ is the set of [*rational numbers*]{}. This formal cover is generated by a family of open subsets $C(\langle p,q\rangle , j )$ indexed on $j\in I(\langle p,q\rangle)$ for $\langle p,q\rangle \in \mathbb Q\, \times\, \mathbb Q$ which is defined as an encoding of the following rules: ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ${\displaystyle\frac{q\leq p}{\langle p,q\rangle\cov_{r} U}}$ ${\displaystyle\frac{\langle p,q\rangle\,\epsilon\,U}{\langle p,q\rangle\cov_{r} U}}$ ${\displaystyle\frac{p'\leq p<q\leq q' \qquad \langle p',q'\rangle\cov_{r} U} {\langle p,q\rangle\cov_{r} U}}$ \[15pt\] ${\displaystyle \frac{p\leq r < s\leq q\qquad \langle p,s\rangle\cov_{r} U \quad \langle r,q\rangle\cov_{r} U } {\langle p,q\rangle\cov_{r} U}}$ $ \mathsf{wc} \ {\displaystyle\frac {wc(\langle p,q\rangle)\cov_{r} U}{\langle p,q\rangle\cov_{r} U}}$ ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- where in the last axiom we have used the abbreviation $$wc(\langle p,q\rangle)\equiv \{\, \langle p',q'\rangle\in \mathbb Q\, \times\, \mathbb Q\ \mid\ p<p'< q'<q\}$$ ($wc$ stands for ‘well-covered’). For relevant applications see for instance [@Palmgren05continuityon; @whyftop] and loc.cit.\ It is worth noting that different presentations of basic covers may yield to the same complete suplattice. For example, any complete suplattice presented by (the collection of fixpoints associated to) a basic cover $\cov_{I,C}$ on a quotient set $B/R$, can be equivalently presented by a cover on the set $B$ itself which behaves like $\cov_{I,C}$ but in addition it considers equal opens those elements which are related by $R$. In order to properly show this fact, which it will be useful in the next, we define a correspondence between subsets of $B/R$ and subsets of $B$ as follows: *In , given a quotient set $B/R$, for any subset $W\in \mathcal{P}(B/R)$ we define $$\mathsf{es}(W)\, \equiv \, \{\ b\in B\ \mid\ [b]\,\epsilon\, W\ \}$$ and given any $V\in \mathcal{P}(B)$ we define $\mathsf{es}^{-}(V)\, \equiv \, \{\ z\in B/R\ \mid\ \exists_{b\in B}\ (\ b\,\epsilon\, V\ \wedge\ z=_{B/R}[b] \ )\}$.* Given an axiom set represented by a set $A\,\equiv\, B/R$ with $ I(x)\ set\ [x\in A]$ and $ C(x,j)\in {\mathcal P}(A) \ \ [x\in A, j\in I(x)]$, we define a new axiom set as follows: $$\begin{array}{l} A^R\, \equiv\, B\qquad \qquad I^R(x)\, \equiv\, I([x]) + (\Sigma y\in B)\ R(x,y)\qquad \qquad \mbox{ for } x\in B\\ \end{array}$$ where $C^R(b,j)$ is the formalization of $$C^R(b,j)\, \equiv\, \begin{cases} \mathsf{es}^{-}(\, C([b],j)\, ) & \mbox{ if } j\in I([b])\\ \{\, \pi_1(j) \, \} & \mbox{ if } j\in (\Sigma y\in B)\ R(b,y) \\ \end{cases}$$ for $b\in B$ and $j\in I^R(x)$. We then call $\cov_{I,C}^R$ the inductive basic cover generated from this axiom set. It is then easy to check that \[eqcov\] *For any axiom set in  represented by a set $A\,\equiv\, B/R$ with $ I(x)\ set\ [x\in A]$ and $ C(x,j)\in {\mathcal P}(A) \ \ [x\in A, j\in I(x)]$, the suplattice defined by $\cov_{I,C}$ is isomorphic to that defined by $\cov_{I,C}^R$ by means of an isomorphism of suplattices.* It is immediate to check that for any subset $W$ of $B/R$ which is a fixpoint for $\cov_{I,C}$ the subset $\mathsf{es}(W)$ is a fixpoint for $\cov_{I,C}^R$ and that, conversely, for any subset $V$ of $B$ which is a fixpoint for $\cov_{I,C}^R$ the subset $\mathsf{es}^{-}(V)$ is a fixpoint for $\cov_{I,C}$. Moreover, this correspondence preserves also the suprema defined as in [@cms13]. Alternatively, one could check that the relation $z\, F\, b\, \equiv\ \mathsf{Id} (\, B/R\, ,\, z\, ,\, [b]\, )$ defines a basic cover isomorphism in the sense of [@cms13] between the basic cover $\cov_{I,C}$ and $\cov_{I,C}^R$. The intensional level [[**mTT**]{}$_{ind}$]{} ============================================= Here we describe the extension [[**mTT**]{}$_{ind}$]{} of the intensional level  of [[**MF**]{}]{}capable of interpreting the extension . We recall that in  as well as in  we have the same four kinds of types with the difference that in  power-collections of sets are replaced by the existence of a [*collection of small propositions*]{} $\mathsf{prop_s}$ and function collections $A\rightarrow \mathsf{prop_s}$ for any set $A$. Such collections are enough to interpret power-collections of sets in   within a quotient model of dependent extensional types built over , as explained in [@m09]. Therefore, in order to define [[**mTT**]{}$_{ind}$]{} we cannot simply add the rules of inductively generated basic covers of  but we need to add an [*intensional*]{} version of them. To this purpose in [[**mTT**]{}$_{ind}$]{} in addition to the new small proposition $$a\triangleleft_{I,C} V\ prop_s$$ we need to add new proof-term constructors associated to it in such a way that judgements asserting that some proposition is true in   are turned into judgements of [[**mTT**]{}$_{ind}$]{}producing a proof-term of the corresponding proposition. It is worth noting that the equality rules of the inductive basic covers are driven by those of well-founded sets in Martin-L[ö]{}f’s type theory in [@PMTT] without assuming generic well-founded sets as in the representations given in [@CSSV03; @silviobar]. However, in accordance with the idea that proof-terms of propositions of  represent just a constructive rendering of the proofs of propositions in , we do restrict the elimination rules of inductive basic covers to act toward propositions non depending on their proof-terms, since these proof-terms do not appear at the extension level of . When expressing the rules of inductive basic covers we use the abbreviation $$a\,\epsilon\, V \qquad \mbox{ to mean }\qquad \mathsf{Ap}(\, V\, , \, a\, )$$ for any set $A$, any small propositional function $ V\in A\rightarrow\mathsf{prop_s}$ and any element $a\in A$.\ The precise rules of inductive basic covers extending  to form a new type system [[**mTT**]{}$_{ind}$]{} are the following: [**Rules of inductively generated basic covers in [[**mTT**]{}$_{ind}$]{}**]{}\ $$\begin{array}{l} \mbox{\rm F-$\triangleleft$} \ \displaystyle{\frac{\begin{array}{l} A \ set \ \ \ \ I(x)\ set \ [x\in A] \ \ \ C(x,j) \in A\rightarrow \mathsf{prop_s}\ [x\in A, j\in I(x)]\\ V\in A\rightarrow\mathsf{prop_s}\qquad a\in A \end{array} } {\displaystyle a\triangleleft_{I,C} V\ prop_s }}\\[20pt] \mbox{\rm rf-$\triangleleft$} \ \displaystyle{\frac{\begin{array}{l} A \ set \ \ \ \ I(x)\ set \ [x\in A] \ \ \ C(x,j) \in A \rightarrow \mathsf{prop_s} \ \ [x\in A, j\in I(x)]\\ V\in A \rightarrow \mathsf{prop_s} \qquad a\in A\qquad q\in a\,\epsilon\, V\ \end{array} } {\displaystyle \mathsf{rf}(a,q)\in a\triangleleft_{I,C} V}} \\[20pt] \end{array}$$ $$\begin{array}{l} \mbox{\rm tr-$\triangleleft$} \ \displaystyle{ \frac{\begin{array}{l} A\ set \ \ \ \ I(x)\ set \ [x\in A] \ \ \ C(x,j) \in A \rightarrow \mathsf{prop_s}\ [x\in A, j\in I(x)]\\ V\in A \rightarrow \mathsf{prop_s} \qquad\ \ a\in A\qquad i\in I(a)\\ q\in \forall_{x\in A}\ (\ x\,\epsilon\, C(a,i)\rightarrow x\triangleleft_{I,C} V\ ) \ \ \end{array} } {\displaystyle \mathsf{tr}(a,i,q)\in a\triangleleft_{I,C}V \ }} \\[20pt] \mbox{ind-$\triangleleft$} \displaystyle{\frac{ \begin{array}{l} A\ set \ \ \ \ I(x)\ set \ [x\in A] \ \ \ C(x,j)\ \in {\mathcal P}( A)\ \ [x\in A, j\in I(x)]\\[5pt] P(x)\ prop\ [x\in A]\qquad V\in A\ \rightarrow\ \mathsf{prop_s}\qquad \\[5pt] a\in A\qquad m\in a \cov_{I,C} V\qquad \\[5pt] q_1(x,z)\in P(x)\ [ x\in A, z\in x\,\epsilon\, V]\qquad \qquad \\[5pt] q_2(y,j, f) \in P(y)\ [ y\in A, j\in I(y), f\in\forall_{z\in A}\ (\ z\,\epsilon\, C(y,j) \ \rightarrow \ P(z)\ )] \end{array} }{\mathsf{ind}(m, q_1,q_2) \in P(a) }} \\[20pt] \end{array}$$ $$\begin{array}{l} \mbox{C$_1$-ind} \displaystyle{\frac{ \begin{array}{l} A\ set \ \ \ \ I(x)\ set \ [x\in A] \ \ \ C(x,j)\ \in A\ \rightarrow\ \mathsf{prop_s}\ \ [x\in A, j\in I(x)]\\[5pt] P(x)\ prop\ [x\in A]\qquad V\in A\ \rightarrow \mathsf{prop_s}\qquad \\[5pt] a\in A\qquad q \in a \,\epsilon\, V\qquad \\[5pt] q_1(x,z)\in P(x)\ [ x\in A, z\in x\,\epsilon\, V]\qquad \qquad \\[5pt] q_2(y,j, f) \in P(y)\ [ y\in A, j\in I(y), f\in\forall_{z\in A}\ (\ z\,\epsilon\, C(y,j) \ \rightarrow \ P(z)\ )] \end{array} }{\mathsf{ind}(\mathsf{rf}(a,q) , q_1,q_2)=q_1(a, q) \in P(a) }} \\[20pt] \mbox{C$_2$-ind} \displaystyle{\frac{ \begin{array}{l} A\ set \ \ \ \ I(x)\ set \ [x\in A] \ \ \ C(x,j)\ \in A\ \rightarrow\ \mathsf{prop_s}\ \ [x\in A, j\in I(x)]\\[5pt] P(x)\ prop\ [x\in A]\qquad V\in A\ \rightarrow \mathsf{prop_s}\qquad \\[5pt] a\in A\qquad i\in I(a)\qquad q\in\forall_{x\in A}\ (\ x\,\epsilon\, C(a,i)\ \rightarrow\ x\triangleleft_{I,C} V\ ) \\[5pt] q_1(x,z)\in P(x)\ [ x\in A, z\in x\,\epsilon\, V]\qquad \qquad \\[5pt] q_2(y,j, f) \in P(y)\ [ y\in A, j\in I(y), f\in\forall_{z\in A}\ (\ z\,\epsilon\, C(y,j) \ \rightarrow \ P(z)\ )] \end{array} } {\mathsf{ind}(\mathsf{tr}(a,i, q) , q_1,q_2)=q_2 (\, a\, ,i\, ,\, \lambda z.\lambda u.\, \mathsf{ind}(\, \mathsf{Ap}(\mathsf{Ap}(q,z),u)\, ,q_1,q_2)\, ) \in P(a) }} \end{array}$$\ Note that the cover relation preserves extensional equality of subsets represented as small propositional functions thanks to the induction principle: *For any axiom set in [[**mTT**]{}$_{ind}$]{} represented by a set $A$ with $ I(x)\ set\ [x\in A]$ and $ C(x,j)\in A\ \rightarrow \mathsf{prop_s} \ \ [x\in A, j\in I(x)]$, for any propositional functions $ V_1\in A\ \rightarrow \mathsf{prop_s}$ and $ V_2\in A\ \rightarrow \mathsf{prop_s}$, there exists a proof-term $$q\in V_1=_{ext} V_2\ \rightarrow\ a\triangleleft_{I,C} V_1=_{ext} a \triangleleft_{I,C} V_2$$ where for any small propositional functions $W_1$ and $W_2$ on a set $A$ we abbreviate $$W_1=_{ext}W_2 \, \equiv\, \forall_{x\in A}\ (\ W_1(x)\ \leftrightarrow\ W_2(x)\ )$$* Recalling that the interpretation of  in  in [@m09] interprets a set $A$ as an extensional quotient defined in as a set $A^J$ of  equipped with an equivalence relation $=_{A^J}$ over $A^J$, as well as families of sets are interpreted as families of extensional sets preserving the equivalence relations in their telescopic contexts, it is crucial to use lemma \[eqcov\] to interpret basic covers of on a base $A$ as basic covers $\cov_{I^J,C^J}^{=_{A^J}}$ of [[**mTT**]{}$_{ind}$]{}  defined on the support $A^J$: \[fullcv\] The interpretation of  in  in [@m09] extends to an interpretation of  in [[**mTT**]{}$_{ind}$]{} by interpreting $a \cov_{I,C} V$ for $a\in A$ and $V\in {\mathcal P}(A)$ in the corresponding basic cover in [[**mTT**]{}$_{ind}$]{} induced over the support of $A^J$ which is an extensional proposition in the sense of [@m09]. The fragment  of intensional Martin-L[ö]{}f’s type theory with inductive basic covers ===================================================================================== We here briefly describe the theory   obtained by adding the rules of inductive basic covers to the first order fragment of intensional Martin-L[ö]{}f’s type theory in [@PMTT] with one universe. This is essentially a fragment of intensional Martin-L[ö]{}f’s type theory which interprets [[**mTT**]{}$_{ind}$]{} as soon as propositions are identified as sets following the Curry-Howard correspondence in [@PMTT] but with the warning that we strengthen the elimination rule of inductive basic covers to act towards sets depending on their proof-terms according to inductive generation of types in Martin-L[ö]{}f’s type theory. Actually the interpretation of [[**mTT**]{}$_{ind}$]{} into  validates also the axiom of choice  as formulated in the introduction. Therefore in order to show the consistency of [[**mTT**]{}$_{ind}$]{} with +  (with  formulated as in the introduction) is enough to show the consistency of  extended with (the translation of) . Here we adopt the notation of types and terms within the first order fragment  of intensional Martin-L[ö]{}f’s type theory with one universe $U_0$ à la Tarsky in [@IMMS] and we just describe the rule of inductive basic covers added to it. To this purpose we add to   the code $$a\, \widehat{\triangleleft}_{s,i,}\, v\in U_0\qquad \mbox{ for } a\in \mathsf{T}(s) \mbox{ and } v\in \mathsf{T}(s)\ \rightarrow \ U_0$$ meaning that [*the element $a$ of a small set $\mathsf{T}(s)$ represented by the code $s\in U_0$ is covered by the subset $v$*]{} represented by a small propositional function from $\mathsf{T}(s)$ to the (large) set of small propositions identified with $U_0$ by the propositions-as-sets correspondence. Moreover, we use the abbreviations $$a\triangleleft_{s,i,c} v\, \equiv\, \mathsf{T}(a\,\widehat{\triangleleft}_{s,i,c}\, v) \qquad \qquad x\,\epsilon\, y\, \equiv\, \mathsf{T}(\mathsf{Ap}(y,x))$$ and the notation $$\mathsf{axcov}(s,i,c)$$ to abbreviate the following judgements $$s\in U_0 \ \ \ \ i(x)\in U_0 \ [x\in \mathsf{T}(s)] \ \ \ c(x,y)\in \mathsf{T}(s)\rightarrow U_0 \ [x\in \mathsf{T}(s), y\in \mathsf{T}(i(x))]$$ Then, the precise rules of inductive basic covers extending  to form a new type system  are the following:\ [**Rules of inductively generated basic covers in** ]{} $$\begin{array}{l} \mbox{\rm F-$\triangleleft$} \ \displaystyle{\frac{\begin{array}{l} s\in U_0\qquad i(x)\in U_0\ [x\in \mathsf{T}(s)] \ \ \ c(x,y) \in \mathsf{T}(s)\rightarrow U_0\ \ [x\in \mathsf{T}(s), y\in \mathsf{T}(i(x))]\\ a\in \mathsf{T}(s)\qquad v\in \mathsf{T}(s)\rightarrow U_0\ \end{array} } {\displaystyle a\,\widehat{\triangleleft}_{s,i,c}\, v\in U_0 }}\\ \\[20pt] \end{array}$$ $$\begin{array}{l} \mbox{\rm rf-$\triangleleft$} \ \displaystyle{\frac{\begin{array}{l} s \in U_0 \qquad i(x)\in U_0\ [x\in \mathsf{T}(s)] \ \ \ c(x,y)\in \mathsf{T}(s)\rightarrow U_0 \ \ [x\in \mathsf{T}(s), y\in \mathsf{T}(i(x))]\\ a\in \mathsf{T}(s)\qquad v\in \mathsf{T}(s)\rightarrow U_0\qquad\qquad r\in a\,\epsilon\, v\ \end{array} } {\displaystyle\mathsf{rf}(a,r)\in a \triangleleft_{s,i,c} v}}\\ \\[20pt] \end{array}$$ $$\begin{array}{l} \mbox{\rm tr-$\triangleleft$} \ \displaystyle{ \frac{\begin{array}{l} s\in U_0\ \ \ \ i(x)\in U_0 \ [x\in \mathsf{T}(s)] \ \ \ c(x,y)\in \mathsf{T}(s)\rightarrow U_0 \ \ [x\in \mathsf{T}(s), y\in \mathsf{T}(i(x))]\\ a\in \mathsf{T}(s)\qquad j\in \mathsf{T}(i(a))\qquad v\in \mathsf{T}(s)\rightarrow U_0 \\ r\in (\Pi x\in \mathsf{T}(s))(x\,\epsilon\,c(a,j)\rightarrow x \triangleleft_{s,i,c}v) \end{array} } {\displaystyle \mathsf{tr}(a,j,r) \in a\triangleleft_{s,i,c}v }}\\ \\[10pt] \end{array}$$ $$\begin{array}{l} \mbox{ind-$\triangleleft$} \displaystyle{\frac{ \begin{array}{l} \mathsf{axcov}(s,i,c)\\ v\in \mathsf{T}(s)\rightarrow U_0 \qquad P(x,u)\ type\ [x\in \mathsf{T}(s), u\in x\triangleleft_{s,i,c}v]\qquad \\ a \in \mathsf{T}(s) \qquad m\in a\triangleleft_{s,i,c}v\\ q_1 (x,z)\in P(x,\mathsf{rf}(x,z))\ [x\in \mathsf{T}(s), z\in x\,\epsilon\, v]\\[5pt] q_2(x,j ,f,k)\in P(x,\mathsf{tr}(x,j,k))\ \ \\ \quad \qquad [x\in \mathsf{T}(s), j\in \mathsf{T}(i(x)),\\ \qquad \qquad f\in (\Pi z\in \mathsf{T}(s))(z\,\epsilon\,c(x,j) \rightarrow P(z)), k\in (\Pi x\in \mathsf{T}(s))(x\,\epsilon\,c(a,j)\rightarrow x \triangleleft_{s,i,c}v)] \end{array} }{\mathsf{ind}(m, q_1,q_2)\in P(a,m) }}\\[20pt] \end{array}$$ $$\begin{array}{l} \mbox{C$_1$-ind-$\triangleleft$} \displaystyle{\frac{ \begin{array}{l} \mathsf{axcov}(s,i,c)\\ v\in \mathsf{T}(s)\rightarrow U_0 \qquad P(x,u)\ type\ [x\in \mathsf{T}(s),u\in x\triangleleft_{s,i,c}v ]\\ a \in \mathsf{T}(s) \qquad r\in a\,\epsilon\, v\\ q_1 (x,z)\in P(x,\mathsf{rf}(x,z))\ [x\in \mathsf{T}(s), z\in x\,\epsilon\, v]\\[5pt] q_2(x,j ,f,k )\in P(x,\mathsf{tr}(x,j,k))\ \ \\ \quad \qquad [x\in \mathsf{T}(s), j\in \mathsf{T}(i(x)),\\ \qquad \qquad f\in (\Pi z\in \mathsf{T}(s))(z\,\epsilon\,c(x,j) \rightarrow P(z)), k\in (\Pi x\in \mathsf{T}(s))(x\,\epsilon\,c(a,j)\rightarrow x \triangleleft_{s,i,c}v)] \end{array} }{\mathsf{ind}(\mathsf{rf}(a,r), q_1,q_2)= q_1(a,r )\in P(a,\mathsf{rf}(a,r)) }}\\[20pt] \end{array}$$ $$\begin{array}{l} \mbox{C$_2$-ind-$\triangleleft$} \displaystyle{\frac{ \begin{array}{l} \mathsf{axcov}(s,i,c)\\ v\in \mathsf{T}(s)\rightarrow U_0 \qquad P(x,u)\ type\ [x\in \mathsf{T}(s),u\in x\triangleleft_{s,i,c}v ]\\ a \in \mathsf{T}(s) \qquad j\in \mathsf{T}(i(a)) \qquad r\in (\Pi x\in \mathsf{T}(s))(x\,\epsilon\,c(a,j)\rightarrow x \triangleleft_{s,i,c}v)\\ q_1 (x,z)\in P(x,\mathsf{rf}(x,z))\ [x\in \mathsf{T}(s), z\in x\,\epsilon\, v]\\[5pt] q_2(x,j ,f,k )\in P(x,\mathsf{tr}(x,j,k))\ \ \\ \quad \qquad [x\in \mathsf{T}(s), j\in \mathsf{T}(i(x)),\\ \qquad \qquad f\in (\Pi z\in \mathsf{T}(s))(z\,\epsilon\,c(x,j) \rightarrow P(z)), k\in (\Pi x\in \mathsf{T}(s))(x\,\epsilon\,c(a,j)\rightarrow x \triangleleft_{s,i,c}v)] \end{array} }{\mathsf{ind}(\mathsf{tr}(a,j,r), q_1,q_2)= q_2(a,j,\lambda z.\lambda u.\mathsf{ind}(\mathsf{Ap}(\mathsf{Ap}(r,z),u),q_1,q_2),r)\in P(a,\mathsf{tr}(a,j,r))) }}\\[20pt] \end{array}$$ A crucial difference from the ordinary versions of Martin-Löf’s type theory is that for ${\mbox{\bf MLtt$_{ind}$}}$ we *postulate just the replacement rule repl)* $$\begin{array}{l} \mbox{repl)} \ \ \displaystyle{ \frac { \displaystyle \begin{array}{l} c(x_1,\dots, x_n)\in C(x_1,\dots,x_n)\ \ [\, x_1\in A_1,\, \dots,\, x_n\in A_n(x_1,\dots,x_{n-1})\, ] \\[2pt] a_1=b_1\in A_1\ \dots \ a_n=b_n\in A_n(a_1,\dots,a_{n-1}) \end{array}} {\displaystyle c(a_1,\dots,a_n)=c(b_1,\dots, b_n)\in C(a_1,\dots,a_{n}) }} \end{array}$$ in place of the usual congruence rules which would include the $\xi$-rule in accordance with the rules of  in [@m09], and hence of [[**mTT**]{}$_{ind}$]{}. The motivation for this restriction in [[**mTT**]{}$_{ind}$]{} and in   is due to the fact that the realizability semantics we present in the next sections, based on that in [@IMMS] and hence on the original Kleene realizability in [@DT88], does not validate the $\xi$-rule[^2] of lambda-terms $$\mbox{ $\xi$} \ \displaystyle{\frac{ \displaystyle c=c'\in C\ [x\in B] } { \displaystyle \lambda x^{B}.c=\lambda x^{B}.c' \in (\Pi x\in B) C}}$$ which is instead valid in [@PMTT]. It is indeed an open problem whether the original intensional version of Martin-L[ö]{}f’s type theory in [@PMTT], including the $\xi$-rule of lambda terms, is consistent with . It worth noting that the lack of the $\xi$-rule does not affect the possibility of adopting  as the intensional level of a two-level constructive foundation as intended in [@mtt], since its term equality rules suffice to interpret an extensional level including extensionality of functions, as that represented by , by means of the quotient model as introduced in [@m09] and studied abstractly in  [@elqu; @qu12; @uxc]. Furthermore our realizability semantics interprets terms as applicative terms in the first Kleene algebra and their equality as numerical equality turning into an extensional equality in the context-dependent case. Hence we need a suitable encoding of lambda-terms which validates the replacement rule under the interpretation. As observed in [@IMMS] not each translation of pure lambda calculus in the first Kleene algebras satisfies this requirement (see pp.881-882 in [@IMMS]). \[intttmtt\] The interpretation of  into  given in [@m09] extends to that of [[**mTT**]{}$_{ind}$]{} in  by interpreting each basic cover $\cov_{I,C}$ of [[**mTT**]{}$_{ind}$]{} associated to an axiom set $I(-)$ and $C(-,-)$ in the corresponding basic cover of  associated to the interpreted axiom set. Note that small propositions are encoded in the universe $U_{0}$ as well as axiom sets generating a basic cover inductively in [[**mTT**]{}$_{ind}$]{}. *It is worth recalling that for any axiom set represented by a set $A$ with $ I(x)\ set\ [x\in A]$ and $ C(x,j)\in A\ \rightarrow \ U_0 \ [x\in A, j\in I(x)]$ and any propositional function $V\in A\ \rightarrow \ U_0$ representing a subset of $A$, the propositional function representing the subset $$\cov_{I,C}(V)\, \equiv\, \{ x\in A\ \mid\ x\cov_{I,C} V\ \}$$ is definable in the extension with well-founded sets as shown in [@silviobar]. A direct representation of $\cov_{I,C}(V)\in A\rightarrow U_0$ is obtained as the well founded set $(\mathsf{W}x\in D)\ B(x)$ where $$D\, \equiv\, (\Sigma x\in A)\ (\, a\,\epsilon\, V\ + I(x)\ ) $$ and $B(x)\ [x\in D]$ is the inductive type defined by recursion on $D$ toward the first universe $U_{0}$ satisfying the following conditions:* $$B(x)\, \equiv\, \begin{cases} N_0 & \mbox{ if } x=\langle a, \mathsf{inl}(z)\rangle \ \mbox{ for } z\in a\,\epsilon\, V\\ C(\ a\, ,\, j\ ) & \mbox{ if } x=\langle a, \mathsf{inr}(j)\rangle \ \mbox{ for } j\in I(a) \end{cases}$$ where we recall that $\mathsf{inl}$ and $\mathsf{inr}$ are the injections in the sum and $\pi_1$ and $\pi_2$ are the projections of the indexed sum. Then the terms of the introduction and elimination rules for basic covers can be represented by means of those of well founded sets. For example we can put $$\begin{array}{rcl} \mathsf{rf}(a,r)\, &\equiv\, &\mathsf{sup}(\ \langle a, \mathsf{inl}(q)\rangle\, \ ,\ \lambda x. r_o(x)\ ) \\ \mathsf{tr}(a,i,q)\,& \equiv\, &\mathsf{sup}(\ \langle a, \mathsf{inr}(j)\, \rangle \, ,\, \lambda y. q (a, j,y) \ ) \end{array}$$ where $r_0(x)$ is the eliminator of the empty set $N_0$. A realizability interpretation of  with Formal Church’s Thesis ============================================================== Here we are going to describe a realizability model of  with extending that of  in [@IMMS]. A main novelty here is that we formalize such a model in the (generalized) predicative and constructive theory $\mathbf{CZF+REA}$ where $\mathbf{CZF}$ stands for Constructive Zermelo-Fraenkel Set Theory and $\mathbf{REA}$ stands for the regular extension axiom (for details see [@czf; @czf2]). Since the interpretation in [@IMMS] is performed in ${\widehat{ID_1}}$ which is a classical theory of fixed points, we cannot follow the proof technique in [@IMMS] to fulfill our purpose. Moreover ${\widehat{ID_1}}$ is a too weak theory to accommodate inductively defined topologies as it can be gleaned from [@CR]. The solution is to adopt the proof-technique in [@R93; @RG94] to fulfill our goal. As usual in set theory we identify the natural numbers with the finite ordinals, i.e. $\mathbb{N}:=\omega$. To simplify the treatment we will assume that $\mathbf{CZF}$ has names for all (meta) natural numbers. Let $\overline{n}$ be the constant designating the $n^{th}$ natural number. We also assume that $\mathbf{CZF}$ has function symbols for addition and multiplication on $\mathbb{N}$ as well as for a primitive recursive bijective pairing function $p:\mathbb{N}\times \mathbb{N}\rightarrow \mathbb{N}$ and its primitive recursive inverses $p_0$ and $p_1$, that satisfy $p_0(p(n,m))=n$ and $p_1(p(n,m))=m$. We also assume that $\mathbf{CZF}$ is endowed with symbols for a primitive recursive length function $\ell:\mathbb{N}\rightarrow \mathbb{N}$ and a primitive recursive component function $(-)_{-}:\mathbb{N}\times \mathbb{N}\rightarrow \mathbb{N}$ determining a bijective encoding of finite lists of natural numbers by means of natural numbers. $\mathbf{CZF}$ should also have a symbol $T$ for Kleene’s $T$-predicate and the result extracting function $U$. Let $P(\{e\}(n))$ be a shorthand for $\exists m(T(e,n,m)\wedge P(U(m)))$. Further, let $p(n,m,k):=p(p(n,m),k)$, $p(n,m,k,h):=p(p(n,m,k),h)$, etc…. We use $a,b,c,d,e,d,f,n,m,l,k,s,t,j,i$ as metavariables for natural numbers. We first need to introduce some abbreviations: 1. $\mathsf{n}_0$ is $p(0,0)$, $\mathsf{n}_1$ is $p(0,1)$ and $\mathsf{n}$ is $p(0,2)$. 2. $\sigma(a,b)$ is $p(1,p(a,b))$, $\pi(a,b)$ is $p(2,p(a,b))$ and $+(a,b)$ is $p(3,p(a,b))$ 3. $\mathsf{list}(a)$ is $p(4,a)$ and $\mathsf{id}(a,b,c)$ is $p(5,p(a,b,c)))$ 4. $a\widetilde{\triangleleft}_{c,d,e}b$ is $p(6,p(a,b,c,d,e))$ 5. $\rho(a,r)$ is $p(7,p(a,r))$ 6. $\tau(a,j,r)$ is $p(8,p(a,j,r))$ Recall that, in intuitionistic set theories, ordinals are defined as transitive sets all of whose members are transitive sets, too. Unlike in the classical case, one cannot prove that they are linearly ordered but they are perfectly good as a scale along which one can iterate various processes. The trichotomy of $0$, successor, and limit ordinal, of course, has to be jettisoned. \[fixdef\] By transfinite recursion on ordinals (cf. [@czf2], Proposition 9.3.3) we define simultaneously two relations $\mathsf{Set}_\alpha(n)$ and $n\,\varepsilon _\alpha\,m$ on $\mathbb{N}$ in $\mathbf{CZF+REA}$. In the following definition we use the shorthand $\mathsf{Fam}_\alpha(e,k)$ to convey that $\mathsf{Set}_\alpha(k)$ and $\forall j(j\,\varepsilon _\alpha\,k\rightarrow \mathsf{Set}_\alpha(\{e\}(j)))$ and we shall write $\mathsf{Set}_{\in \alpha}(n)$ for $\exists \beta\in \alpha(\mathsf{Set}_\beta (n))$, $n\,\varepsilon _{\in\alpha}\,m$ for $\exists \beta\in \alpha(n\,\varepsilon _\beta\,m)$ and $\mathsf{Fam}_{\in\alpha}(e,k)$ for $\exists \beta\in \alpha(\mathsf{Fam}_\beta(e,k))$. 1. 2. $\mathsf{Set}_\alpha(\mathsf{n}_j)$ iff $j=0$ or $j=1$, and $m\,\varepsilon _\alpha\,\mathsf{n}_j$ iff $m<j$; 3. $\mathsf{Set}_\alpha(\mathsf{n})$ holds, and $m\,\varepsilon _\alpha\,\mathsf{n}$ iff $m\in \mathbb{N}$. 4. 5. If $\mathsf{Fam}_{\in \alpha}(e,k)$, then $\mathsf{Set}_\alpha(\pi(k,e))$ and $\mathsf{Set}_\alpha(\sigma(k,e))$; 6. if $\mathsf{Fam}_{\in \alpha}(e,k)$, then 1. $n\,\varepsilon _\alpha\,\pi(k,e)$ iff there exists $\beta\in \alpha$ such that $\mathsf{Fam}_{\beta}(e,k)$ and $\forall i(i\,\varepsilon _\beta\,k\rightarrow \{n\}(i)\,\varepsilon _\beta \{e\}(i))$ 2. $n\,\varepsilon _\alpha\,\sigma(k,e)$ iff there exists $\beta\in \alpha$ such that $\mathsf{Fam}_{\beta}(e,k)$, $p_0(n)\,\varepsilon _\beta\,k\,\wedge\, p_1(n)\varepsilon _\beta \{e\}(p_0(n))$ 7. 8. If there exists $\beta\in\alpha$ such that $\mathsf{Set}_\beta(n)$ and $\mathsf{Set}_\beta(m)$, then $\mathsf{Set}_\alpha(+(n,m))$, and 9. $i\,\varepsilon _\alpha\, +(n,m)$ iff there exists $\beta\in\alpha$ such that $\mathsf{Set}_\beta(n)$, $\mathsf{Set}_\beta(m)$ and $$(p_0(i)=0\wedge p_1(i)\,\varepsilon _\beta\,n)\vee(p_0(i)=1\wedge p_1(i)\,\varepsilon _\beta\,m)$$ 10. 11. If there exists $\beta\in\alpha$ such that $\mathsf{Set}_\beta(n)$, then $\mathsf{Set}_\alpha(\mathsf{list}(n))$, and 12. $i\,\varepsilon _\alpha\, \mathsf{list}(n)$ iff there exists $\beta\in\alpha$ such that $\mathsf{Set}_\beta(n)$ and $\forall j(j<\ell(i)\rightarrow (i)_j \,\varepsilon _\beta\,n)$. 13. 14. If $\mathsf{Set}_{\in \alpha}(n)$, then $\mathsf{Set}_\alpha(\mathsf{id}(n,m,k))$, and 15. $s\,\varepsilon _\alpha\,\mathsf{id}(n,m,k)$ iff there exists $\beta\in \alpha$ such that $\mathsf{Set}_{\beta}(n)$, $m\,\varepsilon _\beta\,n$ and $s=m=k$. 16. 17. Let $\beta\in \alpha$. Suppose that the following conditions (collectively called $*_\beta$) are satisfied: 1. $\mathsf{Set}_\beta(s)$, 2. $a\,\varepsilon _\beta\,s$, 3. $\mathsf{Fam}_\beta(v,s)$, 4. $\mathsf{Fam}_\beta(i,s)$ and 5. $\forall x\forall y(x\,\varepsilon _\beta s\,\wedge\, y\,\varepsilon _\beta \{i\}(x)\rightarrow \mathsf{Fam}_\beta(\{\{c\}(x)\}(y),s))$, then $\mathsf{Set}_\alpha(a\widetilde{\triangleleft}_{s,i,c} v)$; 18. assuming $*_\beta$, let $C_\beta(a\widetilde{\triangleleft}_{s,i,c} v)$ be the smallest subsets of $\mathbb{N}$ such that whenever $r\,\varepsilon _\beta\{v\}(a)$ then $\rho(a,r)\in C_\beta(a\widetilde{\triangleleft}_{s,i,c} v)$ and whenever $j\,\varepsilon _\beta\{i\}(a)$ and $$\forall z\forall s(z\,\varepsilon _\beta a\wedge s\,\varepsilon _\beta \{\{\{c\}(a)\}(j)\}(z)\rightarrow \{r\}(z,s)\in C_\beta(z\widetilde{\triangleleft}_{s,i,c} v)))$$ then $\tau(a,j,r)\in C_\beta(a\widetilde{\triangleleft}_{s,i,c} v)$. 19. The existence of the set $C_\beta(a\widetilde{\triangleleft}_{s,i,c} v)$ is guaranteed by the axiom $\mathbf{REA}$. 20. Finally we define $q\,\varepsilon _\alpha\,a\widetilde{\triangleleft}_{s,i,c} v$ iff $\exists \beta\in \alpha(*_\beta \wedge q\in C_\beta(a\widetilde{\triangleleft}_{s,i,c} v))$. *It is worth noting that in the above definition the interpretation of the Propositional Identity $\widehat{\mathsf{Id}}(s, a,b)\in U_0$ for $s\in U_0$ and $a\in \mathsf{T}(s)$ and $b\in \mathsf{T}(s)$ agrees with that in [@IMMS] which validates the rules of the [*extensional Propositional Identity*]{} in [@PMTT]. Then also our realizability semantics actually validates the extensional version of . Hence the elimination rule of inductive basic covers can be equivalently weakened to act towards types non dipendenting on proof-terms of basic covers, as soon as we add a suitable $\eta$-rule in a similarly way to what happens to the rules of first-order types (like disjoint sums or natural numbers or list types) in the extensional type theories in [@tumscs].* Here we have a crucial lemma. \[fix\] In $\mathbf{CZF}+\mathbf{REA}$, for all $m\in \mathbb{N}$, if $\mathsf{Set}_\alpha(m)$, then for all $\rho$ such that $\mathsf{Set}_\rho(m)$, $$\forall i\in \mathbb{N}(i\,\varepsilon _\alpha\,m\leftrightarrow i\,\varepsilon _\rho\,m).$$ We proceed by induction on $\alpha$. Suppose $\mathsf{Set}_\alpha(m)$ and $\mathsf{Set}_\rho(m)$. We look at the forms $m$ can have. If $m$ is $\mathsf{n}_0$, $\mathsf{n}_1$ or $\mathsf{n}$, then the claim is immediate in view of clauses $(1)$ and $(2)$ in the previous definition. If $m$ is of the form $\pi(k,e)$, then there exists $\beta\in \alpha$ such that $\mathsf{Fam}_\beta(e,k)$. The induction hypothesis applied to $\beta$ yelds that whenever $\mathsf{Fam}_\xi(e,k)$, then $$\forall j\in \mathbb{N}(i\,\varepsilon _\beta\,m\leftrightarrow i\,\varepsilon _\xi\,m)$$ $$\forall i\in \mathbb{N}\forall j\in \mathbb{N}(i\,\varepsilon _\beta\,m\rightarrow (j\,\varepsilon _\beta\,\{e\}(i)\leftrightarrow j\,\varepsilon _\xi\,\{e\}(i)))$$ The thesis follows from these. If $m$ is either $\sigma(k,e)$, $+(a,b)$, $\mathsf{list}(a)$ or $\mathsf{id}(a,b,c)$ the argument proceeds as in the previous case. If $m$ is of the form $a\widetilde{\triangleleft}_{s,i,c}v$, the proof is similar, although more involved. We define in $\mathbf{CZF}+\mathbf{REA}$ the formula $\mathsf{Set}(n)$ as $\exists \alpha(\mathsf{Set}_\alpha(n))$ and $x\,\varepsilon \,y$ as $\exists \alpha(x\,\varepsilon _\alpha\, y)$. \[real\] The theory  is consistent with the formal Church thesis . We outline a realizability semantics in [**CZF**]{}+$\mathbf{REA}$. Every preterm is interpreted as a $\mathcal{K}_1$-applicative term (that is, a term built with numerals and Kleene application) as it is done in [@IMMS]. We only need to interpret the new preterms of  that is: 1. $(a\widehat{\triangleleft}_{s,i,c}v)^{I}$ is defined as $\{\mathbf{p}\}(6,\{\mathbf{p}_5\}(a^I,v^I,s^I,\Lambda x.i^I,\Lambda x.\Lambda y.c^I))$, where $\mathbf{p}$ and $\mathbf{p}_5$ are numeral representing the encoding of pairs of natural numbers and of $5$-tuples of natural numbers, respectively[^3]; 2. $(\mathsf{rf}(a,r))^I:=\{\mathbf{p}\}(7,\{\mathbf{p}\}(a^I,r^I))$; 3. $(\mathsf{tr}(a,j,r))^I:=\{\mathbf{p}\}(8,\{\mathbf{p}_3\}(a^I,j^I,r^I))$, where $\mathbf{p}_3$ is a numeral representing the encoding of triples of natural numbers; 4. $(\mathsf{ind}(m,q_1,q_2))^I$ is $\{\mathbf{ind}_{q_1,q_2}\}(m^I)$ where $\mathbf{ind}_{q_1,q_2}$ is the code of a recursive function such that 1. $\mathbf{ind}_{q_1,q_2}(\rho(a,r))\simeq\{\{\Lambda x.\Lambda z. q_1^I\}(a^I)\}(r^I)$ 2. $\mathbf{ind}_{q_1,q_2}(\tau(a,j,r))\simeq\{\Lambda x.\Lambda k.\Lambda f.\Lambda k.q_{2}^I\}(a,j,\Lambda y.\Lambda s.\mathbf{ind}_{q_1,q_2}(\{\{r\}(y)\}(s)),r )$ If $\tau$ is an $\mathcal{K}_1$-applicative term, we will define $\tau\,\varepsilon\, A$ as an abbreviation for $\phi[\tau/x]$. We will interpret pretypes as definable subclasses of $\mathbb{N}$ in $\mathbf{CZF}+\mathbf{REA}$ as follows: 1. $\mathsf{N}_{0}^{I}:=\{x\in \mathbb{N}|\,\bot\}$ 2. $\mathsf{N}_{1}^{I}:=\{x\in \mathbb{N}|\,x=0\}$ 3. $((\Sigma y\in A)B)^{I}:=\{x\in \mathbb{N}|\,p_0(x)\in A^{I}\wedge p_1(x)\in B^{I}[p_0(x)/y] \}$ 4. $((\Pi y\in A)B)^{I}:=\{x\in \mathbb{N}|\,\forall y\in \mathbb{N}\,(y\in A^{I}\rightarrow \{x\}(y)\in B^{I})\}$ 5. $(A+B)^{I}:=\{x\in \mathbb{N}|\,(p_0(x)=0 \wedge p_1(x)\in A^{I})\vee(p_0(x)=1 \wedge p_1(x)\in B^{I})\}$ 6. $(\mathsf{List}(A))^{I}:=\{x\in \mathbb{N}|\,\forall i\in \mathbb{N}\,(i<\ell(x)\rightarrow (x)_{i}\in A^{I})\}$ 7. $(\mathsf{Id}(A,a,b))^{I}:=\{x\in \mathbb{N}|\,x=a^{I}\wedge a^{I}=b^{I}\, \wedge\, a^{I}\in A^{I}\}$ 8. $U_0^{I}:=\{x|\,\mathsf{Set}(x)\}$ 9. $\mathsf{T}(a)^{I}:=\{x|\,x\,\varepsilon \,a^{I}\}$ Precontexts are interpreted as conjunctions of formulas of $\mathbf{CZF}+\mathbf{REA}$ as follows. 1. $[\;]^{I}$ is the formula $\top$; 2. $[\Gamma, x\in A]^{I}$ is the formula $\Gamma^{I}\,\wedge\, x^{I}\in A^{I}$. Validity of judgements $J$ in the model is defined as follows: 1. $A\, type\,[\Gamma]$ holds if $\Gamma^{I}\vdash_{\mathbf{CZF}+\mathbf{REA}} \forall x\,(x\in A^I\rightarrow x\in \mathbb{N})$ 2. $A=B\, type\,[\Gamma]$ holds if $\Gamma^{I}\vdash_{\mathbf{CZF}+\mathbf{REA}}\forall x\,(x\in A^{I}\leftrightarrow x\in B^{I})$ 3. $a\in A\,\,[\Gamma]$ holds if $\Gamma^{I}\vdash_{\mathbf{CZF}+\mathbf{REA}}a^{I}\in A^{I}$ 4. $a=b\in A\, \,[\Gamma]$ holds if $\Gamma^{I}\vdash_{\mathbf{CZF}+\mathbf{REA}}a^{I}\in A^{I}\wedge a^{I}=b^{I}$ where $x$ is a fresh variable. The encoding of lambda-abstraction in terms of $\mathcal{K}_1$-applicative terms can be chosen (see [@IMMS]) in such a way that if $a$ and $b$ are terms and $x$ is a variable which is not bounded in $a$, then the terms $(\,a[b/x]\,)^{I}$ and $a^{I}[\,b^{I}/x^{I}]$ coincide. The proof that for every judgement if ${\mbox{\bf MLtt$_{ind}$}}\vdash J$, then $J$ holds in the realizability model is a long, but straightforward verification. We just prove for the sake of example that the rules for the inductively generated covers (rf-$\triangleleft$) and (tr-$\triangleleft$) preserve the validity of judgments in the model in the empty-context case. 1. Suppose the premisses of the following rule are valid in the model. $$\mbox{\rm rf-$\triangleleft$} \ \displaystyle{\frac{\begin{array}{l} s \in U_0 \qquad i(x)\in U_0\ [x\in \mathsf{T}(s)] \ \ \ c(x,y)\in \mathsf{T}(s)\rightarrow U_0 \ \ [x\in \mathsf{T}(s), y\in \mathsf{T}(i(x))]\\ a\in \mathsf{T}(s)\qquad v\in \mathsf{T}(s)\rightarrow U_0\qquad\qquad r\in a\,\epsilon\, v\ \end{array} } {\displaystyle\mathsf{rf}(a,r)\in a \triangleleft_{s,i,c} v}}$$ Then, in particular $a^I\,\varepsilon \, s^I$ and $r^I\,\varepsilon \, \{v^I\}(a^I)$ hold in $\mathbf{CZF}+\mathbf{REA}$. As a consequence of definition \[fixdef\], we hence have that $\mathsf{rf}(a,r)^I=\rho(a^I,r^I)\,\varepsilon \,a^I\widetilde{\triangleleft}_{s^I,i^I,c^I}v^I$ holds in $\mathbf{CZF}+\mathbf{REA}$, but this is equivalent to the validity of the judgement $\mathsf{rf}(a,r)\in a \triangleleft_{s,i,c} v$ in the model. 2. Suppose the premisses of the following rule are valid in the model. $$\mbox{\rm tr-$\triangleleft$} \displaystyle{ \frac{\begin{array}{l} s\in U_0\ \ \ \ i(x)\in U_0 \ [x\in \mathsf{T}(s)] \ \ \ c(x,y)\in \mathsf{T}(s)\rightarrow U_0 \ \ [x\in \mathsf{T}(s), y\in \mathsf{T}(i(x))]\\ a\in \mathsf{T}(s)\qquad j\in \mathsf{T}(i(a))\qquad v\in \mathsf{T}(s)\rightarrow U_0 \\ r\in (\Pi x\in \mathsf{T}(s))(x\,\epsilon\,c(a,j)\rightarrow x \triangleleft_{s,i,c}v) \end{array} } {\displaystyle \mathsf{tr}(a,j,r) \in a\triangleleft_{s,i,c}v }}$$ Then, in $\mathbf{CZF}+\mathbf{REA}$, $a^{I}\,\varepsilon \, s^{I}$, $j^{I}\,\varepsilon \,\{i^{I}\}(a^{I})$, $\forall x\in \mathbb{N}\,(x\,\varepsilon \,s^{I}\rightarrow \mathsf{Set}(\{v^{I}\}(x)))$ and $$\forall x\in \mathbb{N}\,\forall y\in \mathbb{N}\,(x\,\varepsilon \,s^{I}\wedge y\,\varepsilon \,\{c^{I}(a^{I},j^{I})\}(x)\rightarrow \{r^I\}(x,y)\,\varepsilon \,\triangleleft(s^I,i^I,c^I;x,v^I))$$ Thus in particular, by definition \[fixdef\], $(\mathsf{tr}(a,j,r))^I=\tau(a^I,j^I,r^I)\,\varepsilon \,a^I\widetilde{\triangleleft}_{s^I,i^I,c^I}v^I$, which means that $\mathsf{tr}(a,j,r) \in a\triangleleft_{s,i,c}v $ is valid in the model. The theory [[**mTT**]{}$_{ind}$]{} is consistent with the axiom of choice, $\mathbf{AC}$, and Formal Church thesis, $\mathbf{CT}$. This follows from theorems  \[intttmtt\] and \[real\]. The theory [[**mTT**]{}$_{ind}$]{}$+\mathbf{AC}+\mathbf{CT}$ has an interpretation in the intensional version of the type theory $\mathbf{ML}_{\mathrm{1W}}\mathbf{V}$ in Definition 5.1 of [@R93] (or [@RG94]). This is a consequence of the proof of the above Theorem \[real\] and Proposition 5.3 in [@R93], namely the interpretability of $\mathbf{CZF}+\mathbf{REA}$ in $\mathbf{ML}_{\mathrm{1W}}\mathbf{V}$. [*In a certain sense there is nothing special about inductively generated basic covers in that the interpretation of  in $\mathbf{CZF}+\mathbf{REA}$ would also work if one added further inductive types such as generic well founded sets to . In the same vein one could add more universes or even superuniverses (see [@pasu; @rasu]) after beefing up the interpreting set theory by adding large set axioms. As a consequence one can conclude that intensional Martin-Löf type theory with some or all these type constructors added, but crucially missing the $\xi$-rule, is compatible with Church’s thesis.*]{}  and $\mathbf{CZF}+\mathbf{REA}$ have the same proof-theoretic strength. It follows from [@R93], Theorem 5.13, Theorem 6.9, Theorem 6.13 (or the same theorems in [@RG94]) together with the observation that the theory $\mathbf{IARI}$ of [@R93] in Definition 6.2 can already be interpreted in  using the interpretation of [@R93] in Definition 6.5. We just recall that $\mathbf{IARI}$ is a subsystem of second order intuitionistic number theory. It has a replacement schema and an axiom of [*inductive generation*]{} asserting that for every binary set relation $R$ on the naturals the well-founded part of this relation is a set. The interpretation for the second order variables are the propositions on the naturals with truth conditions in $U_0$. The crucial step is to interpret the axiom of [*inductive generation*]{} of $\mathbf{IARI}$ in . To this purpose one has to show that if $s\in U_0$ and $R\in \mathsf{T}(s)\times\mathsf{T}(s)\to U_0$ then the well-founded part of $R$, $\mathsf{WP}(R)$, can be given as a predicate $\mathsf{WP}(R)\in \mathsf{T}(s)\to U_0$. To this end define $i\in \mathsf{T}(s)\to U_0$ by $i(x):=s$, $v\in \mathsf{T}(s)\to U_0$ by $v(p):=\mathsf{n}_0$, $c(x,y)\in \mathsf{T}(s)\to U_0$ by $c(x,y)(z) := R(z,x)$ (so $y$ is dummy) for $x\in \mathsf{T}(s)$ and $y\in \mathsf{T}(s)$. Now let $\mathsf{WP}(R)(a):=a\triangleleft_{s,i,c}v$ for $a\in \mathsf{T}(s)$. Then it follows that $a$ is in the well-founded part exactly when $\mathsf{WP}(R)(a)$ is inhabited. To see this, suppose we have a truth maker $r$ for $(\Pi x\in \mathsf{T}(s))(R(x,a) \to \mathsf{WP}(R)(x))$. Then $r\in (\Pi x\in \mathsf{T}(s))(x\,\epsilon\, c(a,a) \to x\triangleleft_{s,i,c}v)$, hence $\mathsf{tr}(a,a,r)\in a\triangleleft_{s,i,c}v$ by ($\mathrm{tr}\mbox{-}\triangleleft)$, whence $\mathsf{tr}(a,a,r)\in \mathsf{WP}(R)(a)$. Thus $\mathsf{WP}(R)$ satisfies the appropriate closure properties characterizing the well-founded part of $R$. The pertaining induction principle is then a consequence of $(\mathrm{ind}\mbox{-}\triangleleft)$. *As an evidence of the validity of the previous theorem, one can notice that well founded sets of small sets in ${\mbox{\bf MLtt$_{ind}$}}$ can be represented by suitable inductive basic covers. Hence the claim essentially follows thanks to theorem 6.13 in [@R93].* Indeed, given a small set $s\in U_0$ and a family of small sets $b(x)\in U_0 \ [x\in T(a)]$ then the well founded set $(\mathsf{W}x\in\mathsf{T}(s))\mathsf{T}(b(x))$ on this family can be interpreted as the open cover on the empty subset $\emptyset\, \equiv\, \lambda x. \widehat{\mathsf{N}}_0\in \mathsf{T}(s)\ \rightarrow U_0$ $$(\mathsf{W}x\in\mathsf{T}(s))\mathsf{T}(b(x))\, \equiv\, \cov_{s,i,c}\,\emptyset$$ of the inductive basic cover generated by $$i(x)=\widehat{\mathsf{N}}_1 \qquad c(x,j)\, \equiv\, b(x)$$ for $x\in \mathsf{T}(s)$ and $j\in \mathsf{N}_1$. Then the term $\mathsf{sup}(a,f)$, for $a\in \mathsf{T}(s)$ and $f(x)\in (\mathsf{W}x\in\mathsf{T}(s))\mathsf{T}(b)\ [x\in \mathsf{T}(s)]$ - with the notation of p.98 in [@PMTT] - can be defined to be $\mathsf{tr}(a,\star,\lambda x.\lambda y.f (a)\, )$. Moreover, as one could expect, there is no term of the form $\mathsf{rf}(a,j)$ since $\mathsf{N}_0$ is the empty set. The elimination constructor of well-founded sets $\mathsf{wrec}(e,f)$ is defined as the term $\mathsf{ind}(e,\mathsf{r}_0,f)$ where $\mathsf{r}_0$ is the elimination constructor of the empty set. Conclusions {#conclusions .unnumbered} ----------- In the future we aim to further extend the realizability semantics presented here to model [[**MF$_{ind}$**]{}]{} enriched with coinductive definitions capable of representing generated Positive Topologies in [@somepoint]. A further goal would be to study the consistency strength of [[**mTT**]{}$_{ind}$]{} or of   extended with specific inductive formal topologies such as that of the real line. Acknowledgments {#acknowledgments .unnumbered} --------------- The first author acknowledges very helpful discussions and suggestions with F. Ciraulo, P. Martin-L[ö]{}f, G. Sambin and T. Streicher. The third author was supported by a grant from the John Templeton Foundation (“A new dawn of intuitionism: mathematical and philosophical advances," ID 60842). The opinions expressed in this publication are those of the authors and do not necessarily reflect the views of the John Templeton Foundation. [^1]: Projects EU-MSCA-RISE project 731143 “Computing with Infinite Data” (CID), MIUR-PRIN 2010-2011 and Correctness by Construction (EU 7th framework programme, grant no. PIRSES-GA-2013-612638) provided support for the research presented in the paper. [^2]: Notice that a trivial instance of the $\xi$-rule is derivable from repl) when $c$ and $c'$ don’t depend on $x^B$. [^3]: when we write $\{b\}(a_{1},...,a{n})$, we mean $\{...\{b\}(a_{1})\}(a_{2})...\}(a_{n})$
--- abstract: 'We show that a finite permutation group containing a regular abelian self-normalizing subgroup is soluble.' address: - 'Enrico Jabara, Dipartimento di Filosofia e Beni Culturali, University Cá Foscari, Dorsoduro 3484/D – 30123 Venezia, I-30100 Venezia, Italy. ' - 'Pablo Spiga, Dipartimento di Matematica Pura e Applicata, University Milano-Bicocca, Via Cozzi 53, 20126 Milano, Italy.' author: - Enrico Jabara - Pablo Spiga title: Abelian Carter subgroups in finite permutation groups --- Introduction {#intro} ============ Let $G$ and $A$ be finite groups with $A$ acting on $G$ as a group of automorphisms. When $\cent G A = 1$, it is customary to say that $A$ acts fixed-point-freely on $G$, that is, $1$ is the only element of $G$ invariant by every element of $A$. From the seminal work of Thompson [@Th], fixed-point-free groups of automorphisms have attracted considerable interest and have shown to be remarkably important within finite group theory. It is well-established that in many cases the condition $\cent G A=1$ forces the group $G$ to be soluble. One of the main contributions is the result of Rowley [@Ro], showing that groups admitting a fixed-point-free automorphism are soluble. This result was then generalized by Belyaev and Hartley [@BH], showing that if a nilpotent group acts fixed-point-freely on $G$, then $G$ is soluble. In this paper, as an application of this remarkable result, we prove the following. \[thm\] If $G$ is a finite transitive permutation group with a regular abelian self-normalizing subgroup, then $G$ is soluble. (A self-normalizing nilpotent subgroup of a finite group is called a *Carter* subgroup.) Theorem \[thm\] is provoked by some recent investigations [@DPS] on Cayley graphs over abelian groups. In fact, Dobson, Verret and the second author have recently proved a conjecture of Babai and Godsil [@BG Conjecture $2.1$] concerning the enumeration of Cayley graphs over abelian groups. In the approach in [@DPS], at a critical juncture [@DPS proof of Theorem $1.5$], it is necessary to have structural information on finite permutation groups admitting a regular abelian Carter subgroup. In view of [@DPS], any improvement in our understanding of the structure of the automorphism group of a Cayley graph over an abelian group requires a detailed description of the finite permutation groups containing a regular abelian Carter subgroup. Theorem \[thm\] is a first step in this direction. The hypothesis of $A$ being abelian is rather crucial in Theorem \[thm\]. For instance, for each Mersenne prime $p=2^\ell-1$, the group $\mathrm{PSL}_2(p)$ acts primitively on the points of the projective line and contains a regular Carter subgroup isomorphic to the dihedral group of order $2^\ell$. A well-known theorem of Carter [@Carter] shows that every finite soluble group contains exactly one conjugacy class of Carter subgroups. On the other hand, a finite non-soluble group may have no Carter subgroup, as witnessed by the alternating group ${\mathop{\mathrm{Aut}}}(5)$. However, using the Classification of the Finite Simple Groups, Vdovin [@Vdovin] has extended the result of Carter by showing that, if a finite group $G$ contains a Carter subgroup, then every two distinct Carter subgroups of $G$ are conjugate. The main ingredients in our proof of Theorem \[thm\] are two results of fundamental importance in finite group theory. The first is the classification of Li [@chli] of the finite primitive groups containing an abelian regular subgroup. This classification will allow us to prove Theorem \[thm\] for primitive groups by a simple direct inspection (see Proposition \[prim\]). The second is the remarkable theorem of Vdovin [@Vdovin] showing that any two Carter subgroups of a finite group are conjugate. This result will allow us to use induction for proving Theorem \[thm\]. In particular, as [@BH; @chli; @Vdovin] depend upon the Classification of the Finite Simple Groups, so does Theorem \[thm\]. Proof of Theorem \[thm\] {#proofs} ======================== Before proving Theorem \[thm\] we need an auxiliary result. \[prim\]Let $G$ be a finite primitive permutation group with a regular abelian subgroup $A$. Then either $\norm G A\neq A$, or $G=A$ has prime order. The finite primitive groups containing an abelian regular subgroup are classified by Li in [@chli]. In fact, from [@chli Theorem $1.1$], we see that either the socle $N$ of $G$ is abelian, or $N$ is the direct product of $\ell\geq 1$ pairwise isomorphic non-abelian simple groups and $G$ is endowed of the primitive product action. We consider these two cases separately. Assume that $N$ is abelian. Then $N$ is an elementary abelian $p$-group, for some prime $p$. Since $N$ and $A$ both act regularly, we have $|N|=|A|$ and hence $A$ is a $p$-group. Therefore $AN$ is also a $p$-group. Suppose $AN>A$. Then (from basic properties of $p$-groups) we have $\norm{AN}A>A $ and hence $\norm{G}A\neq A$. Suppose $AN=A$. Then $A=N\lhd G$ and, if $G>A$, then $\norm G A=G\neq A$, and if $G=A$, then $A$ must have prime order. Assume that $N$ is non-abelian. Let $T_1,\ldots,T_\ell$ be the simple direct factors of $N$. So $N=T_1\times \cdots\times T_\ell$. From [@chli Theorem $1.1$ (2)], we see that $G$ contains a normal subgroup $\widetilde{N}=\widetilde{T}_1\times \cdots \times \widetilde{T}_\ell$ with $\widetilde{T}_i$ having socle $T_i$ for each $i\in \{1,\ldots,\ell\}$, with $A\leq \widetilde{N}$ and with $$A=(A\cap \widetilde{T}_1)\times \cdots\times (A\cap \widetilde{T}_\ell).$$ Furthermore, for $i\in \{1,\ldots,\ell\}$, each of the possible pairs $(\widetilde{T}_i,A\cap \widetilde{T}_i)$ is listed in [@chli Theorem $1.1$ (2) (i),…,(iv)]. An easy inspection on each of these four cases reveals that $\norm{{\widetilde{T}_i}}{{A\cap \widetilde{T}_i}}>(A\cap \widetilde{T}_i)$, for each $i\in \{1,\ldots,\ell\}$. Therefore $$\norm G A\geq \norm {\widetilde{N}}A=\norm{\widetilde{T}_1}{A\cap \widetilde{T}_1}\times \cdots \times \norm{\widetilde{T}_\ell}{A\cap \widetilde{T}_\ell}>(A\cap\widetilde{T}_1)\times \cdots\times (A\cap\widetilde{T}_\ell)=A.$$ We argue by contradiction and among all possible counterexamples we choose $G$ and $A$ with $|G|+|A|$ as small as possible. We let $\Omega$ be the transitive set acted upon by $G$. As $G$ is insoluble and $A$ is self-normalizing, from Proposition \[prim\] we deduce that $G$ is imprimitive. Among all non-trivial systems of imprimitivity for $G$ choose $\mathcal{B}$ with blocks of minimal size. We denote by $G_{(\mathcal{B})}$ the pointwise stabilizer of $\mathcal{B}$, that is, $G_{(\mathcal{B})}=\{g\in G\mid B^g=B, \,\textrm{ for each }B\in \mathcal{B}\}$. In particular, $G_{(\mathcal{B})}$ is the kernel of the action of $G$ on $\mathcal{B}$. Also, for $B\in \mathcal{B}$, we write $G_{\{B\}}=\{g\in G\mid B^g=B\}$ for the setwise stabilizer of $B$ and $G_{(B)}=\{g\in G\mid \alpha^g=\alpha, \,\textrm{for each }\,\alpha\in B\}$ for the pointwise stabilizer of $B$. Moreover, given a subgroup $X$ of $G$, we let $X^{\mathcal{B}}$ denote the permutation group induced by $X$ on $\mathcal{B}$. In particular, $X^{\mathcal{B}}\cong XG_{(\mathcal{B})}/G_{(\mathcal{B})}$. Observe that $A^{\mathcal{B}}$ is a transitive abelian subgroup of $G^{\mathcal{B}}$, and hence $A^{\mathcal{B}}$ acts regularly on $\mathcal{B}$. We show that $A^{\mathcal{B}}$ is a Carter subgroup of $G^{\mathcal{B}}$. As $A^{\mathcal{B}}\cong AG_{(\mathcal{B})}/G_{\mathcal{B}}$, it suffices to show that $\norm G {AG_{(\mathcal{B})}}=AG_{(\mathcal{B})}$. Let $g\in \norm {G}{AG_{(\mathcal{B})}}$. Now $(AG_{(\mathcal{B})})^g=AG_{(\mathcal{B})}$ and hence $A,A^g\leq AG_{(\mathcal{B})}$. As $A$ and $A^g$ are Carter subgroups of $G$, they are also Carter subgroups of $AG_{(\mathcal{B})}$. In particular, by [@Vdovin], $A$ and $A^g$ are conjugate in $AG_{(\mathcal{B})}$. Therefore there exists $h\in G_{(\mathcal{B})}$ with $A^{h}=A^g$. Thus $gh^{-1}\in \norm G A=A$ and $g\in AG_{(\mathcal{B})}$. Since $A^{\mathcal{B}}$ is a Carter subgroup of $G^{\mathcal{B}}$, we see that $G^{\mathcal{B}}$ and $A^{\mathcal{B}}$ satisfy the hypothesis of Theorem \[thm\]. As $\mathcal{B}$ is a non-trivial system of imprimitivity, $|A^{\mathcal{B}}|<|A|$ and $|G^{\mathcal{B}}|\leq |G|$ and hence, by minimality, $G^{\mathcal{B}}\cong G/G_{(\mathcal{B})}$ is soluble. In particular, $G_{(\mathcal{B})}$ is insoluble. Suppose that $AG_{(\mathcal{B})}<G$. Then $A$ is a regular abelian Carter subgroup of $AG_{(\mathcal{B})}$ with $|AG_{(\mathcal{B})}|<|G|$. By minimality, $AG_{(\mathcal{B})}$ is soluble and hence so is $G_{(\mathcal{B})}$, a contradiction. Thus $$\label{eq1} G=AG_{(\mathcal{B})}.$$ From , we have $G^{\mathcal{B}}=A^{\mathcal{B}}$ and hence $G$ acts regularly on $\mathcal{B}$. In particular, $G_\omega\leq G_{(\mathcal{B})}$, for every $\omega\in \Omega$, and $G_{\{B\}}=G_{(\mathcal{B})}$, for every $B\in \mathcal{B}$. Moreover, by the minimality of $|B|$, the group $G_{(\mathcal{B})}=G_{\{B\}}$ acts primitively on $B$. We show that $$\label{eq2-} \norm {G}{A_0}=\cent {G}{A_0},\qquad \textrm{for every }A_0\leq A.$$ Clearly the right hand side is contained in the left hand side. Let $g\in \norm {G}{A_0}$. Now $A_0$ is centralized by $A$ and by $A^g$, and so by $\langle A,A^g\rangle$. As $A$ and $A^g$ are Carter subgroups of $\langle A,A^g\rangle$, by [@Vdovin] there exists $x\in \langle A,A^g\rangle$ with $A^g=A^x$. Thus $gx^{-1}\in \norm G A=A$. In particular, $gx^{-1}$ and $x$ centralize $A_0$, and hence so does $g$. Thus $g\in \cent {G}{A_0}$. Applying  with $A_0=A\cap G_{(\mathcal{B})}$ we obtain $$\label{eq2} \norm {G}{A\cap G_{(\mathcal{B})}}=\cent {G}{A\cap G_{(\mathcal{B})}}.$$ Fix $B_1\in \mathcal{B}$ and write $H=G_{\{B_1\}}^{B_1}$, that is, the permutation group induced by $G_{\{B_1\}}$ on the block $B_1$. By the Embedding Theorem [@meldrum Theorem $1.2.6$], the group $G$ is permutation isomorphic to a subgroup of the wreath product $W=H{\mathop{\mathrm{wr}}}A^{\mathcal{B}}$ (in its natural imprimitive product action on the cartesian product $B_1\times\mathcal{B}$). Write $\ell=|\mathcal{B}|$. In particular, under this embedding $G_{(\mathcal{B})}$ is identified with a subgroup of $H^\ell$. As $G_{(\mathcal{B})}$ is insoluble and primitive in its action on $B_1$, it follows that $$\label{eq3} H\quad\textrm{is an insoluble primitive permutation group.}$$ Let $N$ be a minimal normal subgroup of $G$ with $N\leq G_{(\mathcal{B})}$. We now divide the proof in two cases: depending on whether $N$ is abelian or non-abelian. <span style="font-variant:small-caps;">Case I:</span> $N$ is non-abelian. Observe that $A$ is a regular abelian Carter subgroup of $AN$ and that $AN$ is not soluble, hence by minimality $G=AN$. The modular law gives $G_{(\mathcal{B})}=(A\cap G_{(\mathcal{B})})N$ and thus $H=G_{(\mathcal{B})}^{B_1}=(A\cap G_{(\mathcal{B})})^{B_1}N^{B_1}$. It is easy to check that $(A\cap G_{(\mathcal{B})})^{B_1}$ is a regular abelian subgroup of $H$ and that $N^{B_1}$ is a characteristically simple normal subgroup of $H$. We now appeal to the classification of Li [@chli] of the primitive groups with a regular abelian subgroup. Using the notation in [@chli], we see that in all the groups $X$ in [@chli Theorem $1.1$(2)] the regular abelian subgroup $G$ of $X$ acts trivially by conjugation on the simple direct factors of the socle of $X$. In particular, as $(A\cap G_{(\mathcal{B})})^{B_1}$ acts transitively on the simple direct factors of $N^{B_1}$, we deduce that $N^{B_1}$ is a non-abelian simple group, that is, $H$ is an almost simple group. Furthermore, $(H,(A\cap G_{(\mathcal{B})})^{B_1})$ is one of the pairs (denoted by $(\widetilde{T}_i,G_i)$) in [@chli Theorem $1.1$(2) (i), …, (iv)]. Let $T_1,\ldots,T_{\ell'}$ be the simple direct factors of $N$ with $T_1$ acting faithfully on $B_1$ and with $N_{(B_1)}=T_2\times\cdots \times T_{\ell'}$. Observe that $T_1,\ldots,T_{\ell'}$ are pairwise isomorphic and that $A$ acts transitively by conjugation on $\{T_1,\ldots,T_{\ell'}\}$. Let $\pi_1:N\to T_1$ be the projection on the first coordinate and set $A_1=\pi_1(N \cap A)$. In particular, replacing $G$ by a suitable conjugate in ${\mathop{\mathrm{Aut}}}(N)$, we may suppose that $N\cap A=\{(a,\ldots,a)\mid a\in A_1\}\leq A_1^{\ell'}$. Now, a direct inspection on each of the pairs in [@chli Theorem $1.1$(2) (i), …, (iv)] shows that there exists $h\in \norm {T_1}{A_1}\setminus \cent {T_1}{A_1}$. The element $g=(h,\ldots,h)\in T_1\times \cdots\times T_{\ell'}=N$ and $g\in \norm G {A\cap N}\setminus \cent G {A\cap N}$, contradicting  (applied with $A_0=A\cap N$). <span style="font-variant:small-caps;">Case II:</span> $N$ is an elementary abelian $p$-group for some prime $p$. Now $N^{B_1}$ is a normal elementary abelian $p$-subgroup of $G_{(\mathcal{B})}^{B_1}=H$. As $H$ is primitive, $N^{B_1}$ is the socle of $H$ and $H$ is a primitive group of affine type, that is, $H$ is isomorphic to a primitive subgroup of the affine general linear group ${\mathop{\mathrm{AGL}}}_d(p)$ for some $d\geq 1$. Let $V$ be the socle of $H$, let $V^\ell$ be the socle of $H^\ell$ (as a subgroup of $W$) and let $U=V^\ell\cap G$. Let $\pi_1:G_{(\mathcal{B})}\to H$ be the projection onto the first coordinate and set $A_1=\pi_1(A\cap G_{(\mathcal{B})})$. Since $A_1=(A\cap G_{\mathcal{B}})^{B_1}$, we see that $A_1$ is a regular abelian subgroup of $H$. Moreover, as $A$ acts transitively by conjugation on the $\ell$ coordinates of $H^\ell$, replacing $G$ by a suitable conjugate, under the embedding $G\leq W$ we have $$\label{eq4} A\cap G_{(\mathcal{B})}=\{(a,\ldots,a)\mid a \in A_1\}.$$ Suppose that $A_1=V$. Then $A\cap G_{(\mathcal{B})}\leq V^\ell\cap G=U$. Moreover, $G_{(\mathcal{B})}/U$ is isomorphic to a subgroup of the direct product ${\mathop{\mathrm{GL}}}_d(p)^\ell$. Since $G_{(\mathcal{B})}$ is insoluble, so is $X=G_{(\mathcal{B})}/U$. Furthermore, $A$ acts as a group of automorphisms on $X$. From [@BH Theorem 0.11], we have $\cent X A\neq 1$ and hence there exists $g\in G_{(\mathcal{B})}\setminus U$ with $[g,A]\leq U$. As $A$ acts transitively on the elements of $\mathcal{B}$, $g=(v_1h,v_2h,\ldots,v_\ell h)$ for some $h\in H$ and some $v_1,\ldots,v_\ell\in V$. As $g\notin U$, we have $h\notin V$ (if $h\in V$, then $g\in V^\ell\cap G=U$, a contradiction). As $V=\cent H V$, we get $h\notin \cent H V$ and $h\in H=\norm H V$. From  it is clear that $g\in \norm G{A\cap G_{(\mathcal{B})}}\setminus \cent G{A\cap G_{(\mathcal{B})}}$, which contradicts . Suppose that $A_1\neq V$. We show that there exists $u\in \norm U{A\cap G_{(\mathcal{B})}}\setminus\cent U{A\cap G_{(\mathcal{B})}}$, from which the theorem will immediately follow by contradicting . Since $A_1 V>A_1$ and $\cent H{A_1}={A_1}$, there exists $\bar{v}\in \norm V {A_1}\setminus \cent H{A_1}$. We label the elements of $\mathcal{B}$ by $B_1,B_2,\ldots,B_{\ell}$, where the action of $g=(g_1,\ldots,g_\ell)\in G_{(\mathcal{B})}$ on $B_i$ is given by the element $g_{i}$ on the $i^{\mathrm{th}}$ coordinate of $g$. Assume that $G_{(B_1)}=1$. Let $u\in U$ with $u^{B_1}=\bar{v}$, that is, $u=(\bar{v},v_2,\ldots,v_\ell)$, for some $v_2,\ldots,v_\ell\in V$. By construction $u$ normalizes $A\cap G_{(\mathcal{B})}$ modulo $G_{(B_1)}$ and hence $u$ normalizes $A\cap G_{(\mathcal{B})}$ because $G_{(B_1)}=1$. Assume that $G_{(B_1)}\neq 1$. Let $B\in \mathcal{B}\setminus\{B_1\}$ with $(G_{(B_1)})^{B}\neq 1$. Relabelling the set $\mathcal{B}$ if necessary, we may assume that $B=B_2$. As $G_{(B_1)}\lhd G_{(\mathcal{B})}$ and $G_{(\mathcal{B})}$ acts primitively on $B_2$, we get $V\leq (G_{(B_1)})^{B_2}$. We show that $$U=U_{(B_1)}U_{(B_2)}.$$ Let $z=(v_1,v_2,v_3,\ldots,v_\ell)\in U$. As $V=(U_{(B_1)})^{B_2}$, there exists $x=(1,v_2,v_3',\ldots,v_\ell')\in U_{(B_1)}$, for some $v_3',\ldots,v_\ell'\in V$. Clearly, $zx^{-1}=(v_1,1,v_3v_3'^{-1},\ldots,v_\ell v_\ell'^{-1})\in U_{(B_2)}$. Now we show that $$(\bar{v},\bar{v},v_3,\ldots,v_\ell)\in U,\qquad \textrm{for some }v_3,\ldots,v_\ell\in V.$$ As $U=U_{(B_1)}U_{(B_2)}$, we get $V=(U_{(B_1)})^{B_2}$ and hence there exists $y=(1,\bar{v},v_3',\ldots,v_\ell')\in U_{(B_1)}$. Similarly, as $V=(U_{(B_2)})^{B_1}$, there exists $x=(\bar{v},1,v_3'',\ldots,v_\ell'')\in U_{(B_2)}$. Now, $xy\in U$ and the first two coordinates of $xy$ are $\bar{v},\bar{v}$. Assume that $G_{(B_1)}\cap G_{(B_2)}=1$, that is, $G_{(B_1\cup B_2)}=1$. Let $u=(\bar{v},\bar{v},v_3,\ldots,v_\ell)\in U$. Now, from , we see that $u$ normalizes $A\cap G_{(\mathcal{B})}$ modulo $G_{(B_1\cup B_2)}$ and hence $u$ normalizes $A\cap G_{(\mathcal{B})}$ because $G_{(B_1\cup B_2)}=1$. Assume that $G_{(B_1)}\cap G_{(B_2)}\neq 1$. Let $B\in \mathcal{B}\setminus\{B_1,B_2\}$ with $(G_{(B_1\cup B_2)})^{B}\neq 1$. Relabelling the set $\mathcal{B}$ if necessary, we may assume that $B=B_3$. As $G_{(B_1\cup B_2)}\lhd G_{(\mathcal{B})}$ and $G_{(\mathcal{B})}$ acts primitively on $B_3$, we get $V\leq (G_{(B_1\cup B_2)})^{B_3}$. We show that $$U=U_{(B_1\cup B_2)}U_{(B_3)}.$$ Let $z=(v_1,v_2,v_3,\ldots,v_\ell)\in U$. As $V=(U_{(B_1\cup B_2)})^{B_3}$, there exists $x=(1,1,v_3,v_4',\ldots,v_\ell')\in U_{(B_1\cup B_2)}$, for some $v_4',\ldots,v_\ell'\in V$. Clearly, $zx^{-1}\in U_{(B_3)}$. Now we show that $$(\bar{v},\bar{v},\bar{v},v_4,\ldots,v_\ell)\in U,\qquad \textrm{for some }v_4,\ldots,v_\ell\in V.$$ Indeed, from above there exists $z=(\bar{v},\bar{v},w_3,\ldots,w_\ell)\in U$, for some $w_3,\ldots,w_\ell\in V$. Moreover, as $U=U_{(B_1\cup B_2)}U_{(B_3)}$, we get $V=U^{B_3}=(U_{(B_1\cup B_2)})^{B_3}$ and hence there exists $y=(1,1,w_3^{-1}v,w_4',\ldots,w_\ell')\in U_{(B_1\cup B_2)}$. Now, $zy\in U$ and the first three coordinates of $zy$ are $\bar{v},\bar{v},\bar{v}$. Assume that $G_{(B_1\cup B_2)}\cap G_{(B_3)}=1$, that is, $G_{(B_1\cup B_2\cup B_3)}=1$. Let $u=(\bar{v},\bar{v},\bar{v},v_4,\ldots,v_\ell)\in U$. Now, from , we see that $u$ normalizes $A\cap G_{(\mathcal{B})}$ modulo $G_{(B_1\cup B_2\cup B_3)}$ and hence $u$ normalizes $A\cap G_{(\mathcal{B})}$ because $G_{(B_1\cup B_2\cup B_3)}=1$. Finally, the case $G_{(B_1\cup B_2\cup B_3)}\neq 1$ follows easily by induction applying the same two-step argument as in the previous paragraphs. L. Babai, C. D. Godsil, On the automorphism groups of almost all Cayley graphs, *European J. Combin.* **3** (1982), 9–15. V. V. Belyaev, B. Hartley, Centralizers of finite nilpotent subgroups in locally finite groups, *Algebra and Logic* **35** (1996), 217–228. R. W. Carter, Nilpotent self-normalizing subgroups of soluble groups, *Math. Z.* **75** (1961), 136–139. J. H. Conway, R. T. Curtis, S. P. Norton, R. A. Parker, R. A. Wilson, *Atlas of Finite groups*, Claredon Press, Oxford 1985. E. Dobson, P. Spiga, G. Verret, Cayley graphs on abelian groups, submitted. 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--- abstract: 'We show that for a locally free $C^{\infty}$-action of a connected and simply connected nilpotent Lie group on a compact manifold, if every real valued cocycle is cohomologous to a constant cocycle, then the action is parameter rigid. The converse is true if the action has a dense orbit. Using this, we construct parameter rigid actions of simply connected nilpotent Lie groups whose Lie algebras admit rational structures with graduations. This generalizes the results of dos Santos [@dS] concerning the Heisenberg groups.' author: - | Hirokazu Maruhashi [^1]\ Department of Mathematics, Kyoto University title: Parameter rigid actions of simply connected nilpotent Lie groups --- Introduction ============ Let $G$ be a connected Lie group with Lie algebra ${{\mathfrak{g}}}$ and $M$ a $C^{\infty}$-manifold without boundary. Let $\rho:M\times G\to M$ be a $C^{\infty}$ right action. We call $\rho$ [*locally free*]{} if every isotropy subgroup of $\rho$ is discrete in $G$. Assume that $\rho$ is locally free. Then we have the orbit foliation ${{\mathcal{F}}}$ of $\rho$ whose tangent bundle $T{{\mathcal{F}}}$ is naturally isomorphic to a trivial bundle $M\times{{\mathfrak{g}}}$. The action $\rho$ is [*parameter rigid*]{} if any action $\rho^{\prime}$ of $G$ on $M$ with the same orbit foliation ${{\mathcal{F}}}$ is $C^{\infty}$-conjugate to $\rho$, more precisely, there exist an automorphism $\Phi$ of $G$ and a $C^{\infty}$-diffeomorphism $F$ of $M$ which preserves each leaf of ${{\mathcal{F}}}$ and homotopic to identity through $C^{\infty}$-maps preserving each leaf of ${{\mathcal{F}}}$ such that $$F(\rho(x,g))=\rho^{\prime}(F(x),\Phi(g))$$ for all $x\in M$ and $g\in G$. Parameter rigidity of actions has been studied by several authors, for instance, Katok and Spatzier [@KS], Matsumoto and Mitsumatsu [@MM], Mieczkowski [@Mi], dos Santos [@dS] and Ramírez [@R]. Most of known examples of parameter rigid actions are those of abelian groups and nonabelian actions have not been considered so much. Parameter rigidity is closely related to cocycles over actions. Now suppose $G$ is contractible and $M$ is compact. Let $H$ be a Lie group. A $C^{\infty}$-map $c:M\times G\to H$ is called a [*$H$-valued cocycle over $\rho$*]{} if $c$ satisfies $$c(x,gg^{\prime})=c(x,g)c(\rho(x,g),g^{\prime})$$ for all $x\in M$ and $g,g^{\prime}\in G$. A cocycle $c$ is [*constant*]{} if $c(x,g)$ is independent of $x$. A constant cocycle is just a homomorphism $G\to H$. $H$-valued cocycles $c,c^{\prime}$ are [*cohomologous*]{} if there exists a $C^{\infty}$-map $P:M\to H$ such that $$c(x,g)=P(x)^{-1}c^{\prime}(x,g)P(\rho(x,g))$$ for all $x\in M$ and $g\in G$. The action $\rho$ is [*$H$-valued cocycle rigid*]{} if every $H$-valued cocycle over $\rho$ is cohomologous to a constant cocycle. \[cp\] If $\rho$ is $G$-valued cocycle rigid, then it is parameter rigid. In [@MM] Matsumoto and Mitsumatsu assume that $\rho$ has at least one trivial isotropy subgroup, but this assumption is not necessary. When $G={{\mathbb{R}}}^n$, the following are equivalent: 1. $\rho$ is ${{\mathbb{R}}}$-valued cocycle rigid. 2. $\rho$ is ${{\mathbb{R}}}^n$-valued cocycle rigid. 3. $\rho$ is parameter rigid. The equivalence of the first two conditions is obvious. In this paper we consider actions of simply connected nilpotent Lie groups. In [@dS], dos Santos proved that for actions of a Heisenberg group $H_n$, ${{\mathbb{R}}}$-valued cocycle rigidity implies $H_n$-valued cocycle rigidity and using this, he constructed parameter rigid actions of Heisenberg groups. To the best of my knowledge these are the only known nontrivial parameter rigid actions of nonabelian nilpotent Lie groups. We prove the following. \[main\] Let $N$ be a connected and simply connected nilpotent Lie group, $M$ a compact manifold and $\rho$ a locally free $C^{\infty}$-action of $N$ on $M$. Then, the following are equivalent: 1. $\rho$ is ${{\mathbb{R}}}$-valued cocycle rigid. \[m1\] 2. $\rho$ is $N$-valued cocycle rigid. 3. $\rho$ is parameter rigid and every orbitwise constant real valued $C^{\infty}$-function of $\rho$ on $M$ is constant on $M$. \[m2\] This theorem enables us to construct parameter rigid actions of nilpotent Lie groups. The most interesting one is the following. Let $N$ denote the group of all upper triangular real matrices with $1$ on the diagonal, $\Gamma$ a cocompact lattice of ${\rm SL}(n,{{\mathbb{R}}})$ and $\rho$ the action of $N$ on $\Gamma\backslash{\rm SL}(n,{{\mathbb{R}}})$ by right multiplication. If $n\geq 4$, $\rho$ is ${{\mathbb{R}}}$-valued cocycle rigid. In [@R], Ramírez proved more general theorems. The above action $\rho$ is parameter rigid. In Section \[const\] we construct parameter rigid actions of nilpotent groups using Theorem \[main\]. It is a generalization of dos Santos’ example. Let $N$ be a connected and simply connected nilpotent Lie group and $\Gamma$, $\Lambda$ be lattices in $N$. Consider the action of $\Lambda$ on ${{\Gamma\backslash N}}$ by right multiplication and let $\tilde{\rho}$ be its suspended action of $N$. \[main2\] If $\Lambda$ is Diophantine with respect to $\Gamma$, then the action $\tilde{\rho}$ of $N$ is parameter rigid. For the definition of Diophantine lattices, see Section \[const\]. Preliminaries ============= Let $G$ be a contractible Lie group with Lie algebra ${{\mathfrak{g}}}$, $M$ a compact manifold and $\rho$ a locally free action of $G$ on $M$ with orbit foliation ${{\mathcal{F}}}$. Let $H$ be a Lie group with Lie algebra ${{\mathfrak{h}}}$. We denote by $\Omega^p({{\mathcal{F}}},{{\mathfrak{h}}})$ the set of all $C^{\infty}$-sections of $\Hom(\bigwedge^pT{{\mathcal{F}}},{{\mathfrak{h}}})$. The exterior derivative $${{d_{\mathcal{F}}}}:\Omega^p({{\mathcal{F}}},{{\mathfrak{h}}})\to\Omega^{p+1}({{\mathcal{F}}},{{\mathfrak{h}}})$$ is defined since $T{{\mathcal{F}}}$ is integrable. By differentiating, $H$-valued cocycles over $\rho$ are in one-to-one correspondence with ${{\mathfrak{h}}}$-valued leafwise one forms $\omega\in\Omega^1({{\mathcal{F}}},{{\mathfrak{h}}})$ such that $${{d_{\mathcal{F}}}}\omega+[\omega,\omega]=0.$$ \[mat\] Let $c_1,c_2$ be $H$-valued cocycles over $\rho$ and let $\omega_1,\omega_2$ be corresponding differential forms. For a $C^{\infty}$-map $P:M\to H$, the following are equivalent: 1. $c_1(x,g)=P(x)^{-1}c_2(x,g)P(\rho(x,g))$ for all $x\in M$ and $g\in G$. 2. $\omega_1=\Ad(P^{-1})\omega_2+P^*\theta$ where $\theta\in\Omega^1(H,{{\mathfrak{h}}})$ is the left Maurer-Cartan form on $H$. \[comat\] The following are equivalent: 1. $\rho$ is $G$-valued cocycle rigid. 2. For each $\omega\in\Omega^1({{\mathcal{F}}},{{\mathfrak{g}}})$ such that ${{d_{\mathcal{F}}}}\omega+[\omega,\omega]=0$, there exist a endomorphism $\Phi:{{\mathfrak{g}}}\to{{\mathfrak{g}}}$ of Lie algebra and a $C^{\infty}$-map $P:M\to G$ such that $$\omega=\Ad(P^{-1})\Phi+P^*\theta.$$ Proposition \[mat\] is obtained by examining the proof of Corollary \[comat\] in [@MM]. In this paper, we will identify a cocycle with its corresponding differential form. Let us consider real valued cocycles. A real valued cocycle over $\rho$ is given by $\omega\in\Omega^1({{\mathcal{F}}},{{\mathbb{R}}})$ satisfying ${{d_{\mathcal{F}}}}\omega=0$. Two real valued cocycles $\omega_1,\omega_2$ are cohomologous if and only if $\omega_1=\omega_2+{{d_{\mathcal{F}}}}P$ for some $C^{\infty}$-function $P:M\to{{\mathbb{R}}}$. Leafwise cohomology $H^*({{\mathcal{F}}})$ of ${{\mathcal{F}}}$ is the cohomology of the cochain complex $(\Omega^*({{\mathcal{F}}},{{\mathbb{R}}}),{{d_{\mathcal{F}}}})$. Thus $H^1({{\mathcal{F}}})$ is the set of all equivalence classes of real valued cocycles. The identification $T{{\mathcal{F}}}\simeq M\times{{\mathfrak{g}}}$ induces a map $H^*({{\mathfrak{g}}})\to H^*({{\mathcal{F}}})$ where $H^*({{\mathfrak{g}}})$ is the cohomology of the Lie algebra ${{\mathfrak{g}}}$. By the compactness of $M$, this map is injective on $H^1({{\mathfrak{g}}})$. Hence we identify $H^1({{\mathfrak{g}}})$ with its image. Note that $H^1({{\mathfrak{g}}})$ is the set of all equivalence classes of constant real valued cocycles. Thus real valued cocycle rigidity is equivalent to $H^1({{\mathcal{F}}})=H^1({{\mathfrak{g}}})$. Notice that $H^0({{\mathcal{F}}})$ is the set of leafwise constant real valued $C^{\infty}$-functions of ${{\mathcal{F}}}$ on $M$ and $H^0({{\mathfrak{g}}})$ consists of constant functions on $M$. Therefore the equivalence of \[m1\] and \[m2\] in Theorem \[main\] can be stated as follows: $H^1({{\mathcal{F}}})=H^1({{\mathfrak{n}}})$ if and only if $\rho$ is parameter rigid and $H^0({{\mathcal{F}}})=H^0({{\mathfrak{n}}})$. Proof of Theorem \[main\] ========================= Let $N$ be a simply connected nilpotent Lie group with Lie algebra ${{\mathfrak{n}}}$, $M$ a compact manifold and $\rho$ a locally free action of $N$ on $M$ with orbit foliation ${{\mathcal{F}}}$. We first prove that $N$-valued cocycle rigidity implies real valued cocycle rigidity. There exist closed subgroups $N^{\prime}$ and $A$ of $N$ such that $N^{\prime}\triangleleft N$, $N=N^{\prime}\rtimes A$ and $A\simeq{{\mathbb{R}}}$. Let $c$ be any real valued cocycle over $\rho$. We regard $c$ as a $N$-valued cocycle over $\rho$ via the inclusion ${{\mathbb{R}}}\simeq A\hookrightarrow N$. By $N$-valued cocycle rigidity, there exist an endomorphism $\Phi$ of $N$ and a $C^{\infty}$-map $P:M\to N$ such that $c(x,g)=P(x)^{-1}\Phi(g)P(\rho(x,g))$ for all $x\in M$ and $g\in N$. Applying the natural projection $\pi:N\to A\simeq{{\mathbb{R}}}$, we obtain $c(x,g)=(\pi\circ P)(x)^{-1}(\pi\circ\Phi)(g)(\pi\circ P)(\rho(x,g))$. Thus $c$ is cohomologous to a constant cocycle $\pi\circ\Phi$. Next we assume $H^1({{\mathcal{F}}})=H^1({{\mathfrak{n}}})$ and prove $N$-valued cocycle rigidity. We need the following two lemmata. \[lem1\] Let $V$ be a finite dimensional real vector space. Assume that $\omega\in \Omega^1({{\mathcal{F}}},V)$ satisfies the equation ${{d_{\mathcal{F}}}}\omega=\varphi$, where $\varphi\in \Hom(\bigwedge^2{{\mathfrak{n}}},V)$ is a constant leafwise two form. Then there exists a constant leafwise one form $\psi\in\Hom({{\mathfrak{n}}},V)$ with $\varphi={{d_{\mathcal{F}}}}\psi$. Since $N$ is amenable, there exists a $N$-invariant Borel probability measure $\mu$ on $M$. Define $\psi\in\Hom({{\mathfrak{n}}},V)$ by $$\psi(X)=\int_M\omega(X)d\mu$$ where $X\in{{\mathfrak{n}}}$. Since $\varphi(X,Y)=X\omega(Y)-Y\omega(X)-\omega([X,Y])$ for all $X,Y\in{{\mathfrak{n}}}$, we obtain $$\varphi(X,Y)=-\int_M\omega([X,Y])d\mu.$$ Thus $${{d_{\mathcal{F}}}}\psi(X,Y)=-\psi([X,Y])=-\int_M\omega([X,Y])d\mu=\varphi(X,Y),$$ hence ${{d_{\mathcal{F}}}}\psi=\varphi$. Set ${{\mathfrak{n}}}^1={{\mathfrak{n}}}$, ${{\mathfrak{n}}}^i=[{{\mathfrak{n}}},{{\mathfrak{n}}}^{i-1}]$. Then ${{\mathfrak{n}}}^s\neq 0$, ${{\mathfrak{n}}}^{s+1}=0$ for some $s$. For each $1\leq i\leq s$, choose a subspace $V_i$ with ${{\mathfrak{n}}}^i=V_i\oplus {{\mathfrak{n}}}^{i+1}$, so that ${{\mathfrak{n}}}=\bigoplus_{i=1}^s V_i$. \[induction\] Let $\omega\in\Omega^1({{\mathcal{F}}},{{\mathfrak{n}}})$ be such that ${{d_{\mathcal{F}}}}\omega+[\omega,\omega]=0$. Decompose $\omega$ as $$\omega=\xi+\omega_k+\omega_{k+1}$$ where $\xi\in\Omega^1({{\mathcal{F}}},\bigoplus_{i=1}^{k-1}V_i)$, $\omega_k\in\Omega^1({{\mathcal{F}}},V_k)$ and $\omega_{k+1}\in\Omega^1({{\mathcal{F}}},{{\mathfrak{n}}}^{k+1})$. If $\xi$ is constant, then there exists $\omega^{\prime}\in\Omega^1({{\mathcal{F}}},{{\mathfrak{n}}})$ with ${{d_{\mathcal{F}}}}\omega^{\prime}+[\omega^{\prime},\omega^{\prime}]=0$ which is cohomologous to $\omega$ and such that $$\omega^{\prime}=\xi^{\prime}+\omega_{k+1}^{\prime}$$ where $\xi^{\prime}\in\Omega^1({{\mathcal{F}}},\bigoplus_{i=1}^kV_i)$ is constant and $\omega_{k+1}^{\prime}\in\Omega({{\mathcal{F}}},{{\mathfrak{n}}}^{k+1})$. By cocycle equation, $$0={{d_{\mathcal{F}}}}\xi+{{d_{\mathcal{F}}}}\omega_k+{{d_{\mathcal{F}}}}\omega_{k+1}+[\xi,\xi]+ \mbox{an element of}\ \Omega^2({{\mathcal{F}}},{{\mathfrak{n}}}^{k+1}).$$ Comparing $V_k$ component, we see that ${{d_{\mathcal{F}}}}\omega_k$ is constant. Hence by Lemma \[lem1\], ${{d_{\mathcal{F}}}}\omega_k={{d_{\mathcal{F}}}}\psi$ for some $\psi\in\Hom({{\mathfrak{n}}},V_k)$. Since we are assuming that $H^1({{\mathcal{F}}})=H^1({{\mathfrak{n}}})$, there exists $\psi^{\prime}\in\Hom({{\mathfrak{n}}},V_k)$ and $C^{\infty}$-map $h:M\to V_k$ such that $$\omega_k=\psi+\psi^{\prime}+{{d_{\mathcal{F}}}}h.$$ Put $P=e^h:M\to N$. Let $x\in M$ and $X\in T_x{{\mathcal{F}}}$. Choose a path $x(t)$ such that $X=\frac{d}{dt}x(t)\big|_{t=0}$. Let $\theta\in\Omega^1(N,{{\mathfrak{n}}})$ be the left Maurer-Cartan form on $N$. Then $$\begin{aligned} P^*\theta(X)&=\frac{d}{dt}P(x)^{-1}P(x(t))\Big|_{t=0} =\frac{d}{dt}e^{-h(x)}e^{h(x(t))}\Big|_{t=0} \\ &=\frac{d}{dt}\exp(-h(x)+h(x(t))+ \mbox{an element of }{{\mathfrak{n}}}^{k+1})\Big|_{t=0} \\ &={{d_{\mathcal{F}}}}h(X)+\mbox{an element of }{{\mathfrak{n}}}^{k+1}. \end{aligned}$$ Thus $P^*\theta={{d_{\mathcal{F}}}}h+ \mbox{an element of }\Omega^1({{\mathcal{F}}},{{\mathfrak{n}}}^{k+1})$. Note that $\Ad(P^{-1})=\exp\ad(-h)$ is identity on $\bigoplus_{i=1}^kV_i$ and preserves ${{\mathfrak{n}}}^{k+1}$. Hence $$\begin{aligned} \omega-P^*\theta&=\xi+\psi+\psi^{\prime}+ \mbox{an element of }\Omega^1({{\mathcal{F}}},{{\mathfrak{n}}}^{k+1}) \\ &=\Ad(P^{-1})(\xi+\psi+\psi^{\prime}+ \mbox{an element of }\Omega^1({{\mathcal{F}}},{{\mathfrak{n}}}^{k+1})). \end{aligned}$$ Let $\omega$ be any $N$-valued cocycle. Using Lemma \[induction\], we can exchange $\omega$ for a cohomologous cocycle whose $V_1$-component is constant. Applying Lemma \[induction\] repeatedly, we eventually get a constant cocycle cohomologous to $\omega$. This proves $N$-valued cocycle rigidity. Next we assume that $\rho$ is parameter rigid and $H^0({{\mathcal{F}}})=H^0({{\mathfrak{n}}})$. Let ${{\mathfrak{n}}}^i$ and $V_i$ be as above. Note that ${{\mathfrak{n}}}^s$ is central in ${{\mathfrak{n}}}$. Fix a nonzero $Z\in{{\mathfrak{n}}}^s$. Let $[\omega]\in H^1({{\mathcal{F}}})$. Let $\omega_0$ be the $N$-valued cocycle over $\rho$ corresponding to the constant cocycle $\mathrm{id}:N\to N$. We call $\omega_0$ [*the canonical $1$-form of $\rho$*]{}. Fix a $\epsilon>0$ and put $\eta:=\omega_0+\epsilon\omega Z$. $\eta$ is an $N$-valued cocycle over $\rho$ since $${{d_{\mathcal{F}}}}\eta+[\eta,\eta]= {{d_{\mathcal{F}}}}\omega_0+\epsilon({{d_{\mathcal{F}}}}\omega)Z+[\omega_0,\omega_0]=0.$$ Since $M$ is compact, we can assume $\eta_x:T_x{{\mathcal{F}}}\to{{\mathfrak{n}}}$ is bijective for all $x\in M$ by choosing $\epsilon>0$ small. There exists a unique action $\rho^{\prime}$ of $N$ on $M$ whose orbit foliation is ${{\mathcal{F}}}$ and canonical $1$-form is $\eta$. See [@A]. By parameter rigidity $\rho^{\prime}$ is conjugate to $\rho$. Thus there exist a $C^{\infty}$-map $P:M\to N$ and an automorphism $\Phi$ of $N$ satisfying $$\label{star} \omega_0+\epsilon\omega Z=\Ad(P^{-1})\Phi_*\omega_0+P^*\theta.$$ Note that $\log:N\to{{\mathfrak{n}}}$ is defined since $N$ is simply connected and nilpotent. Let us decompose $\omega_0=\sum_{i=1}^s\omega_{0i}$, $\Phi_*\omega_0=\sum_{i=1}^s\omega_{0i}^{\prime}$ and $\log P=\sum_{i=1}^sP_i$ according to the decomposition ${{\mathfrak{n}}}=\bigoplus_{i=1}^sV_i$. \[pp\] Assume that $P_1=\dotsm=P_{k-1}=0$ i.e. $\log P\in{{\mathfrak{n}}}^k$. 1. If $k<s$, then there exist a $C^{\infty}$-map $Q:M\to N$ and an automorphism $\Psi$ of $N$ such that $$\omega_0+\epsilon\omega Z=\Ad(Q^{-1})\Psi_*\omega_0+Q^*\theta$$ and $Q_1=\dotsm=Q_k=0$ where $\log Q=\sum_{i=1}^sQ_i$. 2. If $k=s$, then $\omega$ is cohomologous to a constant cocycle. For all $X=\frac{d}{dt}x(t)\big|_{t=0}\in T_x{{\mathcal{F}}}$, $$\begin{aligned} P^*\theta(X)&=\frac{d}{dt}P(x)^{-1}P(x(t))\Big|_{t=0} =\frac{d}{dt}\exp\left(-\sum_{i=k}^sP_i(x)\right) \exp\left(\sum_{i=k}^sP_i(x(t))\right)\Big|_{t=0} \\ &=\frac{d}{dt}\exp\left\{\sum_{i=k}^s\left(P_i(x(t))-P_i(x)\right)+ \mbox{an element of }{{\mathfrak{n}}}^{k+1}\right\}\Big|_{t=0} \\ &=\frac{d}{dt}\exp\left(P_k(x(t))-P_k(x)+ \mbox{an element of }{{\mathfrak{n}}}^{k+1}\right)\Big|_{t=0} \\ &={{d_{\mathcal{F}}}}P_k(X)+\mbox{an element of }{{\mathfrak{n}}}^{k+1}. \end{aligned}$$ We have $$\begin{aligned} \Ad(P^{-1})\Phi_*\omega_0 &=\exp\left(\ad\left(-\sum_{i=k}^sP_i\right)\right) \sum_{i=1}^s\omega_{0i}^{\prime} \\ &=\sum_{i=1}^s\omega_{0i}^{\prime}+\mbox{an element of }{{\mathfrak{n}}}^{k+1}.\end{aligned}$$ Comparing the $V_k$-component of we get $$\omega_{0k}+\delta_{ks}\epsilon\omega Z =\omega_{0k}^{\prime}+{{d_{\mathcal{F}}}}P_k.$$ When $k=s$ the equation $$\omega Z=\epsilon^{-1}(\omega_{0s}^{\prime}-\omega_{0s})+ {{d_{\mathcal{F}}}}(\epsilon^{-1}P_s)$$ shows that $\omega$ is cohomologous to a constant cocycle. If $k<s$, then ${{d_{\mathcal{F}}}}P_k=\phi\circ\omega_0$ for some linear map $\phi:{{\mathfrak{n}}}\to V_k$. For any $X\in{{\mathfrak{n}}}$, let $\tilde{X}$ denote the vector field on $M$ determined by $X$ via $\rho$. We have $\tilde{X}P_k=\phi(X)$ and by integrating over an integral curve $\gamma$ of $\tilde{X}$ we get $P_k(\gamma(T))-P_k(\gamma(0))=\phi(X)T$ for all $T>0$. Since $M$ is compact, $\phi(X)=0$. Therefore ${{d_{\mathcal{F}}}}P_k=0$, so that $P_k$ is constant on each leaf of ${{\mathcal{F}}}$. Thus $P_k$ is constant on $M$ by our assumption. Put $g:=\exp(-P_k)$ and $Q:=gP=\exp\left(\sum_{i=k+1}^sP_i+ \mbox{an element of }{{\mathfrak{n}}}^{k+1}\right)$. Then $$\begin{aligned} \omega_0+\epsilon\omega Z &=\Ad(Q^{-1}g)\Phi_*\omega_0+(L_{g^{-1}}\circ Q)^*\theta \\ &=\Ad(Q^{-1})\Psi_*\omega_0+Q^*\theta\end{aligned}$$ where $\Psi_*:=\Ad(g)\Phi_*$. Applying Lemma \[pp\] repeatedly, we see that $\omega$ is cohomologous to a constant cocycle. Finally we assume $H^1({{\mathcal{F}}})=H^1({{\mathfrak{n}}})$ and prove that $\rho$ is parameter rigid and $H^0({{\mathcal{F}}})=H^0({{\mathfrak{n}}})$. Parameter rigidity of $\rho$ follows from Proposition \[cp\]. Let $f\in H^0({{\mathcal{F}}})$. Fix a nonzero $\phi\in H^1({{\mathfrak{n}}})$ and denote the corresponding leafwise $1$-form on $M$ by $\tilde{\phi}$. Then $f\tilde{\phi}\in H^1({{\mathcal{F}}})=H^1({{\mathfrak{n}}})$. Thus there exist $\psi\in H^1({{\mathfrak{n}}})$ and a $C^{\infty}$-function $g:M\to{{\mathbb{R}}}$ such that $f\tilde{\phi}-\tilde{\psi}={{d_{\mathcal{F}}}}g$ where $\tilde{\psi}$ is the leafwise $1$-form corresponding to $\psi$. Choose $X\in{{\mathfrak{n}}}$ satisfying $\phi(X)\neq 0$. Let $x(t)$ be an integral curve of $\tilde{X}$ where $\tilde{X}$ is the vector field corresponding to $X$. We have $f(x(t))\phi(X)-\psi(X) =\tilde{X}_{x(t)}g=\frac{d}{dt}g(x(t))$. By integrating over $[0,T]$, we get $T(f(x(0))\phi(X)-\psi(X))=g(x(T))-g(x(0))$. Since $g$ is bounded, $f(x(0))\phi(X)-\psi(X)$ must be zero. Then $f(x(0))=\frac{\psi(X)}{\phi(X)}$ and $f$ is constant on $M$. This completes the proof of Theorem \[main\]. A construction of parameter rigid actions {#const} ========================================= Let us now construct real valued cocycle rigid actions of nilpotent groups. For the structure theory of nilpotent Lie groups, see [@CG]. A basis $X_1,\dots,X_n$ of ${{\mathfrak{n}}}$ is called a [*strong Malcev basis*]{} if $\spn_{{{\mathbb{R}}}}\{X_1,\dots,X_i\}$ is an ideal of ${{\mathfrak{n}}}$ for each $i$. If $\Gamma$ is a lattice in $N$, there exists a strong Malcev basis $X_1,\dots,X_n$ of ${{\mathfrak{n}}}$ such that $\Gamma=e^{{{\mathbb{Z}}}X_1}\cdots e^{{{\mathbb{Z}}}X_n}$. Such a basis is called a [*strong Malcev basis strongly based on $\Gamma$*]{}. Let $\Gamma$ and $\Lambda$ be lattices in $N$. $\Lambda$ is [*Diophantine with respect to $\Gamma$*]{} if there exists a strong Malcev basis $X_1,\dots,X_n$ of ${{\mathfrak{n}}}$ strongly based on $\Gamma$ and a strong Malcev basis $Y_1,\dots,Y_n$ of ${{\mathfrak{n}}}$ strongly based on $\Lambda$ such that $Y_i=\sum_{j=1}^ia_{ij}X_j$ for every $1\leq i\leq n$, where $a_{ii}$ is Diophantine. Let $\rho$ be the action of $\Lambda$ on ${{\Gamma\backslash N}}$ by right multiplication. First we will prove the following. \[constant\] If $\Lambda$ is Diophantine with respect to $\Gamma$, then every real valued $C^{\infty}$ cocycle $c:{{\Gamma\backslash N}}\times\Lambda\to{{\mathbb{R}}}$ over $\rho$ is cohomologous to a constant cocycle. Note that $X_1$ is in the center of ${{\mathfrak{n}}}$. Let $\pi:N\to\bar{N}:={{e^{{{\mathbb{R}}}X_1}}}\backslash N$ be the projection. Since $\Gamma\cap{{e^{{{\mathbb{R}}}X_1}}}=e^{{{\mathbb{Z}}}X_1}$ is a cocompact lattice in ${{e^{{{\mathbb{R}}}X_1}}}$, $\bar{\Gamma}:=\pi(\Gamma)={{e^{{{\mathbb{R}}}X_1}}}\backslash\Gamma{{e^{{{\mathbb{R}}}X_1}}}$ is a cocompact lattice in $\bar{N}$. Let $\bar{{{\mathfrak{n}}}}={{\mathbb{R}}}X_1\backslash{{\mathfrak{n}}}$, then $\bar{X_2},\dots,\bar{X_n}$ is a strong Malcev basis of $\bar{{{\mathfrak{n}}}}$ strongly based on $\bar{\Gamma}$. We will see that the naturally induced map $\bar{\pi}:{{\Gamma\backslash N}}\to{{\bar{\Gamma}\backslash\bar{N}}}$ is a principal $S^1$-bundle. Indeed, $$\Gamma\backslash\Gamma{{e^{{{\mathbb{R}}}X_1}}}\hookrightarrow{{\Gamma\backslash N}}\twoheadrightarrow\Gamma{{e^{{{\mathbb{R}}}X_1}}}\backslash N$$ is a principal $\Gamma\backslash\Gamma{{e^{{{\mathbb{R}}}X_1}}}$-bundle and we have $$\Gamma\backslash\Gamma{{e^{{{\mathbb{R}}}X_1}}}\simeq\Gamma\cap{{e^{{{\mathbb{R}}}X_1}}}\backslash{{e^{{{\mathbb{R}}}X_1}}}=e^{{{\mathbb{Z}}}X_1}\backslash{{e^{{{\mathbb{R}}}X_1}}}\simeq{{\mathbb{Z}}}\backslash{{\mathbb{R}}}$$ and $$\xymatrix{ {{e^{{{\mathbb{R}}}X_1}}}\backslash\Gamma{{e^{{{\mathbb{R}}}X_1}}}\ar@{^{(}->}[r] & {{e^{{{\mathbb{R}}}X_1}}}\backslash N \ar@{->>}[r] \ar@{->>}[d] & \Gamma{{e^{{{\mathbb{R}}}X_1}}}\backslash N \\ & {{\bar{\Gamma}\backslash\bar{N}}}\ar[ur]^{\sim} & }.$$ Since $\Lambda\cap{{e^{{{\mathbb{R}}}X_1}}}=\Lambda\cap e^{{{\mathbb{R}}}Y_1}=e^{{{\mathbb{Z}}}Y_1}$ is a cocompact lattice in $e^{{{\mathbb{R}}}X_1}$, $\bar{\Lambda}:=\pi(\Lambda)$ is a cocompact lattice in $\bar{N}$. $\bar{Y_2},\dots,\bar{Y_n}$ is a strong Malcev basis of $\bar{{{\mathfrak{n}}}}$ strongly based on $\bar{\Lambda}$ and $\bar{Y_i}=\sum_{j=2}^ia_{ij}\bar{X_j}$ where $a_{ii}$ is Diophantine. Therefore $\bar{\Lambda}$ is Diophantine with respect to $\bar{\Gamma}$. Since $\bar{\pi}$ is $\Lambda$-equivariant, the action $\rho$ of $\Lambda$ when restricted to $e^{{{\mathbb{Z}}}Y_1}$, preserves fibers of $\bar{\pi}$. Let $z\in{{\bar{\Gamma}\backslash\bar{N}}}$. Choose a point $\Gamma x$ in ${{\bar{\pi}^{-1}(z)}}$. Then we have a trivialization $$\iota_{\Gamma x}:{{\mathbb{Z}}}\backslash{{\mathbb{R}}}\simeq{{\bar{\pi}^{-1}(z)}}$$ of ${{\bar{\pi}^{-1}(z)}}$ given by $\iota_{\Gamma x}(s)=\Gamma e^{sX_1}x$. Note that if we take another point $\Gamma y\in{{\bar{\pi}^{-1}(z)}}$, $\iota_{\Gamma y}^{-1}\circ\iota_{\Gamma x}: {{\mathbb{Z}}}\backslash{{\mathbb{R}}}\to{{\mathbb{Z}}}\backslash{{\mathbb{R}}}$ is a rotation. Let $Y_1=aX_1$ where $a$ is Diophantine. If we identify ${{\bar{\pi}^{-1}(z)}}$ with ${{\mathbb{Z}}}\backslash{{\mathbb{R}}}$ by $\iota_{\Gamma x}$, then the action of $e^{Y_1}$ on ${{\mathbb{Z}}}\backslash{{\mathbb{R}}}$ is $s\mapsto s+a$. Let $\mu_z$ be the normalized Haar measure naturally defined on $\bar{\pi}^{-1}(z)$, $\mu$ the $N$-invariant probability measure on ${{\Gamma\backslash N}}$ and $\nu$ the $\bar{N}$-invariant probability measure on ${{\bar{\Gamma}\backslash\bar{N}}}$. For any $f\in C({{\Gamma\backslash N}})$, $$\label{Weil} \int_{{{\Gamma\backslash N}}}fd\mu=\int_{{{\bar{\Gamma}\backslash\bar{N}}}}\int_{\bar{\pi}^{-1}(z)}fd\mu_zd\nu.$$ $\rho$ is ergodic with respect to $\mu$. We use induction on $n$. If $n=1$, $\rho$ is an irrational rotation on ${{\mathbb{Z}}}\backslash{{\mathbb{R}}}$, hence the result is well known. In general, Let $f:{{\Gamma\backslash N}}\to{{\mathbb{C}}}$ be a $\Lambda$-invariant $L^2$-function with $\int_{{{\Gamma\backslash N}}}fd\mu=0$. Since the action of $e^{{{\mathbb{Z}}}Y_1}$ on ${{\bar{\pi}^{-1}(z)}}$ is ergodic, $f|_{{{\bar{\pi}^{-1}(z)}}}$ is constant $\mu_z$-almost everywhere. We denote this constant by $g(z)$. Then $g:{{\bar{\Gamma}\backslash\bar{N}}}\to{{\mathbb{C}}}$ is $\bar{\Lambda}$-invariant measurable function. By induction, $g$ is constant $\nu$-almost everywhere. By , this constant must be zero. Therefore $f$ is zero $\mu$-almost everywhere. Let $c:{{\Gamma\backslash N}}\times\Lambda\to{{\mathbb{R}}}$ be a $C^{\infty}$-cocycle over $\rho$. We must show that $c$ is cohomologous to a constant cocycle $c_0:\Lambda\to{{\mathbb{R}}}$ where $c_0(\lambda):=\int_{{{\Gamma\backslash N}}}c(x,\lambda)d\mu(x)$. Therefore we may assume that $\int_{{{\Gamma\backslash N}}}c(x,\lambda)d\mu(x)=0$ for all $\lambda\in\Lambda$, and we will show that $c$ is a coboundary. We prove this by induction on $n$. When $n=1$, $\rho$ is a Diophantine rotation on ${{\mathbb{Z}}}\backslash{{\mathbb{R}}}$, hence the result is well known. \[int0\] For all $m\in{{\mathbb{Z}}}$, $$\int_{{{\bar{\pi}^{-1}(z)}}}c(s,e^{mY_1})d\mu_z(s)=0.$$ Fix $m$ and put $g(z)=\int_{{{\bar{\pi}^{-1}(z)}}}c(s,e^{mY_1})d\mu_z(s)$. For any $\lambda\in\Lambda$, cocycle equation gives $c(x,\lambda)+c(x\lambda,e^{mY_1})=c(x,e^{mY_1})+c(xe^{mY_1},\lambda)$. By integrating this equation on ${{\bar{\pi}^{-1}(z)}}$, we get $g(z\pi(\lambda))=g(z)$. Since the action of $\bar{\Lambda}$ on ${{\bar{\Gamma}\backslash\bar{N}}}$ is ergodic, $g$ is constant. By , $g$ must be zero. Let $f:{{\mathbb{Z}}}\backslash{{\mathbb{R}}}\xrightarrow{\iota_{\Gamma x}} {{\bar{\pi}^{-1}(z)}}\xrightarrow{c(\cdot,e^{Y_1})}{{\mathbb{R}}}$. We define $$h_z(\iota_{\Gamma x}(s))=\sum_{k\in{{\mathbb{Z}}}\setminus\{0\}} \frac{\hat{f}(k)}{-1+e^{2\pi ika}}e^{2\pi iks}.$$ Then $h_z:{{\bar{\pi}^{-1}(z)}}\to{{\mathbb{R}}}$ is $C^{\infty}$, since $f$ is $C^{\infty}$ and $a$ is Diophantine. By Lemma \[int0\], we have $$c(\iota_{\Gamma x}(s),e^{Y_1}) =-h_z(\iota_{\Gamma x}(s))+h_z(\iota_{\Gamma x}e^{Y_1}).$$ If we choose another point $\Gamma e^{s_0X_1}x\in{{\bar{\pi}^{-1}(z)}}$ to define $h_z$, $$\begin{aligned} h_z(\iota_{\Gamma x}(s))&=h_z(\Gamma e^{sX_1}x) =h_z(\iota_{\Gamma e^{s_0X_1}x}(s-s_0)) \\ &=\sum_{k\in{{\mathbb{Z}}}\setminus\{0\}}\frac{1}{-1+e^{2\pi ika}} \int_0^1c(\Gamma e^{(u+s_0)X_1}x,e^{Y_1})e^{-2\pi iku}du\ e^{2\pi ik(s-s_0)} \\ &=\sum_{k\in{{\mathbb{Z}}}\setminus\{0\}}\frac{1}{-1+e^{2\pi ika}} \int_0^1f(u+s_0)e^{-2\pi iku}du\ e^{2\pi ik(s-s_0)} \\ &=\sum_{k\in{{\mathbb{Z}}}\setminus\{0\}}\frac{\hat{f}(k)}{-1+e^{2\pi ika}} e^{2\pi iks}, \end{aligned}$$ so that $h_z$ is determined only by $z$. Define $h:{{\Gamma\backslash N}}\to{{\mathbb{R}}}$ by $h|_{{{\bar{\pi}^{-1}(z)}}}=h_z$. Then for all $x\in{{\Gamma\backslash N}}$ and $m\in{{\mathbb{Z}}}$, $c(x,e^{mY_1})=-h(x)+h(xe^{mY_1})$. Let $U\subset{{\bar{\Gamma}\backslash\bar{N}}}$ be open and $\sigma:U\to\bar{\pi}^{-1}(U)$ a section of $\bar{\pi}$. Then we have a trivialization ${{\mathbb{Z}}}\backslash{{\mathbb{R}}}\times U\simeq\bar{\pi}^{-1}(U)$ which sends $(s,z)$ to $\iota_{\sigma(z)}(s)=\Gamma e^{sX_1}\sigma(z)$. Hence $$h(\iota_{\sigma(z)}(s))=\sum_{k\in{{\mathbb{Z}}}\setminus\{0\}} \frac{1}{-1+e^{2\pi ika}} \int_0^1c(\iota_{\sigma(z)}(u),e^{Y_1})e^{-2\pi iku}du\ e^{2\pi iks}$$ on $\bar{\pi}^{-1}(U)$. The following lemma shows $h$ is $C^{\infty}$ on ${{\Gamma\backslash N}}$. Let $U\subset{{\mathbb{R}}}^n$ be open and let $f:{{\mathbb{Z}}}\backslash{{\mathbb{R}}}\times U\to{{\mathbb{R}}}$ be a $C^{\infty}$-function. Define $$h(s,z)=\sum_{k\in{{\mathbb{Z}}}\setminus\{0\}}\frac{1}{-1+e^{2\pi ika}} \widehat{f_z}(k)e^{2\pi iks}$$ where $f_z(u)=f(u,z)$. Then $h:{{\mathbb{Z}}}\backslash{{\mathbb{R}}}\times U\to{{\mathbb{R}}}$ is $C^{\infty}$. Let $V$ be open such that $\bar{V}\subset U$ and $\bar{V}$ is compact. We will show that $h$ is $C^{\infty}$ on ${{\mathbb{Z}}}\backslash{{\mathbb{R}}}\times V$. Choose constants $C,\alpha>0$ such that $\lvert -1+e^{2\pi ika}\rvert\geq C\lvert k\rvert^{-\alpha}$ for all $k\in{{\mathbb{Z}}}\setminus\{0\}$. We will first prove that $h$ is continuous. Since for any $m\in{{\mathbb{Z}}}_{>0}$, $$\frac{\partial^mf_z}{\partial s^m}(s)= \sum_{k\in{{\mathbb{Z}}}}(2\pi ik)^m\widehat{f_z}(k)e^{2\pi iks}$$ in $L^2({{\mathbb{Z}}}\backslash{{\mathbb{R}}})$, $$\begin{aligned} \Bigl\lVert\frac{\partial^mf_z}{\partial s^m}\Bigr\rVert_2^2 &=\sum_{k\in{{\mathbb{Z}}}}\lvert(2\pi ik)^m\widehat{f_z}(k)\rvert^2 \\ &\geq(2\pi)^{2m}\lvert k\rvert^{2m}\lvert\widehat{f_z}(k)\rvert^2 \geq\lvert k\rvert^{2m}\lvert\widehat{f_z}(k)\rvert^2. \end{aligned}$$ Since $\Bigl\lVert\frac{\partial^mf_z}{\partial s^m}\Bigr\rVert_2 =\left(\int_0^1\Bigl\lvert\frac{\partial^m}{\partial s^m}f(s,z) \Bigr\rvert^2ds\right)^{\frac{1}{2}}$ is continuous in $z$, there exists $M>0$ such that $\Bigl\lVert\frac{\partial^mf_z}{\partial s^m}\Bigr\rVert_2<M$ for every $z\in\bar{V}$. Hence for all $k\in{{\mathbb{Z}}}$ and $z\in\bar{V}$, $\lvert k\rvert^m\lvert\widehat{f_z}(k)\rvert\leq M$. Therefore, for any $z\in\bar{V}$, $$\begin{aligned} \sum_{k\in{{\mathbb{Z}}}\setminus\{0\}}\Bigl\lvert\frac{1}{-1+e^{2\pi ika}} \widehat{f_z}(k)e^{2\pi iks}\Bigr\rvert &\leq C^{-1}\sum_{k\in{{\mathbb{Z}}}\setminus\{0\}}\frac{1}{\lvert k\rvert^2} \lvert k\rvert^{\alpha+2}\lvert\widehat{f_z}(k)\rvert \\ &\leq C^{-1}M\sum_{k\in{{\mathbb{Z}}}\setminus\{0\}}\frac{1}{\lvert k\rvert^2} <\infty. \end{aligned}$$ This implies continuity of $h$ on ${{\mathbb{Z}}}\backslash{{\mathbb{R}}}\times\bar{V}$. We have $$\frac{\partial h}{\partial s}(s,z)= \sum_{k\in{{\mathbb{Z}}}\setminus\{0\}}\frac{2\pi ik}{-1+e^{2\pi ika}} \widehat{f_z}(k)e^{2\pi iks}.$$ Thus a similar argument shows that $\frac{\partial h}{\partial s}$ is continuous. Let $z=(z_1,\dots,z_n)$. For any $z\in\bar{V}$, $$\begin{aligned} \Bigl\lvert\frac{\partial}{\partial z_j} \left(\frac{1}{-1+e^{2\pi ika}} \widehat{f_z}(k)e^{2\pi iks}\right)\Bigr\rvert &=\Bigl\lvert\frac{1}{-1+e^{2\pi ika}} \widehat{\frac{\partial f}{\partial z_j}(\cdot,z)}(k)e^{2\pi iks} \Bigr\rvert \\ &\leq C^{-1}\frac{1}{\lvert k\rvert^2}\lvert k\rvert^{\alpha+2} \Bigl\lvert\widehat{\frac{\partial f}{\partial z_j}(\cdot,z)}(k) \Bigr\rvert \\ &\leq C^{-1}M^{\prime}\frac{1}{\lvert k\rvert^2}\ \in L^1({{\mathbb{Z}}}\setminus\{0\}). \end{aligned}$$ Thus $$\frac{\partial h}{\partial z_j}(s,z)= \sum_{k\in{{\mathbb{Z}}}\setminus\{0\}}\frac{1}{-1+e^{2\pi ika}} \widehat{\frac{\partial f}{\partial z_j}(\cdot,z)}(k)e^{2\pi iks}.$$ Hence $\frac{\partial h}{\partial z_j}$ is continuous by an argument similar to those above. For higher derivatives of $h$, continue this procedure. Set $c_1(x,\lambda)=c(x,\lambda)+h(x)-h(x\lambda)$. $c_1:{{\Gamma\backslash N}}\times\Lambda\to{{\mathbb{R}}}$ is a $C^{\infty}$-cocycle and $c_1(x,e^{mY_1})=0$. Thus for any $\lambda\in\Lambda$, cocycle equation implies $c_1(x,\lambda)=c_1(xe^{Y_1},\lambda)$. Since the action of $e^{{{\mathbb{Z}}}Y_1}$ on ${{\bar{\pi}^{-1}(z)}}$ is ergodic, $c_1(x,\lambda)$ is constant on ${{\bar{\pi}^{-1}(z)}}$. Therefore we can define a cocycle $\bar{c}:{{\bar{\Gamma}\backslash\bar{N}}}\times\bar{\Lambda}\to{{\mathbb{R}}}$ by $\bar{c}(\bar{\pi}(x),\pi(\lambda))=c_1(x,\lambda)$. Indeed, if $\bar{\pi}(x)=\bar{\pi}(y)$ and $\pi(\lambda)=\pi(\lambda^{\prime})$, then there exists a $m\in{{\mathbb{Z}}}$ with $\lambda=e^{mY_1}\lambda^{\prime}$, so that $$c_1(x,\lambda)=c_1(x,e^{mY_1}\lambda^{\prime}) =c_1(xe^{mY_1},\lambda^{\prime})=c_1(y,\lambda^{\prime}).$$ Furthermore, $$\begin{aligned} \int_{{{\bar{\Gamma}\backslash\bar{N}}}}\bar{c}(x,\pi(\lambda))d\nu(z) &=\int_{{{\bar{\Gamma}\backslash\bar{N}}}}\int_{{{\bar{\pi}^{-1}(z)}}}c_1(s,\lambda)d\mu_z(s)d\nu(z) \\ &=\int_{{{\Gamma\backslash N}}}c_1(x,\lambda)d\mu(x)=0. \end{aligned}$$ By induction, there exists a $C^{\infty}$-function $P:{{\bar{\Gamma}\backslash\bar{N}}}\to{{\mathbb{R}}}$ such that $\bar{c}(z,\pi(\lambda))=-P(z)+P(z\pi(\lambda))$. Put $Q=P\circ\bar{\pi}$. Then $c_1(x,\lambda)=\bar{c}(\bar{\pi}(x),\pi(\lambda))=-Q(x)+Q(x\lambda)$. This proves Theorem \[constant\]. Let $\tilde{\rho}:M\times N\to M$ be the suspension of $\rho:{{\Gamma\backslash N}}\times\Lambda\to{{\Gamma\backslash N}}$ where $M={{\Gamma\backslash N}}\times_{\Lambda}N$ is a compact manifold. Then $\tilde{\rho}$ is locally free and let ${{\mathcal{F}}}$ be its orbit foliation. We have $$H^1({{\mathcal{F}}})\simeq H^1(\Lambda,C^{\infty}({{\Gamma\backslash N}}))$$ by [@PdS] where the right hand side is the first cohomology of $\Lambda$-module $C^{\infty}({{\Gamma\backslash N}})$ obtained by $\rho$. It is easy to prove that $\Hom(\Lambda,{{\mathbb{R}}})\to H^1(\Lambda,C^{\infty}({{\Gamma\backslash N}}))$ is injective. By Theorem \[constant\], $$H^1(\Lambda,C^{\infty}({{\Gamma\backslash N}}))=\Hom(\Lambda,{{\mathbb{R}}}).$$ $\dim\Hom(\Lambda,{{\mathbb{R}}})=\dim H^1({{\mathfrak{n}}}). $ Recall that $[N,N]\backslash\Lambda[N,N]$ is a cocompact lattice in $[N,N]\backslash N$ and that $[\Lambda,\Lambda]\backslash\left(\Lambda\cap[N,N]\right)$ is finite. Since $$0\to[\Lambda,\Lambda]\backslash\left(\Lambda\cap[N,N]\right)\to [\Lambda,\Lambda]\backslash\Lambda\to[N,N]\backslash\Lambda[N,N]\to0$$ is exact, we have $$\rank[\Lambda,\Lambda]\backslash\Lambda =\rank[N,N]\backslash\Lambda[N,N]=\dim[N,N]\backslash N.$$ Thus $$\begin{aligned} \dim\Hom(\Lambda,{{\mathbb{R}}}) &=\dim\Hom([\Lambda,\Lambda]\backslash\Lambda,{{\mathbb{R}}}) \\ &=\rank[\Lambda,\Lambda]\backslash\Lambda \\ &=\dim[N,N]\backslash N \\ &=\dim\Hom_{{{\mathbb{R}}}}([{{\mathfrak{n}}},{{\mathfrak{n}}}]\backslash{{\mathfrak{n}}},{{\mathbb{R}}}) \\ &=\dim H^1({{\mathfrak{n}}}). \end{aligned}$$ Therefore we obtain $$H^1({{\mathcal{F}}})=H^1({{\mathfrak{n}}}).$$ This proves Theorem \[main2\]. Existence of Diophantine lattices ================================= Let ${{\mathfrak{n}}}_{{{\mathbb{Q}}}}$ be a rational structure of ${{\mathfrak{n}}}$. We construct Diophantine lattices when ${{\mathfrak{n}}}_{{{\mathbb{Q}}}}$ admits a graduation. Namely, we assume that ${{\mathfrak{n}}}_{{{\mathbb{Q}}}}$ has a sequence $V_i$ of ${{\mathbb{Q}}}$-subspaces such that ${{\mathfrak{n}}}_{{{\mathbb{Q}}}}=\bigoplus_{i=1}^kV_i$ and $[V_i,V_j]\subset V_{i+j}$. Let $X_1,\dots,X_n$ be a ${{\mathbb{Q}}}$-basis of ${{\mathfrak{n}}}_{{{\mathbb{Q}}}}$ such that $X_1,\dots,X_{i_1}\in V_k$, $X_{i_1+1},\dots,X_{i_2}\in V_{k-1}$$,\dots$, $X_{i_{k-1}+1},\dots,X_n\in V_1$. Then $X_1,\dots,X_n$ is a strong Malcev basis of ${{\mathfrak{n}}}$ with rational structure constants. Multiplying $X_1,\dots,X_n$ by a integer if necessary, we may assume that $\Gamma:=e^{{{\mathbb{Z}}}X_1}\cdots e^{{{\mathbb{Z}}}X_n}$ is a cocompact lattice in $N$. Let $\alpha$ be a root of a irreducible polynomial of degree $k+1$ over ${{\mathbb{Q}}}$. Since $\alpha,\alpha^2,\dots,\alpha^k$ are irrational algebraic numbers, they are Diophantine. If we define a linear map $\varphi:{{\mathfrak{n}}}\to{{\mathfrak{n}}}$ by $\varphi(X)=\alpha^iX$ for $X\in V_i\otimes{{\mathbb{R}}}$, then $\varphi$ is an automorphism of Lie algebra ${{\mathfrak{n}}}$. Put $Y_i=\varphi(X_i)$. $Y_1,\dots,Y_n$ is a strong Malcev basis of ${{\mathfrak{n}}}$ strongly based on $\Lambda:=e^{{{\mathbb{Z}}}Y_1}\cdots e^{{{\mathbb{Z}}}Y_n}$. Thus $\Lambda$ is Diophantine with respect to $\Gamma$. Acknowledgement {#acknowledgement .unnumbered} =============== The author would like to thank his advisor, Masayuki Asaoka, for helpful comments. [99]{} M. Asaoka. [*Deformation of lacally free actions and the leafwise cohomology*]{}. arXiv:1012.2946. L. Corwin and F. P. Greenleaf. [*Representations of nilpotent Lie groups and their applications. Part 1:Basic theory and examples*]{}. Cambridge studies in advanced mathematics, vol. 18, Cambridge University Press, Cambridge, 1990. A. Katok and R. J. Spatzier. [*First cohomology of Anosov actions of higher rank abelian groups and applications to rigidity*]{}. Inst. Hautes Études Sci. Publ. Math. **79**(1994), 131–156. S. Matsumoto and Y. Mitsumatsu. [*Leafwise cohomology and rigidity of certain Lie group actions*]{}. Ergod. Th. & Dynam. Sys. **23**(2003), 1839–1866. D. Mieczkowski. [*The first cohomology of parabolic actions for some higher-rank abelian groups and representation theory*]{}. J. Mod. Dyn. **1**(2007), 61–92. M. S. Pereira and N. M. dos Santos. [*On the cohomology of foliated bundles*]{}. Proyecciones **21**(2)(2002), 175–197. F. A. Ramírez. [*Cocycles over higher-rank abelian actions on quotients of semisimple Lie groups*]{}. J. Mod. Dyn. **3**(2009), 335–357. N. M. dos Santos. [*Parameter rigid actions of the Heisenberg groups*]{}. Ergod. Th. & Dynam. Sys. **27**(2007), 1719–1735. [^1]: Research Fellow of the Japan Society for the Promotion of Science
--- abstract: 'Coupling of two dielectric wires with rectangular cross-section gives rise to bonding and anti-bonding resonances. The latter is featured by extremal narrowing of the resonant width for variation of the aspect ratio of the cross-section and distance between wires when the morphology of the anti-bonding resonant mode approaches to the morphology of the Mie resonant mode of effective circular wire with high azimuthal index. Then plane wave resonant to this anti-bonding resonance gives rise to unprecedent enhancement of the optical forces up to several nano Newtons per micron length of wires. The forces oscillate with angle of incidence of plane wave but always try to repel the wires. If the wires are fixed at the ends the optical forces result in elastic deflection of wires of order $100 nm$ for wires’s length $50\mu m$ and the light power $1.5mW/\mu m^2$.' author: - Evgeny Bulgakov - 'Almas F. Sadreev' bibliography: - 'sadreev.bib' title: Resonant bending of silicon nanowires by light --- Introduction ============ The response of a microscopic dielectric object to a light field can profoundly affect its motion. A classical example of this influence is an optical trap, which can hold a particle in a tightly focused light beam [@Ashkin1986]. When two or more particles are present, the multiple scattering between the objects can, under certain conditions, lead to optically bound states. This is often referred to peculiar manifestation of optical forces as optical binding (OB), and it was first observed by Burns et al. on a system of two plastic spheres in water in 1989 [@Burns1989] and after by other scholars [@Chaumet2001; @Tatarkova2002; @Mohanty2004]. Optical binding belongs to an interesting type of mechanical light-matter interaction between particles at micro-scale mediated by the light scattered by illuminated particles. Depending on the particle separation, OB leads to attractive or repulsive forces between the particles and, thus, contributes to the formation of stable configurations of particles. The phenomenon of OB can be realized, for example, in dual counter propagating beam configurations [@Tatarkova2002; @Gomez-Medina2004; @Metzger2006; @Dholakia2010]. It is clear that excitation of the resonant modes with high $Q$ factor in dielectric structures by light results in large enhancement of near electromagnetic (EM) fields and respectively in extremely large EM forces proportional to squared EM fields. First, sharp features in the force spectrum, causing mutual attraction or repulsion between successive photonic crystal layers of dielectric spheres under illumination of plane wave has been considered by Antonoyiannakis and Pendry [@Antonoyiannakis1997]. It was shown that the normal force acting on each layer as well as the total force acting on both layers including the optical binding force follow these resonances. It was revealed that the lower frequency bonding resonance forces push the two layers together and the higher frequency anti-bonding resonance pull them apart. Later these disclosures were reported for coupled photonic crystal slabs [@Liu09] and two planar dielectric photonic metamaterials [@Zhang2014] due to existence of resonant states with infinite $Q$ factor (bound states in the continuum (BICs)). Recently Hurtado [*et al*]{} [@Hurtado2020] have shown that excitation of quasi BICs in a dimerized high-contrast grating with a compliant bilayer structure stimulates considerable forces capable for structural deformations of the dimer. However it is remarkable, even two particles can demonstrate extremely high $Q$ resonant modes owing to avoided crossings. The vivid example is avoided crossing of whispering-gallery modes (WGM) in coupled microresonators which results in extremely high $Q$ factor [@Povinelli2005; @Benyoucef2011]. As a result an enhancement of the OB force around of hundreds of nano Newtons between coupled WGM spherical resonators takes place in applied power $1mW$ [@Povinelli2005]. After a few years Wiederhecker [*et al*]{} [@Wiederhecker2009] demonstrated a static mechanical deformation of up to 20 nm in double silicon nitride rings of 30 $\mu m$ diameter by illumination of milliwatts optical power. In 2005 Povinelli [*et al*]{} [@Povinelli2005a] calculated forces between two parallel, silicon wires of square cross-section as shown in Fig. \[fig1\] (a) caused by electromagnetic (EM) waves propagating along the wires with frequencies below the light line. Both attractive and repulsive forces, determined by the choice of relative input phase, the forces were found large enough to cause displacements 20 nm of wire with length 30$\mu m$. However these optical binding (OB) forces are caused by evanescent EM fields which are exponentially weak between the wires that requires a considerable input power 1W/$\mu m^2$ [@Povinelli2005a]. In the present letter we consider scattering of plane wave by parallel wires in the resonant regime. Scheme of illumination is shown in Fig. \[fig1\] (a). That demands incident power of only 1.5 mW/$\mu m^2$ in order to result considerable deflection of wires of order 100 nm for the wires of length $50 \mu m$. We show that for two-parametric variation (the aspect ratio and distance between wires) the system of two wires acquires anti-bonding resonant modes with extremely high $Q$ factor. That happens when a morphology of the anti-bonding resonant mode becomes close to the morphology of the Mie resonant modes with high azimuthal index of effective cylindrical wire. If the plane wave with power $1.5 mW/\mu m^2$ is capable to excite such an anti-bonding resonant mode optical forces reach a value till one nano Newton per micron length of wires. As a result wires of enough length are bending as shown in Fig. \[fig1\]. Mie resonance in the system of two parallel nanowires ===================================================== The behavior of resonances of resonances of isolated nanowire as dependent on aspect ratio of the cross-section $a/b$ was studied by Huang [*et al*]{} [@Huang2019] in aim to optimize the $Q$ factor of the nanowire by use the same strategy of avoided crossing as it was used in papers [@Wiersig2006; @Rybin2017]. Therefore it is reasonable to start a consideration of optical binding (OB) force of two wires each of them is optimized by the $Q$ factor which reaches maximal amount 2730 for silicon wire with the refractive index 3.48 at $a=b$. The corresponding resonant mode of each wire shown in inset of Fig. \[fig2\] has a morphology of the Mie resonant mode with rather high azimuthal index $m=6$ (close to whispering gallery mode) that explains so small radiation losses. The corresponding optical forces, acting on each wire are shown in Fig. \[fig2\]. One can see that the OB force tends to repel the wires and reaches a magnitude 70 pico Newtons per micron. In order to achieve unprecedent $Q$ factor the another strategy was proposed in Ref. [@Bulgakov2020a] based on crossing of the resonances which have opposite symmetry in the isolated wire as shown in Fig. \[fig3\] (a). However the presence of the second wire lifts this symmetry restriction giving rise to a new series of avoided crossings of resonances. One of such an avoided crossing is shown in Fig. \[fig3\] (b) with insets showing evolution of the anti-bonding resonant modes. We do not show the bonding resonances because of their undistinguished $Q$ factors. For variation of the distance $L$ between the wires one of the anti-bonding resonant modes of two coupled wires reaches extraordinary small radiation losses. As highlighted in respective inset in Fig. \[fig3\] (b) at the point of minimal imaginary part the anti-bonding resonant mode of two wires with the cross-section $a=1.19\mu m, b=0.592\mu m, a/2b=1.0152$ and the distance between wires $L=1.555\mu m$ acquires morphology close to the morphology of effective isolated wire with cylindrical cross-section with $m=7$ as highlighted by white circle in inset of Fig. \[fig3\] (b). It is remarkable that the $Q$ factor of this anti-bonding resonance reaches unprecedent value around 15000 at precise tuning of wire’s scales: $a/2b=1.0152, L/b=1.5546$ as shown in Fig. \[fig3\] (c). Then one can expect giant OB forces as it was established for two dielectric disks when the frequency of Bessel beams was resonant to the anti-bonding resonance [@Bulgakov2020]. Indeed Fig. \[fig4\] show that the OB force reaches giant values around 4 nano Newtons that exceeds the case shown in Fig. \[fig2\] by two orders in value. Deflection of wires =================== Following to Povinelli [*et al*]{} [@Povinelli2005a] we estimate the deflection of wires $w(z)$ due to optical forces shown in Fig. \[fig4\]. The deflection obeys the equation $$\label{defl} \frac{d^4w_j}{dz^4}=\frac{12f_j}{ab^3E}, ~~j=1,2$$ where $f_j$ are the optical force acting on unit length of the j-th wire, and $E=169GPa$ is the Young modulus of silicon. In what follows we disregard optical properties of plates to which the wires are fixed assuming their refractive index close to air. For $|w(z)|\ll S$ and applying the boundary conditions $w_j(\pm S/2)=0$ and $dw_j(\pm S/2)/dz=0$ one can obtain the solution of Eq. (\[defl\] $$\label{deflsol} w_j(z)=\frac{f_j}{ab^3E}\left[\frac{1}{2}z^4-\frac{S^2}{4}z^2+\frac{S^4}{32}\right].$$ At the specific values $a=1.19\mu m, b=0.592\mu m$ and $f_j\approx \pm 4200 pN$ for the EM power $1.5mW/\mu m^2$ we find that maximal deflection at the center between plates $w(0)=3.1\times 10^{-8}S^4$ where the length of wires $S$ and deflection $w$ is measured in microns. Therefore for $S=50\mu m$ we obtain the maximal deflection is around $150nm$ while the EM power of $1W/\mu m^2$ propagating along the wires result in deflection $20nm$ [@Povinelli2005a] that considerably yields the case of the resonant plane wave illumination with the power by three orders less. However this estimation of deflection of wires is to be considered as upper limit. As soon as the wires start to bend the anti-bonding resonance with extremely high $Q$ factor established for two parallel wires will go away from extremal point shown in Fig. \[fig3\] (b) by, at least, two reasons. The first reason is related to deviation from optimal distance between the wires. As a result the optical forces will be strongly decreased as shown in Fig. \[fig5\]. The second reason which also affects the optical forces is related to the bending of wires which put the problem into the 3d one. Funding ======= The work was supported by Russian Foundation for Basic Research projects No. 19-02-00055. Disclosures =========== The authors declare no conflicts of interest.
--- abstract: | For a compact homogeneous space $G/K$, we study the problem of existence of $G$-invariant Riemannian metrics such that each eigenspace of the Laplacian is a real irreducible representation of $G$. We prove that the normal metric of a compact irreducible symmetric space has this property only in rank one. Furthermore, we provide existence results for such metrics on certain isotropy reducible spaces. address: 'Institut für Differentialgeometrie, Leibniz Universität Hannover Welfengarten 1, 30167 - Hannover - Germany' author: - David Petrecca - Markus Röser bibliography: - '../../allbib/allbib.bib' title: Irreducibility of the Laplacian eigenspaces of some homogeneous spaces --- Introduction {#introduction .unnumbered} ============ Let $(M, g)$ be a compact Riemannian manifold and let $\Delta_g$ be the Laplace operator of $g$ acting on smooth functions. It is an elliptic, self-adjoint and non-negative operator, so its non-zero eigenvalues $\lambda_i$ are positive and the corresponding eigenspaces $E_{\lambda_i}$ are finite-dimensional. By definition we have $E_0 = \ker\Delta_g$, which, if $M$ is connected, consists of only the constant functions. The increasing sequence of real numbers $$0 = \lambda_0 < \lambda_1 \leq \lambda_2 \leq \ldots \nearrow +\infty,$$ where each $\lambda_i$ is repeated according to its multiplicity $\dim E_{\lambda_i}$, is called the *spectrum* of $(M,g)$ and denoted by $\operatorname{Spec}(M,g)$. A metric such that $\dim E_{\lambda_i}=1$, i.e. all eigenspaces are of minimal possible dimension, is called *simple*. If ${\mathcal}M$ is the space of smooth Riemannian metrics on $M$, Uhlenbeck [@Uhlenbeck] proved that the set of simple metrics is residual in ${\mathcal}M$ with respect to an appropriate topology. We recall that a subset of a topological space is called residual if it is the intersection of countably many sets with dense interior. Recently, Guillemin, Legendre and Sena-Dias [@GuilleminLegendreSenaDias] computed the variation of the Rayleigh quotients within a conformal class, reproving the result of Uhlenbeck stating that every Riemannian metric can be conformally deformed into a metric with simple spectrum. The notion of simplicity can be generalized in the following way in presence of symmetries. Assume that $(M, g)$ is acted on by a group $G$ of isometries. The extended action of $G$ on $C^\infty(M,{\mathbb C})$ given by $(g \cdot f)(x) = f(g^{-1}\cdot x)$ commutes with the Laplacian, so it defines a $G$-representation on each eigenspace $E_\lambda$ which is real in the sense that it preserves the real subspace $C^\infty(M,{\mathbb R})\cap E_\lambda$. In this context, the spectrum of $\Delta$ will of course not be simple. An analogous minimality property of the eigenspaces is to be irreducible as $G$-representations. This motivates the following definition. A compact Riemannian manifold $(M, g)$ with an isometric action of a group $G$ is said to have a *complex $G$-simple spectrum* if, for every eigenvalue $\lambda\in\operatorname{Spec}(M,g)$, the complex $G$-representation $E_\lambda\subset C^\infty(M, {\mathbb C})$ is irreducible. If the real $G$-representations $E_\lambda\cap C^\infty(M, {\mathbb R})$ are all irreducible, we say that $(M,g)$ has *$G$-simple spectrum*, or is *$G$-simple* for short. It is clear that this property is invariant under homothetic variations of the metric. This notion first appeared in the monograph [@spectre] by Berger, Gauduchon and Mazet. Recently, Schueth [@schueth] studied the problem for $M=G$ with a left-invariant metric and $G$ acting by left-translations. She established a number of genericity results for $G$ given by a product of a torus and copies of $\operatorname{SU}(2)$ and quotients of such groups by discrete central subgroups. In this work, we focus on $G$-invariant metrics on homogeneous spaces $M = G/K$, where $G$ is a compact connected Lie group and $K$ a closed subgroup. On such spaces, $G$-invariant Riemannian metrics correspond to $\operatorname{Ad}(K)$-invariant inner products on the Lie algebra ${\mathfrak}g$ of $G$, see e.g. [@KN Chap. X]. Fixing a $\mathrm{Ad}(G)$-invariant inner product on ${\mathfrak}g$, we get a $K$-invariant decomposition ${\mathfrak}g = {\mathfrak}k \oplus {\mathfrak}m$, where ${\mathfrak}m = {\mathfrak}k^\perp$ is called the isotropy representation. If ${\mathfrak}m$ is an irreducible representation of $K$, the space $M$ is called isotropy irreducible, otherwise it is isotropy reducible. The space of left-invariant metrics on $G$ can then be identified with the space $\operatorname{Sym}_K^+({\mathfrak}m)$ of $\operatorname{Ad}(K)$-invariant inner products on ${\mathfrak}m$. If ${\mathfrak}m$ decomposes into pairwise inequivalent irreducible representations ${\mathfrak}m_i$, $i=1,\dots,s$, then $\operatorname{Sym}_K^+({\mathfrak}m)$ may be identified with the positive orthant in ${\mathbb R}^s$, which we denote by ${\mathbb R}^s_+$. Otherwise, $\operatorname{Sym}_K⁺({\mathfrak}m)$ is more complicated since equivalent submodules need not be orthogonal for a $K$-invariant inner product. The Peter-Weyl theorem describes the decomposition of the $G$-representation $C^\infty(G/K,{\mathbb C})$ into isotypical components. The representations that occur are exactly the *spherical representations* of the pair $(G,K)$, namely the irreducible complex $G$-representations containing a nonzero subspace of vectors fixed by $K$. The action of the Laplacian on each isotypical component can be described in purely Lie-algebraic terms. Adapting Schueth’s arguments to our setting, we can give a Lie-algebraic characterization of $G$-simplicity and prove the following result, which means that if a $G$-simple metric on $G/K$ exists, then the generic left-invariant metric on $G/K$ has this property. Let $M=G/K$ be a homogeneous space with isotropy representation ${\mathfrak}m$. If the space of $G$-simple Riemannian metrics on $G/K$ is non-empty, then it is a residual set in $\operatorname{Sym}_K^+({\mathfrak}m)$. Of course, this proposition does not say anything about the problem of existence of $G$-simple metrics on $G/K$. In order to tackle this, one needs to understand the set of spherical representations. However, in [@spectre], the following result is obtained by analytic techniques, without determining spherical representations. \[thm:BGM\] Let $G/K$ be a compact homogeneous Riemannian manifold with a $G$-invariant metric and consider the orthogonal splitting ${\mathfrak}g = {\mathfrak}k \oplus {\mathfrak}m$. If the connected component $K^0$ of $K$ acts transitively on the unit sphere of ${\mathfrak}m$ under the isotropy representation, then $G/K$ has a $G$-simple spectrum. A homogeneous space $G/K$ satisfying this sufficient condition must be isotropy irreducible. As any two invariant metrics are constant multiples of each other it follows that in this case every invariant metric on $G/K$ is $G$-simple. Examples are given by the homogeneous presentations of compact rank-one symmetric spaces (CROSS) along with some non-symmetric spaces described in Examples \[ex:G2SU3\] and \[ex:SO7G2\]. For symmetric pairs, spherical representations are well understood due to the Cartan-Helgason theorem. They are given by *restricted dominant weights* (see Section \[sec:restricted\]). In this work, we use this theory to prove that the CROSS’s are the only *irreducible* symmetric spaces with this property. An irreducible symmetric space $G/K$ has simple $G$-spectrum if, and only if, it has rank one. On the other hand, we obtain reducible $G$-simple symmetric spaces $G/K$ of arbitrary rank by considering Riemannian products of CROSS’s. A generic $G$-invariant metric on the product $G/K = \prod_i G_i/K_i$ of CROSS’s is $G$-simple. Non-symmetric isotropy irreducible homogeneous spaces have been classified [@wolf_class], but there is in general no efficient parameterization of their spherical representations. It is therefore not clear whether one can adapt the proof of Theorem \[thmrefl\] or whether the sufficient condition of Theorem \[thm:BGM\] is necessary. Full flag manifolds $G/T$, where $T\subset G$ is a maximal torus, form a class of isotropy reducible spaces for which the spherical representations are well understood [@SpectraFlag]. This is due to the fact that the irreducible spherical representations in this case are exactly those whose highest weight lies in the root lattice. However, the monoid of spherical representations is not freely generated in this case, and this causes difficulties in obtaining sufficient information on the eigenspaces of the Laplacian of a $G$-invariant metric. If the pair $(G,K)$ is spherical, i.e. for any irreducible representation of $G$ the set of $K$-fixed points is at most one dimensional, then more is known. In fact, Krämer [@kraemer] classified spherical pairs $(G,K)$ with $G$ compact, connected and simple and he also gave explicit descriptions of the weights corresponding to spherical representations. Using these results, we are able to perform explicit computations on a class of spherical spaces that are circle bundles over Hermitian symmetric spaces. Combining well-known properties of the Laplacian on fiber bundles with totally geodesic fibers along with an explicit analysis of a special case, we are able to prove the following theorem for total spaces of Hopf fibrations $S^{2n+1}\to {\mathbb{CP}}^n$, $n>1$, by stretching the metric along the fibers. The corresponding result for $n=1$ was established by Schueth in [@schueth] by a different argument. Let $n>1$. Then a generic $\operatorname{SU}(n+1)$-invariant metric on the total space of the Hopf fibration $S^{2n+1} \to {\mathbb{CP}}^n$ is $\operatorname{SU}(n+1)$-simple. We treat in Subsection \[sec:SU2F\] an example of a quotient of $\operatorname{SU}(2)$ by a non-normal, non-spherical finite subgroup and we prove that a generic $\operatorname{SU}(2)$-invariant metric is $\operatorname{SU}(2)$-simple. This space appeared in [@art:bedulli_stab] in the context of homogeneous Lagrangian submanifolds of projective spaces. The paper is organized as follows. In Section \[sec:lapl\] we discuss the required background on the Laplacian on a homogeneous space and the Peter-Weyl theorem. We give a purely Lie-algebraic characterization of $G$-simplicity and establish Proposition \[prop:resultants\]. Then we move to the case of compact symmetric spaces in Section \[sec:symm\] and prove Theorems \[thmrefl\] and \[thm:generic1\]. Finally, we treat the non-symmetric examples in Section \[sec:nonsym\]. Acknowledgements {#acknowledgements .unnumbered} ---------------- The authors are supported by the Research Training Group 1463 “Analysis, Geometry and String Theory” of the DFG and the first author is supported as well by the GNSAGA of INdAM. Moreover, they would like to thank Fabio Podestà for valuable feedback and his interest in this work and Emilio Lauret for pointing out an inaccuracy in an earlier version of this article. The eigenspaces of the Laplacian of a homogeneous space {#sec:lapl} ======================================================= Preliminaries and notations --------------------------- Let $G$ be a compact connected Lie group. Then we have an action of $G\times G$ on $G$ given by $$(g,h) \cdot x = gxh^{-1},\qquad g,h,x\in G.$$ This action induces an action of $G\times G$ on $C^{\infty}(G,{\mathbb C})$ via $$(g,h) \cdot f(x) = f(g^{-1}xh) =: (L_g\circ R_h)(f)(x).$$ The maps $g\mapsto L_g, h\mapsto R_h$ define the *left and right regular representations of $G$ on $C^\infty(G,{\mathbb C})$*. Denote by ${\widehat}G$ a complete set of representatives for the set of isomorphism classes of irreducible complex representations of $G$. Given $V\in {\widehat}G$, we can define smooth functions on $G$ as follows. Consider the tensor product $V^*\otimes V$ and define for a decomposable element ${\varphi}\otimes v\in V^*\otimes V$ the function $f_{{\varphi},v}\in C^\infty(G,{\mathbb C})$ by $$f_{{\varphi},v}(x) := {\varphi}(x \cdot v).$$ In the literature, such a function is called a *matrix coefficient* (see e.g. [@BtD]). By linear extension, we obtain a linear map $$V^*\otimes V\to C^{\infty}(M,{\mathbb C}).$$ This map is equivariant with respect to the natural $G\times G$ actions on $V^*\otimes V$ and on $C^\infty(G,{\mathbb C})$. Since $V$ is irreducible, the $(G\times G)$-representation $V^*\otimes V$ is irreducible, too. Thus, the map $V^*\otimes V\to C^{\infty}(M,{\mathbb C})$ must be injective. We therefore have a natural way of realizing $V^*\otimes V$ as a subrepresentation of the $(G\times G)$-representation $C^\infty(G,{\mathbb C})$. Let $K\subset G$ be a closed subgroup. Then $G$ acts on $M=G/K$ by left translations via $g \cdot xK = (gx)K$. Smooth functions on $M$ are given by smooth functions on $G$ which are $K$-invariant with respect to the right-regular action, i.e. functions $f$ such that $f(xk) = f(x)$ for all $x \in G$ and $k \in K$. For $V\in {\widehat}G$ define the subspace of $K$-invariant vectors $$V^K = \{v\in V\ : kv = v,\ \forall k\in K\}.$$ Recall that a $G$-representation $V$ is *spherical with respect to $(G,K)$* if $\dim V^K > 0$. This property is clearly invariant under the equivalence of representations and we denote by ${\widehat}G_K\subset {\widehat}G$ a complete set of representatives for the equivalence classes of spherical representations with respect to $(G,K)$. It is clear from the above construction, that $$V^*\otimes V^K\subset C^\infty(G,{\mathbb C})^K = C^\infty(G/K,{\mathbb C}).$$ Using the Haar measure on $G$, we can consider the Hilbert space completion $L^2(G,{\mathbb C})$ of $C^\infty(G,{\mathbb C})$. For $g\in G$, the maps $L_g,R_g:C^\infty(G,{\mathbb C})\to C^\infty(G,{\mathbb C})$ extend to bounded linear operators on $L^2(G,{\mathbb C})$. The Peter-Weyl theorem states in particular that the space generated by the matrix coefficients, as $V$ ranges over the spherical representations of $(G,K)$, is dense in $L^2(G/K, {\mathbb C})$. More precisely, the following holds. The subspaces $$\begin{aligned} \bigoplus_{V \in {\widehat}G} V^*\otimes V &\subset & L^2(G,{\mathbb C})\\ \bigoplus_{V \in {\widehat}G_K} V^* \otimes V^K &\subset & L^2(G/K,{\mathbb C})\end{aligned}$$ are dense. For $V\in {\widehat}G$, we denote by $I(V)$ the $V$-isotypical component inside $L^2(G,{\mathbb C})$ (resp. $L^2(G/K,{\mathbb C})$) with respect to the left-regular representation. The Peter-Weyl theorem asserts that $I(V^*) \cong V^*\otimes V\subset L^2(G,{\mathbb C})$ (resp. $I(V^*)\cong V^*\otimes V^K\subset L^2(G/K,{\mathbb C})$). Then $\dim V$ (resp. $\dim V^K$) is the multiplicity of $V^*$ in $L^2(G,{\mathbb C})$ (resp. $L^2(G/K,{\mathbb C})$). We say that $V\in {\widehat}G$ is of *real type* (resp. of *quaternionic type*) if there exists a linear $G$-map $J \in \operatorname{End}(V)$ such that $J^2 = \operatorname{id}$ (resp. $J^2 = -\operatorname{id}$). If $V$ is neither of real nor of quaternionic type, it is called of *complex type*. Let $V$ be an irreducible representation of $G$ and let $V^*$ be its dual representation. 1. $V$ is of real type if, and only if, it is self-dual and equals the complexification of an irreducible real $G$-module; 2. $V$ is of quaternionic type if, and only if, it is self-dual and $V \oplus V^*$ equals the complexification of a real irreducible $G$-module; 3. $V$ is of complex type if, and only if, it is not self-dual. The Laplacian of a $G$-invariant metric on $M=G/K$ -------------------------------------------------- A $G$-invariant metric on $G/K$ corresponds to an $\operatorname{Ad}(K)$-invariant inner product on the Lie algebra ${\mathfrak}g$, which in turn corresponds to a $(G\times K)$-invariant metric on $G$. So let $g$ be a $(G\times K)$-invariant Riemannian metric on $G$. Let $d = \dim {\mathfrak}g$ and $k = \dim {\mathfrak}k$. Then $g$ is in particular a left-invariant metric on $G$ and the Laplacian $\Delta_g$ acts on $f\in C^\infty(G,{\mathbb C})$ by $$\Delta_g(f)(x) = -\sum_{i=1}^{d} \frac{d^2}{dt^2} \biggr |_{t=0}f(x\exp(tY_i)),$$ for an orthonormal basis $\{Y_1,\dots,Y_d\}\subset {\mathfrak}g$. Let $V\in {\widehat}G$, $v\in V,\varphi\in V^*$. Then the Laplacian $\Delta_g$ acts on a matrix coefficient $f_{{\varphi},v}$ by $$\Delta_g(f_{{\varphi},v})(x) = -\sum_{i=1}^{d} \frac{d^2}{dt^2} \biggr |_{t=0}{\varphi}(x\exp(tY_i)v)= -\sum_{i=1}^{d} {\varphi}(x Y_i^2v) = f_{{\varphi},\Delta_g^Vv}(x),$$ where $$\Delta_g^V = -\sum_{i=1}^{d}Y_i^2: V\to V$$ is the Casimir operator of $V$ with respect to the inner product on ${\mathfrak}g$ induced by $g$. Thus, $\Delta_g$ is compatible with the Peter-Weyl decomposition and acts as on a subspace $V^*\otimes V$ by $\operatorname{id}\otimes \Delta_g^V$. Using the metric we can split ${\mathfrak}g$ into orthogonal $K$-invariant vector subspaces ${\mathfrak}g = {\mathfrak}k\oplus {\mathfrak}m$, where ${\mathfrak}m = {\mathfrak}k^\perp$. If we choose an orthonormal basis $\{Y_1,\dots,Y_d\}$ adapted to this splitting, i.e. $\{Y_1,\dots, Y_k\}$ is an orthonormal basis for ${\mathfrak}k$, we get for the action of $\Delta_g$ on $C^\infty(G,{\mathbb C})^K$ $$\Delta_gf(x) = -\sum_{i=k+1}^{d} \frac{d^2}{dt^2}\biggl |_{t=0}f(x\exp(tY_i))$$ and on the subspace $V^*\otimes V^K$ the Laplacian $\Delta_g$ acts by $\operatorname{id}\otimes \Delta^{V^K}_g$ where $$\Delta_g^{V^K} = \Delta_g^V \biggr |_{V^K} = -\sum_{i=k+1}^{d}Y_i^2: V^K\to V^K.$$ If we choose a $G$-invariant inner product on $V$, then the Casimir $\Delta^V_g$ is self-adjoint and non-negative, hence diagonalizable with non-negative real eigenvalues. It follows that the eigenvalues of $\Delta_g$ on the subspace $V^*\otimes V^K$ are given by the eigenvalues of the Casimir $\Delta_g^{V^K}$ on $V^K$. Denote by $V_\lambda^K\subset V^K$ the eigenspace of $\Delta_g^{V^K}$ associated with the eigenvalue $\lambda\in{\mathbb R}_{\geq 0}$. Then we see that the left-regular representation of $G$ on $V^*\otimes V^K\subset L^2(G/K,{\mathbb C})$ decomposes into the eigenspaces of $\Delta_g$ as follows $$V^* \otimes V^K \cong \bigoplus_\lambda V^*\otimes V^K_\lambda.$$ In particular, one necessary condition for the eigenspaces $E_\lambda$ to be *complex* irreducible is that $\dim V^K_\lambda = 1$, i.e. $\Delta_g^{V^K}$ should have simple eigenvalues on $V^K$. Another necessary condition is that $\Delta^{V^K}_g$ and $\Delta^{W^K}_g$ do not share any common eigenvalues if $V$ and $W$ are inequivalent. Note, however, that $(V^*)^K = (V^K)^*$ and that the Casimir of $(V^*)^K$ acts on ${\varphi}\in (V^*)^K$ by $$\Delta^{(V^*)^K}_g({\varphi})(v) = -\sum_{i} Y_i^2({\varphi})(v) = -\sum_{i} {\varphi}((-Y_i)^2v) = {\varphi}(\Delta_g^{V^K}v).$$ Thus, the eigenvalues of $\Delta^{(V^*)^K}_g$ and $\Delta^{V^K}_g$ coincide. In particular, $(M,g)$ is not *complex* $G$-simple as soon as there exists a spherical representation $V$ of complex type, i.e. such that $V\ncong V^*$. So for $(M,g)$ to be complex $G$-simple any spherical representation must be of real or quaternionic type. For a spherical representation $V$ of quaternionic type with quaternionic structure $J$ the eigenspaces of $\Delta^V_g$ will be $J$-invariant, which implies that they must be of even dimension. So if $V$ is of quaternionic type, the eigenspaces of $\Delta_g: V^*\otimes V^K\to V^*\otimes V^K$ cannot be irreducible. We summarize our observations in the following proposition. \[Prop:NecCond\] Let $K\subset G$ be closed subgroup of a compact Lie group and consider the homogeneous space $M=G/K$ equipped with a $G$-invariant Riemannian metric $g$. Then the metric is complex $G$-simple, i.e. the eigenspaces $E_\lambda(M,g)\subset L^2(M,{\mathbb C})$ of the Laplacian $\Delta_g$ are irreducible complex $G$-modules, provided the following hold: 1. \[CI1\] Any spherical representation $V\in {\widehat}G_K$ satisfies $\dim V_\lambda^K=1$ for any eigenspace of the Casimir $\Delta^V_g: V^K\to V^K$. 2. \[CI2\] Any spherical representation $V\in {\widehat}G_K$ is of real type. 3. \[CI3\] For any pair $V,W\in {\widehat}G_K$ with $V\ncong W$ the Casimir operators $\Delta^{V^K}_g:V^K\to V^K$ and $\Delta^{W^K}_g: W^K\to W^K$ do not have any eigenvalues in common. Normal homogeneous spaces and spherical pairs --------------------------------------------- Suppose the metric $g$ on the homogeneous space $M = G/K$ in the previous paragraph is induced from a bi-invariant metric on $G$. If $M$ is isotropy irreducible, then up to scale this is the only choice of $G$-invariant metric on $M$. In this case, for any $V\in {\widehat}G$, the Casimir $\Delta_g^V$ acts as a multiple of the identity on $V$. Hence, $\Delta_g$ acts as a multiple of the identity on $V^*\otimes V^K$. Thus, if we look for pairs $(G,K)$ such that the normal homogeneous space $G/K$ has complex $G$-simple spectrum, then by Proposition \[Prop:NecCond\] we necessarily need $\dim V^K= 1$ for any spherical representation. This motivates the following definition from [@kraemer]. Let $K$ be a subgroup of $G$. The subgroup $K$ is *spherical* if every spherical representation $V\in{\widehat}G_K$ satisfies $\dim V^K = 1$. In this case we call $(G,K)$ a *spherical pair*. When $G$ is compact, connected and simple, Krämer classified all possible connected spherical subgroups $K \subset G$ and the list of the weights generating the subset of spherical representations (the *spherical weights*) is given in the fifth column of his table [@kraemer p. 149]. The spectrum of a normal homogeneous space can be calculated explicitly using Freudenthal’s formula for the eigenvalues of the Casimir operator. If $V_\rho\in{\widehat}G_K$ is the spherical irreducible representation with dominant highest weight $\rho$, then the Laplacian $\Delta^V_g$ acts on $V_\rho^*\otimes V_\rho^K$ with eigenvalue $$\lambda(\rho) = (\rho+2\delta,\rho)$$ where $(\cdot , \cdot)$ denotes the inner product on ${\mathfrak}g$ induced by the bi-invariant metric and $\delta$ is the sum of the fundamental weights [@takeuchi]. An explicit analysis of the function $\rho\mapsto \lambda(\rho)$ will allow us later to determine the $G$-simple compact irreducible symmetric spaces. Real eigenspaces ---------------- The property to require all eigenspaces of the Laplacian $\Delta_g$ associated with a $G$-invariant metric inside $L^2(G/K,{\mathbb C})$ to be complex irreducible representations fails as soon as there is a quaternionic spherical representation of $G$. In this paragraph we will consider the eigenspaces of $\Delta_g$ inside $L^2(G/K,{\mathbb R})$. In what follows, we will adapt Schueth’s [@schueth] line of argument to the homogeneous setting. We first observe that for any $V\in {\widehat}G$ a choice of invariant hermitian metric induces an isomorphism $\bar V\cong V^*$. For $V\in {\widehat}G_K$ we denote the subspace $V^*\otimes V^K\subset L^2(G/K,{\mathbb C})$ by $I(V^*)$, as it is the isotypical component associated with the isomorphism class of $V^*$ with respect to the left-regular representation. Define $$\mathcal C_V = I(V^*)+I(V) = \begin{cases} I(V^*) & \text{if $V$ is of real or quaternionic type,}\\ I(V^*)\oplus I(V) &\text{if $V$ is of complex type.}\end{cases}$$ On $C^\infty(G/K,{\mathbb C})$ we have a natural real structure given by complex conjugation $f\mapsto \bar f$, which induces a map $I(V^*) = I(\bar V)\to I(V)$ and thus preserves $\mathcal C_V$. We further put $$\mathcal E_V = \mathcal C_V\cap C^\infty(G/K,{\mathbb R}),$$ which is invariant under the left-regular action of $G$. Then it follows that $\mathcal C_V = \mathcal E_V\otimes {\mathbb C}$. The Laplacian is a real operator and hence acts on $\mathcal E_V$. The following Lemma gives criteria for the eigenspaces of $\Delta_g|_{\mathcal E_V}$ to be irreducible. Let $\mu$ be an eigenvalue of $\Delta^{V^K}_g|_{\mathcal E_V}$ with multiplicity $m$. Then the corresponding eigenspace is irreducible as a real $G$-representation if and only if $m$ is minimal, i.e. $$m = \begin{cases}1 & \text{if $V$ is of real or complex type}\\ 2 & \text{if $V$ is of quaternionic type.} \end{cases}$$ Let $\mu$ be an eigenvalue of $\Delta^{V^K}_g$ with multiplicity $m$, which is always even if $V$ is of quaternionic type. Let $V^K_\mu\subset V^K, (V^*)^K_\mu\subset V^*, \mathcal C^\mu_V, \mathcal E^\mu_V$ be the corresponding eigenspaces of $\Delta^{V^K}_g,\Delta_g^{(V^*)^K}$. As a complex $G$-representation $\mathcal C^\mu_V$ satisfies $$\mathcal C^\mu_V = \begin{cases} V^*\otimes V^K_\mu \cong (V^*)^{\oplus m}\cong V^{\oplus m} & \text{if $V$ is of real or quaternionic type}\\ V^*\otimes V^K_\mu \oplus V\otimes (V^*)^K_\mu \cong (V^*\oplus V)^{\oplus m} &\text{if $V$ is of complex type.}\end{cases}$$ This implies that there exists in each case an irreducible real $G$-representation $U$ such that $$\mathcal C^\mu_V = \begin{cases} U^{\oplus m}\otimes {\mathbb C}& \text{if $V$ is of real type}\\ U^{\oplus m}\otimes {\mathbb C}&\text{if $V$ is of complex type} \\ U^{\oplus m/2}\otimes {\mathbb C}& \text{if $V$ is of quaternionic type.}\end{cases}$$ Now $C^\mu_V$ is the complexification of $\mathcal E^\mu_V$, i.e. as a real $G$-representation we have $$\mathcal C^\mu_V\cong \mathcal E^\mu_V\oplus i\mathcal E^\mu_V \cong \mathcal (\mathcal E^\mu_V)^{\oplus 2}.$$ Thus the decomposition of $\mathcal E^\mu_V$ into irreducible real representations is $$\mathcal E^\mu_V = \begin{cases} U^{\oplus m}& \text{if $V$ is real}\\ U^{\oplus m} &\text{if $V$ is of complex type} \\ U^{\oplus m/2} & \text{if $V$ is quaternionic.}\end{cases}$$ We see that this is irreducible if $m=1$ in the real and complex cases and $m=2$ in the quaternionic case. \[IrrCond\] Let $g$ be a $G$-invariant metric on $M=G/K$. Then the eigenspaces of the Laplacian $\Delta_g$ inside $L^2(M,{\mathbb R})$ are irreducible $G$-modules if, and only if, the following three conditions are all satisfied: 1. \[irrcondA\] For any pair $V,W\in {\widehat}G_K$ with $V\ncong W$ and $V^*\ncong W$ the Casimir operators $\Delta^{V^K}_g,\Delta^{W^K}_g$ have no common eigenvalues. 2. \[irrcondB\] For any $V\in {\widehat}G_K$ of real or complex type, the eigenvalues of $\Delta^{V^K}_g$ are simple. 3. \[irrcondC\] For any $V\in {\widehat}G_K$ of quaternionic type, the eigenvalues of $\Delta^{V^K}_g$ are of multiplicity exactly two. If $(G,K)$ is a spherical pair, then the Laplacian associated with a $G$-invariant metric on $G/K$ has irreducible real eigenspaces if it has irreducible complex eigenspaces. Conversely, if every spherical representation is of real type then real irreducibility implies complex irreducibility. If $(G,K)$ is a spherical pair, then conditions , and are always satisfied. Let $(G,K)$ have irreducible complex eigenspaces. Then condition implies condition , so $(G,K)$ has also real irreducible eigenspaces. Conversely, if $(G,K)$ has real irreducible eigenspaces and every spherical representation is of real type, then condition is satisfied by assumption and condition implies condition . Suppose we are given $M=G/K$ and a decomposition $${\mathfrak}g = {\mathfrak}k \oplus {\mathfrak}m$$ as above, orthogonal with respect to a fixed $\mathrm{Ad}(G)$-invariant inner product $\langle \cdot, \cdot \rangle$ on ${\mathfrak}g$. Any $\operatorname{Ad}(K)$-invariant inner product $g \in \operatorname{Sym}_K^+({\mathfrak}m)$ on ${\mathfrak}m$ can be written as $g = g_B$, where $g_B:= \langle B^{-1} \cdot, \cdot \rangle$ for a $K$-invariant symmetric positive-definite endomorphism $B \in \operatorname{Aut}_K({\mathfrak}m)$. If $\{ Y_i\}$ is an orthonormal basis of ${\mathfrak}m$ with respect to $\langle \cdot, \cdot \rangle$, then we can find $A\in \operatorname{GL}({\mathfrak}m)$ such that $\{ A Y_i\}$ is an orthonormal basis with respect to $g_B$. This implies that $B = AA^T$, where the transpose is taken with respect to $\langle\cdot,\cdot\rangle$. We write $AY_i = \sum_j A_{ji}Y_j$. For $V\in{\widehat}G_K$, we consider the associated Casimir operator $\Delta^V_B:= \Delta_{g_B}^{V^K}:V^K\to V^K$ given by $$\Delta^V_B = \sum_{i=1}^n (AY_i)^2= \sum_{i,j=1}^n B_{ij}Y_iY_j.$$ To each $V\in{\widehat}G_K$ we associate the map $$p_V:\operatorname{Sym}_K^+({\mathfrak}m)\to {\mathbb C}[t],\quad p_V(B)(t) = \det (t \cdot \operatorname{id}_{V^K}-\Delta^V_B)\in{\mathbb C}[t],$$ i.e. $p_V(B)$ is the characteristic polynomial of $\Delta^V_B$. Using the resultant $$\operatorname{res}: {\mathbb C}[t]\times {\mathbb C}[t]\to {\mathbb C}$$ which is a polynomial in the coefficients of $p$ and $q$ with the property that $$\operatorname{res}(p,q) =0 \qquad \text{$\iff$ $p,q\in{\mathbb C}[t]$ have a common root},$$ we can rephrase the conditions in Proposition \[IrrCond\] as follows. Let $V,W\in {\widehat}G_K$. Then $\Delta_B^V$ and $\Delta^W_B$ have no common eigenvalues if and only if $\operatorname{res}(p_V(B),p_W(B)) \neq 0$. We denote by $p_V',p_V''$ the first and second derivative (with respect to $t$) of $p_V$. Then the polynomial $p_V(B)\in {\mathbb C}[t]$ has only simple roots if, and only if, $\operatorname{res}(p_V(B),p_V'(B)) \neq 0$. Thus, there exists $B \in \operatorname{Sym}_K^+({\mathfrak}m)$ such that $\Delta^V_B$ has only simple eigenvalues (resp. only eigenvalues of multiplicity at most two) iff the polynomial map $\operatorname{res}\circ(p_V,p_V'):\operatorname{Sym}_K^+({\mathfrak}m) \to {\mathbb C},\quad B \mapsto \operatorname{res}(p_V(B),p_V'(B))$ (resp. $\operatorname{res}\circ(p_V,p_V''):\operatorname{Sym}_K^+({\mathfrak}m)\to {\mathbb C}$) is not the zero map. Notice that the formula for $\Delta_B^V$ and hence the polynomial $p_V(B)\in {\mathbb C}[t]$ makes sense for any $B \in \operatorname{Sym}_K({\mathfrak}m)$, although the metric $g_B$ of course does not. Moreover, the polynomial in $B$ given by $\operatorname{res}(p_V(B),p_V'(B))$ vanishes on all of $\operatorname{Sym}_K({\mathfrak}m)$ if and only if it vanishes on the open subset $\operatorname{Sym}_K^+({\mathfrak}m)$. This allows us to make the following observation, which is analogous to Proposition 3.7 in [@schueth]. \[prop:resultants\] The existence of a $G$-invariant metric on $M=G/K$ such that the associated Laplacian has irreducible real eigenspaces is equivalent to the following three conditions being simultaneously satisfied: 1. \[irrcondresA\] For all $V,W\in {\widehat}G_K$ such that $V\ncong W$ and $V^*\ncong W$ the polynomial $$\operatorname{res}\circ (p_V,p_W)\colon \operatorname{Sym}_K({\mathfrak}m) \to {\mathbb C}$$ is not the zero polynomial. 2. \[irrcondresB\] For all $V\in{\widehat}G_K$ of real or complex type the polynomial $$\operatorname{res}\circ (p_V,p_V')\colon \operatorname{Sym}_K({\mathfrak}m) \to {\mathbb C}$$ is not the zero polynomial. 3. \[irrcondresC\] For all $V\in{\widehat}G_K$ of real or quaternionic type the polynomial $$\operatorname{res}\circ (p_V,p_V'') \colon \operatorname{Sym}_K({\mathfrak}m)\to{\mathbb C}$$ is not the zero polynomial. In particular, the space of $G$-simple invariant metrics on $G/K$ is a residual set in $\operatorname{Sym}^+_K({\mathfrak}m)$, provided that it is non-empty. It is obvious that the conditions are just reformulations of the corresponding conditions in Proposition \[IrrCond\]. It follows that the complement of the space of $G$-simple metrics inside $\operatorname{Sym}_K^+({\mathfrak}m)$ is given as the intersection of a countable union of vanishing loci of non-trivial polynomials on $\operatorname{Sym}_K({\mathfrak}m)$ with the open subset $\operatorname{Sym}_K^+({\mathfrak}m)$. Hence the set of $G$-simple metrics on $G/K$ is a residual set, provided it is not empty. If the isotropy representation ${\mathfrak}m$ decomposes into a direct sum of $s>0$ inequivalent irreducible representations ${\mathfrak}m_i$, $i=1,\dots,s$, $${\mathfrak}m = {\mathfrak}m_1\oplus\dots\oplus {\mathfrak}m_s,$$ then one can obtain a simple description of $\operatorname{Sym}_K^+({\mathfrak}m)$. For any $K$-invariant inner product the decomposition must be orthogonal and up to scale each ${\mathfrak}m_i$ carries a unique $K$-invariant inner product by irreducibility. In other words, any metric is of the form $g = \sum_{i=1}^s \beta_i^{-1}\langle\cdot,\cdot\rangle|_{{\mathfrak}m_i}$ for suitable coefficients $\beta_i>0$ and we call such metrics of diagonal type. If we write such a metric in the form $g_B$ as above, then the matrix $B$ must be block diagonal with respect to the above decomposition, where the blocks are just given by $\beta_i\mathrm{id}_{{\mathfrak}m_i}$. Hence in this case we may identify $\operatorname{Sym}_K^+({\mathfrak}m)$ with the positive orthant ${\mathbb R}^s_+\subset {\mathbb R}^s$. If two or more of the ${\mathfrak}m_i$’s are equivalent as $K$-representations, then their sum need no longer be orthogonal with respect to a $K$-invariant inner product, which therefore need not be of diagonal type. This means that the possible matrices $B$ need no longer be block diagonal. Using that the ${\mathbb R}$-algebra $\mathrm{End}_K({\mathfrak}m_i)$ is a division algebra by Schur’s lemma and so isomorphic to ${\mathbb R},{\mathbb C}$ or $\mathbb H$, where $\mathbb H$ denotes the quaternions, one can then in principle describe the possible “off-diagonal” blocks of such $B$ explicitly. We conclude this section with a generalization of Lemma 4.9 of [@schueth]. \[lemma:covering\] Let $G/K$ be a homogeneous space satisfying all the conditions in Proposition \[IrrCond\] with respect to some $G$-invariant metric $g$. Let $\Gamma \subset G$ be a discrete central subgroup and let $\bar G = G/\Gamma$ and $\bar K = K / (K \cap \Gamma)$. Then the metric $g$ descends to a $\bar G$-invariant metric on $\bar G / \bar K$ satisfying the conditions of Proposition \[IrrCond\]. We can identify $\bar G / \bar K$ with $(G/K)/\Gamma$, where the action of $\Gamma$ is by translations. It is then clear that, since $\Gamma$ is discrete and central, every $G$-invariant metric on $G/K$ descends to a $\bar G$-invariant metric on $\bar G / \bar K$. We thus have a bijective correspondence between $G$-invariant metrics on $G/K$ and $\bar G$-invariant metrics on $\bar G/\bar K$. On the Lie algebra level we have $\bar{{\mathfrak}g} = {\mathfrak}g, \bar{{\mathfrak}k} = {\mathfrak}k , \bar{{\mathfrak}m} = {\mathfrak}m$. This gives an identification of $\operatorname{Sym}_K^+({\mathfrak}m)$ with the $\bar G$-invariant metric on $\bar G/\bar K$ We observe that the spherical representations of the pair $(\bar G, \bar K)$ are given by the spherical representations of $(G,K)$ on which $\Gamma$ acts trivially. Such representations are of real, complex or quaternionic type as $\bar G$-representation if and only if they are of respective type as $G$-representations. So if $(G,K)$ satisfies the conditions , , and of Proposition \[prop:resultants\], then they are also satisfied for $(\bar G, \bar K)$, since for the latter space we only have to check the conditions on the subset of representations on which $\Gamma$ acts trivially. Compact symmetric spaces {#sec:symm} ======================== In this section we apply the above results to compact irreducible symmetric spaces $G/K$. It is known that such symmetric pairs $(G,K)$ are spherical, see e.g. [@kraemer; @takeuchi]. Our aim in this section is to show that the only irreducible symmetric spaces whose associated Laplacian has irreducible real or complex eigenspaces are the CROSS’s. Spherical representations of a compact symmetric pair {#sec:restricted} ----------------------------------------------------- At the Lie algebra level, let $({\mathfrak}g, {\mathfrak}k)$ be a compact symmetric pair with Cartan splitting ${\mathfrak}g = {\mathfrak}k \oplus {\mathfrak}p$. Recall that its rank is defined to be the dimension $\ell$ of a maximal Abelian subalgebra of ${\mathfrak}p$. Denote by $r$ the rank of ${\mathfrak}g$. Let ${\mathfrak}h$ be a Cartan subalgebra of the complexification ${\mathfrak}g^{\mathbb C}$. It can be split as ${\mathfrak}h = {\mathfrak}a \oplus {\mathfrak}a_{{\mathfrak}k}$ and note that ${\mathfrak}a$ and ${\mathfrak}a_{{\mathfrak}k}$ are orthogonal with respect to the Killing form. Let $\alpha$ be a root of $({\mathfrak}g^{\mathbb C}, {\mathfrak}h)$. We denote by $\tilde \alpha$ its restriction to ${\mathfrak}a$. If $\beta \in {\mathfrak}a^*$ we consider the space $$\label{resrootspace} \tilde{{\mathfrak}g}_{\beta} = \{ X \in {\mathfrak}g^{\mathbb C}: [A, X] = \beta(A) X \text{ for all $A \in {\mathfrak}a$} \}.$$ A *restricted root* is a functional $\beta \in {\mathfrak}a^*$ such that $\tilde{{\mathfrak}g}_{\beta} \neq 0$ and the *restricted root system* of $({\mathfrak}g, {\mathfrak}k)$ is defined to be $$\Sigma({\mathfrak}g, {\mathfrak}k) := \{ \beta \in {\mathfrak}a^*: \beta \neq 0, \tilde{{\mathfrak}g}_{\beta} \neq 0 \}.$$ Given $\alpha\in {\mathfrak}h^*$, we denote by $\tilde \alpha\in {\mathfrak}a^*$ its restriction to ${\mathfrak}a$. From the root system $\Sigma({\mathfrak}g)$ of $({\mathfrak}g^{\mathbb C}, {\mathfrak}h)$, the restricted root system can be recovered as $$\Sigma({\mathfrak}g, {\mathfrak}k) = \{ \tilde \alpha: \alpha \in \Sigma({\mathfrak}g), \alpha|_{{\mathfrak}a} \neq 0 \}.$$ The set of restricted simple roots is $$\Pi ({\mathfrak}g, {\mathfrak}k) = \{ \tilde \alpha: \alpha \in \Pi({\mathfrak}g), \alpha|_{{\mathfrak}a} \neq 0 \},$$ where $\Pi({\mathfrak}g)$ is a system of simple roots for ${\mathfrak}g$. The set of restricted roots $\Sigma({\mathfrak}g, {\mathfrak}k)$ forms a *restricted root system* in ${\mathfrak}a^*$, namely they span ${\mathfrak}a^*$ and they are closed under reflections. Unlike root systems in the classical sense, if $\alpha$ is in a restricted root system $\Sigma$, then also its multiples $\pm 2 \alpha$ and $\pm \frac 1 2 \alpha$ may occur in $\Sigma$. Moreover, the dimension of the restricted root space can be greater than one and this dimension is called the *multiplicity* of the restricted root $\beta$ and denoted by $m_\beta$. Equivalently $m_\beta$ is given by the number of roots $\alpha\in \Sigma({\mathfrak}g)$ which satisfy $\tilde\alpha = \beta$. Let $\{ \alpha_1, \ldots, \alpha_r \}$ be simple roots of ${\mathfrak}g$ and let $\{\gamma_1, \ldots, \gamma_\ell \}$ be simple restricted roots of $({\mathfrak}g, {\mathfrak}k)$. As in Takeuchi [@takeuchi], let $$\beta_i = \begin{cases} 2 \gamma_i & \text{ if $2\gamma_i$ is a restricted root}\\ \gamma_i & \text{ otherwise}. \end{cases}$$ A *restricted weight* is a functional $\lambda \in {\mathfrak}a^*$ such that $\frac{2(\lambda, \alpha)}{(\alpha, \alpha)} \in {\mathbb Z}$ for all restricted roots $\alpha$. The *fundamental restricted weights* $M_i$ satisfy $$\label{fundwg} \frac{(M_i, \beta_j)}{(\beta_j, \beta_j)} = \delta_{ij}.$$ For the theory of restricted root systems we refer to Helgason [@helgason]. The spherical weights, i.e. the highest weights of spherical representations, of $(G,K)$ can be characterized in the following way. Let $(G,K)$ be a compact symmetric pair and let $\lambda$ be the highest weight of a representation $V$ of $G$. Then $V$ if spherical with respect to $K$ if, and only if, $$\label{eq:CartanHelgason} \lambda|_{{\mathfrak}a_{{\mathfrak}k}} = 0 \text{ and } \frac{2 (\lambda, \alpha)}{(\alpha, \alpha)} \in {\mathbb Z}_+ \text{ for all positive $\alpha \in \Sigma(G,K)$}.$$ For this result, we refer to [@helgason1993geometric], see also [@takeuchi]. This means that spherical representations of $(G,K)$ are parameterized by dominant restricted weights, which can be written uniquely as $\lambda = \sum_k a_k M_k$ with the $M_k$ defined in and coefficients $a_k\in{\mathbb Z}_+$. The data given by the fundamental root system of $({\mathfrak}g, {\mathfrak}k)$ can be represented graphically by means of the *Satake diagram*, see e.g. [@SatakeDiag]. One starts with the Dynkin diagram of ${\mathfrak}g$ and paints the node corresponding to the simple root $\alpha$ in black if it vanishes on ${{\mathfrak}a}$ and in white if it does not. Moreover, two simple roots $\alpha$ and $\beta$ are joined by a double tipped arrow if they are related by the *Satake involution*, see [@takeuchi]. If $\pi_i$ are the fundamental weights of ${\mathfrak}g$, then the fundamental restricted weights are given by $$M_i = \begin{cases} \pi_i \text{ if $\alpha_i$ has no arrows and is connected to a black node}\\ 2 \pi_i \text{ if $\alpha_i$ has no arrows and is not connected to a black node}\\ \pi_i + \pi_j \text{ if $\alpha_i$ and $\alpha_j$ are joined by an arrow}. \end{cases}$$ We isolate the following Lemma for future reference. It can be proven by examining the list of restricted root systems and Satake diagrams for simple Lie algebras from [@SatakeDiag; @Sugiura]. \[lemma:selfdual\] Let $(G,K)$ be symmetric pair and let $\lambda = \sum_k m_k M_k$ be the highest weight of a spherical representation, where the $M_k$ are fundamental restricted weights. Then the dual representation is given by the highest weight $\lambda^* = \sum_k m_{\sigma k} M_k$, where $\sigma$ is the permutation corresponding to the canonical involution of the restricted root system. The weight $\delta$ {#sec:delta} ------------------- Let $\delta \in {\mathfrak}h^*$ be the strictly dominant weight given by the half sum of all positive roots of ${\mathfrak}g$. Its components along ${\mathfrak}a_{{\mathfrak}k}^*$ and ${\mathfrak}a^*$ are given by Lemma 5.1 of [@GinGoo]. The component along ${\mathfrak}a^*$ of $\delta$ is given by $$\label{eq:bardelta} \bar \delta = \frac 1 2 \sum_{\xi \in \Sigma^+({\mathfrak}g, {\mathfrak}h)} m_\xi \xi,$$ where $m_\xi$ is the multiplicity of the positive restricted root $\xi$. We want to compute the coefficients of $\bar \delta$ with respect to the basis of restricted fundamental weights $\{M_i\}$. We have the following Lemma. Let $\bar \delta = \sum_{i=1}^\ell k_i M_i$ be the expression of $\bar \delta$ with respect to the basis of fundamental restricted weights. Then we have $$k_i = \begin{cases} \frac 1 2 m_{\gamma_i} & \text{ if $m_{2 \gamma_i} = 0$} \\ \frac 1 4 ( m_{\gamma_i} + 2 m_{2 \gamma_i}) & \text{ if $m_{2 \gamma_i} \neq 0$}. \end{cases}$$ The proof goes along a similar line as the classical case, but taking into account that we have a restricted root system. Consider the transformation $\sigma_i$ in the restricted Weyl group associated to the simple restricted root $\gamma_i$. It is proved in Helgason [@helgason Lemma 2.21, Chap. VII] that $\sigma_i$ permutes all the positive roots not proportional to $\gamma_i$ and that, in our base, such roots are $\gamma_i$ and, in case, $2 \gamma_i$. Taking $\bar \delta$ as in , arguing as in the classical case, we have that $$\sigma_i \bar \delta = \bar \delta - (m_{\gamma_i}+ 2 m_{2 \gamma_i}) \gamma_i.$$ We then have $$\begin{aligned} ( \bar \delta - (m_{\gamma_i}+ 2 m_{2 \gamma_i}) \gamma_i, \gamma_i) &= (\sigma_i^2 \bar \delta, \sigma_i \gamma_i) \\ &= (\bar \delta, -\gamma_i).\end{aligned}$$ So we have that $$- \frac{(\bar \delta, \gamma_i)}{(\gamma_i, \gamma_i)} = \frac{(\bar \delta, \gamma_i)}{(\gamma_i, \gamma_i)} - (m_{\gamma_i}+ 2 m_{2 \gamma_i}),$$ that means $$\label{eq:prodgamma} \frac{(\bar \delta, \gamma_i)}{(\gamma_i, \gamma_i)} = \frac 1 2 (m_{\gamma_i}+ 2 m_{2 \gamma_i}).$$ Recalling how $\beta_i$ is defined, together with the relation , the expression gives us the claim. We summarize in Table \[tab:delta\] the coefficients of $2 \bar \delta$ for every Cartan type. [cCC]{} Cartan type & 2 |&\ A I & (1,…, 1) &\ A II & (4, …, 4) &\ A III (1) & (2, …, 2, r -2 + 2) & r 4\ A III (2) & (2, …, 2,1) &\ B I & (1, …, 1, 2(r-)+1) & r 2\ C I & (1, …, 1) &\ C II (1) & (4, …, 4, 2(r-2) +3) & r 3\ C II (2) & (4, …, 4,3) &\ D I (1) & (1, …, 1,2) &\ D I (2) & (1, …, 1, 2(r-)) & r 4\ D I (3) & (1,…, 1) &\ D III (1) & (4, …, 4,1) &\ D III (2) & (4, …, 4,3) &\ E I & (1,1,1,1,1,1) &\ E II & (2,1,2,1) &\ E III & (5,6) &\ E IV & (8,8) &\ E V & (1,1,1,1,1,1,1) &\ E VI & (1,1,4,4) &\ E VII & (8,8,1) &\ E VIII & (1,1,1,1,1,1,1,1) &\ E IX & (1,1,8,8) &\ F I & (1,1,1,1) &\ G & (1,1) &\ Rank-two symmetric spaces ------------------------- In Table \[tab:rank2\] we isolate, for future reference, the rank two symmetric spaces whose weight $2 \bar \delta$ has different coefficients. [cLLC]{} Cartan type & 2 |& &\ & (2, r - 2) &r 4 &C\_2\ [A III (2)]{} & (2,1) & &C\_2\ [B I]{} & (1, 2r-3) &r2 &B\_2\ [C II (1)]{} & (4, 2r-5) &r5 &B\_2\ [C II (2)]{} & (4,3) & &B\_2\ [D I (1)]{} & (1,2) & &B\_2\ [D I (2)]{} & (1, 2(r-2)) &r 4 &B\_2\ [D III (1)]{} & (4,1) & &B\_2\ [D III (2)]{} & (4,3) & &C\_2\ & (5,6) & &B\_2\ Using relation , it follows that the Gram matrix of the Killing form on ${\mathfrak}a^*$ with respect to the basis $\{M_1, M_2\}$ is given, up to a multiple, by $$\label{eq:gram} g = 2 \bar m^{-T} \begin{pmatrix}(\beta_1, \beta_1)&0\\0& (\beta_2, \beta_2) \end{pmatrix},$$ where $\bar m$ denotes the Cartan matrix of the restricted root system. For the cases described by Table \[tab:rank2\], we have $$\label{eq:gram2} g = \begin{cases} \begin{pmatrix} 2&2\\2&4 \end{pmatrix} & \text{ if the type is $B_2$}\\ & \\ \begin{pmatrix} 4&2\\2&2 \end{pmatrix} & \text{ if the type is $C_2$.}\\ \end{cases}$$ The main theorem ---------------- We are now in a position to state and prove one of our main results. \[thmrefl\] Let $(G, K)$ be a compact irreducible symmetric pair with simple $G$-spectrum (in either the complex or real sense). Then $(G,K)$ has rank one. We split the proof in the two following Proposition and an analysis of the rank-two case. Let $(G, K)$ be a compact irreducible symmetric pair of rank $\ell \geq 3$. Then $G/K$ is not $G$-simple. Our aim is to show that a pair $(G,K)$ of rank greater than one violates condition in Proposition \[IrrCond\]. Let ${\mathfrak}g = {\mathfrak}k \oplus {\mathfrak}m$ be a Cartan splitting of ${\mathfrak}g$, with maximal Abelian subalgebra ${\mathfrak}a \subseteq {\mathfrak}m$ and denote by $\ell\geq 2$ the rank of $(G,K)$. Let $\{\beta_1, \ldots, \beta_\ell \}$ be simple restricted roots, with associated fundamental weights $\{ M_1, \ldots, M_\ell \}$. The irreducible spherical representations are parameterized by the cone of dominant restricted weights. Each such weight is of the form $$\rho = \sum_{i=1}^\ell n_iM_i,$$ where $n_i$ are non-negative integers. By Freudenthal’s formula the representation $V_\rho$ is contained in the eigenspace of $\Delta$ with eigenvalue $$\lambda_\rho = (\rho+2\delta,\rho).$$ Thus, the symmetric space $G/K$ will have $G$-simple spectrum if the function $\rho\mapsto (\rho+2\delta,\rho)$ is injective on the cone of dominant restricted weights. Note that if $R:{\mathfrak}a^*\to{\mathfrak}a^*$ is an isometry fixing $\delta$, then $(R\rho+2\delta,R\rho) = (\rho+2\delta,\rho)$. Thus, in order to show that $G/K$ does not have $G$-simple spectrum we will construct an isometry $R$ and a dominant restricted weight $v$ such that $R\delta = \delta$ and $Rv$ is again a dominant restricted weight. Our strategy is to construct, for rank $\ell \geq 3$, a suitable restricted weight $\alpha$ which is perpendicular to $\delta$ and to take the reflection $R_\alpha: v\mapsto v-\frac{2(\alpha,v)}{(\alpha,\alpha)}\alpha$. This clearly fixes $\delta$. Let $(c_{ij})$ be the Gram matrix of the basis $\{\beta_1, \ldots, \beta_\ell \}$. Note that since for any $i,j$ the number $\frac{2(\beta_i,\beta_j)}{(\beta_i,\beta_i)}$ is an integer and also the square norms $(\beta_i,\beta_i)$ are rational, the numbers $c_{ij}$ must also be *rational*. The components of the projection on ${\mathfrak}a^*$ of the dominant weight $2\delta$, that we call still $2\delta$ in this proof, of ${\mathfrak}g$ were computed in Table \[tab:delta\]. From such table we can identify, for all irreducible symmetric spaces, a pair of fundamental restricted roots that we relabel $\beta_1$ and $\beta_2$, such that $2 \delta = M_1 + M_2 + \sum_i k_i M_i$. From that computation, together with [@takeuchi §. 5, Lemma 4], we also see that $\beta_1$ and $\beta_2$ have the same square norm, say, $2$. Consider $\alpha = \beta_1 - \beta_2\in{\mathfrak}a^*$. By the above remark, we have $(\alpha, \delta)=0$. Of course, this relation only depends on the direction defined by $\alpha$, i.e. we may multiply it by a non-zero real number. In the basis $\{M_i\}$ we can write $$\alpha = \sum_{i=1}^\ell \frac{(\alpha,\beta_i)}{(\beta_i,\beta_i)}M_i.$$ Since $$(\alpha,\beta_i) = c_{1i}-c_{2i},$$ we may thus rescale $\alpha$ by some positive integer in such a way that $(\alpha,\beta_i)$ and $\frac{(\alpha,\beta_i)}{(\beta_i,\beta_i)}$ are also integers for any $i=1,\dots,\ell$. The square length of $\alpha$ is $$(\alpha, \alpha) = 4 - 2 c_{12},$$ which is a positive integer. We notice that, by construction, $R_\alpha$ fixes not only $\delta$ but also the fundamental weights $M_3, \ldots, M_\ell$. Our goal is to find some dominant $v$ such that $R_\alpha(v)$ is dominant. We make an ansatz for $v$ to be the form $v= M_1 + \sum_{k=3}^\ell m_kM_k$ with non-negative integers $m_k$ to be chosen later. We have $$R_\alpha v = R_\alpha M_1 + \sum_{k=3}^\ell m_kM_k = \sum_{k=1}^\ell \frac{(R_\alpha v,\beta_k)}{(\beta_k,\beta_k)}M_k$$ and for this to be a dominant restricted weight, we need the coefficients to be $$\frac{(R_\alpha v,\beta_k)}{(\beta_k,\beta_k)} \geq 0, \text{ for all $k$.}$$ Note that the component of $R_\alpha v$ along $M_1$ and $M_2$ is the same as the component of $R_\alpha M_1 $ along these vectors. The component of $R_\alpha M_1$ along $M_1$ is $$\frac{(R_\alpha M_1,\beta_1)}{(\beta_1,\beta_1)} = \frac{ (M_1, \beta_1)}{c_{11}} - 2 \frac{ (M_1, \alpha)}{c_{11}} \frac { (\alpha, \beta_1)}{(\alpha, \alpha)}.$$ This quantity is $$\begin{aligned} \frac{ (M_1, \beta_1)}{c_{11}} - 2 \frac{ (M_1, \alpha)}{c_{11}} \frac { (\alpha, \beta_1)}{(\alpha, \alpha)} &= 1-2 \frac{ (M_1, \alpha)}{c_{11}} \frac { (\alpha, \beta_1)}{(\alpha, \alpha)} \\ &= 1-2 \frac{2-c_{12}}{4- 2c_{12}}\\ &= 0,\end{aligned}$$ by our choices of $\beta_1, \beta_2$. The component of $R_\alpha M_1$ along $M_2$ is given by $$\frac{(R_\alpha M_1,\beta_2)}{(\beta_2,\beta_2)} = -2 \frac{(M_1, \alpha)}{c_{22}} \frac{(\alpha, \beta_2)}{(\alpha, \alpha)} = -2 \frac{c_{11}}{c_{22}} \frac{c_{12}-c_{22}}{(4-2c_{12})} = 1.$$ For $k \geq 3$, we have that the component of $R_\alpha v$ along $M_k$ is $$\frac{(R_\alpha v,\beta_k)}{(\beta_k,\beta_k)} = -2 \frac{(M_1, \alpha)}{c_{kk}} \frac{(\alpha, \beta_k)}{(\alpha, \alpha)} + m_k = -2\frac{2 (c_{1k} - c_{2k})}{c_{kk}(\alpha,\alpha)} +m_k.$$ We thus choose the integers $m_k$ such that this quantity becomes non-negative. We see that a suitable positive integral multiple of $v := M_1 + \sum_{k=3}^\ell m_k \cdot M_k$ is mapped to another dominant weight by the reflection $R_\alpha v$. This proves that, as long as $\ell \geq 3$, there are always two dominant restricted weights $v \neq w$ such that $\lambda:= (v + \delta, v + \delta) = (w + \delta, w + \delta)$, proving that the space of spherical functions $V_v$ and $V_w$ are two different $G$-invariant subspaces of $E_\lambda$. In the basis $\{M_i\}$, we have that $v = (1,0,m_3, \ldots, m_\ell)$ and $R_\alpha v = (0,1,*)$ so, by means of Lemma \[lemma:selfdual\], we can choose the $m_k$ also in order to have the representation $V$ induced by $v$ not isomorphic to the dual of the one $W$ induced by $R_\alpha v$. We now move on to the rank-two case. In this case, two dominant restricted weights giving repetitions of eigenvalues need not be related by an isometry of ${\mathfrak}a^*$. We exhibit explicitly some repetitions. We refer to the cases in Table \[tab:rank2\]. Let $\rho = xM_1 + yM_2$. Note that root systems of type $BC_2$ have no outer automorphisms, hence all the spherical representations exhibited are self-dual. AIII (1) : For all $r \geq 4$, we have $$(\rho+2\delta,\rho) = 2r x + 4 x^{2} + 2 r y + 4 x y + 2 y^{2} + 4 x,$$ and repetitions are given by $(\ell+3,0)$ and $(\ell-1,6)$ if $r = 2 \ell$ or by $(\ell, 0)$ and $(\ell-2,3)$ if $r = 2 \ell-1$. A III (2) : We have $$(\rho+2\delta,\rho) = 4 \, x^{2} + 4 \, x y + 2 \, y^{2} + 10 \, x + 6 \, y.$$ Repetitions are given by the weights $(0,3)$ and $(2,0)$. B I : We have $$(\rho+2\delta,\rho) = 4 \, r x + 2 \, x^{2} + 8 \, r y + 4 \, x y + 4 \, y^{2} - 4 \, x - 10 \, y.$$ For all $r \geq 3$, repetitions are given by $(r+3,0)$ and $(r-3,4)$. For $r=2$, the weight $2 \bar \delta$ is $M_1+M_2$ so we can apply the argument in the proof of Theorem \[thmrefl\]. C II (1) : We have $$(\rho+2\delta,\rho) = 4 \, r x + 2 \, x^{2} + 8 \, r y + 4 \, x y + 4 \, y^{2} - 2 \, x - 12 \, y.$$ If $r > 5$, then the weights $(r+3,0)$ and $(r-6,6)$ give repetitions. For $r=5$ we have the weights $(3,0)$ and $(0,2)$. C II (2) : We have $$(\rho+2\delta,\rho) = 2 \, x^{2} + 4 \, x y + 4 \, y^{2} + 14 \, x + 20 \, y.$$The weights $(0,3)$ and $(3,1)$ give a repetition. D I (1) : We have $$(\rho+2\delta,\rho) = 2 \, x^{2} + 4 \, x y + 4 \, y^{2} + 6 \, x + 10 \, y.$$ The weights $(0,2)$ and $(3,0)$ give a repetition. D I (2) : We have $$(\rho+2\delta,\rho) = 4 \, r x + 2 \, x^{2} + 8 \, r y + 4 \, x y + 4 \, y^{2} - 6 \, x - 14 \, y.$$For all $r \geq 4$ we have the weights $(r,0)$ and $(r-3,2)$. D III (1) : We have $$(\rho+2\delta,\rho) = 2 \, x^{2} + 4 \, x y + 4 \, y^{2} + 10 \, x + 12 \, y.$$ The weights $(1,5)$ and $(4,3)$ give a repetition. D III (2) : We have $$(\rho+2\delta,\rho) = 2 \, {\left(2 \, x + y + 11\right)} x + 2 \, {\left(x + y + 7\right)} y.$$ The weights $(0,3)$ and $(2,0)$ give a repetition. E III : We have $$(\rho+2\delta,\rho) = 2 \, x^{2} + 4 \, x y + 4 \, y^{2} + 22 \, x + 34 \, y.$$ The weights $(1,3)$ and $(4,1)$ give a repetition. Irreducible compact symmetric spaces of type II ----------------------------------------------- The argument above works for the special case of pairs $({\mathfrak}g + {\mathfrak}g, {\mathfrak}g)$ where ${\mathfrak}g$ is diagonal. It is known that the symmetric pair is isometric to $G$ with a bi-invariant metric, so our Theorem \[thmrefl\] tells us that the only Lie groups $G$ with a $(G\times G)$-simple metric are of rank one. It follows from Proposition \[IrrCond\] that a bi-invariant metric on $G$ is never $G$-simple, viewed as a $G$-invariant metric on $G = G/\{1\}$. Riemannian products of normal homogeneous spaces {#ProdCROSSs} ------------------------------------------------ Let us now consider Riemannian products of normal homogeneous spaces of the form $M_i = G_i/K_i$, where $G_i$ is a compact semisimple Lie group and $M_i$ is endowed with the Riemannian metric $g_i$ induced from the Cartan-Killing form of ${\mathfrak}g_i$. The following fact about eigenspaces of a product metric is well known, see for example [@spectre]. Let $(M, g)$ and $(M', g')$ be Riemannian manifolds. Then we have $$E_\lambda(M \times M', g \times g') = \bigoplus_{\substack{\mu \in \operatorname{Spec}(M,g) \\ \nu \in \operatorname{Spec}(M',g') \\ \mu + \nu = \lambda}} E_\mu(M, g) \otimes E_\nu(M', g').$$ Let $G,G'$ be compact Lie groups. We observe that if $V,W$ are irreducible representations of $G,G'$ respectively, then the tensor product $V\otimes W$ is an irreducible representation of $G\times G'$. Let $(M,g)$ (resp. $(M',g')$) be Riemannian manifolds with an isometric action of $G$ (resp. $G'$). Thus, for any $\mu\in\operatorname{Spec}(M,g), \nu\in \operatorname{Spec}(M',g')$ the space $E_\mu(M,g)\otimes E_\nu(M',g')\subset E_{\mu+\nu}(M\times M',g\times g')$ is a $(G\times G')$-subrepresentation. Given an array of positive real numbers $\beta = (\beta_1,\dots,\beta_n)\in {\mathbb R}^n_+$, we consider the metric on $M$ given by $g_\beta = \sum_{i=1}^n \beta_i^{-1} g_i$. We shall refer to a metric of this kind as a *weighted Riemannian product*. The aim of this subsection is to prove the following. \[thm:generic1\] Let $(G_i/K_i,g_i)$ be normal $G_i$-simple spherical homogeneous spaces, with $G_i$ semisimple, $i=1,\dots,n$. Then a weighted Riemannian product $G/K = \prod_{i=1}^n G_i/K_i$ with metric $g_\beta$ as above is generically $G$-simple. In particular, a generic $G$-invariant Riemannian metric on a reducible symmetric space $G/K$ with factors of rank one, is $G$-simple. Let $a = (a_1, \ldots, a_n)$ be an array of dominant spherical weights $a_i$ for the pair $(G_i, K_i)$. The weight $a_i$ defines an irreducible representation $V_i=V_{a_i}$ of $G_i$ with $\dim V_{a_i}^{K_i} = 1$ and $V_a = V_{a_1}\otimes\dots\otimes V_{a_n}$ is an irreducible spherical representation of the pair $(G,K)$. Being each factor endowed with the normal metric, each Casimir operator $\Delta_{g_i}^{V_i}$ acts, by the Freudenthal formula, as $(a_i + 2 \delta_i, a_i)_i$, where $( \cdot, \cdot)_i$ is the Cartan-Killing form of ${\mathfrak}g_i$ and $\delta_i$ is the half-sum of the positive roots of ${\mathfrak}g_i$. Denote by $\lambda_a \in {\mathbb Q}^n$ the array whose $i$-th entry is $(a_i + 2 \delta_i, a_i)_i$. The Casimir of $g_\beta$ acts then on the representation $V_a^K$ by $\lambda_a \cdot \beta$ times the identity, where $\lambda_a\cdot \beta = \sum_{i=1}^n \beta_i(a_i + 2 \delta_i, a_i)_i$. Then $(M, g_\beta)$ if $G$-simple if, and only if $$B:= {\mathbb R}^n_+ \setminus \bigcup_{a \neq a'} (\lambda_a - \lambda_{a'})^\perp \neq \emptyset.$$ The space $B$ is the positive orthant in ${\mathbb R}^n$ with a countable set of hyperplanes removed. It is thus empty if, and only if, one of these hyperplanes is the whole space, i.e. if there exist $a \neq a'$ such that $\lambda_a = \lambda_{a'}$. This means that there exists an $i\in\{1,\dots,n\}$ such that $a_i\neq a_i'$ but $(a_i + 2 \delta_i, a_i)_i=(a'_i + 2 \delta_i, a'_i)_i$. In other words, there exists an $i\in\{1,\dots,n\}$ such that $M_i$ is not $G_i$-simple. Towards the case of non-symmetric homogeneous spaces {#sec:nonsym} ==================================================== For simple compact connected $G$, Krämer [@kraemer] provides a classification of possible subgroups $K$ along with a set of dominant weights whose integral non-negative combinations give all spherical representations. Spherical pairs are a generalization of symmetric pairs, but the theory of restricted root systems does not in general work. However, they are weakly symmetric, as proven in [@weaklysymm]. The generators of the dominant spherical weights need not form a free semigroup and the machinery provided by the theory of restricted weights and roots that we used to prove Theorem \[thmrefl\], in particular the facts in Subsection \[sec:delta\], do not hold for non-symmetric pairs (cf. [@GinGoo]). Moreover, the spherical pairs in Krämer’s list are not isotropy irreducible, so the study of the quadratic form given by Freudenthal’s formula only covers the case of the normal metric. The only two examples of spherical non-symmetric isotropy irreducible homogeneous spaces have been treated in [@spectre] and are described below. \[ex:G2SU3\] The group $\operatorname{SU}(3)$ is embedded in the standard way in $\operatorname{G}_2$ and the isotropy representation on ${\mathbb R}^6$ is the classical one of $\operatorname{SU}(3)$ of ${\mathbb C}^3$. It is transitive on the unit sphere, so we apply the result in [@spectre] to conclude that the spectrum of the homogeneous nearly Kähler $S^6 = \operatorname{G}_2 / \operatorname{SU}(3)$ is $\operatorname{G}_2$-simple. This pair is spherical and not symmetric. \[ex:SO7G2\] The standard inclusion $\operatorname{G}_2 \subset \operatorname{SO}(7)$ gives rise to a spherical pair. The homogeneous space $M = \operatorname{SO}(7) / \operatorname{G}_2$ has a $\operatorname{SO}(7)$-simple spectrum with respect to the normal metric, since $\operatorname{G}_2$ acts transitively on the unit $7$-sphere. In this section we discuss two examples. The first is a class of circle bundles that submerge Hermitian symmetric spaces, and the second is a non-spherical example. So, as a by-product, we obtain the following, in contrast with the case of $G$-irreducibility of *complex* Laplacian eigenspaces. Let $G/K$ be a Riemannian homogeneous space with $G$-simple spectrum (over the reals). Then $K$ need not be spherical in $G$. Circle bundles over Hermitian symmetric spaces {#sec:bundles} ---------------------------------------------- In Krämer’s list, there is a class of spherical homogeneous spaces that are circle bundles over Hermitian symmetric spaces, namely 1. \[bdlAIII\] $\operatorname{SU}(n+m)/(\operatorname{SU}(m) \times \operatorname{SU}(n)) \to \operatorname{SU}(n+m)/S(\operatorname{U}(n) \times \operatorname{U}(m))$ where the base is of Cartan type AIII; 2. \[bdlDIII\] $\operatorname{SO}(2n)/\operatorname{SU}(n) \to \operatorname{SO}(2n)/\operatorname{U}(n)$, for odd $n \geq 3$, where the base is of Cartan type DIII; 3. \[bdlEIII\] $\operatorname{E}_6/\operatorname{D}_5 \to \operatorname{E}_6/(\operatorname{U}(1) \times \operatorname{D}_5)$ on a base of Cartan type EIII. Let us unify the notation by writing $M=G/K$ for the total space of the circle bundle, $N=G/H$ for the symmetric base and $H/K$ for the fiber. From [@kraemer_factors; @kraemer], we can see that, in these cases, the isotropy representation of $K$ splits as $$\label{eq:splitting} {\mathfrak}g = {\mathfrak}k \oplus {\mathbb R}H_0 \oplus {\mathfrak}m,$$ where $H_0$ generates the vertical subbundle of $G/K \to G/H$ and, if ${\mathfrak}a_{{\mathfrak}k}$ is a maximal torus of ${\mathfrak}k$, then ${\mathfrak}a_{{\mathfrak}k} \oplus {\mathbb R}H_0$ is a maximal torus of ${\mathfrak}g$. We may scale any invariant metric on $M$ to make these bundles Riemannian submersions onto the symmetric space $N$. Moreover, the fiber $K/H$ is a totally geodesic circle, as one can compute from the Levi-Civita connection of the invariant metric, see e.g. [@besse]. Using the well-known relations between the Laplacians of $M$ and $N$, see [@onishchik2012lie], we can see that every eigenfunction of $N$ is also an eigenfunction on $M$ with the same eigenvalue. Moreover, both eigenspaces are real $G$-modules. We then have the following. If $N$ is not $G$-simple, then neither is $M$. Indeed, the spectrum of $N$ is contained in the spectrum of $M$, and for all $\mu \in \operatorname{Spec}(N)$, we have that the eigenspace $E_\mu(N)$ is a $G$-invariant subspace of $E_\mu(M)$. Since our base spaces $N$ are irreducible symmetric spaces, it is clear from Theorem \[thmrefl\] when they are simple, namely only when $m=1$ for case , $n=3$ for and never for . In case we have the Hopf fibration of $M = S^{2n+1}= \operatorname{SU}(n+1)/\operatorname{SU}(n)$ over ${\mathbb{CP}}^n$ and in case we have $N = \operatorname{SO}(6)/\operatorname{U}(3)$. We denote by ${\widehat}G_K$ the set of maximal weights corresponding to irreducible representations of $G$ spherical with respect to $K$. Generators $M_i$ for such semigroups are given in Krämer’s table. Note that there are some $M_i$ that do not vanish on ${\mathfrak}a_{{\mathfrak}k}$. Since the two summands in are $\operatorname{Ad}(K)$-inequivalent, any $G$-invariant Riemannian metric $g_\gamma$ on $G/K$ is parametrized by two positive real numbers $\gamma_1, \gamma_2$ and equals $\gamma_1^{-1}$ (resp. $\gamma_2^{-1}$) times the Killing form restricted to ${\mathbb R}\cdot H_0$ (resp. ${\mathfrak}m$). So we identify $\operatorname{Sym}_K^+({\mathfrak}m) \simeq {\mathbb R}_+^2$ and $\operatorname{Sym}_K({\mathfrak}m) \simeq {\mathbb R}^2$. Let $X_1, \ldots X_k$ be a basis of ${\mathfrak}m$ orthonormal with respect to the Killing form. For $\gamma = (\gamma_1, \gamma_2)$, the Casimir operator $C_\gamma$ of $g_\gamma$ is $$\label{eq:casimirbdl} \begin{split} C_\gamma &= \gamma_1 H_0^2 + \gamma_2 \sum_k X_k^2 \\ &= (\gamma_1 - \gamma_2) H_0^2 + \gamma_2 C, \end{split}$$ where $C$ is the Casimir operator of ${\mathfrak}g$. Let $V\in {\widehat}G_K$. Since $(G,K)$ is a spherical pair, the fixed point set $V^K$ is a line, which is mapped to itself by $\Delta_\gamma^{V^K} = \Delta_{g_\gamma}^{V^K}$. Let $\lambda$ be the highest weight of $V$ and let $v\in V$ be a generator of $V_\lambda^K$. We have $$-\Delta_\gamma^{V^K}v = (\gamma_1 - \gamma_2) \alpha_\lambda v + \gamma_2 (\lambda + 2 \delta, \lambda)v,$$ since the Casimir $C$ are given by Freudenthal’s formula and $\alpha_\lambda v := (H_0)^2v$. So the system to consider to check property of Proposition \[IrrCond\] is $$\label{sys:alpha} \begin{cases} \alpha_\lambda = \alpha_{\lambda'}\\ (\lambda + 2 \delta, \lambda) = (\lambda' + 2 \delta, \lambda') \end{cases}$$ for two different dominant spherical weights $\lambda$ and $\lambda'$. We consider Case . From Krämer’s list, we have that such a representation of $\operatorname{SU}(n+1)$ is given by the highest weight $p \pi_1 + q \pi_n$ with $p,q \geq 0$. This representation is known to be the space ${\mathcal}H_{p,q}({\mathbb C}^{n+1})$ of harmonic complex polynomials of $n+1$ variables that are homogeneous of degree $p$ in the $z_i$ and of degree $q$ in the $\bar z_i$, see e.g. [@sepanski]. Using the explicit $\operatorname{SU}(n+1)$-invariant projection map $P$ from [@projharm Thm. 1] from the space of all polynomials of bidegree $(p,q)$ to $V={\mathcal}H_{p,q}({\mathbb C}^{n+1})$, one can see that the space $V^K$ is spanned by $v= P(z_{n+1}^p \bar z_{n+1}^q)$ and that $Hv = in(-p+q)v$, so $\alpha_\lambda = -n^2(-p+q)^2$. The Freudenthal formula can be computed by means of the inverse of the Cartan matrix (see e.g. [@onishchik2012lie p. 295]), so the system is equivalent to $$\begin{cases} (-p+q)^2 = (-p' + q')^2\\ n(p^2 + q^2) + 2pq + n(p+q) = n(p'^2 + q'^2) + 2p'q'+ n(p'+q'). \end{cases}$$ Substituting the first equation in the second, we obtain $$\begin{cases} (-p+q)^2 = (-p' + q')^2\\ 2(n+1)pq + n(p+q) = 2(n+1)p'q'+ n(p'+q'). \end{cases}$$ With the substitution $x = 2(n+1)p + n$ and $y=2(n+1)q+n$ and their analog with the prime, the system is equivalent to $$\begin{cases} x^2 + y^2 = x'^2 + y'^2 \\ xy = x'y'. \end{cases}$$ The problem is, in turn, equivalent to finding repetitions in the first quadrant of the quadratic form $x^2 + y^2$, so we know that the only repetition can be obtained by a swap of $x,y$. \[prop:hopf\] Let $n>1$. A generic $\operatorname{SU}(n+1)$-invariant Riemannian metric on the total space of the Hopf fibration is $\operatorname{SU}(n+1)$-simple. Since $(G,K)$ is spherical, i.e. for all $V \in {\widehat}G_K$ the fixed point space $V^K$ is one-dimensional, the polynomial $p_V(\gamma)(t) = t- \Delta_\gamma^{V^K}$ is linear in $t$. Thus, condition and of Proposition \[prop:resultants\] are automatically satisfied. Assume that no $G$-invariant metric is $G$-simple on $G/K$, so condition of Proposition \[prop:resultants\] is violated, i.e. there exist $V, W \in {\widehat}G_K$ such that $V \not\cong W$ and $V^* \not \cong W$ and the polynomial $$\operatorname{res}\circ (p_V, p_W) \colon \operatorname{Sym}_K({\mathfrak}m) \simeq {\mathbb R}^2 \to {\mathbb C}$$ is the zero polynomial. The above computation shows that if such polynomial is zero on $\operatorname{Sym}_K^+({\mathfrak}m)$, then $V \cong W$ or $V^* \cong W$, a contradiction. For $n=1$, the existence of generic $\operatorname{SU}(2)$-simple metrics on $\operatorname{SU}(2)$ is obtained by Schueth [@schueth] but not by stretching the fibers of the Berger sphere, as the conditions of Proposition \[IrrCond\] are not satisfied. In case for $n=3$, note that $\operatorname{SO}(6) = \operatorname{SU}(4)/\{ \pm I\}$, so this induces a double covering $\operatorname{SU}(4)/\operatorname{SU}(3) \to \operatorname{SO}(6)/\operatorname{SU}(3)$. We can then apply Lemma \[lemma:covering\], since the total space is treated in case . A quotient of ${\mathbb{RP}}^3$ {#sec:SU2F} ------------------------------- We now exhibit an example of a non-spherical $G$-simple homogeneous space $G/K$. The low-dimensionality of this example allows us to examine its representations and the action of the Casimir operator. Let $G = \operatorname{SU}(2)$ and let $F$ be the finite subgroup generated by $$\sigma = \begin{pmatrix} e^\frac{i \pi}{3} & 0 \\ 0 & e^{-\frac{i \pi}{3}} \end{pmatrix}; \qquad \tau = \begin{pmatrix} 0 & i \\ i & 0 \end{pmatrix}.$$ The homogeneous space $\operatorname{SU}(2)/F$ has been studied in [@art:bedulli_stab]. Let $$H = \frac{1}{\sqrt 8} \begin{pmatrix} i & 0 \\ 0 & -i \end{pmatrix}; X = \frac{1}{\sqrt 8} \begin{pmatrix} 0 & 1 \\ -1 & 0 \end{pmatrix}; Y = \frac{1}{\sqrt 8} \begin{pmatrix} 0 & i \\ i & 0 \end{pmatrix}$$ be an orthonormal basis of ${\mathfrak}{su}(2)$ with respect to the Cartan-Killing form. The isotropy representation of $F$ splits as irreducible $F$-modules as $${\mathfrak}{su}(2) = {\mathbb R}\cdot H \oplus \operatorname{span}\{ X, Y \},$$ hence all possible $\operatorname{SU}(2)$-invariant metrics are parameterized by two positive real numbers. All the complex irreducible representations of $\operatorname{SU}(2)$ are parameterized by a positive integer $k$ and are given by $V_k = S^k {\mathbb C}^2$, that can be expressed as the space of symmetric polynomials over ${\mathbb C}$ of degree $k$ in two variables $z_1, z_2$. Denote by $v_\ell$ the polynomial $z_1^\ell z_2^{k-\ell}$. It is known that $V_k$ is of real type if $k$ is even and of quaternionic type if $k$ is odd. We want to prove the existence of metrics that satisfy the conditions in Proposition \[IrrCond\]. The pair $(\operatorname{SU}(2), F)$ is not spherical and its spherical representations were computed in [@art:bedulli_stab]. They verify that all spherical representations $V_k$ need to have $k$ even, so they are all of real type and condition is always empty. We focus then on conditions and . Any $\operatorname{SU}(2)$-invariant Riemannian metric on $M$ is induced by a scalar product on ${\mathfrak}{su}(2)$ that equals $a^{-1}$ times the Killing form on ${\mathbb R}\cdot H$ and $b^{-1}$ times the Killing form on ${\mathfrak}m_2 := \operatorname{span}\{X, Y \}$, for some positive reals $a, b$. The Casimir operator is then $$\begin{aligned} C_{a,b} &= \frac{a}{8} H^2 + \frac{b}{8} (X^2 + Y^2) \\ &= \frac{b}{8} (H^2 + X^2 + Y^2) + \frac{a-b}{8} H^2 \\ &= b C + \frac{a-b}{8} H^2,\end{aligned}$$ where $C$ is the standard Casimir operator of ${\mathfrak}{su}(2)$. In [@art:bedulli_stab] the authors explicitly compute a basis of $V_k^F$ for all $k$. They also compute that the action of $H^2$ on $v_\ell$, given by $$H^2 \cdot v_\ell = - (2 \ell - k)^2 v_\ell.$$ Moreover, the Casimir $C$ acts on $V_k$ as $\frac 1 8 k (k+2) \cdot \operatorname{id}$. Hence, the our Casimir $C_{a,b}$ acts on $V_k^F$ by $$C_{a,b} v_\ell = \biggl [ b k(k+2) - \frac{a-b}{8} (2 \ell - k)^2 \biggr ] v_\ell.$$ From this expression and the explicit generators of $V_k^F$ determined in [@art:bedulli_stab], we can say that on every $V_k^F$ the operator $C_{a,b}$ acts with distinct eigenvalues for all $a,b$. So condition in Proposition \[IrrCond\] is satisfied. To verify condition for appropriate $a,b$, we can apply a genericity argument similar to the one in Subsection \[ProdCROSSs\]. Define $$\lambda_{k,\ell} = \left(-\frac{(2\ell-k)^2}{8}, k(k+2) + \frac{(2\ell-k)^2}{8} \right) \in {\mathbb R}^2$$ Then for $v_\ell\in V^F_k$ as above $$C_{a,b}v_\ell = \lambda_{k,\ell}\cdot \begin{pmatrix} a \\ b \end{pmatrix}.$$ Thus, if for any choice of $a,b$ the above metric violates condition we would have $\lambda_{k\ell} = \lambda_{k'\ell'}$ for some $k,\ell,k',\ell'$. But looking at the explicit form of $\lambda_{k\ell}$ we see that this implies $k=k'$ and since we know that $C_{a,b}$ acts on $V_k^F$ with distinct eigenvalues for any choice of $a,b$ it follows that $\ell=\ell'$. So, we can conclude that $\operatorname{SU}(2)/F$ admits $\operatorname{SU}(2)$-simple metrics.
--- abstract: 'The $0^+$ states of $^8$He are studied in a five-body $^4$He+$n$+$n$+$n$+$n$ cluster model. Many-body resonances are treated on the correct boundary condition as Gamow states using the complex scaling method. The $0^+_2$ state of $^8$He is predicted as a five-body resonance in the excitation energy of 6.3 MeV with a width of 3.2 MeV, which mainly has a $(p_{3/2})^2(p_{1/2})^2$ configuration. In this state, number of the $0^+$ neuron pair shows almost two, which is different from the ground state having a large amount of the $2^+$ pair component. The monopole transition of $^8$He from the ground state into the five-body unbound states is also evaluated. It is found that the $^7$He+$n$ component mostly exhausts the strength, while the $0^+_2$ contribution is negligible. The final states are dominated by $^6$He+$n$+$n$, not $^4$He+$n$+$n$+$n$+$n$. The results indicate the sequential breakup process of $^8$He $\to$ $^7$He+$n$ $\to$ $^6$He+$n$+$n$ by the monopole excitation.' address: - 'General Education, Faculty of Engineering, Osaka Institute of Technology, Osaka, Osaka 535-8585, Japan' - 'Research Center for Nuclear Physics (RCNP), Osaka University, Ibaraki 567-0047, Japan' - 'Division of Physics, Graduate School of Science, Hokkaido University, Sapporo 060-0810, Japan' author: - Takayuki Myo - Ryosuke Ando - Kiyoshi Katō title: 'Five-body resonances of $^8$He using the complex scaling method' --- neutron skin ,neutron halo ,resonance ,complex scaling ,monopole strength 21.60.Gx ,21.10.Pc ,27.20.+n The development of experiments using radioactive beam has provided us with much information on unstable nuclei far from the stability. In particular, the light nuclei near the drip-line exhibit new phenomena of nuclear structures, such as the neutron halo structure in $^6$He, $^{11}$Li and $^{11}$Be [@tanihata85]. Recently, many experiments on $^8$He have been reported [@iwata00; @meister02; @chulkov05; @skaza07; @golovkov09]. Its ground state is considered to have a neutron skin structure consisting of four valence neutrons around $^4$He with small binding energy of 3.1 MeV. The recent experiments reported the matter and charge radius of $^8$He in addition to $^6$He [@tanihata92; @mueller07]. For the excited states of $^8$He, most of them can be located above the $^4$He+4$n$ threshold energy [@skaza07]. This fact indicates that the observed resonances of $^8$He can decay into the channels of $^7$He+$n$, $^6$He+2$n$, $^5$He+3$n$ and $^4$He+4$n$. These multiparticle decays of $^8$He are related to the Borromean nature of $^6$He, which breaks up easily into $^4$He+$2n$, and make it difficult to settle the excited states of $^8$He. Similar situation is also occurred for other He isotopes, such as $^6$He and $^7$He [@skaza06]. In the theoretical side, [*ab initio*]{} calculation of Green’s function Monte Carlo [@pieper04] has shown that the calculated energy levels fairly show a good correspondence with the experiments, although the results depend on the choice of the three-nucleon forces. This calculation is based on the bound state approximation and the continuum effect of the open channels is not included, while the excited states of $^8$He are unbound. Several methods have been proposed to treat the continuum effects explicitly, such as the microscopic cluster model [@wurzer97; @adahchour06; @arai09], the continuum shell model [@volya05] and the Gamow shell model [@betan09; @michel07]. It is, however, difficult to satisfy the multiparticle decay conditions correctly for all open channels. The energy spectra of many-body resonances depend on the treatment of open channels. For $^8$He, it is necessary to describe the $^4$He+4$n$ five-body resonances in the theory. Furthermore, it is important to reproduce the threshold energies of subsystems for particle decays. Emphasizing these theoretical conditions, in this study, we employed the cluster orbital shell model (COSM) [@suzuki88; @masui06] of the $^4$He+$4n$ five-body system for He isotopes. In COSM, the effects of all open channels are taken into account explicitly, so that we can treat the many-body decaying phenomena. In our previous works [@myo077; @myo09], we have successfully obtained the $^4$He+$3n$ four-body resonances of $^7$He, including the full couplings with $^{5,6}$He. We have described many-body resonances as Gamow states using the complex scaling method (CSM) [@aoyama06; @ho83; @moiseyev98], under the correct boundary conditions for all decay channels. In CSM, the resonant wave functions are directly obtained by diagonalization of the complex-scaled Hamiltonian using the $L^2$ basis functions. The successful results of He isotopes have been obtained for energies, decay widths, spectroscopic factors, Coulomb breakups and so on. In this study, we proceed our study of He isotopes to the $^8$He structures. It is interesting to see how our model describes $^8$He in addition to $^{5-7}$He and predicts the excited states of $^8$He. The excited states of $^8$He can be a five-body resonance. It is a challenge of CSM to describe the five-body nuclear resonances. For this purpose, in this article, we concentrate on the $0^+$ states of $^8$He. We predict the excited $0^+$ resonances and investigate their structures in comparison with the ground state. We also calculate the monopole strength from the ground state into the unbound states of $^8$He. This is to see the characteristics not only of the resonances, but also of non-resonant continuum states of $^8$He. In the breakups of $^8$He in a low energy region, two kinds of the final states of $^6$He+2$n$ and $^4$He+4$n$ are available. Using the five-body unbound states of $^8$He obtained in COSM, we clarify the breakup processes of $^8$He into the above final states by the monopole excitation. Similar analysis has been performed in the three-body Coulomb breakups of halo nuclei [@myo01; @myo03; @myo0711]. We explain the present model of He isotopes. We use COSM of the $^4$He+$N_{\rm v} n$ systems, where $N_{\rm v}$ is a valence neutron number around $^4$He, namely, $N_{\rm v}=4$ for $^8$He. The Hamiltonian is the same as that used in Refs. [@myo077; @myo01]; $$\begin{aligned} H &=& \sum_{i=1}^{N_{\rm v}+1}{t_i} - T_G + \sum_{i=1}^{N_{\rm v}} V^{\alpha n}_i + \sum_{i<j}^{N_{\rm v}} V^{nn}_{ij} \\ &=& \sum_{i=1}^{N_{\rm v}} \left[ \frac{\vec{p}^2_i}{2\mu} + V^{\alpha n}_i \right] + \sum_{i<j}^{N_{\rm v}} \left[ \frac{\vec{p}_i\cdot \vec{p}_j}{4m} + V^{nn}_{ij} \right] , \label{eq:Ham}\end{aligned}$$ where $t_i$ and $T_G$ are the kinetic energies of each particle ($n$ and $^4$He) and of the center of mass of the total system, respectively. The operator $\vec{p}_i$ is the relative momentum between $n$ and $^4$He. The reduced mass $\mu$ is $4m/5$ using a nucleon mass $m$. The $^4$He-$n$ interaction $V^{\alpha n}$ is given by the microscopic KKNN potential [@aoyama06; @kanada79], in which the tensor correlation of $^4$He is renormalized into the potential based on the resonating group method of the $^4$He+$n$ scattering. We use the Minnesota potential [@tang78] as $V^{nn}$. These interactions reproduce the low-energy scattering of the $^4$He-$n$ and the $n$-$n$ systems, respectively. For the wave function, $^4$He is treated as the $(0s)^4$ configuration of a harmonic oscillator wave function, whose length parameter is 1.4 fm to fit the charge radius of $^4$He as 1.68 fm. The motion of valence neutrons around $^4$He is solved variationally using the few-body technique. We expand the relative wave functions of the $^4$He+$N_{\rm v} n$ system using the COSM basis states [@suzuki88; @masui06]. In COSM, the total wave function $\Psi^J$ with a spin $J$ of the $^A$He=$^4$He+$N_{\rm v} n$ system (mass number $A=4+N_{\rm v}$) is represented by the superposition of the configuration $\Psi^J_c$ as $$\begin{aligned} \Psi^J(^{A}{\rm He}) &=& \sum_c C^J_c \Psi^J_c(^{A}{\rm He}), \label{WF0} \\ \Psi^J_c(^{A}{\rm He}) &=& \prod_{i=1}^{N_{\rm v}} a^\dagger_{\alpha_i}|0\rangle, \label{WF1}\end{aligned}$$ where $^4$He corresponds to a vacuum $|0\rangle$. The creation operator $a^\dagger_{\alpha}$ is for the single particle state of a valence neutron above $^4$He, with the quantum number $\alpha=\{n,\ell,j\}$ in a $jj$ coupling scheme. Here, the index $n$ represents the different radial component. The index $c$ represents the set of $\alpha_i$ as $c=\{\alpha_1,\cdots,\alpha_{N_{\rm v}}\}$. We take a summation over the available configurations in Eq. (\[WF0\]), which give a total spin $J$. The expansion coefficients $\{C_c^J\}$ in Eq. (\[WF0\]) are determined with respect to the total wave function $\Psi^J$ by diagonalization of the Hamiltonian matrix elements. ![Sets of the spatial coordinates in COSM for the $^4$He+$N_{\rm v} n$ system.[]{data-label="fig:COSM"}](He_COSM.eps){width="7.0cm"} The coordinate representation of the single particle state corresponding to $a^\dagger_{\alpha}$ is given as $\psi_{\alpha}(\vc{r})$ as function of the relative coordinate $\vc{r}$ between the center of mass of $^4$He and a valence neutron [@suzuki88; @masui06; @myo09], as shown in Fig. \[fig:COSM\]. We employ sufficient number of radial bases of $\psi_\alpha$ in order to describe the spatial extension of valence neutrons in the weak binding state and also in the resonances. In this model, the radial part of $\psi_\alpha$ is expanded with Gaussian basis functions [@hiyama03] as $$\begin{aligned} \psi_\alpha &=& \sum_{k=1}^{N_{\ell j}} d^k_{\alpha}\ \phi_{\ell j}^k(\vc{r},b_{\ell j}^k), \label{WFR} \\ \phi_{\ell j}^k(\vc{r},b_{\ell j}^k) &=& {\cal N}\, r^{\ell} e^{-(r/b_{\ell j}^k)^2/2} [Y_{\ell}(\hat{\vc{r}}),\chi^\sigma_{1/2}]_{j}. \label{Gauss}\end{aligned}$$ The index $k$ is for the Gaussian basis with the length parameter $b_{\ell j}^k$. A basis number and the normalization factor of the basis are given by $N_{\ell j}$ and ${\cal N}$, respectively. The coefficients $\{d^k_{\alpha}\}$ in Eq. (\[WFR\]) are determined using the Gram-Schmidt orthonormalization, and hence, the basis states $\psi_\alpha$ are orthogonal to each other. The numbers of the radial bases of $\psi_\alpha$ are at most $N_{\ell j}$, and are determined to converge the physical solutions. The length parameters $b_{\ell j}^k$ are chosen in geometric progression [@aoyama06; @hiyama03]. We use at most 17 Gaussian basis functions by setting $b_{\ell j}^k$ from 0.2 fm to around 40 fm with the geometric ratio of 1.4 as a typical one. Due to the expansion of the radial wave function using a finite number of basis states, all the energy eigenvalues are discretized for bound, resonant and continuum states. To obtain the Hamiltonian matrix elements of multi-neutron system, we employ the $j$-scheme technique of the shell model. The antisymmetrization between a valence neutron and $^4$He is treated on the orthogonality condition model [@aoyama06], in which the single particle state $\psi_{\alpha}$ is imposed to be orthogonal to the $0s$ state occupied by $^4$He. In COSM, the asymptotic boundary condition of the wave functions for neutron emissions are correctly described [@myo09; @aoyama06]. For $^8$He, all the channels of $^8$He, $^7$He+$n$, $^6$He+$2n$, $^5$He+$3n$, $^4$He+$4n$ are automatically included in the total wave function $\Psi^J$ in Eq. (\[WF0\]). These components are coupled to each other by the interactions and the antisymmetrization, which depend on the relative distances between $^4$He and a valence neutron and between the valence neutrons. For the single-particle states, we take the angular momenta $\ell\le 2$ to keep the accuracy of the converged energy within 0.3 MeV of $^6$He in comparison with the full space calculation. In this model, we adjust the two-neutron separation energy of $^6$He($0^+$) to the experiment of 0.975 MeV by taking the 173.7 MeV of the repulsive strength of the Minnesota force instead of the original value of 200 MeV. The adjustment of the $nn$ interaction is originated from the pairing correlation between valence neutrons with higher angular momenta $\ell>2$ [@aoyama06]. Hence, the present model reproduces the observed properties of $^{5,6}$He, as shown in Fig. \[fig:5678\], namely, the threshold energies of the particle emissions of He isotopes. We explain CSM, which describes resonances and nonresonant continuum states [@aoyama06]. Hereafter, we refer to the nonresonant continuum states as simply the continuum states. In CSM, we transform the relative coordinates of the $^4$He+$N_{\rm v} n$ system, as $\vc{r}_i \to \vc{r}_i\, e^{i\theta}$ for $i=1,\cdots,N_{\rm v}$, where $\theta$ is a scaling angle. The Hamiltonian in Eq. (\[eq:Ham\]) is transformed into the complex-scaled Hamiltonian $H_\theta$, and the corresponding complex-scaled Schrödinger equation is given as $$\begin{aligned} H_\theta\Psi^J_\theta &=& E\Psi^J_\theta, \label{eq:eigen}\end{aligned}$$ The eigenstates $\Psi^J_\theta$ are obtained by solving the eigenvalue problem of $H_\theta$ in Eq. (\[eq:eigen\]). In CSM, we obtain all the energy eigenvalues $E$ of bound and unbound states on a complex energy plane, governed by the ABC theorem [@aoyama06]. In this theorem, it is proved that the boundary condition of Gamow resonances is transformed to the damping behavior at the asymptotic region. This condition makes it possible to use the same method of obtaining the bound states for resonances. For a finite value of $\theta$, every Riemann branch cut is commonly rotated down by $2\theta$. Hence, the continuum states such as $^7$He+$n$ and $^6$He+2$n$ channels are obtained on the branch cuts rotated with the $2\theta$ dependence [@myo077; @myo01]. On the contrary, bound states and resonances are obtainable independently of $\theta$ (see Fig. \[fig:ene8\]). Hence, we can identify the resonance poles with complex eigenvalues: $E=E_r-i\Gamma/2$, where $E_r$ and $\Gamma$ are the resonance energies and the decay widths, respectively. In the wave function, the $\theta$ dependence is included in the coefficients in Eq. (\[WF0\]) as $\{C_c^{J,\theta}\}$. The value of the angle $\theta$ is determined to search for the stationary point of each resonance in a complex energy plane[@aoyama06; @ho83; @moiseyev98]. We take $\theta$ as 20 degree in the $^8$He calculation. In CSM, the amplitudes of the obtained resonances are finite and normalized to be unity totally, as $\sum_{c} \left(C_c^{J,\theta}\right)^2=1$. Here, the Hermitian product is not applied due to the biorthogonal relation [@berggren68]. In this study, we calculate the monopole strength function of $^8$He into the unbound states. To calculate the strength function, one needs the extended completeness relation (ECR) of $^8$He consisting of bound, resonant, and continuum states, which are constructed using the complex-scaled eigenstates $\Psi^J_\theta$ in Eq. (\[eq:eigen\]). We briefly explain ECR of $^8$He using CSM [@myo09; @myo01; @berggren68]. When we take a large $\theta$ sufficiently, five-body unbound states of $^8$He are decomposed into several classes of the state, which consist of the five-body ECR of $^8$He as $$\begin{aligned} {\bf 1} &=& \sum_{~\nu} \kets{\Psi^\theta_\nu}\bras{\wtil{\Psi}^\theta_\nu} \nonumber \\ &=& \{\mbox{bound state of $^8${He}}\} ~+~ \{\mbox{resonances of $^8${He} }\} \nonumber \\ &+& \{\mbox{two-body continuum states of $^7${He}$^{(*)}$+$n$}\} \nonumber \\ &+& \{\mbox{three-body continuum states of $^6${He}$^{(*)}$+$2n$}\} \nonumber \\ &+& \{\mbox{four-body continuum states of $^5${He}$^{(*)}$+$3n$}\} \nonumber \\ &+& \{\mbox{five-body continuum states of $^4${He}+$4n$}\} , \label{eq:ECR}\end{aligned}$$ where $\{ \Psi_\nu^\theta,\wtil{\Psi}_\nu^\theta \}$ forms a set of biorthogonal bases with a state $\nu$. For simplicity, we here do not write the spin index explicitly. The expressions of $2n$, $3n$ and $4n$ in Eq. (\[eq:ECR\]) mean no-interacting states of multi-neutrons. We explain how to calculate the strength function using ECR within CSM. To do this, we define the complex-scaled Green’s function ${\cal G}^\theta(E)$ with the energy $E$ of the system as $$\begin{aligned} {\cal G}^\theta(E) &=& \frac{ {\bf 1} }{ E-H_\theta } ~=~ \sum_{~\nu} \frac{|\Psi^\theta_\nu\rangle \langle \wtil{\Psi}^\theta_\nu|}{E-E_\nu^\theta} , \label{eq:green1}\end{aligned}$$ where, the complex-scaled eigenvalue $E_\nu^\theta$ is associated with the wave function $\Psi^\theta_\nu$. The strength function ${\cal S}_\lambda(E)$ for the operator  $\hO_\lambda$ with rank $\lambda$ is defined in terms of Green’s function without CSM as $$\begin{aligned} {\cal S}_\lambda(E) &=& \sum_{~\nu} \bras{\wtil{\Psi}_0}\hO_\lambda^\dagger\kets{\Psi_\nu}\bras{\wtil{\Psi}_\nu}\hO_\lambda\kets{\Psi_0}\ \delta(E-E_\nu) \label{eq:strength_org} \\ &=& -\frac1{\pi}\ {\rm Im}\left[ \langle \wtil{\Psi}_0 | \hO^\dagger_\lambda {\cal G}(E) \hO_\lambda | \Psi_0 \rangle \right] , \label{eq:strength1}\end{aligned}$$ where $\Psi_0$ is the initial state. We operate the complex scaling on the strength function of Eq. (\[eq:strength1\]) and insert the complex-scaled Green’s function in Eq. (\[eq:green1\]). $$\begin{aligned} {\cal S}_\lambda(E) &=& \sum_{~\nu} {\cal S}_{\lambda,\nu}(E) , \\ {\cal S}_{\lambda,\nu}(E) &=& -\frac1{\pi}\ {\rm Im}\left[ \frac{ \bras{\wtil{\Psi}_0^\theta} (\hO_\lambda^\dagger)^\theta \kets{\Psi_\nu^\theta} \bras{\wtil{\Psi}_\nu^\theta} \hO_\lambda^\theta \kets{\Psi_0^\theta} }{E-E_\nu^\theta} \right] . \label{eq:strength3}\end{aligned}$$ From the decomposed strength function $S_{\lambda,\nu}(E)$, we can identify the contributions of each state $\nu$ in the total strength $S_\lambda(E)$. It is noted that the functions $S_{\lambda}(E)$ and $S_{\lambda,\nu}(E)$ are independent of $\theta$ [@myo09; @myo01]. This is because any matrix elements are obtained independently of $\theta$ in CSM, and also because the state $\nu$ of $^8$He is uniquely classified according to ECR in Eq. (\[eq:ECR\]). As a result, $S_{\lambda,\nu}(E)$ is uniquely obtained. This method has been applied to the calculations of scattering amplitudes of the breakup reactions [@suzuki05; @kruppa07; @kikuchi09]. Here, we discuss the properties of the function $S_{\lambda,\nu}(E)$. The total strength function $S_{\lambda}(E)$ is an observable being positive definite for every energy. On the other hand, the decomposed one $S_{\lambda,\nu}(E)$ is not necessarily positive definite at all energies, because $S_{\lambda,\nu}(E)$ cannot be directly observed, similar to resonant poles. This means that $S_{\lambda,\nu}(E)$ can sometimes have negative values. This property of the decomposed strength has been generally discussed in Refs. [@myo01; @myo03]. In this study, we discretize the continuum states in terms of the basis expansion, as shown in Fig. \[fig:ene8\]. The reliability of the continuum discretization in CSM has already been shown using the continuum level density [@suzuki05; @kruppa07; @kikuchi09]. ![Energy levels of He isotopes measured from the $^4$He energy. Units are in MeV. Black and gray lines are theory and experiments, respectively. Small numbers are decay widths. For $^8$He, the experimental data are taken from Ref. [@golovkov09], and only the $0^+$ states are shown in theory.[]{data-label="fig:5678"}](8He_lev6.eps){width="8.5cm"} ![Energy eigenvalue distribution of $^8$He($0^+$) in complex energy plane.[]{data-label="fig:ene8"}](ene26_4.eps){width="8.5cm"} We show the obtained energy spectra of He isotopes in Fig. \[fig:5678\]. One can see a good agreement between theory and experiment. For $^6$He, the position of the $2^+$ state is good and its decay width of 0.132 MeV agrees with the experiment of 0.113(20) MeV [@aj89]. For $^7$He, the ground state is located by 0.40 MeV above the $^6$He ground state, which agrees with the recent experiments of 0.44 MeV and 0.36 MeV [@skaza06]. For $^8$He, the ground state binding energy is obtained as 3.22 MeV from the $^4$He+$4n$ threshold, which agrees with 3.11 MeV of the experiment. We predict the $0^+_2$ state with the 6.29 MeV excitation energy and the 3.19 MeV decay width. We demonstrate the example of the $0^+$ eigenvalue distribution using CSM in Fig. \[fig:ene8\] with $\theta$ being 26 degree. We successfully obtain the $0^+_2$ state of $^8$He as a five-body resonance and confirm other continuum states such as $^6$He+$2n$ and $^7$He+$n$. The obtained matter and charge radius of $^6$He and $^8$He for their ground states are shown in Table \[tab:radius\] and reproduce the experiments. Hence, the present model well describes the neutron halo and skin structures in He isotopes. The proton and neutron radius are obtained as 1.82 fm and 2.60 fm for $^6$He and 1.80 fm and 2.72 fm for $^8$He, respectively. \[1\] Present Experiments ---------------- --------- --------------------------------------------------------------    $R_{\rm m}$ 2.37 2.33(4)$^{\rm a}$    2.45(10)$^{\rm b}$    2.37(5)$^{\rm c}$ $R_{\rm ch}$ 2.01 2.068(11)$^{\rm d}$    $R_{\rm m}$ 2.52 2.49(4)$^{\rm a}$    2.53(8)$^{\rm b}$    2.49(4)$^{\rm c}$ $R_{\rm ch}$ 1.92 1.929(26)$^{\rm d}$ : Matter ($R_{\rm m}$) and charge ($R_{\rm ch}$) radius of $^6$He and $^8$He in comparison with the experiments; a[@tanihata92], b[@alkazov02], c[@kiselev05], d[@mueller07]. Units are in fm.[]{data-label="tab:radius"} Orbit     $0^+_1$     $0^+_2$ ----------- ------------- --------------------- $p_{1/2}$   $0.14$    $1.94 -i 0.02$ $p_{3/2}$   $3.71$    $2.04 -i 0.02$ $s_{1/2}$   $0.02$ $-0.02 +i 0.003$ $d_{3/2}$   $0.02$    $0.04 +i 0.04$ $d_{5/2}$   $0.10$    $0.01 -i 0.0004$ : Occupation numbers of valence neutrons in $^8$He.[]{data-label="tab:occupy"} ![Pair numbers of the $0^+_{1,2}$ states of $^8$He.[]{data-label="fig:pair"}](pair20.eps){width="8.0cm"} We discuss the structure of the $0^+_2$ state of $^8$He in comparison with the ground state. We list the occupation numbers of four neutrons in each orbit for $^8$He in Tables \[tab:occupy\]. Since the $0^+_2$ state is a Gamow state, its quantities become complex values with a relatively small imaginary part, while their summation conserves the valence neutron number. In the $^8$He ground state, the $p_{3/2}$ orbit is dominant and its number is close to four. In fact, the $(p_{3/2})^4$ configuration dominates the total wave function with a mixing of 86.0%. The next dominant configurations are $(p_{3/2})^2(p_{1/2})^2$ with 6.9%, $(p_{3/2})^2(d_{5/2})^2$ with 4.2%, $(p_{3/2})^2(d_{3/2})^2$ with 0.8% and $(p_{3/2})^2(1s_{1/2})^2$ with 0.6%. This result means that the $jj$ coupling scheme is well established in the ground state of $^8$He. In the $0^+_2$ state, the $p_{1/2}$ orbit is dominant with the number being around two. In this state, the $(p_{3/2})^2(p_{1/2})^2$ configuration dominates the total wave function with a mixing of 96.9%, while $(p_{3/2})^4$ is given as 2.0%. Hence, the $0^+_2$ state of $^8$He corresponds to the $2p2h$ excited state of the ground state. We also calculate the pair number of four valence neutrons in $^8$He, which is defined by the matrix element of the operator $\sum_{\alpha \le \beta} A^\dagger_{J^\pi}(\alpha\beta)A_{J^\pi}(\alpha\beta)$. Here the quantum number $\alpha$ and $\beta$ are for the single particle state and $A^\dagger_{J^\pi}$ ($A_{J^\pi}$) is the creation (annihilation) operator of a neutron pair with spin-parity $J^\pi$. This pair number is useful to understand the structures of four neutrons from the viewpoint of pair coupling. The summation of the pair number over all $J^\pi$ satisfies six from the total pair number of four neutrons. In Fig. \[fig:pair\], we show the results of the pair number up to the $3^-$ component. In the ground state, it is found that the $2^+$ neutron pair is dominant with about 4.5 and the $0^+$ pair is almost unity. This is consistent with the main configuration of $(p_{3/2})^4$ from the CFP decomposition ( 1 and 5 for $0^+$ and $2^+$, respectively). The importance of the $2^+$ neutron pair is suggested in the experiment [@korsheninnikov03], and is also obtained in the $^6$He+$n$+$n$ model [@adahchour06]. On the other hand, the $0^+_2$ state has much $0^+$ neutron pair, about two, in addition to the large $2^+$ pair number. This is also consistent with the $(p_{3/2})^2(p_{1/2})^2$ configuration, which is decomposed into the pairs of $0^+$, $1^+$ and $2^+$ with the occupations of $2$, $1.5$ and $2.5$, respectively. The result of $0^+$ pair of neutrons in the $0^+_2$ state is interesting in relation with the dineutron-like cluster correlation in $^8$He suggested in AMD [@enyo07]. The analysis of the spatial correlation of neutrons in the $0^+_2$ state will be performed elsewhere. Finally, we calculate the monopole transition of $^8$He into unbound states and see the effect of the $0^+_2$ resonance, because the monopole strength is useful to investigate the configuration properties of the states [@schwerdtfeger09]. Recently, Yamada et al. [@yamada08] discussed the relation between the clustering excited state and its monopole strength from the ground state. They mentioned that the enhancement of the monopole strength can be seen in the clustering state, because of the concentration of the strength into the relative motion of the intercluster. In $^8$He, hence, it is interesting to investigate the monopole strength of $0^+_2$ in relation with the dineutron-like structure. In the monopole strength, it is also important to see the effects of the continuum states in addition to that of resonance. We take care of not only the $0^+_2$ resonance, but also all of the residual continuum states of $^7$He+$n$, $^6$He+$2n$, $^5$He+$3n$ and $^4$He+$4n$ using ECR in Eq. (\[eq:ECR\]). The angle $\theta$ in the complex scaling is taken as 20 degree to describe ECR. ![Monopole strengths of $^8$He for isoscalar (IS) and isovector (IV) transitions as functions of the excitation energy of $^8$He. The threshold energies of $^7$He+$n$, $^6$He+$2n$, $^5$He+$3n$ and $^4$He+$4n$ are indicated by arrows with $S_n$, $S_{2n}$, $S_{3n}$ and $S_{4n}$, respectively.[]{data-label="fig:monopole"}](mono14_5S2.eps "fig:"){width="8.3cm"} ![Monopole strengths of $^8$He for isoscalar (IS) and isovector (IV) transitions as functions of the excitation energy of $^8$He. The threshold energies of $^7$He+$n$, $^6$He+$2n$, $^5$He+$3n$ and $^4$He+$4n$ are indicated by arrows with $S_n$, $S_{2n}$, $S_{3n}$ and $S_{4n}$, respectively.[]{data-label="fig:monopole"}](mono14_5V2.eps "fig:"){width="8.3cm"} In Fig. \[fig:monopole\], the monopole strengths for isoscalar (IS) and isovector (IV) responses are shown. It is found that two strengths exhibit a similar shape showing the low energy enhancement just above 3 MeV in the excitation energy. There is no clear signature of the $0^+_2$ state around its excitation energy of 6.29 MeV in both strengths. In fact, the transition matrix elements from the ground state into the $0^+_2$ state are obtained as $1.78-i0.38$ fm$^4$ for IS and $-0.003+i0.018$ fm$^4$ for IV, respectively. These values are so small in comparison with the total strengths. This result is understood from the single particle structures of the $0^+_2$ state. In the $0^+_2$ state, the $p_{1/2}$ orbit is largely mixed shown in Table \[tab:occupy\], and this orbit cannot be excited from the $p_{3/2}$ orbit in the ground state by the monopole operator. As a result, the monopole strength into $0^+_2$ becomes negligible. Instead, the continuum strength gives a main contribution, which makes it difficult to observe the $0^+_2$ state via the monopole transition. Our results do not support the enhancement of the strength into the $0^+_2$ state, however, the possibility of the dineutron-like structure in $^8$He should be investigated carefully, in addition to the monopole strength. It is also necessary to search for the observables which are responsible for the $0^+_2$ state. Since the present $^8$He wave functions contain not only resonances but also continuum states, the application to the sophisticated reaction analysis including continuum coupling would be promising. One of the candidates of the reactions is considered to be the two-neutron transfer into $^6$He to produce the excited states of $^8$He. Experimentally, the $^6$He($t$,$p$)$^8$He reaction was reported [@golovkov09] and the observed cross section shows some peaks without spin assignment, around the resonance energy of $0^+_2$ obtained in this study. We decompose the monopole strengths in Fig. \[fig:monopole\] into several continuum components using ECR and see the individual contribution. From the analysis, it is found that the IS and IV strengths both dominantly come from the $^7$He($3/2^-_1$)+$n$ components. This selectivity of the continuum states is related with the properties of the monopole operator, which are one-body concerning with $\vc{r}_i$ in Fig. \[fig:COSM\]. By the monopole operator, one of the relative motions of $^8$He can be strongly excited. As a result, the intercluster motion between the $^7$He cluster and a valence neutron is strongly coupled with the ground state by the monopole excitation. The obtained result also indicates the sequential breakup process of $^8$He(G.S.) $\to$ $^7$He($3/2^-_1$)+$n$ $\to$ $^6$He(G.S.)+$n$+$n$ in the monopole excitation. Experimentally, the large contribution of the sequential process via $^7$He+$n$ was also reported in the Coulomb breakup of $^8$He [@iwata00], which is dominated by the $E1$ transition. It is interesting to see the components of the final states of $^6$He+$n$+$n$ and $^4$He+$n$+$n$+$n$+$n$ in the monopole strengths. In Table \[tab:ratio\], the ratios of the integrated strengths into two final states with respect to the total ones are shown. Dominance of the $^6$He+$n$+$n$ state is commonly found in the IS and IV cases and more significant in the IS case. This is because one of the relative coordinates of $\{\vc{r}_i\}$ in $^8$He can be excited independently by the IS response, and then the $^7$He component largely remains and decays into $^6$He+$n$. In the IV case, protons are included only in $^4$He, so that the IV response excites the relative motion between the $^4$He core and the center of mass of $4n$ as a recoil effect. Owing to the excitation of the relative motion, the components of $^6$He and $^7$He in $^8$He are relatively dissolved than the IS case and the transition into $^4$He+$n$+$n$+$n$+$n$ increases in the IV response. $^6$He+$n$+$n$ $^4$He+$n$+$n$+$n$+$n$ ---- ---------------- ------------------------ IS 0.939 0.061 IV 0.690 0.310 : Ratios of the integrated monopole strengths into $^6$He+$n$+$n$ and $^4$He+$n$+$n$+$n$+$n$ final states.[]{data-label="tab:ratio"} In summary, we have investigated the structures of the $0^+$ states of $^8$He in a five-body cluster model. The boundary condition for many-body resonances is accurately treated using CSM. We successfully obtain the five-body $0^+_2$ resonance of $^8$He in CSM. This state dominantly has a $(p_{3/2})^2(p_{1/2})^2$ configuration and mainly consists of two of the $0^+$ neutron pairs. We further investigate the monopole strengths into five-body unbound states, which are described by using ECR within CSM. It is found that the $0^+_2$ contribution is negligible in the strength, so that it is difficult to observe the $0^+_2$ state from the monopole strength. It is dominant that the sequential breakup process of $^8$He via the $^7$He+$n$ states into the $^6$He+$2n$ three-body final states, instead of the $^4$He+$4n$ five-body states. Acknowledgments {#acknowledgments .unnumbered} =============== We would like to thank Prof. Kiyomi Ikeda for fruitful discussions. This work was supported by a Grant-in-Aid for Young Scientists from the Japan Society for the Promotion of Science (No. 21740194). Numerical calculations were performed on a supercomputer (NEC SX8R) at RCNP, Osaka University. References {#references .unnumbered} ========== [00]{} I. Tanihata [*et al.*]{}, . Y. Iwata [*et al.*]{}, . M. Meister [*et al.*]{}, . L.V. Chulkov [*et al.*]{}, . F. Skaza [*et al.*]{}, , and references therein. M.S. Golovkov [*et al.*]{}, . I. Tanihata [*et al.*]{}, . P. Mueller [*et al.*]{}, . F. Skaza [*et al.*]{}, , and references therein. S. C. Pieper, R. B. Wiringa, J. Carlson, . J. Wurzer and H. M. Hofmann, . A. Adahchour and P. Descouvemont, . K. Arai and S. Aoyama, . A. Volya and V. Zelevinsky, . R. Id Betan, A. T. Kruppa, and T. Vertse, . N. Michel, W. Nazarewicz and M. Ploszajczak, . Y. Suzuki and K. Ikeda, . H. Masui, K. Katō, K. Ikeda, . T. Myo, K. Katō and K. Ikeda, . T. Myo, R. Ando and K. Katō, . S. Aoyama, T. Myo, K. Katō, K. Ikeda, . Y. K. Ho, . N. Moiseyev, . H. Kanada, T. Kaneko, S. Nagata, M. Nomoto, . T. Myo, K. Katō, S. Aoyama and K. Ikeda, . T. Myo, S. Aoyama, K. Katō and K. Ikeda, . T. Myo, K. Katō, H. Toki and K. Ikeda, . Y. C. Tang, M. LeMere and D. R. Thompson, . E. Hiyama, Y. Kino and M. Kamimura, . T. Berggren, . R. Suzuki, T. Myo and K. Katō, . A. T. Kruppa, R. Suzuki and K. Katō, . Y, Kikuchi, T. Myo, M. Takashina, K. Katō and K. Ikeda, . F. Ajzenberg-selove, . G. D. Alkhazov [*et al.*]{}, . O. A. Kiselev [*et al.*]{}, Eur. Phys. J. 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--- author: - 'M. Baes' - 'D. Herranz' - 'S. Bianchi' - 'L. Ciesla' - 'M. Clemens' - 'G. De Zotti' - 'F. Allaert' - 'R. Auld' - 'G. J. Bendo' - 'M. Boquien' - 'A. Boselli' - 'D. L. Clements' - 'L. Cortese' - 'J. I. Davies' - 'I. De Looze' - 'S. di Serego Alighieri' - 'J. Fritz' - 'G. Gentile' - 'J. González-Nuevo' - 'T. Hughes' - 'M. W. L. Smith' - 'J. Verstappen' - 'S. Viaene' - 'C. Vlahakis' bibliography: - 'planck.bib' subtitle: 'XV. Planck submillimetre sources in the Virgo Cluster' title: 'The Herschel Virgo Cluster Survey[^1]' --- Introduction ============ Roughly half of all the energy emitted by stars and AGNs in the Universe has been absorbed by dust and re-emitted at far-infrared (FIR) and submillimetre (submm) wavelengths. For many years, this important wavelength regime was one of the least explored windows. Several space missions have been operational in the FIR wavelength region since the mid 1980s, including the Infrared Astronomical Satellite [IRAS, @1984ApJ...278L...1N], the Infrared Space Observatory , the Spitzer Space Telescope [@2004ApJS..154....1W] and the Akari mission [@2007PASJ...59S.369M]. At the same time, the submm window has been explored using bolometer instruments on ground-based telescopes including SCUBA and SCUBA-2 on the James Clerck Maxwell Telescope [@1999MNRAS.303..659H; @2006SPIE.6275E..45H] and LABOCA on the Atacama Pathfinder Experiment , as well as balloon experiments such as the Balloon-borne Large Aperture Submillimeter Telescope [BLAST, @2008ApJ...681..400P]. All these missions suffered from relatively poor resolution and/or sensitivity, and left the wavelength region between 200 and 800 $\mu$m virtually unexplored. The simultaneous launch of the Herschel Space Observatory and the Planck mission in May 2009 has finally opened the FIR/submm window in earnest. Herschel was a FIR/submm observatory facility that has offered for the first time imaging and spectroscopic capabilities from space in the wavelength range between 55 and 671 $\mu$m. It contained three instruments onboard: the FIR imager and spectrometer PACS , the submm imager and spectrometer SPIRE , and the high-resolution heterodyne spectrometer HIFI . The PACS camera had three photometric bands, centered at 70, 100 and 160 $\mu$m, the corresponding SPIRE camera operates at three bands centered at 250, 350 and 500 $\mu$m. The angular resolution of the Herschel imaging data varies from 6 arcsec at the PACS 70 $\mu$m band to 36 arcsec at the SPIRE 500 $\mu$m band. The main goal of Herschel was to study the cool interstellar dust and gas in the Universe, with pointed observations of all types of objects, ranging from the Solar System to the deep extragalactic sky . Planck is an all sky survey mission that has been simultaneously and continuously scanning the entire sky in nine submm/mm wavebands. It has two instruments onboard: the High Frequency Instrument contains bolometers operational at six bands centered at 857, 545, 353, 217, 143 and 100 GHz (corresponding to wavelengths of 350, 550, 850 $\mu$m and 1.38, 2.1 and 3 mm, respectively) and the Low Frequency Instrument contains radiometers that cover three bands at 70, 44 and 30 GHz (corresponding to 4.3, 6.8 and 10 mm, respectively). The angular resolution decreases from 4.33 arcmin at the highest frequency to 32.88 arcmin at the lowest frequency. The prime objective of Planck is to measure the spatial anisotropies of the temperature and the polarization of the cosmic microwave background with an unprecedented sensitivity and resolution [@2013arXiv1303.5075P; @2013arXiv1303.5076P; @2013arXiv1303.5083P]. In addition, the Planck mission has produced a treasure of valuable information on the submm/mm properties of foreground objects, both in our Milky Way and extragalactic objects . A preliminary catalogue of Planck compact sources, the Early Release Compact Source Catalogue (ERCSC) was publicly released in January 2011 and contains more than 15,000 unique sources . A similar, but more complete catalogue (more than 25,000 sources), the Planck Catalogue of Compact Sources [PCCS, @2013arXiv1303.5088P], was released in March 2013 as part of the major Planck 2013 data and scientific results release [@2013arXiv1303.5062P]. The complementarity and overlap of the Planck and Herschel missions offers a number of advantages. Planck has the advantage of observing the entire sky, and is likely to discover the most rare and extreme submm sources, including strongly lensed high-redshift starburst galaxies and proto-clusters of luminous submm galaxies. With its much better spatial resolution, Herschel has been ideal to follow up these sources, to determine their nature and to characterize their physical properties (the ERCSC was released a few months before Herschel’s second in-flight announcement of opportunity for observing time proposals). More generally, Herschel observations of Planck sources are useful to quantify the boosting of Planck flux densities due to noise peaks and confusion, to provide more accurate positions, and to distinguish between Galactic foreground cirrus and genuine extragalactic point sources. Herschel can also take advantage of the absolute calibration of the Planck HFI instrument. Finally, while both missions have a limited overlap in wavelength coverage, they can take advantage of the complementarity beyond this common range. For Herschel sources, Planck flux densities at frequencies of 545 GHz and lower (i.e. wavelengths of 550 $\mu$m and beyond) can be useful to investigate a change in the slope of the spectral energy distribution (SED) of galaxies at about 500 $\mu$m that might be related to very cold dust or might originate from dust with a shallow emissivity function. In turn, SPIRE 250 $\mu$m and PACS data are useful to bridge the gap in the SED between the Planck HFI and the 100 $\mu$m band as observed by IRAS. The complementarity of Herschel and Planck can be exploited in different ways. A first obvious way is to consider a set of galaxies observed by Herschel and complement the Herschel data with Planck observations. This approach has been applied to several samples of nearby galaxies , either as a check on the Herschel photometry or to add additional submm data to the SED beyond the SPIRE 500 $\mu$m limit. The disadvantage of this approach is that the population of galaxies is fixed beforehand (typically selected in the optical), and hence does not allow for a characterization of the Planck population or a discovery of peculiar Planck sources. An alternative approach is to search for and characterize the Planck sources in large-area “blind” extragalactic surveys with Herschel. A first effort along this line is the recent work by . They cross-correlate the Planck ERCSC catalogue with Herschel observations taken as part of the Herschel Astrophysical Terahertz Large Area Survey [H-ATLAS, @2010PASP..122..499E]. H-ATLAS is a Herschel open time key programme that has surveyed about 550 deg$^2$ of extragalactic sky in five FIR/submm bands, using the PACS and SPIRE instruments in parallel mode. analyzed the compact Planck submm sources in the so-called Phase 1 fields, a set of equatorial fields with a total survey area of 134.5 deg$^2$. They detected 28 compact Planck sources, 16 of which are most probably high-latitude Galactic cirrus features. 10 of the Planck sources correspond to bright, low-redshift galaxies and one source corresponds to a pair of nearby galaxies. Interestingly, one compact Planck source is resolved by Herschel into a condensation of about 15 faint point sources surrounding an unusually bright submm source, which turned out to be a strongly lensed submm galaxy at $z=3.259$ [@2012ApJ...753..134F; @2012ApJ...752..152H]. In this paper, we investigate the Planck compact submm sources in the survey field of another Herschel extragalactic survey, the Herschel Virgo Cluster Survey . HeViCS is a Herschel open time key program and has conducted a deep survey of four 16 deg$^2$ fields in the Virgo Cluster. The prime aim of HeViCS is a detailed study of the FIR/submm properties of the galaxies in the Virgo Cluster and a study of the environmental effects through comparison with nearby galaxies in less dense environments, such as in the HRS or KINGFISH [@2011PASP..123.1347K; @2012ApJ...745...95D] surveys. However, given that HeViCS has observed a large area of extragalactic sky, it is very suitable for a cross-correlation between Herschel and Planck. Similar to H-ATLAS, HeViCS is a fully-sampled, blind survey and adopts parallel mode observations with both PACS and SPIRE at five wavelengths between 100 and 500 $\mu$m. The main difference between both surveys, apart from the choice of the survey fields and the specific scientific objectives, is the depth of the observations: while H-ATLAS uses two cross-linked scans of every field, HeViCS has performed eight cross-linked scans of every field, with the aim to reach the 250 $\mu$m confusion limit. Due to the time span between the different observations and the corresponding orientation of the survey fields and the spatial offset between the PACS and SPIRE instruments on the sky, HeViCS has observed a total area of 84 deg$^2$, of which 55 deg$^2$ has been observed to the full depth [for details, see @2013MNRAS.428.1880A]. Combined with the fact that the Virgo Cluster has been observed extensively at virtually all wavelengths from X-rays to radio waves , the HeViCS fields are ideal for a flux density comparison between Planck and Herschel and a characterisation of the nature of the detected Planck submm sources within the survey field. The content of this paper is as follows: in Section 2 we search for and characterize the PCCS 857 and 545 GHz sources in the HeViCS survey fields and we search for their Herschel FIR/submm and optical counterparts. In Section 3, we analyze different characteristics of these populations: we compare the various flavors of the flux densities from the PCCS of the Planck detected sources in the HeViCS fields to the corresponding SPIRE flux densities at 350 and 500 $\mu$m, discuss the nature of sources without obvious counterparts, we investigate the completeness and positional accuracy of the PCCS, and we discuss the overdensity of the Virgo Cluster at submm wavelengths. Finally, in Section 4 we present our conclusions and summary. Identification and characterization of the sources ================================================== We searched the Planck Catalogue of Compact Sources (PCCS) for all sources within the boundaries of the 84 deg$^2$ extended HeViCS fields. In this section we identify these sources and discuss the global properties of the sample of detected 857 and 545 GHz sources. 857 GHz sources {#857.sec} --------------- ------------------------- ------------------------- ------------------------- ------------------ -------------- ---------- ---------- ------------------ PCCS source $\alpha_{\text{J2000}}$ $\delta_{\text{J2000}}$ PCCS 857 GHz counterpart VCC type SPIRE 350 $\mu$m (deg) (deg) (Jy) (Jy) PCCS1 857 G261.44+74.24 182.12855 14.94011 $0.746\pm0.136$ $\cdots$ $\cdots$ $\cdots$ $\cdots$ PCCS1 857 G260.38+75.42 182.66101 16.03927 $2.111\pm0.147$ NGC4152 25 Sc $1.996\pm0.149$ PCCS1 857 G266.31+73.24 182.68154 13.34018 $0.591\pm0.110$ IC3029 27 Sc $0.317\pm0.029$ PCCS1 857 G269.69+72.43 183.12050 12.12778 $0.980\pm0.154$ IC769 58 Sb $0.877\pm0.072$ PCCS1 857 G276.80+67.94 183.26859 07.04378 $1.770\pm0.158$ NGC4180 73 Sb $1.582\pm0.114$ PCCS1 857 G268.39+73.70 183.44470 13.41055 $3.140\pm0.175$ NGC4189 89 Sc $2.687\pm0.196$ PCCS1 857 G265.42+74.97 183.46055 14.91470 $7.332\pm0.194$ M98 92 Sb $11.571\pm0.840$ PCCS1 857 G268.93+73.52 183.48450 13.17543 $1.610\pm0.144$ NGC4193 97 Sc $1.592\pm0.125$ PCCS1 857 G278.93+66.94 183.64987 05.81024 $1.749\pm0.129$ NGC4197 120 Scd $1.574\pm0.118$ PCCS1 857 G270.21+73.54 183.81944 13.01505 $1.942\pm0.151$ NGC4206 145 Sc $2.205\pm0.149$ PCCS1 857 G268.89+74.36 183.91888 13.90904 $4.512\pm0.168$ NGC4212 157 Sc $4.847\pm0.346$ PCCS1 857 G270.46+73.72 183.97418 13.13815 $8.083\pm0.188$ NGC4216 167 Sb $9.207\pm0.656$ PCCS1 857 G270.58+73.91 184.10005 13.29014 $1.586\pm0.188$ NGC4222 187 Scd $1.559\pm0.115$ PCCS1 857 G267.17+75.77 184.29755 15.33868 $2.645\pm0.129$ NGC4237 226 Sc $2.919\pm0.210$ PCCS1 857 G282.23+65.16 184.30274 03.67542 $0.878\pm0.129$ NGC4234 221 Sc $0.857\pm0.068$ PCCS1 857 G279.82+68.01 184.35336 06.68030 $0.814\pm0.149$ NGC4223 234 Sa $0.521\pm0.052$ PCCS1 857 G279.69+68.53 184.49064 07.18604 $1.055\pm0.124$ NGC4246 264 Sc $1.000\pm0.079$ PCCS1 857 G270.45+75.20 184.71895 14.42515 $20.680\pm0.251$ M99 307 Sc $24.229\pm1.720$ PCCS1 857 G282.51+66.97 184.98252 05.35636 $3.772\pm0.146$ NGC4273 382 Sc $3.776\pm0.277$ PCCS1 857 G280.56+69.18 185.01716 07.68557 $0.985\pm0.144$ NGC4276 393 Sc $0.651\pm0.059$ PCCS1 857 G284.34+65.49 185.24478 03.73035 $1.156\pm0.140$ NGC4289 449 Sbc $1.059\pm0.081$ PCCS1 857 G277.07+72.84 185.31732 11.50735 $1.937\pm0.153$ NGC4294 465 Sc $1.939\pm0.144$ PCCS1 857 G277.41+72.86 185.41840 11.48644 $1.176\pm0.160$ NGC4299 491 Scd $1.112\pm0.093$ PCCS1 857 G272.51+75.68 185.42477 14.59761 $8.296\pm0.168$ NGC4298/4302 483/497 Sc/Sc $12.573\pm0.651$ PCCS1 857 G284.35+66.28 185.47457 04.47981 $17.872\pm0.226$ M61 508 Sc $20.528\pm1.455$ PCCS1 857 G280.61+70.62 185.53208 09.03536 $2.147\pm0.136$ NGC4307 524 Sbc $1.757\pm0.127$ PCCS1 857 G282.47+68.84 185.55513 07.13962 $0.828\pm0.160$ NGC4309 534 Sa $0.603\pm0.060$ PCCS1 857 G284.61+66.39 185.60479 04.55969 $0.822\pm0.137$ NGC4301 552 Sc $0.454\pm0.049$ PCCS1 857 G271.42+76.60 185.63878 15.53121 $1.516\pm0.168$ NGC4312 559 Sab $1.532\pm0.113$ PCCS1 857 G277.71+73.24 185.65326 11.80437 $1.706\pm0.144$ NGC4313 570 Sab $1.586\pm0.119$ PCCS1 857 G280.71+70.96 185.67829 09.34033 $2.502\pm0.177$ NGC4316 576 Sbc $1.941\pm0.142$ PCCS1 857 G270.81+76.99 185.71130 15.93892 $1.098\pm0.200$ $\cdots$ $\cdots$ $\cdots$ $\cdots$ PCCS1 857 G271.17+76.89 185.73530 15.81529 $17.639\pm0.257$ M100 596 Sc $25.968\pm1.840$ PCCS1 857 G284.50+67.11 185.77029 05.26741 $0.933\pm0.146$ NGC4324 613 Sa $0.722\pm0.061$ PCCS1 857 G278.76+72.90 185.80843 11.36855 $1.634\pm0.131$ NGC4330 630 Sd $1.547\pm0.118$ PCCS1 857 G282.91+69.24 185.83429 07.47657 $1.941\pm0.177$ NGC4334 638 $1.736\pm0.127$ PCCS1 857 G283.52+68.77 185.90001 06.96069 $2.263\pm0.164$ NGC4343 656 Sb $1.517\pm0.113$ PCCS1 857 G282.53+70.30 186.03782 08.52197 $0.821\pm0.153$ NGC4356 713 Sc $0.539\pm0.047$ PCCS1 857 G283.94+69.33 186.21012 07.44889 $0.963\pm0.155$ NGC4370 758 S0 $0.734\pm0.057$ PCCS1 857 G286.12+66.93 186.33071 04.92117 $1.192\pm0.134$ NGC4378 785 Sa $1.298\pm0.105$ PCCS1 857 G285.56+67.74 186.33387 05.75341 $0.904\pm0.155$ NGC4376 787 Scd $0.506\pm0.044$ PCCS1 857 G281.96+71.82 186.35291 10.02117 $2.201\pm0.156$ NGC4380 792 Sab $2.001\pm0.149$ PCCS1 857 G284.72+69.17 186.43134 07.21570 $2.705\pm0.162$ UGC7513 827 Sc $2.171\pm0.157$ PCCS1 857 G279.14+74.33 186.45118 12.66357 $3.153\pm0.176$ NGC4388 836 Sab $3.272\pm0.234$ PCCS1 857 G281.83+72.27 186.46056 10.45480 $0.892\pm0.154$ NGC4390 849 Sbc $0.833\pm0.070$ PCCS1 857 G284.56+69.49 186.46772 07.53835 $0.860\pm0.162$ IC3322 851 Sc $0.785\pm0.061$ PCCS1 857 G287.41+65.54 186.48927 03.44221 $1.990\pm0.184$ UGC7522 859 Sc $1.167\pm0.087$ PCCS1 857 G274.37+77.10 186.49855 15.66868 $1.950\pm0.170$ NGC4396 865 Sc $1.823\pm0.135$ PCCS1 857 G278.75+74.77 186.51927 13.10641 $5.638\pm0.161$ NGC4402 873 Sc $5.606\pm0.396$ PCCS1 857 G279.83+74.35 186.63556 12.60895 $1.243\pm0.171$ NGC4407 912 Sbc $1.058\pm0.081$ PCCS1 857 G287.47+66.09 186.64485 03.96705 $1.378\pm0.142$ NGC4412 921 Sbc $1.040\pm0.079$ PCCS1 857 G284.86+69.93 186.69124 07.92894 $0.939\pm0.123$ NGC4416 938 Sc $1.011\pm0.081$ PCCS1 857 G284.05+70.86 186.69953 08.90471 $1.128\pm0.147$ UGC7546 939 Sc $1.115\pm0.108$ PCCS1 857 G276.41+76.63 186.72600 15.04733 $4.248\pm0.195$ NGC4419 958 Sa $3.090\pm0.220$ PCCS1 857 G285.63+69.32 186.78597 07.27420 $0.879\pm0.172$ UGC7557 975 Scd $0.692\pm0.069$ PCCS1 857 G283.86+71.40 186.79590 09.43363 $1.308\pm0.147$ NGC4424 979 Sa $0.931\pm0.073$ PCCS1 857 G286.66+67.97 186.79940 05.87693 $0.772\pm0.142$ NGC4423 971 Sd $0.509\pm0.043$ PCCS1 857 G286.52+68.35 186.84390 06.25747 $1.927\pm0.162$ NGC4430 1002 Sc $1.654\pm0.125$ PCCS1 857 G282.36+73.01 186.85696 11.11080 $0.904\pm0.137$ NGC4429 1003 S0/a $0.672\pm0.052$ ------------------------- ------------------------- ------------------------- ------------------ -------------- ---------- ---------- ------------------ ------------------------- ------------------------- ------------------------- ------------------ -------------- ----------- ------- ------------------ PCCS source $\alpha_{\text{J2000}}$ $\delta_{\text{J2000}}$ PCCS 857 GHz counterpart VCC type SPIRE 350 $\mu$m (deg) (deg) (Jy) (Jy) PCCS1 857 G280.30+74.83 186.92881 13.01208 $3.006\pm0.153$ NGC4438 1043 Sb $3.373\pm0.257$ PCCS1 857 G285.11+71.32 187.16366 09.24766 $1.104\pm0.138$ NGC4451 1118 Sc $0.847\pm0.068$ PCCS1 857 G289.14+65.84 187.25201 03.57333 $1.763\pm0.135$ NGC4457 1145 Sb $1.600\pm0.116$ PCCS1 857 G286.11+70.88 187.35918 08.73663 $1.198\pm0.150$ NGC4469 1190 Sa $0.788\pm0.061$ PCCS1 857 G287.55+69.00 187.36934 06.79160 $0.778\pm0.143$ IC3414 1189 Sc $0.367\pm0.036$ PCCS1 857 G286.91+70.02 187.39922 07.83258 $1.031\pm0.142$ NGC4470 1205 Sc $0.977\pm0.076$ PCCS1 857 G289.65+66.56 187.60654 04.23901 $1.607\pm0.136$ NGC4480 1290 Sb $1.394\pm0.104$ PCCS1 857 G283.74+74.48 187.69562 12.38577 $1.041\pm0.169$ M87 1316 E $0.941\pm0.074$ PCCS1 857 G287.69+70.33 187.73649 08.07343 $0.961\pm0.145$ NGC4492 1330 Sa $0.597\pm0.054$ PCCS1 857 G290.53+66.33 187.90427 03.93704 $4.098\pm0.174$ NGC4496 1375 Sc $3.641\pm0.260$ PCCS1 857 G282.29+76.51 187.99116 14.43010 $18.381\pm0.246$ M88 1401 Sbc $22.165\pm1.564$ PCCS1 857 G283.78+75.59 188.04652 13.44229 $0.775\pm0.141$ NGC4506 1419 Sa $0.178\pm0.026$ PCCS1 857 G283.51+76.21 188.17680 14.05463 $1.165\pm0.142$ IC3476 1450 Sc $1.085\pm0.092$ PCCS1 857 G289.13+71.03 188.36439 08.64580 $2.403\pm0.152$ NGC4519 1508 Sc $2.268\pm0.170$ PCCS1 857 G288.89+71.57 188.40485 09.19277 $1.873\pm0.145$ NGC4522 1516 Sc $1.434\pm0.107$ PCCS1 857 G290.13+70.13 188.50485 07.69646 $3.638\pm0.163$ NGC4526 1535 S0 $2.857\pm0.207$ PCCS1 857 G286.15+75.39 188.57616 13.08787 $1.077\pm0.167$ NGC4531 1552 Sa $0.590\pm0.051$ PCCS1 857 G290.06+70.64 188.58279 08.20291 $8.869\pm0.185$ NGC4535 1555 Sc $14.235\pm1.024$ PCCS1 857 G291.04+68.93 188.58782 06.46346 $3.274\pm0.166$ NGC4532 1554 Sm $2.893\pm0.206$ PCCS1 857 G290.87+69.64 188.65963 07.16808 $0.886\pm0.142$ IC3521 1575 Sm $0.631\pm0.055$ PCCS1 857 G285.72+76.81 188.86411 14.47983 $4.691\pm0.173$ M91 1615 Sb $6.128\pm0.482$ PCCS1 857 G289.78+73.72 189.13379 11.23932 $15.958\pm0.216$ NGC4567/4568 1673/1676 Sc/Sc $15.196\pm1.281$ PCCS1 857 G288.41+75.62 189.19517 13.17062 $7.469\pm0.162$ M90 1690 Sab $8.169\pm0.581$ PCCS1 857 G287.53+76.64 189.23359 14.21160 $3.749\pm0.197$ NGC4571 1696 Sab $3.597\pm0.275$ PCCS1 857 G290.38+74.34 189.42769 11.81137 $6.354\pm0.189$ M58 1727 Sab $8.286\pm0.616$ ------------------------- ------------------------- ------------------------- ------------------ -------------- ----------- ------- ------------------ The dominant emission mechanism at 857 GHz (350 $\mu$m) is thermal dust emission. For realistic dust temperatures, 857 GHz is already significantly down the Rayleigh-Jeans tail of the blackbody spectrum, so we primarily expect emission by large amounts of cold dust, rather than warm dust emission linked to star formation . This frequency is still too high for free-free emission or synchrotron emission to start playing a significant role. On the other hand, thanks to the negative K-correction, galaxies at intermediate or high redshift can have the peak of their SED around 857 GHz or even lower frequencies, and star-forming galaxies at $z>0.5$ form a substantial contribution to the galaxy population at the frequency of 857 GHz . Strongly lensed ultra-luminous infrared galaxies at redshifts $z>1$ could also contribute to the population [@2007MNRAS.377.1557N; @2012ApJ...755...46L]. So we expect that the counterparts of the Planck sources at 857 GHz would be dominated by nearby galaxies, with a potential contribution of intermediate to high redshift galaxies (in particular if they tend to cluster within the relatively large Planck beam). We used the online Planck Legacy Archive (PLA) tool to select from the PCCS 857 GHz catalogue those sources located in the extended HeViCS survey field. The search resulted in 84 compact Planck sources. In Table [\[857.tab\]]{} we list each of these sources, with the central positions and flux densities as listed in the PCCS. Subsequently, we searched for optical and FIR/submm counterparts for each of these sources, using a simple query in the NASA/IPAC Extragalactic Database (NED) for sources within the 857 GHz beam FWHM and a search in the HeViCS Herschel maps. For 82 of the 84 sources detected at 857 GHz, we could immediately match the Planck source to an optical and Herschel counterpart. Examples of such identifications of Planck sources are shown on Figure [\[DetectedSources.pdf\]]{}. ![image](DetectedSources-1.pdf){width="\textwidth"}\ ![image](DetectedSources-2.pdf){width="\textwidth"}\ ![image](DetectedSources-3.pdf){width="\textwidth"}\ ![image](DetectedSources-4.pdf){width="\textwidth"} In 80 cases the counterpart was a single nearby galaxy (top two rows of Figure [\[DetectedSources.pdf\]]{}), and 77 of these belong to the Virgo Cluster. The three remaining ones are the background galaxies IC3029, NGC4246 and NGC4334, at distances of about 98, 56 and 66 Mpc respectively [@2005ApJS..160..149S; @2009ApJS..182..474S; @2000ApJ...529..786M]. The majority of these 80 nearby galaxies are late-type spiral galaxies, with the Sc class accounting for slightly more than 40% of all the Planck detected galaxies. Only three galaxies (NGC4370, 4429 and 4526) are classified as lenticular in the Galaxy On Line Database Milano Network , and the first of these three is sometimes classified as Sa [e.g., @1991rc3..book.....D]. The only elliptical galaxy detected at 857 GHz is M87, the massive radio galaxy at the centre of the Virgo Cluster. Whether or not this galaxy has a substantial interstellar dust reservoir detectable at FIR/submm wavelengths is still a matter of debate, even with the Herschel data at hand . It is clear, however, that the emission at frequencies of 857 GHz and lower is dominated by synchrotron emission that goes as $S_\nu \propto \nu^\alpha$ with a spectral index $\alpha\approx-0.75$ . Two 857 GHz PCCS sources, PCCS1 857 G272.51+75.68 and G289.78+73.72, correspond to a galaxy pair rather than a single bright galaxy (bottom two rows of Figure [\[DetectedSources.pdf\]]{}). The former corresponds to the close pair KPG332, formed by the edge-on galaxy NGC4302 and the inclined spiral NGC4298, with a projected separation of 2.4 arcmin (recall that the Planck’s beam size at 857 GHz is 4.33 arcmin). The position of the Planck source is centered between the two galaxies, closer to NGC4302, which is the most luminous of both sources at 350 $\mu$m [@2012MNRAS.419.3505D; @2013MNRAS.428.1880A]. The latter Planck source PCCS1 857 G289.78+73.72 corresponds to the close pair VV219, also known as the Siamese Twins. It consists of the two spiral galaxies NGC4567 and 4568, with a separation of 1.3 arcmin. Here the position of the Planck source is close to the centre of NGC4568, which is significantly more luminous than its companion at 350 $\mu$m [@2012MNRAS.419.3505D; @2013MNRAS.428.1880A]. The remaining two Planck detections at 857 GHz, PCCS1 857 G261.44+74.24 and G270.81+76.99, do not have any nearby galaxy as optical counterpart. Their nature will be discussed in Section [\[NoCounterpart.sec\]]{}. 545 GHz sources {#ghz-sources} --------------- ------------------------- ------------------------- ------------------------- ----------------- -------------- ----------- ---------- ------------------ PCCS source $\alpha_{\text{J2000}}$ $\delta_{\text{J2000}}$ PCCS 545 GHz counterpart VCC type SPIRE 500 $\mu$m (deg) (deg) (Jy) (Jy) PCCS1 545 G260.37+75.43 182.66285 16.04584 $0.922\pm0.114$ NGC4152 25 Sc $0.735\pm0.061$ PCCS1 545 G276.77+67.93 183.25488 07.03960 $0.533\pm0.104$ NGC4180 73 Sb $0.562\pm0.044$ PCCS1 545 G268.38+73.71 183.44879 13.41927 $0.498\pm0.097$ NGC4189 89 Sc $0.911\pm0.072$ PCCS1 545 G265.43+74.97 183.46130 14.91524 $2.640\pm0.107$ M98 92 Sb $4.301\pm0.318$ PCCS1 545 G268.93+73.49 183.47034 13.15472 $0.548\pm0.111$ NGC4193 97 Sc $0.646\pm0.060$ PCCS1 545 G278.92+66.95 183.64901 05.81566 $0.654\pm0.111$ NGC4197 120 Scd $0.670\pm0.054$ PCCS1 545 G278.07+68.11 183.77733 07.01443 $0.493\pm0.094$ $\cdots$ $\cdots$ $\cdots$ $\cdots$ PCCS1 545 G270.22+73.54 183.81908 13.01133 $0.838\pm0.103$ NGC4206 145 Sc $0.954\pm0.076$ PCCS1 545 G268.88+74.36 183.91488 13.90592 $0.148\pm0.106$ NGC4212 157 Sc $1.650\pm0.123$ PCCS1 545 G270.44+73.74 183.97938 13.16155 $2.269\pm0.099$ NGC4216 167 Sb $3.454\pm0.252$ PCCS1 545 G270.53+73.94 184.09985 13.31559 $0.814\pm0.103$ NGC4222 187 Scd $0.694\pm0.057$ PCCS1 545 G267.22+75.46 184.13517 15.07150 $0.559\pm0.105$ $\cdots$ $\cdots$ $\cdots$ $\cdots$ PCCS1 545 G267.21+75.76 184.29853 15.32441 $0.807\pm0.105$ NGC4237 226 Sc $0.994\pm0.077$ PCCS1 545 G270.36+75.19 184.69368 14.43079 $4.538\pm0.116$ M99 307 Sc $7.927\pm0.564$ PCCS1 545 G282.53+66.97 184.99044 05.36161 $0.942\pm0.109$ NGC4273 382 Sc $1.368\pm0.108$ PCCS1 545 G277.04+72.84 185.30844 11.50868 $0.707\pm0.139$ NGC4294 465 Sc $0.805\pm0.064$ PCCS1 545 G272.43+75.68 185.41151 14.61110 $2.796\pm0.120$ NGC4298/4302 483/497 Sc/Sc $4.454\pm0.325$ PCCS1 545 G284.36+66.28 185.47873 04.48218 $4.789\pm0.116$ M61 508 Sc $7.050\pm0.502$ PCCS1 545 G280.54+70.63 185.50988 09.04869 $0.569\pm0.114$ NGC4307 524 Sbc $0.623\pm0.050$ PCCS1 545 G271.16+76.88 185.72922 15.81079 $4.626\pm0.114$ M100 596 Sc $8.825\pm0.638$ PCCS1 545 G278.76+72.88 185.80236 11.35158 $0.523\pm0.100$ NGC4330 630 Sd $0.635\pm0.053$ PCCS1 545 G281.96+71.79 186.34195 09.99236 $0.708\pm0.115$ NGC4380 792 Sab $0.709\pm0.059$ PCCS1 545 G284.73+69.17 186.43614 07.21787 $1.125\pm0.129$ UGC7513 827 Sc $0.908\pm0.071$ PCCS1 545 G279.13+74.34 186.45220 12.66788 $0.964\pm0.111$ NGC4388 836 Sab $1.218\pm0.093$ PCCS1 545 G274.40+77.10 186.50366 15.65977 $0.640\pm0.114$ NGC4396 865 Sc $0.771\pm0.062$ PCCS1 545 G287.46+65.53 186.50593 03.43199 $0.664\pm0.133$ UGC7522 859 Sc $0.526\pm0.045$ PCCS1 545 G278.76+74.77 186.52223 13.10600 $1.747\pm0.116$ NGC4402 873 Sc $1.951\pm0.142$ PCCS1 545 G284.05+70.82 186.68484 08.86340 $0.655\pm0.114$ UGC7546 939 Sc $0.538\pm0.060$ PCCS1 545 G276.42+76.65 186.73352 15.05677 $0.834\pm0.125$ NGC4419 958 Sa $1.074\pm0.934$ PCCS1 545 G286.60+68.36 186.87344 06.25631 $0.664\pm0.103$ NGC4430 1002 Sc $0.599\pm0.052$ PCCS1 545 G280.36+74.81 186.93803 12.98807 $1.026\pm0.134$ NGC4438 1043 Sb $1.208\pm0.110$ PCCS1 545 G289.65+66.58 187.61114 04.25924 $0.503\pm0.097$ NGC4480 1290 Sb $0.551\pm0.045$ PCCS1 545 G283.76+74.47 187.69775 12.37910 $1.639\pm0.096$ M87 1316 E $1.244\pm0.094$ PCCS1 545 G290.54+66.32 187.90694 03.92769 $1.329\pm0.119$ NGC4496 1375 Sc $1.423\pm0.106$ PCCS1 545 G282.29+76.51 187.99024 14.42921 $5.887\pm0.120$ M88 1401 Sbc $7.886\pm0.558$ PCCS1 545 G287.49+72.83 188.27660 10.51415 $0.548\pm0.112$ $\cdots$ $\cdots$ $\cdots$ $\cdots$ PCCS1 545 G289.12+71.03 188.35920 08.65074 $0.877\pm0.109$ NGC4519 1508 Sc $0.907\pm0.075$ PCCS1 545 G288.90+71.54 188.40228 09.16123 $0.537\pm0.090$ NGC4522 1516 Sbc $0.539\pm0.044$ PCCS1 545 G290.14+70.13 188.50820 07.69494 $0.780\pm0.103$ NGC4526 1535 S0 $0.929\pm0.071$ PCCS1 545 G289.64+71.15 188.55205 08.73088 $0.612\pm0.122$ $\cdots$ $\cdots$ $\cdots$ $\cdots$ PCCS1 545 G291.04+68.92 188.58393 06.44702 $0.827\pm0.088$ NGC4532 1554 Sm $1.099\pm0.081$ PCCS1 545 G290.09+70.63 188.58932 08.18835 $2.967\pm0.110$ NGC4535 1555 Sb $5.387\pm0.408$ PCCS1 545 G285.69+76.83 188.86145 14.49705 $1.525\pm0.096$ M91 1615 Sb $2.065\pm0.154$ PCCS1 545 G286.36+76.69 188.97390 14.32142 $0.568\pm0.115$ $\cdots$ $\cdots$ $\cdots$ $\cdots$ PCCS1 545 G289.82+73.73 189.14824 11.24395 $4.231\pm0.113$ NGC4567/4568 1673/1676 Sc/Sc $5.197\pm0.447$ PCCS1 545 G288.44+75.62 189.20407 13.16827 $2.241\pm0.106$ M90 1690 Sab $2.849\pm0.210$ PCCS1 545 G287.49+76.66 189.22770 14.22749 $0.947\pm0.130$ NGC4571 1696 Sc $1.182\pm0.103$ PCCS1 545 G290.37+74.35 189.42504 11.82011 $1.887\pm0.100$ M58 1727 Sab $2.877\pm0.218$ ------------------------- ------------------------- ------------------------- ----------------- -------------- ----------- ---------- ------------------ The dominant emission mechanism at 545 GHz (550 $\mu$m) is still thermal emission by cool, large dust. Due to the steep spectral slope of a modified blackbody at submm wavelengths, $F_\nu \propto \nu^{2+\beta}$ with $\beta\sim1.5-2$ the emissivity index , the fluxes of nearby dusty galaxies at 545 GHz are typically a factor 6 lower compared to 857 GHz, and we therefore expect a correspondingly lower number of HFI detections corresponding to local dusty galaxies. On the other hand, the negative K-correction is stronger at 545 GHz and the relative contribution of intermediate and high redshift galaxies to the population counts is expected to be larger. Querying the Planck PCCS catalogue through the PLA tool, we found 48 compact Planck sources within the extended HeViCS fields at 545 GHz. The list of sources can be found in Table [\[545.tab\]]{}. The characterization of the sample is similar as for the 857 GHz sample: the vast majority (43 out of 48) of the Planck sources can be clearly identified with nearby galaxies. All of the 43 sources with clear optical counterparts are in common with the 857 GHz list, including the two galaxy pairs NGC4298/4302 and NGC4567/4568. All of the 545 GHz detected galaxies belong to the Virgo Cluster, i.e. the three background galaxies detected at 857 GHz are not detected anymore at 545 GHz. Concerning the type of the galaxies detected, we note that the relative number of late-type galaxies has even increased compared to the 857 GHz detected population: more than half of all the galaxies detected at 545 GHz are of the Sc type. Remarkably fewer early-type spiral galaxies are detected: of the 10 Sa galaxies detected at 857 GHz, only one single system, NGC4419, is also detected at 545 GHz. This galaxy at the centre of the Virgo Cluster is classified as a galaxy ’at the end of the Sa sequence’ [@1994cag..book.....S], and is characterized by an unusually high ratio between molecular versus atomic gas [@1990ApJ...353..460K]. Concerning the early-type galaxy population, we are left with only two galaxies: the synchrotron-dominated elliptical M87, and the lenticular galaxy NGC4526, known to host a prominent dust disc [@2006ApJS..164..334F; @2006MNRAS.366.1151S; @2009AJ....137.3053Y]. The PCCS contains five 545 GHz sources in the extended HeViCS fields without an obvious optical counterpart. Their nature will be discussed in Section [\[NoCounterpart.sec\]]{}. Analysis and discussion ======================= Flux density comparison {#FluxComparison.sec} ----------------------- ![image](FluxComparison857.pdf){width="60.00000%"} Frequency Flux density $a$ $b$ $\chi^2_{\text{red}}$ ----------- -------------- ------- -------- ----------------------- 857 GHz APERFLUX 0.944 0.0828 0.852 DETFLUX 0.833 0.0953 1.643 PSFFLUX 0.927 0.0723 1.778 GAUFLUX 0.852 0.271 4.314 545 GHz APERFLUX 0.884 0.0656 1.194 DETFLUX 0.792 0.0262 1.431 PSFFLUX 0.864 0.0483 1.968 GAUFLUX 0.815 0.165 2.950 : Results of the linear regression fits of the SPIRE versus Planck flux densities. $a$ and $b$ represent the slope and intercept of the linear relation in the form $\log F_{\text{Planck}} = a \log F_{\text{SPIRE}} + b$. The last column gives the reduced $\chi^2$ value of the fit. \[FluxComparison.tab\] The Planck 857 GHz and SPIRE 350 $\mu$m filters were designed to have virtually the same transmission curve, with nearly identical central wavelength and bandwidth. Therefore, we can immediately compare the PCCS 857 GHz fluxes to the integrated SPIRE 350 $\mu$m fluxes, without significant color corrections. SPIRE flux densities, both at 350 and 500 $\mu$m, are based on the results from @2013MNRAS.428.1880A, who performed automated flux density determination of all Virgo Cluster VCC galaxies in the HeViCS fields. Their flux density determination method consists of several steps, including an advanced determination of the background, an iterative determination of the ideal aperture within which the flux is determined, and an elaborate determination of the uncertainty, taking into account contributions from calibration uncertainty, aperture uncertainty and zero-point uncertainty. The flux densities are presented in their Table B1. We have applied a number of additional corrections to these values. First, the fluxes were still based on the “old” SPIRE beam sizes, whereas these have been updated since. Secondly, @2013MNRAS.428.1880A did not contain a correction for extended fluxes, known as the $K_{4E}/K_{4P}$ correction. And finally, the new SPIRE calibration, based on Neptune as a primary calibration source [@2013MNRAS.433.3062B], has been taken into account. The final correction factor comes down to a multiplication by a factor 0.90 and 0.87 at 350 and 500 $\mu$m, respectively. The final integrated flux densities we use for our comparison are listed in the last column of Tables [\[857.tab\]]{} and [\[545.tab\]]{} for 350 and 500 $\mu$m respectively. As mentioned in Section [\[857.sec\]]{}, three galaxies detected at 857 GHz (IC3029, NGC4246 and NGC4334) do not belong to the Virgo Cluster (even though they all have a number in the VCC). These background galaxies were not considered in Table B1 of @2013MNRAS.428.1880A. For these galaxies, we have measured the SPIRE flux densities from the HeViCS maps using exactly the same technique. Each source in the PCCS has four different measures of the flux density, depending on the source detection algorithm. The detection pipeline flux density (DETFLUX) is directly obtained from the filtered maps and assumes that the sources are pointlike. The three other measures are estimated from the full-sky maps at the positions of the sources. The aperture photometry flux density (APERFLUX) is estimated by integrating the data in a circular aperture centred at the position of the source, with the average FWHM of the effective beam as radius for the aperture. The PSF fit photometry (PSFFLUX) is obtained by fitting a model of the PSF at the position of the source [@2011ApJS..193....5M]. Finally, the Gaussian fit photometry flux density (GAUFLUX) is obtained by fitting a Gaussian model, centred at the position of the source, in which the size, shape and background offset are allowed to vary. @2013arXiv1303.5088P present a comparison between the different measures for the Planck PCCS 857 GHz flux densities and the SPIRE 350 $\mu$m flux densities for a set of galaxies from four different catalogues[^2]. They found that, at low flux densities, the smallest dispersion between the SPIRE and Planck flux densities is achieved by the DETFLUX photometry because the filtering process removes structure not associated with compact sources. At the highest levels, corresponding to the most extended galaxies, the flux densities are underestimated by DETFLUX, APERFLUX and PSFFLUX. GAUFLUX accounts for the size of the source and is therefore able to estimate the flux density correctly. The general recommendation from the PCCS is that the appropriate photometry to be used depends on the nature of the source, with GAUFLUX to be used for the brightest and most resolved sources. A similar conclusion was obtained by . In Figure [\[FluxComparison857.pdf\]]{} we show the comparison between the Herschel SPIRE 350 $\mu$m flux density and the Planck PCCS 857 GHz flux density, for each of the four different methods used in the PCCS to measure flux densities. The solid blue lines in the different panels correspond to a one-to-one correlation, the solid green lines represent a linear regression fit. These fits were made using the IDL MPFITEXY tool [@2010MNRAS.409.1330W], based on the non-linear least-squares fitting package MPFIT [@2009ASPC..411..251M], and the parameters of these fits can be found in Table [\[FluxComparison.tab\]]{}. It is immediately obvious that the GAUFLUX is not an appropriate choice: only the highest flux densities at $S_{350}\gtrsim10$ Jy are recovered fairly reliably, whereas lower flux densities are typically strongly overestimated. Somewhat surprisingly, the DETFLUX flux densities, which showed the smallest dispersion in the comparison of @2013arXiv1303.5088P, also show systematic deviations from the SPIRE flux densities. At high levels, the flux densities are systematically underestimated, as also found by @2013arXiv1303.5088P. Also at the lowest flux density levels, however, the DETFLUX measure shows a systematic overestimation compared to the SPIRE flux densities. This might be due to the flux boosting also noted by for faint ERCSC sources in the H-ATLAS fields. We find the smallest scatter between the Planck and SPIRE flux densities for the APERFLUX flux density estimate. The flux densities agree very well, even up to the highest values where the DETFLUX estimate breaks down. The reduced $\chi^2$ of this correlation is 0.852, which indicates that the error bars on the APERFLUX flux densities might be slightly overestimated. The PSFFLUX flux densities are a close second best, although the scatter at the lowest flux densities is a bit larger. Comparing Planck 545 GHz and SPIRE 500 $\mu$m fluxes is less obvious, as the former filter has a significantly longer central wavelength (550 $\mu$m). In order to make a comparison possible, we converted the observed SPIRE 500 $\mu$m fluxes by multiplying it with a conversion factor. The FIR/submm SEDs of nearby galaxies in the Virgo Cluster can be approximated by a modified blackbody, $F_\nu \propto \nu^\beta\,B_\nu(T)$, with a temperature of $T\approx20$ K and $\beta\approx2$ [@2012MNRAS.419.3505D; @2013MNRAS.428.1880A]. Taking into account the color corrections of the SPIRE 500 $\mu$m and Planck HFI 545 GHz bands, a color correction of 0.83 has to be applied to the SPIRE flux densities. A similar conversion factor (0.87) was obtained by , based on a modified blackbody SED with $\beta=1.5$. Only for M87, for which the FIR/submm emission is completely dominated by synchrotron emission rather than by thermal dust emission , we applied a different conversion term. For a power-law synchrotron spectrum with $\alpha=-0.75$, we find a conversion term of 1.07. ![image](FluxComparison545.pdf){width="60.00000%"} The comparison between the observed Planck 550 $\mu$m flux densities and the converted SPIRE 500 $\mu$m flux densities is shown in Figure [\[FluxComparison545.pdf\]]{}, where each panel again corresponds to the different measures for the flux densities in the PCCS. The results of the linear regression fits can be found in Table [\[FluxComparison.tab\]]{}. In general, we find the same behavior as for the comparison of the Planck 857 GHz and SPIRE 350 $\mu$m flux densities. For the highest flux density levels, DETFLUX and PSFFLUX underestimate the true flux densities, whereas GAUFLUX and APERFLUX do not suffer from this effect. Except for the brightest sources, the GAUFLUX estimates are the worst, with an exceptionally spurious flux density estimate more than an order of magnitude off for the otherwise unremarkable Sc galaxy NGC4571. In general, the APERFLUX flux density estimates show least dispersion compared to the SPIRE flux densities. In this case, the PSFFLUX flux densities tend to underestimate the true flux densities significantly at the lowest levels. Modified blackbody fits {#DustProperties.sec} ----------------------- The FIR/submm traced by Herschel and Planck is dominated by the emission from cold dust. The physical properties of the dust can in principle be determined by fitting SED models to the observed flux densities. While complex multi-component methods have been developed that take into account the entire range from mid-infrared to mm wavelengths , the most common approach is a simple modified blackbody fit. This approach enables to determine the most fundamental properties of the dust medium: the mean dust temperature and the amount of dust. In the Virgo Cluster, modified blackbody fits have been applied to the FIR/submm SED in several different studies: @2012MNRAS.419.3505D fitted modified blackbodies to a sample of SPIRE 500 $\mu$m selected galaxies, @2013MNRAS.428.1880A and @2013arXiv1311.1774D used them to study the dust characteristics of a samples of optically selected galaxies, and , and @2013MNRAS.436.1057D applied modified blackbody fits to smaller samples of star-forming dwarfs, early-type galaxies and transition-type galaxies, respectively. All of these studies relied on Herschel flux densities. One important question is whether the availability of additional Planck flux densities at 857 and 545 GHz substantially modifies the derived parameters. In order to test this, we considered all galaxies in the HeViCS fields detected at 545 GHz, and constructed FIR/submm SEDs by combining the Planck HFI with Herschel/SPIRE observations at 250, 350 and 500 $\mu$m, Herschel/PACS data at 100 and 160 $\mu$m. The PACS and SPIRE fluxes and corresponding error bars were taken from @2013MNRAS.428.1880A, where an automatic flux density measurement was applied to the five HeViCS maps for all galaxies from the Virgo Cluster Catalogue [@1985AJ.....90.1681B]. Only M87 was omitted from the sample as its SED is dominated by synchrotron emission rather than cold dust emission. We fitted the observed broadband fluxes using a simple modified blackbody function, i.e.$$F_\nu = \frac{M_{\text{d}}}{D^2}\,\kappa_\nu\,B_\nu(T_{\text{d}})$$ with $M_{\text{d}}$ the dust mass, $D$ the distance to the galaxy, $T_{\text{d}}$ the dust temperature and $\kappa_\nu$ the emissivity. We used a common distance of 16.5 Mpc to all galaxies [@2007ApJ...655..144M], and, as customary, we assume a power-law dust emissivity in the FIR/submm wavelength range, i.e. $\kappa_\nu \propto \nu^\beta$, where we fixed $\beta$ to 1.8 , and the zero-point to $\kappa_\nu = 0.192$ m$^2$kg$^{-1}$ at 350 $\mu$m . Note that both the value of the emissivity index $\beta$ and the absolute calibration of the emissivity are notoriously uncertain; actually, the quoted value of the zeropoint at $\lambda = 350$ $\mu$m has been derived assuming a dust model with $\beta=2$, so it would in principle not be applicable to models with a different value of $\beta$ . However, the goal of the present exercise is not to determine absolute dust masses, but rather to compare SED fits with and without Planck data, so the absolute normalization of the emissivity (and the distance to the galaxies) is of less importance in our case. The two remaining free parameters in our fitting routine are the dust mass and the dust temperature. The fits were done by performing a $\chi^2$ minimization using a simple gradient search method. Error bars on the derived parameters were derived using a Monte Carlo bootstrapping method. ![image](CompareTemperature.pdf){width="35.00000%" height="35.00000%"} ![image](CompareDustMass.pdf){width="35.00000%" height="35.00000%"} Fitting modified blackbody fits to the PACS, SPIRE and HFI APERFLUX data, we find a mean temperature $\langle T_{\text{d}}\rangle = (21.3\pm3.2)$ K and dust mass $\langle\log M_{\text{d}}\rangle = 7.10\pm0.17$. These values are consistent with previous studies of the dust properties in late-type galaxies in the Virgo Cluster [@2012MNRAS.419.3505D; @2013arXiv1311.1774D; @2013MNRAS.428.1880A], especially if we take into account that sometimes other assumptions have been made on the value of $\beta$ and the distances. More important here than the absolute values, however, is whether the inclusion of Planck data has a systematic effect on the derived dust characteristics. Figure [\[ComparePlanck.pdf\]]{} compares the dust mass and temperature for all galaxies in our sample, fitted with and without the Planck 857 and 545 GHz flux densities. The consistency between both fits is convincing: for every individual galaxy in our sample, the fitted modified blackbody parameters are fully consistent within the error bars. This implies that the addition or omission of the Planck 857 and 545 GHz flux densities does not imply a systematic bias, which is not unexpected given the good agreement between the SPIRE and Planck HFI flux densities (Section \[FluxComparison.sec\]). Note that we have, at this stage, limited ourselves to the 857 and 545 GHz data, whereas Planck covers wavelengths up to the mm range. Our conclusion that the addition of 857 and 545 GHz data to the Herschel PACS and SPIRE data points does not affect the derived dust parameters does not necessarily imply that the same applies when longer wavelength Planck data are also taken into account. In particular, there has been quite some evidence for a submm/mm excess beyond a simple modified blackbody, in particular for dwarf galaxies . A study of the population of Planck sources detected at longer wavelengths, including their extended spectral energy distribution, the implied dust properties and a possible submm/mm excess, will be considered in a separate HeViCS study. Nature of the sources without counterparts {#NoCounterpart.sec} ------------------------------------------ The PCCS contains two 857 GHz sources and five 545 GHz sources without an obvious nearby galaxy as counterpart. These Planck sources are particularly interesting as their submm flux density might originate from the common contribution of intermediate redshift galaxies or high-redshift ultra-luminous infrared galaxies. An example of such a compact 857 GHz Planck source is PLCKERC857 G270.59+58.52, linked to the strongly lensed hyperluminous starburst galaxy H-ATLAS J114637.9+001132 [@2012ApJ...753..134F]. ![image](UndetectedSources-1.pdf){width="\textwidth"}\ ![image](UndetectedSources-2.pdf){width="\textwidth"}\ ![image](UndetectedSources-3.pdf){width="\textwidth"}\ ![image](UndetectedSources-4.pdf){width="\textwidth"} ![image](HeViCS-map.pdf){width="1.2\columnwidth"} Within the beam area around the position of the first unidentified 857 GHz source in the HeViCS fields, PCCS1 857 G261.44+74.24, NED lists 55 SDSS galaxies and one NVSS radio source. All of these galaxies are very dim at optical wavelengths: one galaxy has a $g$-band magnitude of 15.8, the others all are fainter than $m_g = 17.7$ mag. The top panel of Figure [\[UndetectedSources.pdf\]]{} compares the HeViCS SPIRE 350 $\mu$m image of the region around PCCS1 857 G261.44+74.24 with an optical SDSS image, near- and mid-infrared images from WISE and a radio continuum image from FIRST. The SPIRE image contains a number of discrete sources, all of them with flux densities below 75 mJy. Only one of the sources, with $S_{350}\sim40$ mJy, has an obvious galaxy as counterpart in the SDSS map, the $m_g=17.7$ mag galaxy SDSS J120835+145750 at redshift $z=0.083$. This Planck source is located at the edge of a prominent cirrus feature that extends from the centre of the cluster in the northwestern direction (Figure [\[HeViCS-map.pdf\]]{}), and it is marked as an extended source in the PCCS, i.e. it has the EXTENDED flag equal to one. Putting all this information together, it seems logical that the Planck detection corresponds to a Galactic cirrus feature (with potentially a non-negligible contribution from background galaxies). The second unidentified 857 GHz PCCS source, PCCS1 857 G270.81+76.99 (Figure [\[UndetectedSources.pdf\]]{}, second row), is located in a region less affected by Galactic cirrus and is not marked as an extended source in the PCCS. NED lists 37 sources with the beam centered around the position of the source, most of them SDSS galaxies fainter than $m_g = 19$ mag. This might hence be a more promising candidate for an intermediate or high-redshift counterpart. However, the source is located very close ($<3$ arcmin) to M100, with a flux density of almost 26 Jy the brightest source in the Virgo Cluster at 350 $\mu$m [@2013MNRAS.428.1880A]. As there are no SPIRE 350 $\mu$m sources with flux densities above 50 mJy within the Planck beam at this position (second row on Figure [\[UndetectedSources.pdf\]]{}), we presume that this source is a spurious detection. The five 545 GHz PCCS sources without obvious optical counterpart are different from the two 857 GHz sources without optical counterpart. Looking at the position of these sources on the HeViCS map (Figure [\[HeViCS-map.pdf\]]{}), it seems obvious for at least two sources (PCCS1 545 G287.49+72.83 and G286.36+76.69) that they are associated with Galactic cirrus emission, as they are located on top of the notorious Galactic cirrus ring that surrounds the central regions of the Virgo Cluster [e.g., @2005ApJ...631L..41M; @2010ApJ...720..569R; @2010MNRAS.403L..26C]. Also for the other sources, we expect them to be dominated by cirrus emission. In all cases, the corresponding fields in the HeViCS SPIRE maps show a collection of faint sources, none of them bright enough to contribute a substantial part of the submm flux density, and most of them not associated to obvious sources in the corresponding SDSS optical maps (bottom two rows of Figure [\[UndetectedSources.pdf\]]{}). Altogether, the number of PCCS sources without optical counterpart is very modest, especially at 857 GHz (2 out of 84). This is particularly striking when we compare these numbers to the results from the equatorial fields of the H-ATLAS survey by . Of the 28 Planck sources they found at 857 GHz, only 11 correspond to bright low-redshift galaxies (one of them a pair of nearby galaxies) and as many as 17 have no nearby galaxy as counterpart. Of these 17 unidentified sources, one source is resolved into a condensation of high-redshift point sources clustered around a strongly lensed galaxy, while the remaining 16 are probably related to Galactic cirrus features. A similar situation, but with fewer detected sources, is seen at 545 GHz. Even though it is logical that we detect many more nearby galaxies compared to (as we are looking at a nearby cluster), the number of sources without optical counterpart remains modest. Based on the number of sources and the area of the H-ATLAS equatorial fields (134.55 deg$^2$), we would expect 10 to 11 sources at 857 GHz without obvious optical counterpart. The obvious reason that explains this difference is the high Galactic latitude ($b=67-76$ deg) of the HeViCS fields compared to the equatorial H-ATLAS fields. In particular, all but one of the Planck detections by are located in the GAMA-09 field, located at a Galactic latitude around 30 deg and known to be strongly contaminated by cirrus [@2011MNRAS.412.1151B]. In spite of the clear signatures of cirrus in the HeViCS fields (see Figure [\[HeViCS-map.pdf\]]{}), this difference in Galactic latitude is probably responsible for the much lower number of Planck cirrus features. An additional factor could have been the difference between the Planck catalogues used for both studies: we use the PCCS, whereas use the ERCSC. Compared to the ERCSC, the PCCS uses a different extraction method that is better at excluding slightly extended objects. Finally, we did not find any Planck source in the HeViCS fields that corresponds to a proto-cluster of intermediate or high-redshift dusty galaxies or strongly lensed submm sources. Based on the extrapolation of the results, at most one such source would be expected. One factor to take into account is again the source extraction method used by PCCS, which is better at excluding slightly extended objects (not only cirrus, but also high-redshift clumps). We can also attempt to make an estimate based on theoretical studies whether it is expected that no high-redshift Planck sources are detected in the HeViCS fields. @2005MNRAS.358..869N made predictions on the surface density of proto-clusters of dusty galaxies detectable by Planck. For the evolution indicated by simulations, they expect a surface density at the Planck detection limit below $10^{-2}$ deg$^{-2}$. This corresponds to at most one detected proto-cluster in the HeViCS fields. These predictions are, however, endowed with very large uncertainties both because the predicted counts of proto-clusters are very steep (and therefore are critically dependent on the detection limit which is hard to determine because the protoclusters are expected to be resolved by Planck), and because there is a considerable uncertainty on the evolution of key quantities such as the amplitude of the three-point correlation function. It is expected that, for luminous matter, this amplitude behaves as $b^{-1}$ or $b^{-2}$, where $b$ is the bias parameter [@1993ApJ...413..447F; @2001ApJ...548..114S], and in this case the expected surface density decreases by more than an order of magnitude compared to the case without three-point correlation amplitude redshift evolution. Moreover, due to the relatively small field size and the effect of cosmic variance, it is difficult to make a firm estimate of the expected number of sources. On the other hand, the expected number counts of the sources as a function of flux density are quite steep [@2005MNRAS.358..869N], so it is likely that substantially more submm bright proto-clusters can be found at fainter fluxes. @2007MNRAS.377.1557N and @2012ApJ...755...46L present predictions of the surface density of lensed spheroids. Constrained by the number of detected and candidate strongly lensed submm galaxies in the H-ATLAS Science Demonstration Field [@2010Sci...330..800N; @2012ApJ...749...65G], the models of @2012ApJ...755...46L predict a surface density around $8\times10^{-4}$ deg$^{-2}$ at a 500 $\mu$m flux density limit of 1 Jy. Again, this estimation is very uncertain and could in principle be considered as a lower limit, as the number of lensed spheroids decreases very quickly with flux density and the well-known clustering of submm sources is not taken into account in these estimates. In any case, these numbers are so low that we do not expect any high-$z$ lensed systems in the 84 deg$^2$ HeViCS fields above the Planck detection limit. Completeness and positional accuracy of the PCCS ------------------------------------------------ According to @2013arXiv1303.5088P, the PCCS catalogue should be 90% complete at 857 GHz down to a flux density level of 680 mJy. To verify this, we checked the SPIRE 350 $\mu$m flux densities of all Virgo Cluster galaxies in the catalogue of @2013MNRAS.428.1880A. This catalogue contains 72 galaxies with a $S_{350}>680$ mJy (taking into account the correction factors for the old beam size, extended flux density correction and updated SPIRE calibration). 69 of these sources are detected by Planck in the PCCS (as 67 sources, as two pairs of galaxies are blended into single Planck sources). The three non-detected sources (IC3061, NGC4351, and NGC4435) all have SPIRE 350 $\mu$m flux densities below 720 mJy. Below $S_{350}\sim650$ mJy the detection rate drops quickly, although there are still occasional PCCS detections with even lower SPIRE flux densities. The most extreme case is the peculiar Sa galaxy NGC4506 with $S_{350}=178$ mJy, which stands out in Figure [\[FluxComparison857.pdf\]]{} as the isolated point at the extreme left of the panels. Flux boosting and confusion with Galactic cirrus emission are probably responsible for the fact that this galaxy made it to the PCCS: it is located on top of the previously mentioned prominent cirrus ring in the centre of the Virgo Cluster. The PCCS 545 GHz catalogue is reported to have a 90% completeness limit of 570 mJy [@2013arXiv1303.5088P]. Using the conversion factor 0.83 as derived in Section 3, this corresponds to a 500 $\mu$m flux density level of 687 mJy. There are 37 galaxies in the @2013MNRAS.428.1880A catalogue with a SPIRE 500 $\mu$m flux density above 687 mJy (again taking into account the previously mentioned correction factors). All but one of these sources, the edge-on spiral NGC4316 with $S_{500}=694$ mJy, are detected by Planck at 545 GHz. The faintest source detected has $S_{500} = 502$ mJy. Overall, our results are in good agreement with the PCCS completeness estimates: in the HeViCS fields, the PCCS recovered $\sim$95% of the sources with $S_{350}>680$ mJy and $\sim$95% of the sources with $S_{550}>570$ mJy. This is the first empirical confirmation of the estimated completeness of the PCCS. For the HFI bands, the PCCS completeness estimates were based on simulations in which unresolved point sources of different intrinsic flux density levels were injected in the real maps and convolved with the effective beam. The fact that we confirm these numbers is a strong support for the internal PCCS validation procedure. ![Positional accuracy of the PCCS at 857 GHz (left panel) and 545 GHz (right panel). The plots show the difference in right ascension and declination between the position of the PCCS source and the coordinates of the centre of the optical counterpart, for all sources with a single nearby galaxy as counterpart.[]{data-label="PositionalAccuracy.pdf"}](PositionalAccuracy.pdf){width="46.00000%"} Finally, in order to test the positional accuracy of the PCCS, we checked the positions of the sources as listed in the PCCS catalogue (and listed in Tables 1 and 2) with the central positions of the galaxies obtained from NED. The results are shown in Figure [\[PositionalAccuracy.pdf\]]{}. There is no systematic offset, and the general positional accuracy is $0.71\pm0.29$ arcmin for the 857 GHz band, and $0.79\pm0.36$ arcmin for the 545 GHz band. These numbers are some 10% larger than, but still fully consistent with, the positional uncertainties quoted by [@2013arXiv1303.5088P] based on the PCCS sources from the KINGFISH, HRS, H-ATLAS and HeViCS BGS samples (0.65 at 857 GHz and 0.72 at 545 GHz). Conclusion and summary ====================== We have cross-correlated the Planck Catalogue of Compact Sources (PCCS) with the 84 deg$^2$ survey fields of the Herschel Virgo Cluster Survey (HeViCS), with the goals of identifying and characterizing the counterparts of the compact Planck sources at 857 and 545 GHz, and searching for possible high-redshift gravitational lenses or proto-clusters of star-forming galaxies. The results of this investigation are the following: 1. We find 84 compact Planck sources at 857 GHz in the HeViCS survey fields. 77 of these sources correspond to individual Virgo Cluster galaxies, 3 to nearby background galaxies, and 2 to galaxy pairs in the Virgo Cluster. Only 2 sources do not have a clearly identifiable counterpart at optical and FIR/submm wavelengths. 2. At 545 GHz, where the dominant emission mechanism is still continuum emission by cold and large dust grains, we find 48 sources in the PCCS. 41 of these sources have a single Virgo Cluster galaxy as counterpart, two correspond to a Virgo Cluster galaxy pair, and 5 sources have no nearby galaxy as a clearly identifiable counterpart at optical or submm wavelengths. No background galaxies are detected, and all sources with a counterpart, including the two galaxy pairs, are also detected at 857 GHz. 3. The vast majority of all Planck detected galaxies are late-type spiral galaxies, with the Sc class dominating the numbers (more than half of all galaxies detected at 545 GHz). The number of early-type spirals reduces sharply when moving from 857 to 545 GHz: of the 43 galaxies detected at 545 GHz, there is only one Sa galaxy, compared to 10 out of 82 at the 857 GHz. Early-type galaxies are virtually absent from the PCCS, with only three lenticular galaxies (NGC4370, 4429 and 4526) and one elliptical galaxy (M87) detected. In the latter galaxy, detected at both 857 and 545 GHz, the submm emission is dominated by synchrotron rather than dust emission. 4. The PCCS presents four different estimates for the flux density, reflecting the different methods used to detect sources and measure their flux density. We have compared these different flux density measures to the SPIRE 350 and 500 $\mu$m flux density measures from the HeViCS Virgo Cluster catalogue [@2013MNRAS.428.1880A]. The Planck pipeline source detection algorithm estimate, DETFLUX, which showed the smallest dispersion in the comparison of @2013arXiv1303.5088P, shows systematic deviations both at the highest and the lowest flux densities. We find the best correlation between the SPIRE flux densities and the Planck aperture photometry flux densities (APERFLUX), even at the highest flux density levels. At 857 GHz, a linear regression fit results in a reduced $\chi^2$ compatible with a relation without intrinsic scatter. 5. Based on the number of detected Virgo Cluster galaxies, we have estimated the completeness and positional accuracy of the PCCS at 857 and 545 GHz. Our estimates are in agreement with the results from @2013arXiv1303.5088P, who report 90% completeness limits of 680 and 570 mJy at 857 and 545 GHz respectively. Our study is the first empirical confirmation of the claimed completeness of the PCCS at submm wavelengths, which was estimated based on simulations, and hence provides a strong support for the internal PCCS validation procedure. 6. We have found only seven PCCS sources in the HeViCS fields without a nearby galaxy as obvious counterpart (two sources at 857 GHz and five sources at 545 GHz). Based on their location in the cluster and the SDSS and HeViCS sources at their positions, we conclude that all of these are probably linked to Galactic cirrus features or spurious detections. The number of sources without counterpart is remarkably low compared to the study by , who found that more than half of the compact Planck sources in the H-ATLAS equatorial fields correspond to cirrus features. This might be due to a combination of the higher galactic latitude of the Virgo Cluster compared to the equatorial H-ATLAS fields and the different Planck catalogue used. Unlike , we find no Planck sources in the HeViCS fields associated to high-redshift proto-clusters of dusty galaxies or strongly lensed submm sources. Acknowledgements {#acknowledgements .unnumbered} ================ MB, JF and TH acknowledge the financial support of the Belgian Science Policy Office (BELSPO) through the PRODEX project “Herschel-PACS Guaranteed Time and Open Time Programs: Science Exploitation” (C90370). MC and GDZ acknowledge financial support from ASI/INAF Agreement I/072/09/0. MB, FA, IDL, GG, SV and JV acknowledge the support from the Flemish Fund for Scientific Research (FWO-Vlaanderen). SPIRE has been developed by a consortium of institutes led by Cardiff University (UK) and including Univ. Lethbridge (Canada); NAOC (China); CEA, LAM (France); IFSI, Univ. Padua (Italy); IAC (Spain); Stockholm Observatory (Sweden); Imperial College London, RAL, UCL-MSSL, UKATC, Univ. Sussex (UK); and Caltech, JPL, NHSC, Univ. Colorado (USA). This development has been supported by national funding agencies: CSA (Canada); NAOC (China); CEA, CNES, CNRS (France); ASI (Italy); MCINN (Spain); SNSB (Sweden); STFC (UK); and NASA (USA). The development of Planck has been supported by: ESA; CNES and CNRS/INSU-IN2P3-INP (France); ASI, CNR, and INAF (Italy); NASA and DoE (USA); STFC and UKSA (UK); CSIC, MICINN and JA (Spain); Tekes, AoF and CSC (Finland); DLR and MPG (Germany); CSA (Canada); DTU Space (Denmark); SER/SSO (Switzerland); RCN (Norway); SFI (Ireland); FCT/MCTES (Portugal); and The development of Planck has been supported by: ESA; CNES and CNRS/INSU-IN2P3-INP (France); ASI, CNR, and INAF (Italy); NASA and DoE (USA); STFC and UKSA (UK); CSIC, MICINN and JA (Spain); Tekes, AoF and CSC (Finland); DLR and MPG (Germany); CSA (Canada); DTU Space (Denmark); SER/SSO (Switzerland); RCN (Norway); SFI (Ireland); FCT/MCTES (Portugal); and PRACE (EU). 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 www.sdss.org. The SDSS is managed by the Astrophysical Research Consortium for the Participating Institutions. The Participating Institutions are the American Museum of Natural History, Astrophysical Institute Potsdam, University of Basel, University of Cambridge, Case Western Reserve University, University of Chicago, Drexel University, Fermilab, the Institute for Advanced Study, the Japan Participation Group, Johns Hopkins University, the Joint Institute for Nuclear Astrophysics, the Kavli Institute for Particle Astrophysics and Cosmology, the Korean Scientist Group, the Chinese Academy of Sciences (LAMOST), Los Alamos National Laboratory, the Max-Planck-Institute for Astronomy (MPIA), the Max-Planck-Institute for Astrophysics (MPA), New Mexico State University, Ohio State University, University of Pittsburgh, University of Portsmouth, Princeton University, the United States Naval Observatory, and the University of Washington. This research has made use of 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. This research has made use of NASA’s Astrophysics Data System Bibliographic Services. This research has made use of SAOImage DS9, developed by Smithsonian Astrophysical Observatory. [^1]: Herschel is an ESA space observatory with science instruments provided by European-led Principal Investigator consortia and with important participation from NASA. [^2]: Comparisons between SPIRE and Planck ERCSC fluxes have also been presented previously for individual galaxy samples, including the HeViCS Bright Galaxy Sample [@2012MNRAS.419.3505D], the Herschel Reference Survey , and a sample of Planck-selected star forming galaxies [@2013MNRAS.429.1309N].
--- author: - | Jiangchuan Huang$^{1,3*}$, Jianghui Ji$^{2*}$, Peijian Ye$^{1}$, Xiaolei Wang$^{4}$,\ Jun Yan$^{5}$, Linzhi Meng$^{3}$, Su Wang$^{2}$, Chunlai Li$^{5}$,\ Yuan Li$^{6}$, Dong Qiao$^{7}$, Wei Zhao$^{8}$, Yuhui Zhao$^{2}$, Tingxin Zhang$^{1}$,\ Peng Liu$^{8}$, Yun Jiang$^{2}$, Wei Rao$^{3}$, Sheng Li$^{9}$,\ Changning Huang$^{10}$, Wing-Huen Ip$^{6,11}$, Shoucun Hu$^{2}$, Menghua Zhu$^{6}$,\ Liangliang Yu$^{2}$, Yongliao Zou$^{5}$, Xianglong Tang$^{8}$, Jianyang Li$^{12}$,\ Haibin Zhao$^{2}$, Hao Huang$^{3}$, Xiaojun Jiang$^{5}$, & Jinming Bai$^{13}$\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ \ bibliography: - 'scibib.bib' title: 'The Ginger-shaped Asteroid 4179 Toutatis: New Observations from a Successful Flyby of Chang’e-2' --- On 13 December 2012, Chang’e-2 conducted a successful flyby of the near-Earth asteroid 4179 Toutatis at a closest distance of 770 $\pm$ 120 meters from the asteroid’s surface. The highest-resolution image, with a resolution of better than 3 meters, reveals new discoveries on the asteroid, e.g., a giant basin at the big end, a sharply perpendicular silhouette near the neck region, and direct evidence of boulders and regolith, which suggests that Toutatis may bear a rubble-pile structure. Toutatis’ maximum physical length and width are (4.75 $\times$ 1.95 km) $\pm$10$\%$, respectively, and the direction of the +$z$ axis is estimated to be (250$\pm$5$^\circ$, 63$\pm$5$^\circ$) with respect to the J2000 ecliptic coordinate system. The bifurcated configuration is indicative of a contact binary origin for Toutatis, which is composed of two lobes (head and body). Chang’e-2 observations have significantly improved our understanding of the characteristics, formation, and evolution of asteroids in general. Introduction {#introduction .unnumbered} ============ Chang’e-2, the second Chinese probe dedicated to the exploration of the Moon, was launched on 1 October 2010 at 10:59:57 UTC. After it accomplished its primary objective, an extended mission was designed for Chang’e-2 to travel to the Sun-Earth Lagrangian point (L2) and then to an asteroid. On 9 June 2011, Chang’e-2 departed its lunar orbit and headed directly to L2, where it arrived on 25 August 2011 and began to perform space-environment exploration. The asteroid’s subsequent flyby mission remained under consideration, and eventually, asteroid 4179 Toutatis was selected as the primary target for the flyby after considering the leftover fuel of the spacecraft, the capability of the tracking and control network, and the fact that the asteroid’s closest approach to Earth would occur on 12 December 2012. An undisclosed story is that Chang’e-2 was intentionally controlled to move near L2 for more than 230 days to favor the Toutatis-flyby injection maneuvers. On 15 April 2012, Chang’e-2 left L2. On 1 June 2012, the spacecraft began its mission to Toutatis. Chang’e-2 implemented the flyby on 13 December 2012 at 8:29:58.7 UTC at a closest distance of $770\pm 120$ ($3 \sigma$) meters from Toutatis’ surface at a high relative velocity of 10.73 km s$^{-1}$, with highest-resolution images of better than 3 meters per pixel [@Huang13] . As of 14 July 2013, Chang’e-2 is still traveling into outer space over 50 million kilometers away from Earth, and it is the first successful multiple-type-objective probe in history that has ever visited the Moon, a Lagrangian point, and an asteroid. Asteroids are remnant building blocks of the formation of the Solar System. They provide key clues to understanding the process of planet formation, the environment of the early Solar nebula, and even the occurrence of life on Earth. Toutatis, as an Apollo near-Earth asteroid in an eccentric orbit that originates from the main belt, was a good target for Chang’e-2’s flyby mission because this asteroid is full of fascinating puzzles. The 3-D shape of Toutatis has been well reconstructed, and its spin state has been determined, from extensive radar measurements taken at Arecibo and Goldstone [@Hudson95; @Ostro95; @Ostro99; @Hudson03; @Busch10; @Busch11; @Busch12; @Takahashi13] since 1992 during its closest approaches to Earth, once every four years. The results indicate that Toutatis is characterized as a non-principal-axis rotator; this tumbling asteroid rotates about its long axis with a period of 5.4 days, and the long-axis precession period is 7.4 days [@Busch11] ; this precession may be induced by the Earth’s tides during each close-in flyby [@Scheeres07] and the Yarkovsky-O’Keefe-Radzievskii-Paddack (YORP) effect [@Bottke02] . Moreover, detailed geological features, such as complex linear structures and concavities, have been revealed in Hudson et al.’s model [@Hudson03] at an average spatial resolution of $\sim$ 34 m. Hence, the primary issue of concern is whether the derived radar models accurately represent the real appearance of Toutatis. Furthermore, some features such as boulders and regolith on its surface have been speculated to exist in previous radar models [@Hudson03; @Busch10] . It was hoped that this mystery could be solved by obtaining optical images during a spacecraft’s close flyby. Furthermore, small asteroids at the km scale are believed to have a so-called rubble-pile structure, which means that they consist of a loose collection of fragments under the influence of gravity. Therefore, the second major question arises — whether Toutatis is also an agglomeration of gravitationally bound chunks with such a fragile structure. The optical images of Toutatis were obtained using one of the onboard engineering cameras, which were originally designed to monitor the deployment of solar panels on the spacecraft and to photograph other objects (such as the Earth and Moon) in space. The camera that is capable of color imaging has a lens with a focal length of 54 mm and a 1024-by-1024-pixel CMOS detector, and it has a field of view (FOV) of 7.2$^{\circ}$ by 7.2$^{\circ}$. A solar panel shields the camera from sunlight to help minimize stray light. The FOV was partly occulted by the solar panel, and thus some of the images were cut off from the early gallery. As part of the imaging strategy, Chang’e-2 maneuvered to adjust the camera’s optical axis to lie antiparallel to the direction of the relative velocity vector between the probe and Toutatis. Imaging during the inbound trajectory was not ideal because there was a large Sun-Toutatis-Chang’e-2 phase angle of $\sim 143.6^{\circ}$. Therefore, Chang’e-2 completed an attitude adjustment approximately an hour ahead of the flyby epoch to prepare for the outbound imaging. The overall imaging time was approximately 25 minutes, and the interval for each snapshot was 0.2 seconds. During the mission, the spacecraft collected more than 400 useful images totaling 4.5 GB of data. Figure \[fig\_all\] shows several outbound images of Toutatis. Panel e, which was the first panoramic image, was taken at a distance of $\sim 67.7$ km at a resolution of $\sim 8.30$ m (see Supplementary Figure 1), whereas panel a, which was the third in a sequence of images, corresponds to an imaging distance of $\sim 18.3$ km at a local resolution of $\sim 2.25$ m, as shown in Fig. \[fig\_all\]. The present highest-resolution optical image has a resolution that is significantly finer than the 3.75 m resolution of the known radar model [@Busch12; @Takahashi13] , although the radar images cover nearly the entire surface of Toutatis, whereas the optical images cover only one side of Toutatis’ surface. Results {#results .unnumbered} ======= We tested several methods to directly establish the 3-D model of Toutatis based on the optical images collected by Chang’e-2. However, there is no satisfactory outcome because of the narrow viewing angle and the local high-resolution images. Therefore, 3-D models developed based on delay-Doppler radar measurements [@Hudson03; @Busch12; @Takahashi13] were adopted to discern the attitude of Toutatis in the approaching epoch. Based on the attitude of the probe and its camera, the relative position between the asteroid and the probe, and optical images, in combination with the models of Hudson et al. and Busch et al. [@Hudson03; @Busch12; @Takahashi13] , which were utilized to match the spin state of Toutatis (see Supplementary Figure 2), the direction of the principal axis was measured to be ($250 \pm 5^\circ, 63 \pm5^\circ$) with respect to the J2000 ecliptic coordinate system [@Scheeres00; @Hudson03] , and the fundamental parameters are summarized in Table 1. Our result is in good agreement with the previous prediction [@Busch12] , which indicates that Toutatis’ spin status was greatly changed because of tidal interaction with the Earth during the 2004 flyby, which had a closest distance of $\sim$ 0.01 AU [@Scheeres07; @Busch12] . Furthermore, we again measured Toutatis’ physical size from the images, and its maximum length and width were estimated to be (4.75 $\times$ 1.95 km) $\pm$10$\%$, respectively (see Supplementary Section 3). New Geological Features from Chang’e-2 Observation {#new-geological-features-from-change-2-observation .unnumbered} -------------------------------------------------- The shape of Toutatis resembles that of a ginger root (Fig. \[fig\_all\]), where the bifurcated asteroid mainly consists of a head (small lobe) and a body (large lobe). The two major parts are not round in shape, and their surfaces have a number of large facets. In particular, at the end of the body, there appears to be a huge square concavity, which could be of large-impact origin. Regarding the asteroid’s appearance, craters are more prominent on Toutatis than on Itokawa. In contrast, we clearly see fewer boulders on Toutatis, although many smaller blocks may have been missed because of the resolution of Chang’e-2 flyby images. In comparison with the radar models [@Ostro95; @Ostro99; @Hudson03] , the proximate observations from Chang’e-2’s flyby have revealed several remarkable discoveries concerning Toutatis, among which the presence of the giant basin at the big end appears to be one of the most compelling geological features, and the sharply perpendicular silhouette in the neck region that connects the head and body is also quite novel. A large number of boulders and several short linear structures (Fig. \[fig\_all\]) are also apparent on the surface, although the lack of sharp contact in the radar model may be an effect of the SHAPE software [@Magri07] . Concavities {#concavities .unnumbered} ----------- The giant basin at the big end of Toutatis has a diameter of $\sim$ 805 m, and we surmise that one or more impactors may have collided with the asteroid in this region, thereby gouging out such an enormous basin. Judging by the contrast of its relatively subdued relief with large impact basins on other asteroids, Mathilde and Eros, this large-end depression may be attributable to the internal structure of the large lobe, with smaller impact features overlying it. The most significant feature is the ridge around the largest basin. The wall of this basin exhibits a relatively high density of lineaments, some of which seem to be concentric to the basin. There are also several linear structures outside the basin that are roughly parallel to the basin rim (Fig. \[fig\_zoom\]a). These ridges are indicative of an internal structure on small bodies [@Prockter02; @Thomas12] . Most of the ridges near the largest basin at the big end are most likely related to the huge stress energy during impact. The interior material in the deep region would have been scattered across Toutatis’ surface by the large impact that formed the basin, leading to the fracturing of the asteroid. Hence, one may speculate that the asteroid most likely has a rubble-pile internal structure. More than fifty craters have been identified from the flyby images, and they have diameters ranging from 36 to 532 m (with an average of $\sim$ 141 m). $\sim$ 44% of the craters exceed 100 m in diameter, which is consistent with Hudson et al.’s model. From the close-up image, a crater of high morphological integrity (with a diameter of $\sim$ 62 m) can be clearly observed, whereas tens of boulders (with lengths of $\sim$ 20-30 m) are randomly distributed nearby (Fig. \[fig\_zoom\]b). Two concavities are closely carved into the surface: the larger (with a diameter of $\sim$ 368 m) has a relatively vague outline, whereas the smaller (with a diameter of $\sim$ 259 m) has a relatively clear edge. Based on their appearance, we suspect that the relatively smaller, young crater (marked with the letter B) is superimposed on the larger, ancient crater (marked with the letter A) (Fig. \[fig\_zoom\]a). Although the images show that parts of the observed surface areas have a high degree of exposure, the topographical features of the craters were identified based on visual inspection and automatic annotation. Therefore, we identified craters on the large and small lobes individually, including vaguely shaped candidate craters on the highly exposed areas, for the investigation of the cumulative and relative size-frequency distributions (see Supplementary Section 4) [@Zou13] . Figure \[fig\_fre\] indicates that the surfaces of the two lobes are very likely to share a similar cratering history because their relative size-frequency profiles for craters [@Arvidson79] look alike throughout a range of diameters. Furthermore, the craters on the two lobes are not in saturation equilibrium, as the two profiles do not tend toward stability with increasing crater diameter (Fig. \[fig\_fre\]a). An abundance of craters would suggest that Toutatis suffered from many impacts of interplanetary projectiles in the past. Boulders {#boulders .unnumbered} -------- From the flyby images, more than thirty boulders can be clearly discerned. They have lengths that range from 12 to 81 m, with an average of $\sim$ 32 m. Approximately 90% of the boulders are less than 50 m in diameter. The largest boulder, along with other large boulders, is located near the neck region (Fig. \[fig\_zoom\]a). It is believed that a large ejected fragment is empirically associated with a large crater [@Gault63; @Thomas01] . The boulders are suggested to be those non-escaping ejecta that re-impact onto the asteroid’s surface within a short timescale [@Scheeres98] . Recently, the Goldstone radar images of 12 December 2012 have also suggested that several 10-m-scale bright features may be boulders [@Busch12] . However, the optical images, although they were collected on only one side of the asteroid, have an advantage over the radar models [@Hudson03; @Busch12] in uncovering local fine features on the surface of Toutatis. Regolith {#regolith .unnumbered} -------- Two craters in the neck area are indicative of the granular flows in the inner flanks. The upper regolith near the craters is fine grained, whereas the lower is relatively coarse, and very many fragments are present (Fig. \[fig\_zoom\]a). Such a structure may indicate the redistribution of regolith via downslope or resurfacing processes, which may be related to the inclined terrain and the orientation of gravity [@Scheeres98] . The results are in good agreement with those of polarimetric observations [@Mukai97; @Hudson98] , thereby indicating that Toutatis is covered with fine regolith made up of light-transmitting materials. The same features have also been reported on Lutetia’s surface [@Vincent12] . Furthermore, the characteristics once again suggest that Toutatis’ surface may consist of a fine granular or regolith layer with the porosity of lunar soils [@Ostro99] . The latest results imply that this asteroid may bear an undifferentiated L-chondrite surface composition [@Reddy12] . There is additional evidence that supports the existence of regolith on Toutatis’ surface. The average thermal inertia for Toutatis may have a low profile, according to an empirical relation between the thermal inertia and the effective diameter [@Delbo07] . For a non-principal-axis rotator such as Toutatis, the global redistribution of regolith that results from lofting may also be one of the factors that contributes to the low surface thermal inertia [@Delbo07] . A considerable number of mm-sized grains may constitute Toutatis’ regolith (Fig. \[fig\_zoom\]b), and the regolith depth on the surface might range from several centimeters to several meters. Clearly, Toutatis’ thermal inertia seems to be much less than that of Itokawa [@Fujiwara06] ; thus, the terrain of Toutatis is a bit smoother than that of Itokawa, and the boulders on Toutatis’ surface are less numerous than those on Itokawa’s surface, which is mostly embedded with numerous pebbles and gravel in all regions. Linear Structures {#linear-structures .unnumbered} ----------------- Several types of linear structures, which are primarily composed of troughs and ridges that appear to be common for small asteroids [@Thomas10; @Robinson02] , can be easily observed on Toutatis’ surface. Notable linear structures are apparent in the radar models [@Hudson03] ; however, the flyby images do not identify any linear features that have been previously reported because the relevant part of the surface was invisible to Chang’e-2. The present linear structures, which are similar to the short linear structures of Itokawa, are clearly observed [@Demura06] to have a length of 120-330 m in the visible regions of the images. The troughs, which are similar in appearance to groove-like features, are primarily scattered over the small lobe; they have an average length of $\sim$ 170 m, and they are approximately orientated along the long axis of the asteroid (Fig. \[fig\_zoom\]a). Generally, troughs are related to extensions resulting from tensile stress in the plane of the surface [@Thomas12] . Consequently, they may be produced by the tensile stresses that arise from nearby impact cratering or other geological processes. Discussion {#discussion .unnumbered} ========== As mentioned above, the existence of the giant basin and its surroundings provides direct evidence that Toutatis is likely to be a rubble-pile asteroid. In this case, the asteroid could reassemble itself into a weak aggregate of large fragments via a heavy impact or many smaller impacts; in this manner, the large interior voids could absorb the collision energy and further resist huge collisions in the formation process. Furthermore, a vast majority of S-type asteroids appear to possess rubble-pile structures with an average porosity of $\sim$ 15% - 25% [@Britt02] . Recent investigations have demonstrated that the typical bulk density of L ordinary chondrites is $\sim$ 3.34 g cm$^{-3}~~$ [@Britt02; @Reddy12] . Assuming a density of 2.1 - 2.5 g cm$^{-3}~~$ [@Scheeres98] , Toutatis may have a porosity in the range $\sim$ 25% to $\sim$ 37%, thereby implying that it may be an intermediate body between Eros ($\sim$ 20% [@Britt02]) and Itokawa ($\sim$ 41% [@Fujiwara06]) with respect to surface features. Moreover, strong evidence from N-body simulations indicates that two km-sized objects with rubble-pile structures would produce Toutatis-like objects [@Richardson02] . In summary, we may conclude that Toutatis is not a monolith but most likely a coalescence of shattered fragments. The bifurcated configuration means that Toutatis is comprised of two major lobes, similar to Itokawa, thereby implying that Toutatis is a contact binary, which is an asteroid type that may constitute more than 9% of the NEA population. Km-sized contact binaries, such as 4486 Mithra and 4769 Castalia, share an irregular, significantly bifurcated shape [@Benner05] . The following question then arises: how do these contact binaries form? Several formation mechanisms have been proposed to produce such bifurcated configurations. The first scenario supposes that the two major lobes were two separate objects, which were moving at a sufficiently low relative speed to yield a contact binary [@Fujiwara06] . The difference in the orientations of head and body may be indicative of their diverse origins and further suggests that the two parts were once detached. The second scenario includes the YORP and binary YORP (BYORP) effects, which are likely to form contact binaries [@Steinberg11] , especially for those objects with slow rotational periods. In this scenario, two components, which may be major fragments of a parent body, suffer re-impact and recombination, thereby leading to the formation of a bifurcated configuration. The third scenario suggests that catastrophic collisions may have gouged out the giant basin and a number of craters on Toutatis’ surface. The boulders near three potential arc depression terrains (Figure \[fig\_zoom\]a) imply that the head may have experienced severe impacts. Last but not least, tidal disruption from a close encounter with a terrestrial planet is also likely to play a role in the formation of such a contact binary. In short, the above-mentioned scenarios may have actually occurred during Toutatis’ history. Hence, we infer that Toutatis may be a reconstituted contact binary. The Chang’e-2 flyby has revealed several new discoveries regarding Toutatis, e.g., the giant basin at the big end, the sharply perpendicular profile about the neck region, and direct images of distributed boulders and surface regolith. It is the first time that geological features on Toutatis’ surface have been observed so clearly. All these observations suggest that Toutatis may bear a rubble-pile structure, similar to other S-type asteroids, such as Itokawa. Moreover, the close-up imaging results for the asteroid provide further indication that the bifurcated configuration that is related to the formation of contact binaries appears to be quite common among the NEA population. The observations of Chang’e-2 have significantly improved our understanding of the formation and evolution of these building blocks of the Solar System. Methods {#methods .unnumbered} ======= To adapt to Chang’e-2’s characteristics of high control accuracy and the wide FOV of 7.2$^{\circ}$ by 7.2$^{\circ}$ of the engineering camera, a close-approach strategy and a specific imaging design were developed. As shown in Supplementary Figure 1, L, S, and $\Delta t$ represent the minimum distance of rendezvous, the imaging distance, and the time interval between two sequential images, respectively. The camera optical axis pointed in the direction of the relative velocity between the probe and Toutatis, which remained nearly unchanged during the flyby. After a short time at the closest approach, the camera captured Toutatis; the asteroid was contained in the FOV, but the imaging size decreased as the departure distance increased. By solving the geometric relation between Image 2 and Image 3, the distances L and S and the resolution of each image were determined for the Chang’e-2 observations. Taking into account the various characteristics of the specific approach design and imaging strategy for the flyby, a physical size of (4.75 $\times$ 1.95 km) $\pm$10$\%$ for the illuminated region of the asteroid surface was statistically estimated from the sequence of images recorded by the CMOS detector during the flyby. The determined size is in good agreement with that of the radar model [@Hudson03] (see Supplementary Sections 1 and 3). The attitude of a rigid body in space is determined by its rotations around three axes of an orthogonal coordinate system. To define the attitude of Toutatis in each optical image (Supplementary Figure 2a), the three axes $l_1$, $l_2$, and $l_3$ are defined as follows: $l_1$ and $l_2$ extend through Toutatis’ centroid in the images along the directions of the long axis and short axis, respectively, and are perpendicular to each other. $l_3$ is perpendicular to the imaging plane and constitutes a right-handed coordinate system with $l_1$ and $l_2$. To match its shape to the optical images, we rotated the model of Busch et al. in intervals of $5^{\circ}$ for each Euler angle around the three axes of the body-fixed coordinate system in space. Next, we chose three criteria of the orientation of the asteroid’s long-axis, the ratio of length to width, and the obvious topographical feature at the joint of the two lobes of Toutatis — to ensure that the radar model agreed with the optical images from Chang’e-2, as shown in Supplementary Figure 2a. The most approximate attitude was finally obtained by rotating the radar model and comparing it with the optical image, as shown in Supplementary Figure 2b. The coordinate transformation from the asteroid’s body-fixed system to the J2000.0 equatorial coordinate system were expressed in terms of three components of Euler angles. The directions of the principal axes were then obtained according to the attitude of the radar model. The craters and other geological features were identified via both visual inspection and automatic annotation. In this case, dozens of low-lying regions were noted, and their positions, contours, and sizes were obtained using surface-structure-analysis techniques. The feature points on the edge of each crater were labeled manually based on the treatment of the imaging process and the radar model. After the points were marked, a piecewise-quadratic curve was fitted to them, and closed curves were obtained as contour profile. 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The orbit determination of (4179) Toutatis from optical and radar data. [*Planet. Space Sci.*]{} [**42**]{}, 741-745 (1994). Milani, A., & Gronchi, G. Theory of orbit determination. (Cambridge University Press, Cambridge, UK, 2010). {#section .unnumbered} **Supplementary Information** is available in the online version of the paper. {#section-1 .unnumbered} **Acknowledgements** The authors thank Michael W. Busch for his constructive comments that greatly helped to improve the original content of this manuscript. We thank the whole Chang’e-2 team to make the mission a success. We appreciate B. Yang for target observations before the flyby mission. We are grateful to R.P. Butler for English improvement of this manuscript. This research was supported by National Science and Technology Major Project, National Natural Science Foundation of China, the Innovative and Interdisciplinary program by CAS, and Key Laboratory of Planetary Sciences (2013DP173302), CAS. JYL’s contribution to this work is not supported by any funds. {#section-2 .unnumbered} **Authors Contributions** J. C. H. was the leader of the asteroid mission. J. H. J. wrote the manuscript with contributions from other co-authors. J.C.H., P.J.Y. and T.X.Z. were responsible for the overall project design of the flyby mission to Toutatis, imaging strategy, and proposed the method of fundamental parameter analysis. H.B.Z., J.H.J., S.C.H., S.W., Y.H.Z., Y.J. and L.L.Y. contributed to ground-based observation, refined the trajectory solution for Toutatis, completed data analysis, prepared for figures and wrote the manuscript. L.Z.M, H.H. and W.R. took charge of overall design of probe system and data analysis. X.L.W. was responsible for controlling the attitude and the spacecraft’s orbit, and contributed to the engineering parameters analysis. J.Y., C.L.L., Y.L.Z. and X.J.J. took charge of data collection, data pretreatment and the ground-based observation. Y.L. and M.H.Z. partly contributed to contents and prepared for figure. D.Q. contributed to the target selection and the design of transfer orbit. W.Z., P.L. and X.L.T. were in charge of the algorithm of optical images and obtained the image information. S.L. was responsible for imaging process. C.N.H. developed the optical camera and determined the parameters of the camera. W.H.I and J.Y.L. contributed ideas and discussions in preparation for the manuscript. J.M.B. partly participated in the target observation. All authors contributed to the manuscript. {#section-3 .unnumbered} **Authors information** The authors declare no competing financial interests. To whom correspondence should be addressed. E-mail:jijh@pmo.ac.cn. . [**Table 1. Parameters for Toutatis.**]{} Property Value ----------------------------- ------------------------------------------------------------------------------ Osculating Orbital elements $a$=2.5336 AU,  $e$=0.6301 $i$=0.4466$^\circ$ ,       $\Omega$=124.3991$^\circ$ $\omega$=278.6910$^\circ$ ,  $M$=6.7634$^\circ$ Special type S (IV) [@Hudson03] Size (diameter) Major axes (x=4.60$\pm$0.10 km, y=2.29$\pm$0.10 km, z=1.92$\pm$ 0.10 km) [@Hudson03] (x=4.46$\pm$0.10 km, y=2.27$\pm$0.10 km, z=1.88$\pm$ 0.10 km) [@Takahashi13] Rotational properties Rotation 5.4 day [@Hudson95; @Hudson03] Precession 7.4 day [@Hudson95; @Hudson03] Pole position in the space $\beta=250 \pm 5^\circ$,    $\lambda=63 \pm5^\circ$ $\beta=54.6^\circ$,         $\lambda=60.6^\circ$ [@Busch12; @Takahashi13] $\alpha=258\pm5^\circ$, $\delta=40\pm5^\circ$ Density 2.5 g cm$^{-3}$    [@Scheeres98] \[param\] Using the data released by the Minor Planet Center[@Krivova94; @Milani10] and optical data from the ground-based observational campaign sponsored by the Chinese Academy of Sciences, we determined Toutatis’ orbit with uncertainties on the order of several kilometers. Osculating orbital elements were calculated for the flyby epoch of 13 December 2012 at 8:30 UTC, where $a,~ e, ~i,~ \Omega, ~\omega$, and $M$ are the semi-major axis, the eccentricity, the inclination, the longitude of the ascending node, the argument of perihelion, and the mean anomaly, respectively. $\beta, ~\lambda$ and $\alpha,~ \delta$ are the longitudes and latitudes of the long axis of the asteroid in the J2000.0 ecliptic and equatorial coordinate systems, respectively.
--- abstract: 'Particle diffusion in a two dimensional curved surface embedded in $R_3$ is considered. In addition to the usual diffusion flow, we find a new flow with an explicit curvature dependence. New diffusion equation is obtained in $\epsilon$ (thickness of surface) expansion. As an example, the surface of elliptic cylinder is considered, and curvature dependent diffusion coefficient is calculated.' author: - 'Naohisa Ogawa [^1]' title: Curvature Dependent Diffusion Flow on Surface with Thickness --- Motivation ========== The particle motion on a given curved surface is old but interesting problem in wide range of physics. Especially the diffusion process of particles on such a manifold is still an open problem, and related to various kinds of phenomena. For example the motion of protein on cell membrane has great importance in biophysics. There are several research papers discussing on this problem. Some of them are treating this problem by using usual diffusion equation with curved coordinate, and discuss the curvature (Gauss curvature) dependence of its solution [@diffusion_equation]. Other of them use the Langevin equation on curved surface and calculating the curvature dependence of diffusion coefficient [@langevin_equation]. The quantum mechanics of particle motion on such a curved manifold is also considered by many authors. This problem is usually explained by the Schroedinger equation with Laplace-Beltrami operator. However, when we treat the curved surface as embedded one in 3 dimensional Euclidean space, situation is changed and then we have a quantum potential term related to the curvature additional to the kinetic operator [@da; @Costa],[@ogawa_fujii],[@fujii]. Another example is in larger scale physics in which our consideration is devoted. Patterns of animal skins are well expressed by the reaction diffusion equation [@Turing]. But the patterns are different for each parts even in one individual. For example, Char fish, the side part has white spot pattern, but the back part has labyrinth pattern. (For these two patterns, see for example [@Shoji_Iwasa].) One of the reasons might come from the curvature difference between side part and back part. If the diffusion is influenced by the curvature, this difference of patterns might be explained. Furthermore, the cross section of fish has form of ellipsoid and the surface can be approximated as the one of elliptic cylinder. In two dimensional space, we have only two kinds of curvature, one is Gauss curvature and other is mean curvature. Both are constructed from second fundamental tensor by taking determinant or trace. Gauss curvature can also be constructed only by metric tensor and its derivatives, but this is not the case for the mean curvature. The elliptic cylinder, in which we have much interest, has zero Gauss curvature and non-zero mean curvature. Therefore to explain the pattern change of Char fish, solution of the diffusion equation should depend on mean curvature. This is impossible if we start from usual diffusion equation because it depends only on metric but not on second fundamental tensor. Therefore we need some new diffusion equation, which bring not only Gauss curvature but also mean curvature. In this article, we discuss how to construct such curvature dependent diffusion equation. Coordinate and Metric ===================== The simple extension of diffusion equation in Euclidean space to Riemannian space can be done by changing Laplacian with Cartesian coordinate to the one with curved coordinate, i.e. Laplace Beltrami operator. This coordinate change is not enough for our purpose, however. The way of construction of new diffusion equation in this paper is the followings. We re-identify the two dimensional diffusion as the limiting process from three dimensional diffusion. We place the curved surface $\Sigma$ in three dimensional Euclidean space $R_3$, and we put two similar copies of $\Sigma$, called $\tilde{\Sigma}$ and $\Sigma'$ at a small distance of $\epsilon/2$. Our particles can only move between these two surfaces, and later we take a limit $\epsilon \to 0$. We look for the form of diffusion equation in this limit. The coordinates we use hereafter is the followings. (See fig.1) $\vec{X}$ is the Cartesian coordinate in $R_3$. $\vec{x}$ is the Cartesian coordinate which specifies only the points on $\Sigma$. $q^i$ is the curved coordinate on $\Sigma$. (Small Latin indices $i,j,k,\cdots$ runs from 1 to 2.) $q^0$ is the coordinate in $R_3$ normal to $\Sigma$. Further by using the normal unit vector $\vec{n}(q^1,q^2)$ on $\Sigma$ at point $(q^1,q^2)$, we can identify any points between two surfaces $\Sigma'$ and $\tilde{\Sigma}$ by the following thin-layer approximation [@fujii]. $$\vec{X}(q^0,q^1,q^2) = \vec{x}(q^1,q^2) + q^0 \vec{n}(q^1,q^2),$$ where $ -\epsilon/2 \leq q^0 \leq \epsilon/2 $. ![Embedding and Coordinate](fig1.eps){width="5cm"} From this relation we can obtain the curvilinear coordinate system between two surfaces ($\subset R_3$) by the coordinate $q^{\mu}=(q^0,q^1,q^2)$, and metric $G_{\mu\nu}$. (Hereafter Greek indices $\mu, \nu,\cdots$ runs from 0 to 2.) $$G_{\mu\nu}=\frac{\partial \vec{X}}{\partial q^{\mu}} \cdot \frac{\partial \vec{X}}{\partial q^{\nu}}.$$ Each part of $G_{\mu\nu}$ is the following. $$G_{ij}=g_{ij} + q^0 (\frac{\partial \vec{x}}{\partial q^{i}} \cdot \frac{\partial \vec{n}}{\partial q^{j}} + \frac{\partial \vec{x}}{\partial q^{j}} \cdot \frac{\partial \vec{n}}{\partial q^{i}}) + (q^0)^2 \frac{\partial \vec{n}}{\partial q^{i}} \cdot \frac{\partial \vec{n}}{\partial q^{j}},$$ where $$g_{ij} = \frac{\partial \vec{x}}{\partial q^{i}} \cdot \frac{\partial \vec{x}}{\partial q^{j}}$$ is the metric on $\Sigma$. Hereafter indices $i,j,k \cdots$ are lowered or rised by $g_{ij}$ and its inverse $g^{ij}$. We also obtain $$G_{0i}=G_{i0}=0, ~~G_{00}=1.$$ We can proceed the calculation by using the new variables. We first define the tangential vector to $\Sigma$ by $$\vec{B}_k = \frac{\partial \vec{x}}{\partial q^{k}}.$$ Note that $\vec{n} \cdot \vec{B}_k =0$. Then we obtain two relations. Gauss equation: $$\frac{\partial \vec{B}_i}{\partial q^j} = - \kappa_{ij} \vec{n} + \Gamma^k_{ij} \vec{B}_k,$$ Weingarten equation: $$\frac{\partial \vec{n}}{\partial q^j} = \kappa_j^m \vec{B}_m,$$ where $$\Gamma^k_{ij} \equiv \frac{1}{2} g^{km}(\partial_i g_{mj} + \partial_j g_{im} - \partial_m g_{ij}).$$ $\kappa_{ij}$ is called Euler-Schauten tensor, or second fundamental tensor defined as $$\kappa_{ij} = \frac{\partial \vec{n}}{\partial q^i}\cdot \vec{B}_j.$$ The second fundamental tensor $\kappa_{ij}$ is the projection of $\partial \vec{n}$ into the surface. Furthermore, the mean curvature is given by $$\kappa = g^{ij} \kappa_{ij},$$ and Ricci scalar curvature $R$ is obtained by $$R/2 = \det(g^{ik} \kappa_{kj})= \det(\kappa^i_j) = \frac{1}{2}(\kappa^2 -\kappa_{ij} \kappa^{ij}).$$ Then we have the formula for metric of curvilinear coordinate in a neighborhood of $\Sigma$. $$G_{ij}=g_{ij} + 2 q^0 \kappa_{ij} + (q^0)^2 \kappa_{im} \kappa^m_j.$$ Under the inversion $q^0 \to -q^0$, we have $\kappa_{ij} \to - \kappa_{ij}$ as well as $\vec{n} \to -\vec{n}$ from $\vec{n}=\partial_0 \vec{X} / \mid \partial_0 \vec{X} \mid$. Therefore $G_{ij}$ is invariant under $q^0 \to -q^0$.\ Now we have the total metric tensor such as, $$G_{\mu\nu} = \left( \begin{array}{cc} 1 & ~~~0~~~\\ 0 &G_{ij} \end{array} \right).$$ Embedding of Diffusion field ============================ Let us denote 3 dimensional diffusion field as $\phi^{(3)}$, and Laplacian as $\Delta^{(3)}$. Then we have the equation with normalization condition $$\begin{aligned} && \frac{\partial \phi^{(3)}}{\partial t} = D \Delta^{(3)} \phi^{(3)},\label{eq:diff}\\ 1 &=& \int \phi^{(3)}(q^0,q^1,q^2) \sqrt{G}~ d^3 q,\end{aligned}$$ where $D$ is the diffusion constant, and $G = \det(G_{\mu\nu}) = \det (G_{ij})$. Our aim is to construct the effective two dimensional diffusion equation from 3D equation above. $$\begin{aligned} && \frac{\partial \phi^{(2)}}{\partial t} = D \Delta^{(eff)} \phi^{(2)},\\ 1 &=& \int \phi^{(2)}(q^1,q^2) \sqrt{g}~ d^2 q,\end{aligned}$$ where $\phi^{(2)}$ is the two dimensional diffusion field, $g = \det(g_{ij})$, and $\Delta^{(eff)}$ is unknown effective 2D diffusion operator which might not be equal to simple 2D Laplace Beltrami operator. From two normalization conditions, we obtain $$\begin{aligned} 1 &=& \int \phi^{(3)}(q^0,q^1,q^2) \sqrt{G}~ d^3 q,\nonumber \\ &=& \int [\int_{-\epsilon/2}^{\epsilon/2} d q^0 (\phi^{(3)} \sqrt{G/g})] ~ \sqrt{g} ~ d^2 q, \nonumber\\ &=& \int \phi^{(2)}(q^1,q^2) \sqrt{g}~ d^2 q.\nonumber\end{aligned}$$ Therefore we obtain the relation, $$\phi^{(2)}(q^1, q^2) = \int_{-\epsilon/2}^{\epsilon/2} \tilde{\phi}^{(3)} d q^0, \label{eq:relation}$$ where $$\tilde{\phi}^{(3)} \equiv \phi^{(3)} \sqrt{G/g}.$$ We multiply $\sqrt{G/g}$ to equation (\[eq:diff\]) and integrate by $q^0$, then we obtain $$\frac{\partial \phi^{(2)}}{\partial t} = D \int_{-\epsilon/2}^{\epsilon/2} \tilde{\Delta}^{(3)} \tilde{\phi}^{(3)} dq^0,\label{eq:laplace}$$ where $$\tilde{\Delta}^{(3)} \equiv \sqrt{G/g} ~~ \Delta^{(3)}\sqrt{g/G}.$$ Next we analyze new operator $\tilde{\Delta}^{(3)}$. From the form of Laplace Beltrami operator $$\Delta^{(3)} = G^{-1/2} \frac{\partial}{\partial q^\mu} G^{1/2} G^{\mu\nu} \frac{\partial}{\partial q^\nu},$$ we have $$\begin{aligned} \tilde{\Delta}^{(3)} &=& g^{-1/2} \frac{\partial}{\partial q^\mu} G^{1/2} G^{\mu\nu} \frac{\partial}{\partial q^\nu}(g/G)^{1/2}\nonumber\\ &=& \tilde{\Delta}^{(2)} + \tilde{\Delta}^{(1)}, \label{eq:laplace2}\end{aligned}$$ where $$\tilde{\Delta}^{(2)} \equiv g^{-1/2} \frac{\partial}{\partial q^i} G^{1/2} G^{ij} \frac{\partial}{\partial q^j}(g/G)^{1/2},$$ and $$\tilde{\Delta}^{(1)} \equiv \frac{\partial}{\partial q^0} G^{1/2} \frac{\partial}{\partial q^0}G^{-1/2}.$$ Then our diffusion equation has form $$\frac{\partial \phi^{(2)}}{\partial t} = D \int_{-\epsilon/2}^{\epsilon/2} \tilde{\Delta}^{(2)} \tilde{\phi}^{(3)} dq^0. \label{eq:new}$$ The contribution from $ \tilde{\Delta}^{(1)} $ vanishes because $$\begin{aligned} \int_{-\epsilon/2}^{\epsilon/2} \tilde{\Delta}^{(1)} \tilde{\phi}^{(3)} d q^0 &=& g^{-1/2} \int_{-\epsilon/2}^{\epsilon/2} \frac{\partial}{\partial q^0} (G)^{1/2} \frac{\partial }{\partial q^0} \phi^{(3)} ~ dq^0 \nonumber\\ &=& g^{-1/2} [~ (G)^{1/2} \frac{\partial \phi^{(3)}}{\partial q^0}] \mid _{-\epsilon/2}^{\epsilon/2} =0. \end{aligned}$$ The last equality is the requirement that diffusion flow does not pass through the surface: $\Sigma'$ and $\tilde{\Sigma}$.\ Now we calculate r.h.s of (\[eq:new\]) up to ${\cal O}(\epsilon^2)$. Since we have $$\tilde{\phi}^{(3)} = {\cal O}(\epsilon^{-1}),$$ from (\[eq:relation\]), we need to expand $\tilde{\Delta}^{(2)}$ up to ${\cal O}(\epsilon^2)$. The following relations are useful $$\begin{aligned} G_{ij} &=& g_{ij} + 2 q^0 \kappa_{ij} + (q^0)^2 \kappa_{im} \kappa^m_j,\\ G^{ij} &=& g^{ij} - 2 q^0 \kappa^{ij} + 3 (q^0)^2 \kappa^i_{m} \kappa^{mj} + {\cal O}(\epsilon^3),\\ G_{~~} &=& g~ \{1 + 2 q^0 \kappa + (q^0)^2 (\kappa^2 + R) + {\cal O}(\epsilon^3)\},\\ G^{1/2}&=& g^{1/2} \{1 + q^0 \kappa + \frac{1}{2} (q^0)^2 R + {\cal O}(\epsilon^3)\},\end{aligned}$$ where $R = \kappa^2 - \kappa_{ij} \kappa^{ij}$ is used. Then the operator $\tilde{\Delta}^{(2)}$ can be expanded as follows $$\tilde{\Delta}^{(2)} = \Delta^{(2)} + q^0 \hat{A} + (q^0)^2 \hat{B} + {\cal O}(\epsilon^3),$$ where, $$\hat{A} = - g^{-1/2} \frac{\partial}{\partial q^i} g^{1/2} (2 \kappa^{ij} \frac{\partial}{\partial q^j} + g^{ij} \frac{\partial \kappa }{\partial q^j}),$$ $$\begin{aligned} \hat{B} &=& g^{-1/2} \frac{\partial}{\partial q^i} g^{1/2} (3 \kappa^{im} \kappa_m^j \frac{\partial}{\partial q^j} \nonumber\\ &+& \frac{1}{2} g^{ij} \frac{\partial (\kappa^2 -R) }{\partial q^j} + 2 \kappa^{ij} \frac{\partial \kappa }{\partial q^j} ).\end{aligned}$$ Then our two dimensional effective diffusion equation up to ${\cal O}(\epsilon)$ is, $$\begin{aligned} \frac{\partial \phi^{(2)}}{\partial t} &=& D \Delta^{(2)} \phi^{(2)} \nonumber\\ &+& D \hat{A} \int_{-\epsilon/2}^{\epsilon/2} q^0 \tilde{\phi}^{(3)} dq^0 \nonumber\\ &+& D \hat{B} \int_{-\epsilon/2}^{\epsilon/2} (q^0)^2 \tilde{\phi}^{(3)} dq^0 + {\cal O}(\epsilon^3).\end{aligned}$$ To proceed the $q^0$ integration, we suppose there is no diffusion flow in normal direction in layer , that is, $$0 = \frac{\partial \phi^{(3)}}{\partial q^0} = g^{1/2} \frac{\partial G^{-1/2} \tilde{\phi}^{(3)}}{\partial q^0}.$$ Solution is, $$\begin{aligned} \tilde{\phi}^{(3)} &=& \frac{1}{N} (G/g)^{1/2} \phi^{(2)}(q^1,q^2),\\ N &\equiv & \int_{-\epsilon/2}^{\epsilon/2} (G/g)^{1/2} dq^0.\end{aligned}$$ Each integration can be explicitly performed, and we obtain $$\begin{aligned} N ~~~~~ &=& \epsilon +\frac{R}{24} \epsilon^3 + {\cal O}(\epsilon^5),\\ <q^0> ~ &=& \frac{\kappa_{} \epsilon^2}{12} + {\cal O}(\epsilon^4),\\ <(q^0)^2> &=& \frac{\epsilon^2}{12} + {\cal O}(\epsilon^4),\end{aligned}$$ where we have used the definition $$<f(q^0)> \equiv \frac{1}{N} \int_{-\epsilon/2}^{\epsilon/2} f(q^0) (G/g)^{1/2} dq^0.$$ We obtain the final form of equation up to $ {\cal O}(\epsilon^2)$ as $$\begin{aligned} \frac{\partial \phi^{(2)}}{\partial t} &=& D \Delta^{(2)} \phi^{(2)} + \tilde{D} (\hat{A} \kappa + \hat{B}) \phi^{(2)}\nonumber \\ &=& D \Delta^{(2)} \phi^{(2)} + \tilde{D} g^{-1/2} \frac{\partial}{\partial q^i} ~ g^{1/2} \nonumber\\ &\times& \{ (3 \kappa^{im} \kappa_m^j -2 \kappa \kappa^{ij}) \frac{\partial}{\partial q^j} - \frac{1}{2} g^{ij} \frac{\partial R}{\partial q^j} \} \phi^{(2)},~~~\end{aligned}$$ where $\tilde{D} = \frac{\epsilon^2}{12} D$.\ We give two comments here. First, ${\cal O}(\epsilon)$ term disappears. Since $\epsilon$ has the dimension of length, it always appears with curvature $\kappa$. Therefore the 1st order term, if it exists, it contains 1st order of curvature $\kappa$. But this curvature depends on unphysical choice of normal unit vector $\vec{n}$, and so it does not appear. Second, additional potential term disappears. In quantum mechanics, the similar embedding techniques leads to the appearance of additional potential term written by curvature. But in our classical case we have no such terms. Because in diffusion equation, potential term breaks probability conservation law, i.e. $$\partial \phi / \partial t =( D \Delta + V(x) )\phi,$$ $$\to ~ \frac{d}{dt} \int d^3x ~\phi = \int d^3x V(x) \phi \neq 0.$$ The normal diffusion flow can be written in general coordinate, $$J_N^i = -D g^{ij} \frac{\partial \phi^{(2)}}{\partial q^j}.$$ The anomalous diffusion flow is, $$J_A^i = -\tilde{D} \{ (3 \kappa^{im} \kappa_m^j - 2\kappa \kappa^{ij})\frac{\partial \phi^{(2)}}{\partial q^j} - \frac{1}{2} g^{ij} \frac{\partial R }{\partial q^j}\phi^{(2)}\}. \label{eq:anom}$$ The Diffusion equation is written as $$\begin{aligned} -\frac{\partial \phi^{(2)}}{\partial t} &=& \nabla_i (J_N^i + J_A^i),\nonumber\\ &=& g^{-1/2} \frac{\partial }{\partial q^j} ~g^{1/2} (J_N^i + J_A^i),\end{aligned}$$ where $\nabla_i$ is the covariant derivative. By using a suitable boundary condition, we can prove $$\frac{d}{dt} \int \phi^{(2)} g^{1/2} d^2 q =0.$$ Properties of curvature dependent flow ====================================== The anomalous flow equals to zero for the flat surface. The last term in equation (\[eq:anom\]) shows that curvature gradient generate the flow without particle density gradient. From the signature of this term, this flow goes from the smaller Ricci scalar point to the larger Ricci scalar point. Ricci scalar can take the negative, zero, and positive values. (Ricci scalar $R$ is related to Gauss curvature by $R/2 = \det[\kappa^i_j]$.) Let us work with the coordinate which satisfies $$g_{ij} = \delta_{ij},~~ \kappa^i_j = ~ \mbox{diag}[1/r_1,~1/r_2],$$ at point $P$, where $r_i$ is the curvature radius along the $q^i$ coordinate and it takes positive or negative value for convex or concave. (The metric can be diagonalized by choosing the two coordinates as to satisfy orthogonality, and it can be normalized by using the re-parametrization. The second fundamental tensor is diagonalized by rotation of coordinate.) Then at point $P$, we have $R= \frac{2}{r_1r_2}$ and, - $R > 0$   if the surface is convex or concave. - $R = 0$   if the surface is essentially flat. - $R < 0$   if the surface is hyperbolic. Therefore the flow goes from hyperbolic or flat points to convex or concave points with positive larger Ricci scalar value.\ Next we consider the first term in (\[eq:anom\]). We have positive or negative value for $$f^{ij} \equiv 3 \kappa^{im} \kappa_m^j - 2\kappa \kappa^{ij},$$ depending on the value of curvature. In our coordinate, we can immediately write it in the simple form $$f^{ij} = \delta^{ij} ( \frac{1}{(r_i)^2} - \frac{2}{r_1 r_2}).$$ When the surface is hyperbolic ($R<0$), $$f^{11} = \frac{1}{(r_1)^2} + \frac{2}{\mid r_1 r_2 \mid}>0,~~ f^{22} = \frac{1}{(r_2)^2} + \frac{2}{\mid r_1 r_2 \mid}>0,$$ usual diffusion occurs (See fig. 2). ![Wave packet on hyperbolic surface diffuses in two directions.](fig2.eps){width="4cm"} When the surface is convex or concave ($R>0$), $$f^{11} = \frac{1}{(r_1)^2} - \frac{2}{\mid r_1 r_2 \mid} \\ = \frac{\mid r_2\mid -2 \mid r_1\mid}{\mid r_1\mid^2 \mid r_2\mid},$$ $$f^{22} = \frac{1}{(r_2)^2} - \frac{2}{\mid r_1 r_2 \mid}\\ = \frac{\mid r_1\mid -2 \mid r_2\mid}{\mid r_2\mid^2 \mid r_1\mid}.$$ In this case, we have three possibilities. One possibility is that both are negative, if $$1 /2 < \mid \frac{r_2}{r_1}\mid < 2.$$ Then we have no diffusion but concentration occurs (See fig. 3). ![Wave packet on convex surface concentrates.](fig3.eps){width="4cm"} The second possibility is that one of two is positive and the other is negative, if $$\mid \frac{r_2}{r_1}\mid < 1/2, ~~ \mbox{or} ~\mid \frac{r_2}{r_1}\mid > 2.$$ Then we have diffusion in one direction, but concentration in another direction (See fig. 4). The third possibility for $R>0$ is, $$\mid \frac{r_2}{r_1}\mid = 1/2, ~~ \mbox{or} ~ \mid \frac{r_2}{r_1}\mid =2.$$ This is critical point, where the flow stops for larger curvature direction and flow concentrates for smaller curvature direction.\ ![Wave packet on convex surface with one direction curvature is over two times higher than another. Packet diffuses in higher curvature direction and concentrates in smaller curvature direction.](fig4.eps){width="4cm"} When the Ricci scalar is zero ($R =0$), for example $r_2 = \infty$, $f^{22}=0$ and $f^{11}>0$ follows. The diffusion occurs only in $q^1$ direction but not in another direction. (See fig. 5). ![Wave packet on elliptic cylinder. Packet diffuses only in curved direction but not in another direction.](fig5.eps){width="4cm"} In this way, this anomalous diffusion flow has much varieties depending on the curvature.\ One Example: Elliptic Cylinder ============================== Let us consider one simple example where diffusion coefficient depends on curvature. The surface of elliptic cylinder is the case of $R=0$ just as figure 5, but the surface has non zero mean curvature. Ellipsoid is given by the equation $$(\frac{x}{a})^2 + (\frac{y}{b})^2 = 1 .$$ ![Elliptic cylinder](fig6.eps){width="3cm"} Then any points on cylinder are specified by curved coordinate $\theta$ and $z$; $$x = a \cos \theta, ~~ y = b \sin \theta.$$ Another choice of coordinate instead of $\theta$ is, $$du = \sqrt{dx^2+ dy^2} = f(\theta) ~d\theta.$$ where $f(\theta)$ is defined as $$f(\theta) \equiv \sqrt{ a^2 \sin^2\theta + b^2 \cos^2 \theta} .$$ The length of $u$ is given by $$\begin{aligned} u(\phi) &=& \int_0^{\phi} f(\theta) d\theta \nonumber\\ &=& b \int_0^{\phi} \sqrt{1-k^2 \sin^2 \theta} d\theta \equiv b E(k,\phi),\label{eq:u}\end{aligned}$$ with $k= \sqrt{1-(a/b)^2},~ a \leq b.$ $E(k,\phi)$ is the Elliptic integral of the second kind.  (See appendix)\ The total length of $u$ is given by $$U \equiv 4b E(k, \pi/2).$$ We use the normalized value for $u$ hereafter such that, $$\tilde{u} = u/U, ~~ 0 \leq \tilde{u} \leq 1.$$ The normal unit vector $\vec{n}$ is given as $$\begin{aligned} \vec{n} &=& (\frac{x}{a^2 \sqrt{(x^2/a^4) + (y^2/b^4)}}, \frac{y}{b^2 \sqrt{(x^2/a^4) + (y^2/b^4)}})\nonumber\\ &=& \frac{1}{f(\theta)} (b \cos \theta, a \sin \theta).\end{aligned}$$ $$\frac{\partial \vec{n}}{\partial \theta} = \frac{1}{f} (-b \sin \theta, a \cos \theta) - \frac{\partial_\theta f}{f} \vec{n}.$$ $$\vec{B}_\theta = \frac{\partial \vec{x}}{\partial \theta} = (-a \sin \theta, b \cos \theta).$$ Then we obtain the second fundamental tensor. $$\kappa_{\theta\theta} =\vec{B}_\theta \cdot \frac{\partial \vec{n}}{\partial \theta} = \frac{ab}{f} .$$ Then we collect all the necessary quantities as follows $$\begin{aligned} && g_{\theta\theta}=f^2, ~~ g_{zz}=1, ~~ g_{\theta z}=0, \nonumber\\ && \kappa_{\theta\theta} = \frac{ab}{f}, ~~\kappa_{zz} = \kappa_{\theta z}=0, ~~ \kappa = \frac{ab}{f^3}. \label{eq:mean}\end{aligned}$$ Then we obtain the total diffusion equation expressed by the parameters $\theta$ and $z$. $$\frac{\partial \phi^{(2)}}{\partial t} = (\frac{1}{f}\frac{\partial}{\partial \theta}) D_{\theta} (\frac{1}{f}\frac{\partial}{\partial \theta}) \phi^{(2)} + D \frac{\partial^2}{\partial z^2} \phi^{(2)},$$ where the effective diffusion coefficient depends on mean curvature. $$D_{\theta} = D (1 + \frac{\epsilon^2 \kappa^2}{12}) = D (1 + \varepsilon^2 (b\kappa)^2) ,$$ where $\varepsilon \equiv \epsilon /(2\sqrt{3}b)$. Under this equation, we obtain the following particle number conservation law. $$\frac{d}{dt} \int dz \int_0^{2\pi} d\theta ~f(\theta)~ \phi^{(2)}(\theta, z) ~= 0$$ with suitable Neumann boundary condition. By using the variable $\tilde{u}$ instead of $\theta$ we have simple dimensionless equation, $$\frac{\partial \phi^{(2)}}{\partial \tau} = \frac{\partial}{\partial \tilde{u}} (1 + V) \frac{\partial}{\partial \tilde{u}} \phi^{(2)} + \frac{\partial^2}{\partial \eta^2} \phi^{(2)}$$ where $ \tau = t D/U^2, ~\eta = z/U, ~ V = \varepsilon^2 (b\kappa)^2, ~ U= 4b E(k,\pi/2)$.\ By using the approximation of elliptic function given in appendix, curvature dependent potential $V$ can be written as function of $\tilde{u}$. The simulation can be done as usual diffusion equation. For $0 \leq \tilde{u} \leq 1$ and $0 \leq \eta \leq 4$ using periodic boundary condition, since the length of $\eta$ is larger than one of $\tilde{u}$, the diffusion in $u$ direction occurs fastly and then the diffusion in $\eta$ direction follows slowly just like $\phi \sim a + \sum_k b(k) e^{- k^2 \tau} \cos(k \eta)$. The $u$-directional diffusion can not occur uniformly, because at $\tilde{u}=0.25,$ and $0.75$ the diffusive coefficient is higher than other points. Therefore the slope of diffusion field is small especially at these two points during the diffusion process. (fig. 7) ![Snap shot of diffusion process starting from the wave packet $\phi = \sin^{10} (\pi \tilde{u})$ as the initial condition. At two points (0.25 and 0.75), diffusion occurs quickly and its slope is smaller than other.](fig7.eps){width="9cm"} ![Mean curvature as a function of $\tilde{u}$ when $b/a=2$](fig8.eps){width="4cm"} Conclusion ========== We have discussed on the diffusion equation on curved surface embedded in $R_3$. We obtained the new diffusion equation up to ${\cal O}(\epsilon^2)$ which includes anomalous diffusive flow additional to the usual one. This anomalous flow depends on the second fundamental tensor, and it has not only diffusion but also concentration properties depending on the curvature of its manifold. At the point with negative Ricci scalar $R<0$, surface is hyperbolic, and diffusion in both direction occurs. (fig.2) When Ricci scalar is positive $R>0$, we have three possibilities. $r_i$ appearing below is curvature radius in each direction ($i=1, 2$). - Concentration in both direction (fig.3), when $$1 /2 < \mid \frac{r_2}{r_1}\mid < 2.$$ - Concentration in smaller curvature direction, and diffusion in higher curvature direction (fig.4), when $$\mid \frac{r_2}{r_1}\mid < 1/2, ~~ \mbox{or} ~\mid \frac{r_2}{r_1}\mid > 2.$$ - Concentration in smaller curvature direction, and no flow in higher curvature direction, when $$\mid \frac{r_2}{r_1}\mid = 1/2, ~~ \mbox{or} ~\mid \frac{r_2}{r_1}\mid = 2.$$ When Ricci scalar is zero $R=0$, surface is essentially flat, but we have finite curvature radius in one direction. Then we have diffusion only in this direction (fig.5). In the case of surface of elliptic cylinder, we gave a concrete form of equation and we showed the curvature dependent diffusion coefficient. $$D_{u} = D (1 + \frac{\epsilon^2 \kappa^2}{12}), ~~~ D_{z} = D,$$ where $\kappa$ is the mean curvature. In this case curvature dependence is simply included into diffusion coefficient. However this is not true in general case, where situation is much more complicated, and this can be seen from the form of anomalous flow. The application to the pattern formation by reaction diffusion using this obtained equation is not yet finished. This will be done in further publication. Appendix ======== We approximate the elliptic integral of the second kind. $$E(k,\phi) \equiv \int_0^\phi \sqrt{1-k^2 \sin^2 \theta} ~ d\theta$$ with $$k = \sqrt{1-(a/b)^2}.$$ Under the expansion in powers of $k^2$, we obtain the power series of Elliptic integral of the second kind. $$E(k,\phi) = \phi - \sum_{n=1}^{\infty} \frac{k^{2n} (2n-3)!!}{n! ~ 2^n} \int_0^\phi \sin^{2n}\theta d\theta.$$ Since the integration part can be expanded by $\phi$ and $\sin 2n \phi$, we have $$E(k,\phi) = a_0 \phi + \sum_{n=1}^{\infty} a_n \sin 2n \phi.$$ with the relation $$\begin{aligned} a_0 &=& \frac{2}{\pi} E(k,\pi/2),\\ a_n &=& (-1)^n \frac{2E(k,\pi/2)}{n \pi} \nonumber\\ && + \frac{4}{\pi} \int_0^{\pi/2} \sin 2n\phi ~E(k,\phi) d\phi. ~~(n \geq 1)\end{aligned}$$ For the real Char fishes, $b/a$ takes values $1.5 \sim 2.5$. Then the value of $k$ takes $0.75 \sim 0.92$. When $b/a=2$, each values of $a_n$ is given numerically $$a_0=0.771,~~a_1=0.123,~~a_2=-0.00506,~~a_3=0.000558.$$ Now we have for $u$ given in (\[eq:u\]), $$u/b = E(k,\phi) = a_0 \phi + a_1 \sin 2\phi + a_2 \sin 4\phi + \cdots.$$ And we rewrite it by using normalized $u$, $$\phi = 2\pi \tilde{u} - \frac{a_1}{a_0} \sin 2\phi - \frac{a_2}{a_0} \sin 4\phi - \cdots,$$ where $\tilde{u} = u/(4b E(k, \pi/2)).$ The iteration method up to order $(a_1/a_0)^1$ gives $$\phi = 2\pi \tilde{u} - \frac{a_1}{a_0} \sin (4 \pi \tilde{u}).$$ Then we take the derivative by $u$ in both hand sides. $$\frac{1}{f} = \frac{1}{b a_0} (1 - \frac{2 a_1}{a_0} \cos (4 \pi \tilde{u})),$$ where the following relation is used. $$\frac{du}{d\phi} = f(\phi) \equiv \sqrt{a^2 \sin^2\phi + b^2 \cos^2\phi}.$$ Then the mean curvature given by (\[eq:mean\]) is obtained as function of $u$. $$b \kappa = \frac{a b^2}{f^3} = \frac{1}{\beta (a_0)^3} (1 - \frac{2 a_1}{a_0} \cos (4 \pi \tilde{u}))^3,$$ where $\beta = b/a$. 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--- abstract: 'The alternating-current optimal power flow [(ACOPF)]{} is one of the best known non-convex non-linear optimisation problems. [We present a]{} novel re-formulation of ACOPF, [which is based on lifting the rectangular power-voltage rank-constrained formulation,]{} and makes it possible to derive [alternative]{} SDP relaxations. For those, we develop a first-order method based on the [parallel]{} coordinate descent [with a novel closed-form step based on roots of cubic polynomials]{}.' author: - Jakub Mareček and Martin Takáč bibliography: - 'acopf.bib' - 'literature.bib' title: 'A Low-Rank Coordinate-Descent Algorithm for Semidefinite Programming Relaxations of Optimal Power Flow ' --- Introduction ============ Alternating-current optimal power flow problem (ACOPF) is one of the best known non-linear optimisation problems [@wood1996power]. Due to its non-convexity, deciding feasibility is NP-Hard even for a tree network with fixed voltages [@Lehmann2015]. Still, there has been much recent progress [[@6756976; @6815671]]{}: Bai et al. [@Bai2008] introduced a semidefinite programming (SDP) relaxation, which turned out to be [particularly]{} opportune. Lavaei and Low [@lavaei2012zero] have shown [that]{} the SDP relaxation produces exact solutions, under certain spectral conditions. More generally, it can be strengthened so as to obtain a hierarchy of SDP relaxations whose optima are asymptotically converging to the true optimum of ACOPF [@Ghaddar2015; @RTE2013; @josz2015moment]. Unfortunately, the run-times of [even]{} the best-[performing]{} solvers for the SDP relaxations [[@sturm1999; @permenter2015solving] remain]{} much higher than that of commonly used methods without global convergence guarantees such as Matpower [@Matpower]. Following a brief overview of our notation, we introduce a novel rank-constrained reformulation of the ACOPF problem [in Section \[sec:reform\]]{}, where all constraints, except for the rank constraint, are coordinate-wise. [Based on this reformulation, we derive novel SDP relaxations.]{} Next, we present a parallel coordinate-descent algorithm [in Section \[sec:algo\]]{}, which solves a sequence of convex relaxations with coordinate-wise constraints, using a novel closed-form step considering roots of cubic polynomials. We can show: - the algorithm converges to the exact optimum of the non-convex problem, when the optimum of the non-convex problem coincides with the optimum of the novel SDP relaxations and certain additional assumptions are satisfied, as detailed in Section \[sec:analysis\] - the algorithm suggests a strengthening of the novel SDP relaxations is needed, whenever it detects the optimum of the non-convex problem has not been found, using certain novel sufficient conditions of optimality - our pre-liminary implementation reaches a precision comparable to the default settings of Matpower [@Matpower], the commonly used interior-point method without global convergence guarantees, on certain well-known instances including the IEEE 118 bus test system, within comparable times, as detailed in Section \[sec:numerical\] [although much more work remains to be done, especially with focus on large-scale instances. Also, as with most solvers, the proposed assumes feasibility and does not certify infeasibility, when encountered. ]{} The proofs of convergence rely on the work of Burer and Monteiro [@BurerMonteiro2005], Grippo et al. [@grippo2009necessary], and our earlier work [@RT2015; @marecek2015distributed]. The Problem =========== Informally, within the optimal power flow problem, one aims to decide where to generate power, such that the demand for power is met and costs of generation are minimised. In the alternating-current model, one considers the complex voltage, complex current, and complex power, although one may introduce decision variables representing only a subset thereof. The constraints are non-convex in the alternating-current model, and a particular care hence needs to be taken when modelling those, and designing the solvers to match. Formally, we start a network of buses ${N}$, connected by branches $L \subseteq {N}\times {N}$, modeled as $\Pi$-equivalent circuits, with the input comprising also of: - $G \subseteq {N}$, which are the generators, [with the associated coefficients $c^0_k, c^1_k, c^2_k$ of the quadratic cost function at $k \in G$]{}, - $P^d_k + jQ^d_k$, which are the active and reactive loads at each bus $k \in {N}$, - $P_k^{\min}$, $P_k^{\max}$, $Q_k^{\min}$ and $Q_k^{\max}$, which are the limits on active and reactive generation capacity at bus $k \in G$, where $P_k^{\min}=P_k^{\max}=Q_k^{\min}=Q_k^{\max} =0$ for all $k \in {N}\setminus G$, - $y \in \mathbb{C}^{|N|\times|N|}$, which is the network admittance matrix capturing the value of the shunt element $\bar{b}_{lm}$ and series admittance $g_{lm}+jb_{lm}$ at branch $(l,m) \in L$, - $V_k^{\min}$ and $V_k^{\max}$, which are the limits on the absolute value of the voltage at a given bus $k$, - $S_{lm}^{\max}$, which is the limit on the absolute value of the apparent power of a branch $(l,m)\in L$. In the rectangular power-voltage formulation, the variables are: - $P^g_k + j Q^g_k$, which is the power generated at bus $k \in G$, - $P_{lm} + j Q_{lm}$, which is the power flow along branch $(l,m) \in L$, - $\Re{V_k}+ j \Im{V_k}$, which is the voltage at each bus $k \in N$. The power-flow equations at generator buses $k \in G$ are: $$\begin{aligned} P^g_k &= P_k^d+\Re{V_k} \sum_{i=1}^n ( { \Re{y_{ik}} \Re{V_i} - \Im{y_{ik}} \Im{V_i} }) + \Im{V_k} \sum_{i=1}^n ({ \Im{y_{ik}} \Re{V_i} + \Re{y_{ik}} \Im{V_i} }), \label{eqn:Pk} \\ Q^g_k &= Q_k^d+\Re{V_k} \sum_{i=1}^n ({ - \Im{y_{ik}} \Re{V_i} - \Re{y_{ik}} \Im{V_i} } ) + \Im{V_k} \sum_{i=1}^n ({ \Re{y_{ik}} \Re{V_i} - \Im{y_{ik}} \Im{V_i} }), \label{eqn:Qk}\end{aligned}$$ [while at all other buses $k \in N \setminus G$ we have:]{} $$\begin{aligned} 0 &= P_k^d+\Re{V_k} \sum_{i=1}^n ( { \Re{y_{ik}} \Re{V_i} - \Im{y_{ik}} \Im{V_i} }) + \Im{V_k} \sum_{i=1}^n ({ \Im{y_{ik}} \Re{V_i} + \Re{y_{ik}} \Im{V_i} }), \\ 0 &= Q_k^d+\Re{V_k} \sum_{i=1}^n ({ - \Im{y_{ik}} \Re{V_i} - \Re{y_{ik}} \Im{V_i} } ) + \Im{V_k} \sum_{i=1}^n ({ \Re{y_{ik}} \Re{V_i} - \Im{y_{ik}} \Im{V_i} }).\end{aligned}$$ [Additionally, the power flow at branch $(l,m) \in L$ is expressed as]{} $$\begin{aligned} P_{lm} &= b_{lm} ( \Re{V_l} \Im{V_m} - \Re{V_m} \Im{V_l}) \label{eqn:Plm} + g_{lm}( \Re{V_l}^2 +\Im{V_m}^2- \Im{V_l}, \Im{V_m}- \Re{V_l} \Re{V_m}), \\ Q_{lm} &= b_{lm} ( \Re{V_l} \Im{V_m} - \Im{V_l} \Im{V_m}-\Re{V_l}^2 -\Im{V_l}^2) \notag \\ & \ + g_{lm}( \Re{V_l}\Im{V_m}- \Re{V_m} \Im{V_l}- \Re{V_m} \Im{V_l}) -\frac{\bar b_{lm}}{2}(\Re{V_l}^2 +\Im{V_l}^2). \label{eqn:Qlm}\end{aligned}$$ [Considering the above,]{} the alternating-current optimal power flow (ACOPF) is: $$\begin{aligned} \min &\sum_{k\in G}\left( c^2_k (P_k^g + P_k^d )^2 + c^1_k (P_k^g + P_k^d) + c^0_k \right)\hspace{-0.2cm}& \tag*{[ACOPF]} \label{ACOPF} \\ {\mbox{s.t.} \ }& P^{\min}_k \le P^g_k + P^d_k \le P^{\max}_k & \forall k \in G\\ & Q^{\min}_k \le Q^g_k + Q^d_k \le Q^{\max}_k & \forall k \in G \\ & (V_k^{\min})^2 \le\Re{V_k}^2+\Im{V_k}^2 \le ( V_k^{\max})^2 \hspace{-0.3cm}& \forall k \in N \\ & P_{lm}^2+Q_{lm}^2 \le (S_{lm}^{\max})^2 & \forall (l,m) \in L\\ & \eqref{eqn:Pk}-\eqref{eqn:Qlm},&\end{aligned}$$ [where $c^2_k$ is the coefficient of the leading term of the quadratßic cost function at generator $k$.]{} In line with recent work [@Bai2008; @lavaei2012zero; @molzahn2011; @Ghaddar2015], let $e_k$ be the $k^{th}$ standard basis vector in $\mathbb{R}^{2|N|}$ and define: $$\begin{aligned} y_k&=e_ke_k^T y, \\ y_{lm}&=(j\frac{\bar b_{lm}}{2}+g_{lm}+jb_{lm})e_le_l^T - (g_{lm}+jb_{lm})e_le_m^T, \\ Y_k&= \frac{1}{2} \left[ \begin{matrix} \Re(y_k + y_k^T) &\Im(y_k^T -y_k) \\ \Im(y_k - y_k^T)& \Re(y_k + y_k^T) \end{matrix} \right], \\ \bar{Y}_k&= -\frac{1}{2} \left[ \begin{matrix} \Im(y_k + y_k^T)& \Re(y_k -y_k^T) \\ \Re(y_k^T - y_k) & \Im(y_k + y_k^T) \end{matrix} \right], \\ M_k&= \left[ \begin{matrix}e_ke_k^T & 0 \\ 0 & e_ke_k^T \end{matrix} \right], \\ Y_{lm}&= \frac{1}{2} \left[ \begin{matrix} \Re(y_{lm} + y_{lm}^T)& \Im(y_{lm}^T -y_{lm}) \\ \Im(y_{lm} - y_{lm}^T) & \Re(y_{lm} + y_{lm}^T) \end{matrix} \right], \\ \bar Y_{lm} &= \frac{{ - 1}}{2}\left[ {\begin{array}{*{20}c} {\Im ( y_{lm} + y_{lm}^T ) } & {\Re ( y_{lm} - y_{lm}^T ) } \\ {\Re ( y_{lm}^T - y_{lm} ) } & {\Im ( y_{lm} + y_{lm}^T ) } \\ \end{array}} \right]. \end{aligned}$$ Using these matrices, we can rewrite \[ACOPF\] as a real-valued polynomial optimization problem of degree four in variable $x \in \R^{2|N|}$ comprising of $\Re{V}_k \in \R^{|N|}$ and $\Im{V}_k \in \R^{|N|}$, stacked: $$\begin{aligned} \min &\sum_{k\in G} \left(c^2_k(\text{tr}(Y_k xx^T) + P_k^d)^2+c^1_k(\text{tr}(Y_k xx^T) + P_k^d)+c^0_k\right) \tag*{[PP4]} \label{PP4} & \\ {\mbox{s.t.} \ }&P_k^{\min}\leq \text{tr}(Y_kxx^T) + P_k^d \leq P_k^{\max} & \forall k \in G \label{eqn:pp1} \\ &P_k^{d} = \text{tr}(Y_kxx^T) & \forall k \in N \setminus G \label{eqn:pp2} \\ & Q_k^{\min}\leq \text{tr}(\bar{Y}_kxx^T) + Q_k^d \leq Q_k^{\max} & \forall k \in G \\ &Q_k^{d} = \text{tr}(Y_kxx^T) & \forall k \in N \setminus G \\ &(V_k^{\min})^2 \leq \text{tr}(M_kxx^T) \leq (V_k^{\max})^2 & \forall k \in N \label{eqn:pp3} \\ &(\text{tr}(Y_{lm}xx^T))^2+(\text{tr}(\bar{Y}_{lm}xx^T))^2 \leq (S_{lm}^{\max})^2 & \forall (l,m) \in L, \label{eqn:pp4}\end{aligned}$$ Henceforth, we use $n$ to denote $2|N|$, i.e., the dimension of the real-valued problem \[PP4\]. Further, the problem can be lifted to obtain a rank-constrained problem in $W=xx^T \in \R^{n \times n}$: $$\begin{aligned} \min &\sum_{k\in G} f_k(W) \tag*{[R1]} \label{R1} & \\ {\mbox{s.t.} \ }&P_k^{\min} \leq \text{tr}(Y_kW) + P_k^d \leq P_k^{\max} & \forall k \in G \label{R1-Constraint1} \\ & P_k^{d} = \text{tr}(Y_kW) & \forall k \in N \setminus G \\ & Q_k^{\min}\leq \text{tr}(\bar{Y}_kW) +Q_k^d \leq Q_k^{\max} & \forall k \in G \\ & Q_k^d = \text{tr}(\bar{Y}_kW) & \forall k \in N \setminus G \\ &(V_k^{\min})^2 \leq \text{tr}(M_kW) \leq (V_k^{\max})^2 & \forall k \in N \\ &(\text{tr}(Y_{lm}W))^2+(\text{tr}(\bar{Y}_{lm}W))^2 \leq (S_{lm}^{\max})^2 & \forall (l,m) \in L \label{R1-Constraint4} \\ & W \succeq 0, \quad {\textrm{rank}}(W) = 1, &\end{aligned}$$ for a suitable definition of $f_k$. This problem \[R1\] is still NP-Hard, but where one can drop the rank constraint to obtain a strong and efficiently solvable SDP relaxation: $$\begin{aligned} \min &\sum_{k\in G} f_k(W) \; {\mbox{s.t.} \ }\; (\ref{R1-Constraint1}-\ref{R1-Constraint4}), \quad W \succeq 0, \tag*{[LL]} \label{LL}\end{aligned}$$ as suggested by [@Bai2008]. Lavaei and Low [@lavaei2012zero] studied the conditions, under which the relaxation \[LL\] (Optimization 3 of [@lavaei2012zero]) is equivalent to \[R1\]. We note that for traditional solvers [@molzahn2011; @lavaei2012zero; @Ghaddar2015], the dual of the relaxation (Optimization 4 of [@lavaei2012zero]) is much easier to solve than \[LL\][, as documented in Table \[tabComparisonWithGP\]]{}. Ghaddar et al. [@Ghaddar2015] have shown the relaxation \[LL\] to be equivalent to a first-level $[OP_4-H_1]$ of a certain hierarchy of relaxations, and how to obtain the solution to \[R1\] under much milder conditions than those of [@lavaei2012zero]. The Reformulation {#sec:reform} ================= The first contribution of this paper is a lifted generalisation of \[R1\]: $$\begin{aligned} \min & \sum_{k\in G} f_k(W) \tag*{[R$r$BC]} \label{RrBC} \\ {\mbox{s.t.} \ }& t_k = \text{tr}(Y_kW) & \forall k \in N \label{eq:bc0}\\ P_k^{\min} - P_k^d \leq \;& t_k \leq P_k^{\max} - P_k^d & \forall k \in G \label{eq:bc1}\ \\ & P_k^{d} = t_k & \forall k \in N \setminus G \\ & g_k = \text{tr}(\bar{Y}_kW) & \forall k \in N \\ Q_k^{\min} - Q_k^d \leq \; & g_k \leq Q_k^{\max} - Q_k^d & \forall k \in G \label{eqn:bc2} \\ & Q_k^{d} = g_k & \forall k \in N \setminus G \\ & h_k = \text{tr}(M_kW) & \forall k \in N \\ (V_k^{\min})^2 \leq \; & h_k \leq (V_k^{\max})^2 & \forall k \in N \label{eqn:bc3} \\ & u_{lm} = \text{tr}(Y_{lm}W) & \forall (l,m) \in L \\ & v_{lm} = \text{tr}(\bar{Y}_{lm}W) & \forall (l,m) \in L \label{eqn:barLimLM}\\ & z_{lm} = (u_{lm})^2+(v_{lm})^2 & \forall (l,m) \in L \\ & z_{lm} \leq (S_{lm}^{\max})^2 & \forall (l,m) \in L \label{eqn:bc4} \\ & W \succeq 0, {\textrm{rank}}(W) \le r. \label{eqn:bc5}\end{aligned}$$ There, we still have: \[prop:1\] \[R1BC\] is equivalent to \[PP4\]. Even in the special case of $r = 1$, however, we have lifted the problem to a higher dimension by adding variables $t, g, h, u,v,z$, which are box-constrained functions of $W$. Subsequently, we make four observations to motivate our approach to solving the \[RrBC\]: - [constraints , , , and are box constraints, while the remainder of (\[eq:bc0\]–\[eqn:bc4\]) are linear equalities]{} - using elementary linear algebra: $$\begin{aligned} \{ W \in \mathcal{S}^n \; \textrm{s.t.} \; W \succeq 0, \quad {\textrm{rank}}(W) \le r \} \\ = \{ RR^T \; \textrm{s.t.} \; R \in \R^{n \times r} \},\end{aligned}$$ where $\mathcal{S}^n$ is the set of symmetric $n \times n$ matrices. - if ${\textrm{rank}}(W^*) > 1$ for the optimum $W^*$ of \[LL\], there are no known methods for extracting the global optimum of \[R1\] from $W$, except for [@Ghaddar2015]. - zero duality gap at any SDP relaxation in the hierarchy of [@Ghaddar2015] does not guarantee the solution of the SDP relaxation is exact for \[PP4\], c.f. [@Josz2015]. Note that Lavaei and Low [@lavaei2012zero] restate the condition in Observation $O_3$ in terms of ranks using a related relaxation (Optimization 3), where the rank has to be strictly larger than 2. The Algorithm {#sec:algo} ============= Broadly speaking, we use the well-established Augmented Lagrangian approach [@powell1978fast; @conn1991globally], with a low-rank twist [@BurerMonteiro2005], and a parallel coordinate descent with a closed-form step. Considering Observation $O_2$, we replace variable $W \in \R^{n \times n}$ by $RR^T \in \R^{n \times n}$ for $R \in \R^{n \times r}$ to obtain the following augmented Lagrangian function: $$\begin{aligned} \label{AugLagrangian} &\mathcal{L}(R, t,h,g,u,v,z,\lambda^t, \lambda^g, \lambda^h, \lambda^{u},\lambda^{v},\lambda^z):= \\ & \sum_{k\in G} f_k(RR^T) \notag \\&- \sum_k \lambda^t_k (t_k - {\textrm{tr}}(Y_k RR^T)) + \tfrac{\mu}{2} \sum_k (t_k - {\textrm{tr}}(Y_k RR^T)) ^2 \notag \\&- \sum_k \lambda^g_k (g_k - {\textrm{tr}}(\bar Y_k RR^T)) + \tfrac{\mu}{2} \sum_k (g_k - {\textrm{tr}}(\bar Y_k RR^T)) ^2 \notag \\&- \sum_k \lambda_k^h (h_k - {\textrm{tr}}(M_k RR^T)) + \tfrac{\mu}{2} \sum_k (h_k - {\textrm{tr}}(M_k RR^T)) ^2 \notag \\&- \sum_{(l,m)} \lambda^u_{(l,m)} (u_{(l,m)} - {\textrm{tr}}(Y_{(l,m)} RR^T)) + \tfrac{\mu}{2} \sum_{(l,m)} (u_{(l,m)}- {\textrm{tr}}(Y_{(l,m)} RR^T)) ^2 \notag \\&- \sum_{(l,m)} \lambda^v_{(l,m)} (v_{(l,m)} - {\textrm{tr}}(\bar Y_{(l,m)} RR^T)) + \tfrac{\mu}{2} \sum_{(l,m)} (v_{(l,m)}- {\textrm{tr}}(\bar Y_{(l,m)} RR^T)) ^2 \notag \\&- \sum_{(l,m)} \lambda^z_{(l,m)} (z_{(l,m)} - u_{(l,m)}^2 -v^2_{(l,m)} ) + \tfrac{\mu}{2} \sum_{(l,m)} (z_{(l,m)} - u_{(l,m)}^2 -v^2_{(l,m)}) ^2 + \nu \mathcal{R}. \notag\end{aligned}$$ Note that constants $\mu, \nu > 0$ pre-multiply regularisers, where $\mathcal{R}$ can often be 0 in practice, although not in our analysis, where we require $\mathcal{R} = \det(R^T R)$, which promotes low-rank solutions. As suggested in Algorithm \[alg:SCDM\], we increase the rank $r = 1, 2, \ldots$ allowed in $W$ in an outer loop of the algorithm. In an inner loop, we use coordinate descent method to find an approximate minimizer of $\mathcal{L}(R, t,h,g,u,v,z,\lambda^t, \lambda^g, \lambda^h, \lambda^{u},\lambda^{v},\lambda^z)$, and denote the $k$-th iterate ${R^{(k)}}$. Note that variables $t,h,g,z$ have simple box constraints, which have to be considered outside of $\mathcal{L}$. In particular: \[alg:stp:initialpoint\] choose $R \in \R^{m\times r}$ compute corresponding values of $t,h,g,u,v,z$ project $t,h,g,u,v,z$ onto the box constraints \[inner-loop\] **in parallel**, minimize $\mathcal{L}$ in $t, g, h, u, v, z$, coordinate-wise \[updateT\] **in parallel**, minimize $\mathcal{L}$ in $R$, coordinate-wise \[updateR\] update Lagrange multipliers $\lambda^t, \lambda^g, \lambda^h, \lambda^{u},\lambda^{v},\lambda^z$ update $\mu$ terminate, if criteria are met \[inner-loop-end\] ### The Outer Loop {#the-outer-loop .unnumbered} The outer loop (Lines 1-12) is known as the “low-rank method” [@BurerMonteiro2005]. As suggested by Observation $O_3$, in the case of \[LL\], one may want to perform only two iterations $r = 1, 2$. In the second iteration of the outer loop, one should like to test, whether the numerical rank of the iterate $R_k$ in the inner iteration $k$ has numerical rank $1$. If this is the case, one can conclude the solution obtained for $r = 1$ is exact. This test, sometimes known as “flat extension”, has been studied both in terms of numerical implementations and applicability by Burer and Choi [@burer2006]. ### The Inner Loop {#the-inner-loop .unnumbered} The main computational expense of the proposed algorithm is to find an approximate minimum of $\mathcal{L}(R, t,h,g,u,v,z,\lambda^t, \lambda^g, \lambda^h, \lambda^{u},\lambda^{v},\lambda^z)$ with respect to $R,t,h,g,u,v,z$ within the inner loop (Lines 6–7). Note that $\mathcal{L}$ as a function of $R$ is – in general – non-convex. The inner loop employs a simple iterative optimization strategy, known as the coordinate descent. There, two subsequent iterates differ only in a single block of coordinates. In the very common special case, used here, we consider single coordinates, in a cyclical fashion. This algorithm has been used widely at least since 1950s. Recent theoretical guarantees of random coordinate descent algorithm are due to Nesterov [@Nesterov:2010RCDM] and the present authors [@RT2015; @marecek2015distributed]. See the survey of Wright [@Wright:ABCRRO] for more details. ### The Closed-Form Step {#the-closed-form-step .unnumbered} An important ingredient in the coordinate descent is a novel closed-form step. Nevertheless, if we update only one scalar of $R$ at a time, and fix all other scalars, the minimisation problem turns out to be the minimisation of a fourth order polynomial. In order to find the minimum of a polynomial $ax^4 + bx^3 + cx^2 + d x + 0$, we need to find a real root of the polynomial $4ax^3 + 3b x^2 + 2c x + d = 0$. This polynomial has at most 3 real roots and can be found using closed form formulae due to Cardano [ [@cardano1968ars]]{}. Whenever you fix the values across all coordinates except one, finding the best possible update for the one given coordinate requires either the minimisation of a quadratic convex function with respect to simple box constraints (for variables $t,g,h,z$) or minimisation of a polynomial function of degree 4 with no constraints (for variables $R, u, v$), either of which can be done by checking the objective value at each out of 2 (for variables $t,g,h,z$) or 3 (for variables $R, u, v$) stationary points and choosing the best one. ### The Parallelisation {#the-parallelisation .unnumbered} For instances large enough, one can easily exploit parallelism. Notice that minimization of coordinates of $t,g,h,u,v,z$ can be carried out in parallel without any locks, as there are no dependences. One can also update coordinates of $R$ in parallel, although some degradation of the speed-up thus obtainable is likely, as there can be some dependence in the updates. The degradation is hard to bound. Most analyses, c.f. [@Wright:ABCRRO], hence focus on the uniformly random choice of (blocks of) coordinates, although there are exceptions [@marecek2015distributed]. Trivially, one could also parallelise the outer loop, for each $r$ that should be considered. ### Sufficient Conditions for Termination of the Inner Loop {#sufficient-conditions-for-termination-of-the-inner-loop .unnumbered} For both our analysis and in our computational testing, we use a “target infeasibility” stopping criterion for the inner loop, considering squared error: $$\begin{aligned} {T_k({R^{(k)}})} = & \sum_k (t_k - {\textrm{tr}}(Y_k {R^{(k)}}({R^{(k)}})^T))^2 + \label{eq:T} \sum_k (g_k - {\textrm{tr}}(\bar Y_k {R^{(k)}}({R^{(k)}})^T))^2 + \\ & \sum_k (h_k - {\textrm{tr}}(M_k {R^{(k)}}({R^{(k)}})^T))^2 + \notag \sum_{(l,m)} (u_{(l,m)} - {\textrm{tr}}(Y_{(l,m)} {R^{(k)}}({R^{(k)}})^T))^2 + \notag \\ & \sum_{(l,m)} (v_{(l,m)} - {}\bar Y_{(l,m)} {R^{(k)}}({R^{(k)}})^T))^2 + \notag \sum_{(l,m)} (z_{(l,m)} - u_{(l,m)}^2 -v^2_{(l,m)} )^2. \notag \end{aligned}$$ We choose the threshold to match the accuracy in terms of squared error obtained by Matpower using default settings on the same instance. ${T_k({R^{(k)}})} \le 0.00001$ is often sufficient. ### The Initialisation {#the-initialisation .unnumbered} In our analysis, we assume that the instance is feasible. This is difficult to circumvent, considering Lehmann et al. [@Lehmann2015] have shown it is NP-Hard to test whether an instance of ACOPF is feasible. In our numerical experiments, however, we choose $R$ such that each element is independently identically distributed, uniformly over \[0, 1\], which need not be feasible for the instance of ACOPF. Subsequently, we compute $t,h,g,u,v,z$ to match the $R$, projecting the values onto the intervals given by the box-constraints. Although one may improve upon this simplistic choice by a variety of heuristics, it still performs well in practice. ### The Choice of $\mu_k, \nu_k$ {#the-choice-of-mu_k-nu_k .unnumbered} The choice of $\mu_k, \nu_k$ may affect the performance of the algorithm. In order to prove convergence, one requires $\lim_{k \to \infty} \nu_k \to 0$, but computationally, we consider $\mathcal{R} = 0$, which obliterates the need for varying $\nu_k$ and computation of the gradient, $\nabla \det(R^T R)$, as suggested by [@BurerMonteiro2005]. In order to prove convergence, any choice of $\mu_k > 0$ is sufficient. Computationally, we use $\mu_k = \mu = 0.0001$ throughout much of the reported experiments. On certain well-known pathological instances, c.f., Section \[sec:pathological\], other choices may be sensible. Considering that many of those instances are very small, the use of large step-sizes, such as $\mu_k = \mu = 0.01$ may seem justified. We have also experimented with an adaptive strategy for changing $\mu_k$, as detailed in Section \[sec:adaptivemu\]. There, we require the infeasibility to shrink by a fixed factor between consecutive iterations. If such an infeasibility shrinkage is not feasible, one ignores the proposed iteration update and picks $\mu_{k+1} < \mu_{k}$ such that the infeasibility shrinkage is observed. Although the performance does improve slightly, we have limited the use of this strategy to Section \[sec:adaptivemu\]. We have also experimented with decreasing $\mu_k$ geometrically, i.e. $\mu_k = c \cdot \mu_{k-1}$, but we have observed no additional benefits. The fact one does not have to painstakingly tune the parameters is certainly a relief. Implementation Details {#sec:details} ====================== In order to understand the details of the implementation, it is important to realise that - the quadratic forms such as $Y_k xx^T$ in the polynomial optimisation problem and $Y_k RR^T$ in the augmented Lagrangian are used for the simplicity of presentation. Indeed, the evaluation of the quadratic forms can be simplified considerably, when one does not strive for the brevity of expression - [many well-known benchmarks, including the Polish network, consider an extension of the polynomial optimisation problem .]{}\ We elaborate upon these points in turn. The Simplifications ------------------- Let us consider the example of ${\textrm{tr}}(Y_k xx^T)$ in more detail. The evaluation of the quadratic form can be simplified to $16 \; \|Y_k\|_0$ multiplications, i.e., performed in time linear in the number $\|Y_k\|_0$ of non-zero elements of the matrix $Y_k$. Moreover, the evaluation of the trace of the quadratic form requires only $2 \; \|Y_k\|_0$ multiplications, as one needs to consider only the diagonal elements. The number of non-zero elements varies in each power system, and is quadratic in the number of buses for hypothetical systems, where there would be a branch between each pair of buses, but the number of non-zero elements is linear in the number of buses, in practice. Consequently, one can simplify the first step of the inner loop (Lines 6–7), where one minimises the augmented Lagrangian , i.e., $\mathcal{L}(R, t,h,g,u,v,z,\lambda^t, \lambda^g, \lambda^h, \lambda^{u},\lambda^{v},\lambda^z)$, with respect to $t$, as follows. At iteration $k$, for each coordinate $j$, in parallel, one computes the update: $$\begin{aligned} t_j^{(k)} := & - \frac{ \pi^t_j + c^1_j S_b + 2 c^2_j S_b^2 (P_j^g)^2 - \frac{{\textrm{tr}}(Y_k {R^{(k)}}({R^{(k)}})^T)}{\mu}}{ \frac{1}{\mu} + 2 c^2_j S_b^2 },\end{aligned}$$ where $S_b$ is the base power of the per-unit system, such as 100 MVA, and $\pi^t$ are the residuals: $$\begin{aligned} \pi_j^t := & - \frac{ {\textrm{tr}}(Y_k R^{(k-1)}(R^{(k-1)})^T) - t_j^{(k-1)} }{\mu}. \label{residual-pit}\end{aligned}$$ Notice that the terms $c^1_j S_b$ and $2 c^2_j S_b^2$ are constant throughout the run and can hence be pre-computed, while the residuals have to be precomputed only once per iteration, e.g., just before the termination criteria are evaluated (Line 10), and hence there are only 3 multiplications and 1 division involved, in addition to the run-time of the evaluation of the quadratic form. Subsequently, one projects onto the box-constraints, i.e., if $t_j$ is large than the upper bound, it is set to the upper bound. If $t_j$ is smaller than the lower bound, it is set to the lower bound. Similar simplifications can be made for minimisation of the augmented Lagrangian with respect to other variables.\ The Extensions -------------- [In an often considered, but rarely spelled out]{} [@molzahn2011] extension of the ACOPF problem , [one allows]{} for tap-changing and phase-shifting transformers, as well as parallel lines and multiple generators connected to one bus. [Let us denote the the total line charging susceptance (p.u.) by $b_{lm}$, the transformer off nominal turns ratio by $t_{lm}$, and the transformer phase shift angle by $\phi_{lm}$. Then, the thermal limits become:]{} $$\begin{aligned} \left[ {\begin{array}{*{20}c} {(S_{lm}^{\max})^2 } & {-{\textrm{tr}}(Z_{lm} xx^T)} & {-{\textrm{tr}}(\bar{Z}_{lm} xx^T)} \\ {-{\textrm{tr}}( Z_{lm} xx^T)} & {1} & {0} \\ {-{\textrm{tr}}( \bar{Z}_{lm} xx^T)} & {0} & {1} \\ \end{array}} \right] & \succeq 0 \\ \left[ {\begin{array}{*{20}c} { (S_{lm}^{\max})^2 } & {-{\textrm{tr}}(\Upsilon_{lm} xx^T)} & {-{\textrm{tr}}(\bar{\Upsilon}_{lm} xx^T)} \\ {-{\textrm{tr}}( \Upsilon_{lm} xx^T)} & {1} & {0} \\ {-{\textrm{tr}}( \bar{\Upsilon}_{lm} xx^T)} & {0} & {1} \\ \end{array}} \right] & \succeq 0 \end{aligned}$$ where: $$\begin{aligned} Z_{lm} &= \frac{g_{lm}}{t_{lm}^2} (e_l e_l^T+e_{l+|N|} e_{l+|N|}^T) \label{ZlmLong} \\ & - \left(\frac{g_{lm} \cos(\phi_{lm})+b_{lm} \cos(\phi_{lm}+\frac{\pi}{2})}{2 t_{lm}}\right) (e_l e_{m}^T+e_{m} e_l^T+e_{l+|N|} e_{m+|N|}^T+e_{m+|N|} e_{l+|N|}^T) \notag \\ & + \left(\frac{g_{lm} \sin(\phi_{lm})+b_{lm} \sin(\phi_{lm}+\frac{\pi}{2})}{2 t_{lm}}\right) (e_l e_{m+|N|}^T+e_{m+|N|} e_l^T-e_{l+|N|} e_{m}^T-e_{m} e_{l+|N|}^T) \notag \\ \Upsilon_{lm} &= g_{lm} (e_{m} e_{m}^T+e_{m+|N|} e_{m+|N|}^T) \\ & - \left(\frac{g_{lm} \cos(-\phi_{lm})+b_{lm} \cos(-\phi_{lm}+\frac{\pi}{2})}{2 t_{lm}}\right) (e_l e_{m}^T+e_{m} e_l^T+e_{l+|N|} e_{m+|N|}^T+e_{m+|N|} e_{l+|N|}^T) \notag \\ & + \left(\frac{g_{lm} \sin(-\phi_{lm})+b_{lm} \sin(-\phi_{lm}+\frac{\pi}{2})}{2 t_{lm}}\right) (e_{l+|N|} e_{m}^T+e_{m} e_{l+|N|}^T-e_l e_{m+|N|}^T-e_{m+|N|} e_l^T) \notag \\ \bar{Z}_{lm} &= -\frac{2 b_{lm}+\bar{b}_{lm}}{2 t_{lm}^2} (e_l e_l^T+e_{l+|N|} e_{l+|N|}^T) \\ & + \left(\frac{g_{lm} \cos(\phi_{lm})+b_{lm} \cos(\phi_{lm}+\frac{\pi}{2})}{2 t_{lm}}\right) (e_l e_{m+|N|}^T+e_{m+|N|} e_l^T-e_{l+|N|} e_{m}^T-e_{m} e_{l+|N|}^T) \notag \\ & + \left(\frac{g_{lm} \sin(\phi_{lm})+b_{lm} \sin(\phi_{lm}+\frac{\pi}{2})}{2 t_{lm}}\right) (e_l e_{m}^T+e_{m} e_l^T+e_{l+|N|} e_{m+|N|}^T+e_{m+|N|} e_{l+|N|}^T) \notag \\ \bar{\Upsilon}_{lm} &= -\frac{2 b_{lm}+\bar{b}_{lm}}{2} (e_{m} e_{m}^T+e_{m+|N|} e_{m+|N|}^T) \\ & + \left(\frac{g_{lm} \cos(-\phi_{lm})+b_{lm} \cos(-\phi_{lm}+\frac{\pi}{2})}{2 t_{lm}}\right) (e_{l+|N|} e_{m}^T+e_{m} e_{l+|N|}^T-e_l e_{m+|N|}^T-e_{m+|N|} e_l^T) \notag \\ & + \left(\frac{g_{lm} \sin(-\phi_{lm})+b_{lm} \sin(-\phi_{lm}+\frac{\pi}{2})}{2 t_{lm}}\right) (e_l e_{m}^T+e_{m} e_l^T+e_{l+|N|} e_{m+|N|}^T+e_{m+|N|} e_{l+|N|}^T) \notag \end{aligned}$$ where $e_k$ be the $k^{th}$ standard basis vector in $\mathbb{R}^{n}$, one can perform similar simplifications. For instance, following the pre-computation of vectors $c^A, c^C, c^D \in \R^{|M|}$ prior to the outer loop of the algorithm, the evaluation of the trace of the quadratic form, ${\textrm{tr}}(Z_{lm} xx^T)$, considering can be implemented using 4 look-ups into the vector $x$ and 13 float-float multiplications: $$\begin{aligned} {\textrm{tr}}(Z_{lm} xx^T) = & \; c^A_{lm} (x_l^2 + x_{l+|N|}^2) \\ & + c^C_{lm} (2 x_l x_m + 2 x_{l+|N|} x_{m+|N|}) \notag \\ & + c^D_{lm} (2 x_l x_{m + |N|} - 2 x_{m} x_{l+|N|}). \notag \end{aligned}$$ One can simplify the multiplication in a similar fashion for the remaining expressions involving (traces of) quadratic forms of $\Upsilon_{lm}, \bar{Z}_{lm}$, and $\bar{\Upsilon}_{lm}$ as well.\ An Analysis {#sec:analysis} =========== The analysis needs to distinguish between optima of the semidefinite programming problem \[LL\], which is convex, and stationary points, local optima, and global optima of \[RrBC\], which is the non-convex rank-constrained problem. Let us illustrate this difference with an example: Consider the following simple rank-constrained problem: $$\begin{aligned} \label{exrank} \max_{W \in \mathcal{S}_+^2, {\textrm{rank}}(W)=1} {\textrm{tr}}(\diag(3,1) W), \end{aligned}$$ subject to ${\textrm{tr}}(I W ) =1$ and $$\begin{aligned} W^* & := \begin{pmatrix} 1&0\\0&0\end{pmatrix} = (1,0) (1,0)^T, \notag \\ \tilde W & := \begin{pmatrix} 0&0\\0&1\end{pmatrix} = (0,1) (0,1)^T =: \tilde R \tilde R^T. \notag \end{aligned}$$ $W^*$ is the unique optimal solution of , as well as the optimal solution of the SDP relaxation, where the rank constraint is dropped. There exists a Lagrange multiplier such that $\tilde W$ is a stationary point, though. Let us define a Lagrange function $\mathcal{L}(R, \lambda) = {\textrm{tr}}(\diag(3,1) RR^T) + \lambda ({\textrm{tr}}(I RR^T ) - 1)$. Then $\nabla_R \mathcal{L}( RR^T,\lambda) = 2\diag(3,1) R +2 \lambda R $. If we plug in $\tilde R$ and $\tilde\lambda = -1$ we obtain $\nabla_R \mathcal{L}( \tilde R\tilde R^T,\tilde \lambda) = 2\diag(3,1) (0,1)^T +2 \tilde\lambda (0,1)^T = (0,0)^T $. Hence $(\tilde R, \tilde \lambda)$ is a stationary point of a Lagrange function. However, one can follow the proof of Proposition \[prop:3\] to show that $[\tilde R, 0]$ is not a local optimum solution of SDP relaxation. Indeed, in generic non-convex quadratic programming, the test whether a stationary point solution is a local optimum [@murty1987] is NP-Hard. For ACOPF, there are a number of sufficient conditions known, e.g. [@MolzahnDeMarco2014]. Under strong assumptions, motivated by Observation $O_3$, we provide necessary and sufficient conditions based on those of Grippo et al. [@grippo2009necessary]. We use $\otimes$ for Kronecker product and $I_r$ for identity matrix in $\R^{r \times r}$. Consider the SDP relaxation obtained from \[RrBC\] by dropping the rank constraint and writing it down as $\min {\textrm{tr}}(QW)$ such that ${\textrm{tr}}(A_i W) = b_i$ for constraints $i = 1, \ldots, m$ in variable $W \succeq 0, W \in \mathcal{S}^n$. If there exists an optimum $W^*$ with rank $r$ for the SDP relaxation, for any point $R \in \R^{n \times r}$, $RR^T$ is a global minimiser of \[RrBC\] if and only if there exists a $\lambda^* \in \R^m$ such that: $$\begin{aligned} \left[ {\left( {Q + \sum\limits_{i = 1}^m {\lambda^*_i A_i} } \right) \otimes I_r} \right] R & = 0 & \\ Q + \sum\limits_{i=1}^{m} \lambda_{i}^{*} A_i & \succeq 0 & \notag \\ R^{T}(A_i \otimes I_r) R & = b_i & \forall i = 1, \ldots, m. \notag\end{aligned}$$ The proof is based on Proposition 3 of Grippo et al. [@grippo2009necessary], which requires the existence of rank $r$ optimum and strong duality of the SDP relaxation. The existence of an optimum of \[LL\] with rank $r$ is assumed. One can use Theorem 1 [@Josz2015] and Theorem 1 of [@Ghaddar2015] to show that a ball constraint is sufficient for strong duality in the SDP relaxation, c.f. Observation $O_4$. Notice that the test for whether there exists an optimum $W^*$ with rank $r$ for the SDP relaxation is suggested Proposition 5, i.e. by solving for rank $r+1$. Next, let us consider the convergence: \[prop:2\] There exists an instance-specific constant $r'$, such that for every $r \ge r'$, whenever Algorithm \[alg:SCDM\] with ${\{ {R^{(k)}}\} \in \R^{n \times r}}, \mathcal{R} = \det(R^T R)$ and $\lim_{k \to \infty} \nu_k \to 0$ produces solution with $\lim_{k \to \infty} {T_k({R^{(k)}})} \to 0$ and a local optimum ${R^{(k)}}$ is generated within the inner loop (5–11), ${R^{(k)}}({R^{(k)}})^T$ is an optimal solution to \[LL\]. Moreover, $r'$ depends on the number of constraints $m$ in the optimisation problem and is $\mathcal{O}(\sqrt m)$. The proof follows from Theorem 3.3 of Burer and Monteiro [@BurerMonteiro2005]. One can rephrase Theorem 3.3 to show that if $\{ {R^{(k)}}\} \in \R^{n \times r}$ is a bounded sequence such that: - $\lim_{k \to \infty} T_k = 0$ - $\lim_{k \to \infty} \nabla \mathcal{L}({R^{(k)}}) = 0$ - $\lim \inf_{k \to \infty} \nabla^2 \mathcal{L} ({R^{(k)}})(H^k, H^k) \ge 0$ for all bounded sequences $\{ H^k \}$, $H^k \in \R^{n \times r}$ - ${\textrm{rank}}({R^{(k)}}) < r$ for all $k$ every accumulation point of ${R^{(k)}}({R^{(k)}})^T$ is an optimal solution of \[LL\]. Let us show that these four conditions are satisfied, in turn. Condition $C_1$, which effectively says that $W={R^{(k)}}({R^{(k)}})^T$ should be feasible with respect to constraints (\[R1-Constraint1\]–\[R1-Constraint4\]), is affected by the termination criteria of the inner loop, albeit only approximately for a finite $k$ and finite machine precision. Conditions $C_2$ and $C_3$ follow from our assumption that ${R^{(k)}}$ is a local optimum. The satisfaction of Condition $C_4$ can be shown in two steps: First, there exists an $r'$, such that for every feasible semidefinite programming problem with $m$ constraints, there exists an optimal solution with a rank bounded from above by $r'$. This $r'$ is $\mathcal{O}(\sqrt m)$. This follows from Theorem 1.2 of Barvinok [@barvinok2001], as explained by Pataki [@pataki1998]. Second, the $\mathcal{R} = \det(R^T R)$ regularisation forces the lower-rank optimum to be chosen, should there be multiple optima with different ranks. This can be seen easily by contradiction. Finally, one can remove the requirement on the sequence to be bounded by the arguments of Burer and Monteiro [@BurerMonteiro2005], as per Theorem 5.4. Further, \[prop:3\] Consider an instance of ACOPF such that there exists an optimum solution $W^*$ of \[LL\] with rank 1 and $\forall k: c^2_k\geq 0$. Let ${R^{(k)}}$ be a iterate produced by the Algorithm \[alg:SCDM\] when run with $r=1$ and moreover it is such that $T_k = 0$. Then if $R R^T$ (where $R = [ {R^{(k)}}, {\bf 0}]$) is a local optimum of \[R2BC\], then ${R^{(k)}}({R^{(k)}})^T$ is a global optimum solution of \[R1BC\] and \[PP4\]. We will prove this proposition by contradiction. For the sake of contradiction, we assume that $R$ is a local optimum and ${R^{(k)}}$ is not a global optimum. Therefore, we know that objective function of \[R1BC\] for $W_1 = W^*$ is smaller than for $W_2 = {R^{(k)}}({R^{(k)}})^T$ and both $W_1$ and $W_2$ are feasible. By the assumption on optimum solution $W^*$ of \[LL\], we know that $W^*$ can be written as $W^*= w w^T$. Now, it is easy to observe that if we define $\tilde R^\lambda=[(1-\sqrt{\lambda}) {R^{(k)}}, \sqrt{\lambda} w^*]$ then for any $\lambda\in [0,1]$ this vector is feasible for \[R2BC\]. Moreover, for $\lambda = 0$ we have $\tilde R^0 = R$. Because the objective function $F$ of \[RrBC\] is convex in $W$ with $c_k^2 \geq 0$, we have that $$F( \tilde R^\lambda (\tilde R^\lambda)^T ) \leq (1-\lambda) F(W_2) + \lambda F(W_1) < F(W_2)$$ and hence for all $\lambda \in (0,1]$ we have that $F( \tilde R^\lambda (\tilde R^\lambda)^T ) < F(W_2)$, which is a contradiction with the assumption that $R$ is a local optimum. Overall, The first iteration of the outer loop of Algorithm \[alg:SCDM\] may produce a global optimum to \[R1\] and \[RrBC\] and \[PP4\], as suggested in Propositions \[prop:1\]–\[prop:3\]. The second and subsequent iterations of the outer loop of Algorithm \[alg:SCDM\] may find the optimum of \[LL\], the semidefinite programming problem, as suggested in Proposition \[prop:2\], but for $\mathcal{R} = 0$, they are not guaranteed to find the global optimum of \[R1\] nor \[RrBC\] nor \[PP4\]. One should also like contrast the ability to extract low-rank solutions with other methods: Whenever there are two or more optima of \[LL\] with two or more distinct ranks, the maximum rank solutions are in the relative interior [of the optimum face]{} of the feasible set, as per Lemma 1.4 in [@laurent2009sums]. Primal-dual interior-point algorithms for semidefinite programming, such as SeDuMi [@sturm1999], in such a case return a solution with maximum rank. Put bluntly, one may conclude that as long as one seeks the exact optimum of \[ACOPF\], it does not make sense to perform more than two iterations of the outer loop of Algorithm \[alg:SCDM\]. When it becomes clear by the second iteration that the rank-one optimum of \[LL\] has not been extracted, one should like to consider stronger convexifications [@Ghaddar2015]. Notice that this advice is independent of whether one uses $\mathcal{R} = \det(R^T R)$ or $0$. Although $\mathcal{R} = 0$ or $\nu = 0$ does not guarantee the recovery of low-rank solutions of \[LL\], [in some cases [@bandeira2016low; @boumal2016non],]{} $\mathcal{R} = \det(R^T R)$ and $\nu > 0$ does not guarantee that the low-rank solution of the augmented Lagrangian coincides with the optimum of \[ACOPF\], in some other cases. It may hence be preferable to use $\mathcal{R} = 0$, as it allows [for more iterations per second]{} and post hoc testing of global optimality. Numerical Experiments {#sec:numerical} ===================== We have implemented Algorithm \[alg:SCDM\] in C++ with GSL and OpenMP and tested it on a collection [@Matpower] of well-known instances. For comparison, we have used three solvers specialised to ACOPF: - the MATLAB-based Matpower Interior Point Solver (MIPS) version 1.2 (dated March 20th, 2015), which has been developed by Zimmerman et al. [@Matpower] - the C-based Semidefinite Programming optimal Power Flow ([sdp\_pf]{}) version 1.0 (dated January 17th, 2014), which has been developed by Mohlzahn et al. [@molzahn2011] using SeDuMi of Sturm et al. [@sturm1999] - the MEX-based [OPF\_Solver]{} beta version dated December 13th, 2014, listed as the most current as of June 1st, 2016[^1], which has been developed by Lavaei et al. [@7065336] using and SDPT3 of T[ü]{}t[ü]{}nc[ü]{} et al. [@toh1999sdpt3; @tutuncu2003solving]. Notice that [OPF\_Solver]{} produces [feasible points with objective values that are near the values of the global optima]{} across all instances tested, for some, non-default settings; we have used the per-instance settings of epB, epL, and line\_prob, as suggested by the authors. For the comparison presented in Table \[tab1\], we have used a standard laptop with Intel i5-2520M processor and 4 GB of RAM. We believe this is fair, as the solvers we compare with cannot make a good use of a more powerful machine. For the presentation of scalability of our code, we have used a machine with 24 Intel E5-2620v3 clocked at 2.40GHz and 128GB of RAM, but used only the numbers of cores listed. We have also tested six general-purpose semidefinite-programming (SDP) solvers: - CSDP version 6.0.1, which has been developed by Borchers [@borchers1999csdp] - MOSEK version 7.0, which has been developed by MOSEK ApS [@Andersen2003; @permenter2015solving] - SeDuMi version 1.32, which has been developed by Sturm et al. [@sturm1999] - SDPA version 7.0, which has been developed by the SDPA group [@yamashita2012latest] - SDPT3 version 4.0, which has been developed by T[ü]{}t[ü]{}nc[ü]{} et al. [@toh1999sdpt3; @tutuncu2003solving] - SDPLR version 1.03 (beta), which has been developed by Burer and Monteiro [@BurerMonteiro2003] Five of the codes (CSDP, SeDuMi, SDPA, SDPLR, and SDPT3) have been tested at the NEOS 7 facility [@714603] at the University of Wisconsin in Madison, where there are two Intel Xeon E5-2698 processors clocked at 2.3GHz and 192 GB of RAM per node. MOSEK has been used at a cluster equipped with one AMD Opteron 6128 processor clocked with 4 cores at 2.0 GHz and 32 GB of RAM per node, as per the license to Lehigh University. IEEE Test Cases --------------- Our main focus has been on the IEEE test cases. In Table \[tab1\], we compare the run-time of [our implementation of Algorithm \[alg:SCDM\]]{} with the run-time of the three leading solvers for the ACOPF listed above, two of which ([sdp\_pf]{}, [OPF\_Solver]{}) use elaborate tree-width decompositions. In order to obtain the numbers, we ran Matpower first using default settings, record the accuracy with respect to squared error $T_k$ , ran our solver up to the same accuracy, and record the time and the objective function. [ In Table \[tabComparisonWithGP\], we present the run-time of six popular general-purpose SDP solvers for comparison. We note that we have used the default parameters and tolerances of each code, so the precision may no longer match Matpower. We list the reported CPU time rounded to one decimal digit, if that yields a non-zero number, and to one significant digit, otherwise. When we display a dash, no feasible solution has been found; the severity of the constraint violation and abruptness of the termination of the solver vary widely. For example, MOSEK terminates very abruptly with fatal error stopenv on five of the instances, while SDPT3 often runs into numerical difficulties. Outside of SDPLR on the largest instance, no solver ran out of memory, iteration limit, or time limit. In this comparison, SeDuMi seems to be most robust solver. SDPLR is also rather robust, but three orders of magnitude slower. Throughout, all interior-point methods (CSDP, Mosek, SeDuMi, SDPA, and SDPT3) perform much better on the dual of the SDP, than on the primal. Please note that direct comparison with the run-time of our implementation of Algorithm \[alg:SCDM\] reported in Table \[tab1\] is no possible, due to the use of three different platforms: Five of the codes (CSDP, SeDuMi, SDPA, SDPLR, and SDPT3) have been tested at the NEOS facility, which does not allow for our code to be run, while Mosek has been tested at a Lehigh University facility. Still, either facility has machines considerably more powerful than the machine used in the tests above, and we report the CPU time reported by the individual solvers, rather than the wall-clock time, so we believe it is fair to claim our specialised solver outperforms the general-purpose solvers. ]{} NESTA Test Cases ---------------- Next, we have tested our approach on the recently introduced test cases from the NICTA Energy System Test Case Archive (NESTA) [@NESTA]. There, bounds on commonly used IEEE test cases have been carefully tightened to make even the search for a feasible solution difficult for many solvers. For example, in the so called Active Power Increase (API) test cases, the active power demands have been increased proportionally throughout the network so as to make thermal limits active. In Table \[tab2\], we present the results. Out of the 11 API instances tested, Matpower and [sdp\_pf]{} fail to find feasible solutions for three instances each. Our implementation of Algorithm \[alg:SCDM\], using default settings, including $\mathcal{R} = 0$, obtains feasible solutions across all the 11 instances, while improving over the objective function values obtained by either Matpower or [sdp\_pf]{}. For example, on the instance nesta\_case30\_as\_\_api, we improve the objective function value by about 10on the instance nesta\_case30\_fsr\_\_api, we improve the objective by considerably more than 10we have not tested all instances of the NESTA archive, we are very happy with these preliminary results. Pathological Instances {#sec:pathological} ---------------------- Additionally, we have tested our approach on a number of recently introduced pathological instances [@Lesieutre2011; @molzahn2013application; @Bukhsh2013; @Ghaddar2015; @kocuk2016inexactness]. Depending on the choices of $R, \nu,$ and $\mu$, Algorithm \[alg:SCDM\] may perform better or worse than the relaxation of Lavaei and Low . Let us illustrate this on the suggested settings of $\mathcal{R} = 0$, $\nu$ arbitrary, and single-thread execution. In the example of Bukhsh et al. [@Bukhsh2013], known as case2w, there are two local optima. Whereas many heuristics may fail or find the local optimum with cost 905.73, we are able to find the exact optimum with cost 877.78 in 0.49 seconds up to the infeasibility of $9.38 \cdot 10^{-6}$ with $\mu = 0.01$ and up to infeasibility of $7.23 \cdot 10^{-6}$ in 0.94 seconds with $\mu = 0.0001$. When we replace $V_2^{\max} = 1.05$ with 1.022, as in [@Ghaddar2015], the instance becomes harder still, and the optimum of the relaxation of Lavaei and Low [ is not rank-1. (Although one could project onto the feasible set of \[R1BC\] and extract a feasible solution of \[PP4\], it would not be the optimum of \[PP4\].) ]{} With $\mu=0.01$, we find only a local optimum with cost 888.05 up to infeasibility of $8.29 \cdot 10^{-6}$, but with $\mu = 0.0001$, we do find the local optimum, which in our evaluation has cost 905.66 up to infeasibility of $1.46 \cdot 10^{-9}$. We stress that this optimum is not the solution of plain , but at the same time that, we do not provide any guarantees of improving upon the relaxation of Lavaei and Low . In the example case9mod of Bukhsh et al. [@Bukhsh2013], $\mu=0.01$ does not converge within 10000 iterations, but using $\mu = 0.0001$, we are able to find the exact optimum of 3087.84 and infeasibility $9.43 \cdot 10^{-12}$ in 12.16 seconds. In the example case39mod2 of Bukhsh et al. [@Bukhsh2013], we are able to find only a local optimum with cost 944.71 and infeasibility $5.63 \cdot 10^{-8}$ after 45.03 seconds, whereas the present-best known solution has cost 941.74. In the example of Molzahn et al. [@Lesieutre2011; @molzahn2013application], which is known as LMBM3, we find a solution with cost 5688.10 up to infeasibility of $9.98 \cdot 10^{-6}$ in 0.88 seconds using $\mu = 0.01$ and a solution with cost 5694.34 up to infeasibility of $1.14 \cdot 10^{-6}$ in 0.56 using $\mu = 0.0001$. This illustrates that the choice of $\mu$ is important and the default value, albeit suitable for many instances, is not the best one, universally. ------------ ------------------ ----------- ------------ -------------- ------------ -------------- ------------ -------------- ------------ Name Ref. Obj. Time \[s\] Obj. Time \[s\] Obj. Time \[s\] Obj. Time \[s\] case2w [@Bukhsh2013] — — 877.78 17.08 877.78 2.52 877.78 0.077 case3w [@Bukhsh2013] — — 560.53 0.56 560.53 2.70 560.53 0.166 case5 [@li2010small] 1.755e+04 21.80 1.482e+03 120.73 1.482e+03 25.93 1.482e+03 0.263 case6ww [@wood1996power] 3.144+03 0.114 3.144e+03 0.74 3.144e+03 2.939 3.144e+03 0.260 case14 [@Matpower] 8.082e+03 0.201 8.082e+03 0.84 – – 8.082e+03 0.031 case30 [@li2010small] 5.769e+02 0.788 5.769e+02 2.70 5.765e+02 6.928 5.769e+02 0.074 case39 [@Matpower] 4.189e+04 0.399 4.189e+04 3.26 4.202e+04 7.004 4.186e+04 0.885 case57 [@Matpower] 4.174e+04 0.674 4.174e+04 2.69 – – 4.174e+04 0.857 case57Tree [@kocuk2015new] 12100.86 6.13 \* 1.045e+04 4.48 \* 1.046e+04 342.00 \* 1.046e+04 3.924 case118 [@Matpower] 1.297e+05 1.665 1.297e+05 6.57 – – 1.297e+05 1.967 case300 [@Matpower] 7.197e+05 2.410 – 17.68 – – 7.197e+05 90.103 ------------ ------------------ ----------- ------------ -------------- ------------ -------------- ------------ -------------- ------------ \ \*: We note that the precision here is approximately $10^{-6}$, which is the case for all three SDP-based solvers, Algorithm \[alg:SCDM\], as well as [sdp\_pf]{} and [OPF\_Solver]{}. --------- -------- ------------- ---------- ------------ ------------ ----------- ----------- Name SDP SeDuMi 1.32 SDPA 7.0 SDPLR 1.03 CSDP 6.0.1 SDPT3 4.0 Mosek 7.0 case2w primal 0.3 0.003 0 0.02 0.4 0.1 case2w dual 0.3 0.004 0 0.02 0.4 0.1 case5w primal 0.4 – 0 0.04 0.5 0.2 case5w dual 0.4 – 0 0.03 0.5 0.2 case9 primal 1.0 0.05 25 0.1 0.5 0.3 case9 dual 1.0 0.07 25 0.1 0.6 0.3 case14 primal 0.7 0.05 58 0.1 – 0.2 case14 dual 0.7 0.04 17 0.1 – 0.3 case30 primal 2.8 – 829 0.8 – – case30 dual 6.1 – 282 1.0 – – case30Q primal 2.8 – 831 0.8 – – case30Q dual 6.2 – 195 1.0 – – case39 primal 4.4 – 2769 1.0 – 0.7 case39 dual 7.7 – 723 1.3 – – case57 primal 3.2 – 1930 0.5 – 0.7 case57 dual 4.0 – 1175 1.0 – 1.8 case118 primal 10.3 0.9 4400 3.4 – 1.7 case118 dual 17.1 – – 13.0 – – case300 primal 27.5 1.7 – 109.6 – – case300 dual 133.7 – – 66.7 – – --------- -------- ------------- ---------- ------------ ------------ ----------- ----------- Instance ---------------------------------- ----------------- ------------ ----------------- ------------ ----------- ------------ Name Obj. Time \[s\] Obj. Time \[s\] Obj. Time \[s\] nesta\_case3\_lmbd 5.812e+03 0.946 5.789e+03 2.254 5.757e+03 0.149 nesta\_case4\_gs 1.564e+02 1.019 1.564e+02 2.392 1.564e+02 0.139 nesta\_case5\_pjm [(1.599e-01)]{} 0.811 [(1.599e-01)]{} 2.708 2.008e+04 0.216 nesta\_case6\_c 2.320e+01 0.825 2.320e+01 2.392 2.320e+01 0.379 nesta\_case6\_ww 3.143e+03 0.884 3.143e+03 2.776 3.148e+03 0.242 nesta\_case9\_wscc 5.296e+03 1.077 5.296e+03 2.621 5.296e+03 0.211 nesta\_case14\_ieee 2.440e+02 0.762 2.440e+02 3.164 2.440e+02 0.267 nesta\_case30\_as 8.031e+02 0.861 8.031e+02 3.916 8.031e+02 0.608 nesta\_case30\_fsr 5.757e+02 0.898 5.757e+02 6.201 5.750e+02 0.613 nesta\_case30\_ieee 2.049e+02 0.818 2.049e+02 4.335 2.049e+02 0.788 nesta\_case39\_epri 9.651e+04 0.863 9.649e+04 5.676 9.651e+04 0.830 nesta\_case57\_ieee 1.143e+03 1.119 1.143e+03 8.053 1.143e+03 0.957 nesta\_case118\_ieee [(4.433e+02)]{} 1.181 [(4.433e+02)]{} 18.097 4.098e+03 4.032 nesta\_case162\_ieee\_dtc [(3.123e+03)]{} 0.975 [(3.123e+03)]{} 32.153 4.215e+03 8.328 nesta\_case3\_lmbd\_\_api 3.677e+02 1.113 3.333e+02 2.459 3.333e+02 0.029 nesta\_case5\_pjm\_\_api [(5.953e+00)]{} 1.002 [(5.953e+00)]{} 2.426 3.343e+03 1.059 nesta\_case6\_c\_\_api 8.144e+02 1.034 8.144e+02 2.456 8.139e+02 0.722 nesta\_case9\_wscc\_\_api 6.565e+02 1.135 6.565e+02 2.869 6.565e+02 0.708 nesta\_case14\_ieee\_\_api 3.255e+02 1.035 3.255e+02 3.569 3.245e+02 0.214 nesta\_case30\_as\_\_api 5.711e+02 1.156 5.711e+02 5.433 5.683e+02 0.615 nesta\_case30\_fsr\_\_api 3.721e+02 0.862 3.309e+02 5.276 2.194e+02 0.604 nesta\_case30\_ieee\_\_api 4.155e+02 1.076 4.155e+02 5.293 4.155e+02 0.618 nesta\_case39\_epri\_\_api [(1.540e+02)]{} 0.965 [(1.540e+02)]{} 5.109 7.427e+03 1.978 nesta\_case57\_ieee\_\_api 1.430e+03 1.041 1.429e+03 7.625 1.429e+03 2.437 nesta\_case162\_ieee\_dtc\_\_api [(1.502e+03)]{} 1.215 [(1.502e+03)]{} 43.339 6.003e+03 5.833 Adaptive Updates of $\mu$ {#sec:adaptivemu} ------------------------- As it has been stressed above, it is important to pick an appropriate $\mu$. Alternatively, one can adapt a strategy to change $\mu$ adaptively. This can be achieved by requiring the infeasibility to shrink by a fixed factor between consecutive iterations. If such an infeasibility shrinkage is not feasible, one ignores the proposed iteration update and decreases $\mu$. Figures \[fig:adaptiveMu1\] and \[fig:adaptiveMu2\] show the evolution of objective function value and the infeasibility for 400 iterations with fixed values [ $\mu\in \{10^{-1}, 10^{-2}, 10^{-3}, 10^{-4}, 10^{-5}, 10^{-6}\}$ ]{} and for the adaptive strategy on the instance case5. Figure \[fig:adaptiveMu3\] details the evolution of $\mu$ within the adaptive strategy [on the same instance]{}. One can see that choosing very small $\mu$ forces the algorithm to “jump” close to a feasible point, and get “stuck”. On the other hand, larger fixed values of $\mu$ lead to the same objective value. One can also see that the adaptive strategy is rejecting the first 80 updates, until the value of $\mu$ is small enough; subsequently, it starts to make rapid progress. ![The evolution of the objective function value [over time on instance case5]{}, for a number of choices of $\mu$ and the adaptive updates of $\mu$.[]{data-label="fig:adaptiveMu1"}](adaptive.eps){width="3in"} ![The evolution of the infeasibility [over time on instance case5]{}, for a number of choices of $\mu$ and the adaptive updates of $\mu$.[]{data-label="fig:adaptiveMu2"}](adaptivefeas.eps){width="3in"} ![The evolution of $\mu$ with adaptive updates [over time on instance case5]{}.[]{data-label="fig:adaptiveMu3"}](adaptivemu.eps){width="3in"} Large Instances --------------- We have also experimented with the so called Polish [network]{} [@Matpower], [also known]{} as case2224 and case2736sp, as well as the instances collected by the Pegase project [@6465783], such as case9241peg. There, we cannot obtain solutions with the same precision as Matpower, within comparable run-times. However, we can decrease the initial infeasibility by factor of $10^2$ for case2224 in 200 seconds, by factor of $10^6$ for case2736sp in 200 seconds, and by factor of $10^4$ for case9241peg, again in 200 seconds. Although these results are not satisfactory, yet, they may be useful, when a good initial solution is available. As discussed in Section \[conclusions\], we aim to improve upon these results by using a two-step method, where first-order methods on the convex problem are combined with a second-order method on the non-convex problem. These results also motivate the need for parallel computing. Parallel Computing {#sec:parallel} ------------------ As it has been discussed above, Algorithm \[alg:SCDM\] is easy to implement in a parallel fashion. We have implemented our multi-threading variant using OpenMP, in order to achieve portability. The significant overhead of using OpenMP makes it impossible to obtain a speed-up on very small instance. For example, in case3w, there is not enough work to be shared across (any number of) threads to offset the overhead. In Figure \[fig:scaling\], we present the scaling properties of Algorithm \[alg:SCDM\]. In particular, we illustrate the speed-up (the inverse of the multiple of single-threaded run-time) as a function of the number of cores used for a number of larger instances. We can also observe that even case118 is too small to benefit from a considerable speed-up. On the other hand, for large datasets, such as the Polish case2224 or larger, we observe a significant speed-up. ![The speed-up as a function of the number of cores employed.[]{data-label="fig:scaling"}](speedup.eps){width="3in"} Conclusions =========== Our approach seems to bring the use of SDP relaxations of ACOPF closer to the engineering practice. A number of authors have recently explored elaborately constructed linear programming [@Bienstock2015] and second-order cone programming [@7285718; @MolzahnHiskens2015a; @MolzahnHiskens2015b; @kocuk2015new; @kocuk2016strong] relaxations, many [@7285718; @kocuk2016strong] of which are not comparable to the SDP relaxations in the sense that they may be stronger or weaker, depending on the instance, but aiming to be solvable faster. Algorithm \[alg:SCDM\] suggests that there are simple first-order algorithms, which can solve SDP relaxations of ACOPF faster than previously, at least on some instances. A major advantage of first-order methods over second-order methods [@Matpower; @sturm1999] is the ease of their parallelisability. This paper presents considers a symmetric multi-processing, where memory is shared, but a distributed variant, where $\K$ agents perform the iterations, is clearly possible, c.f. [@marecek2015distributed]. The $\K$ agents may represent companies, each of whom owns some of the generators and does not want to expose the details, such as cost functions. If $\K$ agents are $\K$ computers, a considerable speed-up can be obtained. Either way, power systems analysis could benefit from parallel and distributed computing. The question as to how far could the method scale, remains open. In order to improve the scalability, one could try to combine first- and second-order methods [@liu2015hybrid]. We have experimented with an extension, which uses Smale’s $\alpha$-$\beta$ theory [@Blum1997] to stop the computation at the point $z_0$, where we know, based on the analysis of the Lagrangian and its derivatives, that a Newton method or a similar algorithm with quadratic speed of convergence [@chen1994approximate] will generate sequence $z_i$ to the correct optimum $z^*$, i.e. $$\begin{aligned} |z_i - z^* | \le (1/2)^{2^i - 1} |z_0 - z^*|.\end{aligned}$$ This should be seen as convergence-preserving means of auto-tuning of the switch to a second-order method for convex functions. One should also study infeasibility detection. [Considering the test of feasibility of an SDP is not known to be in NP [@ramana1997exact],]{} we assumed the instance are feasible, throughout. [Nevertheless, one may need to test their feasibility first, in many practical applications.]{} There are some very fast heuristics [@6629370] available already, but one could also use Lagrangian methods in a two-phase scheme, common in robust linear-programming solvers. Both in methods based on simplex and feasible interior-point, one first considers a variant of the problem, which has constraints relaxed by the addition of slack variables, with objective minimising a norm of the slack variables. This makes it possible to find feasible solutions quickly, and by duality, one can detect infeasibility. One could also apply additional regularisations, following [@6736636; @7402079]. In \[R1\], one could drop the rank-one constraint and modify the objective function to penalise the solutions with large ranks, e.g., by adding the term $\lambda \|W\|^*$ to the objective function, where $\|\cdot\|_*$ is a nuclear norm [@fazel2002matrix] and $\lambda>0$ is a parameter. Alternatively, one can replace the rank constraint by the requirement that the nuclear norm of the matrix should be small, i.e. $\|W\|_* \leq \lambda$. However, both approaches require a search for a suitable parameter $\lambda$ such that the optimal solution has indeed rank 1. Moreover, the penalised alternative may not produce an optimal solution of \[PP4\], necessitating further algorithmic work. Finally, one could extend the method to solve relaxations a variety of related applications, such as security-constrained problems, stability-constrained problems, network expansion planning [@marecek2016minlp], and unit commitment problems. The question as to whether the method could generalise to the higher-order relaxations, c.f., Ghaddar et al. [@Ghaddar2015], also remains open. First steps [@Ma2015PhD] have been taken, but much work remains to be done. [^1]: <http://ieor.berkeley.edu/~lavaei/Software.html>
--- abstract: | We show a break down of the conventional partition of optical states into its radiative and non-radiative parts. Large divergence of experimental observations from current theory in the case of emitters interacting with fully absorbing plasmonic nanoparticles only a few nanometers in dimensions, are now evident. A model of fluctuation-dissipation demands non-local behavior from $limiting$ small metal nanoparticles and proximal metal surfaces. We point that widely used techniques to enhance optical sensing such as surface-enhanced-Raman-spectroscopy (SERS), may not have been viable but for this effect. Qualitatively, this quantum effect seems to present itself only when the classical probability of scattering of an emitted photon by a near-by absorbing nanostructure approaches zero. Hence, though different in origin and scale, this has an interesting analogy with quantum effects resulting in Hawking radiation near a black-hole.\ author: - Kritika Jain - Murugesan Venkatapathi title: 'Emitter-Vacuum coupling through a leaky metal nanostructure and the role of dynamics in density of optical states' --- Introduction {#intro} ============ Spontaneous emission is typically elucidated as a function of vacuum fluctuations and proximal matter, and its nature in the weak vacuum-coupling regime is predicted by the $density$ of optical states. Note that the strong vacuum-coupling regime becomes relevant when only a few optical states (modes) of vacuum are available for the emission, as in a cavity. In most other applications, increasing density of radiative states relative to the non-radiative states using appropriate nanostructures placed near emitters can result in significant gains in power and even efficiency of emission; and this is broadly referred to as Purcell enhancement [@purcell1946purcell; @drexhage1970influence; @kuhn2006enhancement; @bharadwaj2007spectral; @cheng2007separation; @kim2013precise; @dulkeith2002fluorescence; @dulkeith2005gold]. This work shows a break down of the current theory of local density of optical states when the dynamics of fluctuations between the emitter and a nanostructure play a significant role in the emission process. Especially, the significant gains in emission observed in presence of resonant and off-resonant fully absorbing metal nanoparticles of limiting small dimensions ($<$ 15 nm)[@tripathi2013plasmonic; @kang2011fluorescence; @schneider2006distance; @haridas2013photoluminescence; @haridas2011photoluminescence; @haridas2010photoluminescence; @haridas2010controlled], directly contradicts our current understanding. A few theoretical evaluations were available in their original reports but this divergence was probably overlooked, while our evaluations are presented here. In conjunction, we also report predictions of our proposed theory that remove any contradictions and divergence. We also show that widely used techniques such as surface-enhanced-Raman-scattering (SERS) exhibit gains up to $10^{10}$ in magnitude [@kovacs1986distance; @ye1997surface; @mcfarland2005wavelength; @dieringer2006introductory; @wang2013wafer; @gabudean2012gold; @ivanov2016strong; @kennedy1999determination] due to this effect discussed in this work, which otherwise would be restricted to a factor of $10^3$ at most. In many cases of sensing fluorescence and Raman signals, near-field enhancement of incident radiation exciting the emitter, accompanies a possible enhancement of its emission. Typically a metallic surface or a larger nanostructure hosts smaller (sharper) nano features that result in this near-field enhancement of exciting radiation, but by conventional theory, this surface adds non-radiative states significantly more than radiative states for emission. The observed factors of enhancements due to a metallic surface are much greater than the combined amplification possible due to relative increase in exciting radiation and density of radiative states, for both resonant and off-resonant emissions. The large enhancements of Raman signals observed has so far lead to tentative mechanisms proposed that do not submit to classical electrodynamics and quantum models [@le2006rigorous]. This divergence of experiments from theory was mostly attributed to a $chemical$ enhancement of unknown origin [@fromm2006SERS]. But later studies with varying chemical properties of dyes showed that these introduce variations of at most an order of magnitude in the surface enhanced Raman signals [@sharma2012SERSreview; @kneipp2016SERSrigorous]. Further, measurements resolved with a varying distance from the nanostructure [@kovacs1986distance; @ye1997surface] have shown that the SERS effect has a longer range than that is possible by any chemical effect. In both strong vacuum-coupling [@raimond1982statistics; @zhu1990vacuum; @boca2004observation; @yoshie2004vacuum; @bernardot1992vacuum] and recently explored strong matter-coupling [@bellessa2004matter; @zengin2015matter] regimes of emission, experiments establish the splitting of the emitted energy spectrum due to Rabi oscillations. This reversible exchange of the excitation between two oscillators, and appearance of the two possible modes of the coupled system, is observable when decay rates of the two oscillators are relatively small. But a related and significant unknown has been the degree of absorption possible in matter strongly coupled to the emitter. We show that when probability of dissipation of the excitation in the metal nanostructure is significantly larger than the probability of its exchange with the emitter, conventional partition of optical states holds. Else, an inhibited dissipation results, with an equivalent increase in stimulation of the emitter by virtual photons. This behavior can also be modeled as fluctuations in the metal nanostructure leaking to the emitter due to the relatively high rate of exchange of virtual excitations i.e a leaky metal particle. It is predicted for limiting small dimensions of a metal nanoparticle, and a metal structure separated by a few nanometers from the emitter. Thus this regime is well distinguished from Purcell enhancement where the emitter is at resonance with a relatively large plasmonic particle 50 - 200 nm in dimensions, and is separated by distances on the order of its dimensions or larger. There, the interaction is dominated by elastic scattering, adding significantly to the relative density of radiative states as in the conventional theory. From a classical electrodynamics point of view, immediate near-field zones of irradiated metallic structures and the dissipation by extremely small metal nanoparticles are dominated by evanescent fields. Originally thought to have no physical significance, such fields were found to play a dominant role in phenomena ranging from tunneling across a thin potential barrier [@adam93evanescent; @meixner94evanescent; @baena05evanescent], to the large fluctuation-driven heat transfer very near a surface [@loomis94fluctuation; @rytov53fluctuation]. In this alternate elucidation, an absorbing nanostructure coupled to the proximal emitter by its evanescent fields allows tunneling through of photons from the emitter into vacuum. This is an anomalous case of enhanced proximal emission, where conventional theory predicts large non-radiative absorption and a near-zero probability of classical scattering of the emitted photon by the nanostructure. Hence the described effect and its observations have an interesting analogy with the quantum tunneling resulting in Hawking radiation to reveal an otherwise completely absorbing black-hole [@Hawking_radiation]. Results and Discussion ====================== ![Comparison of theoretical and experimental results: The minimum and maximum gains predicted are given by quantum efficiency $Q$ and power radiated $Q\Gamma^r$, normalized by the values of isolated emitter; note $log$-scale in Y-axis. Experimental data[@kang2011fluorescence] is for a gold nanoparticle of radius 5 nm coated with a bi-polymer and the peak emission wavelength was 830 nm.](fig1) An increase in radiative states was inferred as the increase in vacuum fluctuations coupled to the emitter due to the proximal nanostructure. The smaller nanoparticles can absorb light in the plasmonic range increasing non-radiative states notably, but have a negligible scattering efficiency and do not add to the density of radiative states. But they have been observed to increase rates of spontaneous emission from proximal emitters significantly more than theoretical evaluations, and in cases even notably increase their efficiency of emission in a direct contradiction with theory. Moreover, we point that the same applies in a lesser degree to an emitter separated from surface of a large plasmonic structure by a few nanometers. This effect thus becomes crucial for techniques such as surface-enhanced-Raman-spectroscopy (SERS), which would not be otherwise viable. ![Comparison of theoretical and experimental results: The minimum and maximum gains predicted are given by quantum efficiency $Q$ and power radiated $Q\Gamma^r$, normalized by the values of isolated emitter; note $log$-scale in Y-axis. Experimental data[@schneider2006distance] is for gold nanoparticles of radius 6.5 nm and the peak emission wavelength was 520 nm.](fig2) ![Comparison of theoretical and experimental results: The minimum and maximum gains predicted are given by quantum efficiency $Q$ and power radiated $Q\Gamma^r$, normalized by the values of isolated emitter; note $log$-scale in Y-axis. Experimental data[@schneider2006distance] is for gold nanoparticles of radius 6.5 nm and the peak emission wavelength of was 580 nm.](fig3) First, we begin with some of the experiments of fluorescence and photo-luminescence in materials with metal nanoparticles of limiting small sizes, referred to in the introduction earlier. As this work is not concerned with the type of emitters and preparation of materials, such details of the original experiments are summarized in the supplementary for convenience of the more interested reader. The exciting radiation was off-resonant, and the limiting small absorbing nanoparticle does not significantly alter intensity of light exciting the emitter in these cases. The rates of radiative and non-radiative processes were estimated using independent life-time measurements in many cases, in addition to the measurement of gain in photons emitted. The measured gains are relatively robust and repeatable, while the decay rate is much more sensitive to any uncertainity in distances[@Supplementary]. To compare, we present evaluations of both relative quantum efficiency $Q$ and relative power of emission $Q\Gamma^r$, with respect to isolated emitters not interacting with metal nanostructures. The former represents the increase in probability of radiative decay of the excited emitter, while the latter also includes relative increase in ground-state population for emitters that can be excited continuously. Figure 1 presents the measured gains for varying separations of a low efficiency emitter ($Q_o$ $\approx$ 0.012) from a gold nanoparticle of 5 nm radius. The large gains in the observed emission and its contradiction with conventional theory that predicts quenching, are clear. Whereas the proposed $non$-$local$ theory predicts these measurements reasonably well notwithstanding possible uncertainty in efficiency of the isolated emitters. Figure 2 and Figure 3 present experiments of quenched emission from two different types of emitters. Here the nanoparticles are marginally larger at 6.5 nm radii and the quenching in the experimental results is significantly less than the expected values of the conventional theory. In contrast these experiments have a reasonable agreement with the predicted range of this extended theory. Note that quenching due to polymer molecules binding the emitter-nanoparticle system has not been included in both the theoretical models of relative emissions. When this is included, both theoretical predictions should reduce notably at larger distances due to the larger effect of polymer molecules. Thus one can infer a better agreement of experiments with the proposed theory, than reflected by figure 3. ![Comparison of theoretical and experimental results: The minimum and maximum gains predicted are given by quantum efficiency $Q$ and power radiated $Q\Gamma^r$, normalized by the values of isolated emitter; note $log$-scale in Y-axis. For short distances 1-5 nm, experimental data [@praveena2015plasmon] is for gold nanoparticles of radius 1.75 nm and the peak emission wavelength is 560 nm. For larger distances (7-15 nm) on the right side of the figure, four experimental data points at a mean distance of 12.5 nm [@haridas2013photoluminescence] are for gold nanoparticles of radius 2.5 nm and a peak emission wavelength of 560 nm.](fig4) Multiple experiments by other researchers on even smaller gold nanoparticles using self-assembled films and monolayers with quantum dot emitters, are summarized in Figure 4. Note that for the smaller metal nanoparticles discussed in Figures 1 and 4, the experiments present a direct contradiction; theory predicts quenching while experiments report significant enhancements of emission. Monolayers with smaller separations of emitters and gold nanoparticles of a smaller radii of 1.75 nm, have larger divergence with conventional theory as expected (section on the left in figure 4). But the notable enhancement up to a factor of 3 for the dots embedded in films due to sparsely doped and well separated 2.5 nm radii gold nanoparticles are equally unexpected (on the right in figure 4). In this section on the right, four experiments with nearly overlapping gain values are marked along with the predictions of theory. Our modified theory of local density of optical states predicts these experiments reasonably well. One notable aspect of the experiments with limiting small metal nanoparticles of 1.75 nm radii, is the weak sensitivity to number-ratio of emitters and metal particles in the monolayer, and this is not discussed here. The significant effect of virtual plasmons in this case may necessitate the inclusion of any collective behavior among emitters, that has been suggested in the presence of plasmonic nanoparticles [@pustovit2009cooperative], phonon interactions [@potma1998exciton] and otherwise at low temperatures [@gross1982superradiance]. But these theoretical works involved long-wavelength approximations and more importantly neglected any thermal effects; while other experimental indications of such collective behavior of emitters in the weak vacuum-coupling regime are few so far [@scheibner2007superradiance; @Goban2015superradiance; @Juan2017cooperative]. This weaker additional effect will be addressed elsewhere. ![Comparison of theoretical and experimental results: The minimum and maximum gains predicted are given by effective values of quantum efficiency $Q\frac{E^2}{E_o^2}$ and power radiated $Q\Gamma^r\frac{E^2}{E_o^2}$, normalized by the values of isolated emitter; note $log$-scale in Y-axis. Relative $\epsilon = -1.05+0.035i$ for the half-space forming the surface. Error bars placed alternately for conventional and non-local theories reflect variations in predictions using near-field enhancements by 2 to 50 nm size features on the metallic surface, where the lines represent a 10 nm feature.](fig5) In the case of SERS, we summarize multiple observations to compare predictions of the proposed theory. These enhancements due to metallic nanostructures of various geometries were typically predicted by the power-4 rule i.e. $(E_{local}/E_o)^4$ where subscripts ‘$o$’ and ‘$local$’ differentiate the amplitude of electric field of incident plane wave used for exciting the emitters, and its enhanced value at the location of emitter. Note that difference between excitation and emission energies are not large in SERS measurements, and we ignore this difference in the following interpretation of this rule. The enhancement of the local intensity near a metal structure is given by $(E_{local}/E_o)^2$, and it indicates the increase in probability of excitation of emitters. But for the power-4 rule to be valid in general, we have to infer an additional enhancement of emission by the same factor $(E_{local}/E_o)^2$. Interestingly, the immediate near-field of an absorbing body includes evanescent fields, and this factor represents the cumulative increase in radiative and non-radiative states for such proximal emitters. Thus SERS experiments can be inferred to demonstrate a significant loss of distinction between the radiative and non-radiative states for proximal emitters that are predominantly excited; as predicted by the proposed theory described in the next section. Figure 5 shows the predictions and experimental measurements of a few distance dependent SERS measurements [@kovacs1986distance; @ye1997surface], along with a few reported experiments without any spacers between the SERS surface and the emitters[@wang2013wafer; @gabudean2012gold; @ivanov2016strong; @mcfarland2005wavelength]. The latter are marked closest to a distance of zero where the probability of excitation is highest. We have plotted theoretical predictions of emission near a plane surface at plasmon resonance (relative permittivity $\epsilon \approx -1$) with near-field enhancement of the excitation due to nano features ranging in size from 2 to 50 nm marked by error bars. The effect of size of these nano features on the fluctuations ($\Gamma_{leak}$ in the next section) were ignored. These represent realistic gold or silver surfaces used in SERS. All experiments are within the predicted range of the non-local theory, but note that the very large Raman enhancements observed are prohibited in the conventional theory. A closer view of the short-range results, off-resonant predictions to account for some of these measurements being off-resonant, and a comparison of the quasi-static and full-wave solutions, are relegated to the supplementary[@Supplementary]. Overall, the proposed theory removes the large divergence observed in the measured enhancements of Raman signals over many decades, and is also supported by the more recent experiments with metal nanoparticles of limiting small dimensions. While our detailed evaluations showing this effect involve dynamics of the strong coupling regime, we believe our alternate elucidation that involves tunneling of photons from emitters coupled to evanescent fields, is also relevant. Further sensitive experiments may firmly establish the full theory defining density of optical states, especially in the strong matter-coupling regime. Methods: A non-local theory for density of optical states ========================================================= Before introducing the proposed extension to theory of density of optical states, we first briefly refer to known methods used to evaluate the modified self-energy of a point dipole in an inhomogeneous medium, and relegate the full details to the supplementary material[@Supplementary]. This problem may not have closed-form analytical solutions in general, and a quasi-static solution in the long wavelength limit has been typically preferred [@pustovit2010plasmon]. As noted before, this may result in errors in the estimated energy shifts [@van2012spontaneous], and in a underestimation of non-radiative decay rates in metal nanostructures [@venkatapathi2014collective]. Hence, we have resorted to more computationally intensive approaches to include retardation for finite wavelengths, and evaluate the variables in the proposed theory described below. But we emphasize that the divergence of experiments from conventional theory is evident even in calculations using first approximations. Single dipole approximation of these smaller metal nanoparticles, or an quasi-static approximation of a metal surface in the case of SERS experiments, are also presented in the supplementary [@Supplementary]. The additional decay rate due to the metallic structure is given by the imaginary part of its self-energy contribution $\Sigma$ i.e. $\Gamma=-2\Im(\Sigma)$; in units normalized by the reduced Planck’s constant as in equation . The contribution of the nanostructure to self-energy of an emitter at $\mathbf{r_o}$ is given by: $$\label{self_energy} \Sigma(\omega) = \frac{-2\pi q^2\omega}{mc^2}\mathbf{e}_1 \cdot \mathbf{G} (\mathbf{r_o,r_o};\omega) \cdot \mathbf{e}_1$$ and the above can be integrated over $\mathbf{e}_1$ for an average over all polarization, and over frequency $\omega$ with the relative spectral density of the free-space emitter in case of broad-band emission. Here $q$ is the oscillating charge, $m$ is its mass, and $c$ is speed of light. The evaluation of the required dyadic Green tensors $\mathbf{G}$ for an arbitrary structure is explained in the supplementary. $\Gamma^r_0$ and $\Gamma^{nr}_0$ are known radiative and non-radiative decay rates of the isolated emitter adding to $\Gamma_o$. The total radiative and non-radiative parts are a sum of the free-space and metallic components as below. $$\Gamma^r_{total}=\Gamma^r_0+\Gamma^r$$ Note that a dipole oscillator with an energy of one quantum represents the emitter as a two-level system in this weak vacuum-coupling regime. Then it is convenient to drop charge, mass and amplitude of the oscillator and normalize all self-energy components by $\Gamma^r_0$ for evaluations, where $\Gamma^r_0=\frac{2\sqrt{\epsilon_o}\mu^2k^3}{3\hbar}$ and $\mu$ is the electric dipole moment of the emitter; $k$, $\epsilon$ and $\hbar$ are the wave number, free-space permittivity and reduced Planck’s constant. $$\Gamma^{nr}_{total}=\Gamma^{nr}_0+\Gamma^{nr}$$ where $\Gamma^r$ and $\Gamma^{nr}$ are additional radiative and non-radiative decay rates of emitter in the presence of metal nanostructure, adding to the total metallic contribution $\Gamma$. The optical theorem for a point source establishes that the evaluated self-interaction of an emitter due to a proximal body represents the total radiative (scattering) and non-radiative (absorption) states of the body. The spatial reflection-symmetry of the free-space and the scattering component of the Green dyads, allows us to equally interpret this perturbation to self-energy as additional action of vacuum on the emitter due to presence of the body. In the rotating wave approximation ($|\Re(\Sigma)| << \omega$), the real part of self-energy in equation represents the energy shift in the emission due to the nanostructure [@pustovit2010plasmon]. $$\label{energy_shift} \hbar \Re(\Sigma)=\Delta E$$ More importantly, $|\Delta E|/\hbar$ is the rate of exchange of excitation between emitter and nanostructure i.e. the frequency of Rabi oscillations. This exchange in the form of emission and absorption of virtual photons is accompanied by the creation and annihilation of virtual plasmons in the metal particle. Typically, $\Gamma >> |\Delta E|/\hbar$ resulting in a negligible probability of a virtual plasmon in metals, and this regime is also irrelevant for dielectric materials where $\Gamma^{nr}$ is negligible. But in case of the smallest metal nanoparticles, decay rates and frequency of Rabi oscillations are comparable. Also, for an emitter very close to a large metallic nanostructure the above inequality weakens significantly. Note that this would replace absorption with an equivalent generation of photons by virtual plasmon annihilation, and is especially significant for the dipole mode of a nanostructure that represents most of its coupling to vacuum modes. These fluctuations of the dipole mode can not be distinguished from its scattering of virtual photons from vacuum. Only the latter contribution of elastic scattering is included in the increase of radiative states in conventional theory. Thus, the following extension of the local density of optical states becomes necessary. First, we assume the memory-less probability $density$ function in time $\tau$ for the real decay of plasmon i.e. $\Gamma e^{-\Gamma \tau}$. A typical case of integration of this probability density in the interval \[0,t\], results in a decay of an excited state as $e^{-\Gamma t}$ for all $t$. But here we restrict it to the interval of a Rabi oscillation as shown below. $$\label{eq:large delta} \mathcal{P}_{real} = \int_{\tau=0}^{\tau=\frac{\hbar}{|\Delta E|}} \Gamma e^{-\Gamma \tau} d\tau$$ Since the virtual decay of plasmon is due every Rabi oscillation between the emitter and the nanoparticle, we use the complement of $\mathcal{P}_{real}$ as the probability of generation of virtual photons. This probability of creation and annihilation of virtual plasmons can be non-negligible and thus given by: $$\mathcal{P}_{virtual} = 1 - \mathcal{P}_{real} = e^{-\frac{\hbar \Gamma}{|\Delta E|}}$$ The dipole mode contributions are numbered ’1’ in the subsequent part of this section. The marginally weaker absorption of higher order modes of an arbitrary nanostructure, made possible due to this dynamics is not discussed, considering we are interested only in limiting small nano-spheres and image dipoles due to surfaces. The effect of additional fluctuations of the dipole mode of the nanostructure on the rates of emission is trivially evaluated using work done on the emitter, as in the classical optical theorem: $$\Gamma_{leak}=e^{-\frac{\hbar \Gamma}{|\Delta E|}}\cdot\Gamma^{nr}_1$$ This rate accounts to a vanishing of non-radiative absorption of the dipole mode and its appearance as a stronger radiative mode. The effective decay rates thus become: $$\Gamma^r_{eff}=\Gamma^r_0+\Gamma^r+\Gamma_{leak}$$ and $$\Gamma^{nr}_{eff}=\Gamma^{nr}_0+\Gamma^{nr}-\Gamma_{leak}$$ The observed quantum efficiency is then: $$Q=\frac{\Gamma^r_{eff}}{\Gamma^r_{eff}+\Gamma^{nr}_{eff}}$$ A salient point in the above corrections is the strong effect even a relatively small $\mathcal{P}_{virtual}$ has on the radiative rates and the quantum efficiency, in case of a proximal large metallic nanostructure or surface dominated by non-radiative decay (i.e. $\Gamma^r_{eff}$ $\gg$ $\Gamma^r$+$\Gamma^r_0$). 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--- abstract: | We prove uniform Hölder regularity estimates for a transport-diffusion equation with a fractional diffusion operator, and a general advection field in BMO, as long as the order of the diffusion dominates the transport term at small scales; our only requirement is the smallness of the negative part of the divergence in some critical Lebesgue space. In comparison to a celebrated result by L. Silvestre (2012), our advection field does not need to be bounded. A similar result can be obtained in the super-critical case if the advection field is Hölder continuous. Our proof is inspired by A. Kiselev and F. Nazarov (2010) and is based on the dual evolution technique. The idea is to propagate an atom property (*i.e.* localization and integrability in Lebesgue spaces) under the dual conservation law, when it is coupled with the fractional diffusion operator.\ **Keywords:** Transport-diffusion, Hölder regularity, fractional diffusion equation, non-local operator, $\operatorname{BMO}$ drift, dual equation, conservation law, functional analysis, atoms.\ **MSC primary:** 35B65.\ **MSC secondary:** 35R11, 35Q35. author: - | Ioann Vasilyev[^1] and François Vigneron[^2]\ Université Paris-Est\ LAMA (<span style="font-variant:small-caps;">umr8050</span>), UPEC, UPEM, CNRS\ 61, avenue du Général de Gaulle, F94010 Créteil, France. title: | Variation on a theme by Kiselev and Nazarov:\ Hölder estimates for non-local transport-diffusion,\ along a non-divergence-free BMO field. --- In this article, we are interested in the following transport-diffusion equation: $$\label{TDalpha} \begin{cases} \partial_t \theta + (-\Delta)^{\alpha/2} \theta = (v \cdot \nabla) \theta\\ \theta(0,x) = \theta_0(x). \end{cases}$$ We consider this Cauchy problem where $\theta:[0,T]\times {\mathbb{R}}^d\to {\mathbb{R}}$ is unknown. The vector field $v$ is given and is of bounded mean oscillation, [*i.e. *]{}it belongs to the space $\operatorname{BMO}({\mathbb{R}}^d)$. In what follows, we will **not** assume that ${\operatorname{div}}v=0$. We will also consider the periodic problem on ${\mathbb{T}}^d$. We are interested in the whole range of parameters $0<\alpha < 2$ and in particular in the critical non-local diffusion ([*i.e. *]{}$\alpha=1$) where diffusion and transport are of similar order. We will, in passing, consider the classical local case $\alpha=2$; however, the local case is much better understood and we refer the reader, for exemple, to the recent monograph [@LBL2019] and the references therein. For $\alpha>1$, the equation is called sub-critical because the drift is then of lower order than the diffusion, which means that the diffusion will be stronger at the smallest scales. On the contrary, for $\alpha<1$, the drift will be stronger than the diffusion at smaller scales and the equation is called super-critical. However, as the diffusion operator remains invariant under Galilean transforms, one can still expect a mild smoothing effect, as long as $\alpha>0$, because putative advected singularities cannot ride along a “defect” of the diffusion operator, simply because the homogeneous and isotropic operator does not have any. The fractional derivative $(-\Delta)^{\alpha/2}$ is the Fourier multiplier by $|\xi|^{\alpha}$. It admits the following kernel expansion on ${\mathbb{R}}^d$ (see [*e.g. *]{}[@CC2004 §2]): $$\label{kernelPseudoLaplace} (-\Delta)^{\alpha/2} \theta (x) = -c_{d,\alpha} \operatorname{pv} \left( \int_{{\mathbb{R}}^d} \frac{\theta(y)-\theta(x)}{|y-x|^{d+\alpha}} dy \right)$$ with $c_{d,\alpha}>0$. As $(-\Delta)^{\alpha/2}$ is a non-local derivative of order $\alpha$ that does not induce phase-shift, it performs a sort of “graphical” interpolation between the graph of $\theta$ and that of $-\Delta \theta$. The goal is to establish uniform Hölder regularity estimates of the solution of  at a later time $t>0$. As demonstrated by A. Kiselev and F. Nazarov [@KN2009] and M. Dabkowski [@D2010] for $\alpha=1$, an elegant but powerful technique consists in using the atomic characterization of Hölder classes (see below). Multiplying the equation by a test-function $\psi(t-s)$ where $\psi$ solves the dual evolution equation, one can then exchange an atomic estimate of $\theta(t)\psi_0$ against an atomic estimate of $\theta_0\psi(t)$. Provided that the dual evolution propagates the atomic property in a controlled way, one then gets the desired Hölder regularity. Another powerful approach, used [*e.g. *]{}by L.A. Caffarelli and A. Vasseur [@CV2010] and L. Silvestre [@Silv2010], consists in using the parabolic De Giorgi method. The idea is to first establish a set of energy estimates. Then, one uses the rigidity given by the equation to deduce scalable uniform bounds. More precisely, one computes the energy cost of one oscillation of the solution between its maximum and minimum value. Given that the total amount of energy available is finite, this limits the size of the oscillations that the solution can perform at a given scale, which ultimately translates as regularity in the Hölder classes. Finally, the method of the modulus of continuity was successfully introduced by A. Kiselev, F. Nazarov and A. Volberg [@KNV2007] for SQG and later used by L. Silvestre, V. Vicol [@SV2018] to study related PDEs. The case of a divergence-free transport field in  is of particular interest as it closely relates to the quasi-geostrophic equation where $d=2$ and $u = \nabla^\perp (-\Delta)^{-1/2} \theta$; see [@CCW2001], [@CV2010], [@KNV2007]. At a technical level, the computations of [@KN2009], [@D2010] rely in a crucial way on the assumption that ${\operatorname{div}}v=0$. First, as they use the non-conservative form for the dual evolution equation $$\partial_s \psi + (-\Delta)^{\alpha/2} \psi = - (v(t-s) \cdot \nabla) \psi$$ each integration by part requires a divergence-free field. Second, they invoque the maximum-principle and the decay of the $L^p$-norms for this kind of equation, which was established by A. Cordoba, D. Cordoba [@CC2004], but either for the particular vector field of SQG, or for a divergence-free transport field. The key of our article is that we allow ${\operatorname{div}}v \neq 0$. The sign of ${\operatorname{div}}v$ then becomes crucial because convergent characteristics will tend to create shocks and break down the regularity while divergent characteristics (rarefaction waves) have a natural smoothing effect and tend to reduce steep gradients. The presence of the diffusion operator fixes the issues of either non-existence or non-uniqueness that the pure transport would induce. In that context, the pertinent question is thus to establish quantitative estimates of the regularity norms. Our main result is the following statement that provides uniform Hölder bounds for solutions of . \[mainThm\] We consider either ${\mathbb{R}}^d$ or ${\mathbb{T}}^d$ as the ambiant space, which is of dimension $d>\alpha$, and an advection field $$\begin{cases} v\in C^{1-\alpha} & \text{if}\quad 0<\alpha<1,\\ v\in\operatorname{BMO}& \text{if}\quad 1\leq \alpha\leq 2. \end{cases}$$ In both cases, one requires: $$\label{critical} {\| ({\operatorname{div}}v)_- \|_{L^{d/\alpha}}} \leq S_{\alpha/2} = \sup_{f\in \dot{H}^{\alpha/2}}\left( \frac{ \displaystyle \int |(-\Delta)^{\alpha/4}f|^2 }{ {\| f \|_{L^{2d/(d-\alpha)}}}^2 } \right)$$ where $({\operatorname{div}}v)_-$ is the negative part of the divergence (see  below) and $\dot{H}^{\alpha/2}$ denotes the homogeneous Sobolev space on ${\mathbb{R}}^d$, or the average-free space on ${\mathbb{T}}^d$. There exist $\beta>0$ and $C>0$ that depend solely on $d$, $\alpha$ and respectively either on ${\| v \|_{\operatorname{BMO}}}$ or ${\| v \|_{C^{1-\alpha}}}$ such that, for any $\theta_0\in L^q$ with $2\leq q \leq \infty$, the corresponding solution of  satisfies: $$\label{mainReg} \forall t\in (0,1], \qquad {\| \theta(t,\cdot) \|_{C^\beta}} \leq C t^{-(\beta+\frac{d}{q})/\alpha} {\| \theta_0 \|_{L^q}}.$$ Moreover, if $\theta_0\in C^\beta$, then: $$\label{mainProp} \forall t\geq0, \qquad {\| \theta(t,\cdot) \|_{C^\beta}} \leq C' {\| \theta_0 \|_{C^\beta}}$$ for some constant $C'$. Among the results quoted in this introduction, that of L. Silvestre [@Silv2010] is the only one that allows for non-divergence-free fields, so we will focus on this one. For $\alpha\geq1$, he assumes that $v\in L^\infty({\mathbb{R}}^d)$ but requires no other size constraint on $v$; for $\alpha<1$, he assumes that $v\in C^{1-\alpha}$ in order to compensate for the super-criticality of the equation. In both cases, L. Silvestre proves the Hölder continuity of the solution of  at positive times, namely that: $${\| \theta(t) \|_{C^\beta({\mathbb{R}}^d)}} \leq C t^{-\beta/\alpha} {\| \theta_0 \|_{L^\infty({\mathbb{R}}^d)}}$$ and he also establishes a similar $C^{\beta/\alpha}$-regularity estimate time-wise. Our result is complimentary and does not quite compare to that of L. Silvestre when $\alpha\geq1$ because, in that case, we do not assume that $v$ is bounded. Instead, we assume that $v$ belongs to $\operatorname{BMO}$ and we require the negative part of its divergence to be small. However, we do not constrain the size of $({\operatorname{div}}v)_+$ and, in particular any BMO divergence-free field is admissible in our result. It is likely that a “universal” result holds true for a general advection field $v=v_1+v_2$ with $v_1\in L^\infty$ and $v_2\in \operatorname{BMO}$ with $({\operatorname{div}}v_2)_-$ small. Results in this spirit are known for general diffusion operators of order $\alpha=2$ (see [@LBL2019]). However, mixing the DeGiorgi and atomic methods is not obvious. The critical value in  can be slightly relaxed but the constants $\beta$ and $C$ will then also depend on the size of $({\operatorname{div}}v)_-$; see remark \[relax\]. When $\alpha=2$, the dimension $d=2$ is excluded from our statement (and similarly the case $\alpha=d=1$) because the corresponding Sobolev Space $H^{\alpha/2}({\mathbb{R}}^d)$ fails to be included in $L^\infty$. Our proof is inspired by A. Kiselev and F. Nazarov [@KN2009] and we use the dual evolution technique. Following M. Dabkowski [@D2010], we also use $L^p$-based atoms for more flexibility. In order to deal with non-divergence-free transport fields, the key is to replace the dual evolution equation by a transport-diffusion equation expressed in a *conservative* form, [*i.e. *]{}as a conservation law. The presence of a general exponent of diffusion $\alpha\neq 1$ also brings about some technicalities that need to be addressed (see the end of §\[par:regularity\]). There are many other takes on the question of regularity for transport-diffusion equations. For example, D. Bresch, P.-E. Jabin [@BJ2018] have studied the regularity of weak solutions to the advection equation set in conservative form. In particular, they investigated the case where both $v$ and ${\operatorname{div}}v$ belong to some Lebesgue space, which allows point-wise unbounded variations of the field and of its compressibility. Another related study, albeit slightly more distant from , is that of the kinetic Fokker-Planck equation. On that mater, we refer the reader to C. Mouhot [@M2018] and the references therein, and, [*e.g.,*]{} to the regularity result by F. Golse, A. Vasseur [@GV2015]. The role of fractional diffusion operators in physical models is growing (see [*e.g. *]{}J. Vaźquez’s book [@Vaz2017]). As for our motivations, we are interested in studying variants of a non-local Burgers equation introduced by C. Imbert, R. Shvydkoy and one of the present authors in [@ISV2016]. The regularity of the solution for unsigned data remains an open problem that seems to share some nontrivial similarities with hydrodynamic turbulence. The present article is structured as follows. In §\[par:atoms\], we recall how Hölder classes can be characterized in terms of atoms. We draft the general ideas on how regularity can be obtained by studying the dual conservation law in §\[par:regularity\]. A weak version ([*i.e. *]{}in Lebesgue spaces) of the maximum principle is established in §\[par:maxPrinciple\]. It is then put to use in §\[par:propAtom\] to quantify how the atomic property is propagated. The proof of theorem \[mainThm\] and, in particular, how the choices of structural constants should be made, is detailed in §\[par:proof\]. #### Notations. In this article, one uses the following common notations for $a,b,x\in{\mathbb{R}}$: $$\begin{gathered} a\wedge b = \min\{a,b\}, \qquad a\vee b = \max\{a,b\}\\\label{PosNegNotation} x_+ = a \vee 0, \qquad x_- = (-x)_+, \qquad x = x_+ - x_-, \qquad |x|= x_+ + x_-.\end{gathered}$$ Balls are denoted by ${\mathcal{B}(x_0,r)}=\left\{x\in{\mathbb{R}}^d \,;\, |x-x_0|<r\right\}$ where $|\cdot|$ denotes the Euclidian distance on ${\mathbb{R}}^d$. Atomic characterization of Hölder classes {#par:atoms} ========================================= Throughout the article, the constant $A\gg1$ is fixed, but is chosen arbitrarily large and $\omega\in(0,1)$. One can check that none of the final estimates actually depend on the particular values of $A$ or $\omega$. For $r>0$ and $p\in (1,\infty]$, the atomic class ${\mathcal{A}}^p_r({\mathbb{R}}^d)$ is defined as a subset of $C^\infty({\mathbb{R}}^d)$ by the following three conditions: $$\begin{gathered} \int_{{\mathbb{R}}^d} \varphi(x) dx =0,\\ {\| \varphi \|_{L^1}} \leq 1 \qquad\text{and}\qquad {\| \varphi \|_{L^p}} \leq A r^{-d(1-\frac{1}{p})},\\ \exists x_0\in{\mathbb{R}}^d, \qquad \int_{{\mathbb{R}}^d} |\varphi(x)| \Omega(x-x_0) dx \leq r^\omega\end{gathered}$$ where $\Omega(z) = |z|^\omega \wedge 1 \in L^\infty({\mathbb{R}}^d)$. If $\lambda^{-1} \varphi(x) \in {\mathcal{A}}^p_r({\mathbb{R}}^d)$ for some $\lambda>0$, then one says that $$\varphi\in \lambda\cdot {\mathcal{A}}^p_r({\mathbb{R}}^d).$$ A typical example of an atom ${\mathcal{A}}^p_r$ of radius $r< 1$ is the function $\varphi = \varphi_r\ast \rho_\epsilon$ where $\rho_\epsilon$ is an standard mollifier supported in ${\mathcal{B}(0,\epsilon)}$ and $$\varphi_r(x) = \begin{cases} - C r^{-d} & \text{if } |x-x_0| \leq r 2^{-1/d} \\ + C r^{-d} & \text{if } r 2^{-1/d} < |x-x_0| \leq r \\ 0 & \text{if } |x-x_0|>r \end{cases}$$ for small enough constants $C < \min\{ |{\mathcal{B}(0,1)}|^{-1}; A |{\mathcal{B}(0,1)}|^{-1/p}; |{\mathcal{B}(0,1)}|/(d+\omega)\}$ and $\epsilon>0$. The volume of the unit ball is $|{\mathcal{B}(0,1)}| = \pi^{d/2}/\Gamma\left(\frac{d}{2}+1\right)$. One can control the $L^q$-norm of atoms for $q\in[1,p]$ by a real interpolation estimate. \[AtomL1LpBound\] If $\varphi \in {\mathcal{A}}^p_r({\mathbb{R}}^d)$, then for any $1\leq q\leq p$: $$\label{AtomLpBound} {\| \varphi \|_{L^q}} \leq A^{\frac{1-1/q}{1-1/p}} r^{-d(1-\frac{1}{q})}.$$ Apply the interpolation inequality ${\| f \|_{L^q}}\leq {\| f \|_{L^1}}^{1-\theta} {\| f \|_{L^p}}^{\theta}$ with $\theta = (1-\frac{1}{q})/(1-\frac{1}{p}) \in [0,1]$. ------------------------------------------------------------------------ Atoms are the “poor man’s wavelet” and offer a very comfortable characterization of Hölder’s classes. For $0<\beta<1$, a bounded function $f$ belongs to $C^\beta({\mathbb{R}}^d)$ if and only if $$\label{atomCbeta} \sup_{\substack{0<r\leq 1\\ \varphi\in {\mathcal{A}}^p_r({\mathbb{R}}^d)}} r^{-\beta} \left| \int_{{\mathbb{R}}^d} f(x) \varphi(x) dx \right| < \infty$$ for some $p\in(1,\infty]$. Moreover, the left-hand side of  is equivalent to the usual semi-norm on $C^\beta({\mathbb{R}}^d)$. #### *Proof.* The proof is a classical exercise. See [@KN2009], [@D2010] (resp. for $p=\infty$ and $p<\infty$) for a short proof that relies on the Littlewood-Paley theory [@STEIN1993]. The key is the equivalent control of $\sup_{j\in{\mathbb{N}}} 2^{\beta j}{\| \Delta_j f \|_{L^\infty}}$ where $\Delta_j$ is a smooth projection on the frequency scale of order $2^j$. ------------------------------------------------------------------------ Regularity through the dual conservation law {#par:regularity} ============================================ As explained in the previous section, in order to obtain estimates of the Hölder regularity of the solution of  at a time $t>0$, one needs to control $$r^{-\beta} \int_{{\mathbb{R}}^d} \theta(t,x) \psi_0(x) dx$$ where $\psi_0\in {\mathcal{A}}^p_r({\mathbb{R}}^d)$ and this control needs to be uniform in $0<r\leq 1$. Let us consider the following test function that solves the dual evolution problem, set in a *conservative* form: $$\label{dualEQ} \begin{cases} \partial_s \psi(s) + (-\Delta)^{\alpha/2} \psi(s) = - {\operatorname{div}}\left( v(t-s) \psi(s) \right)\\ \psi(0,x)=\psi_0(x). \end{cases}$$ One can then immediately check the following result. If $\theta$ is a smooth solution of  and $\psi$ is a smooth solution of , then one has: $$\label{transfert} \forall t\geq0,\qquad \int_{{\mathbb{R}}^d} \theta(t,x) \psi_0(x) dx = \int_{{\mathbb{R}}^d} \theta_0(x) \psi(t,x) dx.$$ #### *Proof.* Let us indeed use $\psi(t-s)$ as a test function for . One gets: $$\iint \partial_s \theta(s) \cdot \psi(t-s) + \iint ((-\Delta)^{\alpha/2} \theta(s)) \cdot \psi(t-s) = \iint (v(s)\cdot \nabla) \theta(s) \cdot \psi(t-s).$$ Here and below, all double integrals are computed for $(s,x)\in [0,t]\times{\mathbb{R}}^d$ unless stated otherwise and the $x$-variable is not made explicit unless it is absolutely necessary. Integrating by part time-wise in the first integral and space-wise in the other two gives: $$\begin{aligned} \iint \theta(s) \cdot (\partial_s\psi)&(t-s) + \left[ \int_{{\mathbb{R}}^d} \theta(s) \psi(t-s) \right]_{0}^{t}\\ &+ \iint \theta(s) \cdot ((-\Delta)^{\alpha/2} \psi)(t-s) = - \iint \theta(s) \cdot {\operatorname{div}}(v(s) \psi(t-s)).\end{aligned}$$ Thanks to , the double integrals cancel each other out and one is left with . ------------------------------------------------------------------------ In [@KN2009], the authors used the non-conservative dual form $-v(t-s)\cdot\nabla\psi(s)$. This choice was harmless and perfectly adapted to their purpose because they assumed $v$ to be divergence-free. Here, on the contrary, it is crucial that the right-hand side of  takes the form of a conservation law. For the sake of the argument, let us assume that one will be able to show subsequently (which is indeed the case if $\alpha=1$) the following infinitesimal propagation property for : $$\psi_0\in{\mathcal{A}}_r^p \qquad\Longrightarrow\qquad \forall s\in [0,\gamma r], \qquad \psi(s) \in (1-h(r) s) {\mathcal{A}}_{r+Ks}^p$$ for a given value of $p\in(1,\infty]$, with universal constants $\gamma, K$ that do not depend on $r$ nor $\psi_0$ and a universal function $h$. One can then immediately infer a global propagation property: $$\psi_0\in{\mathcal{A}}_r^p \qquad\Longrightarrow\qquad \forall s\in \left[0, \frac{1-r}{K}\right], \qquad \psi(s) \in f_r(s) {\mathcal{A}}_{r+Ks}^p$$ with $f_r'(s) \geq -h(r+Ks) f_r(s)$. Let us introduce a function $H$ such that $H'(z)=h(z)$. Then $$f_r(s) = \exp\left( \frac{H(r)-H(r+Ks)}{K} \right)$$ is an acceptable bound for the global propagation property. Coupled with , this means the following: for any solution of  and $\psi_0\in{\mathcal{A}}^p_r$, one has $$\left| \int_{{\mathbb{R}}^d} \theta(t,x) \psi_0(x) dx \right| = \left| \int_{{\mathbb{R}}^d} \theta_0(x) \psi(t,x) dx \right| \leq {\| \theta_0 \|_{C^\beta}} (r+Kt)^\beta f_r(t).$$ One is thus able to propagate $C^\beta({\mathbb{R}}^d)$ bounds of $\theta$ if $f_r(s) \leq C ( \frac{r}{r+Ks} )^\beta$. This is the case if, for example, $h(r) = \delta/r$ with $\delta = K\beta$. The regularization estimate is obtained in the same fashion. Dealing with a general exponent $\alpha$ requires a slightly more careful computation. The fundamental idea remains that the dual conservation law propagates atoms and that a small gain on the amplitude of the atoms can be obtained as a tradeoff with a slight increase in the size of the atoms’ radii. The main technical difficulty is that the radii now grow as $(r^\alpha+Ks)^{1/\alpha}$, which is not linear in $s$ anymore, at least not when $s\gg r^{\alpha}$. This non-linear region invades any neighborhood of $r=s=0$ and the corresponding correction of amplitude will be $h(r)=\delta/r^\alpha$. We found that the simplest workaround is to forfeit the ODE point of view presented here for $\alpha=1$ and to use direct estimates on the corresponding rate of change during a finite increment of an Euler scheme (see §\[par:propAtom\], estimate ). Weak maximum principle for the dual conservation law {#par:maxPrinciple} ==================================================== In this section, we establish the weak maximum principle [*i.e. *]{}the decay of the $L^p$-norms for a non-local transport-diffusion equation written in a conservative form. In this section, one considers thus the following general problem for $0<\alpha\leq2$: $$\label{TDalphaCons} \begin{cases} \partial_s \psi(s) + (-\Delta)^{\alpha/2} \psi(s) = - {\operatorname{div}}\left( \mathbf{v}(s) \psi(s) \right)\\ \psi(0,x)=\psi_0(x) \end{cases}$$ and we will assume, when necessary, that $\displaystyle \int_{{\mathbb{R}}^d} \psi_0 = 0$. Subsequently, the results of this section will be applied to  at a given time $t>0$ by choosing $\mathbf{v}(s) = v(t-s) \in\operatorname{BMO}({\mathbb{R}}^d)$. A brief note on the well-posedness theory ----------------------------------------- For smooth $\mathbf{v}(s,x)$, the well-posedness theory of the scalar conservation law $$\partial_s \psi(s) = - {\operatorname{div}}\left( \mathbf{v}(s) \psi(s) \right)$$ was established by S.N. Kružkov [@Kru1970], in the setting of entropy solutions. The theory was refined and generalized to the non-conservative convective form by R.J. DiPerna and P.L. Lions [@DipL1989]; their theory ensures that assuming a transport field $\mathbf{v} \in L^1([0,T];W^{1,1})$ with $({\operatorname{div}}\mathbf{v})_-\in L^1([0,T];L^\infty)$ is enough to guarantee the existence, uniqueness and stability in the proper function spaces. The key idea is a celebrated commutation lemma: $$\rho_\varepsilon \ast ( \mathbf{v}\cdot\nabla) \psi - \mathbf{v}\cdot\nabla(\rho_\varepsilon \ast \psi)\to 0 \qquad \text{in } L^1([0,T];L^\beta_{\text{loc}}).$$ On ${\mathbb{R}}^d$, an additional assumption of mild growth at infinity is required, [*e.g. *]{}$\mathbf{v}\in (1+|x|)\cdot(L^1+L^\infty)$. To handle unbounded data, the idea is to use renormalization, [*i.e. *]{}to consider $\Phi(\psi)$ for suitable smooth and bounded $\Phi$. For a review of the fundamental ideas and the last developments of the theory, we refer the reader to the monograph [@LBL2019] by C. Le Bris and P.L. Lions, and the references therein. See also the lecture notes by L. Ambrosio and D. Trevisan [@AT2017] or those of C. De Lellis [@DL2007]. Adding the coercive diffusion term $(-\Delta)^{\alpha/2}\psi$ in  with $0<\alpha\leq 2$ obviously does not alter these results. On the contrary, the assumptions on the transport field can even be relaxed. For example, for $\alpha=2$ and even with a fully general second-order elliptic operator, one can accept a field $\mathbf{v}\in L^2 + W^{1,1}$ with $({\operatorname{div}}\mathbf{v})_-\in L^\infty$, as mentioned in [@LBL2019 §3.2]. The local well-posedness of  is thus classical; see [*e.g. *]{}[@BJ2018]. If the transport term takes the conservative form, the equation is called a *conservation law*; if not, it is referred to as a general *convection*. When the Laplace operator has variable coefficients, then the term *conservative* is preferred to describe the equation with the operator written in divergence form $-\partial_i(a_{ij}\partial_j)$, regardless of whether the transport part is a convection or a conservation law. In our present case, however, the fractional power $(-\Delta)^\alpha$ is obviously a conservative operator so our use of the adjective *conservative* concerns only the form of the advection term. Propagation of positivity ------------------------- The classical positivity result for $\alpha=2$ can be generalized for fractional diffusions. If $\psi$ is a solution to , stemming from $\psi_0\geq0$, then $\psi(s)\geq0$ for any $s\geq0$. Let us sketch the argument first. If a solution of  is smooth and positive, then at a first contact point with zero, say $(s_0,x_0)$, it reaches a global minimum. One thus has $\psi(s_0,x_0)=0$ and $\nabla\psi(s_0,x_0)=\mathbf{0}$, and therefore: $${\operatorname{div}}\left( \mathbf{v} \psi \right) = ({\operatorname{div}}\mathbf{v}) \psi(s_0,x_0) + (\mathbf{v}\cdot\nabla) \psi(s_0,x_0) =0.$$ Moreover, for $0<\alpha<2$, by , there exists a positive kernel $K_{d,\alpha}$ such that: $$(-\Delta)^{\alpha/2} \psi (s_0,x_0) = - \int_{{\mathbb{R}}^d} \left(\psi(s_0,y)-\psi(s_0,x_0) \right) K_{d,\alpha}(y-x_0) dy \leq 0$$ and the inequality is strict, unless $\psi(s_0,\cdot) \equiv0$. The equation ensures that $\partial_s \psi (s_0,x_0)\geq 0$ and, in particular, the solution remains positive forever. To make the proof fully rigorous, one proceeds [*e.g. *]{}as in [@ISV2016 §2.1]: for given $T,R>0$ and $\psi_0>0$, one considers the approximation $\psi_R$ where the kernel $K_{d,\alpha}$ is restricted to ${\mathcal{B}(0,R)}$ and $$s_0 = \inf \left\{ s\in(0,T) \,;\, \exists x_0 \in {\mathcal{B}(0,R)}, \quad \psi_R(s,x_0) = 0 \right\}.$$ By compactness, $s_0$ is attained and $s_0>0$. As $\psi_R(s,\cdot)$ is not identically zero, the previous computation ensures that $\partial_s \psi_R(s_0,x_0)>0$ and thus $\psi_R$ had to be negative in the neighborhood of $x_0$ a short time before $s_0$, which is contradictory. For a general $\psi_0\geq0$, the data can be approximated by a strictly positive mollification, whose strict positivity propagates downstream. Passing to the limit at a later time $s>0$ ensures therefore that $\psi(s)\geq0$. ------------------------------------------------------------------------ Propagation of the $L^1$ norm and conservation of momentum ---------------------------------------------------------- The simple structure of  inherited from the underlying conservation law plays in our favor. Let $\psi$ be a solution to . Then $$\begin{gathered} \label{propL1Norm} \|\psi(s,\cdot)\|_{L^1}\leq \|\psi_0\|_{L^1} \\ \intertext{and} \label{consMomentum} \int_{{\mathbb{R}}^d} \psi(s,x) dx = \int_{{\mathbb{R}}^d}\psi_0(x)dx.\end{gathered}$$ For the first statement, let us decompose $\psi_0=\psi_0^+-\psi_0^-$ where both $\psi_0^+$ and $\psi_0^-$ are positive and have disjoint supports. Let $\psi^+$ and $\psi^-$ be the solutions to the equation with initial data $\psi^+_0$ and $\psi^-_0$ correspondingly. Then, by linearity, $\psi(s)=\psi^+(s)-\psi^-(s)$ and therefore $$\|\psi(s,\cdot)\|_{L^1}\leq \|\psi^+(s,\cdot)\|_{L^1}+\|\psi^-(s,\cdot)\|_{L^1}.$$ Equation  and an integration by part ensure that: $$\frac{d}{ds} \int_{{\mathbb{R}}^d} \psi^\pm (s,x) dx = -\int_{{\mathbb{R}}^d} (-\Delta)^{\alpha/2}\psi^\pm -\int_{{\mathbb{R}}^d}{\operatorname{div}}(\mathbf{v}\psi^\pm)dx=0.$$ As the propagation of positivity yields that $\psi^\pm\geq 0$, one gets $\|\psi^\pm(s,\cdot)\|_{L^1}= \|\psi^\pm_0\|_{L^1}$ and finally $\|\psi(s,\cdot)\|_{L^1}\leq \|\psi^+_0\|_{L^1}+ \|\psi^-_0\|_{L^1}= \|\psi_0\|_{L^1}$, hence . The identity  is immediate. ------------------------------------------------------------------------ Note that, because of the diffusion, the functions $\psi^\pm$ of the previous proof will not coincide, in general, with the positive and negative parts $\psi_\pm$ of $\psi$. Estimate of the $L^p$ norm -------------------------- For $h<d/2$ let us introduce the constant in the Sobolev embedding $\dot{H}^{h}({\mathbb{R}}^d) \subset L^{2d/(d-2h)}$ (see [*e.g. *]{}[@T1979]): $$\label{sobolevEmbedding} S_{h}({\mathbb{R}}^d)^{-1} = \sup \left\{ {\| f \|_{L^{2d/(d-2h)}}}^2 \,;\, f\in \dot{H}^{h}({\mathbb{R}}^d), \enspace \displaystyle \int_{{\mathbb{R}}^d} |(-\Delta)^{h/2} f|^2 = 1 \right\} >0.$$ The idea is to relax the uniform control given by the maximum principle for  into a weaker one in the scale of Lebesgue spaces. \[propagationLp\] For any $\alpha\in(0,2]$ and any dimension $d>\alpha$, if the transport field satisfies $$\label{smallnessAssumptiom} (p-1) {\| ({\operatorname{div}}\mathbf{v})_- \|_{L^\infty_t L^{d/\alpha}_x}} \leq S_{\alpha/2}({\mathbb{R}}^d)$$ for some $p\geq2$ (eventually restricted to $p=2^n$ with $n\in{\mathbb{N}}$ if $\alpha<2$), then any solution of  satisfies $$\label{estimPropagationLp1} {\| \psi(s) \|_{L^p}}^p + S_{\alpha/2}({\mathbb{R}}^d) \int_0^s {\| \psi(\tau) \|_{L^\sigma}}^p d\tau \leq {\| \psi_0 \|_{L^p}}^p \qquad\text{with}\qquad \sigma = \frac{dp}{d-\alpha}$$ and in particular $$\label{estimPropagationLp2} \forall q\in[1,p],\qquad \forall s\geq0,\qquad {\| \psi(s) \|_{L^q}} \leq {\| \psi_0 \|_{L^q}}$$ for any $\psi_0\in L^1\cap L^p$. In particular, when ${\operatorname{div}}(\mathbf{v})\geq0$, the estimate holds for $1\leq q\leq \infty$. The following proof also establishes that all solutions of  satisfy: $${\| \psi(s) \|_{L^p}} \leq {\| \psi_0 \|_{L^p}} e^{t (1-\frac{1}{p}) {\| ({\operatorname{div}}\mathbf{v})_- \|_{L^\infty_{t,x}}} }$$ regardless of the diffusion term and independently of . For what follows, we are however interested in getting a better ([*i.e. *]{}non-increasing) control of the $L^p$-norm as given by -. Using $p|\psi|^{p-2}\psi$ as a multiplier for the equation leads to: $$\frac{d}{ds} \int_{{\mathbb{R}}^d} |\psi(s,x)|^p dx + p \int_{{\mathbb{R}}^d} |\psi|^{p-2}\psi \cdot (-\Delta)^{\alpha/2}\psi = - p \int_{{\mathbb{R}}^d} {\operatorname{div}}\left( \mathbf{v} \psi \right) |\psi|^{p-2}\psi.$$ For the integral on the right-hand side, an integration by part gives: $$\begin{aligned} \int_{{\mathbb{R}}^d} {\operatorname{div}}\left( \mathbf{v} \psi \right) |\psi|^{p-2}\psi & = - (p-1)\int_{{\mathbb{R}}^d} |\psi|^{p-2}\psi (\mathbf{v}\cdot\nabla) \psi \\ & = - (p-1)\int_{{\mathbb{R}}^d} {\operatorname{div}}\left( \mathbf{v} \psi \right) |\psi|^{p-2}\psi + (p-1) \int_{{\mathbb{R}}^d}|\psi|^p {\operatorname{div}}\mathbf{v}.\end{aligned}$$ One thus has this identity: $$\label{identityRhsLpEstim} \int_{{\mathbb{R}}^d} {\operatorname{div}}\left( \mathbf{v} \psi \right) |\psi|^{p-2}\psi = \left(1-\frac{1}{p}\right) \int_{{\mathbb{R}}^d}|\psi|^p {\operatorname{div}}\mathbf{v}$$ and thus $$\label{startingLpEstimate} \frac{d}{ds} \int_{{\mathbb{R}}^d} |\psi(s,x)|^p dx + p \int_{{\mathbb{R}}^d} |\psi|^{p-2}\psi \cdot (-\Delta)^{\alpha/2}\psi = - (p-1) \int_{{\mathbb{R}}^d} |\psi|^p {\operatorname{div}}\mathbf{v}.$$ For the integral on the left-hand side of  and when $\alpha=2$, the following identity holds: $$\label{heatLp} p \int_{{\mathbb{R}}^d} |\psi|^{p-2}\psi \cdot (-\Delta)\psi = p(p-1) \int_{{\mathbb{R}}^d} |\psi|^{p-2} |\nabla \psi|^2 = 4\left(1-\frac{1}{p}\right)\int_{{\mathbb{R}}^d} |\nabla (|\psi|^{p/2})|^2 \geq 0.$$ For $0<\alpha<2$, one needs to replace  because the Leibniz formula is no longer valid; instead, one follows the ideas of [@CC2004]. The key is the point-wise inequality [@CC2004 Prop. 2.3]: $$2\psi \cdot (-\Delta)^{\alpha/2}\psi\geq (-\Delta)^{\alpha/2}(|\psi|^2)$$ which follows immediately from the kernel representation  of $(-\Delta)^{\alpha/2}$. Applied recursively $n-1$ times when $p=2^n$ and $n\geq1$ is an integer, it provides for $1\leq k\leq n-1$ (or without intermediary $k$ if $n=1$): $$\label{fractionalHeatLp} p\int_{{\mathbb{R}}^d} |\psi|^{p-2}\psi \cdot (-\Delta)^{\alpha/2}\psi \geq \frac{p}{2^k}\int_{{\mathbb{R}}^d} |\psi|^{p-2^k} (-\Delta)^{\alpha/2}(|\psi|^{2^k}) \geq 2\int_{{\mathbb{R}}^d} |(-\Delta)^{\alpha/4}(\psi^{p/2})|^2.$$ Compared to [@CC2004 Lemma 2.4], the inequality  is improved by a factor of 2. Overall, for $p\geq2$ (restricted to exact powers of $2$ if $0<\alpha<2$), the evolution of the $L^p$-norm of smooth solutions of  obeys the following inequality: $$\label{groundLpEstimate} \frac{d}{ds} {\| \psi \|_{L^p}}^p + 2\int_{{\mathbb{R}}^d} |(-\Delta)^{\alpha/4} (\psi^{p/2})|^2 \leq - (p-1) \int_{{\mathbb{R}}^d} |\psi|^p {\operatorname{div}}\mathbf{v}.$$ Obviously, only the focusing zones ([*i.e. *]{}regions where ${\operatorname{div}}\mathbf{v}<0$) of the transport field can contribute to an increase of the $L^p$ norm; the other just tends to spread $\psi$ out. Using the notation  for the negative part, one thus has the following estimate: $$\label{groundLpEstimateSigned} \frac{d}{ds} {\| \psi \|_{L^p}}^p + 2\int_{{\mathbb{R}}^d} |(-\Delta)^{\alpha/4} (\psi^{p/2})|^2 \leq (p-1) \int_{{\mathbb{R}}^d} |\psi|^p ({\operatorname{div}}\mathbf{v})_-.$$ In dimension $d\geq2$ and for $0<\alpha<2$, one uses the Sobolev embedding . For $\sigma = dp/(d-\alpha)$, one thus has: $${\| \psi \|_{L^\sigma}}^p = {\| \psi^{p/2} \|_{L^{2d/(d-\alpha)}}}^2 \leq \frac{1}{S_{\alpha/2}({\mathbb{R}}^d)} \int_{{\mathbb{R}}^d} |(-\Delta)^{\alpha/4} \psi^{p/2}|^2 .$$ The estimate  becomes: $$\label{groundLpEstimateSignedSimplified} \frac{d}{ds} {\| \psi \|_{L^p}}^p + 2 S_{\alpha/2}({\mathbb{R}}^d) \cdot {\| \psi \|_{L^\sigma}}^p \leq (p-1) \int_{{\mathbb{R}}^d} |\psi|^p ({\operatorname{div}}\mathbf{v})_-.$$ Finally, as the conjugate exponent of $q=d/\alpha>1$ satisfies $p q' = \sigma$, one splits the right-hand side in the following way: $$\int_{{\mathbb{R}}^d} |\psi|^p ({\operatorname{div}}\mathbf{v})_- \leq {\| \psi \|_{L^\sigma}}^p {\| ({\operatorname{div}}\mathbf{v})_- \|_{L^{d/\alpha}}}.$$ Thanks to the smallness assumption , it is then possible to bootstrap the Lebesgue norm into the left-hand side. In that case, ensures that $\frac{d}{ds}{\| \psi \|_{L^p}} + S_{\alpha/2}({\mathbb{R}}^d) \cdot {\| \psi \|_{L^\sigma}}^p\leq 0$, which gives . One can then interpolate with  to control all $L^q$ norms for $1\leq q\leq p$. ------------------------------------------------------------------------ \[LargerV\] If $C_{\alpha,p}(\mathbf{v})=2S_{\alpha/2}({\mathbb{R}}^d)-(p-1) {\| ({\operatorname{div}}\mathbf{v})_- \|_{L^\infty_t L^{d/\alpha}_x}}>0$ then  still holds, but with the constant $S_{\alpha/2}({\mathbb{R}}^d)$ replaced by $C_{\alpha,p}(\mathbf{v})$, which is not uniform in $\mathbf{v}$ anymore. An improved version of  valid for average-free functions is found in [@CGHV2014] or [@CTV2015 Prop. 2.4]: $$\int_{{\mathbb{R}}^d} |\psi|^{p-2}\psi \cdot (-\Delta)^{\alpha/2}\psi \geq \frac{1}{p} {\| (-\Delta)^{\alpha/2}(\psi^{p/2}) \|_{L^2}}^2 + C {\| \psi \|_{L^p}}^p.$$ However, in our case, using a Sobolev embedding for $\psi^{p/2}$ provides some additional integrability and in particular a control of the $L^\sigma$-norm with $\sigma>p$. This gain will be crucial in what follows. It also allows us to put a restriction on the $L^{d/\alpha}$-norm of $({\operatorname{div}}\mathbf{v})_-$, instead of requiring smallness in $L^\infty$. Note that on ${\mathbb{T}}^d$, the Sobolev embedding $\dot{H}^{h}({\mathbb{T}}^d) \subset L^{2d/(d-2h)}$  is only valid for average-free functions. However, $\psi^{p/2}$ is not average-free in general ([*i.e. *]{}$p\neq2$), even if $\psi$ is so. For the periodic case, one will use the following simpler result, whose proof is also contained above. \[periodicCase\] If $\psi$ is an average-free solution of  on ${\mathbb{T}}^d$ with $\alpha\in(0,2]$ and $d>\alpha$ and $${\| ({\operatorname{div}}\mathbf{v})_- \|_{L^\infty_t L^{d/\alpha}_x}} \leq S_{\alpha/2}({\mathbb{T}}^d),$$ then $${\| \psi(s) \|_{L^2}}^2 + S_{\alpha/2}({\mathbb{T}}^d) \int_0^s {\| \psi(\tau) \|_{L^\sigma}}^2 d\tau \leq {\| \psi_0 \|_{L^2}}^2 \qquad\text{with}\qquad \sigma = \frac{2d}{d-\alpha}\cdotp$$ Propagation of the atom property by the dual conservation law {#par:propAtom} ============================================================= As long as the advection field has mildly convergent characteristics (expressed precisely by ), the weak maximum principle implies that the (non-local) diffusion propagates the properties of atoms. It is possible to trade a slow increase in each atomic radius to gain some decay in amplitude. Local propagation ----------------- \[PropagationAtom\] Let us assume that $1\leq \alpha\leq 2$ and $d>\alpha$ and that the velocity field $\mathbf{v} \in \mathrm{BMO}$ satisfies $$\label{smallnessAssumption2} (p-1) {\| ({\operatorname{div}}\mathbf{v})_- \|_{L^\infty_t L^{d/\alpha}_x}} \leq S_{\alpha/2}({\mathbb{R}}^d)$$ for some $p\geq2$ (eventually restricted to $p=2^n$ with $n\in{\mathbb{N}}$ if $\alpha<2$) such that $$p > \frac{d}{d-(\alpha-\omega)} \qquad\text{with}\qquad 0<\omega<\alpha\wedge 1.$$ Then there exist constants $\delta, K$ and $\gamma$, depending only on $d$, $p$, $\alpha$ and $\|\mathbf{v}\|_{\mathrm{BMO}}$, such that for all $r\in (0,1]$, the following implication holds: $$\psi_0\in {\mathcal{A}}^p_{r} \qquad\Longrightarrow\qquad \forall s\in[0,\gamma r^\alpha],\quad \psi(s,\cdot)\in \left(1-\frac{\delta s}{r^{\alpha}}\right) {\mathcal{A}}^p_{(r^\alpha+Ks)^{1/\alpha}}$$ where $\psi$ denotes the solution of the Cauchy problem . The constant $A$, which is implicit in the definition of ${\mathcal{A}}^p_{r}$, has to be chosen large enough. The admissible threshold for $A$, which also depends only on $d$, $p$, $\alpha$ and $\|\mathbf{v}\|_{\mathrm{BMO}}$, is specified in remark \[choiceOfA\]. \[choiceOfp\] The proposition holds with $p=2$ if $d>2(\alpha-\omega)$, which is always possible if one chooses $\omega$ such that $\alpha-1 < \omega<1$ when $\alpha<2$ (and $\omega>1/2$ when $\alpha=2$ and $d\geq3$); in this case  is also the least restrictive. Thanks to proposition \[periodicCase\], the result then also holds, mutatis mutandis, on ${\mathbb{T}}^d$. The proof of proposition \[PropagationAtom\] is inspired by [@KN2009] and [@D2010], though the fractional derivative requires some additional care. Thanks to , the zero-average property of atoms is obviously propagated by . Let $x(s)$ be the solution to the following ODE, which tracks the average flow on a ball of size $r$. It is obviously well defined for $\mathbf{v}\in L^1_{\text{loc}}({\mathbb{R}}_+\times{\mathbb{R}}^d)$: $$\label{trackCenter} \begin{cases} x^\prime(s)=\bar{\mathbf{v}}_{{\mathcal{B}(x(s),r)}}\\ x(0) = x_0 \end{cases} \qquad\text{where}\qquad \bar{\mathbf{v}}_{{\mathcal{B}(x,r)}}(s) = \frac{1}{|{\mathcal{B}(x,r)}|}\int_{{\mathcal{B}(x,r)}} \mathbf{v}(s,y) dy.$$ #### Step 1. Strict decay of the $L^1$-norm. One introduces $S=\psi(s)^{-1}(\{0\})$ and $D_\pm = \operatorname{supp}\psi_\pm(s)$. Arguing as in [@KN2009 §4] and taking advantage of the conservative form of : $$\frac{d}{ds}{\| \psi(s) \|_{L^1}} = -\int_{D_\pm} \frac{\psi}{|\psi|} (-\Delta)^{\alpha/2}\psi + \int_{S} |(-\Delta)^{\alpha/2}\psi|.$$ The kernel formula  allows us to improve  and gives $$\label{propL1NormImproved} \frac{d}{ds}{\| \psi(s) \|_{L^1}} \leq -\frac{c_{d,\alpha}}{2} \iint_{D_\pm^2} \left(\frac{\psi(y)}{|\psi(y)|}-\frac{\psi(x)}{|\psi(x)|}\right) \frac{\psi(y)-\psi(x)}{|y-x|^{d+\alpha}} dydx$$ because (if $S$ is a set of strictly positive measure) $$\int_{S} |(-\Delta)^{\alpha/2}\psi (y)| dy \leq c_{d,\alpha}\int_{x\in D_\pm} \int_{y\in S} \frac{|\psi(x)|}{|y-x|^{d+\alpha}} dydx.$$ The right-hand side of is negative: $$\frac{d}{ds}{\| \psi(s) \|_{L^1}} \leq -c_{d,\alpha}\left[ \int_{D_+} \left(\int_{D_-} \frac{dy}{|x-y|^{d+\alpha}}\right) \psi_+(s,x) dx +\int_{D_-} \left(\int_{D_+} \frac{dy}{|x-y|^{d+\alpha}}\right) \psi_-(s,x) dx \right].$$ Obviously, the domains of integration $D_\pm$ can be reduced to $D_\pm\cap {\mathcal{B}(x(s),100 r)}$. Provided that most of the $L^1$-mass of $\psi$ is localized in ${\mathcal{B}(x(s),100 r)}$, which, as explained in [@KN2009], is ensured by the 3rd part of the proof, it ends up giving: $$\frac{d}{ds}{\| \psi(s) \|_{L^1}} \lesssim - r^{-\alpha}$$ [*i.e. *]{}for $\delta$ and $\gamma>0$ small enough $$\forall s\in[0,\gamma r^\alpha],\qquad {\| \psi(s) \|_{L^1}} \leq 1-\frac{\delta s}{r^{\alpha}}\cdotp$$ #### Step 2. Strict decay of the $L^p$ norm. We have already proven that, under the smallness assumption , the right-hand side of  can be resorbed within the elliptic term, [*i.e. *]{}$$\frac{d}{ds} {\| \psi \|_{L^p}}^p \leq - S_{\alpha/2}({\mathbb{R}}^d) \cdot {\| \psi \|_{L^\sigma}}^p <0.$$ Next, as $\sigma=\frac{dp}{d-\alpha}>p$, one can use an elegant idea of [@CC2004 p. 517], which is to combine the interpolation inequality ${\| \psi \|_{L^p}} \leq {\| \psi \|_{L^1}}^{1-\theta} {\| \psi \|_{L^\sigma}}^\theta$ for $\theta = \frac{(p-1)d}{(p-1)d+\alpha}$ with the propagation of the $L^1$-norm . As $\psi_0$ is an atom ${\mathcal{A}}^p_{r}$, it ensures that: $$\frac{d}{ds} {\| \psi \|_{L^p}}^p \leq - S_{\alpha/2}({\mathbb{R}}^d) \cdot {\| \psi_0 \|_{L^1}}^{-\frac{\alpha p}{(p-1)d}} {\| \psi \|_{L^p}}^{p/\theta} \leq -S_{\alpha/2}({\mathbb{R}}^d) \cdot {\| \psi \|_{L^p}}^{p/\theta}.$$ This is a Riccati-type ODE that can be solved explicitly: $${\| \psi(s) \|_{L^p}}^p \leq \left( {\| \psi_0 \|_{L^p}}^{-\frac{\alpha p}{(p-1)d}} + {\textstyle \frac{\alpha S_{\alpha/2}({\mathbb{R}}^d)}{(p-1)d}} \cdot s \right)^{-\frac{(p-1)d}{\alpha}}.$$ Using the atom property ${\| \psi_0 \|_{L^p}} \leq A r^{-d(1-\frac{1}{p})}$ and rearranging the terms, one gets: $${\| \psi(s) \|_{L^p}} \leq A r^{-d(1-\frac{1}{p})} \left( 1 + {\textstyle \frac{\alpha S_{\alpha/2}({\mathbb{R}}^d)}{(p-1)d}}\cdot A^{\frac{\alpha p}{(p-1)d}} r^{-\alpha} s \right)^{-\frac{d}{\alpha}(1-\frac{1}{p})}$$ [*i.e. *]{}${\| \psi(s) \|_{L^p}} \leq A r(s)^{-d(1-\frac{1}{p})}$ with $$r(s) = \left( r^{\alpha} + C_{d,p,\alpha} A^\mu s \right)^{1/\alpha} \quad\text{and}\quad \mu = \frac{\alpha}{d(1-\frac{1}{p})} \cdotp$$ One chooses $\delta>0$ small enough, then $$\label{largeA1} 0<K<C_{d,p,\alpha} A^\mu - \frac{2\delta}{\frac{d}{\alpha}\left(1-\frac{1}{p}\right)}\cdotp$$ Thanks to the reversed Bernoulli inequality $(1-x)^{-1/\beta}\leq 1+2x/\beta$ for $\beta>1$ and $x\in[0,1/2]$, this choice ensures that $$\forall t\in [0,\gamma],\qquad \frac{1+ C_{d,p,\alpha} A^\mu t}{1+ K t} \geq 1 + \frac{2\delta t}{\frac{d}{\alpha}{(1-\frac{1}{p})}} \geq (1-\delta t)^{-1/[\frac{d}{\alpha}(1-\frac{1}{p})]}$$ with $\gamma = \frac{\frac{d}{\alpha}(1-\frac{1}{p})}{2\delta}\left( \frac{ C_{d,p,\alpha} A^\mu - 2\delta / [ \frac{d}{\alpha}(1-\frac{1}{p}) ]}{K} - 1 \right)$ and thus, after substituting $t=s/r^\alpha$: $${\| \psi(s) \|_{L^p}} \leq A \left( 1-\frac{\delta s}{r^\alpha} \right) (r^\alpha+K s)^{-\frac{d}{\alpha}(1-\frac{1}{p})}$$ for $s\in[0,\gamma r^\alpha]$. #### Step 3. Propagation of the concentration. With $x(s)$ defined by , one considers $$\chi(s) = \int_{{\mathbb{R}}^d}\psi(s,x) \Omega(x-x(s)) dx .$$ Using the equation  and the fact that $(-\Delta)^{\alpha/2}$ is self-adjoint, the derivative of $\chi$ satisfies: $$\label{eqHprime} \chi'(s)=\int_{{\mathbb{R}}^d} (\mathbf{v}-\bar{\mathbf{v}}_{{\mathcal{B}(x(s),r)}} ) \cdot \nabla \Omega(x-x(s)) \psi(s,x)dx -\int_{{\mathbb{R}}^d} \psi(s,x) \cdot (-\Delta)^{\alpha/2}\Omega(x-x(s)) dx.$$ Let us collect obvious estimates for the derivatives of $\Omega$: \[omegaEstimates\] $$\begin{gathered} | \nabla\Omega (z)| \lesssim |z|^{-(1-\omega)} \cdot \mathbf{1}_{{\mathcal{B}(0,2)}}(z),\\ \label{OmegaSecondEstimate} | (-\Delta)^{\alpha/2}\Omega(z) | \lesssim |z|^{-(\alpha-\omega)_+} \cdot \mathbf{1}_{{\mathcal{B}(0,2)}}(z) + |z|^{-2-\alpha} \cdotp \mathbf{1}_{{\mathcal{B}(0,2)}^c}(z).\end{gathered}$$ They follow easily from the scaling properties of the Fourier transform (and thus of $(-\Delta)^{\alpha/2}$) and from the kernel representation . Recall that we assume $\omega<\min\{\alpha,1\}$ throughout the proof. ##### 3a. Transport term in $\chi'$. Let us introduce $E_k(s):=\{x\in {\mathbb{R}}^d: |x-x(s)|\in [2^{k-1}r,2^kr)\}$ to estimate $$I_1 =\left| \int_{{\mathbb{T}}^d} (\mathbf{v}-\bar{\mathbf{v}}_{{\mathcal{B}(x(s),r)}})\cdot \nabla\Omega(x-x(s))\psi(s,x) dx \right|.$$ One has $I_1 \lesssim J_0 + \sum\limits_{k=1}^\infty J_k$ with $$J_0 = \int_{{\mathcal{B}(x(s),r)}}|\mathbf{v}-\bar{\mathbf{v}}_{{\mathcal{B}(x(s),r)}}| |x-x(s)|^{-(1-\omega)} |\psi|$$ and $$J_k = \left( \int_{E_k(s)}|\mathbf{v}-\bar{\mathbf{v}}_{{\mathcal{B}(x(s),r)}}| |\psi | \right)r^{-(1-\omega)} 2^{-k(1-\omega)}.$$ For $J_0$, we use the Hölder inequality with $a^{-1}+b^{-1}+c^{-1}=1$ and the BMO property $$\begin{aligned} J_0 &\leq \|\mathbf{v}-\bar{\mathbf{v}}_{{\mathcal{B}(x(s),r)}}\|_{L^a({\mathcal{B}(x(s),r)})} \|\psi_0\|_{L^b} \||x-x(s)|^{-(1-\omega)}\|_{L^c({\mathcal{B}(x(s),r)})} \\ & \lesssim {\| \mathbf{v} \|_{\operatorname{BMO}}} \cdot r^{d/a} \cdot A^{b_\ast} r^{-d(1-\frac{1}{b})} \cdot r^{\frac{d}{c}-(1-\omega)}\\ & \lesssim r^{-(1-\omega)} A^{b_\ast} {\| \mathbf{v} \|_{\operatorname{BMO}}}\end{aligned}$$ with $$b_\ast = \frac{1-\frac{1}{b}}{1-\frac{1}{p}} = \frac{p'}{b'}\cdotp$$ Here, one should comment on the choice of the powers $a,b,c$. Obviously, we have to take $c<d/(1-\omega)$ for local integrability reasons. Second, as we used the decay of the $L^b$ norm of $\psi$ given by proposition \[propagationLp\] followed by proposition \[AtomL1LpBound\] on $\psi_0$, one needs $p\geq b>d/(d-(1-\omega))$. Since $a$ can be chosen arbitrary large, it is always possible to find a proper triplet $(a,b,c)$ as soon as $$\label{restrictP} p > \frac{d}{d-(1-\omega)}\cdotp$$ For $J_k$ with $k\geq 1$, we apply the Hölder inequality with a pair of conjugate powers $q_1$ and $q'_1$, with $q_1>d/(1-\omega)$. Thanks to , one thus has $q'_1<\frac{d}{d-(1-\omega)}<p$, which ensures again that we have propagation of the $L^{q'_1}$ norm of $\psi$ and that proposition \[AtomL1LpBound\] may be used liberally on $\psi_0$. One also uses that for $\operatorname{BMO}$ functions, the averages of adjacent dyadic balls are comparable and that ${\| \psi \|_{L^1(E_k)}} \leq 2^{kd/q_1} {\| \psi \|_{L^{q'_1}(E_k)}}$ uniformly for $r\in (0,1]$. One thus gets: $$\begin{aligned} \int_{E_k(s)}|\mathbf{v}-\bar{\mathbf{v}}_{{\mathcal{B}(x(s),r)}}| |\psi | &\leq \int_{E_k(s)}|\mathbf{v}-\bar{\mathbf{v}}_{{\mathcal{B}(x(s),2^kr)}}| |\psi | + \int_{E_k(s)}|\bar{\mathbf{v}}_{{\mathcal{B}(x(s),2^kr)}}-\bar{\mathbf{v}}_{{\mathcal{B}(x(s),r)}}| |\psi | \\ & \lesssim {\| \mathbf{v}-\bar{\mathbf{v}}_{{\mathcal{B}(x(s),2^k r)}} \|_{L^{q_1}({{\mathcal{B}(x(s),2^k r)}})}} {\| \psi_0 \|_{L^{q'_1}}} + k {\| \mathbf{v} \|_{\operatorname{BMO}}} {\| \psi \|_{L^1(E_k)}} \\[2pt] &\lesssim (1+k) 2^{kd/q_1} A^{p'/q_1} {\| \mathbf{v} \|_{\operatorname{BMO}}} .\end{aligned}$$ As we choose $d/q_1<1-\omega$, the geometric series in $k\geq1$ is convergent and $p'/q_1<b_\ast$, and thus: $$\label{estimateI1} I_1\lesssim r^{-(1-\omega)} A^{b_\ast} {\| \mathbf{v} \|_{\operatorname{BMO}}}.$$ Let us observe that, due to the admissible range for $b$, the value of $b_\ast$ can be chosen arbitrarily within the interval $$\frac{1-\omega}{d(1-\frac{1}{p})} < b_\ast \leq 1.$$ In the next part of this proof, we will chose $b_\ast$ to be as close as possible to the lowest bound. As $\operatorname{supp} \nabla \Omega \subset {\mathcal{B}(0,2)}$ the series of $J_k$ terms is limited to $k\lesssim |\log r|$. However, this upper bound becomes arbitrarily large when $r\to0$. Our previous estimate is uniform for $r\in(0,1]$. ##### 3b. Nonlocal viscous term in $\chi'$. Let us now consider the second term of : $$I_2=\biggl|\int_{{\mathbb{R}}^d} \psi(s,x) \cdot (-\Delta)^{\alpha/2} \Omega(x-x(s)) dx\biggr|.$$ Recall that we assume $\alpha>\omega$. Thanks to , for any $0< \rho\leq r<1$, one has: $$I_2 \lesssim \int_{{\mathcal{B}(x(s),\rho)}}|x-x(s)|^{-(\alpha-\omega)} |\psi(s,x)| dx + \rho^{-(\alpha-\omega)} {\| \psi_0 \|_{L^1}}.$$ We apply the Hölder inequality with another pair of conjugate powers $q_2$ and $q'_2$, with $1\leq q_2<\frac{d}{\alpha-\omega}$, which is always possible. One also needs $q_2' \leq p$ to ensure the propagation of the $L^{q_2'}$ norm by proposition \[propagationLp\], [*i.e. *]{}$\frac{1}{q_2}\leq 1-\frac{1}{p}$. Such a choice is possible if $$\label{restrictP2} p > \frac{d}{d-(\alpha-\omega)}\cdotp$$ As $\alpha\geq1$, this restriction on $p$ supersedes . One gets $$I_2 \leq {\| |x-x(s)|^{-(\alpha-\omega)} \|_{L^{q_2}({\mathcal{B}(x(s),\rho)})}} {\| \psi_0 \|_{L^{q_2'}}} + \rho^{-(\alpha-\omega)} {\| \psi_0 \|_{L^1}}$$ [*i.e. *]{}$$I_2 \lesssim \rho^{\frac{d}{q_2} - (\alpha-\omega)} A^{p'/q_2} r^{-d/q_2} + \rho^{-(\alpha-\omega)}.$$ The optimal choice for $\rho$ is given by $\rho = r A^{-p'/d}$, which belongs indeed to $(0,r]$ as $A\gg1$. Substituting this value in the previous estimate of $I_2$ gives: $$\label{estimateI2} I_2 \lesssim r^{-(\alpha-\omega)} A^{\mu_\ast} \qquad\text{with}\qquad \mu_\ast = \frac{\alpha-\omega}{d(1-\frac{1}{p})} \cdotp$$ ##### 3c. Conclusion. Putting together  with and , one gets: $$|\chi'(s)|\leq I_1+I_2 \lesssim r^{-(1-\omega)} A^{b_\ast} {\| \mathbf{v} \|_{\operatorname{BMO}}} + r^{-(\alpha-\omega)} A^{\mu_\ast}.$$ After integration and considering that $\chi(0)\leq r^{\omega}$ and $1\leq r^{-1}\leq r^{-\alpha}$ because $\alpha\geq1$: $$\chi(s) \leq r^{\omega}\left( 1 + C_{d,p,\alpha}' \left[ A^{b_\ast} {\| \mathbf{v} \|_{\operatorname{BMO}}} + A^{\mu_\ast} \right] \frac{s}{r^{\alpha}} \right).$$ Provided $A$ is large enough, one may amend the previous choice  of $\delta$ and $K$ to ensure that $$\label{largeA2} K \geq \frac{\alpha}{\omega} \left\{ \delta + C_{d,p,\alpha}' \left[ A^{b_\ast} {\| \mathbf{v} \|_{\operatorname{BMO}}} + A^{\mu_\ast} \right] \right\},$$ which, in turn, ensures that $$\forall t\in[0,\gamma],\qquad 1 + C_{d,p,\alpha}' \left[ A^{b_\ast} {\| \mathbf{v} \|_{\operatorname{BMO}}} + A^{\mu_\ast} \right] t \leq (1-\delta t)(1+Kt)^{\omega/\alpha}$$ [*i.e. *]{}, with $t=s/r^\alpha$: $$\forall s\in[0,\gamma r^{\alpha}],\qquad \chi(s) \leq \left(1-\frac{\delta s}{r^\alpha}\right) (r^\alpha+K s)^{\omega/\alpha}.$$ This concludes the proof of $\psi(s)\in \left(1-\frac{\delta s}{r^{\alpha}}\right) {\mathcal{A}}^p_{(r^\alpha+Ks)^{1/\alpha}}$. ------------------------------------------------------------------------ \[choiceOfA\] Our conditions  and  generalize respectively the conditions 4.6 - 4.15 of [@KN2009]. Both conditions are compatible, provided $A$ is chosen large enough, because $$\mu > b_\ast \vee \mu_\ast.$$ In turn, this condition is satisfied by choosing $b_\ast$ as small as possible and because $\alpha>\omega \vee (1-\omega)$. \[troubleWithL1\] It could be tempting to discard the $L^1$-property from the atom definition and use proposition \[AtomL1LpBound2\] from the appendix to control this norm a-posteriori. In this case, instead of  and , one is led to choose $\delta$ and $K$ such that $$\frac{\alpha}{\omega} \left\{ \delta + C_{d,p,\alpha}' \left[ A^{\tilde{b}} {\| \mathbf{v} \|_{\operatorname{BMO}}} + A^{\tilde\mu} \right] \right\} \leq K<C_{d,p,\alpha} A^{\tilde\mu} - \frac{2\delta}{\frac{d}{\alpha}\left(1-\frac{1}{p}\right)}$$ with $\tilde{b} = \frac{\omega+d(1-\frac{1}{b})}{\omega+d(1-\frac{1}{p})} \in (\tilde\mu/\alpha,1]$ and $\tilde\mu = \frac{\alpha}{\omega+d(1-\frac{1}{p})}$. As both sides are order $A^{\tilde\mu}$, it is not clear anymore that the choice can be resolved for some large value of $A$. This alternate path is thus a subtle deadlock. Global propagation ------------------ \[PropagationAtomGlobal\] In the conditions of Proposition \[PropagationAtom\], the constants $\delta$, $K$ are such that $$\psi_0\in {\mathcal{A}}^p_{r} \qquad\Longrightarrow\qquad \forall s>0,\qquad \psi(s,\cdot)\in \left(\frac{r^\alpha}{r^\alpha+Ks}\right)^{\delta /K} {\mathcal{A}}_{(r^\alpha+Ks)^{1/\alpha}}$$ where $\psi$ denotes the solution of the Cauchy problem . We keep the assumptions and notations of proposition \[PropagationAtom\]. Let us split the time-line in consecutive intervals $[\ell \gamma r^\alpha, (\ell+1)\gamma r^\alpha]$ with $\ell\in{\mathbb{N}}$. For $\ell=0$, one has $$\forall s\in [0,\gamma r^\alpha],\qquad 1-\frac{\delta s}{r^{\alpha}} \leq \left(1+\frac{Ks}{r^\alpha}\right)^{-\delta /K} = \left(\frac{r^\alpha}{r^\alpha+Ks}\right)^{\delta /K}.$$ Let us assume that, for some integer $\ell\in{\mathbb{N}}$, one has: $$\psi(\ell \gamma r^\alpha,\cdot) \in (1+K\ell \gamma)^{-\delta/K} {\mathcal{A}}_{r(1+K \ell \gamma)^{1/\alpha}}.$$ Then for any $s\in [\ell \gamma r^\alpha, (\ell+1)\gamma r^\alpha]$, proposition \[PropagationAtom\] gives: $$\psi(s,\cdot) \in (1+K\ell \gamma)^{-\delta/K} \left(1+\frac{KS}{R^\alpha}\right)^{-\delta /K} {\mathcal{A}}_{(R^\alpha+K S)^{1/\alpha}}$$ with $S = s-\ell \gamma r^\alpha$ and $R=r(1+K \ell \gamma)^{1/\alpha}$. The new radius is an exact match: $$\label{radiusMatch} (R^\alpha+K S)^{1/\alpha} = (r^\alpha + Ks)^{1/\alpha}.$$ Similarly, the amplitude satisfies: $$(1+K\ell \gamma) \left(1+\frac{KS}{R^\alpha}\right) = 1 + \frac{Ks}{r^\alpha} \cdotp$$ The proposition thus follows by induction on $\ell\in{\mathbb{N}}$. ------------------------------------------------------------------------ Modifications in the case $0<\alpha<1$ -------------------------------------- Throughout §\[par:propAtom\], the assumption $\alpha\geq1$ has only been used in the third step of the proof of proposition \[PropagationAtom\], [*i.e. *]{}to ascertain the propagation of the concentration. Let us investigate in this subsection how to deal with the case $0<\alpha<1$. When dealing with the super-critical case, L. Silvestre [@Silv2010] assumes a higher regularity for the advection field. We will do the same here and assume respectively that $v\in C^{1-\alpha}({\mathbb{R}}^d)$ or $C^{1-\alpha}({\mathbb{T}}^d)$. The identity  still holds. To deal with the transport term, one uses $$\left|\mathbf{v}(x)-\bar{\mathbf{v}}_{{\mathcal{B}(x(s),r)}}\right| \leq \frac{1}{|{\mathcal{B}(x(s),r)}|}\int_{{\mathcal{B}(x(s),r)}} |\mathbf{v}(x) - \mathbf{v}(y)| dy \leq {\| \mathbf{v} \|_{C^{1-\alpha}}} \:\rule{10pt}{1pt}\hspace{-12pt}\int_{{\mathcal{B}(x(s),r)}}|x-y|^{1-\alpha} dy,$$ which gives $$\label{improvedVminusVBar} \left|\mathbf{v}(x)-\bar{\mathbf{v}}_{{\mathcal{B}(x(s),r)}}\right| \leq (|x-x(s)|+r)^{1-\alpha} {\| \mathbf{v} \|_{C^{1-\alpha}}}.$$ For $J_0$, one takes $a=\infty$ and the same constraints for the exponents $b$ and $c$, thus $$J_0\lesssim r^{-(\alpha-\omega)} A^{b_\ast} {\| \mathbf{v} \|_{C^{1-\alpha}}}.$$ Similarly, for $J_k$, one takes $q_1=\infty$: $$\int_{E_k(s)}|\mathbf{v}-\bar{\mathbf{v}}_{{\mathcal{B}(x(s),r)}}| |\psi | \lesssim 2^{k(1-\alpha)} r^{1-\alpha} {\| \mathbf{v} \|_{C^{1-\alpha}}}$$ thus $J_k\lesssim 2^{-k(\alpha-\omega)} r^{-(\alpha-\omega)} {\| \mathbf{v} \|_{C^{1-\alpha}}}$ and as $\omega<\alpha$, the geometric series in $k$ is convergent. The estimate  can therefore be replaced by $$\label{estimateI1bis} I_1\lesssim r^{-(\alpha-\omega)} A^{b_\ast} {\| \mathbf{v} \|_{C^{1-\alpha}}} \qquad\text{with}\quad b_\ast = \frac{1-\omega}{d(1-\frac{1}{p})} + \varepsilon, \quad \varepsilon>0.$$ For the non-local viscous term, the estimate  remains valid. The sole difference is that now $$\frac{d}{d-(1-\omega)} > \frac{d}{d-(\alpha-\omega)}$$ and, consequently, the requirement trumps . Putting together  with and , one gets: $$|\chi'(s)| \lesssim r^{-(\alpha-\omega)} \left( A^{b_\ast} {\| \mathbf{v} \|_{C^{1-\alpha}}} + A^{\mu_\ast}\right)$$ and $$\chi(s) \leq r^{\omega}\left( 1 + C_{d,p,\alpha}' \left[ A^{b_\ast} {\| \mathbf{v} \|_{C^{1-\alpha}}} + A^{\mu_\ast} \right] \frac{s}{r^{\alpha}} \right).$$ The conclusion is identical, provided that $A$ is large enough and that the choice of $K$ and $\delta$ ensures $$\label{largeA2bis} K \geq \frac{\alpha}{\omega} \left\{ \delta + C_{d,p,\alpha}' \left[ A^{b_\ast} {\| \mathbf{v} \|_{C^{1-\alpha}}} + A^{\mu_\ast} \right] \right\}$$ instead of . Note that to reconcile with for large $A$, one needs $\alpha>\omega \vee (1-\omega)$, which is always possible if $\alpha>1/2$. However, when $\alpha\leq 1/2$, one needs one final modification, which is to replace the average $\bar{\mathbf{v}}_{{\mathcal{B}(x(s),r)}}$ by the point-wise value $\mathbf{v}(x(s))$, where: $$\label{trackCenterBis} \begin{cases} x^\prime(s)=\mathbf{v}(x(s)),\\ x(0) = x_0. \end{cases}$$ In this case, estimate  is improved one step further into the following one: $$\left|\mathbf{v}(x)-\mathbf{v}(x(s))\right| \leq |x-x(s)|^{1-\alpha} {\| \mathbf{v} \|_{C^{1-\alpha}}}.$$ This changes $J_0$ into $$\widetilde{J_0} = \int_{{\mathcal{B}(x(s),r)}} |x-x(s)|^{-(\alpha-\omega)} |\psi|,$$ which is then estimated in an identical manner to $I_2$. Note that this modification also allows us to drop all requirements concerning $b_\ast$ and in particular , which is beneficial for any $\alpha\in(0,1)$. Let us finally point out that, in the other parts of the proof, the requirement $d>\alpha$ now allows for any dimension $d\geq1$. We have thus established the following statement: \[PropagationAtomBis\] Let us assume that $0<\alpha<1$ and $d\geq 1$ and that the velocity field $\mathbf{v} \in C^{1-\alpha}$ satisfies $$\label{smallnessAssumption2} (p-1) {\| ({\operatorname{div}}\mathbf{v})_- \|_{L^\infty_t L^{d/\alpha}_x}} \leq S_{\alpha/2}({\mathbb{R}}^d)$$ for some $p=2^n$ with $n\in{\mathbb{N}}$ such that $$p > \frac{d}{d-(\alpha-\omega)} \qquad\text{with}\qquad 0<\omega<\alpha.$$ Then there exist constants $\delta, K$ and $\gamma$ (and a lower threshold for $A$), depending only on $d$, $p$, $\alpha$ and $\|\mathbf{v}\|_{C^{1-\alpha}}$, such that for all $r\in (0,1]$, the following implication holds: $$\psi_0\in {\mathcal{A}}^p_{r} \qquad\Longrightarrow\qquad \forall s>0,\qquad \psi(s,\cdot)\in \left(1+\frac{Ks}{r^\alpha}\right)^{-\delta /K} {\mathcal{A}}_{(r^\alpha+Ks)^{1/\alpha}}$$ where $\psi$ denotes the solution of the Cauchy problem . Note that we can take $p=2$ in the previous statement (and thus, using remark \[choiceOfp\], claim a similar one in the case of ${\mathbb{T}}^d$) if $$\frac{d}{2} > \alpha-\omega.$$ Such a choice is always possible. Proof of Theorem \[mainThm\] {#par:proof} ============================ The proof of theorem \[mainThm\] is now straightforward. Given $d\geq2$ and $1\leq \alpha\leq 2$ (with $d\geq 3$ when $\alpha=2$), one chooses $\omega\in(0,1)$ such that $\alpha-1 < \omega<1$ if $\alpha<2$, or $\omega>1/2$ if $\alpha=2$. One checks immediately that $\omega<\alpha<d$ and $d>2(\alpha-\omega)$. Let us now consider an advection vector field $v\in\operatorname{BMO}$ with $${\| ({\operatorname{div}}v)_- \|_{L^{d/\alpha}}} \leq S_{\alpha/2}.$$ One takes $p=2$. One chooses the constant $A$, which is implicit in the definition of atoms, according to the threshold mentioned in remark \[choiceOfA\]; this threshold depends solely on $d$, $\alpha$ and ${\| v \|_{\operatorname{BMO}}}$. One considers the constants $\gamma$, $\delta$ and $K$ given by propositions \[PropagationAtom\] and \[PropagationAtomGlobal\] and sets $$\beta = \alpha\delta/K.$$ The value of $\beta$ depends on $d$, $\alpha$ and ${\| v \|_{\operatorname{BMO}}}$. For $d\geq 1$ and $0<\alpha<1$, one chooses $\omega$ such that $(\alpha-\frac{d}{2})_+<\omega<\alpha$ and $p=2$. In this case, the $\operatorname{BMO}$ norm is replaced by the $C^{1-\alpha}$-norm in all computations. \[relax\] When ${\| ({\operatorname{div}}v)_- \|_{L^{d/\alpha}}} < 2 S_{\alpha/2}$, one can still run the following proof. However, the choice of $A$ and of all constants then depends not only on ${\| v \|_{\operatorname{BMO}}}$ but also on $C(v)=2 S_{\alpha/2} - {\| ({\operatorname{div}}v)_- \|_{L^{d/\alpha}}}>0$ and degenerates as $C(v)\to0$. See remark \[LargerV\]. Propagation of the Hölder regularity ------------------------------------ For any solution $\theta$ of  stemming from $\theta_0\in C^\beta$ and for $\psi_0\in{\mathcal{A}}^2_r$, identity  implies that: $$\int_{{\mathbb{R}}^d} \theta(t,x) \psi_0(x) dx = \int_{{\mathbb{R}}^d} \theta_0(x) \psi(t,x) dx$$ where $\psi$ is solution of the dual equation , which, by proposition \[PropagationAtomGlobal\], is an atom of calibrated size. Using  for $\theta_0$, one gets: $$r^{-\beta} \left| \int_{{\mathbb{R}}^d} \theta(t,x) \psi_0(x) dx \right| \lesssim r^{-\beta}\left(1+\frac{Kt}{r^\alpha}\right)^{-\delta /K} (r^\alpha+Kt)^{\beta/\alpha}{\| \theta_0 \|_{C^\beta}} = {\| \theta_0 \|_{C^\beta}}.$$ A second application of  then ensures that $\theta(t)\in C^\beta$ and that $${\| \theta(t) \|_{C^\beta}} \leq C {\| \theta_0 \|_{C^\beta}}.$$ The constant $C$ is the implicit one in . The same argument also gives ${\| \theta(t) \|_{C^{\beta'}}} \leq C {\| \theta_0 \|_{C^{\beta'}}}$ for any $0\leq \beta'\leq \beta$. Gain in Hölder regularity ------------------------- One can use the Hölder inequality and proposition \[AtomL1LpBound\] with $p=2$ to control $$\left| \int_{{\mathbb{R}}^d} \theta_0(x) \psi(t,x) dx \right| \leq {\| \theta_0 \|_{L^q}} {\| \psi(t) \|_{L^{q'}}} \leq \frac{ A^{2/q} r^\beta {\| \theta_0 \|_{L^q}} }{ (r^\alpha + K t)^{(\beta+\frac{d}{q})/\alpha} } \lesssim r^\beta t^{-(\beta+\frac{d}{q})/\alpha} {\| \theta_0 \|_{L^q}}$$ for any Lebesgue exponent $q$ such that $2\leq q\leq \infty$. One thus gets a regularization estimate: $${\| \theta(t,\cdot) \|_{C^\beta}} \simeq \sup_{\substack{0<r\leq 1\\ \psi_0\in {\mathcal{A}}^2_r}} r^{-\beta} \left| \int_{{\mathbb{R}}^d} \theta(t,x) \psi_0(x) dx \right| \leq C t^{-(\beta+\frac{d}{q})/\alpha} {\| \theta_0 \|_{L^q}}$$ with a constant $C$ that depends on $q$ and $A$ and thus ultimately on $d$, $\alpha$ and ${\| v \|_{\operatorname{BMO}}}$. Appendix: on the $L^1$-control of atoms ======================================= Even without the a-priori constraint ${\| \psi \|_{L^1}}\leq 1$, one can control the $L^1$-norm of atoms (or any $L^q$ norm for $q\leq p$) by a real interpolation estimate. \[AtomL1LpBound2\] If $\varphi$ satisfies $${\| \varphi \|_{L^p}} \leq A r^{-d(1-\frac{1}{p})}, \qquad\text{and}\qquad \exists x_0\in{\mathbb{R}}^d, \quad \int_{{\mathbb{R}}^d} |\varphi(x)| \Omega(x-x_0) dx \leq r^\omega$$ for some $0<r\leq1$ and $p\in (1,\infty]$, then $$\label{AtomL1Bound2} {\| \varphi \|_{L^1}} \leq C_{d,p} A^{\omega/(\omega+d(1-\frac{1}{p}))}$$ and, more generally, for any $1\leq q\leq p$, one has $$\label{AtomLpBound2} {\| \varphi \|_{L^q}} \leq C_{d,p,q} A^{\frac{\omega+d(1-1/q)}{\omega+d(1-1/p)}} r^{-d(1-\frac{1}{q})}.$$ Compared to proposition \[AtomL1LpBound\], these estimates “lose” powers of $A$, which would provoke a critical collision of exponents in the previous proof (see remark \[troubleWithL1\]). For any $\rho\in [0,r]$, one has $${\| \varphi \|_{L^1}} \leq \int_{{\mathcal{B}(x_0,\rho)}} |\varphi| + \rho^{-\omega}\int_{{\mathbb{R}}^d\backslash{\mathcal{B}(x_0,\rho)}} |\varphi(x)| \Omega(x-x_0) dx \leq A \left(\frac{\rho}{r}\right)^{d(1-\frac{1}{p})} |{\mathcal{B}(0,1)}|^{1-\frac{1}{p}}+ \left( \frac{r}{\rho} \right)^{\omega}$$ and follows from choosing the optimal value $\rho=r \big( A |{\mathcal{B}(0,1)}|^{1-\frac{1}{p}} \big)^{-1/(\omega+d(1-\frac{1}{p}))}$. For the second estimate, one proceeds similarly with $\tau\in [0,r]$; for clarity, we do not track the constant related to $|{\mathcal{B}(0,1)}|$: $$\begin{aligned} \int_{{\mathbb{R}}^d} |\varphi|^{q} &\leq \int_{{\mathcal{B}(x_0,\tau)}} |\varphi|^{q} + \int_{{\mathbb{R}}^d \backslash {\mathcal{B}(x_0,\tau)}} |\varphi|^{q} \\ & \lesssim \biggl(\int_{{\mathbb{R}}^d} |\varphi|^{p}\biggr)^{\frac{q}{p}} \tau^{d(1-\frac{q}{p})} + \biggl(\tau^{-\omega}\int_{{\mathbb{R}}^d \backslash {\mathcal{B}(x_0,\tau)}} |\varphi(x)| \Omega(x-x_0) dx\biggr)^{\frac{p-q}{p-1}} \biggl(\int_{{\mathbb{R}}^d} |\varphi|^{p}\biggr)^{\frac{q-1}{p-1}},\end{aligned}$$ thanks to the Hölder inequality (with $p/q\geq1$) for the first term, and the interpolation inequality ${\| f \|_{L^q}}\leq {\| f \|_{L^1}}^{1-\theta} {\| f \|_{L^p}}^{\theta}$ with $\theta = (1-\frac{1}{q})/(1-\frac{1}{p}) \in [0,1]$ for the second. We now use the fact that $\varphi\in {\mathcal{A}}^p_r$ and deduce that $$\int_{{\mathbb{R}}^d} |\varphi|^{q} \lesssim \left(A r^{-d(1-\frac{1}{p})} \right)^{q} \tau^{d(1-\frac{q}{p})} + \biggl(\frac{r}{\tau}\biggr)^{\omega (\frac{p-q}{p-1})} (A^p r^{-(p-1)d})^{\frac{q-1}{p-1}}.$$ The optimal choice for $\tau$ is the one that balances the weight of both terms; it is $\tau=r A^{-p/(d(p-1)+\omega p)}$. The computation then boils down to $$\int_{{\mathbb{R}}^d} |\varphi|^{q}\lesssim r^{-(q-1)d}A^{\frac{d p (q-1) + \omega pq}{d(p-1)+\omega p}} \qquad{\textit{i.e.~}}\qquad {\| \varphi \|_{L^q}} \lesssim A^{\frac{\omega+d(1-1/q)}{\omega+d(1-1/p)}} r^{-d(1-\frac{1}{q})}$$ and the lemma is proven. ------------------------------------------------------------------------ [XX]{} L. Ambrosio, D. Trevisan. *Lecture notes on the DiPerna-Lions theory in abstract measure spaces.* Ann. Fac. Sci. Toulouse, XXVI, 4 (2017), 729-766. 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Sci. Toulouse, XXV, 4 (2016), 723-758. A. Kiselev, F. Nazarov. *Variation on a theme of Caffarelli and Vasseur.* Journal of Mathematical Sciences 166 – 1 (2010), 31-39. A. Kiselev, F. Nazarov and A. Volberg. *Global well-posedness for the critical 2D dissipative quasi-geostrophic equation.* Invent. Math. 167, 3 (2007), 445-453. S.N. Kružkov. *First order quasilinear equations in several independent variables.* Mat. USSR Sbornik 10, 2 (1970), 217-243. C. Mouhot. *De Giorgi-Nash-Moser and Hörmander theories: new interplay.* arXiv:1808.00194 (2018). L. Silvestre. *Hölder estimates for advection fractional-diffusion equations.* Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) Vol. XI (2012), 843-855. L. Silvestre, V. Vicol. *Hölder continuity for a drift-diffusion equation with pressure*. Ann. I.H.P. Analyse non linéaire 29, 4 (2012), 637-652. E.M. Stein. *Harmonic Analysis.* Princeton University Press, 1993. 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--- abstract: 'We investigate the physics that drives the gas accretion rates onto galaxies at the centers of dark matter haloes using the EAGLE suite of hydrodynamical cosmological simulations. We find that at redshifts $z{\le}2$ the accretion rate onto the galaxy increases with halo mass in the halo mass range $10^{10}-10^{11.7}{\, {\rm M_{\odot}}}$, flattens between the halo masses $10^{11.7}-10^{12.7}{\, {\rm M_{\odot}}}$, and increases again for higher-mass haloes. However, the galaxy gas accretion does not flatten at intermediate halo masses when AGN feedback is switched off. To better understand these trends, we develop a physically motivated semi-analytic model of galaxy gas accretion. We show that the flattening is produced by the rate of gas cooling from the hot halo. The ratio of the cooling radius and the virial radius does not decrease continuously with increasing halo mass as generally thought. While it decreases up to ${\sim}10^{13}{\, {\rm M_{\odot}}}$ haloes, it increases for higher halo masses, causing an upturn in the galaxy gas accretion rate. This may indicate that in high-mass haloes AGN feedback is not sufficiently efficient. When there is no AGN feedback, the density of the hot halo is higher, the ratio of the cooling and virial radii does not decrease as much and the cooling rate is higher. Changes in the efficiency of stellar feedback can also increase or decrease the accretion rates onto galaxies. The trends can plausibly be explained by the re-accretion of gas ejected by progenitor galaxies and by the suppression of black hole growth, and hence AGN feedback, by stellar feedback.' bibliography: - 'biblio.bib' title: The impact of feedback and the hot halo on the rates of gas accretion onto galaxies --- cosmology: theory – galaxies: formation, evolution Introduction ============ Cosmological simulations have not only shown that the evolution of galaxies’ gas reservoirs is governed by feedback from stars and black holes (e.g. @Keres09 [@Oppenheimer10; @Dubois12; @Haas13; @Crain17]), but also that it is critically linked to the cosmic web and halo gas flows (e.g. @Dekel09 [@vandeVoort12]), which are responsible for the galaxy mass growth. Evidence that feedback manifests itself in the form of enriched outflows and energetic winds comes from various observations (e.g. @Sharp10 [@Feruglio10; @Cicone14; @Anderson15; @Turner15; @Nielsen16]), but gas accretion from the circumgalactic medium (CGM) is difficult to observe and the physical processes that drive its evolution are not well understood. It has been proposed that outflows in the form of winds join the warm-hot intergalactic medium and may eventually reverse trajectory to re-accrete onto the galaxy. This is generally referred to as wind recycling or a galactic fountain (see e.g. @Oppenheimer10 [@Ubler14; @Angles17]). On the other hand, it has also been suggested that gas accretion is suppressed by outflows generated by black holes located in the center of the active galactic nuclei (AGN), that quench the diffuse accretion rates onto galaxies (e.g. @DiMatteo05 [@Bower06; @Croton; @Sijacki; @Martizzi; @Booth; @Dubois12; @Dubois]). Accretion shocks occur as a result of collisions between gas from the circum-halo medium falling into haloes and the hot halo gas in hydrostatic equilibrium. When accretion shocks occur, the gravitational energy of the infalling gas is converted into thermal energy. The works of @Rees and @Silk77 proposed that shock-heated gas with long cooling times forms a hot hydrostatic halo atmosphere, pressure supported against gravitational collapse. Semi-analytic models of galaxy formation (SAMs) generally assume that gas falls into galaxies either through a rapid or a slow cooling flow, depending on the radial scale below which gas is able to cool (i.e. where the gas has a cooling time shorter than the dynamical time). This radius is typically referred to as the cooling radius ($r_{\rm{cool}}$). When $r_{\rm{cool}}$ becomes smaller than the virial radius ($R_{200}$), a hot hydrostatic atmosphere is formed. Gas accretion onto galaxies then changes from being in the rapid cooling regime to the slow cooling regime (e.g. @White91). Hydrodynamical simulations, however, have shown that both quasistatic and inflowing gas components can exist in the halo at the same radius (see e.g. Figs. 1 and 2 from @Correa17 and references therein). These are usually referred to as hot and cold modes of accretion. Gas not only falls into a galaxy through a cooling flow (hot mode), but also through a cold flow (cold mode). Cold flows tend to be filamentary, clumpy and of higher density than the hot mode gas, and are strongly correlated with the dark matter filaments that feed haloes (e.g. @Keres05 [@Ocvirk; @Dekel09; @vandeVoort11; @Faucher; @vandeVoort12; @Nelson13; @Woods14]). These two modes of accretion are able to coexist in massive haloes at high redshift (@Dekel09 [@Correa17]) and feed the galaxy at the same time. The co-evolution between the different modes of accretion has not been implemented in SAMs until recently. @Cousin15 included a two phase smooth baryonic accretion, with the hot and cold component built over the smooth dark matter accretion, whereas @Lu15 modelled a circum-halo medium, assumed to be preheated up to a certain entropy level, to reduce the baryonic accretion. Once gas crosses the hot halo and cools, feedback from stars and black holes can potentially reheat it and prevent it from falling into the galaxy. Therefore, due to the complexity in the interaction of all the possible mechanisms that modify the manner in which galaxies accrete gas, a physical model of galaxy gas accretion is still missing. In this work, we use the “Evolution and Assembly of GaLaxies and their Environments” (EAGLE) simulations to study the modes of gas accretion onto galaxies and their dependence on feedback variations. We derive a physically motivated model of galaxy gas accretion that aims to explain the underlying physics of the way gas cools from the hot halo and accretes onto galaxies. In @Correa17 [hereafter Paper I], we derived a new criterion to determine when the hot halo forms, based on the fraction of gas accreting onto haloes that shock-heats to the halo virial temperature (i.e. hot accretion) and on the hot halo gas mass. We calculated a heating rate produced by accretion shocks and compared it to the gas cooling rate. We found that haloes with masses above the critical mass threshold of $10^{11.7}{\, {\rm M_{\odot}}}$ are able to develop a hot stable hydrostatic atmosphere, in agreement with previous work (e.g. @Birnboim [@Dekel]). In Paper I we showed that feedback affects the mass-scale of hot halo formation and impacts the distribution of gas in the halo. In this work, we make use of the analytic heating and cooling rates from Paper I and assume that there are two modes of galaxy gas accretion, hot and cold. This work is organized as follows. In Section \[Simulations\_sec\] (as well as in Appendix \[Simulations\_app\]) we describe the numerical simulations used and the methods we employ to measure gas accretion rates. In Section \[gas\_accretion\_sec\] we analyze the total gas accretion rates onto galaxies from the EAGLE simulations as a function of halo mass and redshift, as well as the hot and cold modes of accretion (Section \[hot\_cold\_modes\_of\_accretion\]), and the impact of stellar and AGN feedback (Sections \[stellar\_fb\] and \[AGN\_fb\], respectively). In Section \[AGN\_hot\_halo\_sec\] we derive a physically motivated model of gas accretion onto galaxies, we show that the model reproduces the gas accretion rates from the EAGLE simulations, and we analyze the role of AGN in the rates of gas cooling from the hot halo. Finally, we summarize and conclude in Section \[Conclusion\_sec\]. Simulation name L ($\rm{cMpc}$) N Description --------------------------- ----------------- ------------ --------------------------------------------------------------------- Ref 100 $1504^{3}$ ref. stellar & AGN feedback No AGN FB 50 $752^{3}$ ref. stellar & no AGN feedback More Explosive AGN FB 50 $752^{3}$ ref. stellar & more explosive and intermittent AGN feedback (but same energy injected per unit mass accreted by the BH as Ref) More Energetic Stellar FB 25 $376^{3}$ twice as much energy injected per unit stellar mass with respect to Ref & ref. AGN feedback Less Energetic Stellar FB 25 $376^{3}$ half as much energy injected per unit stellar mass with respect to Ref & ref. AGN feedback No Stellar FB 25 $376^{3}$ no stellar feedback & ref. AGN feedback No Stellar/AGN FB 25 $376^{3}$ no stellar feedback & no AGN feedback Simulations {#Simulations_sec} =========== The EAGLE simulation suite (@Schaye14 [@Crain15]) was run using a modified version of GADGET-3 (@Springelb), an $N$-Body Tree-PM smoothed particle hydrodynamics (SPH) code, but with a new formulation of SPH, new time stepping and new subgrid physics. The simulations assume a $\Lambda$CDM cosmology with the parameters given by [*[Planck-1]{}*]{} data (@Planck), $\Omega_{\rm{m}}=1-\Omega_{\Lambda}=0.307$, $\Omega_{\rm{b}}=0.04825$, $h=0.6777$, $\sigma_{8}=0.8288$, $n_{s}=0.9611$, and are run from redshift $z=127$ to $z=0$. Throughout this work we use simulations with different box sizes (ranging from 25 to 100 comoving ${\, {\rm Mpc}}$) and particle numbers (ranging from $2\times 376^{3}$ to $2\times 1504^{3}$), but the same resolution. For clarity, the simulation names contain strings of the form $L$xxx$N$yyyy, where xxx is the simulation box size in comoving ${\, {\rm Mpc}}$ and yyyy is the cube root of the number of particles per species (where the number of baryonic particles is equal to the number of dark matter particles). For our analysis we mainly use the L100N1504 reference model (hereafter Ref), that contains an initial baryonic particle mass of $1.81\times 10^{6}{\, {\rm M_{\odot}}}$, a dark matter particle mass of $9.70\times 10^{6}{\, {\rm M_{\odot}}}$, a comoving (Plummer equivalent) gravitational softening of 2.66 comoving kpc and a maximum physical softening of 0.7 proper kpc. Although the Ref model was calibrated without considering gas properties, it is able to reproduce the cosmic HI column density distribution and circumgalactic covering fraction profiles (@Rahmati15 [@Rahmati16]) and the neutral gas mass and profiles (@Bahe16). Note, however, that the HI masses of present-day dwarf galaxies are underestimated (@Crain17). In order, to investigate the impact of feedback on the rates of galaxy gas accretion, we additionally use simulations with varying AGN and stellar feedback prescriptions. In the stellar feedback case, we use simulations with ‘Less/More Energetic Stellar FB’, which means that the energy injected per unit mass of stars formed is half/twice the amount used in Ref. In the AGN case, we use ‘No AGN FB’ and ‘More Explosive AGN FB’, meaning that AGN feedback is switched off or is more explosive and intermittent than the Ref model, respectively (but note that in the latter case the energy injected per unit mass accreted by the BH does not change with respect to Ref). See Table \[Table\_sims2\] for reference. In Appendix A (also in Paper I) we include a brief description of the modeling of the EAGLE simulations, but see @Schaye14 and @Crain15 for more details. The EAGLE simulations are publicly available; for details see @McAlpine16. Dark matter haloes in EAGLE (and the self-bound substructures within them associated with galaxies) are identified using the Friends-of-Friends (FoF) and SUBFIND algorithms (@Springel [@Dolag]). The virial masses and radii are determined using a spherical overdensity routine within SUBFIND, centered on the minimum gravitational potential of the main subhalo from the FoF group. Halo masses ($M_{200}$) are defined as the total mass within a radius, $R_{200}$, within which the mean density is 200 times the critical density. In each FoF halo, the ‘central’ galaxy is the galaxy closest to the center (minimum of the potential), which is usually the most massive. The remaining galaxies within the FoF halo are its satellites. Throughout this work we focus on the gas accretion rates onto central galaxies. The gas accretion rates onto satellite galaxies differ from the rates onto centrals because they are strongly suppressed by environmental effects, such as ram pressure or tidal stripping. For details of these effects on gas accretion rates onto satellite galaxies from EAGLE see @vandeVoort17. Methodology {#Methodology} ----------- In this section we describe the methodology we employ to calculate the gas accretion rates onto galaxies at the center of dark matter haloes. We begin by building merger trees across the simulation snapshots[^1]. At each output redshift we select haloes that contain more than 1000 dark matter particles (which corresponds to a minimum halo mass of $M_{200}=10^{9.8}{\, {\rm M_{\odot}}}$ in the Ref-L100N1504 simulation). This particle number-cut is based on the convergence analysis presented in Appendix \[Numerical\_convergence\], where we find that in smaller haloes the accretion onto galaxies does not converge, indicating that the inner galaxies are not well resolved (see Appendix \[Numerical\_convergence\] for a discussion). We refer to these haloes as ‘descendants’, and link each descendant with a unique ‘progenitor’ at the previous output redshift. This is nontrivial due to halo fragmentation: subhalos of a progenitor halo may have descendants that reside in more than one halo. The fragmentation can be spurious or due to a physical unbinding event. To correct for this, we link the descendant to the progenitor that contains the majority of the descendant’s 25 most bound dark matter particles (see @Correa15b for an analysis of halo mass history convergence using the mentioned criteria to connect haloes between snapshots). To calculate the gas accretion onto haloes, we perform a particle ID matching between particles within linked haloes from consecutive snapshots. Particles that are new to the system, and are within the virial radius, are labeled as accreted particles in the redshift range $z_{i} < z < z_{j}$. Different methods have been employed to determine gas accretion onto galaxies. For example, @Faucher measured accretion rates through shells of prescribed radii. In order to differentiate outflows from inflowing material, they added the particles within a given shell and defined the net accretion as $\dot{M}\propto \sum_{p} M_{p}{\bf{v}}_{p}/\Delta r_{p}$ (with $M_{p}$ the particle mass, $\bf{v}_{p}$ the velocity vector and $\Delta r_{p}$ the thickness of shell). They classified the net accretion rates as inward or outward according to the direction of the velocity vector. A different approach was used by @vandeVoort17 who, in order to separate the galaxy from the halo, used a radial cut of 30 pkpc and a cut in the hydrogen number density ($n_{\rm{H}}> 0.1\hspace{1mm}\rm{cm}^{-3}$) to define the interstellar medium (ISM) (see also @vandeVoort11). They considered particles that are star-forming and part of the ISM at $z_{i}$, but which were gaseous and not part of the ISM at $z_{j}$ to have been accreted onto a galaxy at $z_{i} < z < z_{j}$. Similarly, @Nelson13 made use of a density-temperature ($\rho-T$) cut criterion ($\log_{10}(T/{\rm{K}})-0.25\log_{10}(\rho/\rho_{{\rm{crit}},z=0}) < 4.11$, with $\rho_{{\rm{crit}},z=0}$ the critical density today), along with a radial cut (${<0.15\times R_{200}}$). They considered a gas element to have accreted onto a galaxy if it belonged to that galaxy at $z_{j}$, and either crossed the phase space cut in $\rho-T$ or the radial cut during $z_{i} < z < z_{j}$. In this work we closely follow the method of @vandeVoort17 and define the ISM to consist of all particles within a sphere of radius $0.15\times R_{200}$ (centered on the minimum of the gravitational potential) that are either star-forming (i.e. have $n_{\rm{H}}>n^{*}_{\rm{H}}=0.1{\rm{cm}}^{-3}(Z/0.002)^{-0.64}$, with $Z$ metallicity, and $T<10^{0.5}T_{\rm{EoS}}$, with $T_{\rm{EoS}}\propto \rho^{1/3}$ a temperature floor that corresponds to the equation of state and is normalised to $T_{\rm{eos}}=8\times 10^{3}$K at $n_{\rm{H}}=10^{-1}$cm$^{-3}$) or part of the atomic ISM (i.e. have $n_{\rm{H}}>0.1\hspace{1mm}{\rm{cm}}^{-3}$ and $T<10^{5}\hspace{1mm}\rm{K}$). We next calculate three rates of gas accretion onto $0.15\times R_{200}$: gas accretion of all gas particles that cross the $0.15\times R_{200}$ radius between $z_{i} < z < z_{j}$ (hereafter $\dot{M}_{0.15R_{200}}$), gas accretion of only star-forming particles ($\dot{M}_{\rm{SFR}>0}$) and gas accretion of only gas particles that form part of the ISM ($\dot{M}_{\rm{ISM}}$). In all the rates we are including star particles that were gas particles in the previous output but turned into stars during the time step between the snapshots. Note that $\dot{M}_{\rm{SFR}>0}$ and $\dot{M}_{\rm{ISM}}$ refer to gas accretion onto the star forming and ISM components within $0.15\times R_{200}$, and that $\dot{M}_{\rm{ISM}}$ includes $\dot{M}_{\rm{SFR}>0}$. Also note that the accreting gas is possibly, but not necessarily, star forming prior to accretion. We further define $\dot{M}_{\rm{0.15R_{200}}}$ as the rate of gas accretion onto galaxies (hereafter $\dot{M}_{\rm{gas,galaxy}}=\dot{M}_{\rm{0.15R_{200}}}$), because we assume that the ‘galaxy’ extends beyond the ISM. To summarize, $\dot{M}_{\rm{gas,galaxy}}$ refers to the accretion rates of gas that crosses the radial boundary $0.15\times R_{200}$ during $z_{i} < z < z_{j}$, $\dot{M}_{\rm{ISM}}$ considers gas that crosses the radial boundary and the phase space cut $n_{\rm{H}}-T$ or is star-forming during $z_{i} < z < z_{j}$, and $\dot{M}_{\rm{SFR}>0}$ considers gas that crosses the radial boundary during $z_{i} < z < z_{j}$ and is star-forming at $z_{i}$. Note that in each case we calculate the accretion rates by adding the mass of all the accreted particles (following the condition of accretion) onto individual galaxies and dividing by the time interval (that is $\sim 1.34$ Gyr at $z=0$ and $\sim 0.33$ Gyr at $z=2$). The final rates given are the median accretion rates onto galaxies in bins of halo mass. Note that we calculate the ‘net’ rate of gas accretion between two consecutive snapshots, i.e. we do not separate gas that is accreted for the first time or re-accreted. Gas particles that fall into galaxies can be classified as gas introduced to the system by a merger event or through smooth accretion. Mergers have been defined as the accretion of particles that belonged to any resolved subhalo at the previous snapshot (e.g. @Keres09 [@Nelson13]) or to subhalos with masses above 1/10th of main progenitor mass (e.g. @vandeVoort11). It has been found that the specific gas accretion rate onto haloes through mergers is much lower than the specific smooth accretion rate, except for high-mass haloes ($M_{200}\gtrsim10^{14}{\, {\rm M_{\odot}}}$) at $z=0$ (@vandeVoort11). In the case of galaxies at $z=2$, gas from all resolved merger events can contribute as much as $60\%$ to the total galaxy accretion rate at all halo masses (@Nelson13). At lower redshifts the contribution decreases and the gas supply from resolved mergers appears to be only important at the high-mass end (@Keres09). In this work however, we do not focus on the origin of the gas that falls into galaxies. Instead we investigate the physics that prevents gas from cooling. Fig. \[growthrate\_plot\] shows $\dot{M}_{\rm{gas,galaxy}}$ (blue solid line), $\dot{M}_{\rm{ISM}}$ (blue long-dashed line) and $\dot{M}_{\rm{SFR}>0}$ (blue dot-dashed line), as a function of halo mass, taken from the Ref-L100N1504 simulation. For comparison, the figure also shows the gas accretion rate onto haloes (i.e. crossing $R_{200}$, black dashed line). In the halo mass range $10^{10}-10^{12}{\, {\rm M_{\odot}}}$, $\dot{M}_{\rm{gas,galaxy}}$ and $\dot{M}_{\rm{ISM}}$ increase with halo mass at approximately the same rate, with $\dot{M}_{\rm{ISM}}$ having a 0.3 dex (on average) lower normalization than $\dot{M}_{\rm{gas,galaxy}}$. For halo masses greater than $10^{12}{\, {\rm M_{\odot}}}$, $\dot{M}_{\rm{ISM}}$ remains roughly constant. While $\dot{M}_{\rm{gas,galaxy}}$ also flattens in $10^{12}{\, {\rm M_{\odot}}}$ haloes, it increases with halo mass for $\gtrsim 10^{13}{\, {\rm M_{\odot}}}$ haloes. $\dot{M}_{\rm{SFR}>0}$ behaves similarly to $\dot{M}_{\rm{ISM}}$ though it is (on average) a factor of 2 lower. In the figure, the grey and cyan shaded regions enclosing the median values of $\dot{M}_{\rm{gas,galaxy}}$ and $\dot{M}_{\rm{ISM}}$ correspond to the $1\sigma$ scatter (16-84th percentiles). From Fig. \[growthrate\_plot\] it can be seen that in the halo mass range $10^{10}-10^{12}{\, {\rm M_{\odot}}}$, roughly $50\%$ of the gas crossing $0.15R_{200}$ joins the ISM. At higher halo masses, the fraction of gas crossing $0.15R_{200}$ that joins the ISM decreases and most of the gas that reaches the inner halo has either a low density ($n_{H}<0.1$ cm$^{-3}$) or a high temperature ($>10^{5}$ K). We then conclude that massive galaxies accrete warm diffuse gas that does not fall into the ISM. In the following sections we investigate the way gas crosses the CGM and the origin of $\dot{M}_{\rm{gas,galaxy}}(M_{200})$ further by first disentangling the impact of stellar and AGN feedback (Sections \[stellar\_fb\] and \[AGN\_fb\], respectively). In addition, we calculate the fraction of hot/cold modes of gas accretion onto galaxies in Section \[hot\_cold\_modes\_of\_accretion\]. To estimate these modes of gas accretion, previous studies followed the temperature history of the accreted gas (see e.g. @Faucher [@vandeVoort11]), however, in Paper I we concluded that selecting gas particles according to their temperature just after accretion, $T_{\rm{post-shock}}$, is a better method to determine hot/cold accretion. This is because it excludes gas particles that go through a shock but immediately cool afterwards, or that did not pass through an accretion shock but instead were heated in the past by stellar feedback and have since cooled. Then, throughout this work, we define the fraction of gas particles accreted hot, $f_{\rm{acc,hot}}$, as the fraction of particles that after being accreted have temperatures larger than $T_{\rm{post-shock}} = 10^{5.5}\hspace{1mm}\rm{K}$. Note that for low-mass haloes the results are sensitive to this specific temperature threshold (e.g. @vandeVoort11). For an analysis of how the hot mode of accretion depends on the temperature threshold, as well as on other criteria (e.g. $T_{\rm{vir}}$, cooling times), see Sections 2.2 and 4 of Paper I. ![Accretion rate of gas onto the central galaxies of dark matter haloes as a function of halo mass in the redshift range $0\leq z<0.1$. The solid line corresponds to the gas accretion rate calculated by counting all gas particles that crossed $0.15\times R_{200}$ during consecutive snapshots, whereas the dashed and dot-dashed lines correspond to the gas accretion rates calculated by counting gas particles that crossed $0.15\times R_{200}$ radius and the phase-space cut $n_{\rm{H}}-T$, and that crossed the $0.15\times R_{200}$ radius and are star-forming, respectively. The grey and cyan shaded regions enclosing the median values of $\dot{M}_{\rm{gas}}$ correspond to the $1\sigma$ scatter (16-84th percentiles) onto the $0.15\times R_{200}$ region and ISM, respectively. The $1\sigma$ scatter of $\dot{M}_{\rm{SFR}>0}$, not included in the figure, is (on average) 0.3 dex similar to that for $\dot{M}_{\rm{ISM}}$. For comparison, the black dashed line shows the rate of gas accretion onto haloes (i.e. crossing $R_{200}$). The figure shows that massive galaxies accrete warm diffuse gas that is not accreted onto the ISM.[]{data-label="growthrate_plot"}](./plots/Accretion_rate_gal_ISM_halo_comparison_028.ps "fig:"){width="46.00000%"}\ \ \ Gas accretion rates {#gas_accretion_sec} =================== In this section we calculate the rates of gas accretion onto haloes and galaxies from the EAGLE simulations. For a detailed analysis of the numerical convergence of our results, see Appendix \[Numerical\_convergence\]. Accretion rates onto galaxies and haloes {#accretion_rates_halos_galaxies} ---------------------------------------- Fig. \[accretion\_plot\] shows the total gas accretion rate onto haloes (bottom-left panel) and onto galaxies (bottom-right panel) over many redshift intervals. The solid lines correspond to the median gas accretion rates from the Ref-L100N1504 simulation. The figure shows the median accretion rates of haloes separated in logarithmic mass bins of 0.2 dex. The dashed lines correspond to the analytic accretion rates from @Correa15a [@Correa15c] multiplied by the universal baryon fraction ($f_{\rm{b}}=0.157$). In the figure, all the curves are coloured according to the redshift interval, as indicated in the legends. The top panels show the $1\sigma$ scatter of $\dot{M}_{\rm{gas,halo}}$ (top-left) and $\dot{M}_{\rm{gas,galaxy}}$ (top-right) for each mass bin. We find that the $z=0$ accretion rate onto haloes deviates from the analytic prediction of @Correa15c. The prediction is 0.8 dex too high for $10^{10}{\, {\rm M_{\odot}}}$ haloes, agrees for $10^{11.5}{\, {\rm M_{\odot}}}$ haloes, and is 0.3 dex too low for $10^{13}{\, {\rm M_{\odot}}}$ haloes. In the redshift range $z=1-2$ the prediction is also $\sim 0.8$ dex too high for $10^{10}{\, {\rm M_{\odot}}}$ haloes, but a better agreement is reached for haloes more massive than $10^{12}{\, {\rm M_{\odot}}}$. At higher redshifts the disagreement between the analytic prediction and the simulation output increases, with the prediction being up to 0.5 dex too high for all halo masses at $z=8$. A possible explanation for the difference in the halo accretion rates between the Ref model (solid lines in left panel of Fig. \[accretion\_plot\]) and the analytic prediction of @Correa15c (dashed lines) is that the analytic model does not consider the impact of baryon physics (such as gas pressure, cooling, reionization and stellar and AGN feedback), that reduces the halo mass by a factor of 0.7 for $10^{10}{\, {\rm M_{\odot}}}$ haloes at $z=0$ (@Schaller15) and $\sim 0.6$ for all haloes at $z>6$ (@Qin17). However, in low-mass haloes the disagreement between the analytic model and the simulation output is expected, because at these masses the extragalactic UV/X-ray background radiation heats the surrounding gas and prevents it from falling into the halo (e.g. @Sawala13 [@Benitez17]). We look for the best-fit expression that reproduces the halo gas accretion rates in the presence of feedback from the Ref model. We find it to be $$\begin{aligned} \label{halo_accr1} \log_{10}\dot{M}_{\rm{gas,halo}} &=& a_{z\le 4}(z)+b_{z\le 4}(z)x+c_{z\le 4}(z)x^{2},\\\label{halo_accr2} a_{z\le 4}(z)&=&0.830+0.553z-0.0523z^2,\\\label{halo_accr3} b_{z\le 4}(z)&=&1.436-0.149z+0.007z^2,\\\label{halo_accr4} c_{z\le 4}(z)&=&-0.134+0.099z-0.023z^2,\\\label{halo_accr5} x &=& \log_{10}(M_{200}/10^{12}{\, {\rm M_{\odot}}}),\end{aligned}$$ if $z \le 4$ and $$\begin{aligned} \label{halo_accr6} \log_{10}\dot{M}_{\rm{gas,halo}}&=& a_{z>4}(z)+b_{z>4}(z)x,\\\label{halo_accr7} a_{z> 4}(z)&=&3.287-0.401z+0.045z^2,\\\label{halo_accr8} b_{z> 4}(z)&=&1.016+0.003z+0.002z^2.\end{aligned}$$ if $z>4$. We show a comparison between the best-fit expression and simulation output in Appendix \[Comparison\]. The bottom-right panel of Fig. \[accretion\_plot\] shows that the gas accretion rate onto galaxies is much lower than that onto haloes. The dependence on mass is also quite different. Although the galaxy accretion rate initially increases with halo mass, it flattens in the halo mass range $10^{11.7}-10^{12.7}{\, {\rm M_{\odot}}}$, particularly at $z\le 2$. We believe that the flattening is produced by the presence of the hot halo atmosphere, which forms in $10^{11.7}{\, {\rm M_{\odot}}}$ haloes (Paper I). In Section \[model\_section\] we investigate this further by deriving an analytic model that includes the heating and cooling rates of gas from the hot halo. The galaxy accretion rates calculated in this work differ from those found in other simulations (@Keres05 [@Ocvirk; @Faucher; @vandeVoort11; @Nelson13]). To mention a few estimates, @vandeVoort11 obtained accretion rates onto the ISM of $\sim 1$, $10$ and $15{\, {\rm M_{\odot}}}$/yr in $10^{12}$, $10^{12.5}$ and $10^{13}{\, {\rm M_{\odot}}}$ haloes, respectively, at $z=0$. @Faucher found rates of cold accretion (gas particles with hydrogen number density lower than $0.13$ $\rm{cm}^{-3}$ and past maximum temperature lower than $2.5\times 10^{5}$ K) onto $0.2R_{200}$ of 0.3 and 1${\, {\rm M_{\odot}}}$/yr in $10^{12}$ and $10^{13}{\, {\rm M_{\odot}}}$ haloes at $z=0$, respectively. Similarly, @Nelson13 obtained total rates of galaxy smooth gas accretion of 1 and 5${\, {\rm M_{\odot}}}$/yr in $10^{11}$ and $10^{12}{\, {\rm M_{\odot}}}$ haloes, respectively, at $z=0$. As discussed in Section 2.1, these works differ on the method employed to calculate the rates of gas accretion, therefore we do not expect good agreement, but we find that our results closely follow those of @vandeVoort11. Note that the rates of galaxy and halo gas accretion in this work are calculated counting new gas particles within $0.15\times R_{200}$ and $R_{200}$, respectively. It is then possible that halo pseudo-evolution (the growth in halo mass due to the redshift evolution of the reference density, e.g. @Diemer) affects the rates of accretion. This occurs if gas particles not inflowing relative to the physical density profile of the halo appear to accrete when $R_{200}$ increases due to pseudo-evolution from one snapshot to the next. We next analyse this possibility. In this work, we calculate the gas accretion rates within time intervals of up to $\approx 1.3$Gyr (e.g. between redshifts 0 and 0.1). We test this by calculating the gas accretion rates onto a fixed proper radius given by $r_{\rm{gal}}=0.15R_{200}(z=0)$. We obtain that when the radius is kept fixed, the accretion rate decreases (on average) $9\%$ for $10^{10}-10^{13}{\, {\rm M_{\odot}}}$ haloes, to a maximum of $13\%$ for $10^{11.6}{\, {\rm M_{\odot}}}$ haloes. We conclude that the change in radius due to pseudo-evolution is not large enough to affect our results significantly. \ Hot and cold modes of accretion {#hot_cold_modes_of_accretion} ------------------------------- In Paper I we calculated the fractions of hot and cold modes of gas accretion onto haloes using the EAGLE simulations, and found that the hot fraction increases smoothly with halo mass (from 0.1 for $10^{11.5}{\, {\rm M_{\odot}}}$ haloes to 0.7 for $10^{12}{\, {\rm M_{\odot}}}$ haloes at $z=0$), and decreases with increasing redshift (from 0.5 in $10^{12}{\, {\rm M_{\odot}}}$ haloes at $z=1$ to 0.2 at $z=2$). In this section we extend this analysis by focusing on the modes of gas accretion onto galaxies. We select gas particles using the radial cut ($0.15\times R_{200}$) and separate the hot and cold modes by applying a temperature cut of $10^{5.5}$ K (see Section \[Methodology\] for a description on how we calculate hot/cold gas accretion). Throughout this work we label the fractions of hot/cold modes of gas accretion onto galaxies as $f^{\rm{galaxy}}_{\rm{acc,hot/cold}}$ and onto haloes as $f^{\rm{halo}}_{\rm{acc,hot/cold}}$. The top panel of Fig. \[accretion\_hotcold\_plot\] shows the median fraction of gas accretion onto galaxies, $f^{\rm{galaxy}}_{\rm{acc,hot}}$, as a function of halo mass for different redshifts. The fractions were taken from the Ref-L100N1504 simulation and the error bars in the figure show the $1\sigma$ scatter. We find that in $>10^{12.8}{\, {\rm M_{\odot}}}$ haloes at $z=0$ ($>10^{13.3}{\, {\rm M_{\odot}}}$ haloes at $z=1$), $\dot{M}_{\rm{gas,galaxy}}$ changes from being cold-mode dominated to hot-mode dominated, and at fixed halo mass $f^{\rm{galaxy}}_{\rm{acc,hot}}$ decreases with increasing redshift. This is expected, since at low redshift there are fewer cold filaments penetrating the hot halo and delivering cold gas within galaxies (see Paper I, Section 3). In Paper I we compared two different methods to select particles accreted hot or cold based on the maximum temperature ($T_{\rm{max}}$) ever reached by the gas particle and the temperature ($T_{\rm{post-shock}}$) of the gas particle after being accreted. By applying the $T_{\rm{max}}$ method, which is the most commonly used (e.g. @Keres05 [@Keres09; @Faucher; @vandeVoort11; @Nelson13]), we find that the fraction of hot accretion onto haloes is in very good agreement with @vandeVoort11, as it decreases with increasing redshift at fixed halo mass (see Paper I for a detailed comparison). However, the hot accretion fraction onto galaxies calculated in this work appears to deviate strongly from previous works that claimed that there is almost no cold accretion onto galaxies at $z\sim 2$. For example, @Nelson15a (as well as @Keres09) found that at $z=2$ cold accretion of external diffuse gas accounts for only $10\%$($30\%$) of the total accretion onto central galaxies of $10^{12}{\, {\rm M_{\odot}}}$ haloes without (with) AGN/stellar feedback. @Nelson15a used simulations run with the AREPO code (Springel 2010) and defined gas to be in the hot mode of accretion if the maximum past temperature of the gas was larger than the virial temperature of the host halo at the accretion time (time of the most recent radial crossing). In Paper I, as well as throughout this work, we apply the $T_{\rm{post-shock}}$ method, based on the temperature of the gas particle after accretion, to calculate the hot/cold modes of gas accretion. We find that cold accretion onto galaxies in $10^{12}{\, {\rm M_{\odot}}}$ haloes accounts for $50\%$ ($70\%$) of the total at $z=0$ ($z=2$) using the $T_{\rm{max}}$ criteria, but it accounts for $95\%$ ($98\%$) using the $T_{\rm{post-shock}}$ criteria. We believe that $T_{\rm{max}}$ is less suitable for identifying cold gas accretion for the following reasons. Firstly, it is possible for gas to go through a shock but immediately cool afterwards. In this case if gas is mostly cold except at a point in space and for a short period of time, numerical studies using $T_{\rm{max}}$ would label it as hot accretion but observations would indicate a cold flow. Secondly, outflows can heat the surrounding gas particles, which can reach high temperatures while being expelled from the galaxy. Such particles eventually cool and are re-accreted onto the galaxy. However, even if they do so via the cold mode they are classified as hot mode accretion by the $T_{\rm{max}}$ criterion (for more details see Paper I). The bottom panel of Fig. \[accretion\_hotcold\_plot\] shows a comparison between the fraction of gas accreted hot onto haloes (solid line) and onto galaxies (dashed line) at $z=0$. It can be seen that while $70\%$ of the gas crosses $R_{200}$ in hot mode for $10^{12}{\, {\rm M_{\odot}}}$ haloes, less than $5\%$ crosses $0.15\times R_{200}$ in hot mode and reaches the galaxy. However this changes in higher mass haloes. In $10^{13}{\, {\rm M_{\odot}}}$ haloes, for instance, while $98\%$ of the gas crosses $R_{200}$ in hot mode, $80\%$ crosses $0.15\times R_{200}$ in hot mode. The increasing amount of warm low-density gas that reaches the galaxies in high-mass haloes seems to indicate that while the hot halo forms in $10^{11.7}{\, {\rm M_{\odot}}}$ haloes (see Paper I for details), the cooling flow from the hot halo develops in haloes with masses between $10^{12}-10^{13}{\, {\rm M_{\odot}}}$. Note that a change in the temperature threshold not only modifies the fraction of hot mode accretion, but also the mass-scale at which the hot halo forms. In Paper I we develop a semianalytic model to estimate a ‘critical halo mass’ for hot halo formation that depends on the build up of the hot gas mass in the halo as well as on $f_{\rm{acc,hot}}$. We show that changing $f_{\rm{acc,hot}}$ from 1 to 0.5 increases the mass-scale of hot halo formation from $10^{11.4}$ to $10^{11.7}{\, {\rm M_{\odot}}}$, respectively (see Section 5.3.1 and Fig. 12 of Paper I for further details). In the following sections we show that the halo mass at which the hot halo cooling flow develops depends strongly on AGN feedback but not on stellar feedback. Impact of stellar feedback {#stellar_fb} -------------------------- It has been shown that the inflow rate of gas onto galaxies sensitively depends not only on definition (as discussed in Section \[Methodology\]), but also on feedback physics (e.g. @Oppenheimer10 [@vandeVoort11; @Faucher; @Nelson15a; @Ubler14]). Recently, @Nelson15a compared two simulations run with the AREPO code. While one included energetic feedback from star formation driven winds as well as supermassive black holes, the other did not include any treatment of metal line cooling, stellar or black hole feedback. They found that feedback strongly suppresses the [*[net]{}*]{} accretion rate onto central galaxies (counted as the number of gas tracers crossing the radius $0.15R_{200}$): by a factor of $\sim$3 at $z=5$, and a factor of $\sim$10 at $z=1$. A similar conclusion was reached by @vandeVoort11, who showed that the effects of stellar feedback and metal-line cooling are much stronger for accretion onto galaxies than for accretion onto haloes, and can result in differences of an order of magnitude. ![Top panel: Gas accretion rate onto central galaxies as a function of halo mass. Bottom panel: fraction of hot mode gas accretion onto central galaxies as a function of halo mass in the redshift range $0\leq z<0.1$. The panels compare the median accretion rates/hot fractions from simulations with the same resolution (L025N0376 box), but with varying feedback models. These include standard stellar feedback (Ref, solid green line), more energetic stellar feedback (purple dot-dashed line), less energetic stellar feedback (blue dot-dashed line), no stellar feedback (red long-dashed line), and no stellar/AGN feedback (orange dashed line). The error bars show the 16-84th percentiles. Depending on the halo mass, changes in the efficiency of stellar feedback can either increase or decrease the accretion rates onto galaxies.[]{data-label="gal_accr_stellar_fb"}](./plots/Galaxy_accretion_feedback_comparison_028_2.ps "fig:"){width="49.00000%"}\ ![Top panel: Gas accretion rate onto central galaxies as a function of halo mass. Bottom panel: fraction of hot mode gas accretion onto central galaxies as a function of halo mass in the redshift range $0\leq z<0.1$. The panels compare the median accretion rates/hot fractions from simulations with the same resolution (L025N0376 box), but with varying feedback models. These include standard stellar feedback (Ref, solid green line), more energetic stellar feedback (purple dot-dashed line), less energetic stellar feedback (blue dot-dashed line), no stellar feedback (red long-dashed line), and no stellar/AGN feedback (orange dashed line). The error bars show the 16-84th percentiles. Depending on the halo mass, changes in the efficiency of stellar feedback can either increase or decrease the accretion rates onto galaxies.[]{data-label="gal_accr_stellar_fb"}](./plots/HotFrac_galaxy_feedback_comparison_028_2.ps "fig:"){width="49.00000%"}\ In scenarios with energetic stellar feedback, the net galaxy accretion rates can be higher due to ‘recycling accretion’. Stellar driven winds blow gas out of the galaxy, but not out of the halo, as a result the same gas elements are accreted onto the galaxy multiple times (@Oppenheimer10, see also @vandeVoort17b for a recent review). @Ubler14 implemented a hybrid thermal/kinetic stellar feedback scheme, and calculated the gas accretion histories onto discs as a function of cosmic time. They found that the amount of re-accreted gas can be a factor of 10 larger in the strong feedback models with respect to the weak feedback models, and tends to dominate the net accretion at $z<1$. To investigate the effect of stellar feedback on the gas accretion rate onto galaxies from the EAGLE simulations, we compare the reference model with identical resolution simulations (the L025N0376 model) that include more/less energetic stellar feedback, no stellar feedback and no stellar/no AGN feedback (referred to as More/Less Energetic Stellar FB, No Stellar FB and No Stellar/AGN FB, respectively). The Ref, More/Less Energetic Stellar FB and No Stellar FB simulations employ the same feedback prescription and choice of parameters for AGN feedback, but the energy budget expelled by the stellar feedback is varied, and in the last case (No Stellar FB) stellar feedback is switched off. In the case of the No Stellar/AGN FB simulation, both stellar and AGN feedback are switched off. In the EAGLE simulations, the probability for a neighboring SPH particle to be heated by stellar outflows is determined by the fraction of the energy budget available for feedback, $f_{\rm{th}}$, which is defined as $f_{\rm{th}}$=$f_{\rm{th,min}}+f(Z,n_{\rm{H}})(f_{\rm{th,max}}-f_{\rm{th,min}})$ (with $f_{\rm{th,max}}$, $f_{\rm{th,min}}$ the maximum and minimum asymptotic values and $f(Z,n_{\rm{H}})$ a function of the gas metallicity and density). In the Ref model $f_{\rm{th,max}}=3.0$ and $f_{\rm{th,min}}=0.3$, whereas in the More/Less Energetic FB models the thresholds are $f_{\rm{th,max}}=6.0$ and $f_{\rm{th,min}}=0.6$ for the More Energetic Stellar FB case, and $f_{\rm{th,max}}=1.5$ and $f_{\rm{th,min}}=0.15$ for the Less Energetic Stellar FB case, respectively. This does not mean that stellar driven winds are stronger/weaker or blow more/less gas out of the galaxy, but rather that in the More/Less Energetic Stellar FB model the energy injected per unit mass of stars formed is twice/half of the amount used in the Ref model. The top panel of Fig. \[gal\_accr\_stellar\_fb\] shows the gas accretion rate onto galaxies at the centers of dark matter haloes, as a function of halo mass in the redshift range $0\leq z < 0.1$. From the panel it can be seen that $\dot{M}_{\rm{gas,galaxy}}$ from the More Energetic Stellar FB model is (on average) a factor of 2 lower than $\dot{M}_{\rm{gas,galaxy}}$ from Ref, but it increases for galaxies in haloes larger than $10^{12}{\, {\rm M_{\odot}}}$ (by up to a factor 4 larger than Ref for galaxies in $10^{12.7}{\, {\rm M_{\odot}}}$ haloes), suggesting that at these halo masses galaxies re-accrete gas that was ejected by stellar feedback in lower-mass progenitors. When stellar feedback is half as energetic, $\dot{M}_{\rm{gas,galaxy}}$ decreases by up to 0.6 dex compared to Ref for galaxies in $\geq 10^{11}{\, {\rm M_{\odot}}}$ haloes. A possible reason for this difference is a lower rate of re-accreted gas or a more efficient AGN feedback at lower halo masses. The latter is also due to the less energetic stellar feedback. @Bower17 showed that in EAGLE stellar feedback limits BH growth in low-mass haloes. If stellar feedback is half as energetic the central BH is able to start growing earlier, thus producing more efficient AGN feedback. The panel also shows that when stellar feedback is switched off, AGN feedback affects the gas accretion rates onto galaxies residing in low-mass haloes. We find that for galaxies in $\geq 10^{10.5}{\, {\rm M_{\odot}}}$ haloes, $\dot{M}_{\rm{gas,galaxy}}$ decreases by up to 1.5 dex relative to Ref. When there is no stellar feedback, the central black hole in low-mass galaxies is able to grow by more than 1 order of magnitude with respect to the Ref model (@Bower17). Thus the outflows expelled by a much more massive black hole suppress the gas infall rates in low-mass galaxies. Indeed, when both stellar and AGN feedback are turned off, there is no mechanism that prevents gas from cooling. Therefore the rates of gas accretion are higher by up one dex than in Ref. We also analyse whether the trends described in Fig. \[gal\_accr\_stellar\_fb\] depend on redshift. We obtain that they do not, at $z=2$ the trends are consistent with the $z=0$ results. The bottom panel of Fig. \[gal\_accr\_stellar\_fb\] shows the fraction of gas accreted in the hot mode onto central galaxies as a function of halo mass. In the models, stellar feedback can either increase or decrease $f_{\rm{acc,hot}}$ by generating winds that heat the gas (before or after accretion). We find that the trend of hot fraction with feedback variation is very complex and difficult to predict. Interestingly, it is nonetheless the No Stellar/AGN FB model, that has the highest hot fractions, by up to an order of magnitude larger than the Ref model. This suggests that feedback preferentially prevents hot gas from reaching the galaxy. This is expected, because energy-driven winds will take the path of least resistance, thus avoiding the cold streams (e.g. @Theuns02). Note that in this work, the cold/hot temperature cut we used to separate cold from hot accretion is applied after accretion onto the galaxy. Therefore, it could happen that the hot fraction artificially increases due to heating by stellar feedback, even though Fig. \[gal\_accr\_stellar\_fb\] seems to indicate otherwise. ![Median gas accretion rate onto central galaxies as a function of halo mass in the redshift range $0\leq z<0.1$ (top panel) and $2\leq z<2.2$ (middle panel). The top and middle panels compare accretion rates from same-resolution simulations (L050N0752) but with standard AGN feedback (Ref, solid orange line), No AGN feedback (red short-dashed line) and more explosive AGN feedback (blue long-dashed line). The solid grey lines correspond to median gas accretion rate onto haloes as a function of halo mass in the redshift range $0\leq z<0.1$ (top panel) and $2\leq z<2.2$ (middle panel). The bottom panel compares the same simulations, but shows the median fraction of gas accreted onto galaxies in the hot mode as a function of halo mass at $0\leq z<0.1$. Error bars show the 16-84th percentiles. When AGN feedback is switched off, $\dot{M}_{\rm{gas,galaxy}}$ does not flatten at ${\sim}10^{12}{\, {\rm M_{\odot}}}$ and increases with halo mass at the same rate as the gas accretion onto haloes but with a lower normalization. This shows that AGN feedback is the mechanism responsible for preventing hot gas from cooling and falling onto central galaxies in massive haloes.[]{data-label="gal_accr_agn_fb"}](./plots/Galaxy_accretion_with_agn_feedback_028.ps "fig:"){width="49.00000%"}\ ![Median gas accretion rate onto central galaxies as a function of halo mass in the redshift range $0\leq z<0.1$ (top panel) and $2\leq z<2.2$ (middle panel). The top and middle panels compare accretion rates from same-resolution simulations (L050N0752) but with standard AGN feedback (Ref, solid orange line), No AGN feedback (red short-dashed line) and more explosive AGN feedback (blue long-dashed line). The solid grey lines correspond to median gas accretion rate onto haloes as a function of halo mass in the redshift range $0\leq z<0.1$ (top panel) and $2\leq z<2.2$ (middle panel). The bottom panel compares the same simulations, but shows the median fraction of gas accreted onto galaxies in the hot mode as a function of halo mass at $0\leq z<0.1$. Error bars show the 16-84th percentiles. When AGN feedback is switched off, $\dot{M}_{\rm{gas,galaxy}}$ does not flatten at ${\sim}10^{12}{\, {\rm M_{\odot}}}$ and increases with halo mass at the same rate as the gas accretion onto haloes but with a lower normalization. This shows that AGN feedback is the mechanism responsible for preventing hot gas from cooling and falling onto central galaxies in massive haloes.[]{data-label="gal_accr_agn_fb"}](./plots/Galaxy_accretion_with_agn_feedback_015.ps "fig:"){width="49.00000%"}\ ![Median gas accretion rate onto central galaxies as a function of halo mass in the redshift range $0\leq z<0.1$ (top panel) and $2\leq z<2.2$ (middle panel). The top and middle panels compare accretion rates from same-resolution simulations (L050N0752) but with standard AGN feedback (Ref, solid orange line), No AGN feedback (red short-dashed line) and more explosive AGN feedback (blue long-dashed line). The solid grey lines correspond to median gas accretion rate onto haloes as a function of halo mass in the redshift range $0\leq z<0.1$ (top panel) and $2\leq z<2.2$ (middle panel). The bottom panel compares the same simulations, but shows the median fraction of gas accreted onto galaxies in the hot mode as a function of halo mass at $0\leq z<0.1$. Error bars show the 16-84th percentiles. When AGN feedback is switched off, $\dot{M}_{\rm{gas,galaxy}}$ does not flatten at ${\sim}10^{12}{\, {\rm M_{\odot}}}$ and increases with halo mass at the same rate as the gas accretion onto haloes but with a lower normalization. This shows that AGN feedback is the mechanism responsible for preventing hot gas from cooling and falling onto central galaxies in massive haloes.[]{data-label="gal_accr_agn_fb"}](./plots/HotFrac_galaxy_agn_feedback_comparison_028.ps "fig:"){width="49.00000%"} Impact of AGN feedback {#AGN_fb} ---------------------- To understand how AGN feedback alters $\dot{M}_{\rm{gas,galaxy}}$ in massive haloes, we compare simulations that include the same stellar feedback scheme but different prescription for AGN feedback, varying from no AGN feedback (No AGN FB), standard AGN feedback (Ref model), to more explosive AGN feedback (More Explosive AGN FB). In the EAGLE simulations, the difference between the Ref and More Explosive AGN FB simulations is the temperature increment of stochastic AGN heating ($\Delta T_{\rm{AGN}}$), which is $\Delta T_{\rm{AGN}}=10^{8.5}$ K in the Ref model and $\Delta T_{\rm{AGN}}=10^{9.0}$ K in the More Explosive AGN FB model. This means that the AGN feedback is more explosive and intermittent, but the energy injected per unit mass accreted by the BH does not change with respect to the Ref model. Fig. \[gal\_accr\_agn\_fb\] shows $\dot{M}_{\rm{gas,galaxy}}(M_{200})$ for the different models in the redshift range $0\leq z < 0.1$ (top panel) and $2\leq z < 2.24$ (middle panel). For comparison, the panels also show the gas accretion rate onto haloes (i.e. within $R_{200}$ as taken from the Ref model) as grey solid lines. Note that the More Explosive AGN FB and No AGN FB models were run in 50 Mpc volumes, so those do not contain haloes more massive than $10^{13.5}{\, {\rm M_{\odot}}}$. The top panel shows that at $z=0$ AGN feedback suppresses $\dot{M}_{\rm{gas,galaxy}}$ in massive haloes ($>10^{12}{\, {\rm M_{\odot}}}$) by up to 0.6 dex in the Ref model, and up to 0.8 dex in the More Explosive AGN FB model. In the No AGN FB simulation, $\dot{M}_{\rm{gas,galaxy}}$ does not flatten at $\sim10^{12}{\, {\rm M_{\odot}}}$ and increases with halo mass at the same rate as the gas accretion onto haloes but with a 0.5 dex lower normalization. This indicates that AGN feedback is the mechanism responsible for preventing hot gas from cooling and falling onto the central galaxies. However, the situation differs at $\approx 2$. In this case the middle panel shows that $\dot{M}_{\rm{gas,galaxy}}$ flattens with increasing halo mass for $10^{12}-10^{12.5}{\, {\rm M_{\odot}}}$ haloes in both the Ref and the No AGN FB model. For the More Explosive AGN FB model the flattening is however more pronounced. We believe that the flattening of $\dot{M}_{\rm{gas,galaxy}}$ in massive haloes (${>}10^{12}{\, {\rm M_{\odot}}}$) at $z=0$ can be explained by the rate of gas cooling from the hot halo. A hot hydrostatic atmosphere is formed in ${\sim}10^{11.7}{\, {\rm M_{\odot}}}$ haloes (Paper I) as a result of heating by accretion shocks. Some time after the hot halo is formed, gas begins to cool and fall onto the central galaxy, but it can also be reheated or be prevented from accreting by AGN feedback. When there is no AGN feedback preventing the shock-heated gas in the halo from cooling, the hot gas is able to cool over a short time-scale. As a result a larger amount of gas cools from the hot halo raising $\dot{M}_{\rm{gas,galaxy}}$ in the No AGN FB model with respect to the Ref model. This can also be seen from the bottom panel of Fig. \[gal\_accr\_agn\_fb\], which shows the fraction of gas accreted onto galaxies in the hot mode as a function of halo mass for the different simulations. We find that the hot fraction does not depend strongly on the explosiveness of AGN feedback, but it does increase if AGN feedback is turned off. This indicates that a larger fraction of gas cooling from the hot halo is able to reach the galaxy without AGN feedback. We further explore the validity of this hypothesis in the following section, where we develop a semi-analytic model of the gas accretion rate onto galaxies that includes the heating and cooling rates of gas from the hot halo. The hot halo cooling flow {#AGN_hot_halo_sec} ========================= In this section we show that the increase in $\dot{M}_{\rm{gas,galaxy}}$ with respect to $\dot{M}_{\rm{ISM}}$ in haloes larger than $10^{12.5}{\, {\rm M_{\odot}}}$ (see Fig. 1) can be explained by the rate of gas cooling from the hot halo. To do so, we develop a model of gas accretion onto galaxies that depends on the hot/cold fractions of gas accretion onto haloes, and on the shock-heating and cooling rates of gas from the hot halo. We briefly describe the model in the following subsection and in Section \[results\] we compare the result of the model with the simulation output. Semi-analytic model of gas accretion onto galaxies {#model_section} -------------------------------------------------- In Section \[hot\_cold\_modes\_of\_accretion\] we showed that the gas accretion onto galaxies can be decomposed into the sum of two modes of accretion, hot and cold. We consider these two modes in our model and calculate $\dot{M}_{\rm{gas,galaxy}}$ in terms of the rate of gas cooling from the hot halo, $\dot{M}_{\rm{cooling}}$, and the rate of cold gas accretion onto haloes in the form of filaments, $f^{\rm{halo}}_{\rm{acc,cold}}\dot{M}_{\rm{gas,halo}}$, as $$\begin{aligned} \label{Mgalaxy1} \dot{M}_{\rm{gas,galaxy}} &\propto & \dot{M}_{\rm{cooling}}+f^{\rm{halo}}_{\rm{acc,cold}}\dot{M}_{\rm{gas,halo}}.\end{aligned}$$ Here $\dot{M}_{\rm{gas,halo}}$ is the gas accretion rate onto haloes in the presence of feedback (given by eqs. \[halo\_accr1\]-\[halo\_accr8\]), and $f^{\rm{halo}}_{\rm{acc,hot/cold}}(M_{200},z)$ are the hot/cold fraction of gas accretion onto haloes. In eq. (\[Mgalaxy1\]) we assume that the cold accretion onto the galaxy is directly proportional to the cold accretion onto the halo, with the latter given by the following best-fit expressions from Paper I $$\begin{aligned} \label{fhot1} f^{\rm{halo}}_{\rm{acc,hot}}(x) &=& [{\rm{exp}}(-4.3[x+0.15])+1]^{-1},\\\label{fhot2} f^{\rm{halo}}_{\rm{acc,cold}}(x) &=& 1-f^{\rm{halo}}_{\rm{acc,hot}}(x),\\\label{fhot3} x &=& \log_{10}(M_{200}/10^{12}{\, {\rm M_{\odot}}}).\end{aligned}$$ We define $M_{\rm{cooling}}$ as the hot gas mass in the halo contained within the cooling radius, $r_{\rm{cool}}$ (with $r_{\rm{cool}}\leq R_{200}$). Therefore $M_{\rm{cooling}}=M_{\rm{hot}}\frac{r_{\rm{cool}}}{R_{200}}$, where $M_{\rm{hot}}$ is the total hot gas mass in the halo and we assumed an isothermal profile. We then assume that the variation of $M_{\rm{cooling}}$ in time is mainly driven by the variation of $M_{\rm{hot}}$ and obtain $$\begin{aligned} \label{xi0} \dot{M}_{\rm{cooling}} &\approx & \dot{M}_{\rm{hot}}\frac{r_{\rm{cool}}}{R_{200}},\\\label{xi1} \dot{M}_{\rm{cooling}} &\approx & f^{\rm{halo}}_{\rm{acc,hot}}\dot{M}_{\rm{gas,halo}}\frac{r_{\rm{cool}}}{R_{200}},\end{aligned}$$ In eq. (\[xi0\]) we assumed that the time scale over which $M_{\rm{cooling}}$ varies is short enough for the halo not to grow in mass and for $\dot{r}_{\rm{cool}}=\dot{R}_{200}=0$. Note that for this equation we assume an isothermal density profile for simplicity, but to better model the flow of gas onto the galaxy we use the actual density profile in the calculation of the cooling rate. In eq. (\[xi1\]) we assumed that $\dot{M}_{\rm{cooling}}$ is determined by the rate of replenishment from hot accretion onto the halo, $\dot{M}_{\rm{hot}}=f^{\rm{halo}}_{\rm{acc,hot}}\dot{M}_{\rm{gas,halo}}$, and therefore $\dot{f}^{\rm{halo}}_{\rm{acc,hot}}=0$. These are first order approximations accurate enough for our semi-analytic model. Eq. (\[xi1\]) gives the cooling radius a new physical meaning. Besides being the radius within which all gas is able to cool, we now interpret it as the fraction of shock-heated gas that cools (${\frac{r_{\rm{cool}}}{R_{200}}=\frac{\dot{M}_{\rm{cooling}}}{f^{\rm{halo}}_{\rm{acc,hot}}\dot{M}_{\rm{gas,halo}}}}$) and reaches the galaxy. In other words, it is the rate of the hot halo cooling flow. By substituting eq. (\[xi1\]) into eq (\[Mgalaxy1\]), the accretion rate onto galaxies becomes $$\label{Mgalaxy2} \dot{M}_{\rm{gas,galaxy}} = \epsilon(f^{\rm{halo}}_{\rm{acc,hot}}\frac{r_{\rm{cool}}}{R_{200}}+f^{\rm{halo}}_{\rm{acc,cold}})\dot{M}_{\rm{gas,halo}},$$ where $\epsilon$ is a dimensionless correction factor set to be 0.3 so that $\dot{M}_{\rm{gas,galaxy}}$ agrees with the gas accretion rate of galaxies from the Ref model in $10^{11}{\, {\rm M_{\odot}}}$ haloes (when $f^{\rm{halo}}_{\rm{acc,hot}}r_{\rm{cool}}/R_{200}+f^{\rm{halo}}_{\rm{acc,cold}}\approx 1$ and $\dot{M}_{\rm{gas,galaxy}}\approx 0.3\dot{M}_{\rm{gas,halo}}$). Note that $\epsilon$ captures the suppression of accretion onto galaxies due to intrahalo feedback processes that likely depend on simulation. The semi-analytic model given by eq. (\[Mgalaxy2\]) uses as input the gas accretion onto haloes ($\dot{M}_{\rm{gas,halo}}$, given by eqs. \[halo\_accr1\]-\[halo\_accr5\]) and the hot/cold fraction of gas accretion onto haloes ($f^{\rm{halo}}_{\rm{acc,hot/cold}}$, given by eqs. \[fhot1\]-\[fhot3\]). To compute the model we also need the ratio $r_{\rm{cool}}/R_{200}$ as a function of halo mass and redshift. We calculate it by equating the heating ($\Gamma_{\rm{heat}}$) and cooling ($\Gamma_{\rm{cool}}$) rates (energy per unit time) of gas from the hot halo (derived in Paper I) $$\begin{aligned} \label{gamma_heat} \Gamma_{\rm{heat}}(M_{200})= &\frac{3}{2}\frac{k_{\rm{B}}T_{\rm{vir}}}{\mu m_{\rm{p}}}\frac{\Omega_{\rm{b}}}{\Omega_{\rm{m}}}\dot{M}_{200}\left[\frac{2}{3}f_{\rm{hot}}+f^{\rm{halo}}_{\rm{acc,hot}}\right],&\\\label{gamma_cool} \Gamma_{\rmn{cool}}(M_{200},r)= &M_{\rm{hot}}\frac{\Lambda[T_{\rm{hot}},Z_{\rm{hot}},\rho_{\rm{hot-gas}}(r)]}{\rho_{\rm{hot-gas}}(r)}.&\end{aligned}$$ In eqs. (\[gamma\_heat\]) and (\[gamma\_cool\]), $\rho_{\rm{hot}}$, $T_{\rm{hot}}$ and $Z_{\rm{hot}}$ are the characteristic hot gas density, temperature and metallicity respectively, and $f_{\rm{hot}}$ is the fraction of gas in the halo that is hot ($f_{\rm{hot}}\equiv M_{\rm{hot}}/[\frac{\Omega_{\rm{b}}}{\Omega_{\rm{m}}}M_{200}]$), which is given by the following best-fit relation from Paper I $$\begin{aligned} \label{Mhot2} \log_{10}\left(\frac{M_{\rm{hot}}}{(\frac{\Omega_{\rm{b}}}{\Omega_{\rm{m}}})M_{200}}\right)&=& -0.8+0.5x-0.05x^{2},\\\nonumber x &=& \log_{10}(M_{200}/10^{12}{\, {\rm M_{\odot}}}).\end{aligned}$$ Also, in eqs. (\[gamma\_heat\]) and (\[gamma\_cool\]), $\dot{M}_{200}$ is the dark matter accretion rate of the halo and $\Lambda$ is the net cooling rate per unit volume. The only parameter that depends on radius is $\rho_{\rm{hot-gas}}(r)$, which we estimate by extracting the hot gas density profiles from the simulation and interpolating to obtain the gas density as a function of radius and halo mass (see Appendix \[density\_profile\] for details of the density profiles). To compute $r_{\rm{cool}}/R_{200}$, we assume that $T_{\rm{hot}}$ is equal to the halo virial temperature, $Z_{\rm{hot}}=0.1Z_{\odot}$ and obtain $\Lambda$ from the tabulated cooling rates given by @Wiersma09a. In Paper I we explored the relation between the mass-weighted median metallicity of the hot gas with halo mass and redshift, and found that $Z_{\rm{hot}}\sim 0.1Z_{\odot}$ for $10^{12}{\, {\rm M_{\odot}}}$ haloes at $z=0$ and varies by up to a factor of 2.5 in the halo mass range $10^{11}-10^{14}{\, {\rm M_{\odot}}}$. We find that changing the metallicity with halo mass, changes the normalization of the relation $r_{\rm{cool}}/R_{200}-M_{200}$, but the qualitative result remains the same. Finally we obtain $\dot{M}_{200}$ from the analytic model of @Correa15c. Note that the various best-fit expressions presented in this section were derived in Paper I by fitting to the results of the Ref model, therefore they depend on simulations. For further details of the calculation of $\Gamma_{\rm{heat}}$ and $\Gamma_{\rm{cool}}$ see Paper I. In the following section we compare the result of the semi-analytic model with the simulation output for $z=0$ only, but we have found that the model works well in the regime $z=0-4$. \ \ Results ------- In this section we provide insight into the physical mechanisms that drive the gas accretion rates onto galaxies. We do so by comparing the model of galaxy gas accretion derived in the previous section with the simulation output, and analysing the model’s prediction in scenarios with and without AGN feedback. We emphasize that because the semi-analytic model uses input from the simulations, its predictions are not independent. We can however use it to test our physical understanding of gas accretion onto galaxies in the simulations. The top panel of Fig. \[plot\_model\] shows the model’s prediction with AGN feedback (blue solid line) and without AGN feedback (blue dashed line). These are compared with the accretion rates from the Ref-L100N1504 (orange diamonds) and the No AGN FB-L050N0752 simulations (red circles). While the model accurately matches the gas accretion rates onto galaxies in haloes less massive than $10^{13}{\, {\rm M_{\odot}}}$ from the Ref model, at higher halo masses it under predicts the rates by up to 0.3 dex. However, the model does reproduce the qualitative trends of the flattening for $10^{12}-10^{12.5}{\, {\rm M_{\odot}}}$ and the upturn at higher halo masses. In the case of gas accretion rates onto galaxies from the No AGN FB simulation, the model is in excellent agreement. Note that for these models, the same $\epsilon$($=0.3$) correction factor is used. The model’s result when AGN feedback is included can be explained as follows. In haloes with masses lower than $10^{11.7}{\, {\rm M_{\odot}}}$, $\frac{r_{\rm{cool}}}{R_{200}}=1$, and since $f^{\rm{halo}}_{\rm{acc,hot}}+f^{\rm{halo}}_{\rm{acc,cold}}=1$, eq. (\[Mgalaxy2\]) gives $\dot{M}_{\rm{gas,galaxy}} = \epsilon\dot{M}_{\rm{gas,halo}}$. In haloes with masses between $10^{11.7}$ and $10^{13}{\, {\rm M_{\odot}}}$ the hot halo forms. As a result, the cooling radius is smaller than the virial radius, yielding $f^{\rm{halo}}_{\rm{acc,hot}}r_{\rm{cool}}/R_{200}+f^{\rm{halo}}_{\rm{acc,cold}}<1$, so that $\dot{M}_{\rm{gas,galaxy}}$ increases less steeply than $\dot{M}_{\rm{gas,halo}}$, remaining almost constant with halo mass. In haloes with masses larger than $10^{13}{\, {\rm M_{\odot}}}$, $\frac{r_{\rm{cool}}}{R_{200}}$ increases with halo mass, indicating that the hot halo cooling flow becomes more prominent. The bottom panel of Fig. 6 shows the ratio between the cooling radius and the virial radius, $\frac{r_{\rm{cool}}}{R_{200}}(M_{200})$, as a function of halo mass for the simulations with and without AGN feedback (blue solid and dashed lines, respectively). The panel shows that the ratio $\frac{r_{\rm{cool}}}{R_{200}}(M_{200})$ does not continuously decrease towards high halo masses as is commonly thought. Mathematically the upturn in the $r_{\rm{cool}}-M_{200}$ relation can be explained by the radial slope ($\gamma=$dln$\rho_{\rm{hot-gas}}/$dln$r$) of the hot gas density profile (measured between $r=0.15\times R_{200}$ and $r=R_{200}$), which changes with halo mass. The slope is roughly -2 in $10^{11.7}{\, {\rm M_{\odot}}}$ haloes, increases to -0.86 in $10^{12.7}{\, {\rm M_{\odot}}}$ haloes, and decreases to -0.94 and -1.7 in $10^{13.1}$ and $10^{13.9}{\, {\rm M_{\odot}}}$ haloes respectively. The change in the slope of $\rho_{\rm{hot-gas}}(r)$ drives the evolution of $r_{\rm{cool}}/R_{200}$, and describes the evolution in the distribution of the hot halo gas as the halo grows in mass. The increase of the cooling radius towards high halo masses may not be physical but the result of a deficiency of the simulations. Haloes in the Ref model more massive than $10^{13}{\, {\rm M_{\odot}}}$ contain not only 0.2 dex higher gas mass fraction (derived from virtual X-ray emission) than observed group fractions (@Schaye14), but also too massive brightest cluster galaxies (@Bahe17). Since the amount of hot gas (and cooling) in the halo is sensitive to the heating temperature of AGN feedback (@LeBrun14), the disagreement with observations indicates that AGN feedback is insufficiently efficient at high halo masses. There is a significant change in the dependence of $\dot{M}_{\rm{gas,galaxy}}$ on halo mass when there is no AGN feedback, which seems to indicate that the hot halo does not impact the galaxy gas accretion rate. However, we find that this is not the case. When the hot halo forms, it reduces the gas mass that cools, but at a lower rate when AGN feedback is not included. To understand how $\dot{M}_{\rm{gas,galaxy}}$ changes with and without AGN feedback, we refer to our model of gas accretion. As described in the previous section, the model uses as input (1) the gas accretion rate onto haloes ($\dot{M}_{\rm{gas,halo}}$), (2) the hot/cold fraction of gas accretion onto haloes ($f^{\rm{halo}}_{\rm{acc,hot/cold}}$), (3) the total hot gas mass in the halo ($M_{\rm{hot}}$) and (4) the hot gas density profile ($\rho_{\rm{hot-gas}}$). We obtain these inputs from two simulations, Ref-L100N1504 and No AGN FB-L050N0752, and predict $\dot{M}_{\rm{gas,galaxy}}$ as a function of halo mass for both models. Note that the main differences between the No AGN FB and Ref model are $M_{\rm{hot}}$ (where $M_{\rm{hot}}$ from No AGN FB is larger than from Ref in the halo mass range $10^{11.7}-10^{14}{\, {\rm M_{\odot}}}$, by a factor of 1.4 in $10^{12}{\, {\rm M_{\odot}}}$ haloes) and $\rho_{\rm{hot-gas}}$ (where the slope flattens but no as much as in the presence of AGN feedback, reaching a maximum of dln$\rho/$dln$r=-1.14$ in $10^{12.8}{\, {\rm M_{\odot}}}$ haloes from the No AGN FB model in contrast to -0.9 from the Ref model). In Paper I we found that $f^{\rm{halo}}_{\rm{acc,hot/cold}}$ and $\dot{M}_{\rm{gas,halo}}$ are roughly insensitive to feedback (in agreement with @vandeVoort11). The variable from eq. (\[Mgalaxy2\]) that is responsible for the excellent match between $\dot{M}_{\rm{gas,galaxy}}$(no AGN) and the simulation output is $r_{\rm{cool}}/R_{200}$, whose evolution differs strongly from the scenario where AGN feedback is on. In the case of no AGN, $r_{\rm{cool}}/R_{200}$ decreases with halo mass in the mass range $10^{11.5}-10^{13}{\, {\rm M_{\odot}}}$ (shown in the bottom panel of Fig. 6), but it does not reach the same minimum value as in the case with AGN (the minimum value of $r_{\rm{cool,NoAGN}}/R_{200}$ is 0.4 compared with 0.08 for $r_{\rm{cool,AGN}}/R_{200}$). This means that the rate of gas cooling from the hot halo is always larger in the absence of AGN activity. We find that $\dot{M}_{\rm{gas,galaxy}}$(no AGN) steadily increases because the rate of gas cooling from the hot halo, $\dot{M}_{\rm{cooling}}/\dot{M}_{\rm{gas,halo}}=(r_{\rm{cool}}/R_{200})f^{\rm{halo}}_{\rm{acc,hot}}$, remains roughly constant with increasing halo mass. We obtain that $(r_{\rm{cool}}/R_{200})f^{\rm{halo}}_{\rm{acc,hot}}\sim 0.4$ for haloes with masses between $10^{12}-10^{13}{\, {\rm M_{\odot}}}$ ($f^{\rm{halo}}_{\rm{acc,hot}}$ increases with halo mass at roughly the same rate as $r_{\rm{cool}}/R_{200}$ decreases). However, since $f^{\rm{halo}}_{\rm{acc,cold}}$ decreases with halo mass, $\dot{M}_{\rm{gas,galaxy}}$(no AGN) differs from $\epsilon\dot{M}_{\rm{gas,halo}}$ by 0.1 dex for $10^{12}{\, {\rm M_{\odot}}}$ haloes and 0.3 dex for $10^{13}{\, {\rm M_{\odot}}}$ haloes. Physically, the steady increase of $\dot{M}_{\rm{gas,galaxy}}$(no AGN) with halo mass means that when there is no AGN feedback, a larger rate of gas cooling from the hot halo develops. We conclude that the formation of a hot halo alone is not enough to prevent gas from reaching the galaxy (see also @Gabor11 [@Liu17; @Gutke17]). Summary and conclusions {#Conclusion_sec} ======================= We have investigated the physics that drives the gas accretion rates onto galaxies at the centers of dark matter haloes using the EAGLE simulation suite as well as analytic calculations. We began by defining the gas accretion rate onto the galaxy as the rate at which gas crosses the radius $0.15\times R_{200}$ between two consecutive snapshots. We also defined the gas accretion rate onto the ISM as the rate at which gas crosses the radius $0.15\times R_{200}$ and the phase space cut $n_{\rm{H}}>0.1\rm{cm}^{-3}$, $T<10^{5}\rm{K}$. We found that at $z=0$ and in the halo mass range $10^{10}-10^{12}{\, {\rm M_{\odot}}}$ the gas accretion rates onto the galaxy ($\dot{M}_{\rm{gas,galaxy}}$) and ISM ($\dot{M}_{\rm{ISM}}$) increase with halo mass at approximately the same rate, with $\dot{M}_{\rm{ISM}}$ having a 0.3 dex (on average) lower normalization than $\dot{M}_{\rm{gas,galaxy}}$. For halo masses $\gtrsim 10^{12}{\, {\rm M_{\odot}}}$, $\dot{M}_{\rm{ISM}}$ remains nearly constant. While $\dot{M}_{\rm{gas,galaxy}}$ flattens at ${\sim}10^{12}{\, {\rm M_{\odot}}}$, it increases with halo mass for haloes with masses $\gtrsim 10^{13}{\, {\rm M_{\odot}}}$ (Fig. 1). We analysed the dependence of the rates of gas accretion onto galaxies, as well as onto haloes, on halo mass and redshift. We defined the rate of gas accretion onto haloes ($\dot{M}_{\rm{gas,halo}}$) as the rate at which gas crosses the virial radius between two consecutive snapshots, and compared the simulation output with the analytic prediction of @Correa15c for the dark matter accretion rate, scaled by the universal baryon fraction. @Correa15b [@Correa15c] demonstrated that their analytic model reproduces the accretion rates onto haloes in collisionless simulations. We found that the analytic prediction is 0.8 dex too high for $10^{10}{\, {\rm M_{\odot}}}$ haloes at $z=0$, agrees for $10^{11.5}{\, {\rm M_{\odot}}}$ haloes, but is 0.3 dex too low for $10^{13}{\, {\rm M_{\odot}}}$ haloes (Fig. 2). At redshifts $z=1-2$ better agreement is obtained between $\dot{M}_{\rm{gas,halo}}$ and the analytic prediction for haloes with mass $\gtrsim 10^{12}{\, {\rm M_{\odot}}}$. At higher redshifts ($z>2$) the analytic prediction and $\dot{M}_{\rm{gas,halo}}$ increase with halo mass at the same rate. However, the analytic prediction has a higher normalization (by up to $\sim 0.5$ dex for $z\sim 8$). We believe that better agreement would be reached for high-mass haloes if the analytic model included the impact of baryonic physics that reduces the halo mass. The large discrepancy for halo masses of $\sim 10^{10}{\, {\rm M_{\odot}}}$ is expected because the potential wells of these haloes are too shallow to strongly bind photo-heated gas (e.g. @Sawala13 [@Benitez17]). The gas accretion rate onto galaxies is (on average) a factor of 4 (and up to a factor of 16) lower than that onto haloes for halo masses $10^{10}-10^{12}{\, {\rm M_{\odot}}}$ ($\sim 10^{12.5}{\, {\rm M_{\odot}}}$) at $z=0$, and a factor of 2 (5) lower for $10^{10}-10^{12}{\, {\rm M_{\odot}}}$ ($\sim 10^{12.5}{\, {\rm M_{\odot}}}$) haloes at $z=2$. In low-mass haloes, $\dot{M}_{\rm{gas,galaxy}}$ increases with halo mass at nearly the same rate as $\dot{M}_{\rm{gas,halo}}$, but it flattens for halo masses of $\sim 10^{12}{\, {\rm M_{\odot}}}$ at $z=0-2$ (Fig. 2). In high-mass haloes ($>10^{12}{\, {\rm M_{\odot}}}$), two modes of accretion (hot and cold) coexist and contribute to the total rates onto galaxies and haloes. We defined hot gas accretion as the accretion rate of gas that after accretion onto the galaxy or halo has a temperature higher than $10^{5.5}\rm{K}$, and calculated the fraction of gas accreted hot onto the galaxy ($f^{\rm{galaxy}}_{\rm{acc,hot}}$) and halo ($f^{\rm{halo}}_{\rm{acc,hot}}$). We found that while $f^{\rm{halo}}_{\rm{acc,hot}}=0.7$ for $10^{12}{\, {\rm M_{\odot}}}$ haloes at $z=0$, $f^{\rm{galaxy}}_{\rm{acc,hot}}=0.05$ for the central galaxies within them. However, this difference decreases for higher-mass haloes, e.g. $f^{\rm{halo}}_{\rm{acc,hot}}=0.98$ and $f^{\rm{galaxy}}_{\rm{acc,hot}}=0.80$ for $10^{13}{\, {\rm M_{\odot}}}$ haloes at $z=0$. The fraction of gas accreted hot onto galaxies strongly depends not only on halo mass but also on redshift. We found that $\dot{M}_{\rm{gas,galaxy}}$ changes from being cold-mode dominated to hot-mode dominated ($f^{\rm{galaxy}}_{\rm{acc,hot}}=0.5$) for galaxies in $10^{12.7}{\, {\rm M_{\odot}}}$ haloes at $z=0$ but in $10^{13.3}{\, {\rm M_{\odot}}}$ haloes at $z=1$ (Fig. 3). We also investigated the dependence of the rates of gas accretion onto galaxies on feedback variations. It has been shown that in scenarios with energetic stellar feedback, the galaxy accretion rates can increase due to re-accretion of gas that was blown out of the galaxy by stellar-driven winds. In this work, we found that when stellar feedback is twice as energetic, $\dot{M}_{\rm{gas,galaxy}}$ for galaxies in $\leq 10^{12}{\, {\rm M_{\odot}}}$ haloes is lower than $\dot{M}_{\rm{gas,galaxy}}$ from Ref. However we find that $\dot{M}_{\rm{gas,galaxy}}$ increases for galaxies in haloes larger than $10^{12}{\, {\rm M_{\odot}}}$, indicating that at these halo masses galaxies re-accrete gas that was ejected by stellar feedback in lower-mass progenitors. When stellar feedback is half as energetic, $\dot{M}_{\rm{gas,galaxy}}$ decreases for galaxies in $\geq 10^{11}{\, {\rm M_{\odot}}}$ haloes, possibly due to a lower rate of re-accreted gas or because black hole accretion, and hence AGN feedback, becomes efficient at lower halo masses (Fig 4). When stellar feedback is turned off, $\dot{M}_{\rm{gas,galaxy}}$ decreases relative to Ref for galaxies in $>10^{10.5}{\, {\rm M_{\odot}}}$ haloes. This is likely because without stellar feedback the central black holes in low-mass galaxies become much more massive (@Bower17), allowing AGN feedback to suppress the gas infall rates. Indeed, when both stellar and AGN feedback are turned off, the rates of gas accretion are higher than in Ref for galaxies in $\lesssim 10^{12}{\, {\rm M_{\odot}}}$ haloes (Fig. 4). When AGN feedback is more explosive and intermittent, $\dot{M}_{\rm{gas,galaxy}}$ decreases with respect to Ref for galaxies in $\geq 10^{12}{\, {\rm M_{\odot}}}$ haloes, indicating that AGN further suppress $\dot{M}_{\rm{gas,galaxy}}$ in massive haloes. When AGN feedback is switched off, $\dot{M}_{\rm{gas,galaxy}}$ does not flatten at $\sim 10^{12}{\, {\rm M_{\odot}}}$ and increases with halo mass at the same rate as the gas accretion onto haloes but with a lower normalization (Fig. 5). This shows that AGN feedback is the mechanism responsible for preventing hot gas from cooling and falling onto central galaxies in massive haloes. To further understand the behavior of $\dot{M}_{\rm{gas,galaxy}}$ with halo mass, we developed a physically motivated semi-analytic model of galaxy gas accretion. The model postulates that two modes of accretion, cold and hot, contribute to the total gas accretion rate. Cold gas accretion onto galaxies is driven by the rate of cold accretion onto haloes, whereas hot gas accretion is driven by the rate of gas cooling from the hot halo, which depends on the rate of gas accreting onto the halo that is shock-heated and on the location of the cooling radius. To calculate the cooling radius, $r_{\rm{cool}}$, we equated the heating rate produced by accretion shocks (derived in Paper I) with the cooling rate. We found that in the radial range $[0.1-1]R_{200}$ the hot gas density profile of haloes from the EAGLE simulations deviates from the isothermal shape. The logarithmic density slope increases with halo mass for $>10^{12}{\, {\rm M_{\odot}}}$ haloes, reaches a maximum of $-0.9$ in $10^{13}{\, {\rm M_{\odot}}}$ haloes, and decreases at higher masses. The change in the slope of the density profile reflects how the distribution of the hot gas changes as the halo evolves due to continued infall, reheating and cooling. Because the density profile evolves with halo mass, the ratio $r_{\rm{cool}}/R_{200}$ does not decrease monotonically with halo mass. It decreases up to ${\sim}10^{13}{\, {\rm M_{\odot}}}$ haloes and increases towards larger haloes (Fig. 6). The upturn in the ratio of the cooling and virial radii with halo mass may indicate that while AGN-driven outflows reduce the density of the hot halo for $\sim 10^{12}-10^{13}{\, {\rm M_{\odot}}}$ haloes, in higher-mass haloes AGN feedback is insufficiently efficient, causing the hot halo cooling flow to become more prominent, and the accretion rate onto galaxies to increase more steeply with halo mass. When there is no AGN feedback, the density of the hot halo is higher, $r_{\rm{cool}}/R_{200}$ does not decrease as much as when AGN feedback is on, and so the rate of gas cooling from the hot halo is higher. We compared our semi-analytic model of gas accretion with the galaxy accretion rates calculated from the simulation at $z=0$ and found excellent agreement. In future work, we plan to investigate whether the correlation between accretion rates onto nearby galaxies (i.e. galactic conformity, @Weinmann06) is driven by the correlation of dark matter halo formation time with environment. We will test the predictions of the semi-analytic model with the simulation outputs. Acknowledgments {#acknowledgments .unnumbered} =============== We are grateful to the EAGLE team for putting together a great set of simulations. This work used the DiRAC Data Centric system at Durham University, operated by the Institute for Computational Cosmology on behalf of the STFC DiRAC HPC Facility (www.dirac.ac.uk). This equipment was funded by BIS National E-infrastructure capital grant ST/K00042X/1, STFC capital grant ST/H008519/1, and STFC DiRAC Operations grant ST/K003267/1 and Durham University. DiRAC is part of the National E-Infrastructure. The EAGLE simulations were performed using the DiRAC-2 facility at Durham, managed by the ICC, and the PRACE facility Curie based in France at TGCC, CEA, Bruyeres-le-Chatel. This work was supported by the Netherlands Organisation for Scientific Research (NWO) through VICI grant 639.043.409. FvdV acknowledges the Klaus Tschira Foundation. We thank the anonymous reviewer for fruitful comments that improved the original manuscript. Simulations {#Simulations_app} =========== In the EAGLE simulations, star formation is modeled following the recipe of @Schaye08. It is stochastic above a density threshold that depends on metallicity (proposed by @Schaye04). Stellar evolution and mass loss follows the work of @Wiersma09b, where star particles are treated as simple stellar populations with @Chabrier initial mass function, spanning the range $0.1-100 {\, {\rm M_{\odot}}}$. Feedback from star formation follows the stochastic thermal feedback scheme of @DallaVecchia12. Rather than heating all neighbouring gas particles within the SPH kernel, they are selected stochastically based on the available energy, then heated by a fixed temperature difference of $\Delta T = 10^{7.5}$K. The probability that a neighbouring SPH particle is heated is determined by the fraction of the energy budget that is available for feedback, that depends on adjustable maximum and minimum threshold values ($f_{\rm{th,max}}$ and $f_{\rm{th,min}}$, respectively), on the gas density as well as on metallicity. For AGN feedback, black hole seeds (of $\approx 1.4\times 10^{5}{\, {\rm M_{\odot}}}$) are included in haloes with mass greater than $\approx 1.4\times 10^{10}{\, {\rm M_{\odot}}}$ (@Springel05). Black holes can grow through mergers and accretion. The accretion events follow a modified Bondi-Hoyle formula that accounts for the angular momentum of the accreting gas (@Rosas13), and a free parameter that is related to disk viscosity ($C_{\rm{visc}}$). AGN feedback is stochastic, it follows the accretion of mass onto the black hole, where a fraction of the accreted gas is released as thermal energy into the surrounding gas. This method is based on that of @Booth and @DallaVecchia12, where the free parameter is the heating temperature $\Delta T_{\rm{AGN}}$. Finally, radiative cooling and photo-heating are included as in @Wiersma09b. The element-by-element radiative rates are computed in the presence of the cosmic microwave background (CMB) and the @Haardt model for UV and X-ray background radiation from quasars and galaxies. The EAGLE simulations include a new SPH formulation named ‘Anarchy’ which improves the performance on standard hydrodynamical tests when compared to the original SPH implementation in GADGET (see @Schaller15b or @Hu14 for similar results). Anarchy makes use of a pressure-entropy formulation derived in Hopkins (2013), allowing it to avoid spurious jumps at contact discontinuities. It also uses an artificial viscosity switch as in @Cullen10, that allows the viscosity limiter to be stronger when shocks and shear flows are present. In addition, it includes an artificial conduction switch (similar to that of @Price08), the $C^{2}$ @Wendland95 kernel and the time step limiters of @Durier12, which ensure that ambient particles do not remain inactive when a shock is approaching (for a more complete description see @Schaye14). Numerical convergence {#Numerical_convergence} ===================== ![Accretion rate of gas onto haloes (top panel) and their central galaxies (bottom panel) as a function of halo mass in the redshift range $0\leq z<0.1$. The curves show the median values of the total accretion rates of haloes, as well as onto galaxies, in logarithmic mass bins of 0.2 dex, each mass bin contains at least 10 haloes. To analyze numerical convergence, we compare accretion rates from simulations with different box sizes and number of particles and therefore different resolution. We classify the simulations as high-resolution (blue curves) and intermediate-resolution (yellow curves). We find strong convergence with box size and weak convergence between different resolution simulations (see text for details).[]{data-label="accretion_convergence"}](./plots/Accretion_rate_convergence_onto_halos_028.ps "fig:"){width="46.00000%"}\ ![Accretion rate of gas onto haloes (top panel) and their central galaxies (bottom panel) as a function of halo mass in the redshift range $0\leq z<0.1$. The curves show the median values of the total accretion rates of haloes, as well as onto galaxies, in logarithmic mass bins of 0.2 dex, each mass bin contains at least 10 haloes. To analyze numerical convergence, we compare accretion rates from simulations with different box sizes and number of particles and therefore different resolution. We classify the simulations as high-resolution (blue curves) and intermediate-resolution (yellow curves). We find strong convergence with box size and weak convergence between different resolution simulations (see text for details).[]{data-label="accretion_convergence"}](./plots/Accretion_rate_convergence_onto_galaxies_028.ps "fig:"){width="46.00000%"} We investigate how numerical resolution impacts the rates of gas accretion in the EAGLE simulations. Fig. \[accretion\_convergence\] shows the total gas accretion onto haloes (top panel) and their central galaxies (bottom panel) in the redshift range $0\leq z<0.1$. Both panels show that the accretion rates (onto haloes and galaxies) increase with increasing halo mass. We find that in $10^{11}{\, {\rm M_{\odot}}}$ haloes, $\dot{M}_{\rm{gas,galaxy}}$ increases (on average) by a factor of 3 if the particle mass resolution is increased by a factor of 8, whereas $\dot{M}_{\rm{gas,halo}}$ increases by up to a factor of 2.5. This lack of convergence in the gas accretion rates onto galaxies is expected, due to the fact that recycling winds from SN and AGN outflows are resolution dependent. However, we achieve convergence between simulations that use different resolution but for which the parameters of the subgrid feedback have been calibrated to the same observations. We find excellent agreement in the accretion rates between the recalibrated model ‘Recal-L025N0752’ and the reference models Ref-L100N1504/Ref-L025N0376. The top panel from Fig. \[accretion\_convergence\] shows a drop in the accretion rates from the intermediate-resolution simulations (yellow curves) in haloes smaller than $10^{10}{\, {\rm M_{\odot}}}$. This is likely a numerical artifact because such a drop is not seen at this mass in the accretion rates from the high resolution simulation (blue curve). We therefore set the minimum halo mass for accretion onto haloes (and their inner galaxies) to correspond to 1000 dark matter particles ($\sim 10^{10}{\, {\rm M_{\odot}}}$ halo mass for the intermediate-resolution simulations). Throughout this work we use the intermediate-resolution (fiducial) simulations. Density profiles of hot gaseous haloes {#density_profile} ====================================== In this section we investigate how the density profile of hot halo gas evolves due to the continued infall, reheating and cooling of gas in haloes from the EAGLE simulations. We select all haloes from the Ref-L100N1504 and No AGN FB-L050N0752 simulations and separate them in mass bins of $\Delta \log_{10}(M_{200})=0.2$ width. To calculate $\rho_{\rm{hot-gas}}(r)$, we select gas particles that are hot (have cooling times longer than local dynamical times) and define a set of concentric spherical shells of width $\Delta\log_{10}(r)=0.078$. We add the mass of the particles within each shell and divide by the volume. As we are only interested in the density profile of gas in the haloes but not within galaxies, we restrict the analysis of $\rho_{\rm{gas}}(r)$ to the radial range $0.1$ to $1\times R_{200}$. We calculate the dynamical time, $t_{\rm{dyn}}$, of the gas particle as $t_{\rm{dyn}}=r/V_{\rm{c}}(r)$, where $V_{\rm{c}}(r)=[GM(<r)/r]^{1/2}$ is the circular velocity and $M(<r)$ is the mass enclosed within $r$. We calculate the cooling time, $t_{\rm{cool}}$, as $t_{\rm{cool}} = \frac{3nk_{\rm{B}}T}{2\Lambda}$, where $n$ is the number density of the gas particle ($n=\rho_{\rm{gas}}/\mu m_{\rm{p}}$, $\mu m_{ rm{p}}$ is the mean particle mass calculated from the cooling tables of @Wiersma09b), $k_{\rm{B}}$ is the Boltzmann constant, $T$ is the gas temperature and $\Lambda$ is the cooling rate per unit volume with units of erg cm$^{-3}$s$^{-1}$. To calculate $\Lambda$, we use the tabulated cooling function for gas exposed to the evolving UV/X-ray background from @Haardt given by @Wiersma09b, which was also used by the EAGLE simulations. Note that the “standard" definition for the dynamical time of gas within a virialized system depends on $R_{200}$ and $V_{\rm{c}}(R_{200})$, and not on the local radius and local circular velocity as defined here. Fig. \[density\_plot\_1\] shows the median gas density profile at $z=0$ of haloes in the mass range $\log_{10}(M_{200}/{\, {\rm M_{\odot}}})\pm 0.1$, with $\log_{10}(M_{200}/{\, {\rm M_{\odot}}})$ varying from 11.7 to 13.7, as indicated in the legends, from the Ref model (top panel) and from the No AGN FB model (bottom panel). We define $\gamma$ as the logarithmic density slope (${\rm{d}}\ln \rho_{\rm{gas}}/{\rm{d}}\ln r\equiv\gamma$), which we measure between the radial range 0.1 and $1\times R_{200}$, assuming $\rho_{\rm{hot-gas}}(r)\propto r^{\gamma}$. We find that $\gamma$ increases from $-1.8$ in $10^{12}{\, {\rm M_{\odot}}}$ haloes to $-0.8$ in $10^{12.7}{\, {\rm M_{\odot}}}$ haloes, and then decreases again towards larger halo masses. In smaller haloes, $\rho_{\rm{hot-gas}}(r)$ is steeper at small radii, and deviates from the isothermal shape (characterized by $\gamma=-2$). The change in the slope of $\rho_{\rm{hot-gas}}(r)$ with mass indicates how the distribution of the hot gas in the halo evolves, and can be explained as follows. Since a stable hot halo forms for $\sim 10^{12}{\, {\rm M_{\odot}}}$, most gas crossing $R_{200}$ remains hot. As a result the gas density in the range $0.5-1R_{200}$ increases and the density profile flattens. When AGN feedback is turned off, $\rho_{\rm{hot-gas}}(r)$ does not flatten as much as in Ref, suggesting that the cooling flow from the hot halo is more prominent. \ Accretion rate onto haloes {#Comparison} ========================== In this section we show a comparison between the best-fit relations presented in Section \[accretion\_rates\_halos\_galaxies\] given by eqs. (1-8) and the simulation outputs. Fig. \[comparison\_plot\] shows the accretion rates onto haloes as a function of halo mass for different redshifts. The solid curves correspond to the accretion rates taken from the Ref-L100N1504 simulation, whereas the dashed curves correspond to the best-fit expression. We show that the best-fit expressions are able to closely reproduce the gas accretion rate onto haloes in the redshift range 0 to 8 and halo mass range $10^{10}$ to $10^{14}{\, {\rm M_{\odot}}}$. [^1]: The simulation data is saved in 9 discrete output redshifts between redshift 0 to 1, in 8 output redshifts between redshift 1 and 3, and in 8 output redshifts between redshift 3 and 8.
--- abstract: '$B$ meson rare decays ($B\to K(K^{*})l\bar l$ and $B\to K^*\gamma$) are analyzed in the framework of effective field theory of heavy quarks. The semileptonic and penguin type form factors for these decays are calculated by using the light cone sum rules method at the leading order of $1/m_Q$ expansion. Four exact relations between the two types of form factors are obtained at the leading order of $1/m_Q$ expansion. Of particular, the relations are found to hold for whole momentum transfer region. We also investigate the validity of the relations resulted from the large energy effective theory based on the general relations obtained in the present approach. The branching ratios of the rare decays are presented and their potential importance for extracting the CKM matrix elements and probing new physics is emphasized.' address: | $\dagger$ Institute of Theoretical Physics, Academia Sinica, Beijing 100080, China\ $*$ Department of Physics, Tsinghua University, Beijing 100084, China author: - 'M. Zhong$\mbox{}^\dagger$, Y.L. Wu$\mbox{}^\dagger$ and W.Y. Wang$\mbox{}^*$' title: | Exclusive B-meson Rare Decays and General Relations of\ Form Factors in Effective Field Theory of Heavy Quarks --- \#1[[$\backslash$\#1]{}]{} Introduction {#int} ============ $B$ meson rare decays $B\to K^*\gamma$ and $B\to K(K^*)\bar ll$ are induced by transitions $b\to s\gamma$ and $b\to s\bar ll$ via penguin loop diagrams in the quark level, which are usually called flavor-changing-neutral-current (FCNC) processes. In the Standard Model (SM), $B$ meson rare decays may provide a quantitative way to determine the CKM matrix elements $V_{td},V_{ts}$ and $V_{tb}$. In the SM, FCNC transitions are forbidden at tree level. They can be induced only starting at 1-loop order, which makes their rates for decaying be sensitive to probe new physics. For these reasons, they have long been hot subjects in both experimental and theoretical studies. Nevertheless, one remains facing difficulties in studying exclusive $B$ meson rare decays due to the requirment of explicit calculations for the relevant form factors which involve in long distance ingredients that can not be calculated via QCD perturbative theory. Some reasonable nonperturbative methods, such as QCD sum rules, lattice simulations and phenomenological models, have been developped to estimate the long distance effects. The rare decays have been studied by using light cone sum rules (LCSR) in full QCD theory [@abhh; @akos; @aos; @saf]. As B meson can be treated as a heavy meson containing a single heavy quark, it is of interest to apply for the heavy quark effective field theory (HQEFT) to deal with the B meson rare decays. In this paper, we shall use the framework and normalization derived in Refs.[@ylw; @wwy; @yww; @ww], which has been applied in Refs.[@bpi; @brho] to investigate the exclusive semileptionic $B$ decays into $\pi$ and $\rho$ by using the LCSR method and obtained quite reasonable results. A more general study on exclusive semileptonic decays of heavy to light mesons within the framework of HQEFT has recently been carried out in [@wwz]. Note that as we only keep to the leading order contributions in the expansion of $1/m_Q$, the results and conclusions are actually independent of any framework of effective field theory of heavy quarks. We shall focus on in this paper the LCSR calculations of the form factors for the exclusive $B$ meson rare decays $B\to K^*\gamma$ and $B\to K(K^*)\bar ll$ within the framework of HQEFT. In section \[formulation\], we first present the hadronic matrix elements in the framework of HQEFT and derive a set of formulae for the relevant form factors. In section \[formulation two\], LCSR approach is applied to the relevant correlator functions in HQEFT. We then obtain four interesting relations among semileptonic type form factors and penguin type ones. Of interest, these relations are found to hold in whole momentum transfer region in the infinite mass limit of heavy quark or at the leading order of $1/m_Q$ expansion in HQEFT. Obviously, at zero recoil of the final light meson, these relations recover the so-called Isgur-Wise relations[@iw]. In early time, the Isgur-Wise relations were conjectured to be also valid in the region of large recoil in ref.[@BD]. Late on, these relations were really shown in the quark model[@stech; @soares] to hold at large recoil. In particular, one of the relations concerning the form factor in radiative decays was shown to hold in the whole momentum transfer by using QCD sum rule approach[@ABS]. Numerical analysis of form factors is presented in section \[result\]. Recently, one developed the so-called large energy effective theory (LEET) in which more relations were obtained near large recoil due to additional symmetries in large energy limit[@cyopr; @efg], while some of the relations were found to be broken down by QCD corrections[@mbtf]. Since all relations in Ref.[@cyopr] were put forward following from the combination of heavy quark effective theory (HQET) and LEET and moreover LEET is compatible with LCSR, our present method provides a reliable and important way to check the validity of LEET relations. So a detailed discussion and comparison will be presented on the basis of our HQEFT calaulation in section \[leet\]. In section \[branch ratios\], we give the relevant branching ratios for the B meson rare decays $B\to K^*\gamma$ and $B\to K(K^*)\bar ll$. A brief summary is outlined in section \[summary\]. General description of matrix elements in HQEFT {#formulation} =============================================== The transition matrix elements responsible for the $B$ meson rare decays $B\to K^*\gamma$ and $B\to K(K^*)\bar ll$ may be grouped into two types: semileptonic and penguin ones. The semileptonic ones are defined as $$\langle K(p)|\bar s\gamma^\mu b|B(p+q) \rangle =2f_{+}(q^2) p^\mu+(f_{+}(q^2)+f_{-}(q^2)) q^\mu$$ for $B$ to $K$ decays and $$\begin{aligned} &&\langle K^*(p,\epsilon^*)|\bar s\gamma^\mu (1-\gamma^5) b|B(p+q) \rangle =-i(m_B+m_{K^*})A_1(q^2) \epsilon^{*\mu} \nonumber\\ &&\hspace{2cm}+i \frac{A_2(q^2)}{m_B+m_{K^*}} (\epsilon^{*}\cdot (p+q))(2p+q)^\mu +i\frac{A_3(q^2)}{m_B+m_{K^*}} (\epsilon^* \cdot (p+q)) q^\mu \nonumber\\ &&\hspace{2cm} + \frac{2 V(q^2)}{m_B+m_{K^*}} \epsilon^{\mu \alpha \beta \gamma}\epsilon^*_\alpha (p+q)_\beta p_\gamma\end{aligned}$$ for $B$ to $K^*$ decays. In this paper, we take $\epsilon_{0123} = 1$ and $\gamma^5=\gamma_5=i\gamma^0\gamma^1\gamma^2\gamma^3$. For convenience, we may define a form factor $A_0(q^2)$ as $$\begin{aligned} A_3(q^2)&=&\frac{2(m_B+m_{K^*})m_{K^*}}{q^2}(\bar A_3(q^2)-A_0(q^2)) \nonumber\\ \bar A_3(q^2)&=&\frac{(m_B+m_{K^*})A_1(q^2)-(m_B-m_{K^*})A_2(q^2)}{2m_{K^*}}\nonumber\\ A_0(0)&=&\bar A_3(0)\end{aligned}$$ $A_0(q^2)$ will directly enter into contributions to the relevant branching ratios. The penguin matrix elements can be written as $$\langle K(p)|\bar s\sigma^{\mu \nu}q_{\nu}(1+\gamma^5)b|B(p+q) \rangle =i\frac{f_T(q^2)}{m_B+m_K} \{q^2(2p+q)^\mu -(m^2_B-m^2_K)q^\mu \}$$ for $B$ to $K$ decays and $$\begin{aligned} &&\langle K^*(p,\epsilon^*)|\bar s\sigma^{\mu \nu}q_{\nu}(1+\gamma^5) b|B(p+q) \rangle =-i\epsilon^{\mu \alpha \beta \gamma} \epsilon^*_\alpha (p+q)_\beta p_\gamma2T_1(q^2) \nonumber\\ &&\hspace{2cm}+T_2(q^2) \{ (m^2_B-m^2_{K^*}) \epsilon^{*\mu}-(\epsilon^{*}\cdot (p+q))(2p+q)^\mu \} \nonumber\\ &&\hspace{2cm}+T_3(q^2)(\epsilon^* \cdot (p+q))\{ q^\mu-\frac{q^2}{m^2_B-m^2_{K^*}}(2p+q)^\mu \}\end{aligned}$$ for $B$ to $K^*$ decays. In the above definitions, $p$ is the momentum of the light meson $K$ or $K^*$. $\epsilon^*$ is the polarization vector of $K^*$ meson, and $q$ is the momentum transfer. $f_{\pm}$ and $f_T$ are the $B$ to $K$ semileptonic and penguin transition form factors respectively. $A_i$(i=0,1,2), $V$ and $T_i$(i=1,2,3) are the corresponding ones for $B$ to $K^*$ transitions. To be convenient for making Borel transformation which helps to suppress the contributions from the possible higher states of bottom mesons, we may change (2.4) and (2.5) into the following forms $$\langle K(p)|\bar s\sigma^{\mu \nu}p_{\nu}(1+\gamma^5)b|B(p+q) \rangle =i\frac{f_T(q^2)}{m_B+m_K} \{ (q\cdot p)(2p+q)^\mu -((2p+q)\cdot p)q^\mu \}$$ $$\begin{aligned} &&\langle K^*(p,\epsilon^*)|\bar s\sigma^{\mu \nu}p_{\nu}(1+\gamma^5) b|B(p+q) \rangle =\nonumber\\ &&\hspace{2cm}-i\epsilon^{\mu \alpha \beta \gamma} \epsilon^*_\alpha (p+q)_\beta p_\gamma \{ \frac{-m^2_B+m^2_{K^*}+q^2}{q^2}T_1(q^2)+\frac{m^2_B-m^2_{K^*}}{q^2}T_2(q^2) \} \nonumber\\ &&\hspace{2cm}+\epsilon^{*\mu}\{ (q\cdot p)\frac{m^2_B-m^2_{K^*}}{q^2}T_2(q^2) -[(q\cdot p)\frac{m^2_B-m^2_{K^*}}{q^2}-(2p+q)\cdot p ]T_1(q^2) \} \nonumber\\ &&\hspace{2cm}+q^\mu \frac{(\epsilon^* \cdot (p+q))((2p+q)\cdot p)}{m^2_B-m^2_{K^*}}\{ T_3(q^2) +\frac{m^2_B-m^2_{K^*}}{q^2}(T_2(q^2)-T_1(q^2))\} \nonumber\\ &&\hspace{2cm}-(2p+q)^\mu \frac{(\epsilon^* \cdot (p+q))(q\cdot p)}{m^2_B-m^2_{K^*}}\{T_3(q^2) +\frac{m^2_B-m^2_{K^*}}{q^2}(T_2(q^2)-T_1(q^2))\}\end{aligned}$$ When applying for the HQEFT to evaluate the matrix elements, they can be expanded into the powers of $1/m_Q$ and also be simply expressed by a set of heavy spin-flavor independent universal wave functions [@wwy; @ww; @bpi; @brho]. It is convenient to adopt the following normalization which relates matrix elements in full QCD with the ones in HQEFT [@wwy] $$\begin{aligned} \frac{1}{\sqrt{m_B} }\langle \kappa |\bar s \Gamma b|B \rangle = \frac{1}{\sqrt{\bar {\Lambda}_B}} \{ \langle \kappa |\bar s \Gamma b_v|B_v \rangle +O(1/m_b) \}\end{aligned}$$ where $\kappa$ represents $K(p)$ or $K^*(p,\epsilon^*)$. The notation $b_v$ is the effective bottom quark field. And $\bar\Lambda_B=m_B-m_b $ is the binding energy. From heavy quark symmetry, one can obtain the following relations [@bpi; @brho; @kms; @gzmy; @hly] $$\begin{aligned} \langle K(p)|\bar s \Gamma b_v|B_v \rangle &=&-Tr[k(v, p)\Gamma {\cal M}_v]\\ \langle K^*(p,\epsilon^*)|\bar s \Gamma b_v|B_v \rangle &=&-i \mbox{Tr}[\Omega(v, p)\Gamma {\cal M}_v]\end{aligned}$$ with $$\begin{aligned} k(v, p)&=&\gamma^5 [A(v\cdot p,\mu)+ {\hat{p}\hspace{-0.2cm}\slash} B(v\cdot p,\mu)] \\ \Omega(v, p)&=&L_1(v\cdot p) {\epsilon\hspace{-0.2cm}\slash}^* +L_2( v\cdot p)(v\cdot \epsilon^*) +[L_3(v\cdot p) {\epsilon\hspace{-0.2cm}\slash}^* +L_4(v\cdot p) (v\cdot \epsilon^* )]{\hat{p}\hspace{-0.2cm}\slash}\end{aligned}$$ and[@wwy] $$\begin{aligned} {\hat{p}}^\mu &=& \frac{p^\mu}{v\cdot p} \\ {\cal M}_v &=& -\sqrt{\bar \Lambda}\frac{1+v\hspace{-0.2cm}\slash}{2} \gamma ^5\end{aligned}$$ Where $A$, $B$ and $L_i(i=1,2,3,4)$ are the leading order wave functions characterizing the heavy to light transition matrix elements in the effective field theory. ${\cal M}_v$ is the spin wave function associated with the heavy meson state. The vector $v^\mu$ is the four-velocity of $B$ meson satisfying $v^2=1$, and $\bar{\Lambda}$ is the heavy flavor independent binding energy $$\bar\Lambda= \lim_{m_Q\to \infty} \bar\Lambda_B$$ which reflects only the effects arising from the light degrees of freedom in the heavy $B$ meson. With eqs.(2.1-2.14), one arrives at the following expressions for the form factors $$\begin{aligned} f_{\pm}(q^2)&=&\frac{1}{m_B} \sqrt{ \frac{m_B \bar\Lambda} {\bar{\Lambda}_B } } \{ A(v\cdot p)\pm B(v\cdot p) \frac{m_B}{v\cdot p} \} +\cdots \\ f_T(q^2)&=&\frac{m_B+m_K}{m_B}\sqrt{\frac{m_B \bar\Lambda}{\bar{\Lambda}_B}} \frac{B^{\prime}(v\cdot p)}{v\cdot p} +\cdots \\ A_1(q^2)&=&\frac{2}{m_B+m_{k^*}} \sqrt{\frac{m_B \bar\Lambda}{\bar\Lambda_B}} \{ L_1(v\cdot p)+L_3(v\cdot p) \} +\cdots \\ A_2(q^2)&=&2 (m_B+m_{K^*}) \sqrt{\frac{m_B \bar\Lambda}{\bar\Lambda_B}} \{\frac{L_2(v\cdot p)}{2 m^2_B} +\frac{L_3(v\cdot p)-L_4(v\cdot p)}{2m_B (v\cdot p)} \} +\cdots \\ A_3(q^2)&=&2 (m_B+m_{K^*}) \sqrt{\frac{m_B \bar\Lambda}{\bar\Lambda_B}} \{ \frac{L_2(v\cdot p)}{2 m^2_B} -\frac{L_3(v\cdot p) -L_4(v\cdot p)}{2m_B (v\cdot p)} \} + \cdots \\ V(q^2)&=&\sqrt{\frac{m_B \bar\Lambda}{\bar\Lambda_B}} \frac{m_B+m_{K^*}}{m_B (v\cdot p) } L_3(v\cdot p) +\cdots \\ T_1(q^2)&=&\sqrt{\frac{m_B \bar\Lambda}{\bar\Lambda_B}}\{\frac{L^{\prime}_1(v\cdot p)}{m_B}+ \frac{L^{\prime}_3(v\cdot p)}{v\cdot p} \}+\cdots \\ T_2(q^2)&=&2 \sqrt{\frac{m_B \bar\Lambda}{\bar\Lambda_B}} \frac{1}{m^2_B-m^2_{K^*}} \{(m_B-v\cdot p)L^{\prime}_1(v\cdot p) +\frac{m_B v\cdot p-m^2_{K^*}}{v\cdot p} L^{\prime}_3(v\cdot p) \}+\cdots \\ T_3(q^2)&=&\sqrt{\frac{m_B \bar\Lambda}{\bar{\Lambda}_B}}\{-\frac{L^{\prime}_1(v\cdot p)}{m_B}+ \frac{L^{\prime}_3(v\cdot p)}{v\cdot p} -\frac{m^2_B -m^2_{K^*}}{m^2_B v\cdot p} L^{\prime}_4(v\cdot p) \}+\cdots\end{aligned}$$ with $$y\equiv v\cdot p=\frac{m^2_B+m^2_\kappa -q^2}{2m_B}$$ denoting the energy of the final light meson. In the above formulae the dots denote possible higher order $1/m_Q$ contributions that are neglected in this paper. $B^{\prime}(v\cdot p)$ and $L^{\prime}_i(v\cdot p)$ are in general different from $B(v\cdot p)$ and $L_i(v\cdot p) (i=1,2,3,4)$ as they arise from different matrix elements. Light-cone sum rules in HQEFT {#formulation two} ============================= In order to calculate the relevant hadronic matrix elements which contain nonperturbative contributions and thus make QCD perturbative method lose its power, we shall apply for LCSR approach. In LCSR calculation, the relevant correlation functions are expanded near the light cone. The light cone distribution functions are introduced to describe the nonperturbative effects. In searching for reasonable and stable results, the quark-hadron duality and Borel transformation are generally adopted (for a detailed review, one may find in Refs. [@vai; @ar; @pvmisu]). The theoretical calculations can often show a simpler process in the framework of HQEFT than in QCD, which can explicitly be seen in Refs.[@bpi; @brho] where the semileptonic form factors for $B\rightarrow \pi$ and $\rho$ have been evaluated. We may directly adopt the formulae in [@bpi; @brho; @wwz] to $B\rightarrow K$ transitions by simply changing the relevant quantities corresponding to the $K$ meson $$\begin{aligned} A(y)&=&-\frac{f_K}{4Fy} \int^{s_0}_{0} ds e^{\frac{ 2\bar\Lambda_B-s}{T}} \left[\frac{1}{y} \frac{\partial}{\partial u}g_2(u)-{\mu_K} \phi_p(u) -\frac{\mu_K}{6}\frac{\partial}{\partial u}\phi_\sigma(u)\right]_{u=1-\frac{s}{2y}} \\ B(y)&=&-\frac{f_K}{4F} \int^{s_0}_{0} ds e^{\frac{ 2\bar\Lambda_B-s}{T}} \left[-\phi_K (u)+\frac{1}{y^2}\frac{\partial^2}{\partial u^2}g_1(u) -\frac{1}{y^2} \frac{\partial}{\partial u} g_2(u) +\frac{\mu_K}{6y}\frac{\partial}{\partial u} \phi_\sigma(u)\right]_{u=1-\frac{s}{2y}}\end{aligned}$$ As for the $B$ to vector Kaon meson decays, we may just adopt the definitions of $K^*$ meson distribution functions given in [@pvesrb]. $$\begin{aligned} \langle K^*(p,\epsilon^*)|\bar{s}(-x) \sigma_{\mu\nu} d(x)|0 \rangle &=&-i f^T_{K^*} [(\epsilon^*_\mu p_\nu - \epsilon^*_\nu p_\mu) \int^1_0 du e^{-i\xi p\cdot x} (\phi_\bot (u)+\frac{m^2_{K^*}x^2}{4}A_T(u)) \nonumber\\ &+&(p_\mu x_\nu-p_\nu x_\mu)\frac{\epsilon^*\cdot x}{(p\cdot x)^2}m^2_{K^*}\int^1_0 du e^{-i\xi p\cdot x}B_T(u) +\frac{1}{2}(\epsilon^*_\mu x_\nu \nonumber\\ &-& \epsilon^*_\nu x_\mu)\frac{m^2_{K^*}}{p\cdot x} \int^1_0 du e^{-i\xi p\cdot x}C_T(u)] \\ \langle K^*(p,\epsilon^*)|\bar{s}(-x) \gamma_\mu d(x)|0 \rangle &=& f_{K^*} m_{K^*}[ p_\mu \frac{\epsilon^*\cdot x}{p\cdot x} \int^1_0 du e^{-i\xi p\cdot x}( \phi_{\|}(u)+\frac{m^2_{K^*}x^2}{4}A(u)) \nonumber\\ &+& (\epsilon^*_\mu-p_\mu \frac{\epsilon^* \cdot x}{p\cdot x}) \int^1_0 du e^{-i\xi p\cdot x} g^{(v)}_\bot (u) \nonumber\\ &-&\frac{1}{2}x_\mu \frac{\epsilon^*\cdot x}{(p\cdot x)^2}m^2_{K^*}\int^1_0 du e^{-i\xi p\cdot x}C(u)] \\ \langle K^*(p,\epsilon^*)|\bar{s}(-x) \gamma_\mu \gamma_5 d(x)|0 \rangle &=&\frac{1}{2} (f_{K^*}-f^T_{K^*} \frac{m_s+m_d}{m_{K^*}}) m_{K^*} \epsilon_{\mu\nu\alpha\beta} \epsilon^{*\nu} p^\alpha x^\beta \int^1_0 du e^{-i\xi p\cdot x} g^{(a)}_\bot (u) \\ \langle K^*(p,\epsilon^*)|\bar{s}(-x)d(x)|0 \rangle &=& i(f^T_{K^*}-f_{K^*}\frac{m_s+m_d}{m_{K^*}}) m^2_{K^*} ( \epsilon^*\cdot x)\int^1_0 du e^{-i\xi p\cdot x} h^{(s)}_{\|}(u)\end{aligned}$$ with $\xi=2u-1$. In the above definitions $\phi_{\bot}(u)$ and $\phi_{\|}(u)$ give the leading twist two distributions in the fraction of total momentum carried by quarks in the transversely and longitudinally polarized meson respectively. The functions $g^{(v)}_\bot (u)$ and $g^{(a)}_{\|}(u)$, which describe transverse polarizations of quarks in the longitudinally polarized meson, provide contributions of twist three. $h^{(s)}_{\|}$ is also twist three function. $A(u)$, $A_T(u)$ and $C(u)$ are twist four, while $B_T(u)$ and $C_T(u)$ are mixing twist functions. In this paper, we only consider distribution functions from two-particle contributions. Applying the same calculation steps as in [@brho], we arrive at the following results $$\begin{aligned} L_1(y)&=& \frac{1}{4F} e^{2\bar \Lambda_B/T} \int^{s_0}_0 ds e^{-s/T} \frac{1}{y} \left[f_{K^*} m_{K^*}g^{(v)}_\bot(u)\right.\nonumber\\ & &\left.+\frac{1}{4}(f_{K^*}-f^T_{K^*}\frac{m_s+m_d}{m_{K^*}})m_{K^*}\frac{\partial}{\partial u} g^{(a)}_\bot (u) +\frac{f^T_{K^*}m^2_{K^*}}{2y}C_T(u) \right]_{u=1-\frac{s}{2y}} \\ L_2(y)&=& \frac{1}{4F} e^{2\bar\Lambda_B/T} \int^{s_0}_0 ds e^{-s/T} \left[\frac{f^T_{K^*}m^2_{K^*}}{y^2}(\frac{1}{2}C_T(u)+B_T(u)+\frac{1}{2}\frac{\partial}{\partial u}h^{(s)}_\|(u))\right. \nonumber\\ & &\left.+f_{K^*}m^2_{K^*}(\frac{m_{K^*}}{2y^3}C(u)-\frac{m_s+m_d}{2y^2m_{K^*}} \frac{\partial}{\partial u}h^{(s)}_\|(u))\right]_{u=1-\frac{s}{2y}} \\ L_3(y)&=& \frac{1}{4F} e^{2\bar\Lambda_B/T} \int^{s_0}_0 ds e^{-s/T} \left[-\frac{1}{4y} (f_{K^*}-f^T_{K^*}\frac{m_s+m_d}{m_{K^*}}) m_{K^*} (\frac{\partial}{\partial u} g^{(a)}_\bot(u) )\right. \nonumber\\ & &\left.+f^T_{K^*} (\phi_\bot(u)-\frac{m^2_{K^*}}{16y^2} \frac{\partial ^2}{\partial u^2}A_T(u))\right]_{u=1-\frac{s}{2y}} \\ L_4(y)&=& \frac{1}{4F} e^{2\bar\Lambda_B/T} \int^{s_0}_0 ds e^{-s/T} \frac{1}{y} \nonumber\\ & &\left[f_{K^*} m_{K^*} ( \phi_{\|}(u)-g^{(v)}_\bot(u) -\frac{1}{4} \frac{\partial}{\partial u} g^{(a)}_\bot (u)-\frac{m^2_{K^*}}{16y^2} \frac{\partial ^2}{\partial u^2}A(u))\right. \nonumber\\ & &\left.+\frac{f^T_{K^*}m^2_{K^*}}{y}B_T(u) +\frac{1}{4}f^T_{K^*}(m_s+m_d)\frac{\partial}{\partial u}g^{(a)}_\bot (u)\right]_{u=1-\frac{s}{2y}}\end{aligned}$$ For the penguin matrix elements, we begin with considering the following correlations $$\begin{aligned} P_T^\mu(p,q)&=&i\int d^4x e^{iq\cdot x}\langle K(p)|T\{\bar s(x)\sigma^{\mu\nu}p_\nu(1+\gamma^5)b(x), \bar b(0)i\gamma^5d(0)\}|0\rangle \\ V_T^\mu(p,q)&=&i\int d^4x e^{-ip_B\cdot x}\langle K^*(p,\epsilon^*)|T\{\bar s(0)\sigma^{\mu\nu}p_\nu(1+\gamma^5)b(0), \bar b(x)i\gamma^5d(x)\}|0\rangle\end{aligned}$$ where $p_B=p+q$. Making the procedures similar to Refs.[@bpi; @brho], one can easily derive the explicit expressions for $B^{\prime}(y)$ and $L^{\prime}_i(y)$, which are exactly the same as $B(y)$ and $L_i(y)$ in (3.2) and (3.7-3.10) at the leading order of $1/m_Q$ expansion. From (2.15) to (2.23), we obtain the following four interesting relations among the semileptonic type and penguin type form factors. $$\begin{aligned} f_T(q^2) & = & \frac{m_B+m_K}{2m_B}(f_+(q^2)-f_-(q^2)) \\ T_1(q^2)&=&\frac{m^2_B-m^2_{K^*}+q^2}{2m_B}\frac{V(q^2)}{m_B+m_{K^*}}+\frac{m_B+m_{K^*}}{2m_B}A_1(q^2) \\ T_2(q^2)&=&\frac{2}{m^2_B-m^2_{K^*}}[\frac{(m_B-y)(m_B+m_{K^*})}{2}A_1(q^2)+\frac{m_B(y^2-m^2_{K^*})}{m_B+m_{K^*}}V(q^2)] \\ T_3(q^2)&=&\frac{m^2_B-m^2_{K^*}}{m_Bq^2}m_{K^*}A_0(q^2)-\frac{m^2_B-m^2_{K^*}+q^2}{2m_Bq^2} [(m_B+m_{K^*})A_1(q^2)-(m_B-m_{K^*})A_2(q^2)] \nonumber\\ & &+\frac{m^2_B+3m^2_{K^*}-q^2}{2m_B(m_B+m_{K^*})}V(q^2)\end{aligned}$$ Note that these relations hold in the whole momentum transfer region. In fact, they can simply be obtained from eqs.(2.9) and (2.10) by assuming that the light meson wave functions $k(v, p)$ and $\Omega(v, p)$ are universal for different types of the currents, namely they are independent of the choice of $\Gamma$ in the currents. The second relation was actually observed in ref.[@ABS] by using QCD sum rule approach. In the heavy quark limit near zero recoil point ($q^2\rightarrow q^2_{max}$), the above relations were known as Isgur-Wise relations[@iw]. In Refs. [@pvesrb; @pab], some detailed discussions were made to explore the Isgur-Wise relations in both small and large recoil conditions by using full LCSR method, where it was shown that the Isgur-Wise relations are satisfied very well at $q^2\rightarrow 0$ (large recoil) and hold with about $80\%$ accuracy at large $q^2$. In Ref.[@cfss], conclusions were made that the Isgur-Wise relations are valid up to $70\%$ in the whole $q^2$ region by applying for the three point QCD sum rules method. In the quark model, the authors of Refs.[@stech; @soares] concluded that Isgur-Wise relations also hold at large recoil. In here we have further confirmed the existed analyzes in the literature and arrived at a general conclusion that the relations (3.13-3.16) hold for whole allowed region of momentum transfer $q^2$ at the leading order of $1/m_Q$ expansion, which has nothing to do with the energy of the final light meson. This may provide a more general proof in support of the hypothesis made in the early time by the authors in Ref. [@BD]. As a consequence, it becomes remarkable that one can directly read off the penguin form factors $f_T$ and $T_i(i=1,2,3)$ from the relations in eqs. (3.13-3.16) without the tedious calculations from LCSR to Borel transformation. In Ref.[@wwz], we have shown that when applying the HQEFT to heavy to light semileptonic decays, heavy quark expansion and heavy quark symmetry enable us to relate various decay channels, consequently, the theoretical analysis is much simplified and the number of independent functions is greatly abated though the number of independent functions in a single decay channel does not decrease. Here with the four exact relations, the number of independent variables among the form factors $f_{\pm}$ and $f_T$ is straightforwardly reduced to two. The number of independent variables in the rare decay $B\to K^*l\bar l$ is then reduced from seven form factors to four form factors. One sees that for the $B$ meson rare decays the heavy quark expansion exhibits its more powerful advantages, the number of independent functions is found to be largely reduced even in a single decay channel. Numerical analysis and results for form factors {#result} =============================================== The light cone wave functions play an important role for a precise calculation of form factors. They have been studied by several groups. We shall use the results given in [@ar; @arsco; @vvar; @vi] for $K$ and the ones in [@pvesrb] for $K^*$ meson in our following analyses. The asymptotic form and the scale dependence of them are taken from perturbative QCD calculations given in [@va; @bf]. We shall first present all two-particle light cone amplitudes of Kaon appearing in (3.1) and (3.2) $$\begin{aligned} \phi_K(u,\mu)&=&6u(1-u)\left[1+a_1(\mu)(3(2u-1))+a_2(\mu)(\frac{15}{2}(2u-1)^2-\frac{3}{2})\right. \nonumber\\ & &\left.+a_3(\mu)(\frac{35}{2}(2u-1)^3-\frac{15}{2}(2u-1))+ a_4(\mu)\frac{15}{8}(21(2u-1)^4-14(2u-1)^2+1)\right] \nonumber\\ \phi_p(u,\mu)&=& 1+\frac{1}{2}B_2(\mu)[3(2u-1)^2-1]+\frac{1}{8}B_4(\mu) [35 (2u-1)^4-30 (2u-1)^2+3] \nonumber\\ \phi_\sigma(u,\mu)&=& 6u(1-u)\{ 1+\frac{3}{2}C_2(\mu) [5(2u-1)^2-1]+\frac{15}{8}C_4(\mu) [21 (2u-1)^4-14(2u-1)^2+1]\} \nonumber\\ g_1(u,\mu)&=& \frac{5}{2}\delta^2(\mu) u^2 (1-u)^2+\frac{1}{2}\epsilon(\mu) \delta^2(\mu) \left[u(1-u)(2+13u (1-u)+10u^3 (\log u)(2-3u+\frac{6}{5}u^2) \right.\nonumber\\ & &\left.+10(1-u)^3 (\log(1-u))(2-3(1-u)+\frac{6}{5}(1-u)^2))\right] \nonumber\\ g_2(u,\mu)&=&\frac{10}{3} \delta^2(\mu) u(1-u)(2u-1)\end{aligned}$$ For the relevant parameters, we will take the following values in our numerical calculations $$\begin{aligned} &a_1(\mu_b)=0.15, \;\; a_2(\mu_b)=0.16, \;\; a_3(\mu_b)=0.05, \;\; a_4(\mu_b)=0.06, \nonumber\\ &B_2(\mu_b)=0.29, \;\; B_4(\mu_b)=0.58, \;\; C_2(\mu_b)=0.059, \;\; C_4(\mu_b)=0.034, \nonumber\\ &\delta^2(\mu_b)=0.17 \mbox{GeV}^2, \;\; \epsilon(\mu_b)=0.36\end{aligned}$$ where we choose $\mu_b=\sqrt{m^2_B-m^2_b}\approx 2.4$GeV which is the appropriate scale characterizing the typical virtuality of the b quark. The SU(3) flavor violation effects of Kaon has been taken into account in $\phi_K(u,\mu)$ with non-vanishing coefficients $a_1(\mu_b)$ and $a_3(\mu_b)$. The SU(3) flavor breaking effects in the higher twist amplitudes are neglected in this paper as they only have a small contribution [@pb]. For the light vector $K^*$ meson wave functions, we use the expressions given in [@pvesrb]. The light meson SU(3) flavor breaking effects are taken into account in the leading twist distributions and partially in the twist three, but ignored in twist four. Other parameters needed are listed in the following : $$\begin{aligned} &m_B=5.28 \mbox{GeV}, \;\; m_b=4.75 \mbox{GeV}, \;\; m_K=0.49 \mbox{GeV}, \;\; m_{K^*}=0.89 \mbox{GeV}, \nonumber\\ &\bar\Lambda_B=0.53 \mbox{GeV}, \;\; \bar{\Lambda}=(0.53\pm 0.08) \mbox{GeV},\;\; F=(0.30\pm 0.06)\mbox{GeV}^{3/2},\nonumber\\ &f_K=0.16 \mbox{GeV}, \;\; f_{K^{*}}=(226 \pm 28) \mbox{MeV}, \;\; f^{\bot}_{K^{*}}(\mu_b)=(175 \pm 9) \mbox{MeV} ,\nonumber\\ &\mu_K(\mu_b)=\frac{m^2_K}{m_s+m_{u,d}}\approx 2.02 \mbox{GeV}\end{aligned}$$ As for the parameters $s_0$ and $T$, according to LCSR criterion that the contributions from both the higher states and higher twist four distribution functions should not be larger than 30%. Thus we choose the region for $T$ to be $ 1\mbox{GeV}<T<3 \mbox{GeV}$. In this region the curves of $f_{\pm}$ and $f_T$ for $B$ to $K$ transitions and $A_i(i=0,1,2)$, $V$ and $T_i(i=1,2,3)$ for $B$ to $K^*$ transitions become most stable at the threshold energy $s_0=2.1-2.7 (2.4\pm 0.3)$GeV and $s_0=1.8-2.4 (2.1\pm 0.3)$GeV respectively, which may be seen explicitly from Fig.1 to Fig.10. It is known that the light cone expansion and the sum rule method may be broken down at large momentum transfer (practically as $q^2$ approaches near half of $m^2_b$)[@ar], which may be seen explicitly from Fig.11-20. Where the curves of form factors calculated from LCSR likely become unstable at large $q^2$ region. Thus for the behavior of the form factors in the whole kinematically accessible region, we may use the following parametrization for the transfer momentum dependence of the form factors $$F(q^2)=\frac{F(0)}{1-a_F q^2/m^2_B+b_F (q^2/m^2_B)^2}$$ where $F(q^2)$ can be any of the form factors $f_+$, $f_-$, $f_T$, $ A_i(i=0,1,2)$, $V$ and $T_i(i=1,2,3)$. We directly use LCSR predictions to fit the parameters. This is because $K^{\ast}$ and $K$ mesons contain a relative heavy strange quark (in comparison with the $u$ and $d$ quarks), so that they are comparatively heavy and shorten the kinematically allowed ranges of $B$ to $K^{\ast}$ and $K$ decays in comparison with the ranges for $B$ to $\pi$ decays. Therefore the sum rules are expected to yield reasonable values for most allowed region of $q^2$ in the $K^{\ast}$ and $K$ meson cases. With the above analyses, we are then able to fix the three parameters for each form factor. We plot in Fig.11-20 the form factors as functions of $q^2$ with different threshold energy $s_0=2.1, 2.4, 2.7$GeV for $B$ to $K$ transitions and $s_0=1.8, 2.1, 2.4$GeV for $B$ to $K^*$ transitions at $T=2.0$GeV. Our numerical results of form factors at $q^2=0$ are given in Table 1, where the uncertainties of the form factors arise from the uncertainties of the threshold energy $s_0$, the Borel parameter $T$ and the parameters in (4.3). In our values, the uncertainties coming from $s_0$ are about 10%, from $T$ and the parameters in (4.3) are around 15%. The total uncertainties are up to 25%. For comparison, we list in table 1 the numerical results obtained from other approaches: QCD LCSR, LEET, quark model (QM), lattice, three point sum rules (SR) and PQCD calculation. present LCSR[@abhh] LEET[@BH] QM[@MS] lattice[@UKQCD] lattice[@ABA] SR[@cfss] PQCD[@TAI] ---------- ---------------------------- ---------------------------- ---------------- --------- ----------------- --------------- ----------- ------------ $f_+(0)$ 0.454 $^{+0.053}_{-0.075}$ 0.319 $^{+0.052}_{-0.041}$ $---$ 0.36 $---$ 0.30(4) 0.25 $---$ $f_0(0)$ 0.454 $^{+0.053}_{-0.075}$ 0.319 $^{+0.052}_{-0.041}$ $---$ 0.36 $---$ 0.30(4) 0.25 $---$ $f_T(0)$ 0.447 $^{+0.046}_{-0.069}$ 0.355 $^{+0.016}_{-0.055}$ $---$ 0.35 $---$ 0.29(6) 0.14 $---$ $A_0(0)$ 0.468 $^{+0.082}_{-0.112}$ 0.471 $^{+0.127}_{-0.059}$ $---$ 0.45 0.32 $---$ 0.30 0.407 $A_1(0)$ 0.350 $^{+0.068}_{-0.089}$ 0.337 $^{+0.048}_{-0.043}$ 0.27 $\pm$0.03 0.36 0.28 $---$ 0.37 0.266 $A_2(0)$ 0.302 $^{+0.063}_{-0.080}$ 0.282 $^{+0.038}_{-0.036}$ $---$ 0.32 $---$ $---$ 0.40 0.202 $V(0)$ 0.426 $^{+0.070}_{-0.098}$ 0.457 $^{+0.091}_{-0.058}$ 0.36 $\pm$0.04 0.44 0.38 $---$ 0.47 0.355 $T_1(0)$ 0.382 $^{+0.068}_{-0.093}$ 0.379 $^{+0.058}_{-0.045}$ 0.31 $\pm$0.02 0.39 0.32 $---$ 0.38 0.315 $T_2(0)$ 0.382 $^{+0.068}_{-0.093}$ 0.379 $^{+0.058}_{-0.045}$ $---$ 0.39 0.32 $---$ 0.38 0.315 $T_3(0)$ 0.266 $^{+0.045}_{-0.063}$ 0.260 $^{+0.035}_{-0.016}$ $---$ 0.27 $---$ 1.4 0.207 On validity of relations in LEET {#leet} ================================ Recently, it was noted in Ref.[@cyopr] that by applying HQET to the initial heavy meson and meanwhile adopting LEET to describe the final light meson, more relations were obtained at the leading order of heavy quark mass and large energy expansion. Eventually, the total number of independent form factors is reduced to three near the large recoil point. While those symmetry relations were shown[@mbtf] to be broken down when radiative corrections are considered. The contributions of the second order in the ratio of the light meson mass to the large recoil energy were taken into account in Ref.[@efg]. To compare our results with those in Ref.[@cyopr] and also to see how much accuracy of the large energy effective theory would be, we first introduce the following definitions $$\begin{aligned} \zeta(m_B,y) & = & \sqrt{\frac{m_B \bar\Lambda}{\bar\Lambda_B}}\frac{1}{y}B(y) \\ \zeta_A(m_B,y) & = & \sqrt{\frac{m_B \bar\Lambda}{\bar\Lambda_B}}\frac{1}{m_B}A(y) \\ \zeta_{\parallel}(m_B,y) & = & \sqrt{\frac{m_B \bar\Lambda}{\bar\Lambda_B}}(\frac{1}{m_{K^*}} L_4(y)-\frac{y}{m_Bm_{K^*}}L_2(y)) \\ \zeta_{\perp}(m_B,y) & = & \sqrt{\frac{m_B \bar\Lambda}{\bar\Lambda_B}}\frac{L_3(y)}{y} \\ \zeta_1(m_B,y) & = & \sqrt{\frac{m_B \bar\Lambda}{\bar\Lambda_B}}\frac{L_1(y)}{y} \\ \zeta_2(m_B,y) & = & \sqrt{\frac{m_B \bar\Lambda}{\bar\Lambda_B}}\frac{y}{m_Bm_{K^*}}L_2(y)\end{aligned}$$ With these functions, the form factors in eqs.(2.15-2.23) can be expressed as $$\begin{aligned} f_{\pm}(q^2) & = & \zeta_A(m_B,y)\pm \zeta(m_B,y) \\ f_0(q^2) & \equiv & \frac{q^2}{m^2_B-m^2_{K}}f_-(q^2)+f_+(q^2) \nonumber\\ & = & (1-\frac{q^2}{m^2_B-m^2_{K}})\zeta(m_B,y)+(1+ \frac{q^2}{m^2_B-m^2_{K}})\zeta_A(m_B,y) \\ f_T(q^2) & = & (1+\frac{m_{K}}{m_B})\zeta(m_B,y) \\ A_0(q^2) & = & (1-\frac{m^2_{K^*}}{ym_B})\zeta_{\parallel}(m_B,y)+\frac{m_{K^*}}{m_B}\zeta_{\perp}+ \frac{y}{m_{K^*}}\zeta_1(m_B,y)-\frac{q^2}{ym_B}\zeta_2(m_B,y) \\ A_1(q^2) & = & \frac{2y}{m_B+m_{K^*}}(\zeta_{\perp}(m_B,y)+\zeta_1(m_B,y)) \\ A_2(q^2) & = & (1+\frac{m_{K^*}}{m_B})(\zeta_{\perp}(m_B,y)-\frac{m_{K^*}}{y}\zeta_{\parallel}(m_B,y)) \\ V(q^2) & = & (1+\frac{m_{K^*}}{m_B})\zeta_{\perp}(m_B,y) \\ T_1(q^2) & = & \zeta_{\perp}(m_B,y)+\frac{y}{m_B}\zeta_1(m_B,y) \\ T_2(q^2) & = & (1-\frac{q^2}{m^2_B-m^2_{K^*}})\zeta_{\perp}(m_B,y)+(1+ \frac{q^2}{m^2_B-m^2_{K^*}}) \frac{y}{m_B}\zeta_1(m_B,y) \\ T_3(q^2) & = & \zeta_{\perp}(m_B,y)-(1-\frac{m^2_{K^*}}{m^2_{m_B}})\frac{m_{K^*}}{y}(\zeta_{\parallel}(m_B,y)+\zeta_2(m_B,y)) -\frac{y}{m_B}\zeta_1(m_B,y)\end{aligned}$$ Comparing the above forms with the ones given by eqs.(104)-(113) in Ref.[@cyopr], it is obvious that the three functions $\zeta_A(q^2)$, $\zeta_1(q^2)$ and $\zeta_2(q^2)$ should vanish, i.e., $\zeta_A(q^2) = \zeta_1(q^2) = \zeta_2(q^2) = 0$, in order to reproduce the results in LEET. For a quantitative comparison, we plot in Fig.21 three curves $\zeta_{\parallel}(q^2)$, $\zeta_{\perp}(q^2)$ and $\zeta(q^2)$ as functions of momentum transfer $q^2$, and in Fig.22 the three ratios $\zeta_2(q^2)/\zeta_{\parallel}(q^2)$, $\zeta_1(q^2)/\zeta_{\perp}(q^2)$ and $\zeta_A(q^2)/\zeta(q^2)$ as functions of $q^2$. It is seen that the three functions $\zeta_A(q^2)$, $\zeta_1(q^2)$ and $\zeta_2(q^2)$ do have small but sizable contributions relative to the three nonzero functions $\zeta_{\parallel}(q^2)$, $\zeta_{\perp}(q^2)$ and $\zeta(q^2)$ in LEET. Numerically, they are about 20%, 10% and 10% respectively when $q^2 = 0$. For the ratios $\zeta_2(q^2)/\zeta_{\parallel}(q^2)$ and $\zeta_A(q^2)/\zeta(q^2)$, they are almost independent of the momentum transfer $q^2$. For the ratio $\zeta_1(q^2)/\zeta_{\perp}(q^2)$, it ranges from $10\%$ to $40\%$ as $q^2$ increases. To be more clear, we directly plot in Figs.23-29 the ratios between the form factors, i.e., $F_{iLEET}(q^2)/F_i(q^2)$ at center value of $s_0$ with $T=2.0$GeV. Here $F_{i LEET}(q^2)$ denote the form factors obtained from HQET/LEET relations, namely $\zeta_A(q^2) = \zeta_1(q^2) = \zeta_2(q^2) = 0$, and $F_i(q^2)$s are the form factors obtained in present paper, namely the three functions $\zeta_A(q^2)$, $\zeta_1(q^2)$ and $\zeta_2(q^2)$ are not zero and given in eq. (5.2), (5.5) and (5.6). It is seen that, except the ratio $A_{0 LEET}/A_0 \approx 0.78$, the other ratios near large recoil point ($q^2\rightarrow 0$) have the value $F_{i LEET}/F_i = 0.9\sim 1.0$. More explicitly speaking $f_{+ LEET}/f_+ \approx f_{T LEET}/f_T \approx0.9$, $A_{1 LEET}/A_1 \approx 0.92$, $T_{1 LEET} \approx T_{2 LEET}/T_2 \approx 0.96$ and $T_{3 LEET}/T_3 \approx 0.98$. So we can say the form factors $F_{i LEET}$ have the accuracy beter than $90\%$ as comparation with $F_i$ except $A_{0 LEET}$ whose accuracy is only about $78\%$. This exception can be explained as that the third term on the R.H.S. of the equation (5.10) contains the factor $y/m_{K^*}$, which undoubtedly enhances several times the contribution of $\zeta_1(q^2)$ when $q^2 = 0$. So when $B$ decays to more light final mesons, say $\rho$, the LEET relation for $A_0(q^2)$ will work worse. It is particularly noted that ratios $f_{+ LEET}/f_+$, $T_{1 LEET}/T_1$ and $T_{3 LEET}/T_3$, which take value among the area $0.85\sim 0.90$, $0.91\sim 0.95$ and $1.0\sim 1.05$ respectively in the whole $q^2$, are almost independent of $q^2$. The reason is clear since the ratios $\zeta_A(q^2)/\zeta(q^2)$ in (5.7) and $\zeta_2(q^2)/\zeta_{\parallel}(q^2)$ in (5.16) are almost independent of the momentum transfer $q^2$ and the coefficient of the wave function $\zeta_1(m_B, y)$ is associated with the factors $y/m_B$ in (5.14) and (5.16), which have a tendency to suppress the contribution of $\zeta_1(m_B, y)$ as $q^2$ increases. While the other ratios, especially the ratio $A_{0 LEET}/A_0$, get worse results for large $q^2$, say $q^2 \approx 15GeV^2$, which can be considered as far from large recoil. For instance $A_{0 LEET}$ is only about half of $A_0$ at $q^2 \approx 15GeV^2$. This means that LEET becomes not appropriate for large $q^2$ region. The reason lies in that on one hand $\zeta_1(q^2)/\zeta_{\perp}(q^2)$ counts notablly when $q^2$ is large. On the other hand, in the general form factor formulations (5.8), (5.10) and (5.15), there is a factor $q^2$ to enhance the contributions of $\zeta_A(q^2)$, $\zeta_2(q^2)$ and $\zeta_1(q^2)$ when $q^2$ increases. In addition, as can be seen from (5.9), (5.12) and (5.13), $f_{T LEET}$, $A_{2 LEET}$ and $V_{LEET}$ are same as the corresponding one obtained by HQEFT in present paper. Besides the relations (3.13-3.16), LEET leads to three additional relations [@cyopr] $$\begin{aligned} f_T(q^2) & = & (1+\frac{m_{K}}{m_B})f_+(q^2) \\ A_0(q^2) & = & (1-\frac{m^2_{K^*}}{ym_B})\frac{ym_B}{m_{K^*}(m_B+m_{K^*})}(V(q^2)-A_2(q^2)) +\frac{m_{K^*}(m_B+m_{K^*})}{2ym_B}A_1(q^2) \\ A_1(q^2) & = & \frac{2ym_B}{(m_B+m_{K^*})^2}V(q^2)\end{aligned}$$ To check this relations, we plot in Figs.30-32 the three form factors $f_T$, $A_0$ and $A_1$ obtained from the direct calculations (i.e., left-hand side (LHS) ) and from the relations (i.e., right-hand side (RHS)). The numerical comparison of the LHS and RHS of the above equations shows that near the large recoil point ($q^2\rightarrow 0$) the relations (5.17) and (5.19) both hold within 90% accuracy, but the relations (5.18) only hold about 80% accuracy. In particular, the deviations of LHS and RHS in relations (5.17) and (5.19) do not exceed 20% through whole $q^2$, while that of relation (5.18) at best hold with 40%. Results for branching ratios {#branch ratios} ============================ The relevant branching ratios are able to be calculated with the form factors given above. The relevant decay width formulae have the following general forms [@abhh] $$\begin{aligned} \frac{d\Gamma}{d{{\hat{s}}}}&=& \frac{G^2_F \alpha ^2 m^5_B}{2^{11} \pi ^5} |V^*_{ts}V_{tb}|^2 {{\hat{u}}} ({{\hat{s}}})\left\{(|A'({{\hat{s}}})|^2+|C'({{\hat{s}}})|^2) (\lambda ({{\hat{s}}})-\frac{{{\hat{u}}}({{\hat{s}}}) ^2}{3}) \right.\nonumber\\ & & \left.+|C'({{\hat{s}}})|^2 4{{\hat{m}_l}}^2(2+2{{\hat{m}_K}}^2-{{\hat{s}}})+Re(C'({{\hat{s}}})D'({{\hat{s}}})^*)8{{\hat{m}_l}} ^2(1-{{\hat{m}_K}}^2)+|D'({{\hat{s}}})|^24{{\hat{m}_l}}^2{{\hat{s}}}\right\}\end{aligned}$$ for $B\rightarrow K \bar{l}l$, and $$\begin{aligned} \frac{d\Gamma}{d{{\hat{s}}}}&=&\frac{G^2_F \alpha ^2 m^5_B}{2^{11} \pi ^5} |V^*_{ts}V_{tb}|^2 {{\hat{u}}} ({{\hat{s}}})\left\{\frac{|A({{\hat{s}}})|^2}{3}{{\hat{s}}}\lambda ({{\hat{s}}}) (1+2\frac{{{\hat{m}_l}}^2}{{{\hat{s}}}})+|E({{\hat{s}}})|^2{{\hat{s}}}\frac{{{\hat{u}}}({{\hat{s}}})^2}{3} \right.\nonumber \\ & &+\frac{1}{4{{\hat{m}_{K^*}}}^2}\left[|B({{\hat{s}}})|^2 (\lambda ({{\hat{s}}})-\frac{{{\hat{u}}}({{\hat{s}}})^2}{3}+ 8{{\hat{m}_{K^*}}}^2 ({{\hat{s}}}+2{{\hat{m}_l}}^2)) +|F({{\hat{s}}})|^2(\lambda ({{\hat{s}}}) -\frac{{{\hat{u}}}({{\hat{s}}})^2}{3}+8{{\hat{m}_{K^*}}}^2 ({{\hat{s}}}-4{{\hat{m}_l}}^2))\right] \nonumber \\ & &+\frac{\lambda ({{\hat{s}}})}{4{{\hat{m}_{K^*}}}^2}\left[|C({{\hat{s}}})|^2(\lambda ({{\hat{s}}})-\frac{{{\hat{u}}}({{\hat{s}}})^2}{3}) +|G({{\hat{s}}})|^2(\lambda ({{\hat{s}}})-\frac{{{\hat{u}}}({{\hat{s}}})^2}{3}+4{{\hat{m}_l}}^2(2+2{{\hat{m}_{K^*}}}^2-{{\hat{s}}}))\right] \nonumber \\ & &-\frac{1}{2{{\hat{m}_{K^*}}}^2}\left[Re(B({{\hat{s}}})C({{\hat{s}}})^*) (\lambda ({{\hat{s}}})-\frac{{{\hat{u}}}({{\hat{s}}})^2}{3})(1-{{\hat{m}_{K^*}}}^2-{{\hat{s}}})\right.\nonumber \\ & &\left.+Re(F({{\hat{s}}})G({{\hat{s}}})^*) ((\lambda ({{\hat{s}}})-\frac{{{\hat{u}}}({{\hat{s}}})^2}{3}) (1-{{\hat{m}_{K^*}}}^2-{{\hat{s}}})+4{{\hat{m}_l}}^2\lambda ({{\hat{s}}}))\right] \nonumber \\ & &\left.-2\frac{{{\hat{m}_l}}^2}{{{\hat{m}_{K^*}}}^2}\lambda ({{\hat{s}}})\left[Re(F({{\hat{s}}})H({{\hat{s}}})^*)-Re(G({{\hat{s}}})H({{\hat{s}}})^*)(1-{{\hat{m}_{K^*}}}^2)\right] +\frac{{{\hat{m}_l}}^2}{{{\hat{m}_{K^*}}}^2}{{\hat{s}}}\lambda ({{\hat{s}}})|H({{\hat{s}}})|^2 \right\}\end{aligned}$$ for $B\rightarrow K^{\ast} \bar{l}l$, as well as $$\Gamma = \frac{G^2_F \alpha m^2_b m^3_B}{32 \pi ^4} |V^*_{ts}V_{tb}|^2 |C^{eff}_7|^2 |T_1(0)|^2 (1-{{\hat{m}_{K^*}}})^3$$ for $B\rightarrow K^{\ast} \gamma$. The functions appearing in the above decay width formulae are defined as [@abhh; @bm] $$\begin{aligned} A'({{\hat{s}}}) &=& C^{eff}_9({{\hat{s}}})f_+({{\hat{s}}})+\frac{2{{\hat{m}_b}}}{1+{{\hat{m}_K}}}C^{eff}_7f_T({{\hat{s}}}),\\ B'({{\hat{s}}}) &=& C^{eff}_9({{\hat{s}}})f_-({{\hat{s}}})-\frac{2{{\hat{m}_b}}}{{{\hat{s}}}}(1-{{\hat{m}_K}})C^{eff}_7f_T({{\hat{s}}}),\\ C'({{\hat{s}}}) &=& C_{10}f_+({{\hat{s}}}),\\ D'({{\hat{s}}}) &=& C_{10}f_-({{\hat{s}}}),\\ A({{\hat{s}}}) &=& \frac{2}{1+{{\hat{m}_{K^*}}}}C^{eff}_9({{\hat{s}}})V({{\hat{s}}})+\frac{4{{\hat{m}_b}}}{{{\hat{s}}}}C^{eff}_7T_1({{\hat{s}}}),\\ B({{\hat{s}}}) &=& (1+{{\hat{m}_{K^*}}})[C^{eff}_9({{\hat{s}}})A_1({{\hat{s}}})+\frac{2{{\hat{m}_b}}}{{{\hat{s}}}}(1-{{\hat{m}_{K^*}}})C^{eff}_7T_2({{\hat{s}}})],\\ C({{\hat{s}}}) &=& \frac{1}{1-{{\hat{m}_{K^*}}}^2}[(1-{{\hat{m}_{K^*}}})C^{eff}_9({{\hat{s}}})A_2({{\hat{s}}})+2{{\hat{m}_b}}C^{eff}_7(T_3({{\hat{s}}})+\frac{1-{{\hat{m}_{K^*}}}^2}{{{\hat{s}}}}T_2({{\hat{s}}}))],\\ D({{\hat{s}}}) &=& \frac{1}{{{\hat{s}}}}[C^{eff}_9({{\hat{s}}})((1+{{\hat{m}_{K^*}}})A_1({{\hat{s}}})-(1-{{\hat{m}_{K^*}}})A_2({{\hat{s}}})-2{{\hat{m}_{K^*}}}A_0({{\hat{s}}}))-2{{\hat{m}_b}}C^{eff}_7T_3({{\hat{s}}})],\\ E({{\hat{s}}}) &=& \frac{2}{1+{{\hat{m}_{K^*}}}}C_{10}V({{\hat{s}}}),\\ F({{\hat{s}}}) &=& (1+{{\hat{m}_{K^*}}})C_{10}A_1({{\hat{s}}}),\\ G({{\hat{s}}}) &=& \frac{1}{1+{{\hat{m}_{K^*}}}}C_{10}A_2({{\hat{s}}}),\\ H({{\hat{s}}}) &=& \frac{1}{{{\hat{s}}}}C_{10}({{\hat{s}}})[(1+{{\hat{m}_{K^*}}})A_1({{\hat{s}}})-(1-{{\hat{m}_{K^*}}})A_2({{\hat{s}}})-2{{\hat{m}_{K^*}}}A_0({{\hat{s}}})],\\ \lambda ({{\hat{s}}}) &=& 1+{{\hat{m}_{K,K^*}}}^4+{{\hat{s}}}^2-2{{\hat{s}}}-2{{\hat{m}_{K,K^*}}}^2(1+{{\hat{s}}}),\\ {{\hat{u}}}({{\hat{s}}}) &=& \sqrt{\lambda ({{\hat{s}}}) (1-4\frac{{{\hat{m}_l}}^2}{{{\hat{s}}}})},\\ C^{eff}_9({{\hat{s}}}) &=& C_9 +g({{\hat{m}_c}},{{\hat{s}}})(3C_1+C_2+3C_3+C_4+3C_5+C_6)-\frac{1}{2}g(1,{{\hat{s}}})(4C_3+4C_4+3C_5+C_6)\nonumber\\ &-& \frac{1}{2}g(0,{{\hat{s}}})(C_3+3C_4)+\frac{2}{9}(3C_3+C_4+3C_5+C_6),\\ g(0,{{\hat{s}}}) &=& \frac{8}{27}-\frac{8}{9}\ln \frac{m_b}{\mu}-\frac{4}{9}\ln {{\hat{s}}}+\frac{4}{9}i\pi ,\\ g(z,{{\hat{s}}}) &=& \frac{8}{27}-\frac{8}{9}\ln \frac{m_b}{\mu}-\frac{8}{9}\ln z+\frac{4}{9} x \nonumber\\ &-& \frac{2}{9}(2+x)\sqrt{|1-x|}\left\{ \begin{array}{ll} (\ln |\frac{\sqrt{1-x}+1}{\sqrt{1-x}-1}|-i\pi),\hspace{0.7cm} for \hspace{0.2cm} x\equiv \frac{4z^2}{{{\hat{s}}}} <1 \\ 2\arctan \frac{1}{\sqrt{x-1}},\hspace{1.4cm} for \hspace{0.2cm} x\equiv \frac{4z^2}{{{\hat{s}}}} >1. \end{array} \right.\end{aligned}$$ with ${{\hat{s}}}=\frac{q^2}{m_B^2}$,${{\hat{m}_{K,K^*}}}=\frac{m_{K,K^*}}{m_B}$,${{\hat{m}_{b,c}}}=\frac{m_{b,c}}{m_B}$ and ${{\hat{m}_l}}=\frac{m_l}{m_B}$. For the Wilson coefficients $C_i$, we take the results calculated in the naive dimensional regularization (NRD) scheme [@bmmp] and their values are listed in Table 2. $C_1$ $C_2$ $C_3$ $C_4$ $C_5$ $C_6$ $C_7$ $C^{eff}_7$ $C_9$ $C_{10}$ -------- ------- ------- -------- ------- -------- -------- ------------- ------- ---------- -0.248 1.107 0.011 -0.026 0.007 -0.031 -0.342 -0.313 4.344 -4.669 It is seen that once the decay rates and the form factors are precisely determined, one is able to extract the CKM matrix elements. Here we may use the current reasonable values of CKM matrix elements extracted from other processes and unitarity of the CKM matrix to predict the branching ratios for the B meson rare decays. When taking the CKM matrix element $|V_{tb}V_{ts}|=0.0385$, we present our results in Table 3. For comparison, we also list the results given in [@abhh] and the ones from experimental measurements. It is seen that within the uncertainties, our prediction is consistent with the experimental results. -------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- present values values in [@abhh] experiment -------------------------- ------------------------------------------ ------------------------------------------------------------------- -------------------------------------------------------------------------------------- $B\to K e\bar e$ (0.84$^{+0.10}_{-0.24}$)$\times 10^{-6}$ (0.57$^{+0.16}_{-0.10}$)$\times 10^{-6}$ (0.75$^{+0.25}_{-0.21}\pm 0.09$)$\times 10^{-6}$[@kek] $B\to K \mu\bar \mu$ $B\to K \tau\bar \tau$ (1.73$^{+0.21}_{-0.47}$)$\times 10^{-7}$ (1.3$^{+0.3}_{-0.1}$)$\times 10^{-7}$ $------$ (4.55$^{+0.72}_{-0.68}\pm 0.34$)$ \times 10^{-5 \hspace{0.2cm}\triangle \triangle}$[@cleo] $B\to K^* \gamma$ (5.47$^{+2.14}_{-2.34}$)$\times 10^{-5}$ (5.37$^{+1.77}_{-1.20}$)$\times 10^{-5 \hspace{0.2cm} \triangle}$ (4.96$\pm$ 0.67$\pm$0.45)$\times 10^{-5 \hspace{0.2cm} \triangle \triangle}$[@belle] (5.7 $\pm$ 3.3)$\times 10^{-5}$[@group] $B\to K^* e\bar e$ (1.86$^{+0.52}_{-0.73}$)$\times 10^{-6}$ (2.3$^{+0.7}_{-0.4}$)$\times 10^{-6}$ (2.08$^{+1.23+0.35}_{-1.00-0.37}$)$\times 10^{-6}$[@kek] $B\to K^* \mu\bar \mu$ (1.78$^{+0.49}_{-0.70}$)$\times 10^{-6}$ (1.9$^{+0.5}_{-0.3}$)$\times 10^{-6}$ $------$ $B\to K^* \tau\bar \tau$ (1.68$^{+0.18}_{-0.55}$)$\times 10^{-7}$ (1.9$^{+0.1}_{-0.2}$)$\times 10^{-7}$ $------$ -------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- Summary ======= In summary, we have studied the $B$ meson rare decays in the framework of HQEFT. The semileptonic type and penguin type form factors have been derived by using LCSR method in HQEFT. It has been seen that the heavy quark expansion brings a much simplification to $B$ meson rare decays. Isgur-Wise relations among the semileptonic type and penguin type form factors have been proved to hold for the whole momentum transfer region at the leading order $1/m_Q$ expansion. As a consequence, all the form factors can be neatly characterized by a set of wave functions ($A$, $B$ and $L_i (i=1,2,3,4)$) at the leading order of $1/m_Q$ expansion. Furthermore from our quantitative discussion, it is obvious that LEET is a valid method for heavy-to-light transition. LEET relations hold within 80% accuracy at large recoil point on the whole, and most of them even hold better than 90% accuracy. Moreover, our numerical prediction is consistent with the experimental results. 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Groom et al, EPJ C [**15**]{} 1 (2000) (460,150)(0,0) (0,40)[ =7.0cm =4.2cm ]{} (100,14) (0,0) (300,0)(0,0) (160,66)[ =7.0cm =4.2cm ]{} (275,40)[(0,0)[Fig.2 Same as Fig.1 but for $f_-$.]{}]{} (460,150)(0,0) (0,40)[ =7.0cm =4.2cm ]{} (100,14)[(0,0)[Fig.3 Same as Fig.1 but for $f_T$.]{}]{} (300,0)(0,0) (160,66)[ =7.0cm =4.2cm ]{} (275,40) (0,0) (460,150)(0,0) (0,40)[ =7.0cm =4.2cm ]{} (100,14)[(0,0)[Fig.5 Same as Fig.4 but for $A_1$.]{}]{} (300,0)(0,0) (160,66)[ =7.0cm =4.2cm ]{} (275,40)[(0,0)[Fig.6 Same as Fig.4 but for $A_2$.]{}]{} (460,150)(0,0) (0,40)[ =7.0cm =4.2cm ]{} (100,14)[(0,0)[Fig.7 Same as Fig.4 but for $V$.]{}]{} (300,0)(0,0) (160,66)[ =7.0cm =4.2cm ]{} (275,40)[(0,0)[Fig.8 Same as Fig.4 but for $T_1$.]{}]{} (460,150)(0,0) (0,40)[ =7.0cm =4.2cm ]{} (100,14)[(0,0)[Fig.9 Same as Fig.4 but for $T_2$.]{}]{} (300,0)(0,0) (160,66)[ =7.0cm =4.2cm ]{} (275,40)[(0,0)[Fig.10 Same as Fig.4 but for $T_3$.]{}]{} (460,150)(0,0) (0,40)[ =7.0cm =4.2cm ]{} (100,14) (0,0) (300,0)(0,0) (160,66)[ =7.0cm =4.2cm ]{} (275,40)[(0,0)[Fig.12 Same as Fig.11 but for $f_-$.]{}]{} (460,150)(0,0) (0,40)[ =7.0cm =4.2cm ]{} (100,14)[(0,0)[Fig.13 Same as Fig.11 but for $f_T$.]{}]{} (300,0)(0,0) (160,66)[ =7.0cm =4.2cm ]{} (275,40) (0,0) (460,150)(0,0) (0,40)[ =7.0cm =4.2cm ]{} (100,14)[(0,0)[Fig.15 Same as Fig.14 but for $A_1$.]{}]{} (300,0)(0,0) (160,66)[ =7.0cm =4.2cm ]{} (275,40)[(0,0)[Fig.16 Same as Fig.14 but for $A_2$.]{}]{} (460,150)(0,0) (0,40)[ =7.0cm =4.2cm ]{} (100,14)[(0,0)[Fig.17 Same as Fig.14 but for $V$.]{}]{} (300,0)(0,0) (160,66)[ =7.0cm =4.2cm ]{} (275,40)[(0,0)[Fig.18 Same as Fig.14 but for $T_1$.]{}]{} (460,150)(0,0) (0,40)[ =7.0cm =4.2cm ]{} (100,14)[(0,0)[Fig.19 Same as Fig.14 but for $T_2$.]{}]{} (300,0)(0,0) (160,66)[ =7.0cm =4.2cm ]{} (275,40)[(0,0)[Fig.20 Same as Fig.14 but for $T_3$.]{}]{} (460,150)(0,0) (0,40)[ =7.0cm =4.2cm ]{} (100,14) (0,0) (300,0)(0,0) (160,66)[ =7.0cm =4.2cm ]{} (275,40) (0,0) (460,150)(0,0) (0,40)[ =7.0cm =4.2cm ]{} (100,14) (0,0) (300,0)(0,0) (160,66)[ =7.0cm =4.2cm ]{} (275,40) (0,0) (460,150)(0,0) (0,40)[ =7.0cm =4.2cm ]{} (100,14) (0,0) (300,0)(0,0) (160,66)[ =7.0cm =4.2cm ]{} (275,40) (0,0) (460,150)(0,0) (0,40)[ =7.0cm =4.2cm ]{} (100,14) (0,0) (300,0)(0,0) (160,66)[ =7.0cm =4.2cm ]{} (275,40) (0,0) (460,150)(0,0) (0,40)[ =7.0cm =4.2cm ]{} (100,14) (0,0) (300,0)(0,0) (160,66)[ =7.0cm =4.2cm ]{} (275,40) (0,0) (460,150)(0,0) (0,40)[ =7.0cm =4.2cm ]{} (100,14) (0,0) (300,0)(0,0) (160,66)[ =7.0cm =4.2cm ]{} (275,40) (0,0)
--- abstract: | We analyze the collider sensitivity for new colored resonances in $t \bar{t}$, $b \bar{b}$, and $jj$ final states. While searches in the single production channel are model-dependent, the pair production rate is model independent and the existing $(JJ)(JJ)$ and $4t$ searches impose strong constraints on the relevant branching fractions, where $J = j$ or $b$. We point out the missing, complementary searches in the mixed decay modes, $t\bar{t}(jj)$, $t\bar{t}(b\bar{b})$, and $(b\bar{b})(jj)$. We propose analysis strategies for the $t\bar{t}(jj)$ and $t\bar{t}(b\bar{b})$ decays and find their sensivity surpasses that of existing searches when the decay widths to tops and light jets are comparable. If no other decays are present, collective lower limits on the resonance mass can be set at 1.5 TeV using 37 fb$^{-1}$ of 13 TeV data. author: - Malte Buschmann - Felix Yu bibliography: - 'referencelist.bib' title: Collider constraints and new tests of color octet vectors --- Introduction {#sec:Introduction} ============ Searches for new particles and new forces beyond the Standard Model (BSM) are a critical endeavor for the ATLAS and CMS experiments at the Large Hadron Collider (LHC), but such results have thus far come up null. The largest rates come from colored particle pair production, where leading order cross sections can be calculated knowing only the color charge and mass [@Han:2010rf]. Even so, the colored particle decay patterns and corresponding collider signatures are highly model dependent. In models with an extended color gauge symmetry [@Hill:1991at; @Chivukula:1996yr; @Frampton:1987dn; @Martynov:2009en; @Frampton:2009rk; @Simmons:2011aa], massive color octet vectors, such as colorons [@Bai:2010dj], are heavy cousins of the familiar gluon and can decay universally to Standard Model (SM) quark–anti-quark pairs. Axigluons [@Hall:1985wz; @Frampton:1987dn; @Bagger:1987fz] are similarly motivated by ascribing chiral projection operators to the parent gauge groups, which necessitates new fermions transforming non-trivially under the extended gauge group to cancel anomalies. Similar phenomenology occurs in models of universal extra dimensions, where the massive color octet vectors arise as Kaluza-Klein (KK) excitations of the gluon [@Appelquist:2000nn; @Rizzo:2001sd; @Macesanu:2002db]. In Randall-Sundrum models [@Randall:1999ee; @Randall:1999vf] with SM fields propagating in the bulk [@Davoudiasl:1999tf; @Grossman:1999ra; @Pomarol:1999ad; @Chang:1999nh; @Huber:2000ie], the KK gluons exhibit flavor-dependent couplings to quark pairs, and preferentially decay to the heavy flavor quarks as a result of the localization of the bulk fermion wavefunction profile [@Dicus:2000hm; @Davoudiasl:2000wi; @Lillie:2007yh; @Casagrande:2008hr; @Agashe:2013kxa]. At the LHC, CMS and ATLAS have searched for color octet vector resonances in the paired dijet channel [@Chatrchyan:2013izb; @Khachatryan:2014lpa; @CMS:2016pkl; @ATLAS:2012ds; @ATLAS:2016sfd; @ATLAS:2017gsy], most recently constraining colorons with a 100% branching fraction to light jets to be heavier than 1.5 TeV [@ATLAS:2017gsy]. Searches for four tops [@CMS:2013xma; @Sirunyan:2017tep; @ATLAS:2012hpa; @Aad:2015kqa; @TheATLAScollaboration:2016gxs; @ATLAS:2016gqb; @ATLAS:2016btu] also constrain the possible $t\bar{t}$ decays of pair-produced resonances. Searches for TeV-scale dijet resonances [@CMS:2017xrr; @Aaboud:2017yvp] and $t\bar{t}$ resonances [@Chatrchyan:2012cx; @Chatrchyan:2012yca; @Aad:2012dpa; @Aad:2012ans; @Aad:2013nca; @Aad:2015fna; @TheATLAScollaboration:2016wfb] offer complementary probes compared to the pair-production searches, since such searches scale with the individual production coupling [@Dobrescu:2013coa]. Given the possibility that the color octet vector has flavor dependent branching fractions to quark pairs, and because color octet vectors have a model independent pair production rate, the mixed decay signature of a paired ditop and dijet resonance is strongly motivated. This channel is complementary to the existing $(JJ)(JJ)$ and $4t$ searches and even offers superior sensitivity when the branching fractions of the resonance to dijets and ditops are comparable. In [Sec. \[sec:theory\]]{}, we review the theory motivation and collider phenomenology of massive color octet vectors. In [Sec. \[sec:collider\]]{}, we analyze the prospects for discovering color octet vector resonances in the $t\bar{t}(jj)$ and $t\bar{t}(b\bar{b})$ mixed decay modes at the LHC and compare with the current constraints. We conclude in [Sec. \[sec:conclusion\]]{}. Theory background {#sec:theory} ================= Massive color octet vector resonances arise in various different beyond the Standard Model extensions, such as models with extended color gauge groups or models with extra dimensions. In the unbroken phase of electroweak symmetry, the general interaction Lagrangian between quarks and a massive color octet vector $X_\mu$ is $$\mathcal{L} \supset g_s \left( \bar{Q}_L \gamma_\mu \lambda T^a X^{\mu \, a} Q_L + \bar{u}_R \gamma_\mu \kappa T^a X^{\mu \, a} u_R + \bar{d}_R \gamma_\mu T^a \eta X^{\mu \, a} d_R \right) \ , \label{eqn:Lagrangian}$$ where $g_S$ is the strong coupling constant, $T^a$ are the $SU(3)$ generators, and $\lambda$, $\kappa$, and $\eta$ are the flavor-dependent couplings in the quark gauge basis. While these matrices must be symmetric by $CPT$, nonzero off-diagonal entries are possible in principle and correspond to tree-level flavor violation. Since these processes are strongly constrained by low-energy precision flavor measurements, such as meson mixing measurements and $b \to X_s \gamma$ transitions [@Bona:2007vi; @Chivukula:2013kw], the simplest ansatz is to adopt a flavor-universal coupling structure, evading the most stringest flavor bounds. In such a scenario, however, the branching ratio for $X$ to light quarks vs. tops is fixed, and there is no model freedom in the complementarity between searching in dijet vs. ditop resonances. Hence, we will focus on the scenario where the $\lambda$, $\kappa$, and $\eta$ matrices are diagonal but not universal in the quark gauge basis. Moving to the quark mass basis will then induce off-diagonal entries proportional to Cabibbo-Kobayashi-Maskawa (CKM) mixing. We will first briefly review models of massive color octet vectors and then discuss the collider and flavor physics phenomenology of the general Lagrangian, [Eq. (\[eqn:Lagrangian\])]{}. Color octet vectors from extended color gauge groups {#subsec:extendedcolor} ---------------------------------------------------- In models with an extended color symmetry, such as a $SU(3)_1 \times SU(3)_2$ gauge group, the two parent gauge groups are typically broken to the diagonal subgroup by the vacuum expectation value (vev) of a bifundamental, complex scalar field $\Sigma$ [@Chivukula:1996yr; @Bai:2010dj]. The diagonal subgroup is then identified with the SM $SU(3)_c$ gauge group, which imposes the requirement $$\frac{1}{g_s^2} = \frac{1}{h_1^2} + \frac{1}{h_2^2} \ ,$$ where $h_1$ and $h_2$ are the gauge couplings of $SU(3)_1$ and $SU(3)_2$, respectively. The Goldstone modes of the complex scalar field become the longitudinal degrees of freedom for the heavy color octet vector, leaving one real and one pseudoreal color singlet scalar and one real color octet scalar as the dynamical scalar fields below $\langle \Sigma \rangle$. Explicitly, the covariant derivative for $\Sigma$ is $$D^\mu \Sigma = (\partial^\mu - i h_1G_1^{\mu a}T^a + i h_2G_2^{\mu a}T^a) \Sigma \ ,$$ and after Higgsing $SU(3)_1 \times SU(3)_2 \to SU(3)_c$, we get the SM gluon and coloron fields, $$\begin{aligned} g^\mu = \cos\theta \, G_1^\mu + \sin\theta \, G_2^\mu \ , \notag \\ X^\mu = \sin\theta \, G_1^\mu - \cos\theta \, G_2^\mu \ ,\end{aligned}$$ respectively, where $\theta = \tan^{-1} (h_1 / h_2)$ is the mixing angle. In this setup, different possibilities for the couplings in [Eq. (\[eqn:Lagrangian\])]{} originate by considering various charge assignments of the quarks in the parent $SU(3)_1 \times SU(3)_2$ gauge symmetry. The main feature for universally coupled models is that all flavor representations are assigned identically to the same gauge representation, which ensures that the gauge symmetry commutes with the global SM quark flavor symmetry. In the coloron model [@Chivukula:1996yr; @Bai:2010dj], all SM quarks transform as $\square$ under one $SU(3)$ gauge group and singlets under the other. Hence, the coloron has flavor universal, purely vector couplings to the SM quarks as $$g_s \tan \theta \bar{q} \gamma^\mu T^a X_\mu^a q \ . \label{eqn:coloroncoupling}$$ An alternative prescription is the chiral color model [@Hall:1985wz; @Frampton:1987dn; @Frampton:1987ut], where left-handed (LH) and right-handed (RH) quark fields are charged under different $SU(3)$ gauge groups. This construction generally requires new fermions to cancel anomalies, notably $SU(3)_1^2 \times U(1)_Y$ and $SU(3)_2^2 \times U(1)_Y$, but this new matter content can be massive and unobservable at colliders. As a result, if the LH quarks transform as $(\square, \mathbf{1})$ and the RH quarks transform as $(\mathbf{1}, \square)$ under $SU(3)_1 \times SU(3)_2$, for example, the resulting massive color vector interaction with quarks is $$g_s \bar{q} \gamma^\mu T^a X_\mu^a (\tan \theta P_L - \cot \theta P_R) q \ , \label{eqn:axigluoncoupling}$$ where $X_\mu^a$ is commonly referred as an axigluon in the literature. To motivate a non-universal yet diagonal coupling structure in [Eq. (\[eqn:Lagrangian\])]{}, we can straightforwardly assign different quark flavors to different gauge representations under $SU(3)_1 \times SU(3)_2$. For example, the topcolor model charges the third generation quarks differently than the first two generations, with $Q_L^{1, 2} \sim (\square, \mathbf{1})$, $Q_L^3 \sim (\mathbf{1}, \square)$, $u_R^{1, 2} \sim (\square, \mathbf{1})$, $u_R^3 \sim (\mathbf{1}, \square)$, $d_R^{1, 2, 3} \sim (\square, \mathbf{1})$ [@Hill:1991at]. This assignment also requires additional matter to cancel anomalies, for which a minimal solution involves two electroweak singlet quarks transforming as $(\mathbf{1}, \square)$ and $(\square, \mathbf{1})$, each with hypercharge $-2/3$. The corresponding massive color octet vector does not have flavor-changing couplings in the quark gauge basis, but instead features distinct couplings to light and heavy generation quarks: $$g_s \cot \theta (\bar{t} \gamma^\mu T^a X_\mu t + \bar{b}_L \gamma^\mu T^a X_\mu b_L) + g_s \tan \theta (\bar{b}_R \gamma^\mu T^a X_\mu b_R + \sum\limits_{i = 1 \dots 4} \bar{q}_i \gamma^\mu T^a X_\mu q_i) \ ,$$ with $\tan \theta = h_1 / h_2$ as before. Although topcolor models are generally motivated by composite Higgs scenarios triggered by top quark condensation [@Hill:1991at], we will only focus on the motivated possibility that $X_\mu$ has non-universal couplings. Color octet vectors from extra dimensions {#subsec:extradimensions} ----------------------------------------- Models with extra spacetime dimensions provide an alternative framework for realizing massive color octet vector resonances [@Appelquist:2000nn; @Rizzo:2001sd; @Macesanu:2002db]. In such models, the SM fields are the lowest-lying states of a Kaluza-Klein tower, whose masses and dynamics result from solving the five-dimensional equations of motion [@Davoudiasl:1999tf; @Grossman:1999ra; @Pomarol:1999ad; @Chang:1999nh; @Huber:2000ie]. In minimal universal extra dimensions [@Datta:2010us], the level-2 KK gluon obtains a coupling to SM quarks from one-loop diagrams with level-1 KK particles running in the loop. These couplings are generated from boundary conditions on the KK gluon and the bulk masses, which provide the only source of translational invariance breaking, and read $$\begin{aligned} & g_s T^a \gamma^\mu \frac{1}{\sqrt{2}} \frac{1}{16 \pi^2} \log \left( \frac{\Lambda}{\mu} \right)^2 \left[ P_L (\frac{1}{8} g_1^2 + \frac{27}{8} g_2^2 - \frac{11}{2} g_s^2) + P_R (Y_{u, d} \ g_1^2 - \frac{11}{2} g_s^2) \right] \ , \label{eqn:mUEDcoupling}\end{aligned}$$ where $Y_u = 2$ for up-type quarks, $Y_d = 1/2$ for down-type quarks, $\Lambda$ is an ultraviolet scale larger than the inverse size of the extra dimension, and $\mu$ is the renormalization scale to evaluate the coupling. In Randall-Sundrum warped scenarios [@Randall:1999ee; @Randall:1999vf], the fermion mass hierarchy can be explained by allowing fermions to propagate in the bulk, where the observed charged fermion mass hierarchy originates as $\mathcal{O}(1)$ differences in bulk mass parameters. Typically, the KK mass scale must then be $\mathcal{O}(5-10~\text{TeV})$ to satisfy low-energy flavor violation probes, especially $\bar{K}-K$ mixing [@Casagrande:2008hr], but this scale can be lowered in the case that the bulk fermions obey a flavor symmetry [@Chen:2009gy; @Chen:2009hr]. Since KK parity is absent, the first KK gluon decays to SM zero-mode quarks, and this coupling is given by calculating the wavefunction overlap between the zero modes and the KK gluon in the extra dimension. Using the general coupling structure in [Eq. (\[eqn:Lagrangian\])]{} and identifying $X_\mu$ with the first KK gluon, $$\begin{aligned} \lambda^{ij} &\approx \dfrac{m_X}{\sqrt{2} M_{\text{KK}}} \left( \dfrac{1}{\sqrt{2L}} \delta_{ij} - \sqrt{2L} F(c_{Q_i}) F(c_{Q_j}) \right) \ ,\end{aligned}$$ where RH up-type couplings are obtained by replacing $\lambda \rightarrow \kappa$ and $Q \rightarrow u$, RH down-type couplings are obtained by replacing $\lambda \rightarrow \eta$ and $Q \rightarrow d$, $m_X \approx 2.4 M_{\text{KK}}$ is fixed by the boundary conditions of the 5D gluon, $L$ is the length of the extra dimension, and $c_Q$, $c_u$, and $c_d$ are the bulk mass parameters for LH and RH quark fields [@Agashe:2013kxa]. In order to reproduce the known SM quark masses, these bulk mass parameters must be chosen to maximize the top quark wavefunction overlap with the TeV-scale infrared brane while the wavefunctions for the light quarks are skewed towards the ultraviolet brane. As a result, the first KK gluon preferentially decays to top quarks, with branching fractions that can exceed 80% [@Agashe:2013kxa]. Non-universal couplings and flavor constraints {#subsec:flavor} ---------------------------------------------- As we have emphasized, massive color octet vectors arise in numerous models of beyond the Standard Model physics. Their collider and flavor physics phenomenology depends crucially on the particular $\lambda$, $\kappa$, and $\eta$ structure defined in [Eq. (\[eqn:Lagrangian\])]{} that is realized in a given model. Given the stringent constraints on tree-level flavor changing neutral currents [@Bona:2007vi; @Chivukula:2013kw], we adopt flavor-diagonal couplings for $\lambda$, $\kappa$, and $\eta$ in the gauge basis. Nevertheless, flavor violating effects are still induced in interactions of LH down quarks by the rotation to the quark mass basis. From [Eq. (\[eqn:Lagrangian\])]{}, we rotate the quark fields to the mass basis by $V_u u_L = u_L^m$, $V_d d_L = d_L^m$, $U_u u_R = u_R^m$, and $U_d d_R = d_R^m$, giving $$\begin{aligned} \mathcal{L} &\supset \bar{u}_L^m g_s t^a \slashed{X}^a V_u \lambda V_u^\dagger u_L^m + \bar{d}_L^m g_s t^a \slashed{X}^a V_d V_u^\dagger V_u \lambda V_u^\dagger V_u V_d^\dagger d_L^m \nonumber \\ &+ \bar{u}_R^m g_s t^a \slashed{X}^a U_u \kappa U_u^\dagger u_R^m + \bar{d}_R^m g_s t^a \slashed{X}^a U_d \eta U_d^\dagger d_R^m \ .\end{aligned}$$ We see that the $\lambda$, $\kappa$, and $\eta$ matrices can be chosen such that the effective interaction matrices $\tilde{\lambda}$, $\tilde{\kappa}$, and $\tilde{\eta}$ are diagonal in the quark mass basis, $$\begin{aligned} \tilde{\lambda} = V_u \lambda V_u^\dagger \ , \ \tilde{\kappa} = U_u \kappa U_u^\dagger \ , \ \tilde{\eta} = U_d \eta U_d^\dagger \ .\end{aligned}$$ The corresponding LH down quark interactions have small, off-diagonal entries induced by $V_{CKM} \equiv V_u V_d^{\dagger}$ [@Gedalia:2010rj; @Olive:2016xmw], $$\begin{aligned} \tilde{\lambda}_D \equiv V_d V_u^{\dagger} V_u \lambda V_u^\dagger V_u V_d^{\dagger} = V_{CKM}^{\dagger} \tilde{\lambda} V_{CKM} \ ,\end{aligned}$$ and hence $X_\mu$ mediates tree-level flavor-changing neutral currents (FCNCs). Since the strongest FCNC constraints come from $\bar{K}_0-K_0$ mixing [@Bona:2007vi], we minimize the impact of these constraints by assuming $\tilde{\lambda}_{11} = \tilde{\lambda}_{22} \neq \tilde{\lambda}_{33}$, which leads to $\tilde{\lambda}_{D, 12} \approx \tilde{\lambda}_{D, 21} \approx -(3.3 - 1.3i) \times 10^{-4} (\tilde{\lambda}_{33} - \tilde{\lambda}_{11})$, using global fit values for the CKM elements [@Olive:2016xmw]. A tree-level exchange of $X_\mu$ can be matched to the four-fermion operator [@Bona:2007vi; @Casagrande:2008hr; @Bai:2011ed; @Haisch:2011up; @Altmannshofer:2012ur; @Chivukula:2013kw] $$\begin{aligned} O_K^1 = (\bar{d}_\alpha \gamma_\mu P_L s_\alpha) (\bar{d}_\beta \gamma^\mu P_L s_\beta) \ ,\end{aligned}$$ with the Wilson coefficient $$\begin{aligned} C_K^1 = \frac{-1}{6} \frac{\tilde{\lambda}_{D, 12}^2}{M_X^2} \ .\end{aligned}$$ Using the requirement $\Lambda > 1.1 \times 10^3$ TeV from Re $C_K^1$ and $\Lambda > 2.2 \times 10^4$ TeV from Im $C_K^1$ [@Bona:2007vi], we can constrain the maximum size of $\tilde{\lambda}_{11}$, $\tilde{\lambda}_{33}$, assuming the other is vanishing: $$\begin{aligned} \text{Re}~C_K^1: &\quad M_X > 130~\text{GeV} g_s \text{ max} (|\tilde{\lambda}_{11}|, |\tilde{\lambda}_{33}|) \\ \text{Im}~C_K^1: &\quad M_X > 2700~\text{GeV} g_s \text{ max} (|\tilde{\lambda}_{11}|, |\tilde{\lambda}_{33}|) \ .\end{aligned}$$ We see that these constraints are easily satisfied for $\mathcal{O}(\text{TeV})$ scale color octets as long as $\tilde{\lambda} \sim \mathcal{O}(0.1)$. Moreover, with this structure, $\tilde{\lambda}_{11} = \tilde{\lambda}_{22}$, the stronger constraints come from $\bar{B}_d^0-B_d^0$ and $\bar{B}_s^0-B_s^0$ measurements, where $|C_{B_d}^1|$ requires $\Lambda > 1.0 \times 10^3$ TeV and $|C_{B_s}^1|$ requires $\Lambda > 240$ TeV. Using the tree-level, CKM-induced off-diagonal elements of $\tilde{\lambda}_D$, we obtain $$\begin{aligned} |C_{B_d}^1|: &\quad M_X > 3500~\text{GeV} g_s \text{ max} (|\tilde{\lambda}_{11}|, |\tilde{\lambda}_{33}|) \\ |C_{B_s}^1|: &\quad M_X > 4000~\text{GeV} g_s \text{ max} (|\tilde{\lambda}_{11}|, |\tilde{\lambda}_{33}|) \ ,\end{aligned}$$ which are again weakened to the sub-TeV scale for $\tilde{\lambda} \sim \mathcal{O}(0.1)$. As a result, we can realistically expect $\mathcal{O}(\text{TeV})$ color octet vectors with dominant branching fractions to either tops, $\tilde{\lambda}_{33} \gg \tilde{\lambda}_{11}$ or jets, $\tilde{\lambda}_{11} \gg \tilde{\lambda}_{33}$, entirely consistent with flavor bounds. We remark that an alternative assumption of the flavor structure can relax these bounds further. In particular, if instead $V_{CKM}^\dagger \tilde{\lambda} V_{CKM}$ is diagonal, then the tree-level flavor violating couplings are shifted to the LH up-quark couplings [@Bai:2011ed; @Haisch:2011up]. We can define $\tilde{\lambda}' = V_{CKM}^\dagger \tilde{\lambda} V_{CKM}$ as diagonal, and then the LH up-quark couples via $V_{CKM} \tilde{\lambda}' V_{CKM}^\dagger$, which is constrained by $D_0-\bar{D}_0$ mixing, where $\Lambda > 700$ TeV [@Bona:2007vi]. The corresponding bound is $$\begin{aligned} \text{Im}~C_D^1: &\quad M_X > 36~\text{GeV} g_s \text{ max}(|\tilde{\lambda}_{11}|, |\tilde{\lambda}_{33}|) \ .\end{aligned}$$ The $\Delta F = 1$ constraints, including $b \to X_s \gamma$ and $b \to X_s g$ penguin amplitudes, are also immediately satisfied if the $X^\mu$ couplings are aligned in the down-quark sector [@Chivukula:2013kw; @Haisch:2011up]. Collider phenomenology of color octet vectors {#subsec:pheno} --------------------------------------------- Color octet vectors can be singly and doubly produced at colliders. The single-production rate scales with the corresponding partial width into $q \bar{q}$, $q = d, u, s, c$, since single production via gluons is forbidden at tree-level. In our scenario, the induced flavor-violating couplings $\tilde{\lambda}_D$ are at most $5\% \times \tilde{\lambda}_{11}, \tilde{\lambda}_{33}$, which we will neglect and hence approximate $\tilde{\lambda}_D \approx \tilde{\lambda}$. The partial width of $X$ to a pair of quarks $q$ is $$\Gamma (X \to \bar{q} q) = \dfrac{\alpha_s m_X}{12} \left( (g_L^2 + g_R^2)(1 - \dfrac{m_q^2}{m_X^2}) + 6 \frac{m_q^2}{m_X^2} g_L g_R \right) \left( 1 - 4 \frac{m_q^2}{m_X^2} \right)^{1/2} \ , \label{eqn:Xtoqqwidth}$$ where $g_L$ and $g_R$ are the corresponding diagonal entries in $\tilde{\lambda}$, $\tilde{\kappa}$, and $\tilde{\eta}$[^1]. If the masses of the quarks can be neglected compared to the $X$ mass, the partial width simplifies to $$\Gamma (X \to \bar{q} q) = \dfrac{\alpha_s (g_L^2 + g_R^2) m_X}{12} \ .$$ The branching fractions for $jj$, $b\bar{b}$, and $t\bar{t}$ final states are then simply ratios of the corresponding sum of squared couplings. We see that large branching fractions to top quarks can easily be realized by increasing $\tilde{\kappa}_{33}$, while large branching fractions to bottom quarks corresponds to increasing $\tilde{\eta}_{33}$. This is the effective description of the warped extra dimension scenario with $t_R$ and $b_R$ wavefunction profiles peaked close to the infrared brane [@Agashe:2013kxa], and small hierarchies in these third generation couplings are consistent with electroweak precision tests [@Bai:2011ed; @Haisch:2011up]. As previously highlighted, the single dijet resonance constraints scale with the overal partial width into light flavor quarks. Pair-production rates, however, are robustly calculable knowing only $m_X$ and its color octet representation. Hence, all searches in pair-production modes provide important complementary reach compared to single resonance searches. The current LHC analyses focus on the simplest topologies, with $X X^* \to (JJ) (JJ)$, with $J = j$ or $b$ inclusively [@Chatrchyan:2013izb; @Khachatryan:2014lpa; @CMS:2016pkl; @ATLAS:2012ds; @ATLAS:2016sfd; @ATLAS:2017gsy] and $X X^* \to 4t$ [@CMS:2013xma; @Sirunyan:2017tep; @ATLAS:2012hpa; @Aad:2015kqa; @TheATLAScollaboration:2016gxs; @ATLAS:2016gqb; @ATLAS:2016btu]. In the case where the $X \to b\bar{b}$ decay width is preferred, additional signal discrimination is easily gained by requiring $b$-tags. The orthogonal $(b\bar{b})(b\bar{b})$, $(b\bar{b})(jj)$, and $(jj)(jj)$ signal regions would then have enhanced and complementary sensitivity compared to the current $(JJ)(JJ)$ searches. If the coupling to tops is preferred, then the $4t$ search and our proposed searches in the $t\bar{t}(jj)$ and $t\bar{t}(b\bar{b})$ mixed decay channels are critical. In particular, the $t\bar{t}(jj)$ and $t\bar{t}(b\bar{b})$ mixed decay searches offer substantial improvements in covering the sensitivity gap when the $X$ decay widths to tops and light quarks are comparable. In the next section, we describe our collider analyses optimized for the $t\bar{t}(jj)$ and $t\bar{t}(b\bar{b})$ final states. Collider analyses of the mixed channels, $t\bar{t}(jj)$ and $t\bar{t}(b\bar{b})$ {#sec:collider} ================================================================================ We analyze pair-produced resonances in a new mixed decay mode, $p p \to XX^* \to t\bar{t} (jj)$ and $p p \to XX^* \to t\bar{t} (b\bar{b})$. Although there are numerous possibilities for the top decays, we mainly focus on the semi-leptonic and fully leptonic final states, which provide clean handles for tagging the reconstructible $t\bar{t}$ and dijet systems. Of course, the main SM background is the irreducible $t \bar{t} + $ jets background, where other single boson and diboson + jets backgrounds are subleading after requiring multiple $b$-jets. We construct a <span style="font-variant:small-caps;">FeynRules</span> v2.3.26 [@Alloul:2013bka] model using Universal FeynRules Output [@Degrande:2011ua] to perform leading order Monte Carlo event simulation with <span style="font-variant:small-caps;">MadGraph</span> 5 v2.4.3 [@Alwall:2014hca], interfaced with <span style="font-variant:small-caps;">Pythia</span> v8.2 [@Sjostrand:2007gs; @Sjostrand:2014zea] for showering and hadronizations. We remark that the leading order calculation allows a direct comparison to existing $(JJ)(JJ)$ and $4t$ search limits [@Chatrchyan:2013izb; @Khachatryan:2014lpa; @CMS:2016pkl; @ATLAS:2012ds; @ATLAS:2016sfd; @ATLAS:2017gsy; @CMS:2013xma; @Sirunyan:2017tep; @ATLAS:2012hpa; @Aad:2015kqa; @TheATLAScollaboration:2016gxs; @ATLAS:2016gqb; @ATLAS:2016btu]. Since the signal will rely on tagging two top candidates, we simulate the SM background, $t\bar{t}j$ with up to two additional jets at leading order, using <span style="font-variant:small-caps;">Sherpa</span> v.2.1.0 [@Gleisberg:2008ta]. We rescale the background by a flat $K$-factor of 1.5, adopted from <span style="font-variant:small-caps;">Sherpa</span>+<span style="font-variant:small-caps;">BlackHat</span> [@Gleisberg:2008ta; @Berger:2008sj]. The inclusive SM $t\bar{t}$ + jets production cross section is known at next-to-next-to-leading order and next-to-next-to-leading log (NNLL) precision [@Czakon:2013goa], whereas differential cross sections can be obtained at next-to-leading order and NNLL precision [@Guzzi:2014wia]. Since our $X$ resonance should appear as a resonant peak over a smooth continuum background, a $p_T$-dependent $K$-factor should not significantly affect our projected results. Event selection strategy {#subsec:selection} ------------------------ We first discuss the semi-leptonic top decay channel, where our signal process is $XX^* \to b \bar{b} \ell^\pm \nu jj (JJ)$, with $J = j$ or $b$. Jets are clustered using the anti-$k_T$ algorithm [@Cacciari:2008gp] with $R = 0.5$, and we require jets to have $p_T > 50$ GeV and $|\eta| < 4.9$. We identify $b$-jets using a <span style="font-variant:small-caps;">Delphes</span> v3.1.2 [@deFavereau:2013fsa] detector simulation, with a $p_T$- and $\eta$-dependent tagging efficiency of about $70\%$ for displaced tracks from $B$-mesons within $\Delta R < 0.3$ from the main jet axis; the charm misidentification rate is roughly $15\%$ and the light flavor mistag rate is $0.1\%$. Events must have at least 5 jets, at least two of which must be $b$-tagged and one must be untagged. The leading $b$-tagged and untagged jets must have $p_T \geq 250$ GeV, and the subleading untagged jet, if present, must have $p_T \geq 80$ GeV. Events must also have exactly one isolated lepton with $p_T > 20$ GeV and $|\eta| < 2.5$, with isolation using <span style="font-variant:small-caps;">Delphes</span> default parameters. We furthermore cut on missing transverse energy (MET), requiring $\slashed{E}_T > 80$ GeV, as well as signal mass-optimized $H_T = \Sigma_j |p_{T, j}| \geq (4/3) \, m_X$. Naively, the $X \to JJ$ signal resonance is extracted from forming the invariant mass of the two leading jets. The jet combinatorics, however, present a major hurdle against using the dijet and ditop masses to discriminate the signal from the irreducible background. Thus, the main goal of our collider strategy is to solve the combinatorial ambiguity, taking advantage of the resonant high mass dijet signal, the large $p_T$ of the resonant dijet system, and the relevant angular spread between the $b$-jets, lepton, and jets. Difficulties in resolving multijet combinatorial ambiguities are discussed in, [*e.g.*]{}, [@Rajaraman:2010hy]. For this purpose, we define two signal regions. We target the $b\bar{b}$ mode by selecting events with more than two $b$-tagged jets, while the $jj$ mode is optimized by requiring exactly two $b$-tagged jets. For the $b\bar{b}$ decay, we assume that the leading $b$-tagged jet comes from the $X \to JJ$ decay directly, not from $t\bar{t}$. We then reconstruct the $b\bar{b}$ system by adding the remaining leading jet, whether tagged or untagged. We find that this reconstruction best reflects the dijet resonance in spite of the underlying combinatorial ambiguity. For the $jj$ mode, we start with the leading light jet $j_1$ and add to it the hardest light jet $j_2$ which satisfies $\Delta R_{j_1 j_2} \leq \pi$. We also add in the next hardest light jet $j_3$ with $\Delta R_{j_1 j_3}\leq\pi$ if $j_1$ and $j_2$ are not balanced in $p_T$, when $p_{T, j_2} / p_{T, j_1} \leq 0.15$. This follows the hemisphere intuition, where the $X \to jj$ decays should be untagged and relatively hard. Adding the third hardest light jet accounts for the wide-angle final state radiation of our signal quarks. For both $b\bar{b}$ and $jj$ decay modes, we construct the dijet invariant mass as our final kinematic discriminant. The signal and background cut efficiencies for the semileptonic analysis are shown in [Table \[table:semilep\_cutflow\]]{}, while the most salient kinematic distributions for the background stacked with different signal hypotheses are shown in [Fig. \[fig:kinematics\]]{}. The upper left panel in [Fig. \[fig:kinematics\]]{} shows the hardening of the $H_T$ distribution coming from the signal jets. The dijet $p_{T, JJ}$ distribution, however, only offers an overal rate shift from the additional signal events and no strong correlation with the signal $X$ mass, which is because of the combinatorial ambiguity among the jets. The bottom panels show the invariant mass distributions in the $jj$ and $b\bar{b}$ targetted modes. Again, the broad peak structure arises mostly from the combinatorial ambiguity in capturing the correct signal jets to reconstruct the resonance. We note that $b$-tagging efficiency and the combinatorial ambiguity also cause $X \to jj$ events to populate the $m_{bb}$ signal region and vice versa. --------------------------------------------------------------------------------------------- --------- --------- --------- --------- --------- --------- [**Semi-Leptonic Search**]{} Signal mass $m_X$ 1.3 TeV 1.5 TeV 1.7 TeV 1.3 TeV 1.5 TeV 1.7 TeV Event selection \[fb\] ($N_J \geq 5$ with $N_j \geq 1$, $N_b \geq 2$; $N_\ell=1$; $p_{T,j}^\text{leading},\ p_{T,b}^\text{leading}>250$ GeV; $p_{T,\ell}>20$ GeV) 1.52 0.45 0.13 5.8 5.8 5.8 $\slashed{E}_T > 80$ GeV 84% 87% 89% 64% 64% 64% $H_T \geq (4/3)\,m_X$ 83% 79% 76% 32% 19% 10% Remaining cross section \[fb\] 1.06 0.31 0.09 1.23 0.71 0.39 --------------------------------------------------------------------------------------------- --------- --------- --------- --------- --------- --------- : Cut flow for different resonance masses $m_X$ and dominant background $t\bar{t}$+jets for the semi-leptonic search. All branching ratios are applied to signal and background when quoting cross sections. We normalize the signal assuming $\text{Br}(X \to t\bar{t})=\text{ Br}(X \to JJ)=50\%$. []{data-label="table:semilep_cutflow"} ![Differential distributions in the semi-leptonic analysis for $H_T$ (upper left), $p_{T, JJ}$ (upper right), $m_{jj}$ (lower left), and $m_{bb}$ (lower right), where the invariant mass definitions are described in the main text. We show distributions for the SM $t\bar{t} + $ jets background (red, dashed), stacked background + $m_X = 1.3$ TeV signal (green, solid), background + $m_X = 1.5$ TeV signal (blue, solid), background + $m_X = 1.7$ TeV signal (black, solid). Differential distributions are presented after jet selection and $\slashed{E}_T$ cuts but before the $H_T$ to show the unsculpted dijet invariant mass spectra.[]{data-label="fig:kinematics"}](HT.pdf "fig:"){width="48.00000%"} ![Differential distributions in the semi-leptonic analysis for $H_T$ (upper left), $p_{T, JJ}$ (upper right), $m_{jj}$ (lower left), and $m_{bb}$ (lower right), where the invariant mass definitions are described in the main text. We show distributions for the SM $t\bar{t} + $ jets background (red, dashed), stacked background + $m_X = 1.3$ TeV signal (green, solid), background + $m_X = 1.5$ TeV signal (blue, solid), background + $m_X = 1.7$ TeV signal (black, solid). Differential distributions are presented after jet selection and $\slashed{E}_T$ cuts but before the $H_T$ to show the unsculpted dijet invariant mass spectra.[]{data-label="fig:kinematics"}](ptJJ.pdf "fig:"){width="48.00000%"}\ ![Differential distributions in the semi-leptonic analysis for $H_T$ (upper left), $p_{T, JJ}$ (upper right), $m_{jj}$ (lower left), and $m_{bb}$ (lower right), where the invariant mass definitions are described in the main text. We show distributions for the SM $t\bar{t} + $ jets background (red, dashed), stacked background + $m_X = 1.3$ TeV signal (green, solid), background + $m_X = 1.5$ TeV signal (blue, solid), background + $m_X = 1.7$ TeV signal (black, solid). Differential distributions are presented after jet selection and $\slashed{E}_T$ cuts but before the $H_T$ to show the unsculpted dijet invariant mass spectra.[]{data-label="fig:kinematics"}](mj.pdf "fig:"){width="48.00000%"} ![Differential distributions in the semi-leptonic analysis for $H_T$ (upper left), $p_{T, JJ}$ (upper right), $m_{jj}$ (lower left), and $m_{bb}$ (lower right), where the invariant mass definitions are described in the main text. We show distributions for the SM $t\bar{t} + $ jets background (red, dashed), stacked background + $m_X = 1.3$ TeV signal (green, solid), background + $m_X = 1.5$ TeV signal (blue, solid), background + $m_X = 1.7$ TeV signal (black, solid). Differential distributions are presented after jet selection and $\slashed{E}_T$ cuts but before the $H_T$ to show the unsculpted dijet invariant mass spectra.[]{data-label="fig:kinematics"}](mb.pdf "fig:"){width="48.00000%"} The search in the fully leptonic top quark decay channel is very similar. We again select jets and leptons as in the semi-leptonic channel, but we require only 4 jets, at least two of which are $b$-tagged, and also require exactly two isolated leptons of opposite charge. We loosen the MET cut, $\slashed{E}_T > 50$ GeV, and keep the $H_T$ cut, $H_T \geq (4/3) \, m_X$. To avoid the anticipated $Z+$jets background we veto events with $m_{\ell\ell}\leq 115$ GeV. Again we distinguish between $b\bar{b}$ and $jj$ dijet resonance decay modes, using the same method as the semi-leptonic search. The corresponding cut flow is presented in [Table \[table:fullylep\_cutflow\]]{}. The final signal rates lead to the same relative signal to background ratios as the semi-leptonic analysis, but the absolute rate is only an $\approx 5\%$ additional contribution compared to the semi-leptonic channel. ----------------------------------------------------------------------------------------------- --------- --------- --------- --------- --------- --------- [**Fully Leptonic Search**]{} Signal mass $m_X$ 1.3 TeV 1.5 TeV 1.7 TeV 1.3 TeV 1.5 TeV 1.7 TeV Event selection \[fb\] ($N_J \geq 4$ with $N_b \geq 2$; $N_\ell = 2$; $p_{T,j}^\text{leading},\ p_{T,b}^\text{leading}>250$ GeV; $p_{T,\ell}>20$ GeV) 0.148 0.045 0.009 0.75 0.75 0.75 $\slashed{E}_T > 50$ GeV 96% 96% 97% 91% 91% 91% $m_{\ell\ell}\geq 115$ GeV 66% 68% 70% 37% 37 % 37% $H_T \geq (4/3)\,m_X$ 63% 57% 48% 21% 11% 2.7% Remaining cross section \[fb\] 0.059 0.017 0.003 0.053 0.027 0.007 ----------------------------------------------------------------------------------------------- --------- --------- --------- --------- --------- --------- : Cut flow for a different resonance masses $m_X$ and dominant background $t\bar{t}$+jets for the fully leptonic search. All branching ratios are applied to signal and background when quoting cross sections. We normalize the signal assuming $\text{Br}(X \to t\bar{t})=\text{ Br}(X \to JJ)=50\%$. []{data-label="table:fullylep_cutflow"} Comparison with current searches {#subsec:comparison} -------------------------------- We now compare the projected sensitivity from the mixed $t\bar{t} (JJ)$ searches in combination with the recasted exclusions from ATLAS and CMS for $(JJ)(JJ)$ and $4t$ searches. We also show the single production limits for dijet and ditop resonance searches. Since we assume that our new physics state $X$ only decays to quark pairs, we can express the pair production constraints in the branching fraction vs. mass plane, where $$\text{Br}(X \to t\bar{t}) = 1 - \text{ Br}(X \to JJ) \ .$$ So, the pair production limits can be translated according to: $$\left. \sigma_{\text{excl}} (JJ)(JJ) \right|_{m_X} = \left. \sigma (pp \to XX^*)\right|_{m_X} \times \left( \text{Br}_{JJ} \right)^2 \ ,$$ with $\text{Br}_{JJ} = \text{Br}(X \to JJ)$. The $4t$ constraint and the $t\bar{t} (JJ)$ projected exclusion after replacing $(\text{Br}_{JJ})^2$ with $(\text{Br}_{tt})^2$ and $2(\text{Br}_{JJ} \ \text{Br}_{tt})$, respectively, where $\text{Br}_{tt} = \text{ Br}(X \to t\bar{t})$. The single dijet resonance limits are determined only after specifying the total width of the resonance. If the total width is narrow, a reference cross section can be rescaled by the partial width into light quarks. We see that $$\begin{aligned} \left. \sigma_{\text{excl}} (pp \to (JJ)_{\text{res}}) \right|_{m_X} &= \sigma_{\text{width}} \text{ Br}_{JJ} A = \sigma_{\text{ref}} \text{ Br}_{JJ}^2 \frac{\Gamma_{\text{tot, width}}}{\Gamma_{qq, \text{ ref}}} A \end{aligned}$$ In the case of $t\bar{t}$ resonances, the above constraint becomes a bounded requirement on the branching fraction to tops. Hence, the $t\bar{t}$ resonance constraint follows $$\begin{aligned} \left. \sigma_{\text{excl}} (pp \to (t\bar{t})_{\text{res}}) \right|_{m_X} &\geq \sigma_{\text{width}} \text{ Br}_{tt} = \sigma_{\text{ref}} (1 - \text{ Br}_{tt}) \text{ Br}_{tt} \frac{\Gamma_{\text{tot, width}}}{\Gamma_{qq, \text{ref}}} \ . \label{eqn:Brttbound}\end{aligned}$$ Note that here we drop the acceptance factor since the experiments unfold the acceptance when presenting their results. We combine all four orthogonal signal regions of the semi-leptonic and fully leptonic search and compute the $95\%$ C.L. limits based on the respective $JJ$ invariant mass shapes, where we assume a 10% systematic uncertainty on signal and background. The result can be found in [Fig. \[fig:exclusion\]]{}. ![Exclusion limits for different resonance masses as a function of $\text{ Br}(X \to JJ) = 1-\text{ Br}(X \to t\bar{t})$. We show our limit in black for 37 fb$^{-1}$ (dot-dashed) and 100 fb$^{-1}$ (dashed) of integrated luminosity. We also show current limits from $4t$ (red) [@ATLAS:2016btu], $(JJ)(JJ)$ (blue) [@ATLAS:2017gsy], and $(JJ)$ (green) [@CMS:2017xrr; @Aaboud:2017yvp] searches.[]{data-label="fig:exclusion"}](exclusion.pdf){width="95.00000%"} ![Left: Strongest mass limits when the various $t\bar{t}$, $b\bar{b}$, and $jj$ decay channels are compared. Right: Search region that gives the best sensitivity.[]{data-label="fig:reach"}](Reach.png "fig:"){width="45.00000%"} ![Left: Strongest mass limits when the various $t\bar{t}$, $b\bar{b}$, and $jj$ decay channels are compared. Right: Search region that gives the best sensitivity.[]{data-label="fig:reach"}](Search.png "fig:"){width="45.00000%"} We remark that the width of our resonance, $\Gamma_X = 5 \times 10^{-4} m_X$, corresponds to diagonal entries of $\tilde{\lambda}$, $\tilde{\kappa}$, $\tilde{\eta} \lesssim 0.1$. As discussed in [Subsec. \[subsec:flavor\]]{}, such LH quark couplings to $X$ readily satisfy flavor violation bounds from meson oscillation measurements and neutral current transitions given $m_X > 1$ TeV. On the other hand, these couplings generally arise from mixing the SM quarks with heavy vectorlike states, since otherwise perturbativity in the parent extended color gauge symmetry is violated [@Dobrescu:2013coa]. The corresponding flavor violation bounds, collider constraints, electroweak precision observable tests are more model-dependent, but realistic and complete scenarios can be constructed [@Bai:2011ed; @Haisch:2011up; @Chivukula:2013kw]. As mentioned previously, the sensitivity from single production compared to the sensitivity from pair production strongly depends on the ratio $\Gamma_X/m_X$. Any direct resonance bounds from $JJ$ searches could in principle be evaded completely with a suitably small choice of $\Gamma_X$, but choosing $\Gamma_X / m_X = 5 \times 10^{-4}$ allows complementary dijet constraints from CMS and ATLAS [@CMS:2017xrr; @Aaboud:2017yvp]. We remark that the $t\bar{t}$ resonance limits [@TheATLAScollaboration:2016wfb] are absent from [Fig. \[fig:exclusion\]]{} for $\Gamma_X / m_X = 5 \times 10^{-4}$. These constraints are only relevant once $\Gamma_X / m_X \gtrsim 7 \times 10^{-4}$ for $m_X$ around 1 TeV. As shown in [Eq. (\[eqn:Brttbound\])]{}, the $t\bar{t}$ resonance limit is symmetric around $\text{Br}_{tt} = 50\%$ since the maximum rate in this channel corresponds to equal partial widths to dijets and ditops. The strongest existing bounds in [Fig. \[fig:exclusion\]]{} are therefore from $(JJ)(JJ)$ [@ATLAS:2017gsy] and $4t$ [@ATLAS:2016btu] searches, which are clearly optimal for their respective $\text{Br}_{JJ}$ and $\text{Br}_{tt}$ corners. The mass reach of both searches weakens by about 250 GeV in the intermediate regime, however, leaving significant room for our dedicated $t\bar{t}(JJ)$ search to explore. In [Fig. \[fig:exclusion\]]{}, we assume that all quark couplings are flavour universal except for the top. We relax this assumption in [Fig. \[fig:reach\]]{}, allowing both branching fractions $\text{Br}(X \to b\bar{b}) = \text{ Br}_{bb}$ and $\text{Br}_{tt}$ to float. The results are presented in an equilateral triangle since the sum of the $jj$, $b\bar{b}$ and $t\bar{t}$ branching fractions must equal 100% in our model. The shading in the left panel of [Fig. \[fig:reach\]]{} shows the lower limit on the resonance mass $m_X$ as a function of the three branching fractions. The right panel of [Fig. \[fig:reach\]]{} indicates which particular dedicated search yields the corresponding lower mass limit. We see that our new search channels, $t\bar{t}(jj)$ and $t\bar{t}(b\bar{b})$, overtake the sensitivity in the central areas of the triangle compared to the existing $(JJ)(JJ)$ and $4t$ searches. We reiterate that a complete characterization of $t\bar{t}$, $b\bar{b}$, and $jj$ decay channels for pair production colored resonances would necessitate optimizing the current $(JJ)(JJ)$ search into $(b\bar{b})(b\bar{b})$, $(b\bar{b})(jj)$, and $(jj)(jj)$ signal regions. In particular, the $(b\bar{b})(b\bar{b})$ search would bear striking similarities with the current searches for pair production of the 125 GeV Higgs [@Aaboud:2016xco; @CMS:2016foy], where the main novelty would be varying the $(b\bar{b})$ mass window to test for new resonances. The multijet background, however, is very challenging to simulate and thus substantial statistics in the $b$-tagged backgrounds would be required to suitably smooth paired invariant mass spectrum in the $(b\bar{b})(b\bar{b})$ and $(b\bar{b})(jj)$ channels. Conclusion {#sec:conclusion} ========== In this work, we have highlighted the $t\bar{t}(jj)$ and $t\bar{t}(b\bar{b})$ mixed decay channels of a massive, color octet vector as new targets for ATLAS and CMS searches. Hierarchies in the underlying couplings of the $X$ resonance to light quarks, bottom pairs, or top pairs are entirely consistent with low energy FCNC constraints if the $X$ mass is above $1$ TeV and its couplings to quarks are at most $0.1$. As a result, the LHC provides the leading reach to TeV-scale color octet vectors with variable couplings to heavy and light flavor quarks by virtue of the model independent pair production rate. In our $t\bar{t}(JJ)$ analyses, we focused on resolving the jet combinatorial ambiguity to reconstruct the dijet or dibottom resonance. In principle, reconstructing the $(t\bar{t})$ resonance is also possible, but our scenario with its many resolved jets did not afford any additional signal discrimination in this regard. Nevertheless, our results show that new $t\bar{t}(jj)$ and $t\bar{t}(b\bar{b})$ searches will fill a sensitivity gap between the $(JJ)(JJ)$ and $4t$ searches. This gap is clear in the $\text{Br}_{JJ}$ vs. $m_X$ plane, which itself provides a useful tool for easily presenting the results from different collider searches of pair-produced resonances. We stress that a post-discovery scenario of a new resonance greatly benefits from this complementary information, where single and pair production modes combined with different decay channels provide direct information about underlying Lagrangian couplings. Acknowledgments {#sec:acknowledgments .unnumbered} =============== FY would like to thank R. Sekhar Chivukula and Elizabeth Simmons for helpful discussions during the 2015 Les Houches summer session, and also Susanne Westhoff at the Mainz Institute for Theoretical Physics, “The TeV Scale: A Threshold to New Physics?” 2017 scientific program. FY would also like to acknowledge the hospitality of the theory group at Fermi National Accelerator Laboratory while this work was completed. This research is supported by the Cluster of Excellence Precision Physics, Fundamental Interactions and Structure of Matter (PRISMA-EXC 1098). The work of MB is moreover supported by the German Research Foundation (DFG) in the framework of the Research Unit “New Physics at the Large Hadron Collider” (FOR 2239). [^1]: For reference, the generalized flavor violating partial width is $$\begin{aligned} \Gamma (X \to \bar{q}_i q_j, \bar{q}_j q_i) &= \dfrac{\alpha_s m_X}{6} \left( ((g_L^{ij})^2 + (g_R^{ij})^2) (1 - \frac{1}{2} (\frac{m_i^2}{m_X^2} + \frac{m_j^2}{m_X^2}) - \frac{1}{2} (\frac{m_i^2}{m_X^2} - \frac{m_j^2}{m_X^2})^2) + 6 \frac{m_i m_j}{m_X^2} g_L^{ij} g_R^{ij} \right) \nonumber \\ &\times \left( (1 - \frac{m_i^2}{m_X^2} - \frac{m_j^2}{m_X^2})^2 - 4 \frac{m_i^2 m_j^2}{m_X^4} \right)^{1/2} \ , \label{eqn:Xtoqiqjwidth}\end{aligned}$$ where $g_L^{ij} = g_L^{ji}$ and $g_R^{ij} = g_R^{ji}$.
--- abstract: 'We use configuration interaction and many-body perturbation theory techniques to calculate spin-independent and spin-dependent parts of the parity nonconserving amplitudes of the transitions between the $6s^2 \ ^1$S$_0$ ground state and the $6s5d \ ^3$D$_1$ excited state of $^{171}$Yb and $^{173}$Yb. The results are presented in a form convenient for extracting spin-dependent interaction constants (such as, e.g., anapole moment) from the measurements.' author: - 'V. A. Dzuba' - 'V. V. Flambaum' date: title: Calculation of parity nonconservation in neutral ytterbium --- Introduction ============ The use of atomic ytterbium to study parity nonconservation (PNC) in atoms was first suggested by DeMille [@DeMille]. The measurements are in progress at Berkeley and afters years of hard work [@Bowers; @Kimball; @Stalnaker02; @Stalnaker06] the first results of PNC measurements are finally reported [@BudkerYbPNC]. As it was expected the PNC in ytterbium is strongly enhanced, being two orders of magnitude larger than in cesium [@Wood]. The cesium PNC experiment together with its interpretation  [@Breit; @QED; @Cs-cor] in terms of nuclear weak charge provides the best current atomic test of the standard model (see, also a review [@Ginges]). It is also the only measurement of the nuclear [*anapole moment*]{} which is produced by the PNC nuclear forces [@anapole]. The extraction of the weak nuclear charge from the PNC measurements relies on atomic calculations. The interpretation of the PNC measurements in ytterbium similar to what was done for cesium is not possible due to limitations of atomic theory. Ytterbium has complicated electron structure and calculations for it on the same level of accuracy as for cesium are not possible now and in foreseen future. The aims of the PNC measurements in Yb are different [@BudkerYbPNC]: (i) to study the ratio of the PNC amplitudes for different isotopes, and (ii) to measure nuclear spin-dependent PNC effects, such as the effect of nuclear anapole moment. The study of the PNC for a chain of isotopes does not require atomic calculations and can deliver useful information about either neutron distribution or new physics beyond standard model (see, e.g. [@DFK86; @Fortson; @DP02; @BDF09]). The extraction of the anapole moment from the measurements does require atomic calculations, however, high theoretical accuracy is not critical here. Ytterbium is a very good candidate for both types of the experimental studies. It has seven stable isotopes with large difference in neutron numbers $\Delta N_{max} = 8$. Two of the isotopes, $^{171}$Yb and $^{173}$Yb, have non-zero nuclear spin provided by valence neutron. This is especially interesting since it allows one to measure the strength of the neutron-nucleus PNC potential [@anapole] (the anapole moment has been measured only for the $^{133}$Cs nucleus which has valence proton). Calculations of the spin-independent PNC in ytterbium were performed in Refs. [@DeMille; @Porsev95; @Das97]. Calculations of the spin-dependent PNC were reported in [@Singh99; @Porsev00]. The results of [@Singh99; @Porsev00] for the spin-dependent PNC amplitudes are presented as tables of reduced matrix elements of the spin-dependent weak interaction for different hyperfine transitions. This, in our view, leads to some difficulties in interpretation. Reduced matrix elements (RME) are very convenient for intermediate calculations. However, presenting final results in a form of RME may lead to confusion due to their unnatural symmetry properties: $$\langle F_a,a || \hat H|| F_b,b \rangle = (-1)^{F_a-F_b} \langle F_b,b || \hat H|| F_a,a \rangle^*. \label{eq:sym}$$ Here $F_a$ is the total momentum of the state $a$, asterisk means complex conjugation which in the case of PNC amplitudes means the change of sign. There is an apparent disagreement between the signs of different RME in [@Singh99] and [@Porsev00]. The most likely explanation for this in our view is that the authors of [@Singh99] and [@Porsev00] presented different RME, say $\langle a||\hat H|| b \rangle$ in [@Singh99] and $\langle b||\hat H||a \rangle$ in [@Porsev00]. At least all sign differences follow strictly the rule (\[eq:sym\]). Strictly speaking, the sign of an amplitude is not defined (since a wave function may be multiplied by an arbitrary phase factor), only the ratio of two amplitudes between the same states has definite sign. Neither of the work [@Singh99; @Porsev00] provides a link between the spin-dependent (SD) PNC amplitudes and spin-independent (SI) PNC amplitudes calculated earlier in [@Porsev95; @Das97]. This means that it is hard to say whether the spin-dependent effects increase or decrease a particular PNC amplitude. In other words, the sign of the spin-dependent interaction constants, such as the anapole moment, cannot be extracted from the measurements when using the calculations of [@Singh99] or [@Porsev00] and no additional assumptions (note that the apparent disagreement between the signs of the amplitudes in [@Singh99] and [@Porsev00] shows that any guesswork about the relative signs of the SI and SD amplitudes is unreliable). To avoid this problem, in present paper both spin-independent and spin-dependent PNC amplitudes are calculated simultaneously using the same procedure and the same wave functions. In this approach the relative sign of the amplitudes is fixed. This allows for unambiguous determination of the sign of the spin-dependent contribution. The constant of the spin-dependent interaction can be expressed via the ratio of the two amplitudes. This brings an extra advantage of more accurate interpretation of the measurements. The accuracy of the calculations for the ratio of the PNC amplitudes is higher than that for each of the amplitudes. This is because the amplitudes are very similar in nature and most of the theoretical uncertainty cancels out in the ratio. Theory ====== Hamiltonian describing parity-nonconserving electron-nuclear interaction can be written as a sum of spin-independent (SI) and spin-dependent (SD) parts (we use atomic units: $\hbar = |e| = m_e = 1$): $$\begin{aligned} H_{\rm PNC} &=& H_{\rm SI} + H_{\rm SD} \nonumber \\ &=& \frac{G_F}{\sqrt{2}} % (1) \Bigl(-\frac{Q_W}{2} \gamma_5 + \frac{\varkappa}{I} {\bm \alpha} {\bm I} \Bigr) \rho({\bm r}), \label{e1}\end{aligned}$$ where $G_F \approx 2.2225 \times 10^{-14}$ a.u. is the Fermi constant of the weak interaction, $Q_W$ is the nuclear weak charge, $\bm\alpha=\left( \begin{array} [c]{cc}% 0 & \bm\sigma\\ \bm\sigma & 0 \end{array} \right)$ and $\gamma_5$ are the Dirac matrices, $\bm I$ is the nuclear spin, and $\rho({\bf r})$ is the nuclear density normalized to 1. The strength of the spin-dependent PNC interaction is proportional to the dimensionless constant $\varkappa$ which is to be found from the measurements. There are three major contributions to $\varkappa$ arising from (i) electromagnetic interaction of atomic electrons with nuclear [*anapole moment*]{}, (ii) electron-nucleus spin-dependent weak interaction, and (iii) combined effect of spin-independent weak interaction and magnetic hyperfine interaction (see, e.g.  [@Ginges]). In this work we do not distinguish between different contributions to $\varkappa$ and present the results in terms of total $\varkappa$ which is the sum of all possible contributions. Within the standard model the weak nuclear charge $Q_W$ is given by [@PDG] $$%Q_W = -N + Z\,(1-4\,\sin^2\theta_W) . Q_W \approx -0.9877N + 0.0716Z.$$ Here $N$ is the number of neutrons, $Z$ is the number of protons. The PNC amplitude of an electric dipole transition between states of the same parity $|i\rangle$ and $|f \rangle$ is equal to: $$\begin{aligned} E1^{PNC}_{fi} &=& \sum_{n} \left[ \frac{\langle f | {\bm d} | n \rangle \langle n | H_{\rm PNC} | i \rangle}{E_i - E_n}\right. \nonumber \\ &+& \left.\frac{\langle f | H_{\rm PNC} | n \rangle \langle n | d_q | i \rangle}{E_f - E_n} \right], \label{eq:e2}\end{aligned}$$ where ${\bm d} = -e\sum_i {\bm r_i}$ is the electric dipole operator, $|a \rangle \equiv |J_a F_a M_a \rangle$ and ${\bm F} = {\bm I} + {\bm J}$ is the total angular momentum. Applying the Wigner-Eckart theorem we can express the amplitudes via reduced matrix elements $$\begin{aligned} E1^{PNC}_{fi} &=& (-1)^{F_f-M_f} \left( \begin{array}{ccc} F_f & 1 & F_i \\ -M_f & q & M_i \\ \end{array} \right) \nonumber \\ &\times& \langle J_f F_f || d_{\rm PNC} || J_i F_i \rangle .\end{aligned}$$ Detailed expressions for the reduced matrix elements of the SI and SD PNC amplitudes can be found e.g. in Refs. [@Porsev01] and [@JSS03]. For the SI amplitude we have $$\begin{aligned} &&\langle J_f,F_f || d_{\rm SI} || J_i,F_i \rangle = (-1)^{I+F_i+J_f+1}\nonumber \\ && \times \sqrt{(2F_f+1)(2F_i+1)} \left\{ \begin{array}{ccc} J_i & J_f & 1 \\ F_f & F_i & I \\ \end{array} \right\} \label{eq:si0}\\ && \times \sum_{n} \left[ \frac{\langle J_f || {\bm d} || n,J_n \rangle \langle n,J_n || H_{\rm SI} || J_i \rangle}{E_i - E_n}\right. \nonumber \\ && + \left.\frac{\langle J_f || H_{\rm SI} || n,J_n \rangle \langle n,J_n || {\bm d} || J_i \rangle}{E_f - E_n} \right] \nonumber \\ && \equiv c(F_f,J_f,F_i,J_i) E^{\prime}_{fi}. \nonumber\end{aligned}$$ Here $c(F_f,J_f,F_i,J_i)$ is the angular coefficient and the sum over $n$, $E^{\prime}_{fi}$ does not depend on $F_f$ or $F_i$: $$\begin{aligned} E^{\prime} &=& \sum_{n} \left[ \frac{\langle J_f || {\bm d} || n,J_n \rangle \langle n,J_n || H_{\rm SI} || J_i \rangle}{E_i - E_n}\right. \label{eq:si} \\ &+& \left.\frac{\langle J_f || H_{\rm SI} || n,J_n \rangle \langle n,J_n || {\bm d} || J_i \rangle}{E_f - E_n} \right]. \nonumber \end{aligned}$$ For the SD PNC amplitude we have $$\begin{aligned} && \langle J_f,F_f || d_{\rm SD} || J_i,F_i \rangle = \frac{G_F}{\sqrt{2}} \varkappa \nonumber \\ &&\times \sqrt{(I+1)(2I+1)(2F_i+1)(2F_f+1)/I} \nonumber \\ &&\times \sum_{n} \left[ (-1)^{J_f - J_i} \left\{ \begin{array}{ccc} J_n & J_i & 1 \\ I & I & F_i \\ \end{array} \right\} \left\{ \begin{array}{ccc} J_n & J_f & 1 \\ F_f & F_i & I \\ \end{array} \right\} \right. \nonumber \\ &&\times \frac{ \langle J_f || {\bm d} || n, J_n \rangle \langle n, J_n || {\bm \alpha}\rho || J_i \rangle }{E_n - E_i} \label{Eq:dsd} \\ &&+ (-1)^{F_f - F_i} \left\{ \begin{array}{ccc} J_n & J_f & 1 \\ I & I & F_f \\ \end{array} \right\} \left\{ \begin{array}{ccc} J_n & J_i & 1 \\ F_i & F_f & I \\ \end{array} \right\} \nonumber \\ &&\times \left. \frac{\langle J_f || {\bm \alpha}\rho ||n,J_n \rangle \langle n,J_n || {\bm d} ||J_i \rangle}{E_n - E_f} \right]. \nonumber\end{aligned}$$ In the case of the $^1$S$_0 - \rightarrow ^3$D$_1$ transition these expressions can be significantly simplified. Substituting $F_i=I$, $J_i=0$, $F_f =I,I \pm 1 \equiv F$, $J_f=1, J_n=1$ we have for the PNC amplitudes ($z$-components) $E_{F_i,F_f}$ of the transitions between specific hfs states of $^{171}$Yb ($I=1/2$) $$\begin{aligned} E_{\frac{1}{2},\frac{1}{2},z} &=& -\frac{1}{3}E^{\prime} Q_W - \sqrt{\frac{2}{27}} E^{\prime\prime}\varkappa, \label{1-2-1-2}\\ E_{\frac{1}{2},\frac{3}{2},z} &=& \sqrt{\frac{2}{9}}E^{\prime} Q_W - \sqrt{\frac{1}{27}} E^{\prime\prime}\varkappa. \label{1-2-3-2} \end{aligned}$$ Similar expressions for $^{173}$Yb ($I=5/2$) are $$\begin{aligned} E_{\frac{5}{2},\frac{3}{2},z} =&-& \sqrt{\frac{4}{45}}E^{\prime} Q_W - \sqrt{\frac{98}{3375}} E^{\prime\prime}\varkappa, \label{5-2-3-2}\\ E_{\frac{5}{2},\frac{5}{2},z} =&-& \sqrt{\frac{5}{21}}E^{\prime} Q_W - \sqrt{\frac{2}{315}} E^{\prime\prime}\varkappa, \label{5-2-5-2} \\ E_{\frac{5}{2},\frac{7}{2},z} =&& \sqrt{\frac{2}{21}}E^{\prime} Q_W - \sqrt{\frac{1}{63}} E^{\prime\prime}\varkappa \label{5-2-7-2}.\end{aligned}$$ Here $E^{\prime\prime}$ is the part of the SD PNC amplitude which is independent on $F_i$ and $F_f$: $$\begin{aligned} E^{\prime\prime} &=& \frac{G_F}{\sqrt{2}}\sum_n \left[ \frac{ \langle J_f || {\bm d} || n, J_n \rangle \langle n, J_n || {\bm \alpha}\rho || J_i \rangle }{E_n - E_i} \right. \nonumber \\ &-& \left. \frac{\langle J_f || {\bm \alpha}\rho ||n,J_n \rangle \langle n,J_n || {\bm d} ||J_i \rangle}{E_n - E_f} \right]. \label{eq:epp}\end{aligned}$$ Note that if at least two PNC amplitudes are measured then the value of $\varkappa$ can be expressed via the ratio $E^{\prime\prime}/E^{\prime}$ of the calculated SD and SI PNC amplitudes. This ratio is much less sensitive to numerical uncertainties than each of the amplitudes. The amplitudes are very similar. Therefore, a greater part of the numerical uncertainty cancels out in the ratio. For example, if amplitudes (\[1-2-1-2\]) and (\[1-2-3-2\]) are measured then $$\varkappa = \sqrt{3}\frac{\sqrt{2}R+1}{S(R-\sqrt{2})}, \label{eq:kappa}$$ where $R= E_{\frac{1}{2},\frac{1}{2},z}/E_{\frac{1}{2},\frac{3}{2},z}$ and $S=E^{\prime\prime}/(E^{\prime}Q_W)$. The ratio of theoretical amplitudes $E^{\prime\prime}/E^{\prime}$ is significantly more stable in the calculations than each of the amplitudes. Calculations ============ We consider ytterbium as an atom with two valence electrons above closed shells and use the combination of the configuration interaction and many-body perturbation theory (CI+MBPT, [@DzuFlaKoz96b]) to perform the calculations. The calculations are very similar to our previous calculations of ytterbium polarizabilities [@DD10]. Below we briefly describe the procedure emphasizing some minor differences. CI+MBPT method -------------- The effective CI+MBPT Hamiltonian for two valence electrons has the form $$\hat H^{\rm eff} = \hat h_1(r_1) + \hat h_1(r_2) + \hat h_2(r_1,r_2), \label{Heff}$$ where $\hat h_1$ is the single-electron part of the relativistic Hamiltonian $$\hat h_1 = c \mathbf{\hat{\alpha}} \mathbf{p} + (\hat{\beta}-1)m_e c^2-\frac{Ze^2}{r} + V^{N-2} + \hat \Sigma_1, \label{h1}$$ and $\hat h_2$ is the two-electron part of the Hamiltonian $$\hat h_2(r_1,r_2) = \frac{e^2}{|\mathbf{r}_1 - \mathbf{r}_2|} + \hat \Sigma_2(r_1,r_2). \label{h2}$$ In these equations, $\mathbf{\hat{\alpha}}$ and $\hat{\beta}$ are the conventional Dirac matrices, $V^{N-2}$ is the Dirac-Hartree-Fock (DHF) potential of the closed-shell atomic core ($N-2=68,Z=70$), and $\hat \Sigma$ is the correlation operator. It represents terms in the Hamiltonian arising due to virtual excitations from atomic core (see Ref. [@DzuFlaKoz96b; @DzuJoh98] for details). $\hat \Sigma \equiv 0$ corresponds to the standard CI method. $\hat \Sigma_1$ is a single-electron operator. It represents a correlation interaction (core-polarization) of a particular valence electron with the atomic core. $\hat \Sigma_2$ is a two-electron operator. It represents screening of the Coulomb interaction between the two valence electrons by the core electrons. We calculate $\hat \Sigma$ in the second order of the MBPT. We use a B-spline technique [@JohSap86] to construct a complete set of single-electron orbitals. We use 40 B-splines in a cavity of radius $R=40\, a_B$ and calculate the eigenstates of the $V^{N-2}$ DHF Hamiltonian up to the maximum value of the angular momentum $l_{max}=5$. The same basis is used in computing $\hat \Sigma$ and in constructing the two-electron states for the valence electrons. 40 out of 60 lowest-energy states for every $l$ up to $l_{max}=5$ are used to calculate $\hat \Sigma$ and 16 lowest states above the core are used for every $l$ up to $l_{max}=4$ to construct the two-electron states. The two-electron valence states are found by solving the eigenvalue problem, $$\hat H^{\rm eff} \Psi_v = E_v \Psi_v \, , \label{CI}$$ using the standard CI techniques. Calculated and experimental energies of a few lowest-energy states of Yb can be found in Ref. [@DD10]. The pure [*ab initio*]{} energies are already close to the experimental values. However, for improving the accuracy further, we re-scale the correlation operator $\hat \Sigma_1$ by replacing $\hat \Sigma_1$ in the effective Hamiltonian (\[Heff\]) in each partial wave $s, p_{1/2}, p_{3/2} , \ldots$ by $f_a\hat \Sigma_{1}$. The rescaling factors are $f_s=0.875$, $f_p=1.268$, $f_d=0.935$, and $f_f=1$. These values are chosen to fit the experimental spectrum of Yb. Some differences in scaling parameters compare to what was used in Ref. [@DD10] is due to the fact that in present work we have fitted exactly the energy of the $^1$P$^o_1$ state while in  [@DD10] we fitted the energy of the $^3$P$^o_1$ state. Dalgarno-Lewis and RPA methods ------------------------------ Matrix elements are found with the random-phase approximation (RPA) [@DzuGin06; @DzuFla07] $$E1_{vw} = \langle \Psi_v || \hat f + \delta V^{N-2} || \Psi_w \rangle, \label{E1}$$ where $\delta V^{N-2}$ is the correction to the core potential due to core polarization by an external field $\hat f$. In present calculations $\hat f$ represents either external electric field, SI weak interaction or SD PNC interaction. Computing PNC requires summing over a complete set of two-electron states (see, e.g. Eq. (\[eq:e2\])). We use the Dalgarno-Lewis method [@DalLew55] for the summation. In this method, a correction $\delta \Psi_v$ to the two-electron wave function of the state $v$ is introduced and the amplitude is reduced to $$A_{vw} = \langle \delta \Psi_v ||\hat f_1|| \Psi_w \rangle \, . \label{eq:deltapsi}$$ The correction $\delta \Psi_v$ is found by solving the system of linear inhomogeneous equations $$(\hat H^{\rm eff} - E_v )\delta \Psi_v = - (\hat f_2+\delta V^{N-2}) \Psi_v. \label{eq:DL}$$ Here $\hat f_1$ and $\hat f_2$ are electric dipole and PNC interaction operators ($\hat f_1 = {\bm d}, \hat f_2 = H_{\rm PNC}$ or vice versa). Accuracy of the calculations ---------------------------- State Calculations Experiment ------------- -------------- ------------ $^3$P$^o_1$ 4460 3958 $^1$P$^o_1$ -819 -1094 : Magnetic dipole hyperfine structure constants $A$ (MHz) for the $^3$P$^o_1$ and $^1$P$^o_1$ states of $^{171}$Yb, comparison with experiment.[]{data-label="t:hfs"} Accuracy of very similar calculations of polarizabilities of ytterbium were studied in detail in our previous work [@DD10] and were found to be about 5%. However, we cannot claim the same accuracy for present calculations due to two important differences. First, there is a resonance contribution to the PNC amplitude involving the $^1$P$^o_1$ state. Energy interval between the $^3$D$_1$ and $^1$P$^o_1$ states is very small. Its experimental values is just 579 cm$^{-1}$. The term in (\[eq:e2\]) involving the $^1$P$^o_1$ state gives more than 80% of the total PNC amplitude. Even very accurate calculations may give significantly different value of small energy interval which would lead to large error in the PNC amplitude. One way around this problem is to separate the resonance term from the rest of the sum and use the experimental energy for the denominator. We use a technically more simple procedure. We have rescaled the correlation operator $\hat \Sigma$ to fit the interval exactly. As a result, the contribution of the error in the energy denominator to the error in the amplitude is small. Another important difference of present calculations from those of Ref. [@DD10] is that we need to calculate matrix elements of weak interaction which are sensitive to the wave functions on short distances. A way to test the wave functions on short distances is to calculate hyperfine structure (hfs) constants. Calculated and experimental values of the magnetic dipole hyperfine structure constants $A$ for the $^3$P$^o_1$ and $^1$P$^o_1$ states of $^{171}$Yb are presented in Table \[t:hfs\]. The first calculated hfs constant is larger than the experimental one by 13%, the second is smaller by 25%. The reason for the calculated hyperfine constant of the $^1$P$^o_1$ state to differ significantly from the the experimental value is the same as for the electric dipole transition amplitude between this and ground state - the admixture of the $4f^{13}5d6s^2 \ (7/2,3/2)^o_1$ state at $E=28857 {\rm cm}^{-1}$ (see Ref. [@DD10] for details). This admixture is small. However, it can change hfs of the $^1$P$^o_1$ state significantly due to the large hfs in the admixed state. In contrast, it cannot change that much the weak matrix element between the $^1$P$^o_1$ and $4f^{14}5d6s \ ^3$D$_1$ states. This is because the transition between the $4f^{13}5d6s^2$ and the $4f^{14}5d6s \ ^3$D$_1$ states in zero approximation is the $ 6s \rightarrow 4f$ transition and corresponding weak matrix element is zero. Therefore, poor accuracy for the hfs of the $^1$P$^o_1$ state is not a good indicator for the accuracy of the PNC calculations. A 13% error in the hfs of the $^3$P$^o_1$ state gives a more realistic estimate for the uncertainty. We stress that the uncertainty in the ratio of SD and SI PNC amplitudes ($E^{\prime\prime}/E^{\prime}$) is significantly lower. Tests show that this ratio is three to five times less sensitive to the variation of the calculation procedure than each of the amplitudes. We believe that 10% is a reasonable estimate for the theoretical uncertainty for this ratio. Results ======= ----- ----- ------- ------- ----------------------------------- ---------------------------- $A$ $I$ $F_1$ $F_2$ 171 0.5 0.5 0.5 $-(1/3)(1-0.0161\varkappa)$ $ 6.15(1-0.0161\varkappa)$ 0.5 1.5 $\sqrt{2/9}(1+0.0081\varkappa)$ $-8.70(1+0.0081\varkappa)$ 173 2.5 2.5 1.5 $-\sqrt{4/45}(1-0.0111\varkappa)$ $5.61(1-0.0111\varkappa)$ 2.5 2.5 $-\sqrt{5/21}(1-0.0032\varkappa)$ $9.18(1-0.0032\varkappa)$ 2.5 3.5 $\sqrt{2/21}(1+0.0079\varkappa)$ $-5.81(1+0.0079\varkappa)$ ----- ----- ------- ------- ----------------------------------- ---------------------------- : PNC amplitudes ($z$-components) for the $|6s^2,^1$S$_0,F_1 \rangle \rightarrow |6s5d,^3$D$_1,F_2\rangle$ transitions in $^{171}$Yb and $^{173}$Yb in units of $E^{\prime}Q_W$ and $10^{-9} iea_0$.[]{data-label="t:1"} Calculations give the following value of the spin-independent PNC amplitude of the $^1$S$_0 \rightarrow ^3$D$_1$ transition in ytterbium: $$E_z^{\rm SI-PNC} = 1.123 \times 10^{-11} Q_W iea_0. \label{ezprime}$$ This corresponds to the following value of the reduced matrix element $$E^{\prime} = 1.945 \times 10^{-11} iea_0. \label{eprime}$$ The electron ($F$-independent) part of the reduced matrix element of the spin-dependent PNC amplitude is found to be $$E^{\prime\prime} = 3.648 \times 10^{-11} iea_0, \label{eprime2}$$ The effect of different nuclear size for $^{171}$Yb and $^{173}$Yb is only 0.1% for both SI and SD PNC amplitudes. It is neglected in (\[ezprime\]), (\[eprime\]) and (\[eprime2\]). We use Fermi-type distribution for nuclear density $\rho$ with nuclear radius $R_N=6.35$ fm for $^{171}$Yb and $R_N=6.37$ fm for $^{173}$Yb [@Angeli]. The ratios of the SD and SI PNC amplitudes are $$\begin{aligned} E^{\prime\prime}/(E^{\prime}Q_W) = -0.0198(20) &{\rm for}& ^{171}{\rm Yb}, \label{eprime4} \\ E^{\prime\prime}/(E^{\prime}Q_W) = -0.0194(20) &{\rm for}& ^{173}{\rm Yb}. \label{eprime5}\end{aligned}$$ The difference in these values is due to different weak nuclear charge $Q_W$ ($Q_W = -94.75$ for $^{171}$Yb and $Q_W =-96.72$ for $^{173}$Yb). The difference is within numerical uncertainty. The results for the specific PNC amplitudes between different hyperfine structure states of $^{171}$Yb and $^{173}$Yb are presented in Table \[t:1\]. These results are obtained by substituting (\[eprime\]) and (\[eprime2\]) into (\[1-2-1-2\]), (\[1-2-3-2\]), (\[5-2-3-2\]), (\[5-2-5-2\]) and (\[5-2-7-2\]). The numerical factors before $\varkappa$ are proportional to ($E^{\prime\prime}/E^{\prime}$). The theoretical uncertainty for these factors is about 10%. The expressions from the table or equations (\[1-2-1-2\]), (\[1-2-3-2\]), (\[5-2-3-2\]), (\[5-2-5-2\]), (\[5-2-7-2\]) together with (\[eprime\]) and (\[eprime2\]) can be used to extract the value of $\varkappa$ from the measurements. For example, for $^{171}$Yb Eq. (\[eq:kappa\]) becomes $$\varkappa = 88(9)\frac{1+\sqrt{2}R}{(\sqrt{2}-R)}. \label{eq:kappa1}$$ Comparison with other calculations ---------------------------------- Calculations of the spin-independent part of the PNC amplitude for ytterbium were performed before in Refs. [@DeMille; @Porsev95; @Das97]. The spin-dependent amplitudes were calculated before in Refs. [@Singh99; @Porsev00]. It is convenient to compare the results in terms of $E^{\prime}$ (\[eq:si\]) and $E^{\prime\prime}$ (\[eq:epp\]) since these values are the same for all hfs transitions. Refs. [@DeMille; @Porsev95; @Das97] present the values of the $z$-component of the SI PNC amplitude. Refs. [@Singh99; @Porsev00] present reduced matrix elements of the SD PNC interaction for each hfs transition. Corresponding values of $E^{\prime}$ and $E^{\prime\prime}$ can be easily extracted from this data using formulas of present paper. The absolute values of the amplitudes are presented in Table \[t:2\]. We have excellent agreement for $E^{\prime}$ with DeMille [@DeMille] and Porsev [*et al*]{} [@Porsev95] while the result of Das [@Das97] is about 30% smaller. We have good agreement with both Porsev [*et al*]{} [@Porsev00] and Singh and Das [@Singh99] for $E^{\prime\prime}$. The difference between Ref. [@Porsev00] and our result is 12% which is within our uncertainty. The difference between our results for $E^{\prime\prime}$ and those of Ref. [@Singh99] is even smaller. But this is probably accidental. Note however that we agree with Singh and Das [@Singh99] on small, practically negligible change of $E^{\prime\prime}$ from $^{171}$Yb to $^{173}$Yb while Porsev [*at al*]{} [@Porsev00] report a 3% increase. Such increase has no physical explanation and must be a numerical effect. In our experience such effect can be a result of just one RPA iteration after a change of nuclear radius from $^{171}$Yb to $^{173}$Yb. Further iterations kill the difference. However, it is up to the authors of [@Porsev00] to explain their results. Note again that in Table  \[t:2\] we present only the absolute values of the amplitudes, ignoring their signs. This is because the sign of an amplitude is not fixed and has no physical meaning. However, the relative sign of the SI and SD PNC amplitudes is not arbitrary. The SD dependent part of the PNC amplitude must either increase or decrease the transition amplitude depending on the sign of $\varkappa$. It is important to know the relative sign of the amplitudes to be able to extract the sign of $\varkappa$ from the measurements. To fix the relative sign of the two PNC amplitudes one should calculate them using the same wave functions. This is how it is done in present work (see Table \[t:1\] and formulas (\[1-2-1-2\]), (\[1-2-3-2\]), (\[5-2-3-2\]), (\[5-2-5-2\]), (\[5-2-7-2\])). Another important advantage of the simultaneous calculation of both amplitudes is that $\varkappa$ can be expressed via the ratio of the amplitudes. This ratio has much smaller theoretical uncertainty than each of the amplitudes (see previous section for discussion). Unfortunately, both previous calculations of the SD PNC amplitude in Yb [@Singh99; @Porsev00] do not compare their results with the earlier calculations of the SI PNC amplitudes [@Porsev95; @Das97] which could be performed using different wave functions. This leads to the uncertainty of the relative signs and larger errors in the ratios of the SI and SD amplitudes which are needed to extract the value of $\varkappa$ from the measurements. -------------------- ------ ------ ------ DeMille [@DeMille] 1.9 Porsev [*et al*]{} 1.97 4.13 4.27 Singh and Das 1.33 3.68 3.67 This work 1.95 3.65 3.64 -------------------- ------ ------ ------ : Spin-independent ($E^{\prime}$) and spin-dependent ($E^{\prime\prime}$) parts of the PNC amplitude (reduced matrix elements) for the $|6s^2,^1$S$_0 \rangle \rightarrow |6s5d,^3$D$_1\rangle$ transition in ytterbium, comparison with other calculations. The signs of the amplitudes are omitted.[]{data-label="t:2"} Comparison with experiment -------------------------- The result of the measurement of the PNC amplitude of the $^1$S$_0 \rightarrow ^3$D$_1$ transition in $^{174}$Yb reported in [@BudkerYbPNC] reads $$|E^{\rm PNC}| = 8.7(1.4) \times 10^{-10} ea_0. \label{eq:exp}$$ Assuming a 13% theoretical uncertainty and substituting weak nuclear charge $Q_W=-97.71$ we get from (\[ezprime\]) the following theoretical value for the amplitude $$|E^{\rm PNC}| = 11.0(1.4) \times 10^{-10} ea_0. \label{eq:theor}$$ The values of (\[eq:exp\]) and (\[eq:theor\]) agree within the declared uncertainty. To measure the constant of spin-dependent PNC interaction ($\varkappa$) more accurate measurements are needed for $^{171}$Yb or $^{173}$Yb. The work is in progress at Berkeley [@BudkerYbPNC]. Conclusion ========== We present simultaneous calculation of the spin-independent and spin-dependent PNC amplitudes of the $6s^2 \ ^1$S$_0 \rightarrow 6s5d \ ^3$D$_1$ transition in ytterbium. The results are to be used for accurate interpretation of future measurements in terms of the parameter of the spin-dependent PNC interaction $\varkappa$. Both, sign and value of $\varkappa$ can be determined. Theoretical uncertainty is at the level of 10%. The authors are grateful to M. G. Kozlov and S. G. Porsev for useful discussions. The work was supported in part by the Australian Research Council. [99]{} D. DeMille, Phys. Rev. Lett. [**74**]{}, 4165 (1995). C. J. Bowers, D. Budker, S. J. Freedman, G. Gwinner, J. E. Stalnaker, and D. DeMille, Phys. Rev. A [**59**]{}, 3513 (1999). D. F. Kimball, D. Clyde, D. Budker, D. DeMille, S. J. Freedman, S. Rochester, J. E. Stalnaker, and M. Zolotorev, Phys. Rev. A [**60**]{}, 1103 (1999). J. E. Stalnaker, D. Budker, D. P. DeMille, S. J. 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--- abstract: 'We present a very general geometrico-dynamical description of physical or more abstract entities, called the *general tension-reduction* (GTR) model, where not only states, but also measurement-interactions can be represented, and the associated outcome probabilities calculated. Underlying the model is the hypothesis that indeterminism manifests as a consequence of unavoidable fluctuations in the experimental context, in accordance with the *hidden-measurements interpretation* of quantum mechanics. When the structure of the state space is Hilbertian, and measurements are of the *universal* kind, i.e., are the result of an average over all possible ways of selecting an outcome, the GTR-model provides the same predictions of the Born rule, and therefore provides a natural completed version of quantum mechanics. However, when the structure of the state space is non-Hilbertian and/or not all possible ways of selecting an outcome are available to be actualized, the predictions of the model generally differ from the quantum ones, especially when sequential measurements are considered. Some paradigmatic examples will be discussed, taken from physics and human cognition. Particular attention will be given to some known psychological effects, like question order effects and response replicability, which we show are able to generate non-Hilbertian statistics. We also suggest a realistic interpretation of the GTR-model, when applied to human cognition and decision, which we think could become the generally adopted interpretative framework in quantum cognition research.' author: - | Diederik Aerts$^1$ and Massimiliano Sassoli de Bianchi$^{2}$\ *$^1$ Center Leo Apostel for Interdisciplinary Studies\ *Brussels Free University, 1050 Brussels, Belgium\ E-Mail: [diraerts@vub.ac.be](diraerts@vub.ac.be)\ *$^2$ Laboratorio di Autoricerca di Base\ *6914 Lugano, Switzerland\ E-Mail: [autoricerca@gmail.com](autoricerca@gmail.com)\ **** title: 'The GTR-model: a universal framework for quantum-like measurements' --- : Probability, Hidden-measurements, Degenerate measurements, Hidden-variables, Born rule, Bloch sphere, Order effects, Response replicability, Quantum cognition.\ Introduction ============ Probability is the key notion used by scientists of different disciplines to quantify, in a meaningful and optimal way, their lack of knowledge regarding certain properties of the systems under study. Before the advent of quantum mechanics, only classical probabilities were taken into consideration, based on the structure of Boolean algebra and obeying the classical Kolmogorovian axioms [@Kolmogoroff1933]. On the other hand, quantum probabilities, which are based on a different structure of the experimental propositions, cannot be represented in a fixed probability space (if multiple measurements are considered), and therefore generally disobey the Kolmogorovian axioms. However, the difference between classical and quantum probabilities is not in a simple correspondence with the difference between macroscopic and microscopic physical entities. Indeed, also macroscopic entities can behave in a quantum-like way, depending on the nature of the experimental actions that an experimenter is considering in relation to them. In other terms, quantum and classical probabilities can always emerge from our experimental investigations, depending on the *structure of possibilities* that are taken into consideration. This means that classic and quantum probabilities should be considered as special cases of more general probability structures, which can simply be called (depending on the context) non-classical, non-quantum, or quantum-like. These more general structures are non-classical in the sense that they do not obey the Kolmogorovian axioms, and they are quantum-like in the sense that, similarly to quantum probabilities, they are based on logical connectives that are *dynamical*, i.e., describing the possible outcomes of *actions* (measurements) that can be performed on the different entities. However, they are also non-quantum, in the sense that they are not necessarily purely quantum, as the structure of the associated state space is not necessarily Hilbertian and the probability values are not necessarily those predicted by the Born rule. From the viewpoint of physics, the interest of investigating more general probability models lies for instance in the possibility of shedding new light onto the problem of the semiclassical limit, i.e., in the understanding of the transition from pure quantum to pure classical regimes. Indeed, if it is true that classical and quantum probabilities are based on different, not commensurable structures, obtained as different limits of more general quantum-like situations, then it is also clear that we need more general models, of a “mixed” quantum-classical kind, if we want to describe the mesoscopic regions of our reality, which cannot be incorporated within the pure quantum or pure classical limit models. A preliminary analysis of these intermediary regions of reality, described by non-classic and non-quantum probabilities, was carried out in some detail by one of us and his collaborators in the past decades, using a paradigmatic model called the $\epsilon$-model [@Aerts1998; @SassolideBianchi2013a]: a generalization of a two-level (qubit) system where an additional real parameter $\epsilon$ can be continuously varied, from $0$ to $1$, so as to produce a non-singular classic-to-quantum transition. Even though the initial motivation in studying these more general probability models came from physics, it became clear early on that their significance went beyond the field of physics, and was of great interest also in the description of human cognitive processes, as today studied in the new emerging field of theoretical and experimental investigation called *quantum cognition* [@Kitto2008; @Khrennikov2010; @BusemeyerBruza2012; @AertsEtal2013a; @AertsEtal2013b; @PothosBusemeyer2013; @WangEtal2013; @BlutnerbeimGraben2014]. Quantum cognition resulted, among other things, from the observation that human concepts, understood as abstract entities interacting with human minds that are sensitive to their meanings, can produce highly contextual dynamics, impossible to explain by only using traditional modelizations in terms of logico-rational thinking processes, and therefore classical probabilities. Similarly, quantum mechanics was historically created, as a mathematical theory, to offer a consistent description of entities whose behavior appeared also to be highly contextual. Therefore, it was quite natural at some point to assume that the quantum formalism could also play a role in the modelization of human cognition and decision-making. This intuition was followed by an increasing number of successful applications of the quantum formalism, and today it is a well-established hypothesis that, in addition to our classical logical layer, describable by classical probability theory, there is an additional quantum conceptual layer, describable by standard quantum mechanics [@AertsSozzoVeloz2015]. This quantum description of the human cognitive behavior, however, has nothing to do with the fact that our human brains would be quantum machines. Quantum cognition is not concerned with the modeling of human brains as quantum computers, but with the possibility of using quantum probabilities, and the structure of Hilbert spaces, to elucidate the working of our mental processes, particularly those mistakenly understood as irrational. Clearly, the contextuality of human concepts mirrors that of elementary quantum entities. Consider for instance the problem of concept combination, i.e., the problem that not all concept combinations will have an intersective semantics, so that the meaning of a combination of concepts will not always be reducible to the meaning of the individual concepts forming the combination, as new meanings are constantly able to emerge. This emergence effect can be naturally described within the quantum formalism by means of the superposition principle and the related constructive and destructive interference effects, which can explain the observed overextension and underextension of the probabilities (with respect to the classical predictions). The superposition principle is in turn a key ingredient in the creation of entangled states that can be used to describe the situation of concepts connected through meaning, which are able to violate Bell’s inequalities in a way similar to quantum microscopic entities [@Aertsetal2000; @AertsSozzo2011]. Other fundamental aspects of the quantum formalism, like for instance quantum field many particles dynamics, can be also be exploited to describe typical situations where human judgments and decisions are at play [@AertsSozzo2015]. This “unreasonable” success of quantum mathematics in the modelization of human cognitive and decision situations requires of course to be widely explained. At the same time, one needs to investigate what are its limits, i.e., to what extent the standard quantum formalism can be used to model all sorts of cognitive situations. These two issues are intimately related. Indeed, there are no a priori reasons for the contextuality built-in in the standard Hilbertian formalism to be exactly the same (in terms of structure) as that incorporated in our human mental processes, also because the latters describe a (non-physical, mental) layer of our reality which, evolutionarily speaking, is much younger than the layer of the fundamental physical processes. This means that, starting from a more general model, containing both the quantum and classical regimes as special situations, one should be able to explain why certain aspects of the quantum formalism, in particular the Born rule, are so effective in describing many empirical data, and at the same time insufficient to model many others, considering that a Hilbert space, equipped with the Born rule, necessarily imposes some specific constraints (like the QQ-equality introduced by @WangBusemeyer2013), that can be violated by our complex cognitive and decisional processes. To provide a convincing explanation of both the success of quantum probabilities, and their lack of universality, the present article is organized as follows. In Sec. \[Measuring a coin\], we use the simple example of a coin flipping experiment to motivate a general description of an entity that gives rise to a general measurement model, called the *general tension-reduction model* (GTR-model). The model was recently derived in the ambit of quantum cognition studies [@AertsSassolideBianchi2015a; @AertsSassolideBianchi2015b]), but is of great interest also for physics, as it offers a non-circular derivation of the Born rule and therefore constitutes a possible solution to the measurement problem [@AertsSassolideBianchi2014a; @AertsSassolideBianchi2015c]. Our strategy here is not that of repeating our previous more formal derivations of the model, but to motivate its construction starting from more qualitative and general considerations. More precisely, inspired by our analysis of the coin flipping measurement of Sec. \[Measuring a coin\], the GTR-model will be introduced in Sec. \[The general tension-reduction (GTR) model\]. In Sec. \[Degenerate measurements in the GTR-model\], it will be shown to allow for the description of experiments where some of the outcomes can be degenerate and, in Sec. \[Composite entities in the GTR-model\], we show that it can naturally handle also the situation of composite entities. In Sec. \[The quantum mechanical example\], we explain how the Born rule can be deduced, when a huge (universal) average is performed over all possible kinds of measurements, showing that the Born rule can be interpreted as a first order approximation of a more general theory, thus explaining its great success also in the description of cognitive experiments. In Sec. \[Non-Kolmogorovian non-Hilbertian structures\], we explicitly show that the GTR-model can describe more general structures than the Kolmogorovian and Hilbertian ones and, in Sec. \[The human cognition example\], we apply the model to human cognition, showing that one needs its full structural richness to described some of the experimental data, like question order effects and response replicability. Finally, in Sec. \[Conclusion\], we offer a few concluding remarks. Measuring a coin {#Measuring a coin} ================ Consider the process that consists in flipping a coin onto the floor. If we are interested in knowing the final upper face of the coin, three possible outcomes have to be distinguished: “head,” “tail” and “edge,” and to these three outcomes three different probabilities can be associated: $P({\rm h})$, $P({\rm t})$ and $P({\rm e})$. In the case of an *American nickel*, their typical experimental values are [@Murray1993]: $$P({\rm h})={2999.5\over 6000},\quad P({\rm t})={2999.5\over 6000}, \quad P({\rm e})={1\over 6000}. \label{American nickel}$$ Flipping a coin onto the floor is a simple action producing a non-predeterminable outcome, and the same is true when we toss a die, draw a ball from a urn, etc. All these simple experiments, called *chance games*, have been largely used in the past to study the logic of probabilities, and culminated in modern classical probability theory, axiomatized by @Kolmogoroff1933. However, the actions associated with chance games are of a very special kind, and one should expect a probability model derived from their analysis to also be a very special model, not necessarily able to describe all the probability structures that can emerge from our experiments. To explain in which sense the random processes traditionally studied by classical probability theory are special, take the coin flipping example. The associated probabilities will depend in part on how the coin is manufactured. For instance, if we have exactly three possible outcomes, this is because there are three distinct faces (two flat faces and a curved one), and the values of the probabilities certainly also depend, in part, on their relative surfaces, on the exact location of the center of mass of the coin, and so on. These are the so-called *intrinsic* properties of the coin, i.e., the attributes it always possesses, in a stable and permanent way (at least for as long as the coin exists). But a coin, as a physical entity, is not only described by its intrinsic (always actual) properties: its condition is also determined by those properties that can contextually change over time, like its position and orientation in space, its linear and angular momentum, its temperature, etc. Now, a process like that of flipping a coin is special because it is usually so conceived that we cannot learn anything about its non-intrinsic properties from the obtained probabilities. In other terms, flipping a coin is an experiment which tells us nothing about the *state* of the coin prior to its execution: there is no *discovery aspect* involved, but only a *creation aspect*. In [@AertsSassolideBianchi2015a; @AertsSassolideBianchi2015b], we have named these special – creation only – processes, *solipsistic measurements*, with the term “solipsistic” used in a metaphorical sense, to express the idea that the measurement tells us nothing about the pre-measurement state of the entity, but only about the measurement process itself. Note that in the following we shall use the terms “measurement” and “experiment” almost interchangeably, being clear that a measurement is an experiment aimed at the observation of a given quantity, and that also the flipping of a coin can be interpreted as a measurement process, whose outcomes are the values taken by a quantity called the “upper face” of the coin. It is worth observing that solipsistic measurements have a truly remarkable property: being totally insensitive to the initial state of the entity,[^1] they are necessarily all *mutually compatible experiments*. Not in the sense that they can be performed at the same time (almost no experiments have this remarkable property), but in the sense that the order with which they are carried out is irrelevant. This because the final state obtained by the first measurement cannot influence the outcome of the second one, and vice versa. It is then clear why the Kolmogorovian probability model, which is founded on the paradigm of the solipsistic measurements, is unable to account for the quantum probabilities: quantum measurements, from which quantum probabilities result, are non-solipsistic indeterministic processes, producing statistics of outcomes which strongly depend on the initial pre-measurement states, and therefore cannot in general be mutually compatible measurements. Let us exploit a bit further the evocative example of the coin to see how we can go from solipsistic measurements to a more general class of measurements, to model more general (non-Kolmogorovian) probability situations. Clearly, an experimenter is free to conceive different measurements, by simply defining different observational protocols. In the case of the coin, one can for instance consider different ways to produce its flipping. Solipsistic measurements, as is known, correspond to a situation where the experimenter has to flip the coin with a vigorous momentum, on a sufficiently hard floor. On the other hand, if the experimenter decides to flip the coin in a less vigorous way, or onto a softer surface, or even a sticky one, it is easy to imagine that the statistics of outcomes will start depending on how the coin is initially positioned before the flipping, for instance on the bottom of the dice cup that is used to produce the shot. To model this possibility, we need to find a representation that allows us to express a dependency of the probabilities on the pre-measurement state. A very simple idea would be to represent the state of the coin directly in terms of the associated outcome probabilities. Of course, by doing so we will not be able anymore to describe solipsistic measurements (as is clear that for them the different initial states are all associated with the same outcome probabilities), but let us explore anyway this idea, as it will show us the path for its natural generalization. So, let us assume that we have chosen a given flipping protocol (i.e., a given measurement), and that by repeating the experiment many times, with the coin always in the same initial state inside the cup, we have obtained the three outcome probabilities $P({\rm h})$, $P({\rm t})$ and $P({\rm e})$. The idea is to describe the initial state of the coin as an abstract point-particle in ${{\mathbb R}}^3$, with position ${\bf x}=(P({\rm h}), P({\rm t}), P({\rm e}))$, i.e., as a point-particle whose coordinates are precisely the outcome probabilities. Since $P({\rm h})+P({\rm t})+P({\rm e})=1$, it follows that ${\bf x}$ belongs to a two-dimensional regular simplex $\triangle_2$, i.e., to an equilateral triangle of side $\sqrt{2}$. We obtain in this way a simple and natural geometric representation of the probabilities characterizing the measurement under investigation. However, as we said, this representation cannot be used to describe solipsistic measurements, as two states ${\bf x}$ and ${\bf x}'$, if different, will necessarily be associated with different outcome probabilities. In fact, this representation doesn’t allow, neither, to describe *deterministic* measurements, such that for a given initial state the outcome would be predetermined. These are experiments such that the protocol allow the experimenter to flip the coin in a perfectly controlled (fluctuation free) way, so as to know in advance what will be the final state, given the initial one. If we want to obtain a representation that can be used to also represent solipsistic and deterministic measurements, we thus need to find a way to describe the state of the entity independently from the outcome probabilities. For this, we need to introduce in our model some additional *elements of reality*, describing the *interactions* between the measured entity and the measuring system, i.e., between the coin and the cup-floor system, in our example. To do so, we start by observing that the point ${\bf x}$ exactly defines three disjoint triangular sub-regions in $\triangle_2$, which we will call $A_{\rm h}$, $A_{\rm t}$ and $A_{\rm e}$ (see the first drawing of Fig. \[triangolo\]). A simple geometric calculation then shows that [@Aerts1986; @AertsSassolideBianchi2014a; @AertsSassolideBianchi2015a; @AertsSassolideBianchi2015b]: $$P({\rm h})={\mu(A_{\rm h})\over \mu(\triangle_2)}, \quad P({\rm t})={\mu(A_{\rm t})\over \mu(\triangle_2)},\quad P({\rm e})={\mu(A_{\rm e})\over \mu(\triangle_2)}, \label{relative Lebesgue}$$ where $\mu$ denotes the Lebesgue measure. In other terms, the relative areas of the three sub-regions defined by the state-vector ${\bf x}$ are exactly the outcome probabilities. To exploit this remarkable geometric property of simplexes, we are now going to describe a *tension-reduction process* that will allow us to represent, in an abstract way, the (possibly) indeterministic part of a measurement process. Assume that $\triangle_2$ is an elastic and disintegrable membrane, stretched between its three vertex points – let us call them ${\bf x}_{\rm h}$, ${\bf x}_{\rm t}$ and ${\bf x}_{\rm e}$, respectively – and that the state of the entity is represented by a point-particle firmly attached to the membrane, at some point ${\bf x}$. Assume also that the line segments separating the three regions $A_{\rm h}$, $A_{\rm t}$ and $A_{\rm e}$ are like “tension lines," along which the membrane can less easily disintegrate. Then, consider the following process: the membrane disintegrates, at some unpredictable point . If $\mbox{\boldmath$\lambda$}\in A_{\rm h}$, the disintegrative process propagates inside the entire sub-region $A_{\rm h}$, but not in the other two sub-regions $A_{\rm t}$ and $A_{\rm e}$, because of the tension lines. This will cause the two anchor points of $A_{\rm h}$, ${\bf x}_{\rm t}$ and ${\bf x}_{\rm e}$, to tear away, and being the membrane elastic, it will consequently collapse toward the remaining anchor point ${\bf x}_{\rm h}$, drawing in this way the point particle (which is attached to it) to the same final position, representative of the outcome “head.” Similarly, if the initial disintegration point happens in $A_{\rm t}$, the final outcome will be “tail,” represented by the vector ${\bf x}_{\rm t}$ (see Fig. \[triangolo\]), and if the initial disintegration point happens in $A_{\rm e}$, the final outcome will be “edge,” represented by the vector ${\bf x}_{\rm e}$. If the membrane is uniform, it is clear that the probability to obtain outcome ${\bf x}_{\rm h}$ is just given by the relative area of sub-region $A_{\rm h}$, which according to (\[relative Lebesgue\]) corresponds to the first component of state vector ${\bf x}$, and similarly for the other two outcomes. Of course, a uniformly disintegrable membrane is a very special situation, and a priori an (uncountable) infinity of different membranes, characterized by different *ways of disintegrating*, can be considered, like for instance those giving rise to solipsistic measurements, whose outcome probabilities are independent of the initial state ${\bf x}$. These can be understood as the limit of membranes that become less and less disintegrable in their interior points and more and more disintegrable in the points belonging only to their three edges, so that the position of the point particle on $\triangle_2$, representative of the initial state, becomes irrelevant in the determination of the outcome probabilities, which will only depend on the probabilities that the initial disintegration happens in one of the three edges of $\triangle_2$. More precisely, if the disintegration probability of the edge opposite to ${\bf x}_{\rm h}$ is $P({\rm h})$, then this will also be the probability for outcome ${\bf x}_{\rm t}$, for (almost) all initial states, and similarly for the other two edges. The membranes describing deterministic measurements, on the other hand, can be understood as the limit of membranes becoming more and more disintegrable in a sub-region that becomes increasingly small and less and less disintegrable everywhere else. Indeed, in this limit we obtain a structure which almost surely will start disintegrating in a single predetermined point . Then, for almost all initial states one can predict the outcome in advance, with certainty. Indeed, if the initial state ${\bf x}$ is such that $\mbox{\boldmath$\lambda$}\in A_{\rm h}$, the outcome will be ${\bf x}_{\rm h}$, with probability $P({\rm h})=1$, and similarly for the other two outcomes. Note that if we have said that outcomes are predetermined for *almost* all, and not all initial states, this is because we cannot exclude the special situation ${\bf x}= \mbox{\boldmath$\lambda$}$, of classical unstable equilibrium, which remains clearly indeterminate (but do not contribute to the probability calculus, being this possibility of zero Lebesgue measure). To describe the most general typology of disintegrable membrane, we only need to introduce a *probability density* $\rho:\triangle_2\to [0,\infty[$, $\int_{\triangle_2}\rho({\bf y})d{\bf y} =1$, characterizing the propensity of the membrane (which will be called $\rho$*-membrane*) to disintegrate in its different possible sub-regions. This means that if the initial state produces the three sub-regions $A_{\rm h}$, $A_{\rm t}$ and $A_{\rm e}$, the probabilities for obtaining the three outcomes ${\bf x}_{\rm h}$, ${\bf x}_{\rm t}$ and ${\bf x}_{\rm e}$, will be given by the integrals: $$P({\bf x}\to {\bf x}_i|\rho) = \int_{A_i} \rho({\bf{y}})d{\bf{y}}, \quad i\in \{{\rm h},{\rm t}, {\rm e}\}, \label{three integrals}$$ and of course, in the special case of a uniform membrane, we simply have $\rho({\bf{y}}) ={1\over \mu(\triangle_2)}={2\over \sqrt{3}}$, for all ${\bf{y}}$, and we recover (\[relative Lebesgue\]). Before continuing in the construction of our model, a remark is in order. It is clear that the ‘tension-reduction’ mechanism associated with the disintegrable elastic membranes can only describe idealized measurements of the *first kind*, i.e., measurements such that, if repeated a second time, will produce exactly the same outcome, with probability 1. This is so because if ${\bf x}$ corresponds to one of the three vertices of $\triangle_2$, then, being already located in one of the end points of the elastic structure, its position cannot be altered by a new membrane’s collapse, when the measurement process is repeated. Of course, not all measurements are of the first kind, but certainly most of them can be made, at least ideally, of the first kind. For instance, to make the coin flipping a measurement of the first kind it is sufficient to specify in the protocol that if the coin is already located on the floor, i.e., if the initial state of the coin is an *on-floor state*, and not an *on-cup state*, then what the experimenter has to do is to simply observe if the upper face is “head,” “tail” or “edge,” and take the result of such observation as the outcome of the measurement. Clearly, the flipping of the coin producing the three on-floor outcomes “head,” “tail” and “edge,” is not the only coin-measurement that we can perform. Imagine for a moment the following *coin shaking* measurement, operationally defined by the following protocol: If the coin is on the floor, then put it at the center of the bottom of the cup, with exactly the same upper face, then shake the cup, following a predetermined procedure (that we don’t need to specify here), and finally look at the bottom of the cup, to see what is the obtained upper face. On the other hand, if the coin is already in the cup, just observe its upper face, which will then be the outcome of the measurement. We now have two different measurements, the *coin-flipping* measurement, which can produce the outcome states “floor-head,” “floor-tail” and “floor-edge,” and the *coin-shaking* measurement, which can produce the outcome states “cup-head,” “cup-tail” and “cup-edge,” and of course we cannot associate the same membrane (i.e., the same measurement simplex) to both measurements. This not only because their outcome states are different, but also because the associated flipping and shaking procedures are different. If different membranes can represent different measurements, belonging to a same state space, then in addition to the tension-reduction process describing the membrane’s collapse we have to introduce a mechanism allowing the states belonging to one membrane to be measured with respect to another membrane. In other terms, we need to describe a process that can transform an off-membrane state into an on-membrane state, in order to be subjected to its (possibly) indeterministic collapse. A process of this kind has to be able, in particular, to bring the state “cup-head” in contact with the ‘potentiality region’ of the flipping-membrane, or the state “floor-head” in contact with the ‘potentiality region’ of the shaking-membrane. Since we are here interested in obtaining a geometrical representation, and that we want the description to be as simple as possible, a very natural choice is to use a (deterministic) *orthogonal projection* process. In other terms, if ${\bf x}$ is an off-membrane state, with respect to the considered measurement, we can describe the latter as a *two-stage process*. The first stage, purely deterministic, would correspond to the point particle orthogonally “falling” onto the membrane, along a rectilinear path, until it reaches its on-membrane position; the second stage, which can either be deterministic or indeterministic (depending on the nature of the membrane), is then the tension-reduction process produced by the disintegration and subsequent contraction of the membrane that we have previously described. The general tension-reduction (GTR) model {#The general tension-reduction (GTR) model} ========================================= In the previous section, we have considered some measurements possibly performed on a coin, generalizing the solipsistic ones usually considered in the classical games of chance. Of course, it was not our intention to describe in a complete and self-consistent way all possible states and measurements that can be described in relation to a coin entity. Our example was just meant to fix ideas and allow introducing some of the basic concepts of a general geometrical description of an entity, which includes not only its states, but also its measurements, and this by means of an interaction mechanism, based on the disintegration of a membrane, which is able to produce the different outcomes and associated probabilities. Based on the intuition we have gained, we are now in a position to reason in more general and abstract terms to identify the fundamental ingredients of what we have called the *general tension-reduction* (GTR) model [@AertsSassolideBianchi2015a; @AertsSassolideBianchi2015b]. We begin by summarizing what we have obtained so far, using now a more formal language. Let $\Sigma$ be the set of *all states* of a given entity $S$. By “all states” we don’t necessarily mean all conceivable states, but more specifically the set of those states that are relevant for the description of the entity in the different measurement contexts it can be meaningfully associated with. If the entity is finite-dimensional, or can be conveniently approximated as such (this is generally the case in all practical experimental situations), which we will assume to be so in the following, then $\Sigma$ can be taken to be a subset of the $M$-dimensional Euclidean space ${{\mathbb R}}^M$, for some given finite integer $2\leq M <\infty$. A measurement on $S$, producing $N$ different outcomes, with $N\leq M$, is then described as a $(N-1)$-dimensional simplex $\triangle_{N-1}$, whose $N$ vertices ${\bf x}_i$, $i=1,\dots, N$, are representative of the $N$ possible outcomes. A measurement of the non-degenerate kind (degenerate measurements will be discussed in Sec. \[Degenerate measurements in the GTR-model\]) is then a two-stage process, bringing the initial state ${\bf x}\in\Sigma$ to one of the outcome states ${\bf x}_i$, $i=1,\dots, N$. The first stage of the process corresponds to the point particle associated with the state ${\bf x}$ orthogonally “falling” onto the $(N-1)$-dimensional $\rho$-membrane, associated with $\triangle_{N-1}$, and firmly attaching to it. If we write ${\bf x}={\bf x}^\parallel + {\bf x}^\perp$, with ${\bf x}^\parallel$ and ${\bf x}^\perp$ the two components of ${\bf x}$ parallel and orthogonal to $\triangle_{N-1}$, respectively, then this first deterministic stage of the measurement corresponds to the transition: ${\bf x}\to{\bf x}^\parallel$. The second stage is a process that can either be deterministic or indeterministic, depending on the nature of the $\rho$-membrane. Its description is a straightforward generalization of what we have explained already in the $N=3$ coin example (see Fig. \[triangolo\]). The presence of the point particle on the $\rho$-membrane, at point ${\bf x}^\parallel$, creates $N$ disjoint sub-regions $A_i$, $i=1,\dots N$ ($A_i$ is the convex closure of $\{ {\bf x}_1,\dots,{\bf x}_{i-1},{\bf x}^\parallel,{\bf x}_{i+1},\dots,{\bf x}_N\}$), such that $\triangle_{N-1}=\cup_{i=1}^N A_i$. These sub-regions are separated by $(N-2)$-dimensional “tension-surfaces,” along which the elastic substance can less easily disintegrate. Then, as soon as the $\rho$-membrane starts disintegrating, at some point , if $\mbox{\boldmath$\lambda$} \in A_i$, the disintegrative process will cause the $N-1$ anchor points ${\bf x}_j$, $j\neq i$, of $A_i$, to tear away, and consequently the $\rho$-membrane will contract toward the remaining anchor point ${\bf x}_i$, drawing the abstract point particle to that position. Thus, we have a two-stage process producing the possible transitions ${\bf x}\to{\bf x}^\parallel\to{\bf x}_i$, with associated probabilities: $$\label{rhoprobabilitymeasurement} P({\bf x}\to {\bf x}_i|\rho) = \int_{A_i} \rho({\bf{y}})d{\bf{y}}, \quad i\in \{1,\dots,N\}.$$ Let us explore a little further the general structure of $\Sigma$. All points belonging to a measurement simplex $\triangle_{N-1}$ are also possible states of the entity under investigation, as is clear that they are all measurable with respect to a $\rho$-membrane associated with $\triangle_{N-1}$. But it is also clear that within the $(N-1)$-dimensional sub-space generated by $\triangle_{N-1}$, there cannot be other points representative of states in addition to those belonging to $\triangle_{N-1}$ itself. Indeed, *bona fide* states are those that, at least in principle, can participate in all well-defined measurements, but the points in that sub-space lying outside of the simplex cannot be orthogonally projected onto the latter, and therefore cannot be measured with respect to its $\rho$-membrane. Of course, this doesn’t mean that the points outside of a simplex, in each simplex sub-space, cannot also be used to describe some kind of states, i.e., some real conditions characterizing the entity under study. However, since these states would describe situations where the entity would not be available in producing any outcome (i.e., in providing an answer when subjected to a measurement’s interrogative process), they have to be considered states of a non-ordinary kind. Let us call them *confined states*, to express the idea that they describe situations where the entity is confined in a “place of reality” which is out of reach for ordinary measurement contexts. In the example of the coin, we can imagine a situation where the coin has been glued to the wall, with a very strong glue, so that it cannot be subjected anymore to the coin-flipping measurement, or the coin-shaking one. In a cognitive psychology experiment, we can consider the situation of a person who is asked to choose one among a set of predetermined responses to a given question, but with the question and responses expressed in a language that the person cannot understand, so that no meaningful answer can be obtained from her. In physics we can also mention the example of color confinement, the well-known difficulty in directly observing single color-charged entities, like quarks, in our Euclidean spatial theater. Having said that, in the following we will limit our discussion to ordinary (non-confined) states, participating to all possible measurements, which means that by the term “state” we will mean (if not mentioned otherwise) a condition in which the entity is available to take part in all well-defined measurements. Consequently, the $(N-1)$-dimensional section of the state space $\Sigma$ that contains the measurement simplex $\triangle_{N-1}$, will be taken to be precisely $\triangle_{N-1}$. As we said, different measurements are associated with different simplexes, i.e., with simplexes having different relative orientations. This means that the dimension $M$ of the state space $\Sigma$ needs to be large enough to accommodate all these different orientations, in a way that the points belonging to the different simplexes are all mutually orthogonally projectable, so that they can all participate to the different possible measurements. For this to be the case, it is easy to imagine that the dimension $M$ of $\Sigma$ will generally have to be considerably larger than $N$, taking into account the fact that in each subspace generated by a simplex no other simplexes can be present. Another issue to be addressed is the center of the simplexes. Of course, they all have to be centered at the same point, say the origin of the system of coordinates considered, otherwise it would not be possible to ensure the overall functioning of the orthogonal projection mechanism. But there is another reason why all measurement simplexes need to share the same origin. The point at the origin corresponds to a state which has quite a remarkable property: it is the state which manifests the same availability in producing whatever outcome, in whatever measurement, and if measurements are described by uniform effective membranes (see Sec. \[The quantum mechanical example\], for the central role played by uniform probability distributions $\rho_u$), it is also the state producing the same probabilities ${1\over N}$, for all outcomes in all measurements. In a sense, it is the most *neutral* state among all possible ones, and if we assume that such condition of *maximum neutrality* should exist, at least in principle, then the different simplexes will have to share the same origin. What about the shape of the state space $\Sigma$ in ${{\mathbb R}}^M$? First of all, what we know is that no states can be at a distance from the origin that is greater than that of the apex points of the different $(N-1)$-dimensional measurement simplexes. Since these points are by definition at distance $1$ from the center of the simplex to which they belong, and that all simplexes share the same center, we have that all the $M$-dimensional vectors of $\Sigma$ are necessarily contained in a $M$-dimensional ball of radius $1$, although of course they will not generally fill such ball. We can also remark that each $(N-1)$-simplex contains an inscribed $(N-1)$-ball of radius ${1\over N-1}$, and since all simplexes have the same origin, within $\Sigma$ there is a $M$-ball of radius ${1\over N-1}$ possessing a maximum density of states, as it contains all the inscribed $(N-1)$-balls associated with the different measurements. Thus, in case the entity under consideration would be associated with a continuity of measurement simplexes, we can expected such $M$-ball to be completely filled with states. Also, considering that for $N=2$ the inscribed 1-sphere (a line segment) has radius $1$, if we have a continuity of two-outcome measurement simplexes $\triangle_{1}$, oriented along all possible directions in ${{\mathbb R}}^M$, the state space will be precisely a $M$-dimensional ball of radius 1, i.e., $\Sigma = B_1({{\mathbb R}}^M)$. Of course, apart being contained in a unit ball, and containing a smaller ball having a maximum density of states, nothing can be said a priori about the shape of $\Sigma$, i.e., about the envelope containing the extremal points of $\Sigma$. These extremal points can be of two kinds: either they are at a distance $1$ from the origin, and thus correspond to one of the vertices of a measurement simplex, or they belong to one of the sub-simplexes of a measurement simplex, and then their distance from the origin will be smaller than $1$. This means that, apart the above mentioned two-dimensional case, $\Sigma$ will generally not have a spherical symmetry. Strictly speaking, the question of the shape of $\Sigma$ is meaningful only if we have a continuity of states, i.e., if $\Sigma$ is a region of ${{\mathbb R}}^M$ completely filled with states. In this case, for consistency reasons, it is reasonable to assume that it will be a *convex* region. Indeed, by definition, a convex region is a set of points such that, given any two points, the line joining them lies entirely within it. This means that the region is connected, in the sense that it is possible to go from one point to another without leaving the region. If we assume, as we did, that a measurement is a process during which the state of the entity changes in a continuous way within $\Sigma$, and that both during the first deterministic stage and the second possibly indeterministic stage the abstract point particle representative of the state follows rectilinear trajectories, then the only way to guarantee that in all circumstances these trajectories are made of states, i.e., that they belong to $\Sigma$, is to require $\Sigma$ to be a convex set. Degenerate measurements in the GTR-model {#Degenerate measurements in the GTR-model} ======================================== To complete our description of the GTR-model, we need to consider the possibility of measurements such that different final states can be associated with a same outcome. These are called *degenerate* measurements in quantum mechanics and we will adopt here the same terminology. However, since our approach is more general, we will have to distinguish between two different possibilities, that we will call *submeasurements of the first type* and *submeasurements of the second type* (not to be confused with von Neumann’s designation of measurements of the first and second kind). Submeasurements of the first type are degenerate measurements in which the experimenter can in principle distinguish between all the possible outcomes, but decides (for whatever reason) not to do so, thus identifying some of them. For instance, in the example of the coin, we can imagine the situation where the “edge” outcome is conventionally identified with the “head” outcome, so that one obtains an effective “head” or “tail” two-outcome measurement, where “head” is now re-interpreted as either point ${\bf x}_{\rm h}$ or point ${\bf x}_{\rm e}$. But apart from this identification, the measurement protocol, and therefore the associated membrane’s collapse mechanism, remains exactly the same. In other terms, a submeasurement of the first type, by only identifying some of the outcomes, produces a change of state of the entity that is identical to that produced by an experiment where such identification is not considered. Submeasurements of the second type, on the other hand, correspond to experimental situations where the distinction between certain outcomes becomes impossible to realize in practice, even in principle. This means that the outcome states are different than those associated with the corresponding non-degenerate situations, as if they were not, the distinction between the different outcomes would always be possible. In other terms, a submeasurement of the second type is characterized by a different experimental context, and therefore the membrane’s mechanism describing its unfolding will also be different. In the example of the coin, we can consider the following modified flipping protocol. Once the coin has reached the floor, before taking knowledge of the value of the upper face, a colleague performs the following additional operations: if she finds that the upper face is “tail,” she does nothing. If instead she finds that the upper face is “edge,” or “head,” she takes the coin and places it on a table, tail up, and then without saying a word leaves the room. Clearly, if the final location of the coin is on the floor, the outcome is “tail,” and more precisely “floor-tail”. On the other hand, if the final location of the coin is on the table, then the outcome is “table-tail,” and evidently such new state contains no information that would allow the experimenter to associate it either with the “floor-head” state or the “floor-edge” state, of the associated non-degenerate measurement. If we denote $P_{\rm deg}({\rm t})$ the probability of obtaining “floor-tail,” and $P_{\rm deg}(\bar{\rm t})$ the probability of obtaining “table-tail,” in the degenerate measurement, we clearly have: $$\label{degenerate-equality} P_{\rm deg}({\rm t}) = P({\rm t}),\quad P_{\rm deg}(\bar{\rm t})=P({\rm h})+P({\rm e}).$$ Degenerate measurements of the second type which obey equalities of the above kind will be said to be *quantum-like*, as is clear that quantum measurements always obey them. Let us now show how we can modify the membrane’s mechanism to describe submeasurements of the second type. Starting from the non-degenerate situation, we must alter the functioning of the membrane in such a way that not only (\[degenerate-equality\]) will be satisfied, but also the collapse will have to produce the two states “floor-tail” (${\bf x}_{\rm t}$) and “table-tail” (${\bf x}_{\bar{\rm t}}$), instead of the three states “floor-tail” (${\bf x}_{\rm t}$), “floor-head” (${\bf x}_{\rm h}$) and “floor-edge” (${\bf x}_{\rm e}$). In the following, we only analyze the $N=3$ situation, the generalization to more general measurements being straightforward. The first deterministic stage of the measurement is exactly the same as for the corresponding non-degenerate situation, with the point particle initially in state ${\bf x}$ orthogonally “falling” onto the membrane and firmly attaching to it. Instead, the second indeterministic stage is different, being that the two sub-regions $A_{\rm h}$ and $A_{\rm e}$ will now be fused together, so as to form a single larger subregion $A_{\bar{\rm t}}=A_{\rm h}\cup A_{\rm e}$. To fix ideas, we can think that a special reactive substance has been applied along the common boundary between $A_{\rm h}$ and $A_{\rm e}$, the effect of this special substance being twofold: firstly, it produces the effective fusion of the two subregions into a single one, so that if the membrane breaks in a point belonging to, say, $A_{\rm h}$, the tearing will now propagate also across the boundary with $A_{\rm e}$ (because of the presence of the reactive substance), causing the collapse of the entire subregion $A_{\bar{\rm t}}$. Secondly, it produces the early detachment of the common anchor point ${\bf{x}}_{\rm t}$, with the consequent contraction of the elastic membrane, drawing the point particle attached to it to a given position on the line segment (the 1-simplex) subtended by ${\bf{x}}_{\rm h}$ and ${\bf{x}}_{\rm e}$. Finally, also the last two anchor points ${\bf{x}}_{\rm h}$ and ${\bf{x}}_{\rm e}$ will detach, causing the membrane to shrink toward the particle, but without affecting its acquired position (see Fig. \[triangolo-degenerate\]). On the other hand, if the membrane breaks in $A_{\rm t}$, then only $A_{\rm t}$ will collapse, producing the final outcome ${\bf x}_{\rm t}$, exactly as in the non-degenerate situation. So, when performing the degenerate measurement, only two transitions are now possibly produced by the membrane mechanism. If the membrane disintegrates in $A_{\rm t}$, the outcome is ${\bf x}_{\rm t}$, which means that the abstract point particle representative of the final state will be at a maximal (unit) distance from the origin of the system of coordinates, that is, from the center of the $M$-dimensional sphere in which $\Sigma$ is inscribed. On the other hand, if the membrane disintegrates in the composite subregion $A_{\bar{\rm t}}$, the membrane’s collapse will not necessarily produce a vector of maximal length within $\Sigma$. Therefore, it is natural to assume in this case that a third deterministic process can possibly occur, to complete the measurement, bringing the point particle to a final position ${\bf x}_{\bar{\rm t}}$ of maximal distance from the origin, at the surface of $\Sigma$, representative of the final outcome-state of the measurement. In view of (\[rhoprobabilitymeasurement\]), it is clear that the degenerate membrane’s mechanism we have just described obeys (\[degenerate-equality\]), considering that the first and third stages of the measurement are purely deterministic. There is however another aspect of interest that needs to be discussed, in relation to this possible third stage of a degenerate measurement of the second type. More precisely, we need to distinguish the following two possibilities. Considering the outcome ${\bf x}_{\bar{\rm t}}$, either (1) its orthogonal projection onto the membrane falls onto some point of the line segment between ${\bf{x}}_{\rm e}$ and ${\bf{x}}_{\rm h}$, or (2) it falls onto whatever other point of the membrane, not lying on that line segment. Condition (1) corresponds to the situation where the corresponding non-degenerate measurement having ${\bf x}_{\bar{\rm t}}$ as the initial state, can only produce the two outcomes ${\bf{x}}_{\rm e}$ and ${\bf{x}}_{\rm h}$, and therefore cannot produce the outcome ${\bf{x}}_{\rm t}$. On the other hand, condition (2) corresponds to the situation where the probability of obtaining ${\bf{x}}_{\rm t}$ will be generally non-zero. We shall say that a submeasurement of the second type, obeying condition (1) above, is a *projection-like submeasurement*, as we know that degenerate quantum measurements do obey this condition as a result of the projection formula. A simple way to automatically implement (1) in our model is to ask the third deterministic stage to be structurally similar to the first one, in the sense that the point particle, when reemerging from the membrane, it will always do so following a rectilinear path, orthogonal to the membrane’s plane. Such final state, when projected back onto the membrane, in a repetition of the measurement, will then land onto the same edge of the triangular membrane, in accordance with condition (1). It is worth observing, however, that a degenerate measurement of the second type will generally not be projection-like, in the above sense. This is immediately clear in our coin example. Indeed, when flipping the coin in a “table state” onto the floor, if we don’t use a very special experimental protocol, all three states “floor-head,” “floor-tail” and “floor-edge” will be easily obtained. Composite entities in the GTR-model {#Composite entities in the GTR-model} =================================== Another important class of measurements we want to describe in our GTR-model is that of measurements performed on *composite entities* (also called *join entities*). To remain within the ambit of our example, consider the situation where instead of a single coin we now have two coins (not necessarily identical). Then, we can perform a first coin-flipping measurement with the first coin, observe the outcome, and do the same with the second coin, observing again the outcome. As each one-entity measurement can produce $3$ different outcomes: “head,” “tail” and “edge,” the combination of the two measurements can produce $9=3\times 3$ different outcomes: “head-head,” “head-tail,” “head-edge,” “tail-head,” “tail-tail,” “tail-edge,” “edge-head,” “edge-tail” and “edge-edge.” It is important to observe that, to be able to perform separately the two measurements, it has to be possible to act separately on the two coins and obtain in a separate way the outcomes of the two experimental actions. We are not saying by this that the two coins must be *experimentally separated*, in the sense that their measurements cannot produce correlations, but that the measurements themselves have to be separable. In physics for instance, when considering measurements on composite entities, these are usually separable, but in more general situations this may not necessarily be the case [@AertsSozzo2014a; @AertsSozzo2014b]. An important characteristic of separable measurements is that they can be performed either sequentially or simultaneously, and that when they are performed in a sequential way the order of the sequence is irrelevant. This is evidently the case in the coin-flipping measurements. Indeed, we can either flip the two coins simultaneously, or one after the other, in whatever order, and we will always obtain the same statistics of outcomes. However, a separable measurement can either produce or not produce correlated outcomes, depending on the nature of the state of the join entity. If such state describes a condition of (experimental) separation of the sub-entities forming the join entity, then no correlations will be observed in the statistics of outcomes. We shall call states of this kind *product states*. The term “product” is here to be understood in the specific sense of a state that when subjected to a separable measurement, the obtained set of outcomes can be described as the *Cartesian product* of the set of outcomes obtained when individual measurements are performed separately, on the different sub-entities, in whatever order. In the GTR-model, the situation of separable measurements performed on product states can be described by simply considering distinct membranes, one for each sub-entity forming the join entity, working independently from one another. In the case of the two-coin entity, its initial state is then described by a couple of initial states, one for each individual coin, and its measurement is described by a couple of membranes, acting separately on these two one-coin initial states. Of course, it is also possible to represent this double-membrane process as a single higher-dimensional membrane process. For this, instead of two three-dimensional triangular membranes, we will have a 8-dimensional hypermembrane (a 8-simplex), with 9 vertices. The initial state of the two coins will then be described by a vector whose first three components define the state of the first coin, the successive three components define the state of the second coin, and the remaining components (the number of which depends on the dimension $M\geq 8$ of the state space) are fully determined by them (as the join entity state, being a product state, it has to be fully determined by the state of its components). This means that the higher-dimensional single-membrane measurement process, when projected onto its first $3+3$ components, will describe two independent processes, in accordance with the fact that, for a separable measurement performed on a product state, “the whole has to be equivalent to the sum of its parts.” The situation is however different if either the state is a non-product state, or the measurement itself is a non-separable measurement, or both. For instance, we may decide to change the operational definition of the flipping measurement by introducing a small rigid rod whose two ends are glued to the two coins (in a way that we don’t need to specify here), before flipping them. The presence of the connecting rod means that the two-coin measurement cannot anymore be conceived as two separate one-coin measurements, as by flipping one coin we will also, inevitably, flip the other one, and because of the connecting rod (which may break or not during the experiment), correlated outcomes can be observed (which one can show are able to violate Bell’s inequalities; see for instance @Aertsetal2000 [@SassolideBianchi2013b; @SassolideBianchi2014]). In other terms, the measurement becomes non-separable. The situation where the state of the two coins is non-product is similar. For instance, one can consider that the two coins are strongly magnetized, and therefore able to attract each other. This means that the description of their state also has to include a description of the magnetic field connecting them, which similarly to the above rigid rod example can produce correlations, under certain conditions. Different from the situation of separable measurements performed on product states, in the case of non-separable measurements and/or non-product states, two separate membranes will clearly not be sufficient to obtain a full description not only of the state of the two-coin entity, but also of the statistics of (correlated) outcomes that the measurement is able to produce. Only a genuinely higher dimensional structure will be able in this case to account for all the experimental possibilities, and in particular the $3+3$ components describing the two initial individual coin states will generally not allow to deduce the value of the remaining components, in accordance with the fact that, for non-product (non-separable) entities, “the whole will be more than just the sum of its parts.” The quantum mechanical example {#The quantum mechanical example} ============================== In the previous sections we have used the coin example as an heuristic to explain the basic ingredients and structure of the GTR-model. In this section, we consider an important implementation of the GTR-model: *quantum mechanics*. Clearly, the GTR-model is much more general than quantum mechanics, in the sense that it is able to account for a much wider class of measurements and states than those usually considered in the standard quantum formalism. Also, even when the model is reduced to quantum mechanics, by means of two assumptions that we are now going to enunciate, it still remains a more general framework than quantum mechanics, in the sense that it describes a completed version of the latter where the Born rule can be derived in a non-circular way. In that sense, the GTR-model also offers a possible solution to the longstanding measurement problem [@AertsSassolideBianchi2014a]. The two additional assumptions that are needed to derive the Born rule of probabilistic assignment from the GTR-model are the following: > [**Hypothesis 1: Metaignorance**]{}. *The experimenter does not control which specific measurement is actualized at each run of the measurement, among those that can actualize the given outcome-states. In other terms, the experimenter lacks knowledge not only about the (almost deterministic) measurement interaction that each time is actualized, but also about the way it is each time selected.* > > [**Hypothesis 2: Hilbertian structure**]{}. *The state space $\Sigma$ is a generalized Bloch sphere.* Let us explain how the above two assumptions can be used to derive the quantum mechanical Born rule. We start by analyzing the consequences of the first one: *metaignorance* (or *metaindifference*, to use the terminology of @Shackel2007; see also @AertsSassolideBianchi2014b). This assumption is natural for the following reason. In a typical quantum measurement (like, say, a Stern-Gerlach spin-measurement), we are in a situation where the experimenter doesn’t want to control in whatsoever way its development. This because a measurement is generally understood as a process of *observation*, and an observation is meant to alter in the least possible way the observed entity. Of course, if the process creates the very property that is observed, then there will be an “intrinsic invasiveness” to it, impossible to remove. This is precisely the situation of quantum measurements, where the entity transitions from an initial to a final state, with the latter being generally different from the former (if the pre-measurement state is not an eigenstate). In other terms, in a measurement context the only controlled aspect is that relative to the definition of the possible outcome-states. But apart from that, the experimenter will avoid as much as possible to interfere with the natural process of actualization of these potential outcomes. Thus, underlying the metaignorance assumption there is the idea that if nothing limits the way the measuring system and the measured entity can interact, they will naturally explore all possible (available) ways of interacting. This means that the quantum statistics, and more generally the statistics of any observational process where the experimenter doesn’t try to influence the outcomes, corresponds to what has been called a *universal average* [@AertsSassolideBianchi2014a; @AertsSassolideBianchi2014b; @AertsSassolideBianchi2015a; @AertsSassolideBianchi2015b], i.e., an average over all possible measurement processes associated with a predetermined set of outcomes. Measurements subtending a universal average are called *universal measurements*, and our point is that quantum measurements are just a special example of universal measurements. More specifically, to each experimental situation characterized by $N$ given outcomes, an uncountable infinity of measurements can be defined, each one characterized by a different $\rho$-membrane, i.e., by a $(N-1)$-simplex associated with a different probability density $\rho$. A universal measurement then corresponds to a two-stage process, where firstly a $\rho$-membrane is selected among the infinity of possible ones, and secondly, from that specific $\rho$-membrane, a -measurement-interaction is also selected, so producing the (almost) deterministic collapse of the membrane. So, to calculate the probabilities associated with a universal measurement one has to perform an average over the probabilities obtained from all these different possible measurements. Of course, if such a huge average is addressed directly, one will be confronted with technical problems related to the foundations of mathematics and probability theory. A good strategy, similar to that used in the definition of the *Wiener measure*, is to proceed as follows. First, one shows that any probability density $\rho$ can be described as the limit of a suitably chosen sequence of *cellular probability densities* $\rho_{n}$, as the number of cells $n$ tends to infinity, in the sense that for every initial state ${\bf x}$ and final state ${\bf y}$, one can always find a sequence of cellular $\rho_{n}$ such that the transition probability $P({\bf x}\to {\bf y}|\rho_{n})$, associated with the $\rho_{n}$-membrane, tends to the transition probability $P({\bf x}\to {\bf y}|\rho)$, associated with the $\rho$-membrane, as $n\to\infty$ [@AertsSassolideBianchi2014a; @AertsSassolideBianchi2015b]. By a cellular probability density we mean here a probability density describing a structure made of a finite number $n$ of regular cells (of whatever shape), tessellating the hypersurface of the $(N-1)$-simplex, which can only be of two sorts: uniformly breakable, or uniformly unbreakable. Then, if one excludes the totally unbreakable case (as it would produce no outcomes), we have a total number $2^{n}-1$ of possible $\rho_{n}$-membranes. This means that for each $n$, one can unambiguously define the average probability: $$\label{average1} \langle P({\bf x}\to {\bf y})\rangle_n \equiv {1\over 2^{n}-1} \sum_{\rho_{n}}P({\bf x}\to {\bf y}|\rho_{n}),$$ where the sum runs over all the possible $2^{n}-1$ probability densities made of $n$ cells. Clearly, $\langle P({\bf x}\to {\bf y})\rangle_n$ is the probability for the transition ${\bf x}\to {\bf y}$, when a probability density $\rho_{n}$ (a cellular $\rho_{n}$-membrane) is chosen at random, in a uniform way. Then, to obtain the transition probabilities of the universal measurement, i.e., of the average over all possible $\rho$-measurements, one has to calculate the infinite cell limit of the above average: $$\label{average-limit} \langle P({\bf x}\to {\bf y})\rangle_{\rm{univ}} =\lim_{n\to\infty} \langle P({\bf x}\to {\bf y})\rangle_n,$$ and it is possible to demonstrate that [@AertsSassolideBianchi2014a; @AertsSassolideBianchi2015b]: $$\label{theoremuniversal} \langle P({\bf x}\to {\bf y})\rangle_{\rm{univ}} =P({\bf x}\to {\bf y}|\rho_u),$$ where $\rho_u$ denotes the uniform probability density, i.e., the probability density associated with a uniform membrane, for which all points have the same probability of disintegrating. A model only contemplating uniform membranes has been called the *uniform tension-reduction* (UTR) model [@AertsSassolideBianchi2015a; @AertsSassolideBianchi2015b]. The result (\[theoremuniversal\]) then tells us that every time a statistics of outcomes is generated by a universal average, an effective UTR-model will naturally emerge. This is however not sufficient to derive the quantum mechanical Born rule. For this, the second assumption, about the structure of the state space, is also needed. Here again we will not go into any technical details of the proof and just explain in broad terms the gist of it, referring the interested reader to @AertsSassolideBianchi2014a [@AertsSassolideBianchi2015a; @AertsSassolideBianchi2015b]. It is well known that the rays of a two-dimensional Hilbert space can be represented as points at the surface of a 3-dimensional unit sphere, called the *Bloch sphere* [@Bloch1946], with its internal points describing the so-called *density operators* (also called *density matrices*). What is less known is that a similar representation can be worked out for general $N$-dimensional Hilbert spaces [@Arvind1997; @Kimura2003; @Byrd2003; @Kimura2005; @Bengtsson2006; @Bengtsson2013; @AertsSassolideBianchi2014a]. The standard 3-d Bloch sphere is then replaced by a generalized $(N^2-1)$-dimensional Bloch sphere, with the only difference that for $N>2$ only a convex portion of it will be filled with states. In other terms, the state space $\Sigma$ is a $M$-dimensional convex set, with $M=N^2-1$, inscribed in a unit sphere of same dimension. It is then possible to show that the set of eigenvectors associated with a given observable, i.e., with a self-adjoint operator, are precisely described, within $\Sigma$, by the $N$ vertexes of a $(N-1)$-simplex inscribed in the generalized Bloch sphere [@AertsSassolideBianchi2014a]. And when the point particle representative of the initial ray-state, located at some point ${\bf x}$ at a unit distance from the origin, plunges into the sphere to reach its on-membrane position ${\bf x}^\parallel$, following a path orthogonal to the simplex, it can also be shown that this deterministic movement precisely causes the off-diagonal elements of the associated density operator (in the measurement’s basis) to gradually vanish, which means that the on-membrane state is precisely a *decohered state*, described by a fully reduced density operator. Different from the usual description of *decoherence theory* [@Schlosshauer2005], the decohered on-membrane state ${\bf x}^\parallel$ does not correspond to the final state of the measurement, but to the state prior to the indeterministic collapse of the membrane. In probabilistic terms, the disintegrative/collapse process is governed by the Born rule, as is clear that the coordinates of the on-membrane (reduced density operator) state ${\bf x}^\parallel$ are precisely the transition probabilities and that, as we already remarked in Sec. \[Measuring a coin\], Eq. (\[relative Lebesgue\]), the relative Lebesgue measure of the different subregions correspond to the values of the corresponding coordinates of the on-membrane vector. Also, in the situation where the measurement is degenerate, a third purely deterministic process can also happen, of the *purification* kind, through which the entity takes a maximal “distance” from the measurement context represented by the membrane, in accordance with the predictions of the Lüders-von Neumann projection formula [@AertsSassolideBianchi2014a]. To complete our description of the quantum mechanical example, we briefly mention the situation with multipartite systems, formed by multiple entities. For this, we recall that the existence of a generalized Bloch representation is based on the observation that one can always find a basis for the density operators acting in ${{\mathbb C}}^N$, made of $N^2$ orthogonal (in the Hilbert-Schmidt sense) operators. One of them is the identity matrix ${\mathbb I}$, and the other $N^2-1$ correspond to a determination of the generators of $SU(N)$, the *special unitary group of degree $N$*. These are traceless and orthogonal self-adjoint matrices $\Lambda_i$, $i=1,\dots, N^2-1$, which can be chosen to be normalized as follows: ${\rm Tr}\, \Lambda_i\Lambda_j=2\delta_{ij}$. Then, any density operator state $D$ can be uniquely determined by a $(N^2-1)$-dimensional real vector ${\bf x}$, by writing [@AertsSassolideBianchi2014a]: $$D({\bf x}) = {1\over N}\left(\mathbb{I} +c_N\, {\bf x}\cdot\mbox{\boldmath$\Lambda$}\right) = {1\over N}\left(\mathbb{I} + c_N\sum_{i=1}^{N^2-1} x_i \Lambda_i\right), \label{formulaNxN}$$ where we have defined the constant: $c_N\equiv \sqrt{N(N-1)\over 2}$. If the entity in question is a joint entity formed by two sub-entities, with Hibert spaces ${{\mathbb C}}^{N_A}$ and ${{\mathbb C}}^{N_B}$, respectively, so that ${{\mathbb C}}^{N}={{\mathbb C}}^{N_A}\otimes {{\mathbb C}}^{N_B}$ and $N=N_AN_B$, it is possible to introduce a tensorial determination of the generators, such that the generators of $SU(N)$ are expressed as tensor products of the generators of $SU(N_A)$ and $SU(N_B)$. In this way, one can show that the vector ${\bf x}$ representative of the state of the joint entity can always be written as a direct sum of three vectors [@AertsSassolideBianchi2015d]: $${\bf x} = d_{N_A}{\bf x}^{A}\oplus d_{N_B}{\bf x}^{B}\oplus {\bf x}^{\rm corr}. \label{direct sum}$$ where $d_{N_A}=({N_A-1\over N-1})^{1\over 2}$, $d_{N_B}=({N_B-1\over N-1})^{1\over 2}$, ${\bf x}^{A}$ belongs to the one-entity Bloch sphere $B_1({{\mathbb R}}^{N_A^2-1})$ and describes the state of the $A$-entity, ${\bf x}^{B}$ belongs to the one-entity Bloch sphere $B_1({{\mathbb R}}^{N_B^2-1})$ and describes the state of the $B$-entity, and ${\bf r}^{\rm corr}$ is that component of the state describing the correlations between the two sub-entities. In accordance with our analysis of Sec. \[Composite entities in the GTR-model\], it is then possible to show that: (1) when the initial state ${\bf x}$ is representative of a *product state*, the components of the two individual vectors ${\bf x}^{A}$ and ${\bf x}^{B}$ are independent of one another, and the components of the correlation vector ${\bf x}^{\rm corr}$ are entirely determined by the components of the latter. This means that the indeterministic process described by the $(N-1)$-dimensional two-entity measurement’s membrane is equivalent to that obtained by two sequential measurements, in whatever order, performed by two $(N_A-1)$- and $(N_B-1)$-dimensional membranes. On the other hand, when the initial state is not of the product form $D=D_A\otimes D_B$, the two vectors ${\bf x}^{A}$ and ${\bf x}^{B}$ are not anymore independent and the components of the correlation vector ${\bf x}^{\rm corr}$ cannot anymore be deduced from the components of ${\bf x}^{A}$ and ${\bf x}^{B}$. This means that in a non-product situation the collapse mechanism of the full $(N-1)$-dimensional membrane is needed to describe the statistics of outcomes of the quantum measurement, which cannot be decomposed into two separate and independent collapse mechanisms [@AertsSassolideBianchi2015d]. Beyond Kolmogorov & Hilbert {#Non-Kolmogorovian non-Hilbertian structures} =========================== In the previous section we have shown that quantum mechanics provides a specific realization of the GTR-model, when the membranes are uniform (a situation we have called the UTR-model) and the state space is a generalized Bloch sphere, which can be seen as the natural completion of the standard Hilbert space structure in which also density matrices are allowed to play the role of pure states. In this section, we provide a few examples of situations where both classical and quantum probabilities structures are violated, showing in this way that the GTR-model can describe more general probability models than those associated with purely classical and purely quantum experimental propositions. Beyond classical ---------------- We start by considering the violation of the classical probability model. More precisely, we will show that the joint probabilities of sequential measurements cannot generally be fitted into a classical probability model equipped with a single sample space. For this, we consider the simplest possible situation: that of an entity whose measurements only have two possible outcomes. The first measurement is characterized by a one-dimensional $\rho_A$-membrane (an elastic band) stretched between the two opposite outcome-states ${\bf a}$ and $-{\bf a}$ (we will call it the $A$-measurement), and a second measurement is characterized by a one-dimensional $\rho_B$-membrane stretched between the two opposite outcome-states ${\bf b}$ and $-{\bf b}$ (we will call it the $B$-measurement). We assume that the entity is initially in eigenstate ${\bf b}$ of the $B$-measurement. This means that the probability $P(\to {\bf b}\to {\bf a}|{\bf b})$ of having first the transition to state ${\bf b}$, then to state ${\bf a}$, knowing that the initial state is ${\bf b}$, is: $P(\to {\bf b}\to {\bf a}|{\bf b})=P(\to {\bf b}|{\bf b})P(\to {\bf a}|{\bf b})=P({\bf b}\to {\bf a})$. Also, the probability $P(\to {\bf a}\to {\bf b}|{\bf b})$ of having first the transition to state ${\bf a}$, then to state ${\bf b}$, knowing that the initial state is ${\bf b}$, is: $P(\to {\bf a}\to {\bf b}|{\bf b})=P(\to {\bf a}|{\bf b})P(\to {\bf b}|{\bf a})=P({\bf b}\to {\bf a})P({\bf a}\to {\bf b})$. We thus find that: $$P(\to {\bf b}\to {\bf a}|{\bf b})-P(\to {\bf a}\to {\bf b}|{\bf b})=P({\bf b}\to {\bf a})[1-P({\bf a}\to {\bf b})]. \label{classicalviolation}$$ This means that whenever $P({\bf b}\to {\bf a})\neq 0$ and $P({\bf a}\to {\bf b})\neq 1$, the right hand side of (\[classicalviolation\]) is different from zero, i.e., $P(\to {\bf b}\to {\bf a}|{\bf b})\neq P(\to {\bf a}\to {\bf b}|{\bf b})$, which is a violation of classical probability, as the (static) propositions of classical probability theory, based on Boolean algebra, always commute. More precisely, in classical theory the probability of the event “${\bf b}$ then ${\bf a}$” has to coincide with the probability of the event “${\bf a}$ then ${\bf b}$,” i.e., $P_{\rm c}(\to {\bf b}\to {\bf a})= P_{\rm c}(\to {\bf a}\to {\bf b})$, for whatever initial state of the entity under study. Thus, the GTR-model easily violates classical probability. As a specific example, we can assume that $\rho_A$ is uniform, whereas $\rho_B$ describes an elastic band uniformly breakable only inside an interval of length $2\epsilon$, centered at the origin of the sphere, and such that $\epsilon <\cos\theta$, with $\theta$ the angle between the two elastic bands. Then, we have the transition probabilities: $P({\bf a}\to{\bf b})=1$, and $P({\bf b}\to{\bf a})={1\over 2}(1+\cos\theta)$, which are clearly different if $\cos\theta\neq 1$ (see Fig. \[twoMes\]). Beyond quantum {#Beyond quantum} -------------- To show that the GTR-model can also easily violate quantum probability, we can still use the example of Fig. \[twoMes\]. Indeed, $P({\bf b}\to{\bf a})\neq P({\bf a}\to{\bf b})$ is already a manifest violation of the Born rule, as is clear that according to the latter we should always have the equality (sometimes called the *reciprocity law*): $P({\bf b}\to{\bf a})= P({\bf a}\to{\bf b})$. Indeed, $\langle \psi|\phi\rangle = \langle \phi|\psi\rangle^*$, implying: $P(|\phi\rangle\to|\psi\rangle)=|\langle \psi|\phi\rangle|^2 = |\langle \phi|\psi\rangle|^2=P(|\psi\rangle\to|\phi\rangle)$. This tells us that the transition probability between two states only depends on their relative orientation in the Hilbert space, as measured by the modulus of their scalar product, and not on the specific direction taken by the transition. One should not conclude, however, that when the reciprocity law is satisfied the probability model would be Hilbertian. Indeed, as soon as $\rho_A=\rho_B$, we have $P({\bf b}\to{\bf a})= P({\bf a}\to{\bf b})$, but this doesn’t mean that the probabilities produced by the elastic bands are necessarily given by the Born rule. For this, as we explained in Sec. \[The quantum mechanical example\], the probability densities have to be uniform. In other terms, testing the reciprocity law is not the same as testing the quantumness of the model, as the reciprocity law can also be satisfied by more general probability models than the Hilbert one. To make this point even more clear, let us introduce a quantity called the *q-test* [@BusemeyerBruza2012; @WangBusemeyer2013; @Wangetal2014]: $$q \equiv [P(\to {\bf a}\to -{\bf b}|{\bf x}) +P(\to -{\bf a}\to {\bf b}|{\bf x})]-[P(\to {\bf b}\to -{\bf a}|{\bf x})+P(\to -{\bf b}\to {\bf a}|{\bf x})], \label{q-test}$$ where ${\bf x}$ is some given initial state. It can be shown that if the probability model is Hilbertian, then independently of the dimension of the Hilbert space we must have $q=0$, which is usually called the “QQ-equality” [@BusemeyerBruza2012; @WangBusemeyer2013; @Wangetal2014; @AertsSassolideBianchi2015e]. We will give a simple proof of the “QQ-equality” in Sec. \[Hilbertian symmetries\]. Let us here calculate explicitly the value of $q$ using the GTR-model, to show that the $q=0$ condition can be easily violated. For this, we observe that: $P(\to {\bf a}\to -{\bf b}|{\bf x})= P({\bf x}\to {\bf a})P({\bf a}\to -{\bf b})$, and similarly for the other terms in (\[q-test\]). Thus, we can write: $$\begin{aligned} \lefteqn{q = [P({\bf x}\to {\bf a})P({\bf a}\to -{\bf b}) +P({\bf x}\to -{\bf a})P(-{\bf a}\to {\bf b})]}\nonumber\\ &&\quad - [P({\bf x}\to {\bf b})P({\bf b}\to -{\bf a}) + P({\bf x}\to -{\bf b})P(-{\bf b}\to {\bf a})]. \label{q-test2}\end{aligned}$$ Limiting our discussion to two-outcome situations, we can use $P({\bf x}\to -{\bf b})= 1-P({\bf x}\to {\bf b})$ and $P({\bf x}\to -{\bf a})= 1-P({\bf x}\to {\bf a})$, so that (\[q-test2\]) becomes: $$\begin{aligned} q&=& q_1+q_2,\\ q_1 &\equiv& P(-{\bf a}\to {\bf b}) - P(-{\bf b}\to {\bf a}) \\ q_2 &\equiv& P({\bf x}\to {\bf a})[P({\bf a}\to -{\bf b}) - P(-{\bf a}\to {\bf b})] + P({\bf x}\to {\bf b})[P(-{\bf b}\to {\bf a}) - P({\bf b}\to -{\bf a})]. \label{q-test3}\end{aligned}$$ The term $q_1$ is called the *relative indeterminism* contribution, and the term $q_2$ is called the *relative asymmetry* contribution [@AertsSassolideBianchi2015e]. To simplify the discussion, we assume that all measurements are described by symmetrical probability densities: $\rho_A(y)=\rho_A(-y)$ and $\rho_B(y)=\rho_B(-y)$, as it is the case in the example of Fig. \[twoMes\]. Then, we have: $P({\bf a}\to -{\bf b}) = P(-{\bf a}\to {\bf b})$ and $P(-{\bf b}\to {\bf a}) = P({\bf b}\to -{\bf a})$, so that $q_2=0$, but: $$q_1 =\int_{\cos\theta}^1[\rho_B(y)-\rho_A(y)]dy. \label{q-test3bis}$$ Clearly, if $\rho_B \neq \rho_A$, then $q_1\neq 0$, so that $q\neq 0$, showing again that the probability model described by the GTR-model can extend beyond quantum. It is interesting to also observe that being the quantum mechanical situation characterized by uniform probability densities, i.e., $\rho_B = \rho_A = \rho_u$, this means that in a pure quantum model both $q_1$ (relative indeterminism) and $q_2$ (relative asymmetry) are zero. But this is not the only way to satisfy the QQ-equality, as also the condition $q_1=-q_2$ can guarantee that $q=0$. In other terms, the QQ-equality is a necessary but not sufficient condition to test the quantumness of a probability model. In fact, even when the stronger condition $q_1=q_2=0$ is satisfied, the model can still be non-Hilbertian. To see this, consider the situation where $\rho_B = \rho_A = \rho$, with $\rho$ a symmetrical (but non-uniform) probability distribution. We then know that, similarly to quantum mechanics, the reciprocal law is satisfied and that $q_1=q_2=0$. As we are now going to show, this doesn’t mean however that the probability model is structurally equivalent to that described by the Born rule. According to the general theorem we have stated in Sec. \[The quantum mechanical example\], only when $\rho$ is a uniform distribution we recover the exact formulae predicted by the Born rule. For example, in the situation of a two-outcome measurement, the Born rule gives: $$P_{\rm Born}({\bf a}\to \pm{\bf b})=|\langle \psi_{\bf b}| \psi_{\bf a}\rangle|^2 = {1\over 2}(1\pm\cos\theta), \label{Born-2-dim}$$ which is very different form the GTR-model expression: $$P({\bf a}\to \pm{\bf b})= \int_{-1}^{\pm\cos\theta}\rho(y)dy. \label{GTR-2-dim}$$ For instance, for the specific choice $\rho_\epsilon(y)={1\over 2\epsilon}\chi_{[-\epsilon,\epsilon]}(y)$, with $\chi_{[-\epsilon,\epsilon]}$ the characteristic function of the interval $[-\epsilon,\epsilon]$, and $\epsilon\in [0,1]$ (the so-called $\epsilon$-model; see @Aerts1998 [@SassolideBianchi2013a]), we can write the more explicit expression: $$P({\bf a}\to \pm{\bf b}) =\delta_{\pm,-1}\Theta(-\cos\theta -\epsilon) + \delta_{\pm,+1}\Theta(\cos\theta -\epsilon)+\frac{1}{2}\left(1\pm {\cos\theta\over \epsilon}\right)\chi_{[-\epsilon,\epsilon]}(\cos\theta), \label{probability2b}$$ which clearly predicts different values than the Born rule (here $\Theta$ denotes the Heaviside step function, equal to 1 when the argument is positive and equal to 0 otherwise, and $\delta$ denotes the Kronecker delta, equal to 1 when the two indices are the same and equal to 0 otherwise). We observe that in the limit $\epsilon\to 0$, $\rho_\epsilon(y) \to \delta(y)$, i.e., the elastic becomes only breakable in its middle point, which corresponds to a measurement with no fluctuations. Then the third term of (\[probability2b\]) vanishes and one recovers an almost classical situation (almost because for $\cos\theta = \pm\epsilon$ we are in a situation of unstable equilibrium). On the other hand, in the opposite uniform limit $\epsilon \to 1$, $\rho_\epsilon(y)\to {1\over 2}$, the first two terms of (\[probability2b\]) vanish and the third term tends to the pure quantum expression (\[Born-2-dim\]). Now, albeit the values of the probabilities predicted by the Born rule (\[Born-2-dim\]) and by the GTR-model (\[GTR-2-dim\]) (or more specifically the $\epsilon$-model (\[probability2b\])) are manifestly different, one may nevertheless ask if the quantum model (i.e., the UTR-model in the Bloch sphere) would nevertheless describe the same experimental situations than a symmetric GTR-model, when all the elastic bands are the same, considering that both models satisfy (at least in the two-outcome situation) the equalities $q_1=q_2=0$. As we are now going to show, the answer is negative. For this, we need to consider three distinct measurements. If they are purely quantum we have, with obvious notation [@AertsSassolideBianchi2014a]: $$\langle \psi_{\bf a}| \psi_{-\bf b}\rangle = \langle \psi_{\bf a}|\left(| \psi_{\bf c}\rangle \langle \psi_{\bf c}|+| \psi_{-\bf c}\rangle \langle \psi_{-\bf c}|\right)| \psi_{-\bf b}\rangle = \langle \psi_{\bf a}| \psi_{\bf c}\rangle \langle \psi_{\bf c}|\psi_{-\bf b}\rangle + \langle \psi_{\bf a}| \psi_{-\bf c}\rangle \langle \psi_{-\bf c}|\psi_{-\bf b}\rangle, \label{resolutionidentity}$$ where for the first equality we have used the resolution of the identity: $| \psi_{\bf c}\rangle \langle \psi_{\bf c}|+| \psi_{-\bf c}\rangle \langle \psi_{-\bf c}|=\mathbb{I}$. It immediately follows that if we can find an experimental situation in the GTR-model with identical and symmetrical elastic bands such that, say, $P({\bf c}\to -{\bf b})=0$ (implying $\langle \psi_{\bf c}|\psi_{-\bf b}\rangle =0$) and $P({\bf a}\to -{\bf c})=0$ (implying $\langle \psi_{\bf a}| \psi_{-\bf c}\rangle =0$), but also $P({\bf a}\to -{\bf b})\neq 0$ (implying $\langle \psi_{\bf a}| \psi_{-\bf b}\rangle \neq 0$), then such situation would clearly be incompatible with the quantum identity (\[resolutionidentity\]), and therefore would be modelizable by the symmetric GTR-model, but not by the Born rule of quantum mechanics. A simple situation of this kind is described in Figure \[threeMes\]. The human cognition example {#The human cognition example} =========================== In Sec. \[The quantum mechanical example\] we have shown that the quantum mechanical (Hilbertian) model is a special case of the GTR-model, obtained by considering only uniform membranes and a Blochean state space. In this section we want to provide another important implementation of the GTR-model: *human cognition*. In that respect, we recall that as from the beginning of the present century the quantum formalism has been applied with success to model a large number of cognitive phenomena, like information processing, human judgment and decision making, perception and memory, as well as concept combinations and conceptual reasoning; see for instance: @Kitto2008 [@Khrennikov2010; @BusemeyerBruza2012; @AertsEtal2013a; @AertsEtal2013b; @PothosBusemeyer2013; @WangEtal2013; @BlutnerbeimGraben2014; @AertsSassolideBianchi2015a; @AertsSassolideBianchi2015b] and the references cited therein. However, different from the situation of the elementary microscopic entities interacting with macroscopic measuring apparatuses, described by standard quantum mechanics, the general structure and the possible symmetries characterizing the human cognitive activity, when human minds interact with different conceptual entities, remain to be identified. Indeed, even though the quantum formalism has proven to work pretty well as a model, to fit many of the existing experimental data, it also fails to do so for many others (some examples will be given in the following). This means that the general probabilistic structures of the data generated by the human minds, in the different cognitive contexts, goes beyond that described by pure classical and pure quantum models. Therefore, to study these more general structures one needs an ampler theoretical framework than just classical or quantum mechanics, able to embrace from the beginning the full complexity of these non-physical (mental) processes. The GTR-model provides this more general framework, both in the Chatton and Occam sense: it is not more complex than necessary (Occam’s razor), but also it is not less complex than necessary (Chatton’s anti-razor). Before explaining why the GTR-model provides a natural, coherent and unitary description of human cognition, at a quite fundamental level, it is useful to briefly address a possible objection, which consists in saying that the GTR-model would contain too many free parameters, allowing it to easily fit all sorts of empirical data, but because of this it would not make it of great interest in providing an explanation for the observed phenomena. In fact, considering that an infinity of different probability densities $\rho$ can be used to describe each measurement (in addition to the choice of the orientations of the measurement simplexes), the model clearly allows for an uncountable infinity of free parameters! To answer this objection (which by the way is usually also addressed in relation to pure Hilbertian models, obeying the Born rule; see for instance the recent discussion by @BlutnerGraben2015), one has to distinguish between phenomenological models, where the different parameters are just introduced *ad hoc*, to obtain a good (or exact) data fit, but only for a reduced set of isolated experimental situations, from more fundamental models, designed with the precise intent of describing [*all possible situations*]{} in a given domain of experimentation, possibly also deriving the observed phenomena from first principles, so providing for them convincing explanations (and whenever possible, predictions). To give an example taken from physics, no one would ever have objected to Einstein that his general relativity theory was a bad explanation because it contained too many free parameters, as for instance the stress-energy tensor appearing in Einstein’s field equations could take any functional dependence on the space-time coordinates. Of course, this was not a weak trait of Einstein’s model, but its intrinsic richness, considering that his equations had to be applicable to all possible densities and fluxes of energy-momentum in spacetime. And this is precisely what made his theory a universal one. What was important in Einstein’s equations is that the free parameters associated with the stress-energy tensor always remained in a clear and logical relation with respect to the other fundamental quantities of the theory, like Einstein’s and Ricci’s tensors and the metric. And as we are now going to explain, the same holds true, [*mutatis mutandis*]{}, for the GTR-model. When the GTR-model is applied to interrogative contexts involving human minds (a measurement can always be understood as an interrogative context, with the different outcomes being the available answers), different from the standard quantum formalism it allows for a clear distinction not only between ‘a question and its possible answers,’ but also between ‘the different *ways* an answer can be selected.’ Indeed, different human subjects, when subjected to a same interrogation (or situation eliciting a decision), will have each, in general, a different ‘way of choosing an answer.’ Different ‘ways of choosing’ can be described within the GTR-model by means of different probability densities $\rho$ (in the same way as different energy-momentum distributions can be described by means of different stress-energy tensors in general relativity). On the other hand, in standard quantum mechanics all measurements are described by the same uniform probability densities $\rho_u$, which means that a pure quantum model can only describe situations where all subjects participating in the experiment act as perfect “Bornian clones,” all selecting an answer exactly in the same way (which is “the way of a uniform membrane”). Also, the disintegration-collapse of a membrane expresses in a very intuitive way what we humans typically perceive when facing a decisional context [@AertsSassolideBianchi2015a; @AertsSassolideBianchi2015b]. Indeed, when we are subjected to an interrogation, or a situation requiring a decision, we know that at the mental level a (neural) state of equilibrium will be built, expressing a sort of balancing of the tensions between the initial state of the conceptual entity we are subjected to, and the available (mutually excluding and competing) answer-states. The building of this equilibrium is described in the GTR-model by the abstract point particle entering the sphere and reaching an on-membrane position, so producing “tension lines” going from its position to the end points representative of the different outcomes. At some moment, always in accordance with what we can subjectively feel, some fluctuations will disturb this mental equilibrium, in a way that we cannot predict in advance, and trigger an irreversible and almost instantaneous process, drawing the abstract point particle to one of the vertices of the mental simplex, reducing in this way the previous tensional equilibrium (hence the “tension-reduction” name given to the model, which was suggested to us by Jerome Busemeyer). So, the different membranes in the GTR-model are representative of aspects of the minds of the different subjects, understood as dynamic memory structures sensitive to meaning. On the other hand, and consequently, the abstract point particle interacting with a membrane is not to be interpreted (as is usually done) as a description of the subject’s beliefs, but as an objective (intersubjective) element of a conceptual reality that is independent of the minds of the individuals that can possibly interact with it. This means that the different locations of the abstract point particle within the generalized Bloch sphere describe the different states a conceptual entity can be in, and that all these states have the same objective status for the different subjects participating in an experiment [@AertsGabora2005a; @AertsGabora2005b]; but different subjects, because of their different *forma mentis* (their different $\rho$), will extract a different meaning from them, in a given cognitive context, i.e., each subject will choose in a different way an outcome, i.e., an answer to the addressed question (and therefore each subject will be generally associated with a different statistics of outcomes). To give an example, consider the concept *Food*. When it is not under the influence of a specific context, we can say that it is in its “ground” state, which can be understood as a sort of basic prototype of the concept. But as soon as *Food* is contextualized, for instance in the ambit of the phrase *This food is very juicy*, its state will change. This means that its previous ground state will stop playing the role of a prototype, which will be played then by its new state, in a sort of new ‘contextualized prototype’. Now, the difference between the concept *Food* in a ground state and in an “excited” state, like the one associated with the above “juicy context,” can be evaluated by subjecting the concept to an additional context: that of a human mind that is asked to select a good representative of the concept, among a number of possible predetermined choices. The difference between the ground state *Food* and the excited state *This food is very juicy* will then manifest in the fact that, assuming for example that *Fruit* and *Vegetable* are among the possible choices of representatives, the former will be chosen much more frequently (i.e., with a higher probability) than the latter, when the concept is in its “juicy” excited state, rather than in its ground state. Replicable measurements ----------------------- As we said, a $\rho$-membrane describes in the GTR-model an aspect of a participant’s mind subjected to a given interrogative (or decisional) context. It is then natural to consider variations of the probability density $\rho$, when measurements are repeated, to account for the replicability effects that are easy to observe in experimental situations. In quantum mechanics, the replicability of an outcome is only predicted by the theory in relation to the repetition of the same measurement, according to von Neumann first kind condition. As we have seen in Sec. \[Measuring a coin\], the ‘tension-reduction’ mechanism associated with a disintegrable elastic membrane does automatically guarantee that if a measurement is repeated a second time, it will produce exactly the same result, with probability $1$. This because the membrane’s collapse cannot alter the position of the point particle when already located in one of the vertices of the measurement simplex. However, if, following a measurement $A$, a second different measurement $B$ is performed, and then, following the $B$-measurement, measurement $A$ is performed once again, one will not generally obtain the same outcome obtained in the first $A$-measurement with probability $1$. Indeed, the intermediary $B$-measurement will generally produce an outcome that is not an eigenstate of the $A$-measurement, so that when $A$ is performed a second time the outcome will not be certain in advance and could be different from the first $A$-outcome (unless the two measurements are compatible, a situation described in quantum mechanics by two commuting self-adjoint operators). When the $A$ and $B$ measurements are interrogative processes, and the measuring apparatus is a human mind, we know however that the situation is different. Indeed, it is to expected in this case that, in most situations, if we have given a certain answer to question $A$, then we will give the same answer to that same question if we are asked it a second time, even if in the meantime we have also answered to question $B$. This can happen for many reasons, like desire of coherence, learning, fear of being judged when we change opinion, etc. In quantum mechanics replicability is easy to model if one assumes that the two measurements $A$ and $B$ are associated with compatible (commuting) observables. The problem is that response replicability is expected to be observed also when the observables $A$ and $B$ are non-compatibles, i.e., when they describe interrogative contexts that, for instance, can give rise to *question order effects*, as commonly observed in social and behavioral research [@SudmanBradburn1974; @SchumanPresser1981; @TourangeauRipsRasinski2000]. In other terms, as @KhrennikovEtal2014 recently emphasized, an experimental situation where both question order effects and response replicability are present cannot be modelized by the standard quantum formalism. What about the GTR-model? Is it able to jointly describe (i.e., jointly model) question order effects and response replicability? The answer is clearly affirmative, as for this it is sufficient to allow the probability density $\rho$, describing the mind aspect of a respondent in relation to a given interrogation, to change in such a way that if the measurement is repeated, following an intermediary measurement, the same answer will be obtained with certainty. In Fig. \[ABA\] we give an example of how the probability densities $\rho_A$ and $\rho_B$ characterizing two two-outcome measurements $A$ and $B$ have to change, to guarantee response replicability. Note that the elastic bands represented in Fig. \[ABA\] are locally uniformly breakable, i.e., uniformly breakable on an interior segment (represented in grey color in the figure) and unbreakable everywhere else (black color in the figure). Note also that the specific structure of these elastics is precisely that required to model in an exact way the data obtained in experiments where subjects were asked to answer the following two incompatible “yes-no” questions, producing some typical question order effects [@Moore2002; @BusemeyerBruza2012; @WangBusemeyer2013]: “Do you generally think Bill Clinton is honest and trustworthy?” ($A$-measurement) and “Do you generally think Al Gore is honest and trustworthy?” ($B$-measurement). We shall not give here the details of this exact modelization, for which we refer the interested reader to @AertsSassolideBianchi2015e. Non-Hilbertian order effects {#Hilbertian symmetries} ---------------------------- Another example of the insufficiency of the quantum formalism in the modeling of psychological experiments is question order effects. This statement may appear a bit surprising, as these effects are considered by many authors to be among the most successful quantitative predictions of quantum theory in social and behavioral sciences. But this depends on the perspective that is taken on the whole issue. To explain what we mean, we start by considering an equality originally derived by @Niestegge2008 and rediscovered by @WangBusemeyer2013, which we have discussed already in Sec. \[Beyond quantum\]. It is very simple to derive: for this, we denote $P_a$ and $P_b$ two orthogonal projection operators acting on some Hilbert space ${\cal H}$, representing two properties $a$ and $b$ of the entity under study. We also consider the complementary projection operators $P_{\bar a}=\mathbb{I}-P_a$ and $P_{\bar b}=\mathbb{I}-P_b$, describing the orthocomplementary properties ${\bar a}$ and ${\bar b}$, and define the following self-adjoint operator: $$Q\equiv P_bP_aP_b - P_aP_bP_a + P_{\bar b}P_{\bar a}P_{\bar b} - P_{\bar a}P_{\bar b}P_{\bar a}. \label{Q-definition}$$ Taking its average $q \equiv{\rm Tr}\, QD$, over an arbitrary state $D$ (which can also be a density matrix), and considering that the probability $P(ba|D)$ of observing, in a sequence, first property $b$ then property $a$, is described in quantum physics by the average [@BusemeyerBruza2012]: $P(ba|D)={\rm Tr}\, P_bP_aP_bD$, and similarly for the other sequential (i.e., conditional) probabilities, then using the additivity of the trace, we can write: $$q=[P(ba|D) - P(ab|D)]+[P({\bar b}{\bar a}|D) -P({\bar a}{\bar b}|D)], \label{QQ-equality2}$$ which apart from the different notation is precisely (\[q-test\]). The so-called QQ-equality consists in observing that, for whatever initial state $D$, $q=0$ [@BusemeyerBruza2012; @WangBusemeyer2013]. This can be easily proven by replacing $P_{\bar a}=\mathbb{I}-P_a$ and $P_{\bar b}=\mathbb{I}-P_b$ into (\[Q-definition\]), then developing the various terms and see that most of them simplify, so that one is left with the equality $Q=(P_b^2 - P_b) -(P_a^2 - P_a)$. Using the idempotency of the orthogonal projection operators, we thus have $Q=0$, and therefore also $q=0$, for whatever state $D$. The condition $q=0$ is generally considered to be a good test of the Hilbertian character of the probabilities involved in an experiment. However, as emphasized in Sec. \[Beyond quantum\], it can also be obeyed by non-Hilbertian models. This because what we have called the relative indeterminism contribution and the relative asymmetry contribution to $q$, can also mutually compensate, whereas these contributions need to be both zero in a Hilbertian model. But as we have seen, even when these contributions are both zero, the model can still be non-Hilbertian. This departure from the Hilbertian model can be better appreciated by performing a detailed analysis of data like the Clinton/Gore ones we have mentioned in the previous section. Indeed, although the associated probabilities appear to almost obey the $q=0$ condition, not only there are no reasons to expect that such condition would be exactly obeyed in the ideal limit of an infinite number of participants, but also, and more important, when these probabilities are described using the GTR-model, one immediately sees that their structure is highly non-Hilbertian. This because the breakable elastic bands that are needed to exactly model the data are very different from the globally uniformly breakable structures that are typical of quantum measurements. For example, the elastics depicted in Fig. \[ABA\] (a) are precisely those required to generate the Clinton/Gore probabilities, and they are manifestly non-uniform and non-symmetric, i.e., non-Bornian (see also the general discussion in @AertsSassolideBianchi2015e, where additional quantum identities are derived and shown to be strongly violated by the Clinton/Gore data, and similar ones). Individual and collective minds ------------------------------- To provide another argument as to why the standard quantum formalism cannot be considered to be sufficient to describe typical psychological measurements, let us come back for a moment to the results of Sec. \[The quantum mechanical example\]. We have shown that when different measurements, all having the same outcomes, are averaged out, in what we have called a universal average, one recovers the Born rule, provided the state space is Hilbertian. A physicists may rightly ask to what exactly these different measurements do correspond. Indeed, when a same measurement is performed a number of times in a physics laboratory, in order to obtain a sufficiently rich statistics of data, and deduce some robust experimental probabilities, the quantum entity is simply prepared every time in the same state, measured by means of the same instrument, in a large number of equivalent runs of the experiment. If the interpretation of a quantum measurement as a universal measurements is correct also for physics, this would mean that even though at each run of the measurement the same apparatus is used, the latter would nevertheless each time select an outcome *in a different way*, i.e., according to a different $\rho$-membrane, but since the experimenter would not know which $\rho$ is each time actualized by the apparatus, all these outcomes generated by the different membranes would be averaged out in the final statistics, yielding an effective description in terms of a uniform membrane, which is the Born rule. The existence of the hidden-membranes and the hidden-interactions associated with their possible breaking points remains of course hypothetical for the time being in physics, considering that we don’t have any direct access to such non-spatial (or pre-spatial) layer of our physical reality, from our limited Euclidean theater [@AertsSassolideBianchi2014a]. This of course does not mean that cleverly designed experiments wouldn’t be able in future to reveal these hidden and multidimensional dynamical structures, but for the time being they only remain a compelling theoretical explanation about how the quantum mechanical Born rule can emerge, as a first order approximation, from a substratum of more general probabilistic theories. What is the situation in psychological measurements? The main difference is that in that ambit there is an aspect of the measurements that, contrary to quantum measurements in physics laboratories, is not at all hidden. Indeed, in a typical psychological measurement the data are obtained from a number of different participants, for instance about a thousand in the previously mentioned Clinton/Gore experiment. And since these participants are all subjected to the same questions, in relation to a conceptual entity presented to them in the same intersubjective initial state, each participant in the measurement is the manifestation of a different $\rho$-membrane, corresponding to that specific mind aspect characterizing the way each of them, different from all the others, will select one of the available answers. In other terms, if the process of actualization of potential $\rho$ remains totally hidden (and also hypothetical for the time being) in physics laboratories, it is instead a perfectly manifest element of reality in psychological measurements. Considering however that the number $n$ of participants is necessarily finite, and in many measurements not necessarily large, it can be expected that in some situations the obtained average will not be well approximated by a universal one, and therefore the final probability model will not be Hilbertian. To put it in a different way, the abstract “collective mind” of the participants may not be representative of a “pure quantum mind,” if some “ways of choosing an outcome” are not actualizable, because the statistical sample of the available $\rho$-membranes is too small. It is however important to emphasize that when we limit our considerations to a single measurement situation, then a Hilbert space probabilistic model, or a Kolmogorovian model, will always be sufficient to fit the experimental data, as these two models are “universal probabilistic machines,” capable of representing all possible probabilities appearing in nature, in a *single* measurement context [@AertsSozzo2012a; @AertsSassolideBianchi2015a]. But when we look for a consistent representation for different non-compatible measurements, this is where classical probabilities become totally inadequate, and pure quantum (Bornian) probabilities become too specific to describe all possible experimental situations. The situation becomes even more problematic when we try not only to devise a consistent model for the description of a collection of different measurements (different questions) associated with different outcomes (different answers), but when we also consider the possibility of combining these different measurements and their outcomes in a sequential way. This introduces an additional difficulty, which is precisely the problem of distinguishing the individual level from the collective one. Indeed, when psychologists consider sequential measurements, to highlight possible question order effects, it is not the abstract collective mind that is subjected to the sequence of measurements, in different orders, but each one of the individual minds of the participants. In other terms, the sequence of measurements is first performed at the individual level (each participant is asked to answer two questions in a given succession), then an average of their obtained answer is considered. Of course, the reason why an overall question order effect is observed is because the effect manifests at the individual level, and is then transferred from the individual to the collective level, in the final statistics of outcomes. If we ask the first question to an individual, then the second question to another individual, no order effects would be observed (and the same holds true for response replicability effects). It is of course essential that a same individual in the sample of respondents replies to the two questions in a given order, for the effect to manifest, being it generated at the level of the individual mind and not of the collective one. All we are saying is of course perfectly evident, but it is important not to mix these two different levels: the individual and the collective. Let us consider the previously mentioned Clinton/Gore example to further clarify our point. Imagine for a moment that we have found a way to perfectly clone the $i$-th individual participating in the opinion pool. If we repeat many times the “$A$ then $B$” and the “$B$ then $A$” sequential measurements, using these $i$-clones (we can only use each clone once, because of response replicability), then calculate the relative frequencies of the observed outcomes and use the GTR-model to fit the data, we would find two elastic bands with a specific orientation in the Bloch sphere, characterized by some generally non-uniform probability densities $\rho_A^{(i)}$ and $\rho_B^{(i)}$. This is the description at the individual level. When we consider the responses obtained from all the $n$ different individuals participating in the pool, the probabilities we end up calculating are equivalent to a uniform average over the different sequential probabilities that would have been generated by these $i$-clones, for $i=1,\dots, n$, i.e., the probabilities we can deduce from the associated $\rho_A^{(i)}$- and $\rho_B^{(i)}$-elastics. These overall probabilities can in turn be modeled by using also two effective elastic bands, characterized by some probability densities $\rho_A$ and $\rho_B$ and a specific orientation. So, at the formal level, the abstract collective mind is described as if it was an individual mind, also performing the sequential measurements, according to its specific “forma mentis.” However, and this is the subtle point we want to clarify with the present discussion, the collective mind does not really perform a sequential measurement. To see this, consider the following sequential measurement performed “at the collective level.” We first ask question $A$ (“Do you generally think Bill Clinton is honest and trustworthy?”), and to obtain an answer we randomly select one of the participants, ask the question and collect the answer. Then, we ask question $B$ (“Do you generally think Al Gore is honest and trustworthy?”) and to obtain the answer we again randomly select one of the participants, ask the question and collect the answer. Similarly, we can perform the same sequential process in reversed order, by asking first $B$ and then $A$. Now, apart the very special circumstance where the same participant would be selected in one or both of the above sequences (a possibility whose probability tends to zero as the number of participants increases), no order effects will be observed in this way. In other terms, at the collective level, if the participants are randomly chosen at each measurement, no order effects will be observed, and of course the same remains true for the response replicability effects. This is because if we randomly chose a new participant every time that we ask a question, all memory effects will be destroyed, and all measurements will become compatible. At the individual level, measurements are generally incompatible because the answer given to a first measurement remains in the field of consciousness of the respondent, changing in this way the state of the conceptual entity when a second question is asked. For example, in the Clinton/Gore experiment, the entity which is measured by each individual mind is the conceptual entity *Honesty and trustworthiness* (which for brevity, we shall simply denote *Honesty*). Prior to a measurement, we can consider that such entity is in its “ground” (most neutral) state, this being true for all the respondents. At the individual level, when performing measurement $A$, a subject is asked if s/he thinks Clinton is honest. Considering this as a measurement of the conceptual entity *Honesty*, the interrogation can be rephrased as follows: “What best represents *Honesty*, between the two possibilities: *Clinton is honest* and *Clinton is not honest*?” It is also worth observing that the outcomes *Clinton is honest* and *Clinton is not honest* are here to be considered as “excited states” of the conceptual entity *Honesty*. Now, when a subject is submitted to the $A$-measurement, the outcome (i.e., the answer) will generally remain in her/his field of consciousness when the same subject is submitted to the subsequent $B$-measurement, corresponding to the question “Do you generally think Al Gore is honest?” This means that when the $B$-measurement is performed immediately after $A$, the measured conceptual entity will not anymore be in its ground state, but in the excited state corresponding to the outcome of the $A$-measurement. If, say, the answer to the $A$-interrogation is “yes,” that is, “Clinton is honest,” the effective subsequent $B$-measurement will be: “What best represents *Clinton is honest* between the two possibilities: ‘*Gore is honest* and *Gore is not honest*?’’ In other terms, the short term memory of a participant is what allows her/him to keep track of the change of the state of the conceptual entity under consideration, in a sequence of different measurements, and this memory effect, manifesting at the individual level, is what in the end produces the order (and replicability) effects. This memory effect would of course be lost at the collective level, if respondents are selected in a random way at each measurement in a sequence. It could however be restored if the experimental protocol would be so designed to keep track of the obtained answer, and use them as a new input state when a successive measurement is performed, on a new randomly chosen subject. Having elucidated this difference between the individual and collective levels, we want now explain how the averaging procedure, when performed on sequential measurements operated at the individual level, can generate a *symmetry breaking process* that can also be held in part responsible for the departure of the experimental probabilities from Hilbertian-like symmetries, like for instance that expressed by the QQ-equality. For this, we consider the simple situation of a collective mind formed by only two individuals and we assume that each of them, when subjected to a sequence of two two-outcome measurements $A$ and $B$, will use the same locally uniform and symmetric probability density, in both measurements. In other terms, we assume that $\rho^{(i)}_A=\rho^{(i)}_B={1\over 2\epsilon_{i}}\chi_{[-\epsilon_{i},\epsilon_{i}]}$, where the index $i=1,2$, denotes the individual subject. Thus, we are here in a situation where both the relative indeterminism contribution and the relative asymmetry contribution are zero and the symmetry expressed by the QQ-equality is obeyed. In other terms, even though the measurements are not purely quantum, they still obey the QQ-equality that is also obeyed by pure quantum systems, and in that sense (but only in that sense) the situation can be considered to be, at the individual level, close to a pure quantum situation. What happens then when we consider a uniform average over these two participants? To see this, we denote ${\bf a}$ and $-{\bf a}$ the two outcomes of measurement $A$, and ${\bf b}$ and $-{\bf b}$ those of measurement $B$, as represented in the Bloch sphere. According to (\[probability2b\]), if we assume that the angle $\theta$ between the two elastic bands ($\cos\theta = {\bf a}\cdot {\bf b}$) is such that $\cos\theta <\epsilon_{i}$, $i=1,2$, and that the angle $\theta_A$ between the unit vector ${\bf x}$ describing the initial state and outcome ${\bf a}$ ($\cos\theta_A = {\bf x}\cdot {\bf a}$) is such that $\cos\theta_A <\epsilon_{i}$, $i=1,2$, then for the sequence of outcomes “${\bf a}$ then ${\bf b}$” we have the $i$-individual probability: $$\begin{aligned} P^{(i)}(\to {\bf a}\to -{\bf b}|{\bf x})&=& P^{(i)}({\bf x}\to {\bf a}) P^{(i)}({\bf a}\to {\bf b})= {1\over 2}(1+ {1\over \epsilon_i}\cos\theta_A){1\over 2}(1+ {1\over \epsilon_i}\cos\theta)\nonumber\\ &=&{1\over 4}[1+{1\over \epsilon_i}(\cos\theta +\cos\theta_A)+{1\over \epsilon_i^2}\cos\theta\cos\theta_A]. \label{symmetricalprob}\end{aligned}$$ If now we consider the uniform average: $P(\to {\bf a}\to -{\bf b}|{\bf x})\equiv{1\over 2}[P^{(1)}(\to {\bf a}\to -{\bf b}|{\bf x})+P^{(2)}(\to {\bf a}\to -{\bf b}|{\bf x})]$, we obtain: $$P(\to {\bf a}\to -{\bf b}|{\bf x})= {1\over 4}\left[1+{\epsilon_1 +\epsilon_2\over 2\epsilon_1 \epsilon_2}(\cos\theta +\cos\theta_A)+{\epsilon_1^2+\epsilon_2^2 \over 2\epsilon_1^2\epsilon_2^2}\cos\theta\cos\theta_A \right]. \label{average-n=2}$$ We observe that (\[average-n=2\]) can be written in the form (\[symmetricalprob\]) only if $(\epsilon_1 +\epsilon_2)^2=2(\epsilon_1^2+\epsilon_2^2)$, or $(\epsilon_1-\epsilon_2)^2 =0$, i.e., if $\epsilon_1=\epsilon_2$. But since by hypothesis $\epsilon_1\neq\epsilon_2$, we immediately see that the effective elastic bands describing the two measurements from the viewpoint of the “collective mind” (here just formed by two individuals), not only they are not anymore the same, but also they are not anymore symmetric (for a specific calculation, see @AertsSassolideBianchi2015e). Also, the obtained averaged probabilities will generally violate the QQ-quality. In other terms, the averaging procedure induces a breaking of possible symmetries in the structure of the probabilities, and a clear departure from the Hilbertian model. A remark is in order. In Sec. \[The quantum mechanical example\], we have explained that when all possible membranes (i.e., all possible ways of choosing an outcome) are allowed to be actualized in a single (non-sequential) measurement context (in what we have called a universal average, or universal measurement), the averaged probabilities are then described by an effective uniform membrane, equivalent to the Born rule, when the state space is considered to be Hilbertian. As we have just seen, the process of averaging over different membranes becomes much more involved when we deal with sequential measurements. Certainly, when considering a finite number of participants (the actual situation in real experiments), not all behaving as “Bornian clones,” the final statistics will be non-Hilbertian. However, it remains an open question to determine what would be the probability model of a ‘universal sequential measurement,’ i.e., of an average over sequential measurements when all possible probability densities are included in the calculation. We plan to come back to this interesting question in future works. Concluding remarks {#Conclusion} ================== In the present work we have reviewed and further illustrated some of the results we have recently obtained in @AertsSassolideBianchi2014a [@AertsSassolideBianchi2014b; @AertsSassolideBianchi2015a; @AertsSassolideBianchi2015b; @AertsSassolideBianchi2015c; @AertsSassolideBianchi2015d; @AertsSassolideBianchi2015e; @AertsSassolideBianchi2015f], to emphasize the interest and role played by the GTR-model (and the associated hidden-measurement approach) in the description of very general measurement situations, extending beyond the pure classical and pure quantum ones. As we have explained, these more general situations, and the associated probability models, are certainly relevant in the description of both physical and psychological experimental situations. Classical (Kolmogorovian) probabilities generally describe “static propositions,” i.e., our lack of knowledge about the actual elements of reality that are present in the system under study. On the other hand, quantum probabilities generally describe “dynamic propositions,” i.e., our lack of knowledge about processes of actualization of potential properties (if we exclude the special situations of measurements performed on eigenstates). Somehow in between these two descriptions, we can consider mixed measurements, where both static and dynamic logics, discovery and creation processes, actuality and potentiality, determinism and indeterminism, play an equivalent role. These more general, hybrid contexts, cannot be described using the too limited Hilbertian or Kolmogorovian models, but require more general structures. In that respect, it is important to realize that our reality, because of its extreme complexity, is able to manifest all sorts of mixtures of creation and discovery processes, also at a fundamental level. Therefore, in our investigations we need to be equipped with probabilistic models that are able to cope with such complexity, beyond the very specific classical and quantum structures. This of course does not mean that, in certain ambits, one will not try to highlight some possible remarkable symmetries, but to correctly describe them we certainly need a general enough theoretical approach, as only in this way we can hope to understand the full logic behind them. The example of the QQ-equality is in that sense paradigmatic: we know that the equality is exactly obeyed by pure Hilbertian models, but we have seen it can also be obeyed by a class of non-Hilbertian models, which are precisely those describing the question order effect in “opinion pool” psychological measurements. This means that if we want to understand the reasons behind these observed regularities, we cannot do so from the limited Hilbertian viewpoint, but need a more general approach. One of the scope of the present article was to point out that the GTR-model precisely provides such needed more general approach. Actually, we think that the GTR-model does more than this. Indeed, if it is correct to say that the Kolmogorovian model is a universal model for the description of situations governed by “static information,” certainly we cannot say that the Hilbertian model, equipped with the Born rule, is a universal model for situations governed by (non-Boolean) “dynamic information,” where also lack of knowledge about processes of actualization of potential properties is considered. This not only because the state space is Hilbertian, which may be a too severe constraint in certain situations, but also because very specific collapsing membranes are considered in quantum mechanics: the uniform ones. When all possible structures of membranes and state spaces are allowed, one certainly obtains the most general possible probabilistic description, i.e., a universal model for the description of both static and dynamic situations, which we have called the GTR-model. As our simple coin example illustrates, the necessity of using the more general GTR-model already manifests when considering experimental situations involving macroscopic objects. This because classical properties, and the associated classical probabilities, are insufficient to describe all possible observations. In fact, macroscopic objects possess more physical properties than those usually accounted for by classical mechanics, like for instance the “upper face” property of a coin, or of a die [@SassolideBianchi2013b]. And when these non-ordinary properties are considered, and tested in an operational way, not only the classical Kolmogorovian probabilities become inadequate, but the Hilbertian (Bornian) ones as well. On the other hand, the universality of the GTR-model allows to properly handle these non-ordinary measurement situations, and we cannot exclude that its structural richness will not also be instrumental in the description of anomalies manifesting in measurements with elementary physical entities (see for instance @AdenierKhrennikov2007 for an example of possible anomalies in the ambit of coincidence measurements). However, if the interest of the GTR-model for elementary (microscopic) physical systems remains to be evaluated (apart of course its theoretical interest in deriving and explaining the Born rule as a universal average), it is already a necessary tool for properly modeling human cognition, as we have illustrated in this work, considering that the data already in our possession cannot be exactly fitted by means of classical and Hilbertian models. 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--- abstract: 'LinkedIn has grown to become a platform hosting diverse sources of information ranging from member profiles, jobs, professional groups, slideshows etc. Given the existence of multiple sources, when a member issues a query like “software engineer”, the member could look for software engineer profiles, jobs or professional groups. To tackle this problem, we exploit a data-driven approach that extracts searcher intents from their profile data and recent activities at a large scale. The intents such as job seeking, hiring, content consuming are used to construct features to personalize federated search experience. We tested the approach on the LinkedIn homepage and A/B tests show significant improvements in member engagement. As of writing this paper, the approach powers all of federated search on LinkedIn homepage.' author: - | Dhruv Arya\ Viet Ha-Thuc\ Shakti Sinha\ - | \ \ bibliography: - 'sigproc.bib' title: Personalized Federated Search at LinkedIn --- ![LinkedIn Federated Search Result Page for Query “Software Engineer”[]{data-label="linkedin_federated_search"}](LinkedInFederatedSearch.png){width="40.00000%"} Introduction ============ LinkedIn has grown over the years from a professional networking site to becoming a platform containing multiple professional information sources such as member profiles, jobs, professional groups, member posts and slideshows. At the same time, the member base has also increased quickly and currently has more than 350 million members. The members visit the site for various reasons ranging from searching and recruiting candidates to looking for jobs or finding professional content etc. As the number of information verticals and member base increase, the problem of serving the right information to fulfill each individual member’s information need becomes more and more critical. This problem contains two subtasks including selecting the right verticals and aggregating the vertical results into a single ranking (See Figure \[linkedin\_federated\_search\]) and typically is referred as federated search. The area of federated search originated from meta search and has been actively researched in the field of information retrieval [@ShokouhiS11] and particularly in the context of Web search. Diaz [@Diaz09] addresses the problem of vertical selection, i.e., whether or not to show a specific item above the Web results given a query. Ponnuswami et. al. [@Ponnuswami11] and Arguello et. al. [@Arguello11] propose machine learning approaches for aggregating vertical results into single search result pages. They demonstrate effectiveness of the approaches on Bing search. More recently, Lefortier et. al. [@LefortierSRR14] present a way to blend individual vertical results and individual Web results with a case study on Yandex video search. However, the problem of federated search on LinkedIn presents unique challenges. First, the level of personalization in a platform like LinkedIn is much deeper than general Web search. For instance, if a member enters a query like “software engineer”, depending on if he or she is a recruiter, job seeker or professional content consumer, the member could expect to see software engineers’ profiles, jobs or slideshows on the topic. Second, individual results and blocks of results from different verticals are often associated with different features. Moreover, even if a feature is common, it is not equally important to them. This challenge is similar to federated Web search. Nonetheless, unlike Web search which typically blends *either* blocks of results [@Arguello11][@Ponnuswami11] or individual results [@LefortierSRR14] from different verticals, our system aggregates *both* individual results (e.g. jobs as shown in Figure \[linkedin\_federated\_search\]) and blocks of vertical results (e.g. people and professional group verticals). Thus, the system has to normalize features and eventually make relevance scores comparable across result verticals and result types (individual vs. block). Third, in Web search, Web results are typically the primary vertical thus they can be used as a “pivot” to normalize scores of the other verticals [@Ponnuswami11]. In our problem, the primary verticals vary depending on queries and other search context which could includes searcher’s data, past activities, location etc. To overcome these challenges, we propose a data-driven approach to personalize federated search. Specifically, we mine members’ data and their recent activities at a large scale to understand whether or not they currently have *intent* of hiring, job seeking or content consuming etc. This insight coupled with other signals are used to select verticals and aggregate vertical results into a single search result page personally relevant to each of our members. To make these signals comparable across different result categories, including verticals and result types (block vs. individual), we construct composite features combining the signals for each of the categories. Then, we let learning algorithms estimate different weights for all of the combinations (i.e., normalize the signals across the result categories) from training data. A/B tests done on the LinkedIn homepage shows improvements in member engagement and downstream traffic to the verticals. At the time of this writing, the work currently powers all of the federated search on the LinkedIn homepage. We organize the rest of the paper as follows. Section 2 details how we formulate federated search problem and our proposed approach. Section 3 describes searcher intent features and other signals used in the system. We discuss experimental results in Section 4. Finally, concluding remarks can be found in Section 5. Overall Approach ================ Problem Statement and Overall Framework --------------------------------------- Given a pair of *(query, searcher)*, our task is to select from a list of all possible verticals including people (members), jobs, companies, professional groups, member posts, slideshows etc. a primary vertical and a set of secondary verticals, then rank the primary individual results and the secondary vertical blocks in a single ranked list as shown in Figure \[linkedin\_federated\_search\], without changing the order in the primary vertical. The reason for this constraint is two fold. First, we believe the base ranker of each vertical is the best one to rank results within its domain. Second, this keeps member experience consistent between federated search and vertical search. ![Federated Search Overall Framework[]{data-label="fed_search_framework"}](Fed_search_framework.png){width="50.00000%"} The overall framework is described in Figure \[fed\_search\_framework\]. When a member issues a query *q*, the query is passed to verticals and triggers the corresponding vertical search engines to get the top *K* results for each. In preliminary vertical selection phase, the federated scorer extracts features (which are described later in Section 3) and computes a relevance score for each of the verticals. The top vertical is selected as the primary one and the rest are selected as *candidates* for secondary verticals. Then, in aggregation phase, these candidates compete with individual results in the primary vertical to form the final ranking. Note that these candidates are not guaranteed to show up in the ranking. Instead, depending on queries, searchers and vertical results, all, some or none of these candidates could be selected. The aggregation algorithm and the process of training the federated scorer are described in the next subsections. Aggregation Algorithm --------------------- In this section we discuss how our aggregation algorithm works. Input to the algorithm includes results from the primary vertical *P* along with all the candidate secondary vertical clusters *C* and a federated scorer $f_s$. We go through each of primary vertical result *$P_i$* computing the relevance score for this result using the federated scorer (the second loop in Algorithm 1). We compare this score with the relevance scores of all candidate secondary vertical clusters. If the former is higher, we pick the primary vertical result for $i^{th}$ position. If on the other hand there exists a candidate secondary vertical cluster that has a higher relevance score than the primary result, we add the secondary vertical cluster to the aggregated rank list and move on to the next primary result. We repeat this process till we position all the primary results. Any secondary vertical results left are dropped. $sortedSecondaryVerticals \rightarrow [ ]; rankList \rightarrow [ ];$ $j \leftarrow 1$; Federated Scorer Training ------------------------- As presented above, the purpose of federated scorer is to provide a *universal* relevance score for each vertical block as well as each vertical individual result. The scores have to be comparable across verticals (for preliminary vertical selection) and between vertical blocks and vertical individual results (for result aggregation). In this work, we train a federated scorer predicting probability that a vertical block or vertical individual result gets clicked if it appears on a search result page (SERP) shown to the member. Traditionally, ground truth data is labeled by human judges. However, this approach is expensive and not scalable. Moreover, it is very hard for the judges to judge the relevance on behalf of some other member, making it challenging to apply the approach for personalized settings. Thus, in this work the training data is collected from click logs via a randomization experiment exposed to a small random fraction of LinkedIn search traffic. In this experiment, given a query we apply some business rules to come up with a few eligible verticals. We randomly pick one as a primary vertical and leave the others as secondary ones. Then, we randomly insert the secondary verticals (as blocks of results) into the primary ranking without re-ordering the primary individual results. Clicked results (either primary individual results or secondary vertical blocks) are labeled positive and skipped results (unclicked ones ranked above the last clicked result in a ranking) are labeled as negative. The results ranked below the last clicked one are dropped since it is unknown if the member ignored these results or simply did not see them. The benefit of randomization is that it avoids the bias towards the original vertical selection and ranking. Given the training data, we apply logistic regression to train a federated scorer. The features used to train the scorer are described in the next section. Features ======== Searcher Intent --------------- A query can be ambiguous in the light of all information that exists in multiple verticals. For example, if a member enters a query like “machine learning”, he or she could be interested in recruiting machine learning people, looking for jobs related to machine learning, finding professional groups on the topic to join, discovering content on the topic or some of the use cases. To tackle this issue, we take a data-driven approach to personalize search results. For instance, if we know that the member is currently looking for a job, he or she is likely to be more interested in job results than the other verticals. Similarly, if the member is hiring machine learning scientists, people results should be more important to him or her. *Intents* of searchers such as job seeking, hiring, content consuming etc. are inferred from their profiles and past behaviors. At a high level, if a user’s title is recruiter, he or she is likely to have hiring intent. Similarly, if a user is a final year student, he or she could have job seeking intent. Another source of data used to infer user intents is their recent activities. For example, if users recently searched or applied for jobs, they tend to have job seeking intent. We train a machine-learned model combining all of the signals to predict intents for all of the member base. The model is run on a daily basic to update members’ intents dynamically. It is worth noting that a member could have multiple intents at the same time. A remaining challenge is that some evidence such as knowing a searcher has job seeking intent might be associated one or a few verticals (e.g. job vertical), but not all of them. Some evidence might be related to multiple verticals but with different levels of importance. To overcome this issue, we construct *composite features*, capturing both searcher intents and result categories including both verticals and result types (individual or block). For instance, the feature below only fires if the searcher has job seeking intent and the result is a block of jobs. Otherwise, it has value of zero. We create all of the combinations of intents and result categories and learn different weights for them. In essence, we let the learning algorithm associate each of the evidence with the categories and normalize them across the categories from training data. $$f = \left\{ \begin{array}{l l} 1 & \quad \text{if searcher has job seeking intent} \\ & \quad \quad \text{and the result is job vertical block} \\ 0 & \quad \text{otherwise} \end{array} \right.$$ Keyword Intent -------------- The feature category aims to capture the intents (result categories) of queries. Specifically, we mine historical click logs to estimate *p(result category| query)*, for instance the probability that members click on professional group vertical block for the query “leadership”. These probabilities are computed offline for every head query and use this insight to construct the features online. One issue of this approach is that the probabilities are biased towards the previous vertical selection and ranking algorithms. Resolving this is a future direction of this work. Base Ranking Features --------------------- We also exploit information provided by vertical first pass rankers (base rankers) to construct features. One example could be relevance scores from the first pass rankers. These features also have an effect of minimizing the inconsistency between the federated scorer and the first pass rankers (recall that the order in the primary vertical is kept unchanged). One issue with this kind of information is that the relevance scores might not be calibrated well across verticals. To resolve this issue, we again construct composite features like relevance score if result is an individual job or a slideshow vertical block. For the later, we compute relevance score for each block by averaging scores of the top results in the block. Then, we let the learning algorithm normalize the scores across result categories by learning different weights for the features from the training data. Experimental Results ==================== ***Baseline*** is a legacy federated search algorithm at LinkedIn. It uses a set of business rules based on past member interaction with verticals associated with keywords and relevance scores returned by vertical ranking functions. It can be viewed as a function on the feature sets described in Sections 3.2 and 3.3. Although the function is manually defined, it has been running on live traffic for a relatively long time and have been iteratively refined. The key difference between the baseline and the new approach is that the later is highly personalized by using searcher intent as a key signal. We conducted A/B test for sufficient duration of time (6 weeks) to account for any novelty effect. A query is first tagged for existence of entities like *names*, *job titles*, *skills* etc. As we are interested in exploratory search, we only experimented with non-name queries. Based on this condition, a random portion of LinkedIn federated search traffic from our world-wide member base is used for A/B testing. The federated search combines results from all of seven verticals available on LinkedIn including people, job, company, university, group, slideshow and members’ posts. The traffic is then randomly split into control and treatment buckets. Each bucket ends up with several hundreds of thousand searches. Searches in the control bucket are processed by the baseline and the treatment bucket uses the proposed approach. We look at three metrics including primary vertical click-throught-rate (CTR), secondary vertical CTR and secondary vertical switches. The difference between the second and the third metrics is that the former is defined on clicks on *individual results* within secondary clusters while the later is based on clicks on *cluster headers* that take users to vertical search pages. Table \[fig:metrics\] shows that the proposed approach is better than the baseline on all of the metrics (all of the improvements are statistically significant). Specifically, the proposed approach is $0.62\%$ better than the baseline on primary vertical CTR. In terms of secondary vertical engagement, the proposed approach largely improves over the baseline: $4.31\%$ and $10.66\%$ improvements on secondary vertical CTR and secondary vertical switches. It is somewhat surprising that the improvement on primary vertical CTR is much lower than on secondary vertical CTR. It is probably because the proposed approach shows more relevant secondary verticals, members are more likely to switch to secondary vertical search result pages ($10.66\%$ higher). Thus, they have less chance to engage in the primary results. A deep dive into log data also reveals that the baseline tends to over-emphasize primary results and on average ranks secondary vertical clusters lower in result pages. Fully understanding and modeling the trade-off between member engagement on primary and secondary results is another future direction of this work. **Metric** **Improvements** ----------------------------- ------------------ -- Primary Vertical CTR Secondary Vertical CTR Secondary Vertical Switches : Metrics improvements of treatment over baseline. Due to business sensitivity, we only show relative improvements instead of absolute metric values.[]{data-label="fig:metrics"} Conclusions =========== In this paper, we present the problem of personalized federated search at LinkedIn and propose a data-driven approach for this problem. Our approach takes into account members’ data and activities to infer their intents such as hiring and job seeking. This insight combined with other signals are used to select primary and candidate secondary verticals and then to aggregate primary individual results and the secondary clusters into a personalized ranking. Though presented in LinkedIn federated search context, the approach is applicable other domains where vertical selection and aggregation are highly personalized. Our A/B tests show that the approach could significantly improve user engagement. The approach is currently serving all of federated search on LinkedIn homepage. One future direction of this work is to remove the bias towards the previous vertical selection and ranking algorithms in the current keyword intent features. Another direction is to understand and model the trade-off between member engagement on primary verticals and engagement on secondary verticals. Given the insight, we will determine the best balance in terms of member experience.
--- abstract: 'This report discusses several topics in both top quark physics and QCD at an International Linear Collider (ILC). Issues such as measurements at the $t\bar{t}$ threshold, including both theoretical and machine requirements, and the determination of electroweak top quark couplings are reviewed. New results concerning the potential of a 500 GeV $e^+e^-$ collider for measuring $Wtb$ couplings and the top quark Yukawa coupling are presented. The status of higher order QCD corrections to jet production cross sections, heavy quark form factors, and longitudinal gauge boson scattering, needed for percent-level studies at the ILC, are reviewed. A new study of the measurement of the hadronic structure of the photon at a $\gamma\gamma$ collider is presented. The effects on top quark properties from several models of new physics, including composite models, Little Higgs theories, and CPT violation, are studied.' author: - 'A. Juste$^a$, Y. Kiyo$^b$, F. Petriello$^c$, T. Teubner$^d$, K. Agashe$^e$, P. Batra$^f$, U. Baur$^g$, C.F. Berger$^h$, J.A.R. Cembranos$^i$, A. Gehrmann-De Ridder$^j$, T. Gehrmann$^k$, E.W.N. Glover$^l$, S. Godfrey$^m$, A. Hoang$^n$, M. Perelstein$^o$, Z. Sullivan$^f$, T. Tait$^f$, S. Zhu$^p$' title: | \ Report of the 2005 Snowmass Top/QCD Working Group --- $\vcenter{ \hbox{\bf FERMILAB-CONF-06-006-T} \hbox{\bf LTH-688} \hbox{\bf MADPH-06-1252} \hbox{\bf SLAC-PUB-11620}} $ $^a$Fermi National Accelerator Laboratory, P.O. Box 500, MS 357, Batavia, IL 60510, USA\ $^b$Institut für Theoretische Physik E, RWTH Aachen, D-52056 Aachen, Germany\ $^c$University of Wisconsin, Madison, WI 53706, USA [*and*]{} Fermi National Accelerator Laboratory, P.O. Box 500, MS 106, Batavia, IL 60510, USA\ $^d$Department of Mathematical Sciences, The University of Liverpool, Liverpool L69 3BX, England, U.K.\ $^e$Johns Hopkins University, Baltimore, MD 21218, USA [*and*]{} Institute for Advanced Studies, Princeton, NJ 08540, USA [*and*]{} Syracuse University, Syracuse, NY 13244, USA\ $^f$High Energy Physics Division, Argonne National Laboratory, Argonne, IL 60439, USA\ $^g$State University of New York at Buffalo, Buffalo, NY 14260, USA\ $^h$Stanford Linear Accelerator Center, Stanford University, Stanford, CA 94309, USA\ $^i$University of California, Irvine, CA 92697, USA\ $^j$Institut für Theoretische Physik, ETH, CH-8093 Zürich, Switzerland\ $^k$Institut für Theoretische Physik, Universität Zürich, CH-8057 Zürich, Switzerland\ $^l$Institute for Particle Physics Phenomenology, University of Durham, Durham DH1 3LE, UK\ $^m$Ottawa Carleton Institute for Physics, Carleton University, Ottawa K1S 5B6, Canada\ $^n$Max-Planck-Institute for Physics, Munich, Germany\ $^o$CIHEP, Cornell University, Ithaca, NY 14853, USA\ $^p$Institute of Theoretical Physics, School of Physics, Peking University, Bejing 100871, China INTRODUCTION ============ The precision study of both new and already discovered particles will be a major component of the experimental programs at both the Large Hadron Collider (LHC) and a future International Linear Collider (ILC). The past two decades in particle physics established the importance of this precision physics program. The outstanding success of the $Z$-pole program at LEP and SLC elevated the global fit to the precision electroweak data into the primary experimental constraint on extensions of the Standard Model (SM). When combined with input from the Tevatron, it probes energy scales far beyond the kinematic limits of current colliders. The enormous production rates for top quarks at future colliders, reaching $10^7$ $t\bar{t}$ pairs in a $10 \, {\rm fb}^{-1}$ year at the LHC and $10^5$ in a $100 \, {\rm fb}^{-1}$ year at a 500 GeV $e^+e^-$ collider, will allow a similar program studying the top quark to be pursued. Rare decays of the top quark, deviations from its chirality structure in the SM, and its electroweak couplings will be studied. A high precision measurement of the top quark mass will greatly reduce the uncertainties in important electroweak parameters. A variety of work is needed for this program to be successful, including the precision calculation of top quark properties in the SM, the determination of experimental capabilities for performing measurements, and finding the most likely deviations predicted by models of new physics. The calculation of higher-order QCD corrections to top quark cross sections can have an important effect on top quark physics; for example, the threshold corrections to the process $e^+e^- \rightarrow t\bar{t}H$ increase its rate by a factor of two, drastically increasing the sensitivity of a 500 GeV ILC to the top quark Yukawa coupling (see Section 2.5 of this report). The precision needed for the $t\bar{t}$ threshold scan imposes strong requirements on the monitoring of the luminosity spectrum, which are discussed in Section 2.7. Detailed analyses can reveal surprising experimental possibilities, such as the measurement of the $Wtb$ coupling below the $t\bar{t}$ threshold (see section 2.1 of this report). The study of the predictions coming from models of new physics show that the top quark can be a powerful discriminator between different theories; for example, composite theories such as the Randall-Sundrum model predict shifts in the coupling of right-handed top quarks to the $Z$, while Little Higgs models generically predict shifts in the left-handed top couplings (see Sections 4.1 and 4.2 of this report). In addition to the study of the top quark, precision programs studying the Higgs boson or Higgs mechanism, the $W$ and $Z$ bosons, and other new particles discovered will be possible. To fully utilize the percent-level experimental precisions for each of these programs, higher order electroweak and QCD corrections in the SM must be included. For example, the two-loop QCD corrections to the $g-2$ of the $b$-quark reach the $2-3\%$ level, and must be included when studying $b$ production during a Giga-$Z$ phase at the ILC (see Section 2.3 of this report). Benchmark processes for studying a strongly-interacting Higgs sector such as $V_L V_L \rightarrow t\bar{t}$ receive QCD corrections reaching $10-20\%$, which can mask the effects of new physics if not taken into account (see Section 3.2 of this report). Finally, fundamental properties of QCD such as the running of the strong coupling constant and the hadronic structure of the photon can be studied with unprecedented precision (see Sections 3.1 and 3.3 of this report, respectively). In this report we discuss several issues in top quark physics and precision QCD, with a focus on the physics program at the ILC. Section 2 discusses precision studies of the top quark at the ILC. Important issues such as the measurement of the $t\bar{t}$ threshold cross section are reviewed, and new results such as the study of the $Wtb$ coupling below the $t\bar{t}$ threshold and the measurement of the top quark Yukawa at $\sqrt{s}=500$ GeV are presented. A preliminary study on the precise determination of the average beam energy and luminosity spectrum at the ILC, required for the $t\bar{t}$ threshold scan, is presented in section 2.7. Section 3 discusses important QCD physics that can be performed at the ILC. Higher order QCD corrections needed for several important measurements are discussed, and new results for the determination of the hadronic structure of the photon are presented. Section 4 studies modifications of top quark properties in several models of new physics, and analyzes the potential of the ILC to measure these shifts and use them to discriminate between different extensions of the SM. PRECISION STUDIES OF THE TOP QUARK AT THE ILC {#sec_ptop} ============================================= Measuring the Coupling Below the Threshold\ {#sbat} ------------------------------------------- A crucial test of the top quark’s electroweak interactions is the strength of the left-handed charged current interaction $W$-$t$-$b$. In the Standard Model, unitarity of the CKM matrix implies that $g_{Wtb} \sim g V_{tb} \sim g$, but in extended models, including the simple extension by a fourth generation of fermions, this interaction can differ significantly from the SM expectation. Currently, it is known that the $W$-$t$-$b$ vertex is sufficiently strong that the dominant top decay is $t \rightarrow W b$, but even an imprecise measurement is lacking. Single top production at the Tevatron and LHC will help fill this gap in our knowledge, and is expected to lead to a measurement at the $10\%$ level, dominated by systematics [@Beneke:2000hk]. Unlike other top measurements, a direct test of the $W$-$t$-$b$ coupling is challenging at a 500 GeV $e^+ e^-$ collider. A scan over the $t \bar{t}$ threshold region is expected to yield precise measurements of many top parameters in the SM, including the top mass, width and Yukawa coupling (see [@Martinez:2002st] and this report for projections), while above-threshold measurements may constrain anomalous, non-SM Lorentz structures [@Boos:1999ca]. Nevertheless, only an indirect measurement of the left-handed $W$-$t$-$b$ coupling is offered from the $t \bar{t}$ threshold region, by inferring its value from the SM relation and a precise value of the top width. If, for example, there is a small non-standard decay mode of top, it will alter the width and distort the inferred coupling. It would be more desirable to have a direct measurement of $W$-$t$-$b$, by making use of a process which is directly proportional to it. Close to the $t \bar{t}$ threshold, sensitivity to the coupling is quite weak, because the rate is essentially the $t \bar{t}$ production cross section times the branching ratios for $t \rightarrow W b$. Since we expect that the BR is very close to one, it does not in fact depend strongly on the magnitude of the $W$-$t$-$b$ interaction. Meanwhile, single-top production above threshold, which is sensitive to the $W$-$t$-$b$ coupling, is swamped by the $t \bar{t}$ background unless a $\gamma e$ collision mode is present [@Boos:2001sj]. ![Inclusive rates for $e^+ e^- \rightarrow W^+ b W^- \bar{b}$ as a function of the center-of-mass energy for $g_{Wtb} = g_{SM}$ (black solid), $g_{Wtb} = 2 g_{SM}$ (blue dashed), and $g_{Wtb} = g_{SM} /2 $ (red dotted).[]{data-label="fig:wtb1"}](wtb1.eps){width="65.00000%"} Just below the $t \bar{t}$ threshold, the reaction $e^+ e^- \rightarrow W^+ b W^- \bar{b}$ still occurs, through a mixture of non-resonant Feynman diagrams as well as through off-shell top quarks. At center-of-mass energies far enough below $2 m_t$ but still above $m_t$, the rate is dominated by contributions from the virtual $t \bar t$ diagrams in a kinematic configuration where one top is on-shell and the other is off-shell. The rate becomes very sensitive to the $W$-$t$-$b$ interaction, by virtue of the off-shell top [@Wtbinprogress]. This is illustrated in Figure 1, which plots the cross section as a function of energy for several values of $g_{Wtb}$, assuming a $175$ GeV top mass and a $115$ GeV Higgs mass. All analysis was performed using the MadEvent package [@Maltoni:2002qb] at tree level. The lines asymptote to the same value at both ends of the energy spectrum, as on-shell $t \bar{t}$ production dominates close to threshold and graphs not involving top dominate far below threshold. Both of these extremes are independent of the $W$-$t$-$b$ coupling. Thus, energies in between these two extremes are suitable to measure $g_{Wtb}$. We avoid the region very close to $2 m_t$ (despite its large rate), because the details of the transition from off-shell to on-shell do depend sensitively on the top width, which could obscure $g_{Wtb}$ if there are non-standard decay modes of the top. Instead, we focus on the energy $\sqrt{s}= 340$ GeV, where good leverage on this coupling appears to be attained with small dependence on the width. We will explore the interplay between $g_{Wtb}$ and $\Gamma_t$ below. Here we restrict ourselves to simple cuts to model the acceptance. To that end, we require the jets to have $p_T > 10$ GeV and rapidities $|y| < 2$. We assume $W$ bosons can be reconstructed with little background and for simplicity assume perfect $b$-tagging efficiency and no mis-tags. We improve the purity by requiring that one of the $b$ quarks and one of the reconstructed $W$’s reconstruct an invariant mass within $m_t \pm 10$ GeV, though we do not assume the charge of either the $b$ or the $W$ can be determined. The dominant background that is independent of the $W$-$t$-$b$ coupling comes from diagrams with an intermediate Higgs, which can be eliminated by subtracting events with $b \bar{b}$ that have an invariant mass close to the Higgs mass, once the mass is known. However, we do not impose such a cut in order to retain the most statistics possible. The number of events will depend strongly on the top mass, the Higgs mass, the top width and $g_{Wtb}$. It is expected that the ILC will determine the top and Higgs masses to order 100 MeV or better, which is enough to render the uncertainty in the rates from these parameters much smaller than the expected statistical uncertainties. The remaining dependence on the width and $g_{Wtb}$ allows us to determine a combination of both these quantities. To illustrate the results, we assume 100 fb$^{-1}$ collected at $\sqrt{s} = 340$ GeV. In Figure 2 we present the contours of constant event numbers in the plane of $g_{Wtb}$ and $\Gamma_t$ which reproduce the expected SM event rate of $\sim 1500$ events. Also shown are the contours corresponding to 1$\sigma$ and 2$\sigma$ deviations from such a measurement (assuming that the SM rate is observed and considering purely statistical uncertainties). The result is the expected bound one would obtain on $g_{Wtb}$ and $\Gamma_t$, which can be combined with the $\Gamma_t$ from the above-threshold scan to extract $g_{Wtb}$ itself (or alternately, one can go to lower energies where the sensitivity to $\Gamma_t$ is less, though at the price of the loss of some statistics). From Figure 2, we see that assuming the width is measured with an uncertainty of 100 MeV, $g_{Wtb}$ can be measured to the $2\%$ level, which would represent better than a factor of 5 improvement compared to the LHC, and a major improvement in our understanding of the $W$-$t$-$b$ interaction. ![Curve corresponding to the SM rate and its 1$\sigma$ and 2$\sigma$ deviations in the plane of $g_{Wtb}$ and $\Gamma_t$. Also overlaid is an expected measurement of $\Gamma_t$ from the on-shell threshold scan with an uncertainty of $100$ MeV.[]{data-label="fig:wtb2"}](wtb2.eps){width="65.00000%"} Many improvements on these rough estimates are possible. Certainly a more detailed and exhaustive study of the background would be interesting, as well as more sophisticated study of the signal, including higher order corrections and theory uncertainties, and detailed modelling of the $W$ decays and the observability of the $b$ quarks. In particular, higher order QCD and EW corrections to the signal will be essential to include in a realistic analysis in order to obtain the desired accuracy in $g_{Wtb}$, but are not likely to strongly change our conclusions as to how accurately one will be able to measure the coupling. Finally, since the cross section is strongly energy dependent, it could also be beneficial to consider the utility of a number of smaller data sets at several different energies below threshold. We leave such refinements for future work. Probing Electroweak Top Quark Couplings at the ILC and the LHC\ {#sec_Baur} --------------------------------------------------------------- Although the top quark was discovered almost ten years ago [@topcdf; @topd0], many of its properties are still only poorly known [@Chakraborty:2003iw]. In particular, the couplings of the top quark to the electroweak (EW) gauge bosons have not yet been directly measured. Current data provide only weak constraints on the couplings of the top quark with the EW gauge bosons, except for the $ttZ$ vector and axial vector couplings which are rather tightly but indirectly constrained by LEP data; and the right-handed $tbW$ coupling, which is severely bound by the observed $b\to s\gamma$ rate [@Larios:1999au]. At an $e^+e^-$ linear collider with $\sqrt{s}=500$ GeV and an integrated luminosity of $100-200$ fb$^{-1}$ one can hope to measure the $ttV$ ($V=\gamma,\,Z$) couplings in top pair production with a few-percent precision [@Abe:2001nq]. However, the process $e^+e^-\to\gamma^*/Z\to t\bar{t}$ is sensitive to both $tt\gamma$ and $ttZ$ couplings and significant cancellations between the various couplings can occur. At hadron colliders, $t\bar{t}$ production is so dominated by the QCD processes $q\bar{q}\to g^*\to t\bar{t}$ and $gg\to t\bar{t}$ that a measurement of the $tt\gamma$ and $ttZ$ couplings via $q\bar{q}\to\gamma^*/Z^*\to t\bar{t}$ is hopeless. Instead, the $ttV$ couplings can be measured in QCD $t\bar{t}\gamma$ production, radiative top quark decays in $t\bar{t}$ events ($t\bar{t}\to\gamma W^+W^- b\bar{b}$), and QCD $t\bar{t}Z$ production [@Baur:2004uw]. $t\bar{t}\gamma$ production and radiative top quark decays are sensitive only to the $tt\gamma$ couplings, whereas $t\bar{t}Z$ production gives information only on the structure of the $ttZ$ vertex. This obviates having to disentangle potential cancellations between the different couplings. In this section we briefly review the measurement of the $ttV$ couplings at the LHC and compare the expected sensitivities with the bounds which one hopes to achieve at an $e^+e^-$ linear collider. The most general Lorentz-invariant vertex function describing the interaction of a neutral vector boson $V$ with two top quarks can be written in terms of ten form factors [@Hollik:1998vz], which are functions of the kinematic invariants. In the low energy limit, these correspond to couplings which multiply dimension-four or -five operators in an effective Lagrangian, and may be complex. If $V$ is on-shell, or if $V$ couples to effectively massless fermions, the number of independent form factors is reduced to eight. If, in addition, both top quarks are on-shell, the number is further reduced to four. In this case, the $ttV$ vertex can be written in the form $$\label{eq:anomvertex} \Gamma_\mu^{ttV}(k^2,\,q,\,\bar{q}) = -ie \left\{ \gamma_\mu \, \left( F_{1V}^V(k^2) + \gamma_5F_{1A}^V(k^2) \right) + \frac{\sigma_{\mu\nu}}{2m_t}~(q+\bar{q})^\nu \left( iF_{2V}^V(k^2) + \gamma_5F_{2A}^V(k^2) \right) \right\} \, ,$$ where $e$ is the proton charge, $m_t$ is the top quark mass, $q~(\bar{q})$ is the outgoing top (anti-top) quark four-momentum, and $k^2=(q+\bar{q})^2$. The terms $F_{1V}^V(0)$ and $F_{1A}^V(0)$ in the low energy limit are the $ttV$ vector and axial vector form factors. The coefficients $F_{2V}^\gamma(0)$ and $F_{2A}^\gamma(0)$ are related to the magnetic and ($CP$-violating) electric dipole form factors. In $t\bar{t}V$ production, one of the top quarks coupling to $V$ is off-shell. The most general vertex function relevant for $t\bar{t}V$ production thus contains additional couplings, not included in Eq. (\[eq:anomvertex\]). These additional couplings are irrelevant in $e^+e^-\to t\bar{t}$, where both top quarks are on-shell. In $e^+e^-\to t\bar{t}$ one often uses the following parameterization for the $ttV$ vertex: $$\label{eq:gordon} \Gamma_\mu^{ttV}(k^2,\,q,\,\bar{q}) = ie \left\{ \gamma_\mu \, \left( \widetilde{F}_{1V}^V(k^2) + \gamma_5\widetilde{F}_{1A}^V(k^2) \right) + \frac{(q-\bar{q})_\mu}{2m_t} \left( \widetilde{F}_{2V}^V(k^2) + \gamma_5\widetilde F_{2A}^V(k^2) \right) \right\} .$$ Using the Gordon decomposition, it is easy to show that Eqs. (\[eq:anomvertex\]) and (\[eq:gordon\]) are equivalent for on-shell top quarks and that the form factors $\widetilde F^V_{iV,A}$ and $F^V_{iV,A}$ ($i=1,\,2$) are related by $$\label{eq:rel1} \widetilde F^V_{1V} = -\left( F^V_{1V}+F^V_{2V} \right) \, , \qquad \widetilde F^V_{2V} = F^V_{2V} \, , \qquad \widetilde F^V_{1A} = -F^V_{1A} \, , \qquad \widetilde F^V_{2A} = -iF^V_{2A} \, . \label{eq:rel4}$$ The most promising channel for measuring the $tt\gamma$ couplings at the LHC is $pp\to \gamma\ell\nu_\ell b\bar{b}jj$ which receives contributions from $t\bar t\gamma$ production and ordinary $t\bar t$ production where one of the top quarks decays radiatively, $t\to Wb\gamma$. In order to reduce the background, it is advantageous to require that both $b$-quarks are tagged. We assume a combined efficiency of $\epsilon_b^2=40\%$ for tagging both $b$-quarks. The non-resonant $pp \to W(\to\ell\nu)\gamma b\bar{b}jj$ background and the single-top backgrounds, $(t\bar{b}\gamma + \bar{t}b\gamma)+X$, can be suppressed by imposing invariant and transverse mass cuts which require that the event is consistent either with $t\bar{t}\gamma$ production, or with $t\bar{t}$ production with radiative top decay [@Baur:2004uw]. Imposing a large separation cut of $\Delta R(\gamma,b)>1$ reduces photon radiation from the $b$ quarks. Photon emission from $W$ decay products can essentially be eliminated by requiring that $m(jj\gamma) > 90~{\rm GeV}$ and $ m_T(\ell\gamma;p\llap/_T) > 90~{\rm GeV,}$ where $m(jj\gamma)$ is the invariant mass of the $jj\gamma$ system, and $m_T(\ell\gamma;p\llap/_T)$ is the $\ell\gamma p\llap/_T$ cluster transverse mass, which peaks sharply at $m_W$. After imposing the cuts described above, the irreducible backgrounds are one to two orders of magnitude smaller than the signal. The potentially most dangerous reducible background is $t\bar{t}j$ production where one of the jets in the final state fakes a photon. ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------ ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------ ![\[fig:fig1\][The differential cross sections as a function of the photon transverse momentum for $\gamma\ell\nu_\ell b\bar{b}jj$ production at the LHC. Part a) shows the SM signal and the various contributions to the background. Part b) shows the SM signal and background, and the signal for various anomalous $tt\gamma$ couplings.]{}](fig1a_lcws05.ps "fig:"){width="8.7cm"} ![\[fig:fig1\][The differential cross sections as a function of the photon transverse momentum for $\gamma\ell\nu_\ell b\bar{b}jj$ production at the LHC. Part a) shows the SM signal and the various contributions to the background. Part b) shows the SM signal and background, and the signal for various anomalous $tt\gamma$ couplings.]{}](fig1b_lcws05.ps "fig:"){width="8.7cm"} ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------ ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------ In Fig. \[fig:fig1\]a we show the photon transverse momentum distributions of the $t\bar{t}\gamma$ signal and the backgrounds discussed above. The $t\bar{t}j$ background is seen to be a factor 2 to 3 smaller than the $t\bar{t}\gamma$ signal for the jet-photon misidentification probability ($P_{j\to\gamma}=1/1600$ [@atlas_tdr]) used. The photon transverse momentum distributions in the SM and for various anomalous $tt\gamma$ couplings, together with the $p_T(\gamma)$ distribution of the background, are shown in Fig. \[fig:fig1\]b. Only one coupling at a time is allowed to deviate from its SM prediction. The process $pp \to t\bar{t}Z$ leads to either ${\ell'}^+{\ell'}^-\ell\nu b\bar{b}jj$ or ${\ell'}^+{\ell'}^- b\bar{b}+4j$ final states if the $Z$-boson decays leptonically and one or both of the $W$ bosons decay hadronically. If the $Z$ boson decays into neutrinos and both $W$ bosons decay hadronically, the final state consists of $p\llap/_Tb\bar{b}+4j$. Since there is essentially no phase space for $t\to WZb$ decays ($BR(t\to WZb)\approx 3\cdot 10^{-6}$ [@Mahlon:1994us]), these final states arise only from $t\bar tZ$ production. In order to identify leptons, $b$ quarks, light jets and the missing transverse momentum in dilepton and trilepton events, the same cuts as for $t\bar t\gamma$ production are imposed. One also requires that there is a same-flavor, opposite-sign lepton pair with invariant mass near the $Z$ resonance, $m_Z - 10~{\rm GeV} < m(\ell\ell) < m_Z + 10~{\rm GeV}$. The main backgrounds contributing to the trilepton final state are singly-resonant $(t\bar{b}Z+\bar{t}bZ)+X$ ($t\bar{b}Zjj$, $\bar{t}bZjj$, $t\bar{b}Z\ell\nu$ and $\bar{t}bZ\ell\nu$) and non-resonant $WZb\bar{b}jj$ production. In the dilepton case, the main background arises from $Zb\bar{b}+4j$ production. To adequately suppress it, one additionally requires that events have at least one combination of jets and $b$ quarks which is consistent with the $b\bar b+4j$ system originating from a $t\bar t$ system. Once these cuts have been imposed, the $Zb\bar{b}+4j$ background is important only for $p_T(Z)<100$ GeV. ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ![\[fig:fig2\][a) The differential cross sections at the LHC as a function of $p_T(Z)$ for ${\ell'}^+{\ell'}^-\ell\nu b\bar{b}jj$ final states. Shown are the SM predictions for $t\bar{t}Z$ production, for several non-standard $ttZ$ couplings, and for various backgrounds. Only one coupling at a time is allowed to deviate from its SM value. b) The differential cross sections as a function of the missing transverse momentum for $p\llap/_Tb\bar b+4$j production at the LHC. Shown are the SM predictions for $t\bar{t}Z$ production and for various backgrounds. ]{}](fig2a_lcws05.ps "fig:"){width="8.7cm"} ![\[fig:fig2\][a) The differential cross sections at the LHC as a function of $p_T(Z)$ for ${\ell'}^+{\ell'}^-\ell\nu b\bar{b}jj$ final states. Shown are the SM predictions for $t\bar{t}Z$ production, for several non-standard $ttZ$ couplings, and for various backgrounds. Only one coupling at a time is allowed to deviate from its SM value. b) The differential cross sections as a function of the missing transverse momentum for $p\llap/_Tb\bar b+4$j production at the LHC. Shown are the SM predictions for $t\bar{t}Z$ production and for various backgrounds. ]{}](fig2b_lcws05.ps "fig:"){width="8.7cm"} ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- The $Z$ boson transverse momentum distribution for the trilepton final state is shown in Fig. \[fig:fig2\]a for the SM signal and backgrounds, as well as for the signal with several non-standard $ttZ$ couplings. Only one coupling at a time is allowed to deviate from its SM prediction. The backgrounds are each more than one order of magnitude smaller than the SM signal. Note that varying $F^Z_{1V,A}$ leads mostly to a cross section normalization change, hardly affecting the shape of the $p_T(Z)$ distribution. For the $p\llap/_Tb\bar{b}+4j$ [@new] final state at least 3 jets with $p_T>50$ GeV and $p\llap/_T>5~{\rm GeV}^{1/2}\sqrt{\sum p_T}$ are required. The largest backgrounds for this final state come from $t\bar t$ and $b\bar b+4j$ production where one or several jets are badly mismeasured, from $pp\to t\bar tjj$ with $t\bar t\to\ell^\pm\nu_\ell b\bar bjj$ and the charged lepton being missed, and from $t\bar tj$ production, where one top decays hadronically, $t\to Wb\to bjj$, and the other via $t\to Wb\to\tau\nu_\tau b$ with the $\tau$-lepton decaying hadronically, $\tau\to h\nu_\tau$. In Fig. \[fig:fig2\]b we show the missing transverse momentum distributions of the SM $t\bar tZ\to p\llap/_Tb\bar{b}+4j$ signal (solid curve) and various backgrounds. The most important backgrounds are $t\bar tjj$ and $t\bar tj$ production. However, the missing transverse momentum distribution from these processes falls considerably faster than that of the signal, and for $p\llap/_T>300$ GeV, the SM signal dominates. The shape and normalization changes of the photon or $Z$-boson transverse momentum distribution can be used to derive quantitative sensitivity bounds on the anomalous $tt\gamma$ and $ttZ$ couplings. For $t\bar{t}Z$ production with $Z\to{\ell'}^+{\ell'}^-$, the $\Delta\Phi({\ell'}{\ell'})$ distribution provides additional information [@Baur:2004uw]. In the following we assume a normalization uncertainty of the SM cross section of $\Delta{\cal N}=30\%$. Even for a modest integrated luminosity of 30 fb$^{-1}$, it will be possible to measure the $tt\gamma$ vector and axial vector couplings, and the dipole form factors, with a precision of typically $20\%$ and $35\%$, respectively. For 300 fb$^{-1}$, the limits improve to $4-7\%$ for $F^\gamma_{1V,A}$ and to about $20\%$ for $F^\gamma_{2V,A}$. At the SLHC, assuming an integrated luminosity of 3000 fb$^{-1}$, one can hope to achieve a $2-3\%$ measurement of the vector and axial vector couplings, and a $10\%$ measurement of $F^\gamma_{2V,A}$, provided that particle identification efficiencies are not substantially smaller, and the reducible backgrounds not much larger, than what we have assumed. To extract bounds on the $ttZ$ couplings, we perform a simultaneous fit to the $p_T(Z)$ and the $\Delta\Phi({\ell'}{\ell'})$ distributions for the trilepton and dilepton final states, and to the $p\llap/_T$ distribution for the $p\llap/_Tb\bar b+4j$ final state. We calculate sensitivity bounds for 300 fb$^{-1}$ and 3000 fb$^{-1}$ at the LHC; for 30 fb$^{-1}$ the number of events expected is too small to yield meaningful results. For an integrated luminosity of 300 fb$^{-1}$, it will be possible to measure the $ttZ$ axial vector coupling with a precision of $10-12\%$, and $F^Z_{2V,A}$ with a precision of $40\%$. At the SLHC, these bounds can be improved by factors of about 1.6 ($F^Z_{2V,A}$) and 3 ($F^Z_{1A}$). The bounds which can be achieved for $F^Z_{1V}$ are much weaker than those projected for $F^Z_{1A}$. As mentioned in Sec. 4, the $p_T(Z)$ distributions for the SM and for $F^Z_{1V,A}=-F^{Z,SM}_{1V,A}$ are almost degenerate. This is also the case for the $\Delta\Phi({\ell'}{\ell'})$ distribution. In a fit to these two distributions, therefore, an area centered at $\Delta F^Z_{1V,A}=-2F^{Z,SM}_{1V,A}$ remains which cannot be excluded, even at the SLHC. For $F^Z_{1V}$, the two regions merge, resulting in rather poor limits. --------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- coupling LHC, 300 fb$^{-1}$ $e^+e^-$ [@Abe:2001nq] coupling LHC, 300 fb$^{-1}$ $e^+e^-$ [@Abe:2001nq] ---------------------------------- ---------------------------------- -------------------------------------- ----------------------------- ----------------------------------- -------------------------------------- $\Delta\widetilde F^\gamma_{1V}$ $\begin{matrix}{ +0.043 \\[-4pt] $\begin{matrix}{ +0.047 \\[-4pt] $\Delta\widetilde F^Z_{1V}$ $\begin{matrix} {+0.24 \\[-4pt] $\begin{matrix} {+0.012 \\[-4pt] -0.041}\end{matrix}$ -0.047}\end{matrix}$ , 200 fb$^{-1}$ -0.62}\end{matrix}$ -0.012}\end{matrix}$ , 200 fb$^{-1}$ $\Delta\widetilde F^\gamma_{1A}$ $\begin{matrix} {+0.051 \\[-4pt] $\begin{matrix} {+0.011 \\[-4pt] $\Delta\widetilde F^Z_{1A}$ $\begin{matrix} {+0.052 \\[-4pt] $\begin{matrix} {+0.013 \\[-4pt] -0.048}\end{matrix}$ -0.011}\end{matrix}$ , 100 fb$^{-1}$ -0.060}\end{matrix}$ -0.013}\end{matrix}$ , 100 fb$^{-1}$ $\Delta\widetilde F^\gamma_{2V}$ $\begin{matrix} {+0.038 \\[-4pt] $\begin{matrix}{ +0.038 \\[-4pt] $\Delta\widetilde F^Z_{2V}$ $\begin{matrix} {+0.27 \\[-4pt] $\begin{matrix} {+0.009 \\[-4pt] -0.035}\end{matrix}$ -0.038}\end{matrix}$ , 200 fb$^{-1}$ -0.19}\end{matrix}$ -0.009}\end{matrix}$ , 200 fb$^{-1}$ $\Delta\widetilde F^\gamma_{2A}$ $\begin{matrix} {+0.16 \\[-4pt] $\begin{matrix} {+0.014 \\[-4pt] $\Delta\widetilde F^Z_{2A}$ $\begin{matrix} {+0.28 \\[-4pt] $\begin{matrix} {+0.052 \\[-4pt] -0.17}\end{matrix}$ -0.014}\end{matrix}$ , 100 fb$^{-1}$ -0.27}\end{matrix}$ -0.052}\end{matrix}$ , 100 fb$^{-1}$ --------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- : Sensitivities achievable at $68.3\%$ CL for the anomalous $ttV$ ($V=\gamma,\,Z$) couplings $\widetilde F^V_{1V,A}$ and $\widetilde F^V_{2V,A}$ of Eq. (\[eq:gordon\]) at the LHC for integrated luminosities of 300 fb$^{-1}$, and the ILC with $\sqrt{s}=500$ GeV (taken from Ref. [@Abe:2001nq]). Only one coupling at a time is allowed to deviate from its SM value.\ \[tab:tab1\] It is instructive to compare the bounds for anomalous $ttV$ couplings achievable at the LHC with those projected for the ILC. The most complete study of $t\bar{t}$ production at the ILC for general $ttV$ ($V=\gamma,\,Z$) couplings so far is that of Ref. [@Abe:2001nq]. It uses the parameterization of Eq. (\[eq:gordon\]) for the $ttV$ vertex function. In order to compare the bounds of Ref. [@Abe:2001nq] with those anticipated at the LHC, the limits on $F^V_{1V,A}$ and $F^V_{2V,A}$ have to be converted into bounds on $\widetilde F^V_{1V,A}$ and $\widetilde F^V_{2V,A}$. Table \[tab:tab1\] compares the bounds we obtain for $\widetilde F^V_{1V,A}$ and $\widetilde F^V_{2V,A}$ with those reported for the ILC in Ref. [@Abe:2001nq]. Note that only one coupling at a time is allowed to deviate from its SM value [@Abe:2001nq]. We show LHC limits only for an integrated luminosity of 300 fb$^{-1}$. For the SLHC, with 3000 fb$^{-1}$, we obtain bounds which are a factor $1.3-3$ more stringent than those shown in Table \[tab:tab1\]. Thus, even if the SLHC operates first, and the $p\llap/_Tb\bar b+4j$ final state is taken into account, a linear collider will still be able to significantly improve the $ttZ$ anomalous coupling limits, with the possible exception of $\widetilde F^Z_{1A}$. The ILC will also be able to considerably strengthen the bounds on $\widetilde F^\gamma_{1A}$ and $\widetilde F^\gamma_{2A}$. It should be noted, however, that this picture could change once correlations between different non-standard $ttZ$ couplings, and between $tt\gamma$ and $ttZ$ couplings, are taken into account. Unfortunately, so far no realistic studies for $e^+e^-\to t\bar t$ which include these correlations have been performed[^1]. The LHC will be able to perform first tests of the $ttV$ couplings. Already with an integrated luminosity of 30 fb$^{-1}$, one can probe the $tt\gamma$ couplings with a precision of about $10-35\%$ per experiment. With higher integrated luminosities one will be able to reach the few percent region. With the exception of $F^Z_{1A}$, the $ttZ$ couplings can only be measured with a precision of $15-50\%$, even at the SLHC. The ILC will be able to further improve our knowledge of the $ttV$ couplings, in particular in the $ttZ$ case. Two-loop Corrections to Heavy Quark Form Factors\ {#sg2} ------------------------------------------------- The international linear collider will produce large numbers of top-antitop quark pairs, thus allowing for precision studies of the properties of the top quark. These experimental precision studies require equally precise theoretical predictions, i.e.higher order corrections in perturbation theory. Up to now, the theoretical effort was focused on a precise description of top quark production at threshold (see [@hoangteubner] for a review), where QCD corrections to next-to-next-to-leading order (NNLO) in perturbation theory are known, while observables other than the total production cross section in the continuum are known only to next-to-leading order (NLO) accuracy. In this section, we present results on the virtual two-loop corrections to vertex functions involving heavy quarks, which are an important ingredient to the NNLO corrections to top quark observables in the continuum. The vertex function coupling an on-shell heavy quark antiquark pair to an external current can be decomposed into so-called form factors, whose coefficients follow from Lorentz invariance and symmetry properties of the current. For the vector and axial vector current (electroweak gauge boson), the vertex function contains, within QCD, four form factors ($F_{1,2}$, $G_{1,2}$): $$\begin{aligned} \parbox{1.5cm}{\includegraphics[width=1.4cm]{ax.ps}} &{ = }&{ (-i) \left(v_Q { F_1(s,m^2)}\gamma^\mu + v_Q\frac{1}{2m}{ F_2(s,m^2)} i \sigma^{\mu \nu} (p_1+p_2)_\nu \right. } \nonumber \\ && \phantom{(-i)}{ \left. + a_Q { G_1(s,m^2)} \gamma^\mu \gamma_5 + a_Q \frac{1}{2m} { G_2(s,m^2)} \gamma_5 (p_1+p_2)^\mu \right) \;. }\end{aligned}$$ The coupling of heavy quarks to Higgs bosons of positive and negative parity contains the scalar and pseudoscalar form factors: $$\parbox{1.5cm}{\includegraphics[width=1.4cm]{h.ps}} { = - i \, \frac{m}{v} \, \left[ S_Q { F_{S}(s,m^2)} + i P_Q { F_{P}(s,m^2)} \gamma_5 \right]}.$$ Here $s$ is the invariant momentum squared of the external current and $m$ the heavy quark mass. ![image](2la.ps){width="120mm"} The two-loop corrections to the form factors are obtained by applying appropriate projections to the Feynman diagrams displayed in Figure \[fd2l\]. As a result, the form factors are expressed in terms of hundreds of different scalar integrals. These integrals are reduced to a small set of master integrals by means of the so-called Laporta algorithm [@Lap] with the help of integration-by-parts identities [@Chet] and Lorentz-invariance identities [@Rem3]. The master integrals themselves were evaluated with the method of differential equations [@Rem3; @Kot; @Rem1; @Rem2] in [@RoPieRem1; @RoPieRem3]. The master integrals, and thus the form factors are represented as series in the regularization parameter $\epsilon$ and expressed in terms of 1-dimensional harmonic polylogarithms up to weight 4 [@Polylog; @Polylog3]. We obtained the complete two-loop corrections for the renormalized vector [@Bernreuther:2004ih], axial-vector [@Bernreuther:2004th; @Bernreuther:2005rw], scalar and pseudoscalar [@scalar] form factors. Agreement was found with earlier partial results and with expansions around special kinematical points. An immediate physical application of the QCD corrections to the heavy quark form factors for arbitrary momentum transfer are predictions for form factors at zero recoil, the so-called static form factors. Measurements of these static form factors for heavy quarks could allow indirect constraints on new physics scenarios. Recently, there has been considerable effort to determine the feasibility of such experimental measurements. Specifically, the couplings to photons and $Z$ bosons have been studied in detail – both for heavy quark production at hadron colliders [@Baur:2004uw; @new; @Baur2] and at a future high-luminosity high-energetic linear electron-positron collider [@tesla; @Abe:2001nq; @new; @Baur3]. At this workshop, this issue was revisited in great depth especially in view of discriminating new physics scenarios (see sections 4.1 and 4.2). The most prominent static form factor is the electromagnetic spin-flipping form factor: the anomalous magnetic moment, which is finite in the zero-recoil limit, and can be obtained from the results of the previous section [@static]. It reads $$F_{2,Q}(s=0) = \frac{\alpha_s}{2\pi}\, C_F + \left(\frac{\alpha_s}{2\pi}\right)^2 F_{2,Q}^{(2l)} \, ,$$ with $$\begin{aligned} F_{2,Q}^{(2l)} &=& \phantom{+}C_F^2\left( -\frac{31}{4}+2\,\zeta_2\,(5-6\,\ln(2))+3\,\zeta_3\right) +C_F\,C_A\left( \frac{317}{36} +3\,\zeta_2\,(-1+2\,\ln(2))-\frac{3}{2}\,\zeta_3\right)\nonumber \\&& +C_F\,T_F\left(\frac{119}{9}-8\,\zeta_2\right) - \frac{25}{9} C_F\,T_F\,N_l + C_F\,\beta_0\,\ln({\mu^2}/{m_Q^2}),\end{aligned}$$ from which we can derive the static magnetic and weak magnetic form factor of a quark $Q$. We consider $$\left(\frac{g-2}{2}\right)^{\gamma,Z}_Q \equiv F_{2,Q}^{\gamma,Z} \left(0 \right) = {v}_Q^{\gamma,Z}\,F_{2,Q}\left(0 \right) \, ,$$ which correspond to the anomalous magnetic (MDM) and weak magnetic (WMDM) moments of $Q$. (Notice that in the literature the WMDM is often associated with $F_{2,Q}^Z(s=m_Z^2)$.) Numerical values for $t$ and $b$ quarks are given in Table \[tab\_anomag\]. $t$ $(\mu=m_t)$ $b$ $(\mu=m_b)$ $b$ $(\mu=m_Z)$ --------------------------- --------------------- --------------------- --------------------- $(g-2)^{\gamma,(1l)}_Q/2$ $1.53\cdot 10^{-2}$ $-1.52\cdot10^{-2}$ $-8.4\cdot10^{-3}$ $(g-2)^{\gamma,(2l)}_Q/2$ $4.7\cdot 10^{-3}$ $-1.00\cdot10^{-2}$ $-6.6\cdot10^{-3}$ $(g-2)^{\gamma}_Q/2$ $2.00\cdot 10^{-2}$ $-2.52\cdot10^{-2}$ $-1.50\cdot10^{-2}$ $(g-2)^{Z,(1l)}_Q/2$ $5.2\cdot 10^{-3}$ $-1.87\cdot10^{-2}$ $-1.03\cdot10^{-2}$ $(g-2)^{Z,(2l)}_Q/2$ $1.6\cdot 10^{-3}$ $-1.24\cdot10^{-2}$ $-8.1\cdot10^{-3}$ $(g-2)^{Z}_Q/2$ $6.8\cdot 10^{-3}$ $-3.11\cdot10^{-2}$ $-1.85\cdot10^{-2}$ : One- and two-loop QCD contributions, and their sums, to the anomalous magnetic and weak magnetic moments of the top and bottom quark, for different values of the renormalization scale $\mu$. \[tab\_anomag\] For the $b$ quark an upper bound on its magnetic moment was derived in [@Escribano:1993xr] from an analysis of LEP1 data, which, in our convention, reads $|\delta (g-2)^{\gamma}_b/2| < 1.5 \times 10^{-2}$ (68 % C.L.). Comparing it with Table \[tab\_anomag\] we see that the QCD-induced contributions to the $b$ quark magnetic moment saturate this bound, which implies that there is limited room for new physics contributions having the same sign as the QCD contributions to this quantity. At a future linear collider [@tesla; @Abe:2001nq], when operated at the $Z$ resonance, the sensitivity to this variable could be improved substantially, either by global fits or by analyzing appropriate angular distributions in $b {\bar b}$ and $b {\bar b} \gamma$ events. As to the static form factors of the top quark, no such tight constraints exist so far on possible contributions from new interactions (see Ref. [@Baur:2004uw] for a review). These quantities are particularly sensitive to the dynamics of electroweak symmetry breaking. For instance, in various models with a strongly coupled symmetry breaking sector one may expect contributions from this sector to the static $t$ quark form factors at the 5 - 10% level [@Berger:2005ht]. The QCD-induced anomalous magnetic moment and the QCD corrections to the axial charge of the top quark are of the same order of magnitude. Future colliders have the potential to reach this level of sensitivity. The form factors presented here have a number of applications, which will be addressed in future work. The vector and axial vector form factors contribute to the NNLO corrections to the forward-backward asymmetry for heavy quarks; they can also be used to compute the differential top quark pair production cross section at the ILC in the continuum, where top quark mass effects are still non-negligible. The scalar and pseudoscalar form factors enter the NNLO corrections to the decay of a Higgs boson into heavy quarks. These would become especially important for a very heavy Higgs boson, decaying into top quark pairs, where the form factors could be used for a differential description of the decay final state. Electroweak Effects at the Threshold\ {#shng} ------------------------------------- The proper treatment of electroweak effects plays an important role in high precision measurements at the $t\bar t$ threshold. In the calculation of the total cross section, one can categorize electroweak effects into three classes. - “Hard” electroweak effects: This class includes electroweak effects related to the $t\bar t$ production mechanism itself or factorizable corrections to various matching conditions of the effective theory that is used to describe the nonrelativistic QCD dynamics of the top pair. In general these corrections are modifications to the hard QCD matching conditions of the effective field theory operators. They can be determined by standard methods and are real numbers. - Electromagnetic effects: Electromagnetic effects are relevant for the luminosity spectrum of an $e^+e^-$ initial state (beam energy spread, beamstrahlung, initial state radiation) and the $t\bar b$ final state (modification to the Coulomb attraction, contributions to hard electroweak effects). - Effects related to the finite top quark lifetime: Apart from the top decay (into $Wb$ for the Standard Model) this class also includes interference contributions with processes having the same final state ($W^+W^-b\bar b$) but only one or even no top quark at intermediate stages. This class also accounts for interactions involving the top decay products (“non-factorizable” effects). For processes involving highly energetic top quarks, where the momentum transfer typically involves scales much larger than $\Gamma_t$, the class (c) effects are small corrections. On the other hand, for the top threshold dynamics all three classes have to be accounted for at leading order, because the typical kinetic energy of the nonrelativistic top quarks is of the same order as the top quark width, $$E_{\rm kin} \sim m_t v^2 \sim m_t \alpha_s^2 \sim \Gamma_t \sim m_t \alpha_{\rm em} \,. \label{ewcounting}$$ The relation in Eq. (\[ewcounting\]) gives the power-counting that has to be employed to systematically account for electroweak effects within the nonrelativistic expansion: $$v^2\sim \alpha_s^2 \sim \alpha_{\rm em} \,.$$ Several analyses concerning electroweak effects belonging to the various classes have been carried out in the past. However, no coherent and comprehensive treatment that systematically accounts for all these electroweak effects beyond leading order currently exists. In the following a brief status report is given concerning top threshold production in $e^+e^-$ annihilation. The leading order (LL) electroweak effects belonging to class (a) describe the production mechanisms of the nonrelativistic top quark pair in $e^+e^-$ annihilation in the various possible spin and angular momentum states. Due to the power-counting, the one-loop hard electroweak corrections already contribute at next-to-next-to-leading order (NNLL) and involve the standard (real parts of the) one-loop electroweak corrections to the $e^+e^-\to t\bar t$ process [@Grzadkowski:1986pm; @Guth:1991ab]. There are also hard electroweak corrections that modify the QCD potentials [@Kummer:1995id], but due to gauge cancellations these corrections do not contribute at NNLL order. The dominant corrections caused by an exchanged Higgs boson through a Yukawa potential can also be considered as a class (a) contribution. The most important contribution belonging to the electromagnetic effects in class (b) is the luminosity spectrum that affects the c.m.energy available in the $e^+e^-$ collision. While beamstrahlung and the beam energy spread are machine-dependent and will have to be measured experimentally, the initial state radiation component is calculable. The luminosity spectrum leads to sizable smearing of the cross section and contributes at LL order. Electromagnetic effects also modify the QCD potential through photon exchange between the top pair and can lead to hard electroweak corrections belonging to class (a). In present analyses QED effects are only accounted for through the luminosity spectrum. In particular there is no coherent treatment of the calculable QED effects that systematically accounts for the effects of initial state radiation, the Coulomb corrections and the hard QED effects. The leading order effect belonging to class (c) is the top decay width, which makes the perturbative treatment to the strong $t\bar t$ dynamics at all possible [@Kuhn:1980gw; @Fadin:1987wz]. Within the nonrelativistic effective theory used to describe QCD effects the top quark width can be implemented through an imaginary mass shift in the top quark propagator $$\frac{i}{k^0-\frac{{\mathbf p}^2}{m_t}+\delta m_t+\frac{i}{2}\Gamma_t} \,,$$ where $\delta m_t$ is related to the top quark mass definition that is used. This effect can be incorporated into the forward $e^+e^-\to e^+e^-$ scattering amplitude results for stable top quarks by simply shifting the c.m.energy into the complex plane: $\sqrt{s}\to \sqrt{s}+i\Gamma_t$. Using the optical theorem one can obtain the total cross section. Beyond leading order the electroweak effects belonging to class (c) can be systematically incorporated into the effective theory by accounting for absorptive parts related to the top decay final states in the matching conditions [@Hoang:2004tg] of the effective theory. This leads to well known effects such as the time dilatation correction [@Kummer:1995id], but can also account for interference contributions with processes where no top pairs are produced, but which have the same final state [@Hoang:2004tg] (see Fig. \[figunstable\]). ![ (a) Cuts in Standard Model diagrams contributing to absorptive parts in the matching conditions for the $t\bar t$ effective theory currents. Within the effective theory the absorptive parts describe the interference of the diagrams shown in (b). \[figunstable\] ](fig1.ps "fig:"){width="8cm"} ![ (a) Cuts in Standard Model diagrams contributing to absorptive parts in the matching conditions for the $t\bar t$ effective theory currents. Within the effective theory the absorptive parts describe the interference of the diagrams shown in (b). \[figunstable\] ](fig2.ps "fig:"){width="8cm"} The gauge cancellation already mentioned for class (a) also applies here and leads to the cancellation of interactions among the top quarks and their decay products caused by ultrasoft gluons for the total cross section at NLL order [@Melnikov:1993np; @Fadin:1993dz] and even at NNLL order [@Hoang:2004tg]. In Ref. [@Hoang:2004tg] it was also shown that within the nonrelativistic effective theory the imaginary matching conditions can lead to ultraviolet phase space divergences that require additional renormalization. A complete treatment of all the class (c) electroweak effects at NLL has not yet been achieved. A crucial prediction of the Higgs mechanism is that the Higgs Yukawa coupling to quarks $\lambda_q$ is related to the quark masses by $m_q=\lambda_q V$. At the $e^+e^-$ Linear Collider the top quark Yukawa coupling can be measured from top quark pair production associated with a Higgs boson, $e^+e^-\to t\bar t H$. This process is particularly suited for a light Higgs boson since the cross section can then reach the $1$-$2$ fb level. Assuming an experimental precision at the percent level, QCD and electroweak radiative corrections need to be accounted for in the theoretical predictions. The Born cross section was already determined some time ago in Ref. [@Borneetth]. For the ${\cal O}(\alpha_s)$ QCD one-loop corrections a number of references in various approximations exist [@Dawson1; @Dawson2; @Dittmaier1]. On the other hand, the full set of one-loop electroweak corrections was obtained in Refs. [@Belanger1; @Denner1] and also in Ref. [@You1]. In Ref. [@Denner1] a detailed analysis of various differential distributions of the cross section $\sigma(e^+e^-\to t\bar t H)$ can be found. For high energies these fixed order predictions are sufficient to reach the required theoretical precision. In Ref. [@Farrell:2005fk] the phase space region where the Higgs energy is large and the $t\bar t$ pair becomes collinear to balance the large Higgs momentum was analyzed (see Fig. \[figtth\]). ![ Typical constellation of momenta for the process $e^+e^-\to t\bar t H$ in the large Higgs energy endpoint region. \[figtth\] ](figtth.ps "fig:"){width="6cm"} 0.0cm It was found that since in this kinematic region the $t\bar t$ invariant mass approaches $2m_t$, it is governed by nonrelativistic dynamics in analogy to the physics relevant for the $t\bar t$ threshold. Thus, the usual fixed-order treatment breaks down and the nonrelativistic effective theory description known from the $t\bar t$ threshold has to be applied. For high c.m.energies above about $700$ GeV the large Higgs energy endpoint region is very small, and a usual fixed-order treatment is sufficient for the theoretical determination of the total cross section. However, for smaller c.m.energies (as available at the first phase of the ILC running), or for larger Higgs masses, the endpoint region increases with respect to the full available phase space and an effective theory treatment for the endpoint region becomes mandatory. In Ref. [@Farrell:2005fk] a QCD endpoint analysis in the framework of the effective theory vNRQCD [@Luke:1999kz] was carried out. The results sum singular terms in the endpoint region $\propto (\alpha_s/v)^n$ and $(\alpha_s\ln v)^n$ at NLL order, where $v$ is the c.m. velocity of the top quarks in the $t\bar t$ c.m. frame. Typical results for the Higgs energy spectrum for small c.m.energies showing the fixed-order QCD predictions and the LL and NLL order nonrelativistic effective theory predictions in the large Higgs energy region are shown in Fig. \[figenH\]. The vertical lines mark the Higgs energy where the top velocity $v=0.2$, which roughly divides the regions where fixed-order and effective theory computations are valid. In Table \[tab3\] the impact of the summation of the Coulomb singularities and the logarithms of the top quark velocity in the endpoint region is analyzed numerically for various choices of the c.m.energy and the Higgs mass. For all cases the top quark mass $m_t^{\rm 1S}=180$ GeV is used and the other parameters are fixed as in Fig. \[figenH\]. Here, $\sigma(\mbox{Born})$ refers to the Born cross section and $\sigma(\alpha_s)$ to the ${\cal O}(\alpha_s)$ cross section in fixed-order perturbation theory using $\mu=\sqrt{s}$ as the renormalization scale. The term $\sigma(\mbox{NLL})$ refers to the sum of the ${\cal O}(\alpha_s)$ fixed-order cross section for $v>0.2$ using $\mu=\sqrt{s}v$ and the NLL nonrelativistic cross section for $|v|<0.2$ with the soft vNRQCD renormalization parameter $\mu_S=0.2 m_t$. The summation of endpoint singularities is particularly important for smaller c.m.energies, since it leads to an additional significant enhancement of the total cross section in a region where the cross section is rather small. This enhancement could improve the prospects of top-Yukawa coupling measurements at the first phase of the ILC (see section 2.5). ![ Higgs energy spectrum at LL (dotted lines) and NLL (solid lines) order in vNRQCD for the soft renormalization scales $\mu_S=(0.1,0.2,0.4)m_t$ and at the Born level and at ${\cal O}(\alpha_s)$ for $\mu=\sqrt{s}, \sqrt s |v|$ for the parameters (a) $\sqrt{s}=500$ GeV, $m_H=120$ GeV and (b) $\sqrt{s}=550$ GeV, $m_H=140$ GeV. The top 1S mass has been set to $m_t^{\rm 1S}=180$ GeV and the other parameters are $\Gamma_t=1.55~\mbox{GeV}$, $M_Z=91.1876$ GeV, $M_W=80.423$ GeV, $c_w=M_W/M_Z$ and $\alpha^{-1}=137.034$. \[figenH\] ](figEnH1.ps "fig:"){width="8cm"} ![ Higgs energy spectrum at LL (dotted lines) and NLL (solid lines) order in vNRQCD for the soft renormalization scales $\mu_S=(0.1,0.2,0.4)m_t$ and at the Born level and at ${\cal O}(\alpha_s)$ for $\mu=\sqrt{s}, \sqrt s |v|$ for the parameters (a) $\sqrt{s}=500$ GeV, $m_H=120$ GeV and (b) $\sqrt{s}=550$ GeV, $m_H=140$ GeV. The top 1S mass has been set to $m_t^{\rm 1S}=180$ GeV and the other parameters are $\Gamma_t=1.55~\mbox{GeV}$, $M_Z=91.1876$ GeV, $M_W=80.423$ GeV, $c_w=M_W/M_Z$ and $\alpha^{-1}=137.034$. \[figenH\] ](figEnH2.ps "fig:"){width="8cm"} -0.3cm ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- $\sqrt{s}$ \[GeV\] $m_H$ \[GeV\] $\mbox{}\;\sigma(\mbox{Born})$ \[fb\] $\mbox{}\;\sigma(\alpha_s)$ \[fb\] $\mbox{}\;\sigma(\mbox{NLL})$ \[fb\] $\mbox{}\quad\frac{\sigma(\mbox{\tiny NLL})}{\sigma(\mbox{\tiny Born})}\quad\mbox{}$ $\mbox{}\quad\frac{\sigma(\mbox{\tiny NLL})}{\sigma(\alpha_s)}\quad\mbox{}$ -------------------- --------------- --------------------------------------- ------------------------------------ -------------------------------------- -------------------------------------------------------------------------------------- ------------------------------------------- $500$ $120$ $ 0.151$ $ 0.263$ $\quad 0.357(20)$ $ 2.362$ $ 1.359$ $550$ $120$ $ 0.984$ $ 1.251$ $\quad 1.342(37)$ $ 1.364$ $ 1.073$ $550$ $160$ $ 0.134$ $ 0.207$ $\quad 0.254(12)$ $ 1.902$ $ 1.226$ $600$ $120$ $ 1.691$ $ 1.939$ $\quad 2.005(30)$ $ 1.185$ $ 1.034$ $600$ $160$ $ 0.565$ $ 0.700$ $\quad 0.745(18)$ $ 1.319$ $ 1.065$ $700$ $120$ $ 2.348$ $ 2.454$ $\quad 2.485(13)$ $ 1.058$ $ 1.012$ $700$ $160$ $ 1.210$ $ 1.303$ $\quad 1.328(11)$ $ 1.098$ $ 1.020$ ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- : Cross sections and K factors for $\sigma_{\rm tot}(e^+e^-\to t\bar t H)$ for various c.m.energies and Higgs masses and top quark mass $m_t^{\rm 1S}=180$ GeV.[]{data-label="tab3"} Towards a Precise Measurement of the Top Quark Yukawa Coupling at the ILC [@Juste:2005vs]\ {#sju} ------------------------------------------------------------------------------------------ The top quark Yukawa coupling ($\lambda_t$) is the largest coupling of the Higgs boson to fermions. A precise measurement of it is very important since it may help unravel the secrets of the Electroweak Symmetry Breaking (EWSB) mechanism, in which the top quark could possibly play a key role. For $m_h<2m_t$, a direct measurement of $\lambda_t$ is possible via associated $t\bar{t}h$ production, both at the LHC and a future ILC. At the LHC, the expected accuracy [@Weiglein:2004hn] is $\delta\lambda_t/\lambda_t\sim 12-15\%$ for $m_h\sim 120-200$ GeV, assuming an integrated luminosity of 300 fb$^{-1}$. Existing feasibility studies at the ILC [@juste1] predict an accuracy of $\delta\lambda_t/\lambda_t\sim 6-10\%$ for $m_h\sim 120-190$ GeV, assuming $\sqrt{s}=800$ GeV and 1000 fb$^{-1}$. However, currently the baseline design for the ILC only contemplates a maximum center-of-mass energy of 500 GeV. Therefore, it is very relevant to explore the prospects of this measurement during the first phase of the ILC, especially in view of the limited accuracy expected at the LHC; for a number of years, the combination of results from the LHC and ILC will yield the most precise determination of $\lambda_t$. A preliminary feasibility study at $\sqrt{s}=500$ GeV was performed in Ref. [@juste2], which we briefly overview here. Indeed, the measurement of $\lambda_t$ at $\sqrt{s}=500$ GeV is more challenging than at $\sqrt{s}=800$ GeV. On the one hand, the reduced phase-space leads to a large reduction in $\sigma_{t\bar{t}h}$ (e.g. $\sigma^{Born}_{t\bar{t}h}\simeq 0.16(2.5)$ fb at $\sqrt{s}=500(800)$ GeV, for $m_h=120$ GeV). On the other hand, the cross section for many background processes is significantly increased. This analysis assumed $m_h=120$ GeV and focused on the $t\bar{t}h\rightarrow (\ell\nu b)(jjb)(b\bar{b})$ decay channel ($BR\sim 30\%$). The dominant background is $t\bar{t}jj$, followed by $t\bar{t}Z$, although other non-interfering backgrounds (e.g. $W^+W^-$) were also considered. Signal and backgrounds were processed through a fast detector simulation. After basic preselection cuts, the signal efficiency was found to be $\simeq 50\%$ and the $S:B\simeq 1:100$. In order to increase the sensitivity, a multivariate analysis using a Neural Network (NN) with 23 variables was performed. The final selection consisted of an optimized cut on the NN distribution. Assuming an integrated luminosity of 1000 fb$^{-1}$, the expected total numbers of signal and background events were 11 and 51, respectively, resulting in $(\delta\lambda_t/\lambda_t)_{stat}\simeq 33\%$. Based on previous experience [@juste1], the addition of the fully hadronic decay channel was expected to ultimately lead to $(\delta\lambda_t/\lambda_t)_{stat}\simeq 23\%$. While this analysis is already rather sophisticated, significant improvements are expected from the usage of a more efficient $b$-tagging algorithm or a more optimal treatment of the kinematic information. In the next sections we discuss additional sources of improvement which are currently under investigation. The precise measurement of $\lambda_t$ requires accurate theoretical predictions for $\sigma_{t\bar{t}h}$. Currently, one-loop QCD and electroweak corrections are available. However, at $\sqrt{s}=500$ GeV and for $m_h\geq 120$ GeV, the kinematic region where the $t\bar{t}$ system is non-relativistic dominates. As discussed in Ref. [@Farrell:2005fk] and Section 2.4, in this regime Coulomb singularities are important and need to be resummed within the framework of the vNRQCD effective theory, leading to large enhancements factors in the cross section relative to the Born level. At the ILC, because of ISR and beamstrahlung (BS), the event-by-event center-of-mass energy ($\sqrt{\hat{s}}$) will be lower than the nominal one, thus bringing the $t\bar{t}$ system deeper into the non-relativistic regime. In order to compute the expected $\sigma_{t\bar{t}h}$ including these effects, the 11-fold Born differential cross section for $e^+e^- \rightarrow t\bar{t}h \rightarrow W^+bW^-\bar{b}h$ was multiplied by a K-factor defined as $K(E_h,\sqrt{\hat{s}})=(d\sigma_{t\bar{t}h}^{NLL}/dE_h)/(d\sigma_{t\bar{t}h}^{Born}/dE_h)$, where $E_h$ stands for the Higgs boson energy in the $e^+e^-$ rest-frame, and then folded with ISR and BS structure functions. The NLL differential cross section was kindly provided by the authors of Ref. [@Farrell:2005fk]. Fig. 9 (left and center) compares the Born (for off-shell top quarks) and NLL differential cross sections for different values of $\sqrt{\hat{s}}$, assuming $m_t^{1S}=180$ GeV and $m_h=120$ GeV. The ratio of these two curves defines $K(E_h,\sqrt{\hat{s}})$ and can be significantly larger than 1, especially for low values of $\sqrt{\hat{s}}$. Since the NLL prediction is only valid for $E_h\leq E^{max}_h$ (where $E^{max}_h$ effectively corresponds to a cut on the top quark velocity in the $t\bar{t}$ rest-frame of $\beta_t<0.2$), we currently set $K(E_h,\sqrt{\hat{s}})=1$ for $E_h>E^{max}_h$, although in practice, it should be possible to use $K(E_h,\sqrt{\hat{s}})=(d\sigma_{t\bar{t}h}^{O(\alpha_s)}/dE_h)/(d\sigma_{t\bar{t}h}^{Born}/dE_h)$. Table IV compares the predicted Born and “NLL-improved” $\sigma_{t\bar{t}h}$ for different scenarios, illustrating the large impact of radiative effects in the initial state. This underscores the importance of being able to predict these effects to the percent level. While the impact of ISR cannot be reduced, it might be possible to find an optimal operating point of the accelerator, as far as this measurement is concerned, in terms of BS and total integrated luminosity. Finally, it is found that, for $m_h=120$ GeV, resummation effects can increase $\sigma_{t\bar{t}h}$ by a factor of $\sim 2.4$ with respect to the Born cross section used in the previous feasibility study. **(ISR,BS)** **$\sigma_{t\bar{t}h}$ (fb) (Born)** **$\sigma_{t\bar{t}h}$ (fb) (“NLL-improved”)** **Enhancement factor** -------------- -------------------------------------- ------------------------------------------------ ------------------------ (off,off) 0.157(1) 0.357(2) 2.27 (off,on) 0.106(1) 0.252(3) 2.38 (on,on) 0.0735(8) 0.179(2) 2.44 : Comparison of the Born and NLL $\sigma_{t\bar{t}h}$ for different scenarios regarding radiative effects in the initial state. \[sigmatth\] ![image](fig1.eps){width="5.5cm"} ![image](fig2.eps){width="5.5cm"} ![image](fig3.eps){width="5.5cm"} So far, all feasibility studies of this measurement have assumed unpolarized beams. Currently, the baseline design for the ILC only includes longitudinal electron beam polarization ($|P_{e^-}|\simeq 0.8$). Positron beam polarization ($|P_{e^+}|\simeq 0.6$) is considered as an option. The ratio of the polarized cross section (for arbitrary longitudinal beam polarization) ($\sigma_{P_{e^-}P_{e^+}}$) to the unpolarized cross section ($\sigma_0$) is given by $\sigma_{P_{e^-}P_{e^+}}/\sigma_0 = (1-P_{e^-}P_{e^+})(1-P_{eff}A_{LR})$, where $P_{eff}=(P_{e^-}-P_{e^+})/(1-P_{e^-}P_{e^+})$ denotes the “effective polarization” and $A_{LR}$ is the “left-right asymmetry” of the process of interest [@gudi]. Therefore, two potential enhancement factors can in principle be exploited: the first one requires having both beams polarized, the second one requires $A_{LR}\neq 0$ and a judicious choice of the signs of $P_{e^-}$ and $P_{e^+}$ in order to have $P_{eff}A_{LR}<0$. In the case of SM $t\bar{t}h$ production, $A_{LR} \simeq 0.44$, essentially independent of $\sqrt{s}$ in the range $\sim 0.5-1.0$ TeV. Assuming $(A_{LR})_{SM}$, Fig. 9 (right) shows the cross section enhancement factor as a function of $P_{e^+}$, for different values of $P_{e^-}$. The optimal (realistic) operating point would be $(P_{e^-},P_{e^+})=(-0.8,+0.6)$, achieving an increase in $\sigma_{t\bar{t}h}$ by a factor of $\simeq 2.1$ with respect to the unpolarized case. Unfortunately, this choice does not help reduce the dominant background, which is increased by a similar factor. Nevertheless, the net result is still an improvement in the statistical precision on $\lambda_t$ by $\sim 45\%$, which would be an argument in favor of including positron polarization in the baseline design. For $(P_{e^-},P_{e^+})=(-0.8,0)$, only a modest increase in $\sigma_{t\bar{t}h}$ by a factor of $\sim 1.3$ would be achieved. It is important to realize that, in order to choose the sign of $P_{e^-}$ and $P_{e^+}$, it is necessary to know the sign of $A_{LR}$. Anomalous couplings in the $tt\gamma$ and $ttZ$ vertices could possibly lead to deviations in $A_{LR}$ from the SM prediction. Unfortunately, due to the limited precision in the measurement of the $ttZ$ couplings [@Baur:2004uw; @new], the LHC is not expected to provide any useful constraints on the sign of $A_{LR}$. Therefore, at the ILC the first step should be to perform measurements of the polarized $\sigma_{t\bar{t}}$ in order to determine the sign of $A_{LR}$, and thus fix the signs of $P_{e^-}$ and $P_{e^+}$ (the magnitudes should be the largest possible). On the other hand, the measurement of $\lambda_t$ requires a percent-level and model-independent determination of the $t\bar{t}\gamma$ and $t\bar{t}Z$ couplings, which typically benefits from changing the beam polarization. Therefore, it would be desirable to optimize the running strategy to maintain the largest possible $\sigma_{t\bar{t}h}$, needed for a precise measurement of $\lambda_t$, while meeting the precision goals for measurements of top quark couplings. We have studied the prospects of a precise measurement of the top quark Yukawa coupling during the first phase of the ILC. Taking into consideration an existing feasibility study, and the additional enhancement factors to $\sigma_{t\bar{t}h}$ discussed here, we anticipate a precision of $(\delta\lambda_t/\lambda_t)_{stat}\sim 10\%$ for $m_h=120$ GeV, assuming $\sqrt{s}=500$ GeV and 1000 fb$^{-1}$. The Threshold at an Collider\ {#skiyo} ----------------------------- It is well known that QCD corrections to $t\bar{t}$ production near threshold develop a Coulomb singularity. The structure of the cross section is $$\begin{aligned} \sigma_{tot}(e^+e^-\rightarrow t\bar{t})=\sigma_{Born} \bigg[1+ c^{(1)}\,\bigg(\frac{\alpha_s}{v}\bigg) + c^{(2)}\,\bigg(\frac{\alpha_s}{v}\bigg)^2 + c^{(3)}\,\bigg(\frac{\alpha_s}{v}\bigg)^3 + \cdots + c^{(n)}\,\bigg(\frac{\alpha_s}{v}\bigg)^n+\cdots \bigg], \label{eq1}\end{aligned}$$ where $\sigma_{Born}$ is the Born cross section and $v=\sqrt{1-4m_t^2/s}$ is the velocity of the top quarks. In Eq. 7, we have pulled out the Coulomb singularity $(\alpha_s/v)^n$ explicitly so that the $n$th coefficient starts with ${\cal O}(v^0)$, $$\begin{aligned} c^{(n)}=c^{(n, 0)}+c^{(n, 1)}\,v+c^{(n, 2)}\, v^2+\cdots. \label{eq2}\end{aligned}$$ Near threshold the kinematics of the top quarks is non-relativistic and $v \sim \alpha_s$ holds. To get a meaningful cross section we thus have to sum up the Coulomb singularities $\sim \alpha_s/v \sim 1$. The leading order (LO) cross section then contains the Coulomb singularities $\sim(\alpha_s/v)^n$ to all orders of the coupling expansion, $$\begin{aligned} \sigma_{tot}^{LO}=\sigma_{Born} \sum_{n=0} c^{\,(n, 0)}\,\bigg(\frac{\alpha_s}{v}\bigg)^n. \label{eq3}\end{aligned}$$ The $c^{n, 1}$ terms are suppressed by $v$ compared with the LO and result in next-to-leading order (NLO) corrections. These terms are known since long. Next-to-next-to leading order (NNLO) calculations have been completed some time ago by several groups [@TWGR]. They sum up all the corrections of $c^{(n,2)}\, v^2\, (\alpha_s/v)^n$ to all orders $n$. ![image](tt01.eps){width="40mm"} The Coulomb singularity originates from potential gluons, which have a typical momentum $k^\mu \sim m_t (v^2,\vec{v})$, exchanged between almost on-shell top and anti-top quarks in Fig. 10. For the kinematics of potential gluons, the corresponding propagator reduces to the Coulomb potential $$\begin{aligned} \widetilde{V_{C}}=-\frac{4\pi C_F \alpha_s}{{\bf q}^2}, \label{eq4}\end{aligned}$$ where $C_F=4/3$. In the gluon propagator, the energy component of $q$ was set to zero because of the potential gluon’s kinematics. Using this gluon propagator in place of normal gluon propagators in the figure, one may reproduce the $c^{(i, 0)}$ coefficients to all orders in Eq. 8. This procedure can be systematically extended to higher orders using effective field theory (EFT) techniques. The constructed effective field theories are versions of non-relativistic QCD called pNRQCD/vNRQCD [@Luke:1999kz]. We do not go into details of the EFT but summarize several features. - The $t\bar{t}$ pair at threshold is created by the production current $\vec{J}=C_J\,[t\vec{\sigma}\bar{t}]+\cdots$ in the EFT, where $C_J$ is the Wilson coefficient of the corresponding current in the EFT, and the dots denote subleading operators. - The produced top quark pair forms bound state resonances by exchanging potential gluons; the binding Coulomb potential is $\displaystyle \widetilde{V}_0={\cal V}_0 \widetilde{V}_C ({\bf q})$, where ${\cal V}_0=1+{\cal O}(\alpha_s)$ includes QCD corrections to the leading order Coulomb potential from loop diagrams. - There are subdominant potentials, e.g. $\displaystyle \widetilde{V}_1 = {\cal V}_1 g_s^2/|{\bf q}|$, which have to be taken into account if one wants to go beyond the NLO cross section. - A yet new type of corrections is known at next-to-next-to-next-to leading order (NNNLO), which is referred to as ultra-soft corrections because of the typical gluon momentum of $k^0\sim|\vec{k}|\sim mv^2$. An important point here is that the EFT scheme is a systematic way to sum up higher order corrections, and it makes practical calculations easier compared to those of full QCD. Currently complete NNLO total cross sections are known in both (semi-) analytical and numerical ways. The results are summarized in Ref. [@TWGR]. In the following we briefly survey recent attempts in going beyond NNLO and in resumming potentially large logarithms for the threshold cross section in the EFT framework.\ In the EFT calculation of the $t\bar{t}$ cross section, higher order QCD corrections enter through the higher order coefficients ${\cal V}_i$ of potentials and through subdominant potentials, e.g. $\widetilde{V}_{1}({\bf q})$. Furthermore, there are corrections from ultra-soft gluons, starting to contribute at NNNLO. They cannot be written in form of potentials, as they include a noninstantaneous, dynamical propagation in time, while the potentials are all instantaneous. The first correction to ${\cal V}_{0}$ is referred to as a Coulomb correction, $$\begin{aligned} {\cal V}^{(n)}_0=1+\frac{\alpha_s}{4\pi}\big(\beta_0 L(q)+\frac{43}{9}\big)+ \cdots +\big(\frac{\alpha_s}{4\pi}\big)^n \cdot \big(\mbox{group factors, $L(q)$, etc.}\big),\end{aligned}$$ where $\beta_0=11-(2/3)n_f$ is the coefficient of the QCD $\beta-$function, $L(q)=\ln(\mu^2/{\bf q}^2)$, and $\mu$ is the QCD renormalization scale. The QCD corrections to all the coefficients can be found in Ref. [@KPSS] up to ${\cal O}(\alpha_s^3)$, except the third order constant term, so-called $a_3$ in the literature. Using $\widetilde{V}_0^{(3)}={\cal V}_0^{(3)}\widetilde{V}_C$, the Coulomb correction to the total cross section was computed in Ref. [@BKS]. The other corrections to the $t\bar{t}$ cross section at NNNLO due to subleading potentials and ultra-soft gluons are not known yet. ![image](cs_order.eps){width="8cm"} In Fig. 11 we show the $t\bar{t}$ threshold total cross section including the Coulomb corrections as calculated in Ref. [@BKS]. The curves are obtained using analytical results of successive Coulomb corrections from LO to NNNLO, and the line denoted by “NNNLO exact” is obtained by numerically solving the Schrödinger equation using ${\cal V}^{(3)}_0$ at NNNLO. The difference between NNNLO and “NNNLO exact” is yet of higher order as the numerical method contains rigid NNNLO plus additional contributions which come from iterations of the potential $\widetilde{V}_0$. Studying the renormalization scale dependence of these predictions, the theoretical uncertainty for the NNNLO Coulomb corrections was estimated to be about 5% of the cross section. Although the full corrections at NNNLO are not known, the $1S$ energy of the $t\bar{t}$ resonance, which corresponds to the peak position of the cross section, is known analytically at NNNLO from Refs. [@KPSS; @ThirdOrder]. The result shows that the uncertainty in the choice of the scale is small and the perturbative series for the $1S$ energy is stabilized (see Ref. [@KS]). In higher order calculations there appear potentially large logarithms of ratios of scales. In the threshold regime we have large logarithms of $v$ in the coefficients $c^{(n, i)}$, arising from ratios of the largely different scales of the problem, namely the energy of the top quarks, $E_t\sim m v^2$, the momentum $p\sim m_t v$ of almost on-shell top quarks, and the c.m. energy $\sqrt{s} \sim 2m_t$. The origin of these logarithms is related to UV divergences in the EFT. They can be resummed using renormalization group (RG) arguments [@Luke:1999kz; @RGI], resulting in RG improved cross section predictions, $\sigma_{LL}, \sigma_{NLL}$, etc. The RG improvement was extensively studied recently, and the next-next-leading-log (NNLL) cross section was calculated in Ref. [@HMST] (apart from the NNLL running of a current for which the anomalous dimension is only fully known at NLL, see also Ref. [@Hoang; @pinedanew]). The RG improved cross section then contains all terms of order $\sum_{n,m} (\alpha_s/v)^n (\alpha_s \ln v)^m$. The resummation of the logarithms leads to a reduction of the scale dependence of the normalization from 20% to about 6% (at NNLL), especially at energies around the 1$S$ peak. This is illustrated in Fig. 12, where the scale dependence of the fixed order (left panel) and RG improved (right panel) cross sections are shown for different orders. The different studies discussed above strengthen our confidence that the estimate of the uncertainty of the top quark mass determination from Ref. [@TWGR], $\Delta m_t < 100$ MeV, is realistic and will not be spoiled by uncalculated higher order corrections. There is even hope to believe that, if different approaches turn out to converge, the estimated error may shrink. For determinations of the top quark total width, the top Yukawa coupling $y_t$ and the strong coupling constant from the threshold scan theoretical uncertainties are comparable to the expected experimental errors. An ultimate theoretical normalization error of at most $3\%$ is desirable. ![image](figa.ps){width="8cm"} ![image](figb.ps){width="8cm"} Differential Distributions and Experimental Aspects of the Threshold Scan\ {#steub} -------------------------------------------------------------------------- The most precise determination of the top quark mass will come from a dedicated threshold scan at $\sqrt{s} \sim 2m_t$. This is mainly a consequence of the fact that by performing a counting experiment of color singlet $t\bar t$ pairs one can avoid most of the systematic uncertainties inherent in a kinematic reconstruction of the decay products of the colored quarks. These uncertainties, which are closely related to the problem of the mass definition, will ultimately limit the determination of $m_t$ as done at the Tevatron and soon at the LHC. Detailed studies have shown that via a threshold scan, measurements with very small statistical and systematic errors will be possible at the ILC [@Martinez:2002st]. A multi-parameter fit for the top quark mass $m_t$, width $\Gamma_t$, the strong coupling $\alpha_s$ and the top Yukawa coupling resulted in experimental errors of about $\Delta m_t \sim 20$ MeV, $\Delta \Gamma_t \sim 30$ MeV, $\Delta \alpha_s \sim 0.0012$, depending on details of the fit. (At threshold, the top Yukawa coupling can only be measured for a light Higgs and even then with less than $30\%$ accuracy.) Such a precision will only be achieved if the accuracy of the theoretical predictions can match the experimental one, and recently a lot of effort has been invested in further improvements of the theory (see sections 2.4 and 2.6). Although most of the information in the fit will come from the precise measurement of the total $t\bar t$ cross section, differential distributions are needed for several reasons. - Experimentally, cuts are needed to discriminate the signal form backgrounds, so the measured cross section can never be the fully inclusive total cross section. - Distributions are required to build realistic (higher order) Monte Carlo generators for the signal process. - Using additional observables beyond $\sigma_{\rm tot}$ adds information, helping to disentangle correlations among the parameters determined from a threshold scan, and increases the sensitivity to possible New Physics in top production and decay. Differential distributions used so far are the top quark momentum distribution ${\rm d}\sigma/{\rm d}p_t$ and the forward-backward asymmetry $A_{\rm FB}$ of the cross section, but theoretical studies exist also for the polarization of the top quarks. In the following we will briefly discuss these observables and their role in the threshold scan, and comment also on the issue of rescattering corrections.\ ![image](dsgdp349.ps){width="8cm"} The top quark momentum distribution ${\rm d}\sigma/{\rm d}p_t$ is available in the framework of the next-to-next-to leading order (NNLO) calculations [@Hoang:1999zc] (see also Ref. [@Nagano:1999nw]). Close to threshold its form resembles a $1S$ Coulomb-like wave function, becoming more symmetric well above threshold. Fig. 13 shows ${\rm d}\sigma/{\rm d}p_t$ at a fixed c.m. energy of $349$ GeV for two different masses $m_t = 174.5, 175.5$ GeV, where the width of the band is obtained by varying $\alpha_s$ between $0.115$ and $0.121$. The peak position of ${\rm d}\sigma/{\rm d}p_t$ is not much dependent on $\alpha_s$ but is very sensitive to $m_t$, and a change of $\alpha_s$ mainly affects the normalization of the distribution. This is in contrast to $\sigma_{\rm tot}$, where $m_t$ and $\alpha_s$ are strongly correlated parameters when fitting theoretical predictions to (simulated) data. ![image](afbn.ps){width="17cm"} Interference of the leading vector current (through $\gamma$ and $Z$ exchange) with the suppressed axial vector contribution (from $Z$ only) leads to a forward-backward asymmetry $A_{\rm FB}$. The size of the asymmetry depends on how much the corresponding $S$ and $P$ wave resonance contributions overlap and hence on the top quark width $\Gamma_t$, but less on $\alpha_s$ (and $m_t$), see Fig. 14. As already demonstrated in experimental studies [@Martinez:2002st], the two observables ${\rm d}\sigma/{\rm d}p_t$ and $A_{\rm FB}$ can be used together with $\sigma_{\rm tot}$ to make best use of the available information and to disentangle the correlations between the parameters $m_t, \Gamma_t, \alpha_s (, y_t)$ in a multi-paramter fit. However, the latest theoretical developments such as NNNLO corrections, renormalization group improvement by summing large logarithms at NNLL order (see section 2.6) or the effect of EW corrections (see section 2.4) are dealing with the total cross section, whereas distributions are available at fixed NNLO only. This situation is quite common, as higher order corrections are increasingly difficult for differential cross sections, and not many distributions are known beyond NLO. Nevertheless, from a pragmatic point of view and for use in Monte Carlos, it is legitimate to use the best available prediction for the total cross section together with distributions available at lower order. By rescaling the distributions using the total cross section, their scale dependence will decrease and a consistent normalization will be ensured. ![image](rescat.ps){width="6cm"} All recent NNLO, NNNLO as well as the renormalization group improved calculations are not taking into account the so-called “rescattering corrections”, i.e. interactions between the $t, \bar t$ quarks and the $b, \bar b$ quarks from the top decays, or between $b$ and $\bar b$ (see Fig. 16 for a typical Feynman diagram). However, such corrections were calculated numerically in Refs. [@Harlander:1996vg; @Peter:1997rk] to NLO accuracy. They are strongly suppressed (vanishing at NLO) in the inclusive $\sigma_{\rm tot}$ but generally important for distributions and typically of order $10\%$. They therefore should be included in a realistic Monte Carlo description. In addition to differential distributions, also the top quark polarization has been studied in the threshold regime [@Harlander:1996vg; @Peter:1997rk; @RefsPolar]. Predictions are available at NLO accuracy for all three polarization components, including rescattering corrections. Even for unpolarized $e^+, e^-$ beams the top quarks are (longitudinally) polarized to $40\%$, and for the foreseen beam polarization the top quarks will be highly polarized. Due to the short life-time the top polarization will be transmitted undisturbed to the decay $b$’s and $W$’s and allow very interesting studies. Making use of the polarization, observables can be defined which are sensitive to the top’s electric dipole moment (probing CP-violation beyond the Standard Model) or to anomalous top couplings like $V+A$ admixtures to the weak interaction. However, more realistic experimental studies are needed in order to fully explore the potential of polarization in the threshold regime.\ [*Experimental aspects of the threshold scan*]{}[^2] Precision measurements from a $t\bar t$ threshold scan require a very precise knowledge of both the average c.m. energy $\langle \sqrt{s} \rangle$ and of the luminosity spectrum ${\rm d}L/{\rm d}\sqrt{s}$ [@Boogert:2002jr]. This is not an easy task, as the beam dynamics at a linear collider is not as constrained as it was e.g. at LEP, and only a single measurement of bunches before collision will be possible. The average c.m. energy can be determined by energy (up- and downstream) spectrometers in dedicated beam line inserts, discussed in Working Group 4 of the Snowmass Workshop. A possible problem here could be a bias between the spectrometer measurements and the collision c.m. energy. Another possibility to measure the beam energy is the use of physics processes like $Z$ pair production, or radiative return (through photon radiation) calibrated to the $Z$ peak [@Hinze:2005kh]. The luminosity spectrum is determined by (a) the beam spread, (b) beamstrahlung, and (c) initial state radiation (ISR). All three effects will lead to a smearing of the $t\bar t$ threshold cross section, resulting in a significant reduction of the effective luminosity and hence the observed cross section, $$\sigma^{\rm obs}(\sqrt{s}) = \frac{1}{L_0} \int_0^1 L(x)\, \sigma(x\sqrt{s})\,{\rm d}x\,.$$ The influence of the three effects is demonstrated in Fig. 16. The beam spread will typically be $\sim 0.1\%$ at the ILC and will cause comparably little smearing (though additional beam diagnostics may be required to measure and monitor the beam spread), but beamstrahlung and ISR are very important. The luminosity spectrum ${\rm d}L/{\rm d}\sqrt{s}$ will lead to a systematic shift in the extracted top mass which must be well understood; otherwise it could become the dominant systematic error. In the past, $t\bar t$ threshold studies were carried out under the assumption that the spectrum is basically known. However it has turned out that the precise determination of ${\rm d}L/{\rm d}\sqrt{s}$ is a challenging task. The proposed method is to analyse the acollinearity of (large angle) Bhabha scattering events, which is sensitive to a momentum mismatch between the beams but insensitive to the absolute energy scale [@Monig:2000bm]. For this, the envisioned high resolution of the forward tracker will be very important to achieve the required accuracy.[^3] [![image](energyLoss.eps){width="8cm"} ![image](circe_e350.eps){width="8cm"}]{} In the simulation of the luminosity spectrum beamstrahlung is the main unknown. Integrated ILC accelerator simulations for these machine dependent effects are becoming available, including effects from the linac beam spread, the beam diagnostics and delivery system and the simulation of the collision dynamics using the package Guinea-pig. At this workshop Stewart Boogert presented first results of spectrum parameterizations (using the package CIRCE), based on new simulations from G. White for $\sqrt{s}=350$ GeV. The new parameterizations show variations from earlier results which were based on beam simulations at higher energies only. The parameterizations will be used as input in the simulation of the Bhabha scattering (based on the package BHWIDE) for the acollinearity analysis. These involved simulations are required to extract ${\rm d}L/{\rm d}\sqrt{s}$ and to understand the uncertainties of the reconstructed spectrum. Only then can one quantify in physics analyses like the $t\bar t$ threshold scan, to which extent the uncertainties in the luminosity spectrum affect the accuracy of the intended measurements like the top quark mass, width, etc. The effects due to ISR are in principle machine independent and calculable to high precision in QED. Nevertheless, ISR effects complicate the measurement of beamstrahlung and accelerator energy spread, and one has to ensure that the precision of the theoretical formulae used in the Monte Carlo codes is sufficient. Similarly, the accuracy of differential distributions from programs used for the simulation of the wide angle Bhabha events must be assessed, and recent theoretical results [@Bhabha] may have to be included. The recent developments discussed here make it clear that much work remains to be done before the sophisticated analysis techniques for the $t\bar t$ threshold scan are understood at the required level of accuracy. For a complete study, which should take into account the total cross section, distributions, full beam effects including asymmetric boosts and detector effects, a full Monte Carlo generator will be needed, and a project for this has been started at Snowmass. With a more detailed experimental study it will also be possible to optimize the scan strategy, i.e. how the available luminosity should be distributed among different energy scan points. The analysis of beamstrahlung and its impact on the physics program at different energies will also be relevant for the optimization of the machine design. Concerning detectors, the demands of the top pair counting experiment are most probably not problematic for the existing detector designs. However, the requirements for Bhabha scattering and radiative $Z$ return might set the requirements of the low angle tracking system and the electromagnetic calorimeter. Finally, it should also be noted, that the understanding of other physics processes like the $WW$ or SUSY thresholds will largely benefit from the top case, and $t\bar t$ at threshold should be regarded as the benchmark. QCD EFFECTS IN ILC PHYSICS {#sec_pQCD} ========================== Precision QCD at the ILC: Jets\ {#sg1} ------------------------------- The production of light quark-antiquark pairs in electron-positron annihilation gives rise to final states containing QCD jets. Depending on the amount of additional hard QCD radiation, one obtains final states with a certain number of jets: if only quark and antiquark are hard, two-jet final states are produced, one additional hard gluon yields a three-jet final state, two extra gluons or a secondary quark-antiquark pair can give rise to four-jet final states and so on. Studying these multi-jet final states, one can probe many aspects of perturbative QCD. Three-jet final states and related event shape observables were studied extensively at LEP in order to determine the strong coupling constant $\alpha_s$, which controls the probability of radiating a hard gluon in these events. The measurement of four-jet-type observables at LEP established the gauge group structure of QCD. Five-jet and higher multiplicities were sometimes considered in new physics searches, where QCD-induced processes form a theoretically calculable background. The LEP measurements of three-jet observables are of a very high statistical precision. The extraction of $\alpha_s$ from these data sets relies on a comparison of the data with theoretical predictions. Comparing the different sources of error in this extraction [@bethke], one finds that the purely experimental error is negligible compared to the theoretical uncertainty. There are two sources of theoretical uncertainty: the theoretical description of the parton-to-hadron transition (hadronization uncertainty) and the theoretical calculation of parton-level jet production (perturbative or scale uncertainty). Although the precise size of the hadronization uncertainty is debatable and perhaps often underestimated, it is certainly appropriate to consider the scale uncertainty as the dominant source of theoretical error on the precise determination of $\alpha_s$ from three-jet observables. This scale uncertainty arises from truncating the perturbative QCD expansion of jet observables at the next-to-leading order (NLO) and can be improved considerably by computing next-to-next-to-leading order (NNLO) corrections. Given the planned luminosity of the ILC, one expects that this collider will deliver jet-production data of a statistical quality similar to LEP. An attractive perspective of such a measurement at the ILC would be to determine the evolution of $\alpha_s$ over a wide energy range, which is potentially sensitive to new physics thresholds. Concerning uncertainties on such a determination, it is worthwhile to note that the hadronization corrections become less important at higher energies, thus leaving the scale uncertainty as dominant source of theoretical error. Experimental aspects of such measurements, especially issues related to the beam energy profile (which were irrelevant at LEP) were not studied up to now, and certainly deserve further attention. In the recent past, many steps towards the NNLO calculation of $e^+e^- \to 3$ jets have been accomplished (see Ref. [@ourant] and references therein). Foremost, the relevant two-loop $1 \to 3$ matrix elements are now available. One-loop corrections to $1\to 4$ matrix elements have been known for longer and form part of NLO calculations of $e^+e^- \to 4$ jets. These NLO matrix elements naturally contribute to $e^+e^- \to 3$ jets at NNLO if one of the partons involved becomes unresolved (soft or collinear). In this case, the infrared singular parts of the matrix elements need to be extracted and integrated over the phase space appropriate to the unresolved configuration to make the infrared pole structure explicit. As a final ingredient, the tree level $1\to 5$ processes also contribute to $e^+e^- \to 3$ jets at NNLO. These contain double real radiation singularities corresponding to two partons becoming simultaneously soft and/or collinear. To compute the contributions from single unresolved radiation at one-loop and double real radiation at tree-level, one has to find subtraction terms which coincide with the full matrix elements in the unresolved limits and are still sufficiently simple to be integrated analytically in order to cancel their infrared pole structure with the two-loop virtual contributions. In the following, we present a new method, named antenna subtraction, to carry out NNLO calculations of jet observables and discuss its application to $e^+e^- \to 3$ jets. In electron-positron annihilation, an $m$-jet cross section at NLO is obtained by summing contributions from $(m+1)$-parton tree level and $m$-parton one-loop processes: $${\rm d}\sigma_{NLO}=\int_{{\rm d}\Phi_{m+1}}\left({\rm d}\sigma^{R}_{NLO} -{\rm d}\sigma^{S}_{NLO}\right) +\left [\int_{{\rm d}\Phi_{m+1}} {\rm d}\sigma^{S}_{NLO}+\int_{{\rm d}\Phi_{m}}{\rm d}\sigma^{V}_{NLO}\right].$$ The cross section ${\rm d}\sigma^{R}_{NLO}$ is the $(m+1)$-parton tree-level cross section, while ${\rm d}\sigma^{V}_{NLO}$ is the one-loop virtual correction to the $m$-parton Born cross section ${\rm d}\sigma^{B}$. Both contain infrared singularities, which are explicit poles in $1/\e$ in ${\rm d}\sigma^{V}_{NLO}$, while becoming explicit in ${\rm d}\sigma^{R}_{NLO}$ only after integration over the phase space. In general, this integration involves the (often iterative) definition of the jet observable, such that an analytic integration is not feasible (and also not appropriate). Instead, one would like to have a flexible method that can be easily adapted to different jet observables or jet definitions. Therefore, the infrared singularities of the real radiation contributions should be extracted using infrared subtraction terms. One introduces ${\rm d}\sigma^{S}_{NLO}$, which is a counter-term for ${\rm d}\sigma^{R}_{NLO}$, having the same unintegrated singular behavior as ${\rm d}\sigma^{R}_{NLO}$ in all appropriate limits. Their difference is free of divergences and can be integrated over the $(m+1)$-parton phase space numerically. The subtraction term ${\rm d}\sigma^{S}_{NLO}$ has to be integrated analytically over all singular regions of the $(m+1)$-parton phase space. The resulting cross section added to the virtual contribution yields an infrared finite result. Several methods for constructing NLO subtraction terms systematically were proposed in the literature [@cs; @singleun; @cullen; @ant]. For some of these methods, extension to NNLO was discussed [@nnlosub] and partly worked out. We focus on the antenna subtraction method [@cullen; @ant], which we extend to NNLO. The basic idea of the antenna subtraction approach at NLO is to construct the subtraction term ${\rm d}\sigma^{S}_{NLO}$ from antenna functions. Each antenna function encapsulates all singular limits due to the emission of one unresolved parton between two color-connected hard partons (tree-level three-parton antenna function). This construction exploits the universal factorization of phase space and squared matrix elements in all unresolved limits, depicted in Fig. 17. The individual antenna functions are obtained by normalizing three-parton tree-level matrix elements to the corresponding two-parton tree-level matrix elements. At NNLO, the $m$-jet production is induced by final states containing up to $(m+2)$ partons, including the one-loop virtual corrections to $(m+1)$-parton final states. As at NLO, one has to introduce subtraction terms for the $(m+1)$- and $(m+2)$-parton contributions. Schematically the NNLO $m$-jet cross section reads, $$\begin{aligned} {\rm d}\sigma_{NNLO}&=&\int_{{\rm d}\Phi_{m+2}}\left({\rm d}\sigma^{R}_{NNLO} -{\rm d}\sigma^{S}_{NNLO}\right) + \int_{{\rm d}\Phi_{m+2}} {\rm d}\sigma^{S}_{NNLO}\nonumber \\ &&+\int_{{\rm d}\Phi_{m+1}}\left({\rm d}\sigma^{V,1}_{NNLO} -{\rm d}\sigma^{VS,1}_{NNLO}\right) +\int_{{\rm d}\Phi_{m+1}}{\rm d}\sigma^{VS,1}_{NNLO} \nonumber \\&& + \int_{{\rm d}\Phi_{m}}{\rm d}\sigma^{V,2}_{NNLO}\;,\end{aligned}$$ where $\d \sigma^{S}_{NNLO}$ denotes the real radiation subtraction term coinciding with the $(m+2)$-parton tree level cross section $\d \sigma^{R}_{NNLO}$ in all singular limits [@doubleun]. Likewise, $\d \sigma^{VS,1}_{NNLO}$ is the one-loop virtual subtraction term coinciding with the one-loop $(m+1)$-parton cross section $\d \sigma^{V,1}_{NNLO}$ in all singular limits [@onelstr]. Finally, the two-loop correction to the $m$-parton cross section is denoted by ${\rm d}\sigma^{V,2}_{NNLO}$. Both types of NNLO subtraction terms can be constructed from antenna functions. In ${\rm d}\sigma^{S}_{NNLO}$, we have to distinguish four different types of unresolved configurations: (a) One unresolved parton but the experimental observable selects only $m$ jets; (b) Two color-connected unresolved partons (color-connected); (c) Two unresolved partons that are not color connected but share a common radiator (almost color-unconnected); (d) Two unresolved partons that are well separated from each other in the color chain (color-unconnected). Among those, configuration (a) is properly accounted for by a single tree-level three-parton antenna function like used already at NLO. Configuration (b) requires a tree-level four-parton antenna function (two unresolved partons emitted between a pair of hard partons) as shown in Fig. 18, while (c) and (d) are accounted for by products of two tree-level three-parton antenna functions. In single unresolved limits, the one-loop cross section $\d \sigma^{V,1}_{NNLO}$ is described by the sum of two terms [@onelstr]: a tree-level splitting function times a one-loop cross section and a one-loop splitting function times a tree-level cross section. Consequently, the one-loop single unresolved subtraction term $\d \sigma^{VS,1}_{NNLO}$ is constructed from tree-level and one-loop three-parton antenna functions, as sketched in Fig. 19. Several other terms in $\d \sigma^{VS,1}_{NNLO}$ cancel with the results from the integration of terms in the double real radiation subtraction term $\d \sigma^{S}_{NNLO}$ over the phase space appropriate to one of the unresolved partons, thus ensuring the cancellation of all explicit infrared poles in the difference $\d \sigma^{V,1}_{NNLO}-\d \sigma^{VS,1}_{NNLO}$. Finally, all remaining terms in $\d \sigma^{S}_{NNLO}$ and $\d \sigma^{VS,1}_{NNLO}$ have to be integrated over the four-parton and three-parton antenna phase spaces. After integration, the infrared poles are rendered explicit and cancel with the infrared pole terms in the two-loop squared matrix element $\d \sigma^{V,2}_{NNLO}$. The subtraction terms $\d \sigma^{S}_{NLO}$, $\d \sigma^{S}_{NNLO}$ and $\d \sigma^{VS,1}_{NNLO}$ require three different types of antenna functions corresponding to the different pairs of hard partons forming the antenna: quark-antiquark, quark-gluon and gluon-gluon antenna functions. In the past [@cullen; @ant], NLO antenna functions were constructed by imposing definite properties in all single unresolved limits (two collinear limits and one soft limit for each antenna). This procedure turns out to be impractical at NNLO, where each antenna function must have definite behaviors in a large number of single and double unresolved limits. Instead, we derive these antenna functions in a systematic manner from physical matrix elements known to possess the correct limits. The quark-antiquark antenna functions can be obtained directly from the $e^+e^- \to 2j$ real radiation corrections at NLO and NNLO [@our2j]. For quark-gluon and gluon-gluon antenna functions, effective Lagrangians are used to obtain tree-level processes yielding a quark-gluon or gluon-gluon final state. The antenna functions are then obtained from the real radiation corrections to these processes. Quark-gluon antenna functions were derived [@chi] from the purely QCD (i.e. non-supersymmetric) NLO and NNLO corrections to the decay of a heavy neutralino into a massless gluino plus partons [@hw], while gluon-gluon antenna functions [@h] result from the QCD corrections to Higgs boson decay into partons [@hgg]. All tree-level three-parton and four-parton antenna functions and three-parton one-loop antenna functions are listed in Ref. [@ourant], where we also provide their integrated forms, obtained using the phase space integration techniques described in Ref. [@ggh]. In Refs. [@ourant; @CFsquare] we derived the $1/N^2$-contribution to the NNLO corrections to $e^+e^- \to 3$ jets. This color factor receives contributions from $\gamma^*\to q\bar q ggg$ and $\gamma^*\to q\bar q q\bar qg$ at tree-level [@tree5p], $\gamma^*\to q\bar q gg$ and $\gamma^*\to q\bar q q\bar q$ at one-loop [@onel4p] and $\gamma^*\to q\bar q g$ at two-loops [@twol3p]. The four-parton and five-parton final states contain infrared singularities, which need to be extracted using the antenna subtraction formalism. In this contribution, all gluons are effectively photon-like, and couple only to the quarks, but not to each other. Consequently, only quark-antiquark antenna functions appear in the construction of the subtraction terms. Starting from the program [EERAD2]{} [@cullen], which computes the four-jet production at NLO, we implemented the NNLO antenna subtraction method for the $1/N^2$ color factor contribution to $e^+e^-\to 3j$. [EERAD2]{} already contains the five-parton and four-parton matrix elements relevant here, as well as the NLO-type subtraction terms. The implementation contains three channels, classified by their partonic multiplicity: (a) in the five-parton channel, we integrate ${\rm d}\sigma_{NNLO}^{R} - {\rm d}\sigma_{NNLO}^{S}$; (b) in the four-parton channel, we integrate ${\rm d}\sigma_{NNLO}^{V,1} - {\rm d}\sigma_{NNLO}^{VS,1}$; (c) in the three-parton channel, we integrate ${\rm d}\sigma_{NNLO}^{V,2} +{\rm d}\sigma_{NNLO}^{S} + {\rm d}\sigma_{NNLO}^{VS,1}$. The numerical integration over these channels is carried out by Monte Carlo methods. By construction, the integrands in the four-parton and three-parton channel are free of explicit infrared poles. In the five-parton and four-parton channel, we tested the proper implementation of the subtraction by generating trajectories of phase space points approaching a given single or double unresolved limit. Along these trajectories, we observe that the antenna subtraction terms converge locally towards the physical matrix elements, and that the cancellations among individual contributions to the subtraction terms take place as expected. Moreover, we checked the correctness of the subtraction by introducing a lower cut (slicing parameter) on the phase space variables, and observing that our results are independent of this cut (provided it is chosen small enough). This behavior indicates that the subtraction terms ensure that the contribution of potentially singular regions of the final state phase space does not contribute to the numerical integrals, but is accounted for analytically. As a final point, we noted in Ref. [@ourant] that the infrared poles of the two-loop (including one-loop times one-loop) correction to $\gamma^*\to q\bar qg$ are canceled in all color factors by a combination of integrated three-parton and four-parton antenna functions. This highly non-trivial cancellation clearly illustrates that the antenna functions derived here correctly approximate QCD matrix elements in all infrared singular limits at NNLO. They also outline the structure of infrared cancellations in $e^+e^-\to 3j$ at NNLO, and indicate the structure of the subtraction terms in all color factors. In this talk, we discussed the theoretical prerequisites for performing precision QCD studies on existing LEP data and at the ILC. In particular, the precise extraction of the strong coupling constant $\alpha_s$ requires improved theoretical predictions to reduce the scale error inherent to calculations in perturbative QCD. At present, this extraction relies on the calculation of $e^+e^- \to 3$ jets at NLO accuracy, and we reported on progress towards the NNLO calculation. This calculation requires a new method for the subtraction of infrared singularities which we call antenna subtraction. We introduced subtraction terms for double real radiation at tree level and single real radiation at one loop based on antenna functions. These antenna functions describe the color-ordered radiation of unresolved partons between a pair of hard (radiator) partons. All antenna functions at NLO and NNLO can be derived systematically from physical matrix elements. Using this method, we implemented the NNLO corrections to the subleading color contribution to $e^+e^- \to 3$ jets into a flexible parton level event generator program, and are currently proceeding with the implementation [@new3j] of the full set of color factors relevant to the NNLO corrections to $e^+e^- \to 3$ jets. Study of at the ILC Including Corrections [@Godfrey:2005gp]\ {#sgod} ------------------------------------------------------------ Understanding the mechanism of electroweak symmetry breaking (EWSB) is a primary goal of the LHC and ILC [@Weiglein:2004hn]. While much effort has been devoted to the weakly interacting weak sector scenario the strongly interacting weak sector (SIWS) remains a possibility. Because the $t$-quark mass is the same order of magnitude as the scale of EWSB it has long been suspected that $t$-quark properties may provide hints about the nature of EWSB and the subprocess $V_LV_L\to t\bar{t}$ has been suggested as a probe. While $V_LV_L\to t\bar{t}$ can be studied at both hadron colliders and $e^+e^-$ colliders, the overwhelming QCD backgrounds will likely make it impossible to study the $V_LV_L\to t\bar{t}$ subprocess at the LHC [@Han:2003pu]. In contrast, the ILC offers a much cleaner environment. But to be able to attach meaning to precision measurements it is necessary to understand radiative corrections, both electroweak and QCD. Here we show ${\cal O} (\alpha_s) $ corrections to the tree level electroweak $V_L V_L \rightarrow t \bar t$ process in the SM at the ILC. Due to space limitations we point the interested reader to Ref. [@Godfrey:2004tj] for a more detailed account and a more complete set of references. We are interested in the subprocesses $VV\to t\bar{t}$ which occur in the processes $e^+e^- \to \ell_1 \ell_2 + V V \to \ell_1 \ell_2 + t\bar{t}$ where $\ell_1 \ell_2$ is $\nu\bar{\nu}$ for the $W^+W^-\to t\bar{t}$ subprocess and $e^+e^-$ for the $ZZ \to t\bar{t}$ subprocess. The vector bosons are treated as partons inside the $e^+$ and $e^-$ using the effective boson approximation [@eva; @Dawson:1986tc]. The total cross section is then obtained by integrating the $W$ (or $Z$) luminosities with the subprocess cross section [@Kauffman:1989aq]. The ${\cal O}(\alpha_s)$ corrections for the processes $W^+W^-\to t\bar{t}$ $ZZ\to t\bar{t}$ are calculated using the FeynArts, FormCalc and LoopTools packages [@fclt]. The QCD corrections to $W^+W^- \to t\bar{t}$ are shown in Fig. 20 (left side). The infrared singularity in the vertex corrections are canceled by the soft contributions from the process $W^+W^-\to t\bar{t}g$ which are shown in Fig. 20 (right side). We regulate the IR-singularity by introducing a gluon mass which is equivalent to standard dimensional regularization for processes with no triple gluon vertex present. This approach has the additional benefit that varying the value of the gluon mass acts as a check of the numerical cancellations between the different contributions. ![${\cal O} (\alpha_s) $ QCD corrections to $W^+W^-\to t\bar{t}$. (a) Virtual QCD contributions to $W^+W^-\to t\bar{t}$. (b) Feynman diagrams for $W^+W^-\to t\bar{t}+g$. []{data-label="diagrams"}](fig4a.ps "fig:"){width="70mm"} ![${\cal O} (\alpha_s) $ QCD corrections to $W^+W^-\to t\bar{t}$. (a) Virtual QCD contributions to $W^+W^-\to t\bar{t}$. (b) Feynman diagrams for $W^+W^-\to t\bar{t}+g$. []{data-label="diagrams"}](fig4b.ps "fig:"){width="70mm"} We include in our results the kinematic cuts $m_{t\bar{t}}>400$ GeV and $p_T^{t,\bar{t}}>10$ GeV. Since the longitudinal scattering cross section is much larger than the $TT$ and $TL$ cases and it is the longitudinal gauge boson processes which corresponds to the Goldstone bosons of the theory we will henceforth only include results for $V_L V_L$ scattering. ![(left side) The K-factors as a function of $\sqrt{s}_{e^+e^-}$ for $e^+e^- \to \nu\bar{\nu} t \bar t$ (via $W^+_LW^-_L$ fusion). The solid line is for $M_H=120$ GeV, the dashed line for $M_H=500$ GeV, the dotted line for $M_H=1$ TeV, and the dot-dashed line for $M_H=\infty$ (LET). (right side) The K-factor as a function of $M_H$ for $e^+e^- \to \nu\bar{\nu} t \bar t$ (via $W^+_LW^-_L$ fusion). The solid line is for $\sqrt{s}_{e^+e^-}=500$ GeV, the dashed line for $\sqrt{s}_{e^+e^-}=1$ TeV, the dotted line for $\sqrt{s}_{e^+e^-}=2$ TeV, and the dot-dashed line for $\sqrt{s}_{e^+e^-}=3$ TeV. See text for an explanation of the K-factor. []{data-label="kfactor"}](fig7a.eps "fig:"){width="70mm"} ![(left side) The K-factors as a function of $\sqrt{s}_{e^+e^-}$ for $e^+e^- \to \nu\bar{\nu} t \bar t$ (via $W^+_LW^-_L$ fusion). The solid line is for $M_H=120$ GeV, the dashed line for $M_H=500$ GeV, the dotted line for $M_H=1$ TeV, and the dot-dashed line for $M_H=\infty$ (LET). (right side) The K-factor as a function of $M_H$ for $e^+e^- \to \nu\bar{\nu} t \bar t$ (via $W^+_LW^-_L$ fusion). The solid line is for $\sqrt{s}_{e^+e^-}=500$ GeV, the dashed line for $\sqrt{s}_{e^+e^-}=1$ TeV, the dotted line for $\sqrt{s}_{e^+e^-}=2$ TeV, and the dot-dashed line for $\sqrt{s}_{e^+e^-}=3$ TeV. See text for an explanation of the K-factor. []{data-label="kfactor"}](fig8b.eps "fig:"){width="70mm"} The QCD corrections to longitudinal scattering are often presented as a K-factor, normally defined as the ratio of the NLO to LO cross sections. Because the ${\cal O} (\alpha_s) $ QCD corrections we calculated are LO corrections to a tree level electroweak result we take the K-factor to be the ratio of the cross section with the ${\cal O} (\alpha_s) $ QCD corrections and the tree level electroweak cross sections. The K-factors for $\sigma(e^+e^- \to \nu\bar{\nu} t \bar t)$ which goes via $W^+_LW^-_L$ fusion and for $\sigma(e^+e^-\to e^+e^- t \bar t)$ is shown in Fig. 21 (left side) as a function of $\sqrt{s}_{e^+e^-}$. The ${\cal O} (\alpha_s) $ QCD corrections are largest for $M_H=500$ GeV with K-factors ranging from over 1.2 for $\sqrt{s}_{e^+e^-}=500$ GeV to 1.15 for $\sqrt{s}_{e^+e^-}=1$ TeV. The corrections decrease as $\sqrt{s}_{e^+e^-}$ increases. The variation of the K-factor with $M_H$ is shown in Fig. 21 (right side). The fact that the K-factor is largest for $M_H=500$ GeV in Fig. 21 (left side) and that it peaks at $M_H\simeq 400$ GeV in Fig. 21 (right side) is a threshold effect which is an artifact of the kinematic cut we imposed on the $t\bar{t}$ invariant mass. The important point is that the QCD corrections are not insignificant compared to the effects we might wish to study such as top Yukawa couplings or anomalous $VVt\bar{t}$ couplings. QCD at a Photon Collider [@Sullivan:2005pb]\ {#ssull} -------------------------------------------- A terascale photon collider will provide a unique opportunity to understand the resolved hadronic structure of light beyond a few GeV. This structure completely dominates the QCD cross section if the invariant mass of hadronic final state particles is less than $\sim W_{\textrm{max}}/3$. This is clear from Fig. 22(a), where we see the cross section for $bb$ production as a function of invariant mass $M_{bb}$. The first calculation of the uncertainty in this cross section is presented in detail in the Proceedings of this workshop [@Sullivan:2005pb]. The result, the dashed band surrounding the upper solid curve in Fig. 22(b), is that the cross section cannot be predicted to better than a factor of 5. There is no way to improve this prediction without measuring the gluon, charm, and bottom parton distributions for the photon *in situ* at this collider. ![image](ALCPG0402_fig1){width="3.25in"}![image](ALCPG0402_fig3){width="3.25in"}\ [ (a)(b) ]{} Loosening the cuts used for Fig. 22(a) will provide an extremely clean sample of events with a gluon in the initial state. Hence, the gluon structure should be quickly well-measured. The peak near 90 GeV is from a $Z$ resonance, for which $40\%$ of the events come from a $c\bar c$ initial state. Therefore, a clean extraction of the charm structure can be made by reconstructing the $Z$ peak in multiple channels (particularly $Z\to\mu^+\mu^-$). Finally, the long resolved-resolved tail above the $Z$ peak is almost entirely due to $bb+\bar b\bar b+b\bar b$ collisions. Therefore, given enough data, even the bottom structure may be accessible. The study [@Sullivan:2005pb] focused on a photon-photon collider, but another option being considered is a photon-electron collider. In general, it is more difficult to cleanly extract the gluon PDF in a $\gamma$–$e$ collision than a $\gamma$–$\gamma$ collision, because the cross section at high invariant mass ($>20$ GeV) is smaller, the decay products tend to be boosted more forward into less-well instrumented regions of the detector, and an additional deconvolution must be performed to remove the effect of extracting an almost-real photon from the electron. Nevertheless, this option should be examined in more detail as it may be simpler to construct a $\gamma$–$e$ collider. NEW PHYSICS EFFECTS ON TOP QUARK PROPERTIES {#sec_np} =========================================== Top Compositeness at Colliders\ {#sagh} ------------------------------- Consider the Randall-Sundrum (RS1) model [@Randall:1999ee] which is a compact slice of AdS$_5$, $$\begin{aligned} ds^2 & = & e^{-2k |\theta| r_c} \eta^{\mu \nu} dx_{\mu} dx_{\nu} + r_c^2 d \theta^2, \; - \pi \leq \theta \leq \pi, \label{metric}\end{aligned}$$ where $k$ is the curvature scale and the extra-dimensional interval is realized as an orbifolded circle of radius $r_c$. The two orbifold fixed points, $\theta = 0, \pi$, correspond to the “UV” (or “Planck) and “IR” (or “TeV”) branes respectively. In warped spacetimes the relationship between 5D mass scales and 4D mass scales (in an effective 4D description) depends on location in the extra dimension through the warp factor, $e^{-k |\theta| r_c}$. This allows large 4D mass hierarchies to naturally arise without large hierarchies in the defining 5D theory, whose mass parameters are taken to be of order the observed Planck scale, $M_{ Pl } \sim 10^{18}$ GeV. For example, the 4D massless graviton mode is localized near the UV brane while the Higgs sector is taken to be localized on the IR brane. In the 4D effective theory one then finds $${\rm Weak ~Scale} \sim M_{ Pl } e^{-k \pi r_c} .$$ A modestly large radius, i.e., $k \pi r_c \sim \log \left( M_{ Pl } / \hbox{TeV} \right) \sim 30$, can then accommodate a TeV-size weak scale. Kaluza-Klein (KK) graviton resonances have masses $m \sim k e^{ - k \pi r_c }$. These masses are at the TeV-scale, since their wave functions are also localized near the IR brane. In the original RS1 model, it was assumed that the entire SM was localized on the TeV brane, and that only gravity propagated in the full 5D space. Thus, the effective UV cut-off for gauge and fermion fields and hence the scale suppressing higher-dimensional operators is at a TeV, the same scale which sets the Higgs sector. However, bounds from electroweak precision tests (EWPT) on this cut-off are approximately 5-10 TeV, while those from flavor changing neutral currents (FCNCs) such as $K - \bar{K}$ mixing are around 1000 TeV. Stabilizing the electroweak scale thus requires fine-tuning; even though RS1 explains the big hierarchy between the Planck and electroweak scales, it has a “little” hierarchy problem between the weak scale and the TeV scale cut-off. An attractive solution to this problem is to allow the SM gauge [@Davoudiasl:1999tf] and fermion [@Grossman:1999ra; @Gherghetta:2000qt] fields to propagate in the extra dimensional bulk. We first explain how bulk fermions enable us to evade flavor constraints. The localization of the wavefunction of the massless chiral mode is controlled by the 5d mass term for each fermion, which in units of $k$ is denoted by the $c$-parameter. In the warped scenario, for $c>1/2$ ($c<1/2$) the zero mode is localized near the Planck (TeV) brane, whereas for $c = 1/2$, the wave function is flat. We therefore choose $c > 1/2$ for light fermions so that the effective UV cut-off at the location of the light fermions is much greater than a TeV, suppressing dangerous FCNCs. This naturally results in a small $4D$ Yukawa coupling to the Higgs on the TeV brane without any hierarchies in the fundamental $5D$ Yukawa couplings [@Grossman:1999ra; @Gherghetta:2000qt; @Huber:2000ie]. Similarly, we choose $c \ll 1/2$ for the top quark to obtain an $O(1)$ Yukawa coupling. We can also show that in this scenario with bulk gauge fields high-scale unification of gauge couplings can be accommodated. Since gauge fields are also in the bulk, their excited KK modes induce additional effects in flavor physics and in EWPT, which are calculable in the 5D effective field theory. For example, the couplings of KK modes to light fermions are flavor-dependent, giving FCNCs. However, this flavor dependence is small. KK modes, just like the Higgs, are localized near the TeV brane, whereas the light fermions are near the Planck brane. The FCNCs are therefore proportional to the Yukawa couplings, resulting in a suppression of flavor violation. This scenario has an analog of the Glashow-Iliopoulos-Maiani (GIM) mechanism of the SM, resulting in the suppression of the calculable FCNCs [@Gherghetta:2000qt; @Huber:2000ie]. Early studies showed that the corrections to EWPT from gauge KK modes are too large, unless the KK mass is greater than 10 TeV [@Huber:2000fh]. Such a large KK scale results in a little hierarchy problem between the weak and KK scales. The localization of the light fermions near the Planck brane reduces the contribution of gauge KK modes to two observables – the $S$ parameter and $4$-fermion operators, but the observable called the $T$ parameter still gives stringent constraints. In Ref. [@Agashe:2003zs], it was shown that this problem can be avoided by extending the electroweak gauge group in the bulk to $SU(2)_L \times SU(2)_R \times U(1)_{ B - L }$. Such an extension provides a custodial isospin symmetry to protect the $T$ parameter from large corrections. Thus, KK masses as low as 3 TeV are allowed by oblique EW data, significantly ameliorating the little hierarchy problem. We now consider the top and bottom quarks. It is clear that we prefer $c \ll 1/2$ for $t_L$ to obtain a top Yukawa coupling of $O(1)$ without too large a 5D Yukawa coupling, but this implies a large shift in coupling of $b_L$ to the $Z$-boson unless the KK scale is larger than a TeV. This shift occurs due to the large coupling of $b_L$ to the KK $Z$ modes, which mix with the zero-mode $Z$ via the Higgs vev [@Agashe:2003zs]: $$\begin{aligned} \frac{ \delta \left( g^{ t_R }_Z \right) }{ g^{ t_R}_Z } \Big|_{ \hbox{ KK gauge } } & \approx & \frac{ m_Z^2 } { \left( 0.41 m_{ KK } \right) ^2 } \frac{ 1 - 2 c_R}{ 3 - 2 c_R } \left( - \frac{ k \pi r_c }{2} + \frac{ 5 - 2 c_R}{ 4 ( 3 - 2 c_R ) } \right).\end{aligned}$$ Here, $m_{ KK } \approx 2.45 \; k e^{ - k \pi r_c }$ is defined to be the mass of the lightest gauge KK mode. Thus, there is a tension between obtaining the top Yukawa and not shifting the coupling of $b_L$ to $Z$. As a compromise, KK masses $\sim 5$ TeV are consistent with a shift in $g_Z^{ b_L } $ of approximately $0.25\%$ for $c_L \sim 0.4$. The $t_R$ must therefore be localized near the TeV brane: $c_R \stackrel{<}{\sim} 0$. Such a profile for $t_R$ leads to sizable shift in its coupling to the $Z$ via exchange of KK $Z$-bosons, and also via KK $t_L$ modes. The additional contribution from KK $t_L$ exchange is due to zero-mode $t_R$ mixing with KK $t_L$ via a Higgs vev which then couples to the $Z$, giving the shift $$\begin{aligned} \frac{ \delta \left( g_Z^{ t_R } \right) }{ g^{ t_R }_Z } \Big|_{ \hbox{ KK fermion } } & \approx & \sum_{ n } \frac{ 1/2 }{ - 2 / 3 \sin^2 \theta_W } \left( \frac{ m_t \sqrt{ 1 / 2 - c_L } }{ m_{ t^{ ( n ) }_L } } \right)^2.\end{aligned}$$ Here $m_t \sqrt{ 1 / 2 - c_l }$ is the mass term coupling zero-mode and KK top quarks, and the KK $t_L$ masses are given by $m_{ t^{ ( n ) }_L } \approx \pi \; k e^{ - k \pi r_c } ( n - c_L /2 ) \approx 0.78 \; m_{ KK } \; ( n - c_L /2 )$. ![Shift in the coupling of $t_R$ to $Z$ as a function of KK mass in GeV on horizontal axis and $c_R$ on vertical axis for the choice $c_L = 0.4$. The solid, long-dashed and short-dashed lines correspond to shifts of $5 \%$, $10 \%$ and $15 \%$, respectively.[]{data-label="shift"}](dashed.eps){width="75mm"} The total shift in the coupling of $t_R$ is plotted in Fig. 23. As seen in the figure, we obtain an $10\%$ shift for KK masses of a few TeV. Smaller KK masses are not allowed by EWPT, and larger masses lead to large fine-tuning. Observability of an effect of this size might be difficult at the LHC since the sensitivity of the LHC is only at the $20 \%$ level for a shift in the axial coupling of the top quark to the $Z$. It should be possible at the ILC, which has sensitivity at the few percent level for shifts in both axial and vector couplings of the top quark to $Z$ [@Abe:2001nq]. There are similar shifts in couplings of Higgs to $W/Z$ due to its profile being localized near TeV brane just like for $t_R$. Finally, an intriguing aspect is that via the AdS/CFT correspondence [@Maldacena:1997re], such a scenario is conjectured to be dual to a purely $4D$ theory with a composite Higgs boson [@Arkani-Hamed:2000ds], with the light fermions being elementary and $t_R$ being composite. This provides an intuitive understanding for the large shifts in the couplings of $t_R$ and Higgs to $Z$. Hence, these signals might be valid for general composite Higgs models as well. Top quark properties in Little Higgs Models [@Berger:2005ht]\ {#sberg} ------------------------------------------------------------- In this section, we study the corrections to the top quark properties in “Little Higgs” models of electroweak symmetry breaking [@review], and compare the expected deviations from the SM predictions with expected sensitivities of experiments at the LHC and the ILC. In the Little Higgs models, electroweak symmetry is driven by the radiative effects from the top sector, including the SM-like top and its heavy counterpart, a TeV-scale “heavy top” $T$. Probing this structure experimentally is quite difficult. While the LHC should be able to discover the $T$ quark, its potential for studying its couplings is limited [@PPP; @ATLAS]. Direct production of the $T$ will likely be beyond the kinematic reach of the ILC. However, we will show below that the corrections to the gauge couplings of the SM top, induced by its mixing with the $T$, will be observable at the ILC throughout the parameter range consistent with naturalness. Measuring these corrections will provide a unique window on the top sector of the Little Higgs. Little Higgs models contain a light Higgs boson which is a [*composite*]{} of more fundamental degrees of freedom. A generic composite Higgs model must become strongly coupled at an energy scale around 1 TeV, leading to unacceptably large corrections to precision electroweak observables. In contrast, Little Higgs models remain perturbative until a higher energy scale, around 10 TeV. The hierarchy between the Higgs mass and the strong coupling scale is natural and stable with respect to radiative corrections. Because of the special symmetry structure of the theory, the Higgs mass vanishes at tree level, as do one-loop quadratically divergent diagrams. The mass term is dominated by the logarithmically divergent one-loop contribution from the top quark, which triggers electroweak symmetry breaking. Many Little Higgs models have been proposed in the literature. We will consider two examples in this study, the “Littlest Higgs” model [@littlest], and its variation incorporating T parity [@LHT]. We will study the effects on the $t\bar{t}Z$ vertex in these models; for a more detailed study, see Ref. [@Berger:2005ht]. ![image](lh_Ztt_final.ps){width="60mm"} ![image](tlh_Ztt.ps){width="60mm"} Corrections to the gauge couplings of the top quark in Little Higgs model arise from two sources: the mixing of the (left-handed) top with the heavy top $T$, and the mixing of the SM gauge bosons $W^\pm, Z^0$ with their heavy counterparts, $W^\pm_H$ and $W_H^3$. Using the superscripts “t” and “g” to denote the contributions from these two sources, the corrections to the $t\bar{t}Z$ coupling can be written as g\^[Z[t]{}]{}\_R = 0,   & &   g\^[Z[g]{}]{}\_R = g\_R\^Z,\ g\^[Z[t]{}]{}\_L = ,   & &   g\^[Z[g]{}]{}\_L = . Here, $g_{L,R}^Z$ are the SM left- and right-handed $t\bar{t}Z$ couplings, $g_V^Z=(g_R^Z+g_L^Z)/2$ and $g_A^Z=(g_R^Z-g_L^Z)/2$ are their vector and axial combinations, $c_W,s_W$ are respectively the cosine and sine of the weak mixing angle, and $s_\psi\equiv \sin\psi$, $c_\psi\equiv\cos\psi$. In the original Littlest Higgs model [@littlest], both the gauge sector shift and the top sector shift occur; in the T-parity model [@LHT], only the top sector shift is present. The predicted shifts in the $t\bar{t}Z$ axial and vector couplings for $m_T=0.5, 1.0,$ and 2.0 TeV, and $\lambda_T=0.5, 1, 2$, are plotted in Fig. 24 (left panel), along with the experimental sensitivities expected at the LHC [@Baur:2004uw] and the ILC [@Abe:2001nq]. The mixing angle $\psi$ is varied between $0$ and $\pi/2$. Note that the shifts have a definite sign. While only a rather small part of the parameter space is accessible at the LHC even with 3000 fb$^{-1}$ integrated luminosity, the ILC experiments will be able to easily observe the shifts in most of the parameter space preferred by naturalness considerations (however, the prospects for observation at the LHC improve when additional final states such as $b\bar{b}+4\,j$ are included; see Ref. [@newbaur]). Similarly, shifts in the $Wtb$ coupling can be probed via deviations in the top quark width at the ILC [@Berger:2005ht]. Testing CPT Symmetry with Top Quark Physics [@CRT]\ {#sjarc} --------------------------------------------------- The viability to observe evidence of CPT violation in the top sector has been analyzed through the measurement of a mass difference between top and anti-top [@CRT]. This study has been focused on the CPT violating ratio of the top quark, $R_{CPT}(t)\equiv 2(m_t-m_{\bar t})/(m_t+m_{\bar t})$. The present constraints from the Tevatron are approximately 10%, and they could be reduced by one order of magnitude at the LHC or ILC. The most promising studied channel is the lepton plus jets channel for top anti-top production. However, other techniques to reconstruct the top mass could also be very interesting, such as the analysis of the J/psi from $b$ decay at the LHC, which improves the systematic uncertainties. Single top production could also be studied, since a combination of different measurements would be necessary in order to consider CPT violation as the explanation of any exotic data. : Di-lepton events originating predominantly from $t\bar t \rightarrow W^+ (\rightarrow \ell^+\nu)\,b\,W^-(\rightarrow \ell^-\bar \nu)\,\bar b $, with $\ell = e$ or $ \mu$, have been used in Tevatron to measure the top quark pole mass supposing an identical mass for the top and anti-top quarks. The same data can be used to study CPT violation through a double peak in the reconstructed invariant mass associated to the lepton and b quark coming from the single decay of the top or anti-top [@CRT]. ![Schematics of the top and anti-top decays in the dilepton channel.[]{data-label="dileptonSch"}](dileptonSch.eps){width="40.00000%"} By using the Tevatron data accumulated at Fermilab from 1992 through 1995 [@CDFdilept1], it is possible to find the bound $|R_{CPT}(t)|<0.13$ at the 95% C.L. [@CRT]. On the other hand, the sensitivity of the LHC can be estimated as $R_{CPT}(t)=0.03$ at the 95% c.l. following an analogous analysis [@CRT]. : A more promising signal is provided when one of the $W$ decays leptonically while the other one decays hadronically: $t\bar t \rightarrow W^+(\rightarrow \ell^+\nu b)\,b\,W^-(\rightarrow q \bar q')\,\bar b$. In fact, the inclusive lepton plus jets channel provides a larger and cleaner sample of top quarks, whose mass can be reconstructed directly using the hadronic part of the decay. The invariant mass of the three jets coming from the top ($m_{jjb}\equiv m_{j_q j_{\bar q} j_{b}}$) or anti-top ($m_{jjb} \equiv m_{j_q j_{\bar q} j_{\bar b}}$) presents a peak at the top ($m_t$) or anti-top ($m_{\bar t}$) mass respectively. The estimate combining the CDF [@CDFl+jets] and D[Ø]{} data [@D0l+jets] gives a more constraining bound of $R_{CPT}(t) < 9.2\times 10^{-2}$. The sensitivity of the LHC is also better in this channel since both statistical and systematic uncertainties are expected to be improved. Indeed, the LHC will be able to test the CPT violation of the top quark to almost one order of magnitude better than the present constraints: $|R_{CPT}(t)|\simeq 0.014$ at the 95% C.L. or equivalently, $m_t-m_{\bar t}\simeq 2.4$ GeV [@CRT]. ![Schematic example of the top and anti-top decays in the lepton plus jets channel.[]{data-label="fig:avto1"}](lplusjetsSch.eps){width="44.00000%"} The same analyses can be performed with the ILC, where an increase of the statistical uncertainties but a decrease of the systematic ones is expected [@Biernacik:2003xv]. The importance of these last uncertainties leads to a small improvement of the sensitivity in relation to the LHC. [**ACKNOWLEDGMENTS**]{} CFB is supported by the US Department of Energy under contract DE-AC02-76SF00515. YK is supported by the DFG Sonderforschungsbereich/Transregio 9 “Computer-gestützte Theoretische Teilchenphysik”. 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--- abstract: 'Fast and accurate integration of geodesics in Kerr spacetimes is an important tool in modeling the orbits of stars and the transport of radiation in the vicinities of black holes. Most existing integration algorithms employ Boyer-Lindquist coordinates, which have coordinate singularities at the event horizon and along the poles. Handling the singularities requires special numerical treatment in these regions, often slows down the calculations, and may lead to inaccurate geodesics. We present here a new general-purpose geodesic integrator, [`GRay2`]{}, that overcomes these issues by employing the Cartesian form of Kerr-Schild coordinates. By performing particular mathematical manipulations of the geodesic equations and several optimizations, we develop an implementation of the Cartesian Kerr-Schild coordinates that outperforms calculations that use the seemingly simpler equations in Boyer-Lindquist coordinates. We also employ the `OpenCL` framework, which allows [`GRay2`]{} to run on multi-core CPUs as well as on a wide range of GPU hardware accelerators, making the algorithm more versatile. We report numerous convergence tests and benchmark results for [`GRay2`]{} for both time-like (particle) and null (photon) geodesics.' author: - 'Chi-kwan Chan, Lia Medeiros, Feryal Özel, and Dimitrios Psaltis' bibliography: - 'ms.bib' - 'my.bib' title: '[`GRay2`]{}: A General Purpose Geodesic Integrator for Kerr Spacetimes' --- Introduction ============ Integrating geodesics of particles and photons in the spacetimes of Kerr black holes is an important aspect of theoretical modeling of various astrophysical phenomena, from the orbits of stars and compact objects around supermassive black holes [see, e.g., @Alexander2017] to the transport of radiation through their accretion flows [see, e.g., @Yuan2014]. Fast geodesic integrators are also critical in fitting data of, e.g., stars in orbit around the black hole in the center of the Milky Way [@Boehle2016; @Gillessen2017], of rotationally broadened fluorescence lines from accreting black holes in the X-rays [@Miller2007], or of interferometric data taken with the Event Horizon Telescope that aims to take the first image of a supermassive black holes with horizon scale resolution [see, e.g., @2008Natur.455...78D; @Doeleman2012]. Calculations of test-particle orbits (time-like geodesics) around black holes has been traditionally done in a post-Newtonian approximation, focusing on N-body effects [see, e.g., @Brem2014; @Hammers2014 for recent work], or by solving simultaneously for the dynamical spacetime of the cluster of particles [see @Shapiro1992 and references therein]. More recently, fast algorithms have been developed that follow the orbits of test particles in stationary black-hole spacetimes, with no approximations [@Yang2014; @Zhang2015]. Integrations of null geodesics (ray tracing) in Kerr spacetimes can be traced back to @Bardeen1973, @Cunningham1975, and , where the images of accretion disks and the outlines of the shadows of Schwarzschild and extreme Kerr black holes were first obtained. More recently, methods of combining polarized radiative transfer with ray tracing [see, e.g., @2003MNRAS.342.1280B; @2004MNRAS.349..994B; @Gammie2012; @Younsi2012; @2013ApJ...777...11S] as well as a variety of open-source algorithms for fast radiative transfer calculations have been developed [see, e.g., @2009ApJ...696.1616D; @2009ApJS..184..387D; @2013ApJ...777...13C; @2013ApJS..207....6Y; @2016ApJ...820..105P; @Dexter2016]. In an earlier article, we described [`GRay`]{} [@2013ApJ...777...13C], the first publicly available numerical algorithm that made explicit use of general-purpose computing on graphics processing units (GPU) for ray tracing in relativistic spacetimes. [`GRay`]{} uses the high computational horsepower of GPUs to speed up this computationally intensive problem. It achieved 1–2 orders of magnitude speed up compared traditional CPU-base algorithms and allowed us to generate large, high-cadence simulations of the observable properties of accreting black holes [@2015ApJ...799....1C; @2015ApJ...812..103C; @2015ApJ...814..115P; @2016ApJ...832..156K; @2016ApJ...826...77B; @2016arXiv161003505M; @2016arXiv160106799M]. Even though [`GRay`]{} is very fast and efficient, it uses a standard physical setup of the ray-tracing problem as well as numerical methods that have a number of limitations. For example, like most of the other algorithms, [`GRay`]{} employs the Boyer-Lindquist (BL) coordinates to take advantage of the symmetry of the Kerr spacetime, which greatly simplifies the derivation and evaluation of the Christoffel symbols. However, the various coordinate singularities in the BL coordinates cause numerical difficulties. Moreover, as in many other algorithms, [`GRay`]{} uses the so called fast-light approximation [see, however, @2009ApJS..184..387D]. This means that, when solving the radiative transfer equation along each ray, the fluid is assumed to be time independent, or equivalently, the speed of each photon is taken to be effectively infinite. This approximation greatly simplifies the algorithms because only a single snapshot of the underlying matter through which radiation propagates is needed in the radiative transfer calculation at each time step. However, this assumption affects the time variability properties of the simulations at the fastest timescales near the black-hole horizons [@2009ApJS..184..387D], which will be important in interpreting the upcoming observations with the Event Horizon Telescope [@2016ApJ...832..156K; @2016arXiv161003505M; @2016arXiv160106799M]. In order to overcome these difficulties, we describe here [`GRay2`]{}, a new open source, hardware accelerated, geodesic integration algorithm. We improve the geodesic integration in [`GRay2`]{} by switching to the Cartesian form of Kerr-Schild (KS) coordinates, overcoming all coordinate singularities and increasing the overall accuracy of the calculations. We also switch to `OpenCL`, which allows [`GRay2`]{} to run on multi-core CPUs as well as on a wide range of hardware accelerators, making the algorithm more versatile. Finally, [`GRay2`]{} can handle both time-like (test particle) and null (photon) geodesics, making it applicable to calculations of both stellar orbits and radiative transfer. In the next section, we discuss the limitations of using the BL coordinates and derive an optimized form of geodesic equations in the Cartesian KS coordinates. In section \[sec:ct\], we provide the details of using coordinate time instead of the affine parameter to integrate the geodesic equations. In section \[sec:implementation\], we summarize the implementation details of [`GRay2`]{}. In section \[sec:tests\], we perform a convergence study using unstable spherical photon orbits and stable particle orbits. In section \[sec:benchmarks\], we report benchmark results of [`GRay2`]{} running on a wide range of CPUs and GPUs and demonstrate that GPUs can be up to two orders of magnitude faster than a single CPU core and that integrating in Cartesian KS coordinates can outperform integrating in BL coordinates. Finally, we summarize our findings in section \[sec:summary\]. The [`GRay2`]{} Algorithm ========================= Implementation of the Cartesian Kerr-Schild Coordinates {#sec:KS} ------------------------------------------------------- Letting $M$ and $a$ be the mass and spin parameter of a Kerr black hole, the Boyer-Lindquist line element reads[^1] $$\begin{aligned} ds^2 &= - \left(1 - \frac{2M\mathcal{r}}{\varrho^2}\right) d\mathcal{t}^2 - \frac{4M\mathcal{r}a\sin^2\!\vartheta}{\varrho^2}d\varphi d\mathcal{t} + \frac{\varrho^2}{\varDelta}d\mathcal{r}^2 \nonumber\\ &\!\!\!+ \varrho^2 d\vartheta^2 + \left(\mathcal{r}^2 + a^2 + \frac{2M\mathcal{r}a^2\sin^2\!\vartheta} {\varrho^2}\right)\sin^2\!\vartheta d\varphi^2,\end{aligned}$$ where $\varrho^2 \equiv \mathcal{r}^2 + a^2\cos^2\vartheta$ and $\varDelta \equiv \mathcal{r}^2 - 2M\mathcal{r} + a^2$. The spherical polar nature of the BL coordinates introduces unnecessary coordinate singularities along the poles, in addition to the coordinate singularities at the event horizons. When integrating geodesics numerically, these coordinate singularities can cause significant difficulties. Although there exist algorithms to overcome these difficulties [see, e.g., the method introduced by @2013ApJ...777...13C], the poles can still cause numerical problems such as slowing down the calculations, leading to inaccurate geodesics, and even crashing the algorithms for extreme cases such as computing a face-on image of a black hole accretion disk at inclination $i = 0^\circ$. In [`GRay2`]{}, we resolve the coordinate singularities by employing the Cartesian form of the KS coordinates $$\begin{aligned} g_{\alpha\beta} &= \eta_{\alpha\beta} + f l_\alpha l_\beta, \label{def:CKS}\end{aligned}$$ where $\eta_{\alpha\beta} \equiv \mathrm{diag}(-1,1,1,1)$ is the Minkowski metric, $$\begin{aligned} f &= \frac{2r^3}{r^4 + a^2 z^2}, \\ l_\alpha &= \left(1, \frac{rx + ay}{r^2 + a^2}, \frac{ry - ax}{r^2 + a^2}, \frac{z}{r}\right),\end{aligned}$$ and $r$ is defined implicitly by $$\begin{aligned} x^2 + y^2 + z^2 = r^2 + a^2\left(1 - z^2 / r^2\right). \label{eq:CKSr}\end{aligned}$$ Its Cartesian nature not only completely avoids the coordinate singularities along the poles but requires no special treatment in the integrator. This is an advantage for implementing numerical integrators on modern hardware accelerators, such as GPUs, because these massive parallel stream processors use the Single-Instruction Multiple-Data (SIMD) paradigm, which are inefficient in handling branch instructions (i.e., conditional statements). Besides the poles, both the spherical and Cartesian KS coordinates are also horizon-penetrating—there is no coordinate singularity at the event horizons. We can, in principle, integrate geodesics through the event horizon into the interior of the black hole. In fact, this property makes the spherical polar form of KS coordinates the default choice for many GRMHD codes [see, e.g., @2003ApJ...589..444G; @2009ApJ...704..937N; @2013MNRAS.429.3533S; @2015ApJ...807...31R; @2016ApJS..225...22W], as no special boundary treatment is required at the horizons. Given that all the elements are non-zero in the Cartesian KS metric, this may seem at first to be a computationally very expensive coordinate system to work with. A rough operation-count goes as following. For the BL coordinates, there are 5 independent, non-zero elements in the metric, 10 independent elements in the metric derivative tensor $g_{\mu\nu,\alpha}$, and 20 independent Christoffel symbols. In contrast, in the Cartesian KS coordinates, we need to compute all 10 independent metric elements. The metric is time-independent but does not use any spatial symmetry, resulting in 30 independent elements in the metric derivative tensor. Since the metric derivative tensor tends to be the most complicated part of the calculation, this suggests that the computation of the 40 Christoffel symbols in the KS coordinates requires roughly 3 times more operations than in the BL coordinates. Therefore, we expect solving the geodesic equations in the Cartesian KS coordinates to be at least 3 times more expensive than in the BL coordinates. For each geodesic, we need to solve all four second-order ordinary differential equations, if we integrate with respect to the affine parameter $\lambda$. Comparing to the methods that use the Killing vectors [see, e.g., @2012ApJ...745....1P; @2013ApJ...777...13C], this is a $$\begin{aligned} \frac{2\times(8~\mbox{variables})} {2\times(6~\mbox{variables}) + (1~\mbox{constant})} - 1 = 23\% \label{eq:bandwidth}\end{aligned}$$ increase in the bandwidth requirement. Nevertheless, as we will show in our benchmarks, geodesic integration is in general *compute-bounded*, meaning that the performance is limited by the speed of the computation and not by the speed of transferring data. Alternatively, we can also integrate the geodesic equations with respect to the coordinate time $t$ (see next section). This way, the number of dynamic variables reduces to six, which is the same as for `GRay` [@2013ApJ...777...13C]. One of the important improvements in [`GRay2`]{} is that, by a series of mathematical manipulations and regrouping, we significantly reduce the operation-count of the geodesic equations in the Cartesian KS coordinates. Let $\lambda$ be the affine parameter and $\dot{x}^\mu \equiv dx^\mu/d\lambda$. Our manipulations start by realizing that, although the Christoffel symbols $\varGamma^\mu_{\alpha\beta}$ provide an elegant form of writing the geodesic equations $$\begin{aligned} \ddot{x}^\mu &= -\varGamma^\mu_{\alpha\beta} \dot{x}^\alpha \dot{x}^\beta,\end{aligned}$$ it is actually more efficient to go back a step and write the equations in terms of the metric derivative tensor as $$\begin{aligned} \ddot{x}^\mu &= -\frac{1}{2} g^{\mu\nu}(g_{\nu\alpha,\beta} + g_{\nu\beta,\alpha} - g_{\alpha\beta,\nu}) \dot{x}^\alpha \dot{x}^\beta \label{eq:geo} \\ &= -g^{\mu\nu} g_{\nu\alpha,\beta} \dot{x}^\alpha \dot{x}^\beta +\frac{1}{2} g^{\mu\nu} g_{\alpha\beta,\nu} \dot{x}^\alpha \dot{x}^\beta. \label{eq:nosym}\end{aligned}$$ We can combine the first two terms in equation (\[eq:geo\]) because the product of an anti-symmetric tensor and a symmetric tensor vanishes—there is no need to explicitly symmetrize $\alpha$ and $\beta$ for $g_{\nu\alpha,\beta}$ from a computational point of view. Furthermore, equation (\[eq:nosym\]) can be written as following by replacing the indices $\nu\rightarrow\beta\rightarrow\alpha\rightarrow\gamma$ for the first term and $\nu\rightarrow\alpha\rightarrow\beta\rightarrow\gamma$ for the second term $$\begin{aligned} \ddot{x}^\mu &= -\left(g^{\mu\beta}\dot{x}^\alpha - \frac{1}{2}g^{\mu\alpha}\dot{x}^\beta\right) g_{\beta\gamma,\alpha}\dot{x}^\gamma. \label{eq:generic}\end{aligned}$$ For each geodesic, $\dot{x}^\mu$ depend only on $\lambda$. Although we still need to evaluate all 30 independent non-zero elements of $g_{\beta\gamma,\alpha}$, we can store their results into the 12 non-zero elements the term outside the parenthesis in the above equation and reuse them in the summation of each $\mu$. Therefore, even in this general form without specifying a metric, the geodesic equations are in fact simpler than how they look, at least in terms of operation-count. Next, by substituting the definition of the Cartesian KS metric (\[def:CKS\]) into equation (\[eq:generic\]), we obtain $$\begin{aligned} \ddot{x}^\mu &= - \left(\eta^{\mu\beta} \dot{x}^\alpha - \frac{1}{2}\eta^{\mu\alpha} \dot{x}^\beta\right) \dot{x}_{\beta,\alpha} + F l^\mu\end{aligned}$$ with $$\begin{aligned} F &\equiv f \left(l^\beta \dot{x}^\alpha - \frac{1}{2}l^\alpha \dot{x}^\beta\right) \dot{x}_{\beta,\alpha}.\end{aligned}$$ In the above equations, the Minkowski metric $\eta^{\mu\nu}$ effectively picks out different components of the derivative tensor and applies different signs. Hence, we can split the equations and optimize them further as $$\begin{aligned} \ddot{x}^0 &= \dot{x}^\alpha \dot{x}_{0,\alpha} - F, \label{eq:opt0}\\ \ddot{x}^i &= - \dot{x}^\alpha \left(\dot{x}_{i,\alpha} - \frac{1}{2}\dot{x}_{\alpha,i}\right) + F l^i. \label{eq:opti}\end{aligned}$$ Note that the positions of the indices 0 and $i$ do not match on the two sides of the above equations. This is not an error; equations (\[eq:opt0\]) and (\[eq:opti\]) are no longer tensor equations. In the above new form, the right hand sides (RHS) of the geodesic equations in the Cartesian KS coordinates have only $\sim 65\%$ more floating-point operations than in the BL coordinates. This is less than half of the operations compared to our rough estimate. Furthermore, the evaluation of the RHS uses many matrix-vector products, which are optimized in modern hardware. Indeed, as we show below, our benchmarks (Table \[tab:elapse\] and \[tab:speedup\]) show that using the Cartesian KS on discrete GPUs can outperform its BL counterpart. Coordinate Time Integration {#sec:ct} --------------------------- In order to efficiently overcome the fast-light approximation, we need to control the integration of a geodesic to a targeted time according to the GRMHD simulations, which are usually performed in the spherical KS coordinates [see, e.g., @2003ApJ...589..444G; @2013MNRAS.429.3533S]. This minimizes both the data reading and memory overhead by requiring only sequential reading with at most two snapshots in memory. In addition, we can take advantage of the fact that GPUs have special purpose hardware for accelerating interpolation, which makes accessing GRMHD simulations essentially free of overhead. In [`GRay2`]{}, we develop two classes of methods to integrate the geodesic equations to a target KS coordinate time: (*i*) by directly integrating the geodesics with respect to the KS coordinate time and (*ii*) by applying different root finders to the numerical solutions to match the targeted time[^2]. In this section, we will limit our discussion to method (*i*) and derive the geodesic equations in terms of the KS coordinate time. Again, these equations are used in [`GRay2`]{} mainly to overcome the fast-light approximation. Although they also reduce the required bandwidth of the geodesic integrator, the performance impact is very minor. Letting $v^\mu \equiv dx^\mu/dt$ and using the chain rule, it is straightforward to derive the geodesic equations in the following form $$\begin{aligned} \frac{dv^i}{dt} &= -\varGamma^i_{\alpha\beta} v^\alpha v^\beta +\varGamma^0_{\alpha\beta} v^\alpha v^\beta v^i.\end{aligned}$$ Note that we only consider the spatial components of the geodesic equations. The time component $dv^0/dt = 0$ is trivial and consistent with the spatial part of the equations [see, e.g., @1993tegp.book.....W]. Substituting the definition of the Christoffel symbols, we get $$\begin{aligned} \frac{dv^i}{dt} &= -\frac{1}{2} g'^{i\nu}(g_{\nu\alpha,\beta} + g_{\nu\beta,\alpha} - g_{\alpha\beta,\nu}) v^\alpha v^\beta, \label{eq:geo_t}\end{aligned}$$ where $g'^{i\nu} \equiv g^{i\nu} - v^i g^{0\nu}$. Equations (\[eq:geo\_t\]) and (\[eq:geo\]) have the same form. Hence, the optimizations we carried out in the last section are still applicable; equation (\[eq:geo\_t\]) can be optimized to the generic form $$\begin{aligned} \frac{dv^i}{dt} &= -\left(g'^{i\beta}v^\alpha - \frac{1}{2}g'^{i\alpha}v^\beta\right) g_{\beta\gamma,\alpha}v^\gamma.\end{aligned}$$ Finally, substituting the definition of the Cartesian KS metric, we obtain $$\begin{aligned} \frac{dv^i}{dt} &= - v^\alpha\left(v_{i,\alpha} - \frac{1}{2}v_{\alpha,i}\right) + F l^i \nonumber\\[-6pt] &\hspace{102pt} - (v^\alpha v_{0,\alpha} - F) v^i.\end{aligned}$$ The first two terms in the RHS of the above equation (i.e., first line) match the RHS of equation (\[eq:opti\]), while the last term (i.e., second line) matches the RHS of equation (\[eq:opt0\]). This is expected and in fact shows that integrating with respect to the affine parameter and coordinate time have the same computational complexity. Therefore, because numerically integrating geodesics is compute bounded, the choice between integrating with respect to the affine parameter or with respect to the coordinate time will depend on the application of [`GRay2`]{} and not on the detailed performance of each method. If one cares only about calculating the shapes of geodesics, then using coordinate time will reduce the number of variables and save some bandwidth as shown by equation (\[eq:bandwidth\]). If, on the other hand, the radiative transfer equation needs to be integrated along a geodesic, then using the affine parameter will give the gravitational redshift with no additional computations[^3]. Additional Implementation Improvements {#sec:implementation} -------------------------------------- In addition to the improvements in the coordinate system and numerical scheme, we have made a number of additional implementation improvements in [`GRay2`]{}. One of them is the adoption of `OpenCL`, an open standard for parallel programming[^4], for executing massively parallel jobs on heterogeneous platforms. Without any modification of the source code, [`GRay2`]{} runs on multi-core CPUs as well as accelerators such as GPUs, Intel Xeon Phi, and potentially Field-Programmable Gate Arrays (FPGA). Another significant change is the adoption of the high performance computing (HPC) framework [`lux`]{} [@lux] for software portability and run time optimizations. With [`lux`]{}, the algorithms in [`GRay2`]{} are broken down into very small modules, with multiple implementations for each of them. This allows the users to easily construct an appropriate compilation of modules that are specific to each application. In addition, [`lux`]{} benchmarks the algorithms at run time and allows [`GRay2`]{} to automatically migrate to the most efficient algorithm. Finally, [`lux`]{} allows [`GRay2`]{} to use all hardware resources on a single computing node. It even automatically balances the work load across the CPU cores and accelerators. A typical application of [`GRay2`]{} is to render millions of mock images of accretion flows onto black holes based on the output of GRMHD simulations, for different model parameters, such as electron number density scale and the electron-to-ion temperature ratio[see, e.g., @2015ApJ...799....1C; @2015ApJ...812..103C]. Because a single mock image takes only a few seconds to render, there is no need to consider inter-node communication. Instead, millions of [`GRay2`]{} jobs can be submitted at the same time and each job runs in parallel, independently from each other. With this “trivially parallelizable” user case in mind, `OpenCL` and [`lux`]{} allows [`GRay2`]{} to run on a wide range of hardware and platforms. For example, one can compute part of the jobs on an Apple desktop, part of the jobs on a local HPC cluster with GPUs, part of the jobs in a supercomputing center with Xeon Phi, and the rest of the jobs in commercial clouds such as the Amazon Web Services. This flexibility allows us to to report benchmarks for the implementation of (*i*) different forms of the equations, (*ii*) different data structures, and (*iii*) different precisions in section \[sec:benchmarks\]. Figure \[fig:fish\] shows a screenshot of [`GRay2`]{} in its interactive mode, which allows for the simultaneous integration and visualization of geodesics in a black-hole spacetime. Convergence Tests {#sec:tests} ================= In this section, we perform a number of tests with [`GRay2`]{}, in situations that resemble expected realistic applications. By default, [`GRay2`]{} uses the classic 4th-order Runge-Kutta scheme, which is very robust and provides fast (4th-order) convergence rates. We are interested here in its long term behavior in the Cartesian KS coordinates [see, e.g., @2005MNRAS.364.1105S for an explanation on the importance of long term behaviors of integrators]. Typical gravitational deflection type of tests are not useful for this purpose because, in the Cartesian KS coordinates, the geodesic equations are trivial at large radii. In such cases, even though we could make the spatial interval of a geodesic arbitrarily long, the numerical error would still be dominated by a short segment of the geodesic near the black hole. This does not help to monitor the long term behavior of the integrators. Instead, we employ in our convergence tests closed, albeit often unstable, photon and particle orbits that can be integrated for long times. Unstable Spherical Photon Orbits -------------------------------- Motivated by the interactive visualization by @sph_orbits, we designed a set of tests using the unstable spherical photon orbits of @2003GReGr..35.1909T. These orbits are non-trivial and are excellent for observing the long term behavior of the integrators. While their instability may seem like a problem at first, it makes the numerical errors accumulate (and grow) instead of canceling out and ensures that the worst numerical scenario is explored in our tests. @2003GReGr..35.1909T showed that any spherical orbits must lie between the radii of the prograde and retrograde circular equatorial orbits, $$\begin{aligned} r_\mathrm{p} &\equiv 2 M \left\{1 + \cos\left[\frac{2}{3} \cos^{-1}\!\left(-\frac{|a|}{M}\right)\right]\right\}, \\ r_\mathrm{r} &\equiv 2 M \left\{1 + \cos\left[\frac{2}{3} \cos^{-1}\!\left(\frac{|a|}{M}\right)\right]\right\},\end{aligned}$$ which satisfy the inequalities $M \le r_\mathrm{p} \le 3 M \le r_\mathrm{r} \le 4M$. Therefore, our test orbits will be very close to the black hole—an ideal place to probe the performance of the integrators. Although no spherical orbits can pass the event horizon, some of them can pass the ergosphere, $$\begin{aligned} r_\mathrm{e} = M + \sqrt{M^2 - a^2\cos^2\theta}. \label{eq:ergosphere}\end{aligned}$$ Note that we use the spherical KS coordinates $r$ and $\theta$ in the above equations because they are equal to the BL coordinates $\mathcal{r}$ and $\vartheta$. [c|cc|cccc]{}  A & 1 & 1.8 & 1.36 & 12.8304 & 0.9387 & 12.0334\ B & 1 & 2 & 1 & 16 & 0.9717 & 10.8428\ C & 1 & $1+ \sqrt{2}$ & 0 & 22.3137 & 1 & 3.1761\ D & 1 & $1+ \sqrt{3}$ & -1 & 25.8564 & 0.9819 & -3.7138\ E & 1 & 3 & -2 & 27 & 0.9352 & -4.0728\ F & 1 & $1+2\sqrt{2}$ & -6 & 9.6274 & 0.4634 & -4.7450\ We list in Table \[tab:tests\] the parameters we use for our convergence study. The first column shows the label of the tests. We consider only the extreme Kerr black hole, $a = M$, and vary the radii of the spherical orbits. These two input parameters are listed in the second and third columns. With these two parameters, we can compute the normalized angular momentum $\varPhi$ and the normalized Carter’s constant $Q$ to initialize the orbits. For each test case, we also list, in the last two columns, the theoretical maximum altitude, $\max(|\cos\theta|)$, that the orbit can reach and the theoretical change in the azimuthal angle, $\Delta\phi$, within one complete polar oscillation. Although the spherical KS angle $\phi$ is different from the BL angle $\varphi$, their difference depends only on $r$ (see the Appendix). Hence, $\Delta\phi = \phi_1 - \phi_0$ and $\Delta\varphi = \varphi_1 - \varphi_0$ are equal to each other for the same spherical orbit. ![image](L=1_36.eps){width="2.75in"} ![image](L=0.eps){width="2.75in"}\ ![image](L=-2.eps){width="2.75in"} ![image](L=-6.eps){width="2.75in"} ![A zoomed-in view of the north pole of Test A in the top-left panel of Figure \[fig:orbits\]. The solid circles are the actual steps of the 4th-order Runge-Kutta integrator with step size $\Delta\lambda = 1/1024$ and the color lines simply join them together for readability. The first pass is the blue line from top-right, the second pass is the orange line from bottom left, etc.](zoom.eps){width="\columnwidth"} \[fig:zoom\] Figure \[fig:orbits\] shows a representative set of unstable spherical photon orbits that we used. For all panels, the blue lines are the photon orbits; the black solid circles are the (physical) singularities; and the red dashed circles mark the radii of the spherical orbits on the equatorial planes. All orbits start by moving upward from the positive $x$ sides of the red dashed circles. The top-left panel shows Test A. In this case, the radius of the orbit is small enough and the polar momentum is large enough that the orbit oscillates in-and-out across the ergosphere, which [`GRay2`]{} in Cartesian KS has no problem handling. Note that, since the orbit is *unstable* and we do not put any constraints on the integrator in [`GRay2`]{}, the small truncation and round off errors in the integrator get amplified as expected. Because of the accumulation of these numerical errors, the photon eventually leaves the $r = 1.8M$ sphere and flies to infinity. The top-right panel shows Test C, for which the whole orbit is now outside the ergosphere. Although the photon does not have any angular momentum in this case ($\varPhi = 0$), the orbit is tilted on the equatorial plane because of frame dragging. Also, this is the special case for which the photon exactly passes the poles multiple times at $x = y = 0$ and $z \approx \pm 2.41$. In Figure \[fig:zoom\] we zoom into the north polar region of Test C. The solid circles are the actual steps of the 4th-order Runge-Kutta integrator with step size $\Delta\lambda = 1/1024$ and the colored lines simply join them together for clarity. The first pass is the blue line from lower-left; the second pass is the orange line from top-right; etc. Again, the integrator has no problem handling the pole because there is no coordinate singularity in the Cartesian KS coordinates. In the bottom-left panel, we plot the unstable spherical photon orbit for Test E. Although the photon angular momentum is negative, it cancels out the frame dragging effect exactly on the equatorial plane so the photon moves vertically when it passes the equator. In the bottom-right panel, we plot the photon orbit for Test F. This time, the initial angular momentum is negative enough that the photon finally moves in the negative $\phi$ direction. ![image](uu.eps){width="\columnwidth"} ![image](dr.eps){width="\columnwidth"} In Figure \[fig:evolr\], we plot two of the error indicators for Test A. The left panel shows the time evolution of $u^2 = u_\alpha u^\alpha$. It is a constant of motion and should remain zero for photons. In [`GRay2`]{}, because we integrate the geodesic equations without explicitly using any constants of motion, $u^2$ is, in principle, unconstrained in the numerical integration. Therefore, its value is a good measure of the numerical errors. We use nine different step sizes in this test, covering the range $\Delta\lambda = 1/4$, 1/8, ..., 1/512, and 1/1024, which are labeled on the left panel. For all step sizes, $u^2$ is initially of order $10^{-16}$ and increases approximately linearly until the numerical solutions blow up. Clearly, a smaller step size leads to smaller and, hence, smaller error. To understand why the numerical solutions blow up, we plot the changes of the radius of the photon orbit as a function of $\lambda$ in the right panel. Recalling that the tests are for unstable spherical photon orbits, $|\Delta r| \equiv |r(\lambda) - r(0)|$ should grow exponentially, with an amplitude proportional to the perturbation—or to the numerical errors in our case—of the orbits. The coloring of the curves is identical to the left panel, namely, blue is for $\Delta\lambda = 1/4$, orange is for $\Delta\lambda = 1/8$, etc. All the curves grow as expected and blow up at around $|\Delta r| \sim 1$. This is not a coincidence. In fact, for all orbits except the gray and yellow ones, the photons approach the physical singularities as they depart from their spherical orbits. When they get very close to the singularities, the fixed time steps fail to integrate the geodesics, resulting in significant jumps in both $u^2$ and $\Delta r$. For the gray and yellow curves, the photons actually move away from the singularities as they depart from the spherical orbits. Therefore, although the photons approach infinity (linearly), always remain small. ![Results of the convergence study using the integral of motion $u^2$. Here, we plot $\max(u^2)$ for $0 \le \lambda < 64$ for the six test problems we performed, as a function of the step $\Delta\lambda$ in the affine parameter. [`GRay2`]{} converges at 4th order it used the 4th-order Runge-Kutta scheme. The amplitude of the error decreases as $r$ increases except for Test E. This is an artifact of the oscillatory behavior of $u^2$, in this test, shown in the inset.[]{data-label="fig:uu-conv"}](uu-conv.eps){width="\columnwidth"} In Figure \[fig:uu-conv\], we summarize the convergence properties of our algorithm as quantified by the integral of motion $u^2$. We plot $\max(|u^2|)$ for $0\le\lambda\le 64$ for all six test problems. For each test problem, we change $\Delta\lambda$ in the range 1/4, 1/8, 1/16, ..., to 1/1024. As Figure \[fig:evolr\] already showed, the errors decrease as we use smaller step sizes and [`GRay2`]{} converges at 4th order—an expected result because of the 4th-order Runge-Kutta scheme we are using. ![image](z-phi.eps){width="\columnwidth"} ![image](z-zoom.eps){width="\columnwidth"} For a more detailed test of the geodesics, in the left panel of Figure \[fig:z-phi\], we unfold the photon orbits in the azimuthal direction and plot the coordinate $z$ as a function of $\phi$. To avoid overlap between the oscillatory curves, we plot the orbits only for Test B and D. The photon in Test B has positive angular momentum, hence it moves toward positive $\phi$. In the same figure, the photon in Test D has a small negative angular momentum. Although its overall orbit points toward negative $\phi$, the photon changes its direction near the equator because of the stronger frame dragging effect there. ![Results of the convergence study using the maximum polar angles a photon reaches during its orbit. The plot is similar to Figure \[fig:uu-conv\]. It shows that the difference between the fitted peak value of the numerical solution converges to the analytical one at the expected 4-th order.[]{data-label="fig:z-conv"}](z-conv.eps){width="\columnwidth"} The peak values of the $z-$coordinate in the left panel of Figure \[fig:z-phi\] can be calculated analytically, are given in equation (8) in @2003GReGr..35.1909T and listed in the sixth column of our Table \[tab:tests\]. We can use these as a second convergence test of [`GRay2`]{}. (Note that, for convenience, we plot in the following figures the maximum cosine of the polar angle, i.e., $\max\vert\cos\theta\vert$, instead of the maximum $z-$coordinate of each photon orbit). The numerical values at the local maxima depend strongly on the resolution because of sampling effect. This is illustrated in the right panel of Figure \[fig:z-phi\], where we zoom into the first peak of Model B as a function of the affine parameter $\lambda$ with different step sizes. The solid circles are the actual steps of the 4th-order Runge-Kutta scheme. It is clear that the circles are offset from the locations of the peaks. To overcome this sampling effect in polluting our convergence test, we fit a quadratic equation to the largest three points for each resolution. The curves in the right panel of Figure \[fig:z-phi\] correspond to these quadratic equations. From the plot, even the red curve with $\Delta\lambda = 1/32$ is visually indistinguishable from the more accurate curves at this scale. Only at $\Delta\lambda = 1/16$, the green curve starts to deviate from the more accurate curves. We compute the differences between the peak values of the fitted quadratic equations and the analytical values. The results are plotted in Figure \[fig:z-conv\], which shows again a 4th-order convergence rate for all the tests. ![Results of the convergence study using the change in the azimuth of the photon orbit during one complete oscillation in latitude. The plot is similar to Figure \[fig:uu-conv\] and \[fig:z-conv\]. It shows that the difference between the numerical and analytic values of $\Delta\phi$ converges at the expected 4-th order.[]{data-label="fig:dphi-conv"}](dphi-conv.eps){width="\columnwidth"} Our final convergence test uses another result from @2003GReGr..35.1909T, who derived the equation to integrate the change in azimuth for one complete oscillation in latitude. Although there is still a sampling effect due to the change of step size, its resolution is much simpler. We simply use linear interpolation to obtain the root of $z(\phi)$ at the first complete cycle and subtract the initial coordinate $\phi$ from it. This numerical value is then compared with the numerical integration of equation (17) in @2003GReGr..35.1909T. The final result is plotted in Figure \[fig:dphi-conv\], which again shows a 4th-order convergence rate for all the tests. Stable Circular Particle Orbits ------------------------------- ![Vertical epicyclic frequency as a function of orbital radius. We use [`GRay2`]{} to integrate a nearly circular time-like geodesics around a black hole of spin $a = 1$. We introduce a small initial vertical velocity $v_z = 10^{-12}$ to induce vertical oscillation and precession. We then compare the numerical vertical epicyclic frequencies (orange circles) with its analytical expression (blue solid line). The fractional difference between the two curves at a radius $r<1.5$ is less than $10^{-13}$.[]{data-label="fig:OmegaV"}](OmegaV.eps){width="\columnwidth"} Since [`GRay2`]{} makes no specific assumption about geodesics, it can integrate orbits for massive test particles, i.e., time-like geodesics, without any modification of the integrator. As a second set of convergence tests, we integrate stable, nearly circular orbits at different radii around a black hole with a spin of $a = 1$. Nearly circular orbits in general relativity precess because of the deviation of the effective gravitational potential from the Newtonian $1/r$ form. One can describe completely their motion along the three polar coordinates using three independent oscillatory frequencies, one azimuthal (the orbital frequency $\Omega$), one radial (the radial epicyclic frequency $\kappa$), and one vertical (the vertical epicyclic frequency $\Omega_\perp$). The radial epicyclic frequency vanishes at the location of the innermost stable circular orbit and becomes imaginary inside that radius. In the same region, the vertical epicyclic frequency is significantly smaller than the orbital frequency. All three frequencies for a nearly circular orbit can be calculated analytically [see, e.g., @2008ApJ...680.1319S]. In order to force a small precession of the orbital plane of a test particle, we introduce a small vertical velocity, $v_z = 10^{-12}$, to the initial conditions that would otherwise lead to a circular orbit. We then use the numerical orbit to calculate the vertical epicyclic frequency (as measured by an observer at infinity) and compare it to the analytic expression. In order to avoid the numerical effects described in the previous section, we perform linear interpolations between the calculated points in $z(t)$ and measure the time interval between successive maxima. In Figure \[fig:OmegaV\], we overplot the numerical measurement from [`GRay2`]{} on the analytic expression, as a function of the radius of the orbit in BL coordinates. The numerical and analytical results are indistinguishable. In fact, for $r \lesssim 1.5$, the difference between the numerical and analytical results is less than $10^{-13}$. For all other convergence properties related to the test-particle orbits, we found no appreciable difference with the results shown earlier for the null geodesics. Profiling and Benchmarks {#sec:benchmarks} ======================== [ccc|cccccccc]{} Single & BL &  AoS & 43.1 & 6.07 & 2.99 & 2.49 & 0.597 & 0.223 & 0.174 & 0.167\ Single & BL & & 49.4 & 6.01 & 2.98 & 2.89 & 0.597 & 0.219 & 0.170 & 0.175\ Single & KS & AoS & 57.5 & 6.27 & 4.53 & 2.69 & 0.995 & 0.420 & 0.311 & 0.297\ Single & KS & SoA & 54.8 & 6.20 & 4.55 & 2.31 & 0.995 & 0.422 & 0.315 & 0.309\ Double & BL & AoS & 55.4 & 16.6 & — & 70.6 & — & 3.90 & 6.44 & 1.65\ Double & BL & SoA & 56.7 & 16.2 & — & 80.6 & — & 3.90 & 6.44 & 1.66\ Double & KS & AoS & 66.0 & 17.4 & — & 59.1 & — & 2.41 & 6.95 & 1.15\ Double & KS & SoA & 63.1 & 17.5 & — & 59.0 & — & 2.40 & 6.96 & 1.15\ In section \[sec:KS\], we reduced the operation-count of the geodesic equations in the Cartesian KS coordinates by a series of mathematical manipulations. This suggests that, theoretically, solving this optimized form of geodesic equations is not much more expensive than in the BL coordinates. In this section, we look at the actual benchmarks on different devices to support our assertions. The most direct method to compare the performances of the geodesic equations in the Cartesian KS and BL forms is to look at the elapsed time for a single 4th-order Runge-Kutta time step of a single ray, which contains four evaluations of the RHS of the geodesic equations. However, modern accelerators such as GPUs are effectively vector processors. They are only efficient when a large number of threads are executed in parallel. In addition, these accelerators are not general purpose devices. Driving them requires a host—usually a full power CPU—to compile the kernels and send instructions and data to the devices. These overheads can sometimes be quite significant compared to the computations. [ccc|cccccccc]{} Single & BL & AoS & 2.25 & 16.0 & 32.5 & 39.0 & 163 & 436 & 558 & 582\ Single & BL & SoA & 1.95 & 16.0 & 32.3 & 33.3 & 161 & 439 & 566 & 549\ Single & KS & AoS & 1.74 & 16.0 & 22.2 & 37.3 & 101 & 239 & 323 & 338\ Single & KS & SoA & 1.81 & 16.0 & 21.8 & 42.9 & 99.7 & 235 & 315 & 321\ Double & BL & AoS & 4.79 & 16.0 & — & 3.76 & — & 68.1 & 41.2 & 161\ Double & BL & SoA & 4.57 & 16.0 & — & 3.22 & — & 66.5 & 40.3 & 156\ Double & KS & AoS & 4.22 & 16.0 & — & 4.71 & — & 116 & 40.1 & 242\ Double & KS & SoA & 4.44 & 16.0 & — & 4.75 & — & 117 & 40.2 & 243\ We design our benchmarks to reduce the impacts of the above factors. Following the approach found in a typical application, we use [`GRay2`]{} to integrate backwards null geodesics from an image at an initial radius $r = 1024M$ toward a black hole of spin $a = 0.999M$. The integration is performed by executing the same `OpenCL` kernel $\sim 64$ times, until all photons pass by or are close enough to the event horizon, or escape to distances larger than the initial radius. The kernel performs host-to-device communication once right after it starts the 4th-order Runge-Kutta methods with fixed step size 1024 times. It also performs device-to-host communications once more just before it ends. We measure the elapsed time between the instants when the kernel starts and ends. Typically, the elapsed time is reduced a bit after the first few kernel executions due to instruction loading and caching and then saturates toward a steady value. Because the performance of GPUs is sensitive to how the computations are grouped and distributed to different sub-processors (for CPUs they are called “cores”; for nVidia GPUs they care called “multiprocessors”), in order to measure the peak performance, we repeat the above process with five different resolutions of images: $64\times64$, $128\times128$, $256\times256$, $512\times512$, and $1024\times1024$, and allow `lux`’s run time performance tuning algorithm to choose the optimal workgroup size. We compute the elapsed time of a single step for each ray by dividing the shortest measured time by 1024 and the total number of rays. We list the benchmark results in Table \[tab:elapse\]. The first three columns are the precision, the coordinate system, and the data-order used in [`GRay2`]{}, respectively. The numbers in all other columns are in nanoseconds. The 4th column is for a mobile/laptop 4-core CPU and the 5th column is for two 8-core server CPUs (i.e., 16 cores in total). The 6th column is for an Intel integrated graphics chip. The 7th–11th columns are for different nVidia GPUs. All the tested processors, except Intel HD4000, support both single and double precisions. For nVidia’s Tesla M2090, our workstation failed to compile [`GRay2`]{}’s `OpenCL` kernel—this may be due to the limitation of the driver or a bug in the `OpenCL` implementation. For CPUs, the performance of single precision is only slightly (or a factor of a few) faster than for double precision. For the mobile and consumer graphics chips GT650M and GTX780, single precision is significantly faster—up to a factor of $\sim 28$—than double precision. This is no surprise. Single precision operations are good enough for computer graphics and gaming applications. Hence, a large number of transistors on these consumer graphics chips are used to perform single precision operations only. For the HPC specific GPU Tesla K20X and high-end graphics cards nVidia Titan Black, the performance difference between single and double precisions is less dramatic. For single precision, integrating in the KS coordinates is slightly more expensive than in the BL coordinates but the difference is usually less than $\sim 70\%$ (except for K20X). This supports our estimate in section \[sec:KS\] that integrating in KS coordinates has roughly 65% more operations compared to integrating in BL coordinates. For double precision, it is interesting to note that, for most GPUs (except GTX780), KS integration is actually faster than BL integration by up to $\sim 40\%$. There are two main reasons for this. First, BL coordinates require evaluations of trigonometric functions, which are expensive in double precision. Second, the equations in KS are highly symmetric, which allows the compiler to optimize them by hardware-accelerated vector instructions. Finally, there is no significant difference between the two different approaches to ordering the structures and arrays for these benchmarks. This is just an indication that our benchmarks successfully overcome memory access overhead and are able to reveal the actual computation performance. In order to more easily read off the performance gain, we convert the elapsed time in Table \[tab:elapse\] to speedups in Table \[tab:speedup\]. Larger numbers mean higher speedups. We use a single core of the E5-2650 CPU as our baseline. Hence, two E5-2650 CPUs give us 16 cores in the 5th column. For single precision, most of the GPUs are $100-600$ times faster than a single E5-2650 core. For double precision, although the speedups are not as large, the HPC specific Tesla K20X and high-end graphics cards nVidia Titan Black are still two orders of magnitude faster than a single E5-2650 core. Summary {#sec:summary} ======= In this paper, we presented [`GRay2`]{}, a new open source, hardware-accelerated, general relativistic integrator for time-like and null geodesics. By using the Cartesian form of Kerr-Schild coordinates, integration in [`GRay2`]{} avoids all coordinate singularities at the pole and at the horizon that are present in Boyer-Lindquist coordinates. Using a rearranged form of the geodesic equations (see equations \[\[eq:opt0\]\] and \[\[eq:opti\]\]) makes the integrator in Cartesian KS coordinates as efficient as in BL coordinates. In addition, by using the `OpenCL` standard and the HPC framework `lux`, [`GRay2`]{} runs optimally on a wide range of software platforms and hardware devices. We carefully examined the properties of the numerical algorithm and showed that, for a number of different problems with known analytic solutions, it converges at the expected rate. We also report extensive performance benchmarks and show the significant (1-2 orders of magnitude) improvement in the efficiency of [`GRay2`]{} when it is run on GPU architectures. This algorithm is optimally suited for massively parallel integration of geodesics. It can be utilized for computing the transport of radiation through black hole accretion flows, the evolution of a cluster of particles in the vicinity of a black hole, or even direct particle or gyrokinetic simulations of plasmas in Kerr spacetimes. C.K.C., D.P., and F.O. are partially supported by NASA TCAN award NNX 14AB48G and NSF grant AST 1312034; L.M. is supported by NFS GRFP grant DGE 1144085. F.O. gratefully acknowledges a fellowship from the John Simon Guggenheim Memorial Foundation. D.P. acknowledges support from the Radcliffe Institute for Advanced Study at Harvard University. All authors acknowledge the hospitality of the Black Hole Initiative at Harvard University, which is supported by a grant from the John Templeton Foundation. The ray tracing calculations were performed on the `El Gato` GPU cluster, the `tesla` and `merope` GPU workstations at the University of Arizona, and C.K.C.’s laptop. `El Gato` is funded by NSF award 1228509; `tesla` is funded by the theoretical astrophysics program at Arizona; and `merope` is provided by Andras Gaspar with nVidia’s hardware donation. We also thank Ramesh Narayan and Jonathan McKinney for valuable comments and discussions. [^1]: We use script symbols $(\mathcal{t}, \mathcal{r}, \vartheta, \varphi)$ to denote the BL coordinates. Standard italic symbols $(t, r, \theta, \phi)$ and $(t, x, y, z)$ are reserved for the spherical polar and Cartesian forms of the KS coordinates. [^2]: These two methods are not mutually exclusive. In principle, we can combine them to integrate with respect to one coordinate and match a target value in another coordinate or variable. This may be useful for performing, e.g., Monte Carlo scattering simulations. [^3]: Note that we can use the constant of motion $v^\alpha v_\alpha$ to solve for $v^0$ from $v^i$. In such a case, however, $v^\alpha v_\alpha$ can no longer be used to monitor the accuracy of the integration. [^4]: `OpenCL` was originally developed by Apple Inc., and currently maintained by the non-profit Khronos Group as an open standard. See `https://www.khronos.org/opencl`.
--- abstract: 'To ensure the discrete maximum principle or solution positivity in finite volume schemes, diffusive flux is sometimes discretized as a conical combination of finite differences. Such a combination may be impossible to construct along material discontinuities using only cell concentration values. This is often resolved by introducing auxiliary node, edge, or face concentration values that are explicitly interpolated from the surrounding cell concentrations. We propose to discretize the diffusive flux after applying a local piecewise linear coordinate transformation that effectively removes the discontinuities. The resulting scheme does not need any auxiliary concentrations and is therefore remarkably simpler, while being second-order accurate under the assumption that the structure of the domain is locally layered.' address: - 'Jaroslav Černi Institute, Jaroslava Černog 80, 11226 Pinosava, Belgrade, Serbia' - 'University of Belgrade, Faculty of Mining and Geology, [0.3ex-0.36em D]{}ušina 7, 11000 Belgrade, Serbia' author: - 'D. Vidović' - 'M. Dotlić' - 'M. Pušić' - 'B. Pokorni' bibliography: - 'biblioT.bib' title: Piecewise linear transformation in diffusive flux discretization --- diffusion equation ,conical combination ,finite volume method ,maximum principle Introduction ============ Diffusion in an anisotropic discontinuous environment plays a role in various fields of engineering, such as subsurface flows. Steady state diffusion of a solute with concentration $C$ in a bounded domain $\Omega\subset\mathbb R^3$ is modeled by the following boundary problem: $$\begin{aligned} \nabla\cdot\mathbf u=g,\label{eq} \\ \mathbf u=-\mathbb D\nabla C, \\ C=g_\mathrm D \quad \text{on } \Gamma_\mathrm D, \label{dirichlet} \\ \mathbf u\cdot\mathbf n=g_\mathrm N \quad \text{on } \Gamma_\mathrm N, \\ \mathbf u\cdot\mathbf n = \Psi(C-g_\mathrm R) \quad \text{on } \Gamma_\mathrm R, \label{robin}\end{aligned}$$ where $\mathbf u$ is the velocity, $g$ is the volumetric source term, $\mathbf n$ is the unit vector normal to $\partial\Omega$ pointing outward, $\Psi$ is the transfer coefficient, $\Gamma_\mathrm D\cup\Gamma_\mathrm R=\overline{\Gamma_\mathrm D\cup\Gamma_\mathrm R}$, $\Gamma_\mathrm D\cup\Gamma_\mathrm N\cup\Gamma_\mathrm R=\partial\Omega$, $\Gamma_\mathrm D\cup\Gamma_\mathrm R\neq\emptyset$, and $\Gamma_\mathrm D$, $\Gamma_\mathrm N$, and $\Gamma_\mathrm R$ are mutually disjoint. Diffusion tensor $\mathbb D$ is symmetric, positive definite, and piecewise continuous. Connected subsets of $\Omega$ in which $\mathbb D$ is continuous are called [*material zones*]{}, and the interfaces between them are referred to as [*material interfaces*]{}. Presumably, mesh faces coincide with material interfaces, i.e. $\mathbb D$ is continuous within mesh cells. Various numerical schemes are used to solve this problem. Most of them produce non-physical oscillations and negative concentration values in particular cases. Nevertheless, certain schemes are specifically designed to address these issues. One such finite volume scheme appeared in [@Lep05] and was further developed in [@Dan09; @Lip07; @Lip09; @Vas08; @Vid11; @Vid13; @Yua08; @Lip13]. These schemes do not satisfy the maximum principle, but they guarantee that the concentration does not become negative. In addition to the primary concentration unknowns associated with mesh cells, these schemes use auxiliary concentration values located in faces and elsewhere. Face concentration values are easily determined from the continuity, but some kind of interpolation must be used to determine other auxiliary values. The interpolation method presented in [@Vid13] performs this task using piecewise linear interpolation and convex combinations. It satisfies the maximum principle and is second order accurate even when it uses interpolation nodes at the opposite sides of a material discontinuity. Since a simple brute force search for collocation points that form a convex combination can result in a combinatorial explosion, in [@Vid13] we proposed a complex but efficient alternative search algorithm based on Delaunay triangulations. In this paper we deploy the piecewise linear transformation introduced in [@Vid13] directly in the velocity decomposition, without any auxiliary concentration variables. Such a scheme is simpler than [@Vid13] because the complicated construction of convex combinations is avoided. In addition, fluxes over discontinuities do not need special treatment, resulting in further simplification. The paper is organized as follows: in §\[sec:fluxes\] we explain how to use the piecewise linear transformation to obtain a one-side flux approximation. In §\[sec:scheme\] the one-side fluxes are combined in the usual fashion to obtain a two-point scheme. Euler implicit temporal discretization of the time-dependent problem is presented in §\[sec:transient\]. Numerical tests presented in §\[sec:examples\] show that the accuracy and the convergence rate do not change much in comparison to [@Vid13], and thus the simpler scheme should be the method of choice. Moreover, it is shown that a single iteration is sufficient to maintain the second order accuracy in a time-dependent problem. Fluxes and piecewise linear transformation {#sec:fluxes} ========================================== The piecewise linear transformation used here to approximate the flux was constructed in [@Vid13]. The main steps are repeated for convenience. ![It is assumed that the structure of the domain is locally layered.[]{data-label="fig:stack"}](fig1.pdf){height="50mm"} We assume that some neighborhood of point $\mathbf x_0\in\Omega$ consists of layers $\Omega_{-m}$, $\dots$, $\Omega_n$, $m\geq0$, $n\geq0$, with interfaces that are either planes or approximated by planes (see Fig. \[fig:stack\]). The diffusion tensor is allowed to have discontinuities between layers, but it is assumed to be constant or almost constant within each layer, thus $\mathbb D|_{\Omega_i}=\mathbb D_i$. This assumption needs to hold only locally — the diffusion tensor is allowed to vary smoothly within layers on a larger scale. We assume that the concentration changes linearly in each layer $$C(\mathbf x)=C_i+\mathbf G_i\cdot(\mathbf x-\mathbf x_i), \qquad \mathbf x_i\in\overline{\Omega}_i. \label{piecewise}$$ This function must satisfy two conditions: 1. It must be continuous, and 2. The flux through each interface must be continuous. The two conditions eliminate all but four degrees of freedom in the piecewise linear function $C(\mathbf x)$, which may be written as $$C(\mathbf x)=C_0+\mathbf G_0\cdot F(\mathbf x),\label{trans}$$ where $F:\mathbb R^3\rightarrow\mathbb R^3$ is a piecewise linear transformation depending on the diffusion tensors and geometry but not on the concentration. In [@Vid13] we derived explicit formulas to compute $F$. In [@Vid13] we evaluate (\[trans\]) to obtain the concentration values at points. In contrast, in this paper (\[trans\]) is not evaluated at points: instead, the gradient of (\[trans\]) is used to approximate the normal velocity component $\mathbf n_f\cdot\mathbf u$ in some mesh face $f$. We assume that this face belongs to cell $T$ and that $\mathbf n_f$ points outside from this cell. For point $\mathbf x_0$ in (\[piecewise\]) we take the centroid $\mathbf x_T$; thus $C_0=C_T$ in (\[trans\]). We need to determine the three components of $\mathbf G_0$. These are found by solving a linear system consisting of three equations of the following types: $$F(\mathbf x_{\mathcal{T}})\cdot\mathbf G_0 =C_{\mathcal{T}}-C_T \quad \text{if } \mathcal{T} \text{ is a mesh cell,}\label{dirichleteq}$$ $$F(\mathbf x_n)\cdot\mathbf G_0=g_\mathrm D(\mathbf x_n)-C_T \quad \text{if } \mathbf x_n \text{ is a node in } \Gamma_\mathrm D,\label{dirichletnodeeq}$$ $$F(\mathbf x_f)\cdot\mathbf G_0=g_\mathrm D(\mathbf x_f)-C_T \quad \text{if } f \text{ is a face in } \Gamma_\mathrm D,\label{dirichletfeq}$$ $$\mathbf n_f^\mathrm T\mathbb D(\mathbf x_f)\nabla F(\mathbf x_f)\mathbf G_0=-g_\mathrm N(\mathbf x_f) \quad \text{if } f \text{ is a face in } \Gamma_\mathrm N,$$ $$\left(F(\mathbf x_f)+\frac1\Psi\mathbf n_f^\mathrm T\mathbb D(\mathbf x_f)\nabla F(\mathbf x_f)\right)\cdot\mathbf G_0=g_\mathrm R(\mathbf x_f)-C_T \quad \text{if } f \text{ is a face in } \Gamma_\mathrm R.\label{robineq}$$ Point $\mathbf x_f$ is the centroid of face $f$. The last equation follows from $$-\mathbf n_f^\mathrm T\mathbb D(\mathbf x_f)\nabla F(\mathbf x_f)\mathbf G_0=\Psi(C_T+\mathbf G_0\cdot F(\mathbf x_f)-g_\mathrm R(\mathbf x_f))$$ and may be written in a form similar to (\[dirichletnodeeq\]) and (\[dirichletfeq\]): $$F\left(\mathbf x_f+\frac1\Psi \mathbb D(\mathbf x_f)\mathbf n_f\right)\cdot\mathbf G_0=g_\mathrm R(\mathbf x_f)-C_T \quad \text{if } f \text{ is a face in } \Gamma_\mathrm R.\label{robineq1}$$ Three equations of the form (\[dirichleteq\])–(\[robineq\]) constitute a linear system $$M\mathbf G_0=\mathbf r.\label{system}$$ If this system has a unique solution $\mathbf G_0=M^{-1}\mathbf r$, then the normal velocity component at face $f$ can be approximated as $$\mathbf n_f\cdot\mathbf u\approx-\mathbf n_f^\mathrm T\mathbb D(\mathbf x_f)M^{-1}\mathbf r=-\boldsymbol\alpha\cdot\mathbf r\label{uapprox}$$ because $f\subset\overline\Omega_0$ and $\nabla F\equiv I$ in $\Omega_0$. Coefficient vector $\boldsymbol\alpha$ is determined by solving $$M^\mathrm T\boldsymbol\alpha=\mathbb D(\mathbf x_f)\mathbf n_f.$$ Equations that make up the system (\[system\]) are chosen in such a way that the obtained coefficients $\boldsymbol\alpha$ are non-negative. The search for these equations is performed by testing all combinations of neighboring cells and boundary conditions belonging to the [*candidate set*]{}, in the same way that the brute force strategy in [@Vid13] works. The candidate set is initialized with cell $T$, cells that share a face with $T$, boundary faces of $T$ and Dirichlet boundary nodes of $T$. If equations leading to non-negative $\boldsymbol\alpha$ are not found, the candidate set is extended by adding all neighboring cells, boundary faces and Dirichlet boundary nodes of the current member cells, until non-negative $\boldsymbol\alpha$ is obtained. If matrix $M$ in (\[system\]) is ill-conditioned, this typically results in large coefficients $\boldsymbol\alpha$. The resulting interpolation is inaccurate, so the combination is discarded if some $\alpha_i\|\mathbf t_i\|/\|\mathbb D\mathbf n_f\|$ is larger than a suitably chosen value $A_\text{max}$ discussed in the next section, where $\mathbf t_i$ is the vector multiplying $\mathbf G_0$ in the corresponding equation (\[dirichleteq\])–(\[robineq\]). If more than one conical combination is found, we apply a strategy similar to the one used in [@Dan09]: starting with combinations containing the cell that shares the face $f$ with the cell $T$, we choose the first encountered conical combination such that $\max_i(\alpha_i m_i)/\|\mathbb D(\mathbf x_f)\mathbf n_f\|_2\leq1$, where $m_i$ is the 2-norm of the $i^{\text{th}}$ row of matrix $M$. If no such combination is encountered, then we choose the one for which this ratio is minimal. Alternative: harmonic averaging points -------------------------------------- Let points $\mathbf x_1$ and $\mathbf x_2$ be separated by a material interface. The same two conditions leading to transformation $F$ were used in [@Age09] to compute the concentration at a suitably chosen point at the interface as a convex combination of concentration values at $\mathbf x_1$ and $\mathbf x_2$. Such so-called [*harmonic averaging point*]{} can be used in (\[dirichleteq\]) in the place of $\mathcal{T}$, as an alternative to transformation $F$. The two approaches are equivalent under certain conditions: - Either $\mathbf x_1$ or $\mathbf x_2$ is equal to $\mathbf x_T$; - The concentration at the harmonic averaging point is evaluated implicitly; - Points $\mathbf x_1$ and $\mathbf x_2$ are in neighboring layers; - Neumann and Robin boundary conditions in faces not belonging to $\overline\Omega_0$ are not used in (\[system\]). The notion of harmonic averaging points could be extended to interpolate over multiple layers and to deploy boundary conditions in a similar way this is done here. Non-linear two-point scheme {#sec:scheme} =========================== As usual, finite volume discretization is performed applying the divergence theorem to the integral of (\[eq\]) over the mesh cell $T$: $$\oint_T \mathbf u\cdot\mathbf n{\rm d}S=\sum_f\chi_{T,f}u_f=\int_Tg{\rm d}T, \qquad u_f=\int_f\mathbf u\cdot\mathbf n{\rm d}S,\label{fv}$$ where $\mathbf n_f$ is the fixed unit normal vector associated with face $f$, and $\chi_{T,f}=1$ if $\mathbf n_f$ points outside of $T$, or $\chi_{T,f}=-1$ otherwise. The two-point scheme is constructed in a similar way as in [@Dan09; @Vid11], but simpler because the discontinuities do not need to be treated separately. Approximation (\[uapprox\]) can be written as $$\mathbf n_f\cdot\mathbf u\approx\sum_i\alpha_i(C_T-C_i)+\sum_j\alpha_jg_\mathrm N(\mathbf x_j).\label{uapprox1}$$ The convex combination of such expressions that correspond to the cells $T^+$ and $T^-$ sharing the internal face $f$ $$\frac{u_f}{|f|}\approx\mu_+\left(\sum_i\alpha_i^+(C_+-C^+_i)+\sum_j\alpha_j^+g_\mathrm N(\mathbf x_j^+)\right)-\mu_-\left(\sum_k\alpha_k^-(C_--C^-_i)+\sum_l\alpha_l^-g_\mathrm N(\mathbf x_l^-)\right)\label{theflux}$$ is chosen in such a way that the contributions of concentration values other than $C_+$ and $C_-$ cancel, together with the contributions of inflow Neumann faces. Here $C_\pm$ denotes the concentration in cell $T^\pm$. Thus weights $\mu_+$ and $\mu_-$ are determined such that $$\mu_++\mu_-=1, \qquad -\mu_+d_++\mu_-d_-=0,\label{mus}$$ where $$d_\pm=\sum_{\substack{i \\ \mathbf x_i\neq\mathbf x_\mp}}\alpha_iC_i^\pm-\sum_{\substack{j \\ g_\mathrm N(\mathbf x_j^\pm)<0}}\alpha_j^\pm g_\mathrm N(\mathbf x_j^\pm).\label{des}$$ If $d_++d_-\neq0$, then $$\mu_+=\frac{d_-}{d_++d_-}, \qquad \mu_-=\frac{d_+}{d_++d_-}.\label{mu}$$ Otherwise we take $\mu_\pm=\tfrac12$. In this way flux approximation (\[theflux\]) reduces to a two-point formula $$u_f\approx M_f^+C_+-M_f^-C_-+r_f,\label{twopoint}$$ where $$M_f^\pm=|f|\left(\mu_\pm\sum_i\alpha_i^\pm+\mu_\mp\sum_{\substack{k \\ \mathbf x_k=\mathbf x_\pm}}\alpha_k^\mp\right), \qquad r_f=|f|\left(\mu_+\sum_j\alpha_j^+g_\mathrm N(\mathbf x_j^+)-\mu_-\sum_l\alpha_l^-g_\mathrm N(\mathbf x_l^-)\right).\label{mdef}$$ Fluxes through Dirichlet faces are discretized in the same way as in [@Dan09; @Vid11], treating the boundary face as a flat cell of zero volume. The same technique is applied to Robin faces by treating the Robin condition at $\mathbf x_f$ as a Dirichlet condition specified at $\mathbf x_f+\tfrac1\Psi \mathbb D(\mathbf x_f)\mathbf n_f$ (see (\[robineq1\])). Experiments have shown that it is better to accept relatively large coefficients $\boldsymbol\alpha$ than to extend the candidate set, which also increases the error due to the increased distances between points. From (\[des\]) and (\[mu\]) we see that large coefficients in a one-side flux cause the weight assigned to this flux to be small and the weight assigned to the opposite flux to be large. In this way the influence of inaccurate interpolation vanishes. Owing to this fortunate circumstance, we only discard combinations if $\alpha_i\|\mathbf t_i\|/\|\mathbb D\mathbf n_f\|$ is larger than $A_\text{max}=1000$ for some $i$. Finite volume discretization with fluxes as in (\[twopoint\]) results in a non-linear system $A(\mathbf C)\mathbf C=\mathbf b(\mathbf C)$, where vector $\mathbf C$ contains concentration values in all cells. Starting with some $\mathbf C^0$, consecutive iterations are computed by solving the linearized system $A(\mathbf C^n)\mathbf C^{n+1}=\mathbf b(\mathbf C^n)$ until the convergence criterion $$\frac{\|A(\mathbf C^n)\mathbf C^n-\mathbf b(\mathbf C^n)\|}{\|\mathbf b(\mathbf C^n)\|}<\varepsilon\label{convergencecriterion}$$ is met. If $\mathbf C^n$ is non-negative, then $A(\mathbf C^n)$ is an M-matrix because coefficients $M^\pm_f$ in (\[twopoint\]) are non-negative, which follows from the non-negativity of $d_\pm$, (\[mu\]), and (\[mdef\]). In the absence of Neumann outflow boundaries and if $g_\mathrm D\geq0$ and $g_\mathrm R\geq0$, the right-hand side is non-negative. Therefore $\mathbf C^{n+1}$ is non-negative as well, and by induction the concentration is non-negative in each iteration. Time-dependent problem {#sec:transient} ====================== A transient problem is obtained by adding a time derivative to the continuity equation (\[eq\]): $$\frac{\partial C}{\partial t}+\nabla\cdot\mathbf u=g$$ when $t>t_0$. For this equation to have a unique solution, in addition to boundary conditions (\[dirichlet\])-(\[robin\]), one also needs to specify an initial condition $C|_{t=t_0}=C_0$. Condition $\Gamma_\mathrm D\cup\Gamma_\mathrm R\neq\emptyset$ is no longer necessary. Using Euler implicit discretization, we obtain $$|T|\frac{C_T^{n+1}-C_T^n}{\Delta t^{n+1}}+\sum_f\chi_{T,f}\left(M_f^{+,n+1}C_+^{n+1}- M_f^{-,n+1}C_-^{n+1}+r_f^{n+1}\right)=\int_Tg^{n+1}{\rm d}T\label{nonlintransient}$$ in the place of (\[fv\]), where $\Delta t$ is the time step and the last superscript indicates the time level. At each time level this non-linear system is solved using Picard linearization $$|T|\frac{C_T^{n+1,m+1}-C_T^n}{\Delta t^{n+1}}+\sum_f\chi_{T,f}\left(M_f^{+,n+1,m}C_+^{n+1,m+1}- M_f^{-,n+1,m}C_-^{n+1,m+1}+r_f^{n+1,m}\right) =\int_Tg^{n+1,m}{\rm d}T, \label{picardtransient}$$ where the superscript after the time level indicates the iteration number. To preserve the second-order accuracy, one must take $\Delta t=\mathcal O(h^2)$, where mesh parameter $h$ is proportional to the longest edge length. Inspired by [@Kav01; @Kav02a; @Kav02b], for the initial approximation we take $$C_T^{n+1,0}=\max (C_T^n+\Delta t^{n+1} \dot C_T^n, 0), \qquad \dot C_T^n = \frac{C_T^n-C_T^{n-1}}{\Delta t^n}.\label{extrapolation}$$ Such an initial value violates the mass conservation property, but it is used only to compute coefficients $M_f^\pm$ in the first iteration. The conservation and positivity of subsequent iterations follow from the matrix structure. Moreover, since the error of the initial value is already $\mathcal O(\Delta t^2)$, a single iteration in each time step is enough to maintain the $\mathcal O(\Delta t)=\mathcal O(h^2)$ accuracy at the end of the time stepping process. Extrapolation (\[extrapolation\]) cannot be used in the first time step because $\dot C_T^0$ is not known. It is suggested in [@Kav01; @Kav02a; @Kav02b] to calculate $\dot C_T^0$ from the truncation error obtained by substituting the initial solution into the stationary problem. However, finite volume schemes are inconsistent in the finite difference sense so $\dot C_T^0$ obtained in this way is inaccurate. One can take $C^{1,0}_T=C^0_T$ instead. If a single iteration is performed, the error $\mathcal O(\Delta t)$ is introduced only at the first time step while all other steps add $\mathcal O(\Delta t^2)$, so the total error is still $\mathcal O(\Delta t)$. Alternatively, one may compute a few iterations in the first time step. Examples {#sec:examples} ======== The following examples demonstrate that, on average, the proposed scheme is not less accurate and does not require more iterations than the more complicated version presented in [@Vid13]. We report the numbers of iterations $n_6$ and $n_9$ until convergence is achieved with $\varepsilon=10^{-6}$ and $\varepsilon=10^{-9}$, respectively. Linear systems are solved using the BiCGStab solver with an accuracy of $10^{-12}$. The following norms are used to quantify the concentration and flux errors: $$\varepsilon^C_2=\left[\frac{\sum_T(C(\mathbf x_T)-C_T)^2|T|}{\sum_T(C(\mathbf x_T))^2|T|}\right]^{1/2}, \quad \varepsilon_\text{max}^C=\frac{\max_T|C(\mathbf x_T)-C_T|}{\left[\sum_T(C(\mathbf x_T))^2|T|/\sum_T|T|\right]^{1/2}},\nonumber$$ $$\varepsilon^{\mathbf u}_2 = \left[\frac{\sum_f(\mathbf u(\mathbf x_f)-u_f/|f|)^2|f|}{\sum_f(\mathbf u(\mathbf x_f)\cdot\mathbf n_f)^2|f|}\right]^{1/2}, \quad \varepsilon^{\mathbf u}_\text{max}=\frac{\max_f|\mathbf u(\mathbf x_f)\cdot\mathbf n_f-u_f/|f||}{\left[\sum_f(\mathbf u(\mathbf x_f)\cdot\mathbf n_f)^2|f|/\sum_f|f|\right]^{1/2}},$$ where $C_T$ is the computed concentration in cell $T$, $u_f$ is the computed flux through face $f$, and $C(\mathbf x)$ and $\mathbf u(\mathbf x)$ are the exact concentration and velocity functions, respectively. \[ex:discontiniousdanilov\] This is a modified version of an example that appeared in [@Dan09]. The domain is the unit cube. The diffusion tensor is $$\mathbb D=\left\{\begin{array}{c}\mathbb D_1 \quad \text {if } x<0.5 \\ \mathbb D_2 \quad \text {otherwise}\end{array}\right., \quad \mathbb D_1=\left[\begin{array}{c c c}3&1&0\\1&3&0\\0&0&1\end{array}\right], \quad \mathbb D_2=\left[\begin{array}{c c c}10&3&0\\3&1&0\\0&0&1\end{array}\right].$$ The source term is chosen such that the exact solution is $$C=\left\{\begin{array}{c}1-2y^2+4xy+2y+6x \quad \text{if } x<0.5,\\3.5-2y^2+2xy+x+3y \quad \text{otherwise.}\end{array}\right.\label{danilovSolution}$$ The exact concentration is set at $z=0$ and $z=1$, the exact flux is specified at $y=0$ and $y=1$, and the Robin condition is prescribed at $x=0$ and $x=1$. We take $g_\mathrm R|_{x=0}\equiv0$ and $g_\mathrm R|_{x=1}\equiv10$, while $\Psi$ is chosen so (\[danilovSolution\]) is the solution. This problem was solved on three types of meshes: cubic, triangular prismatic, and tetrahedral. The errors and iteration numbers given in Table \[disDanErr\] show that there is little difference in the accuracy and convergence rate, compared with the previous version of the scheme [@Vid13]. [|c|c c c c|c c c c|]{} & &\ $h$ & 1/10 & 1/20 & 1/40 & 1/80 & 1/10 & 1/20 & 1/40 & 1/80\ \ $\varepsilon_2^c$ & 7.64e-4 & 1.98e-4 & 5.09e-5 & 1.31e-5 & 7.15e-4 & 1.91e-4 & 4.97e-5 & 1.25e-5\ $\varepsilon_\text{max}^c$ & 3.12e-3 & 9.88e-4 & 3.08e-4 & 9.43e-5 & 3.00e-3 & 1.44e-3 & 5.90e-4 & 2.19e-4\ $\varepsilon_2^{\mathbf u}$ & 4.53e-3 & 1.89e-3 & 7.40e-4 & 2.76e-4 & 3.89e-3 & 1.48e-3 & 5.46e-4 & 1.98e-4\ $\varepsilon_\text{max}^{\mathbf u}$ & 2.88e-2 & 1.71e-2 & 9.18e-3 & 4.61e-3 & 1.85e-2 & 9.94e-3 & 5.07e-3 & 2.60e-3\ $n_6$ & 28 & 52 & 81 & 107 & 38 & 62 & 87 & 115\ $n_9$ & 45 & 84 & 143 & 205 & 61 & 100 & 154 & 220\ \ $\varepsilon_2^c$ & 4.78e-4 & 1.05e-4 & 2.48e-5 & 5.64e-6 & 4.41e-4 & 9.74e-5 & 2.31e-5 & 5.49e-6\ $\varepsilon_\text{max}^c$ & 4.87e-3 & 2.25e-3 & 6.68e-4 & 2.95e-4 & 3.92e-3 & 1.68e-3 & 5.27e-4 & 2.17e-4\ $\varepsilon_2^{\mathbf u}$ & 4.04e-3 & 2.21e-3 & 1.15e-3 & 5.84e-6 & 3.92e-3 & 2.16e-3 & 1.14e-3 & 5.83e-4\ $\varepsilon_\text{max}^{\mathbf u}$ & 4.31e-2 & 2.35e-2 & 1.38e-2 & 7.74e-3 & 2.81e-2 & 1.55e-2 & 8.10e-3 & 4.50e-3\ $n_6$ & 39 & 65 & 88 & 110 & 50 & 72 & 95 & 117\ $n_9$ & 63 & 112 & 164 & 230 & 82 & 126 & 179 & 240\ \ $\varepsilon_2^c$ & 5.70e-4 & 1.41e-4 & 2.94e-5 & 7.45e-6 & 6.83e-4 & 1.65e-4 & 3.20e-5 & 7.42e-6\ $\varepsilon_\text{max}^c$ & 6.15e-3 & 2.32e-3 & 7.59e-4 & 2.61e-4 & 8.96e-3 & 4.30e-3 & 1.02e-3 & 3.68e-4\ $\varepsilon_2^{\mathbf u}$ & 5.15e-3 & 2.37e-3 & 1.15e-3 & 5.59e-4 & 5.06e-3 & 2.37e-3 & 1.15e-3 & 5.58e-4\ $\varepsilon_\text{max}^{\mathbf u}$ & 3.47e-2 & 1.49e-2 & 8.18e-3 & 2.61e-3 & 3.15e-2 & 2.09e-2 & 8.17e-3 & 6.03e-3\ $n_6$ & 96 & 159 & 220 & 279 & 97 & 158 & 226 & 290\ $n_9$ & 147 & 260 & 384 & 534 & 150 & 259 & 393 & 549\ \[ex:sinuslepotier\] This is a modified version of an example used in [@Vid13]. The domain is again a unit cube. The diffusion tensor is $$\mathbb D=(1+0.25\cos (x+y-z))\cdot\left\{\begin{array}{c}\mathbb D_1 \quad \text{if } x<0.5 \\ \mathbb D_2 \quad \text{otherwise,}\end{array}\right.$$ $$\mathbb D_1=\left[\begin{array}{c c c}1&0&0\\0&1&0\\0&0&1\end{array}\right], \quad \mathbb D_2=\left[\begin{array}{c c c}100&0&0\\0&0.01&0\\0&0&1\end{array}\right].$$ The source term is chosen such that the exact solution is $$C=\left\{\begin{array}{c}\cos(\pi x)\sin(\pi y)+1 \quad \text{if } x<0.5,\\0.01 \cos(\pi x)\sin(\pi y)+1 \quad \text{otherwise.}\end{array}\right.$$ The exact concentration is prescribed at $x=1$, $y=0$, and $y=1$, and the exact flux is specified elsewhere. Table \[sinLepErr\] shows the errors and iteration numbers. [|c|c c c c|c c c c|]{} & &\ $h$ & 1/10 & 1/20 & 1/40 & 1/80 & 1/10 & 1/20 & 1/40 & 1/80\ \ $\varepsilon_2^c$ & 2.39e-3 & 5.98e-4 & 1.49e-4 & 3.73e-5 & 5.28e-3 & 1.04e-3 & 2.17e-4 & 5.34e-5\ $\varepsilon_\text{max}^c$ & 6.66e-3 & 1.69e-3 & 4.25e-4 & 1.06e-4 & 3.22e-2 & 8.07e-3 & 2.94e-3 & 7.78e-4\ $\varepsilon_2^{\mathbf u}$ & 4.21e-3 & 1.05e-3 & 2.62e-4 & 6.56e-5 & 3.14e-2 & 1.08e-2 & 3.56e-3 & 1.46e-3\ $\varepsilon_\text{max}^{\mathbf u}$ & 1.28e-2 & 3.20e-3 & 7.99e-4 & 1.99e-4 & 2.79e-1 & 1.60e-1 & 8.50e-2 & 4.13e-2\ $n_6$ & 1 & 1 & 1 & 1 & 2 & 4 & 4 & 4\ $n_9$ & 1 & 1 & 1 & 1 & 4 & 6 & 7 & 7\ \ $\varepsilon_2^c$ & 1.43e-3 & 3.58e-4 & 8.92e-5 & 2.12e-5 & 3.53e-3 & 8.05e-4 & 1.51e-4 & 3.13e-5\ $\varepsilon_\text{max}^c$ & 5.53e-3 & 1.78e-3 & 4.18e-4 & 1.07e-4 & 2.36e-2 & 8.21e-3 & 2.05e-3 & 5.19e-4\ $\varepsilon_2^{\mathbf u}$ & 4.90e-3 & 1.26e-3 & 3.96e-4 & 1.92e-4 & 2.80e-2 & 1.07e-2 & 3.63e-3 & 1.31e-3\ $\varepsilon_\text{max}^{\mathbf u}$ & 5.77e-2 & 2.03e-2 & 1.22e-2 & 8.82e-3 & 3.79e-1 & 2.06e-1 & 1.02e-1 & 5.36e-2\ $n_6$ & 6 & 8 & 11 & 17 & 6 & 8 & 11 & 16\ $n_9$ & 10 & 15 & 22 & 36 & 10 & 14 & 21 & 35\ \ $\varepsilon_2^c$ & 1.03e-3 & 1.74e-4 & 3.45e-5 & 7.08e-6 & 1.02e-3 & 1.74e-4 & 3.48e-5 & 7.10e-6\ $\varepsilon_\text{max}^c$ & 1.21e-2 & 1.50e-3 & 3.92e-4 & 9.44e-5 & 1.23e-2 & 1.50e-3 & 4.00e-4 & 1.02e-4\ $\varepsilon_2^{\mathbf u}$ & 2.49e-2 & 8.98e-3 & 4.19e-3 & 1.99e-3 & 2.31e-2 & 8.56e-3 & 4.09e-3 & 1.96e-3\ $\varepsilon_\text{max}^{\mathbf u}$ & 2.02e-1 & 1.49e-1 & 6.59e-2 & 4.21e-2 & 2.03e-1 & 1.01e-1 & 4.62e-2 & 2.53e-2\ $n_6$ & 54 & 88 & 147 & 231 & 56 & 89 & 154 & 239\ $n_9$ & 91 & 164 & 313 & 571 & 94 & 166 & 323 & 579\ In this example the scheme reduces to a linear one on structured grids, therefore convergence is achieved in a single iteration. This was not the case with scheme [@Vid13], owing to the treatment of Dirichlet boundary conditions. The change also results in smaller errors on structured and prismatic grids. This is only possible because the anisotropy axes are aligned with the grid, which was not the case in the previous example. The present example was also solved on Kershaw meshes [@Ker81]. The errors and iteration numbers are shown in Table \[lepRobKerErr\]. No more than one candidate set extension was necessary to find the cells and boundary conditions that give non-negative coefficients $\boldsymbol\alpha$ in (\[uapprox\]), in comparison to [@Vid13] where up to 11 extension rounds were necessary to find convex combinations on the same meshes. As explained in [@Vid13], to find a convex combination for a mesh node that does not belong to the convex hull of nearby collocation points, a large number of extensions is necessary. Kershaw grids contain such nodes. On the other hand, the new scheme does not use interpolated point values so it does not suffer from this problem. -------------------------------------- --------- --------- --------- --------- --------- $h$ 1/8 1/16 1/32 1/64 1/128 $\varepsilon_2^c$ 2.38e-2 2.08e-2 1.44e-2 6.62e-3 2.21e-3 $\varepsilon_\text{max}^c$ 1.21e-1 9.32e-2 6.06e-2 2.85e-2 9.21e-3 $\varepsilon_2^{\mathbf u}$ 3.67e-1 2.55e-1 1.52e-1 7.56e-2 2.95e-2 $\varepsilon_\text{max}^{\mathbf u}$ 2.52 1.92 9.04e-1 5.58e-1 3.21e-1 $n_6$ 59 118 203 381 635 $n_9$ 91 186 331 670 1239 $\varepsilon_2^c$ 2.11e-1 2.07e-2 1.44e-2 6.62e-3 2.21e-3 $\varepsilon_\text{max}^c$ 7.91e-1 9.38e-2 6.06e-2 2.84e-2 9.21e-3 $\varepsilon_2^{\mathbf u}$ 5.51e-1 2.41e-1 1.52e-1 7.51e-2 2.93e-2 $\varepsilon_\text{max}^{\mathbf u}$ 3.90 1.42 8.37e-1 5.34e-1 3.13e-1 $n_6$ 63 118 206 384 636 $n_9$ 97 199 335 674 1241 -------------------------------------- --------- --------- --------- --------- --------- : Errors and iteration numbers on Kershaw grids in Example \[ex:sinuslepotier\].[]{data-label="lepRobKerErr"} The diffusion tensor is $$\mathbb D=\left\{\begin{array}{c c }\mathbb D_1 & \text{if } x<0.1 \\ \mathbb D_2 & \text{if } x<0.95-0.75y \\ \mathbb D_3 & \text{otherwise,} \end{array}\right.$$ $$\mathbb D_1 = \left[\begin{array}{c c c}3&1&0\\1&3&0\\0&0&1\end{array}\right], \quad \mathbb D_2 = \left[\begin{array}{c c c}10&3&0\\3&1&0\\0&0&1\end{array}\right], \quad \mathbb D_3 = \left[\begin{array}{c c c}1.2&1&0\\1&2&0\\0&0&1\end{array}\right].$$ The exact concentration is a piecewise linear function $$C=\left\{\begin{array}{c c}1+6x+4y & \text{if } x<0.1 \\ 1.5+7x+4y & \text{if } x<0.95-0.75y \\ -2.3+5x+7y & \text{otherwise.}\end{array}\right.$$ The exact concentration is set at $z=0$ and $z=1$, the exact flux is specified at $y=0$ and $y=1$, while the Robin condition is set elsewhere with $g_\text R|_{x=0}\equiv0$, $g_\text R|_{x=1}\equiv11$, and with $\Psi$ chosen accordingly. The problem was solved on the unit cube using tetrahedral, prismatic and hexahedral grids of various sizes. The obtained concentration was exact in all cases. \[noFlow\] This example was taken from [@Dan09]. The domain is a unit cube with two holes $[0,1]^3\backslash S_1\backslash S_2$, $S_1=[3/11,4/11]\times[5/11,6/11]\times[0,1]$, $S_2=[7/11,8/11]\times[5/11,6/11]\times[0,1]$. The diffusion tensor is $$\mathbb D=R_z(-\theta)\diag (1,10^{-3},1) R_z(\theta),\label{rotatedTensor}$$ where $R_z(\theta)$ is the matrix of rotation by angle $\theta=67.5\,^{\circ}$ around the $z$ axis. Concentration 0 is set in $S_1$, concentration 1 is set in $S_2$, and the no-flow condition is specified at the remaining boundary. The example was solved on a range of hexahedral grids. The minimal and maximal concentrations and the iteration numbers are shown in Table \[noFlowTable\]. This example demonstrates that the scheme preserves positivity. The maximum principle violation is somewhat lesser than in [@Dan09], owing to the implicit treatment of Neumann boundary conditions. ---------------- --------- --------- --------- --------- --------- --------- --------- --------- $h$ 1/11 1/22 1/44 1/88 1/11 1/22 1/44 1/88 $C_\text{min}$ 1.50e-2 3.55e-3 5.66e-4 5.36e-5 1.47e-2 3.51e-3 5.61e-4 5.33e-5 $C_\text{max}$ 1.47 1.35 1.08 1.02 1.56 1.38 1.08 1.02 $n_6$ 119 360 816 1063 129 346 813 1063 $n_9$ 230 788 1998 3136 240 777 2001 3138 ---------------- --------- --------- --------- --------- --------- --------- --------- --------- : Concentration span and iteration numbers in Example \[noFlow\].[]{data-label="noFlowTable"} \[noFlowDiscontinuous\] Example \[noFlow\] was modified to demonstrate that positivity is preserved in the presence of discontinuities. The domain is $[0,1]^3\backslash S_1\backslash S_2$, $S_1=[0.2,0.4]\times[0.4,0.6]\times[0,1]$, $S_2=[0.6,0.8]\times[0.4,0.6]\times[0,1]$. The diffusion tensor is given by (\[rotatedTensor\]), where $\theta=22.5\,^{\circ}$ if $(x<0.5\land y<0.5)\lor(x>0.5\land y>0.5)$ and $\theta=67.5\,^{\circ}$ otherwise. The boundary conditions are the same as in the previous example. The example was solved on a range of hexahedral grids. The minimum and maximum concentration values, as well as the iteration numbers, are given in Table \[noFlowDiscontinuousTable\]. Positivity is preserved but, unlike the previous example, it is not evident that the maximum principle violation reduces as the grid is refined. ---------------- --------- --------- --------- --------- --------- --------- --------- --------- $h$ 1/10 1/20 1/40 1/80 1/10 1/20 1/40 1/80 $C_\text{min}$ 2.57e-3 1.06e-3 1.18e-4 7.47e-6 1.20e-3 1.05e-3 1.15e-4 7.23e-6 $C_\text{max}$ 1.17 1.53 1.72 1.33 1.06 1.49 1.731 1.34 $n_6$ 76 188 461 861 103 213 484 877 $n_9$ 140 389 1090 2457 185 432 1133 2475 ---------------- --------- --------- --------- --------- --------- --------- --------- --------- : Concentration span and iteration numbers in Example \[noFlowDiscontinuous\].[]{data-label="noFlowDiscontinuousTable"} \[ex:discontiniousdanilovtransient\] We modified Example \[ex:discontiniousdanilov\] to test the behavior of the new scheme in the transient case. The exact solution is $$C=e^t\left\{\begin{array}{c}1-2y^2+4xy+2y+6x \quad \text{if } x<0.5,\\3.5-2y^2+2xy+x+3y \quad \text{otherwise.}\end{array}\right.$$ The initial solution at $t=0$ and the source term are chosen accordingly. The boundary conditions and the diffusion tensor are as in Example \[ex:discontiniousdanilov\]. The problem was solved on the same series of tetrahedral grids as in Example \[ex:discontiniousdanilov\] using one, two, and three iterations per time step $\Delta t=1.6h^2$. Such a time step was chosen because it generates an error of the same order of magnitude as the spatial discretization error. For comparison, on the coarser grids the problem was also solved using $\Delta t=10^{-5}$ and as many iterations as necessary to reach $\varepsilon = 10^{-9}$. The errors at $t=2$ given in Table \[disDanTranErr\] show that a single iteration is sufficient to maintain second order accuracy, although the maximal flux error is significantly reduced if two or three iterations are performed. $h$ 1/10 1/20 1/40 1/80 1/10 1/20 1/40 1/80 -------------------------------------- --------- --------- --------- ---------- --------- --------- --------- --------- $\varepsilon_2^c$ 2.88e-3 6.48e-4 1.36e-4 3.23e-5 2.34e-3 4.77e-4 8.87e-5 2.20e-5 $\varepsilon_\text{max}^c$ 2.88e-2 7.70e-3 1.93e-3 5.73e-4 2.19e-3 6.39e-3 1.71e-3 5.39e-4 $\varepsilon_2^{\mathbf u}$ 7.71e-3 2.98e-3 1.32e-3 6.02e-4 5.59e-3 2.42e-3 1.17e-3 5.64e-4 $\varepsilon_\text{max}^{\mathbf u}$ 7.46e-2 6.19e-2 3.23e-2 1.93e-02 5.27e-2 3.05e-2 9.84e-3 8.44e-3 $\varepsilon_2^c$ 2.41e-3 5.34e-4 1.07e-4 2.60e-5 1.65e-3 3.85e-4 $\varepsilon_\text{max}^c$ 2.09e-2 6.29e-3 1.71e-3 5.40e-4 1.72e-2 5.47e-3 $\varepsilon_2^{\mathbf u}$ 5.23e-3 2.37e-3 1.15e-3 5.61e-4 5.12e-3 2.36e-3 $\varepsilon_\text{max}^{\mathbf u}$ 3.72e-2 1.44e-2 8.24e-3 5.60e-3 3.60e-2 1.45e-2 : Concentration errors in Example \[ex:discontiniousdanilovtransient\]. \[disDanTranErr\] Discussion and conclusion ========================= We have demonstrated that there is little difference in accuracy and convergence rate between the current scheme and [@Vid13]. They both preserve solution positivity. Since the current scheme is much simpler, it should be used instead of [@Vid13]. To construct conical combinations (\[uapprox\]), a more sophisticated search strategy could be devised, such as the one presented in [@Vid13]. However, the need to do so is lesser than in the case of convex combinations because the computational effort is smaller. There are two reasons for this. The first reason is that we have not encountered a case where the cells and boundary conditions used to construct system (\[system\]) were more than three cells away. For comparison, up to 11 candidate set extension rounds were necessary to represent Kershaw grid node values as convex combinations in [@Vid13]. Second, three vectors suffice to construct a conical combination, while in general four points are necessary for a convex combination — so the systems are smaller and fewer in the case of conical combinations, even for the same number of extensions. In practice, we have never come across a situation in which the proposed algorithm did not yield a conical combination. However, this may happen in the case presented in [@Vid13], where the diffusion tensor varies in a sharp domain corner. In such a case we suggest modifying the mesh. Situations where a discontinuity meets the boundary, which required the introduction of auxiliary variables in the earlier versions of the scheme, do not present a difficulty in the current version. Maximum principle violations are due to the way the one-side fluxes are combined in §\[sec:scheme\] and can be severe, as shown in Example \[noFlowDiscontinuous\]. Nevertheless, this is not related to the technique of using piecewise linear transformation in linear flux reconstruction. The same technique can be used in combination with a variety of other finite volume schemes, such as [@Dro11; @She11]. It was demonstrated that in the transient case a single iteration per time step is enough to maintain the second order spatial accuracy with Euler implicit scheme, provided that sufficiently small time steps and the proposed initial guesses are used. The authors wish to thank the Ministry of Education, Science and Technological Development of the Republic of Serbia for the financial support provided through Technology Development Project TR37014.
==16true cm=24.8true cm plus 1pt =.aux .aux ==1 \#1\#2[[ = [^1][\#2]{}]{}]{} \#1 \#1 \#1 =cmr17 scaled 1 =cmbx10 scaled1 =cmbx10 scaled2 =cmss10 =cmr8 =cmsl8 =cmsy8 =cmmi8 =cmbx8 =cmr6 =cmcsc10 =cmr5 scaled 833 =msbm10=msbm7=msbm5 === =cmmib10 =cmmib8 =cmmib5 === =eufm10 =eufm8 =eufm5 === =eufb10 =eufb8 =eufb5 === \#1\#2[1.5pt/ 1.5pt]{} \#1[\#1]{} =1 \#1[[\#1]{}]{} \#1[[\#1]{}]{} \#1[[\#1]{}]{} \#1[[\#1]{}]{} \#1[[\#1]{}]{} $$#1{\mskip#1mu} \def$$\#1[-\#1mu]{} \#1\#2[(\#1) $\Longrightarrow$ (\#2)]{} \#1[\#1]{} \#1 \#1 \#1[\#1]{} \#1\#2 =0 \#1 by1[. \#1]{} =0 \#1\#2 by1\#1=\#1[. \#2]{}=0 \#1 \#1\#2[1\#1=\#1\#2]{} \#1 \#1 \#1\#2[[**  \#2. **]{}]{} \#1\#2[[ **\#2. **]{}**]{} \#1[[ **Theorem. **]{}**]{} \#1[[ **Proposition. **]{}**]{} \#1[[ **Lemma. **]{}**]{} \#1[[ **Corollary. **]{}**]{} \#1[[ **Question. **]{}**]{} \#1[[ **Problem. **]{}**]{} \#1[[ **Remark. **]{}]{} \#1[[ **Example. **]{}]{} \#1[[ **Definition. **]{}]{} \#1[[[\#1]{}\#1]{}]{} \#1[[()]{}]{} =1 \#1 [.]{}\#1ZZZ\#1=1 \#1[\#1ZZZ\[\]]{} =[incomplete draft of ]{} =[&gt;1Symmetric Cauchy-Riemann manifolds]{} =.aux On symmetric Cauchy-Riemann manifolds $$\vbox{\hsize=0.33\hsize {\obeylines\everypar{\hfil}\parindent=0pt\parskip-3pt \Icke Wilhelm Kaup \vskip5pt \sevenrm Mathematisches Institut Universit\"at T\"ubingen Auf der Morgenstelle 10 D-72076 T\"ubingen Germany ~ kaup@uni-tuebingen.de }}\qquad \vbox{\hsize=0.33\hsize {\obeylines\everypar{\hfil}\parindent=0pt\parskip-3pt \Icke Dmitri Zaitsev \vskip5pt \sevenrm Mathematisches Institut Universit\"at T\"ubingen Auf der Morgenstelle 10 D-72076 T\"ubingen Germany ~ dmitri.zaitsev@uni-tuebingen.de}}$$ The Riemannian symmetric spaces play an important role in different branches of mathematics. By definition, a (connected) Riemannian manifold $M$ is called symmetric if, to every $a\in M$, there exists an involutory isometric diffeomorphism $s_a\colon M\to M$ having $a$ as isolated fixed point in $M$ (or equivalently, if the differential $d_a\!s_a$ is the negative identity on the the tangent space $T_a=T_aM$ of $M$ at $a$). In case such a transformation $s_a$ exists for $a\in M$, it is uniquely determined and is the geodesic reflection of $M$ about the point $a$. As a consequence, for every Riemannian symmetric space $M$, the group $G=G_M$ generated by all symmetries $s_a$, $a\in M$, is a Lie group acting transitively on $M$. In particular, $M$ can be identified with the homogeneous space $G/K$ for some compact subgroup $K\subset G$. Using the elaborate theory of Lie groups and Lie algebras [E.$\,$Cartan]{} classified all Riemannian symmetric spaces. The complex analogues of the Riemannian symmetric spaces are the Hermitian symmetric spaces. By definition a Hermitian symmetric space is a Riemannian symmetric space $M$ together with an almost complex structure on $M$ such that the metric is Hermitian and such that every symmetry $s_a$ is holomorphic (i.e. satisfies the Cauchy-Riemann partial differential equations with respect to the almost complex structure). Also all Hermitian symmetric spaces were completely classified by [E.$\,$Cartan]{}. In particular, every Hermitian symmetric space $M$ can be written in a unique way as an orthogonal direct product $M=M_+\times M_0\times M_-$ of Hermitian symmetric spaces $M_\epsilon$ with holomorphic curvature of sign $\epsilon$ everywhere (possibly of dimension 0, i.e. a single point). $M_+$ is a compact simply connected complex manifold, $M_-$ is a bounded domain in some $\CC^n$ and $M_0$ can be realized as the flat space $\CC^m/\Omega$ for some discrete subgroup $\Omega\subset\CC^m$. In particular, the almost complex structure of $M$ is integrable. Furthermore, there exists a remarkable duality between symmetric spaces which for instance gives a one-to-one correspondence between those of compact type (i.e. $M=M_+$) and those of non-compact type (i.e. $M=M_-$), see for details. A joint generalization of real smooth as well of complex manifolds are the Cauchy-Riemann manifolds (CR-manifolds for short) or, more generally, the CR-spaces, where the integrability condition is dropped and thus also arbitrary almost complex manifolds are incorporated. These objects are smooth manifolds $M$ such that at every point $a\in M$ the Cauchy-Riemann equations only apply in the direction of a certain linear subspace $H_a\subset T_a$ of the tangent space to $M$, see f.i. or section for details. The tangent space $T_a$ is an $\RR$-linear space while $H_a$, also called the [*holomorphic*]{} tangent space at $a\in M$, is a $\CC$-linear space. The two extremal cases $H_a=0$ and $H_a=T_a$ for all $a\in M$ represent the two cases of smooth and of almost complex manifolds respectively. The main objective of this paper is to generalize the notion of symmetry to the category of CR-spaces. It turns out that for symmetries in this more general context the requirement of isolated fixed points is no longer adequate. In fact, this would happen only for Levi-flat CR-spaces (see Proposition ) and hence would not be interesting. Let us illustrate our concept on a simple example (compare also section ). Consider $E\colon=\CC^n$, $n>1$, with the standard inner product as a complex Hilbert space and denote by $S\colon=\{z\in E:\|z\|=1\}$ the euclidean unit sphere with Riemannian metric induced from $E$. Then $M$ is symmetric when considered as Riemannian manifold. But $S$ also has a canonical structure of a CR-manifold – define for every $a\in S$ the holomorphic tangent space $H_a$ to be the maximal complex subspace of $E$ contained in $T_aS\subset E$, i.e. the [*complex*]{} orthogonal complement of $a$ in $E$. Then $T_a$ is the orthogonal sum $H_a\oplus i\RR a$. It can be seen that for every isometric CR-diffeomorphism $\phi$ of $S$, the differential $d_a\phi$ is the identity on $i\RR a$ as soon as it is the negative identity on $H_a$, i.e. there does not exist a CR-symmetry of $S$ at $a$ in the strict sense. On the other hand, the unitary reflection $s_a(z)\colon=2(a|z)-z$ defines an involutory isometric CR-diffeomorphism of $S$ with differential at $a$ being the negative identity on $H_a$. We call this the symmetry of the CR-manifold $S$ at $a$ and take it as a guideline for our general definition . Among all symmetric CR-manifolds we distinguish a large subclass generalizing the above example. This class consists of the Shilov boundaries $S$ of bounded symmetric domains $D\subset\CC^n$ in their circular convex realizations (Theorem ). A remarkable feature of these CR-submanifolds is the fact that various geometric and analytic constructions, hard to calculate in general, can be obtained here in very explicit forms. We illustrate this on the case of polynomial and rational convex hulls. Recall that the polynomial (resp. rational) convex hull of a compact subset $K\subset\CC^n$ is the set of all $z\in\CC^n$ such that $|f(z)|\le \sup_K |f|$ for every polynomial $f$ (resp. every rational function $f$ holomorphic on $K$). If $K$ is a connected real-analytic curve, $p(K)\backslash K$ is a complex analytic subset of $\CC^n\backslash K$, due to [J.$\,$Wermer]{} , where $p(K)$ denotes the polynomial convex hull. Later, the analyticity of $p(K)\backslash K$ was proved by [H.$\,$Alexander]{} , for compact sets of finite length and recently by [T.C.$\,$Dinh]{} for rectifiable closed (1,1)-currents under very weak assumptions (see also [E.$\,$Bishop]{} , [G.$\,$Stolzenberg]{} and [M.G.$\,$Lawrence]{} for related results). On the other hand, if $K$ is not a smooth submanifold, $p(K)\backslash K$ is not analytic in general (see e.g. [G.$\,$Stolzenberg]{} or [J.$\,$Wermer]{} ). In the present paper we calculate the polynomial and the rational convex hulls of $S$, where $S\subset \partial D$ is a Shilov boundary as above (see Corollary ). Here it turns out that $p(S)\backslash S$ is not necessarily analytic but rather a [*submanifold with “real-analytic corners”*]{}, even though $S$ itself is a connected real-analytic submanifold. Similar is the behaviour of the rational convex hull of $S$. Both hulls are canonically stratified into real-analytic CR-submanifolds such that the (unique) stratum of the highest dimension is complex for the polynomial and Levi-flat for the rational convex hull. The paper is organized as follows. Preliminaries on CR-spaces are given in section . In section we introduce symmetric CR-spaces and establish their main properties: [*The uniqueness of symmetries and the transitivity of the spanned group*]{}. Example (a generalized Heisenberg group) shows that there exist symmetric CR-manifolds $M$ of arbitrary CR-dimension and arbitrary CR-codimension having arbitrary Levi form at a given point. In section we study more intensively the unit sphere $S$ in the complex space $\CC^n$, some symmetric domains in $S$ and their coverings. In particular, we obtain uncountable families of pairwise non-isomorphic symmetric CR-manifolds (see Example ). In section we associate to every symmetric CR-space $M$ a canonical fibration and discuss the situation when the base is a symmetric CR-space itself. In fact, every symmetric CR-space can be obtained in this way. As mentioned above, the underlined CR-structure of a symmetric CR-space does not need to be integrable. In section we give a construction principle for symmetric CR-spaces in terms of Lie groups and illustrate this with various examples. In section we give Lie theoretic conditions for $M$ to be embeddable into a complex manifold. We show by an example (see ) that this in general is not possible – in contrast to the case of Hermitian symmetric spaces. Finally, in section , we consider symmetric CR-manifolds arising from bounded symmetric domains $D\subset\CC^n$. To be a little more specific, we assume without loss of generality that $D$ is realized as bounded circular convex domain in $\CC^n$. Then it is known that the Shilov boundary $S\subset \partial D$ of $D$ (which coincides here with the set of all extreme points of the convex set $\overline D$) is an orbit of the group $\Aut(D)$ of all biholomorphic automorphisms of $D$. Furthermore, every maximal compact subgroup $K$ of $\Aut(D)$ still acts transitively on $S$, and, with respect to a suitable $K$-invariant Hermitian metric, $S$ is a symmetric CR-manifold. Our main result states: In case $D$ does not have a factor of tube type, (i) every smooth CR-function on $S$ extends to a continuous function on $\overline D$ holomorphic on $D$, and (ii) the group $\AUT(S)$ of all smooth CR-diffeomorphisms of $S$ coincides with the group $\Aut(D)$. [**Notation.**]{} For every vector space $E$ over the base field $\KK=\RR$ or $\KK=\CC$ we denote by $\5L(E)$ the space of all $\KK$-linear endomorphisms of $E$ and by $\GL(E)\subset\5L(E)$ the subgroup of all invertible operators. More generally, for every total subset $S\subset E$ put $\GL(S)\colon=\{g\in\GL(E):g(S)=S\}$ and denote by $\Aff(S)$ the group of all affine transformations of $E$ mapping $S$ onto itself. $\KK^{n\times m}$ is the $\KK$-Hilbert space of all $n\times m$-matrices over $\KK$ ($n=$ row-index) with the inner product $(u|v)=\tr(u^*v)$ and $u^*=\overline u'\in\KK^{m\times n}$ the adjoint. By a complex structure we always understand a linear operator $J$ on a real vector space with $J^2=-\id$. If misunderstanding is unlikely we simply write $ix$ instead of $J(x)$. For complex vector spaces $U,V,W$ a sesqui-linear map $\Phi\colon U\times V\to W$ is always understood to be conjugate linear in the first and complex linear in the second variable. With $\U(n)$, $\O(n)$, $\Sp(n)$ etc. we denote the unitary, orthogonal and symplectic groups (see for related groups). In particular, we put $\TT:=\U(1)=\exp(i\RR)$ and $\RR^+\colon=\exp(\RR)$. For every topological group $G$ we denote by $G^0$ the connected identity component. A continuous action of $G$ on a locally compact space $M$ is called proper if the mapping $G\times M\to M\times M$ defined by $(g,a)\mapsto (g\Kl a ,a)$ is proper, i.e. pre-images of compact sets are compact. For every set $S$ and every map $\sigma\colon S\to S$ we denote by $\Fix(\sigma)\colon=\{s\in S:\sigma\Kl s=s\}$ the set of all fixed points. For Lie groups $G$, $H$ etc., the corresponding Lie algebras are denoted by small Gothic letters $\7g$, $\7h$ etc. For linear subspaces $\7m,\7n\subset\7g$ we denote by $[\7m,\7n]$ always the linear span of all $[x,y]$ with $x\in\7m$, $y\in\7n$. Suppose that $M$ is a connected smooth manifold of (finite real) dimension $n$. Denote by $T_a\colon=T_aM$, $a\in M$, the tangent space which is a real vector space of dimension $n$. An [*almost Cauchy-Riemann structure*]{} (almost CR-structure for short) on $M$ assigns to every $a\in M$ a linear subspace $H_a=H_aM\subset T_a$ (called the [*holomorphic tangent space*]{} to $M$ at $a$) together with a complex structure on $H_a$ in such a way that $H_a$ and the complex structure depend smoothly on $a$. Smooth dependence can be expressed in the following way: [*Every point of $M$ admits an open neighbourhood $U\subset M$ together with a linear endomorphism $j_a$ of $T_a$ for every $a\in U$ such that $-j_a^2$ is a projection onto $H_a$ with $j_av=iv$ for all $v\in H_a$, and $j_a$ depending smoothly on $a\in U$*]{}. Then, in particular, all $H_a$ have the same dimension. A connected smooth manifold together with an almost CR-structure on it is called in the sequel an [*almost CR-manifold*]{}, or a [*CR-space*]{} for short. For more details on (almost) CR-manifolds see e.g. , , , , . In the following $M$ always denotes a CR-space. Denote by $\7V=\7V(M)$ the Lie algebra of all smooth vector fields on $M$ and by $\7H=\7H(M)$ the subspace of all vector fields $X$ with $X_a\in H_a$ for all $a\in M$. Then, for all $a\in M$, $\7H_a\colon=\{X_a:X\in\7H\}=H_a$ holds and $(JX)_a=i(X_a)$ canonically defines a complex structure $J$ on $\7H$. Define inductively $$\7H^k\colon =\7H^{k-1}+[\7H,\7H^{k-1}]\,,\steil{where}\7H^1\colon=\7H \steil{and}\7H^j\colon=0\;\;\hbox{\rm for}\;\;j\le0\,.\Leqno{HKKK}$$ Then $[\7H^r,\7H^s]\subset\7H^{r+s}$ holds for all integers $r,s$ and $\7H^\infty\colon=\bigcup_k\7H^k$ is the Lie subalgebra of $\7V$ generated by $\7H$. Also, we call the quotient vector spaces $$T_a^r\colon=T_a/H_a\steil{and}T_a^{rr}\colon=T_a/\7H^\infty_a$$ the [*real*]{} and the [*totally real*]{} part of $T_a$. The complex dimension of $H_a$ and the real dimension of $T_a^r$ do not depend on $a\in M$ - they are called the [*CR-dimension*]{} and the [*CR-codimension*]{} of $M$. Finally, $M$ is called [*minimal*]{} (in the sense of [Tumanov]{} ) if $U=N$ holds for every domain $U\subset M$ and every closed smooth submanifold $N\subset U$ with $H_aM\subset T_aN$ for all $a\in N$. It is known that in case $M$ is real-analytic (all CR-spaces we discuss later will have this property, see ) minimality is equivalent to $T_a^{rr}=0$ for all $a\in M$, i.e. to the finite type in the sense of [Bloom-Graham]{} . $M$ is called [*totally real*]{} if $M$ has CR-dimension 0. The CR-spaces of CR-codimension 0, i.e. satisfying $H_aM=T_aM$, are also called almost complex manifolds. The CR-spaces form a category in a natural way. By definition, a smooth map $\phi\colon M\to N$ of CR-spaces is called a CR-[*map*]{} if, for every $a\in M$ and $b=\phi(a)\in N$, the differential $d_a\]1\phi\colon T_aM\to T_bN$ maps $H_aM$ complex linearly into $H_bN$. For every $M$ we denote by $\Aut(M)$ or $\AUT(M)$ the group of all CR-diffeomorphisms $\phi\colon M\to M$ and endow this group with the compact open topology. Suppose, $N$ is a CR-space and $M\subset N$ is a submanifold. We call $M$ a CR-[*subspace*]{} of $N$ if the dimension of $H_aM\colon=(T_aM\cap H_aN)\,\cap\,i(T_aM\cap H_aN)$ does not depend on $a\in M$. Then $M$ is a CR-space with the induced CR-structure. A CR-space $M$ is called [*integrable*]{} or a [*CR-manifold*]{} if the following integrability condition is satisfied: $$Z\colon=[JX,Y]+[X,JY]\in\7H\steil{and} JZ=[JX,JY]-[X,Y]\steil{for all}X,Y\in\7H\,.\Leqno{INTE}$$ In the special case, where $M$ is real-analytic (which includes that the holomorphic tangent space $H_aM$ depends in a real-analytic way on $a\in M$) it is known (compare f.i. or ) that is equivalent to the existence of local realizations of $M$ as a CR-submanifold of some $\CC^n$. In that case there even exist a complex manifold $N$ and a (global) realization of $M$ as a real-analytic CR-submanifold of $N$ which is generic, i.e. $T_aM+iT_aM=T_aN$ for all $a\in M$ (see ). An important invariant of a CR-space $M$ is the so-called [*Levi form*]{} defined at every point $a\in M$ in the following way. Denote by $\pi_a\colon T_a\to T_a/H_a$ the canonical projection. Then it is easy to see that there exists a map $$\omega_a\colon H_a\times H_a\;\to\;T_a/H_a\steil{with}\omega_a(X_a,Y_a)=\pi_a([X,Y]_a) \steil{for all}X,Y\in\7H$$ (in the proof of Proposition a more general satement actually will be shown). Then $\omega_a$ is $\RR$-bilinear and skew-symmetric. For all $\epsilon,\mu=\pm1$, there exist uniquely determined $\RR$-bilinear maps $$\omega_a^{\epsilon\mu}\colon H_a\times H_a\to (T_a/H_a)\otimes_\RR\CC\steil{with} \omega_a^{\epsilon\mu}(sx,ty)=s^\epsilon t^\mu \omega_a^{\epsilon\mu} (x,y)$$ for all $s,t\in\TT$, $x,y\in H_a$ such that $\omega_a=\sum \omega_a^{\epsilon\mu}$. The sesqui-linear part $L_a\colon=\omega_a^{-1,1}$ of $\omega_a$ is called the [*Levi form*]{} of $M$ at $a$, i.e. $$4\,L_a(x,y)=\big(\omega_a(x,y)+\omega_a(ix,iy)\big)\,+\,i\big(\omega_a(ix,y)-\omega_a(x,iy)\big)$$ and in particular $2L_a(x,x)=i\omega_a(ix,x)\in i(T_a/H_a)$ for all $x,y\in H_a$. The convex hull of $\{L_a(x,x):x\in H_a\}$ is called the [*Levi cone*]{} at $a\in M$ and its interior always refers to the linear space $i(T_a/H_a)$. The CR-space $M$ is called [*Levi flat*]{} if $L_a=0$ holds for every $a\in M$. Denote by $^*$ the conjugation of $(T_a/H_a)\otimes_\RR\CC$ given by $(\xi+i\eta)^*\colon=(-\xi+i\eta)$ for all $\xi,\eta\in T_a/H_a$. The following statement is obvious. The Levi form $L_a$ is $^*$-Hermitian, that is, $L_a(x,y)=L_a(y,x)^*$ for all $x,y\in H_a$. In the case, the integrability condition holds, we have $\omega_a^{1,1}=\omega_a^{-1,-1}=0$ and $$2\,L_a(x,y)\;=\;\omega_a(x,y)+ i\omega_a(ix,y)\,.\eqno{\qex}$$ Various authors call $-2iL_a$ the Levi form. In $L=L_a$ is called the [*extrinsic Levi form*]{}. If $M$ is a CR-subspace of a complex manifold $U$, $(T_a/H_a)\otimes_\RR\CC$ can be canonically identified with a complex subspace of $T_aU/H_aM$. Then, for every $x\in H_a$ the vector $L(x,x)\in iT_a/H_a$ is transversal to $M$ and points into the ‘pseudoconvex direction’ of $M$. [Bogges]{} and [Polking]{} proved that all CR-functions on $M$ extend holomorphically to a wedge in $U$ ‘in the direction of the Levi form’. This was generalized by [Tumanov]{} to the case $M$ is minimal, whereas [Baouendi]{} and [Rothschild]{} proved the necessity of the minimality condition. Let us illustrate the Levi form at a simple example: Let $M\colon=\{(z,w)\in\CC^2:z\overline z+w\overline w=1\}$ be the euclidean sphere and put $a\colon=(1,0)$, $e\colon=(0,1)$. Then $M$ is a CR-submanifold with $T_aM=i\RR a\oplus\CC e$ and $H_aM=\CC e$. We identify $(T_aM/H_aM)\otimes_\RR\CC$ in the obvious way with the complex line $\CC a\subset\CC^2$. Consider the vector field $X\in\7H$ defined by $X_{(z,w)}=(-\overline w,\overline z)$ for all $(z,w)\in M$. Then $[JX,X]_a=2ia=\omega_a(ie,e)$ and hence $L_a(e,e)=-a$. This vector points from $a\in M$ into the interior of the sphere $M$. For the rest of the section assume that on $M$ there is given a [*Riemannian metric*]{}, i.e. every tangent space $T_a$ is a real Hilbert space with respect to an inner product $\langle u|v\rangle_a$ depending smoothly on $a\in M$. Then we call $M$ an [*Hermitian CR-space*]{} if the metric is compatible with the CR-structure in the sense that $\|iv\|=\|v\|$ holds for all $a\in M$, $v\in H_a$, where $\|v\|=\sqrt{\langle v,v\rangle}$. In particular, every $H_a$ is a complex Hilbert space. If the CR-structure of a Hermitian CR-space $M$ is integrable, we call $M$ an [*Hermitian CR-manifold*]{}. In the case of vanishing CR-codimension, i.e. $H_aM=T_aM$, Hermitian CR-spaces are usually called [*Hermitian manifolds*]{}. For every $a\in M$ and every integer $k\ge0$, let $\7H^k$ be as in and let $H_a^k\subset T_a$ be the orthogonal complement of $\7H_a^{k-1}$ in $\7H_a^k$. Then $H_a^0=0$ and $H_a^1=H_a$ is the holomorphic tangent space in $a$. If we denote by $H_a^{-1}\cong T_a^{rr}$ the orthogonal complement of $\7H^\infty_a$ in $T_a$ we obtain the following orthogonal decomposition $$T_a=\bigoplus_{k\ge-1}H_a^k\;.\Leqno{DECO}$$ Denote by $\pi_a^k\in\5L(T_a)$ the orthogonal projection onto $H_a^k$. Then $a\mapsto\pi_a^k$ defines a (not necessarily continuous) tensor field $\pi^k$ of type $(1,1)$ on $M$. For later use we define $$T_a^+\colon=\bigoplus_{k{\rm~even}}H_a^k\steil{and} T_a^-\colon=\bigoplus_{k{\rm~odd}}H_a^k\;.\Leqno{DEOR}$$ The Hermitian CR-spaces form a category together with the contractive CR-mappings as morphisms (i.e. $\|d_a\phi\Kl v \|\le\|v\|$ for all $a\in M$, $v\in T_aM$). We always denote by $I_M\subset\AUT(M)$ the closed subgroup of all isometric CR-diffeomorphisms. Then it is known that $I_M$ is a Lie group acting smoothly and properly on $M$. In particular, $I_M$ has dimension $\,\le n(n\]4+\]42)+m(m\]4+\]41)/2$, where $M$ has CR-dimension $n$ and CR-codimension $m$. The full CR-automorphism group $\AUT(M)$ can be infinite-dimensional. In case of vanishing CR-dimension we have the full sub-category of (connected) Riemannian manifolds and in case of vanishing CR-codimension we have the full sub-category of Hermitian manifolds. In both sub-categories there exists the classical notion of a symmetric space. In the following we want to extend this concept to arbitrary Hermitian CR-spaces. Let $M$ be an Hermitian CR-space and let $\sigma\colon M\to M$ be an isometric CR-diffeomorphism. Then $\sigma$ is called a [*symmetry*]{} at the point $a\in M$ (and $a$ is called a [*symmetry point*]{} of $M$) if $a$ is a (not necessarily isolated) fixed point of $\sigma$ and if the differential of $\sigma$ at $a$ coincides with the negative identity on the subspace $H_a^{-1}\!\oplus H_a^1$ of $T_a$. At every point of $M$ there exists at most one symmetry. Furthermore, every symmetry is involutive. The statement is an easy consequence of the following. Let $\phi,\psi$ be isometric CR-diffeomorphisms of the Hermitian CR-space $M$ with $\phi\Kl a=\psi\Kl a $ for some $a\in M$. Then $\phi=\psi$ holds if the differentials $d_a\phi$ and $d_a\psi$ coincide on the subspace $H_a^{-1}\!\oplus H_a^1$ of $T_a=T_aM$. Without loss of generality we may assume that $\psi$ is the identity transformation of $M$. Then $a$ is a fixed point of $\phi$ and by a well known fact (compare f.i. 62) we only have to show that the differential $\lambda\colon=d_a\phi$ is the identity on $T_a$. Now $\lambda(X_a)=(\tau X)_a$ holds for every vector field $X\in\7V$, where $\tau$ is the Lie automorphism of $\7V$ induced by $\phi$. For all $r,s>0$ and $k\colon=r+s$, every smooth function $f$ on $M$ and all $X\in\7H^{r}$, $Y\in\7H^{s}$ the formula $$\pi_a^k\big([fX,Y]_a\big)\;=\;f\Kl a \cdot\pi_a^k\big([X,Y]_a\big)$$ is easily derived, where the orthogonal projection $\pi_a^k$ is defined as above. On the other hand, every $X\in\7H^r$ with $X_a=0$ can be written as finite sum $X=X_0+f_1X_1+\cdots+f_mX_m$ with $X_0\in\7H^{r-1}$, $X_1,\dots,X_m\in\7H^r$ and smooth functions $f_1,\dots,f_m$ on $M$ vanishing in $a$. Therefore, $\pi_a^k\big([X,Y]_a\big)$ only depends on the vectors $\pi_a^r(X_a)$ and $\pi_a^s(Y_a)$ for $X\in\7H^{r}$, $Y\in\7H^{s}$. Since $\lambda$ is an isometry of $T_a$ and $\tau$ leaves invariant all subspaces $\7H^k\subset\7V$ also all $H_a^k$ must be left invariant by $\lambda$. We show by induction on $k$ that actually $\lambda$ is the identity on every $H_a^k$. For $k\le1$ this follows from the assumptions. For $k>1$, fix $X\in\7H$, $Y\in\7H^{k-1}$ and consider the vector $v\colon=\pi_a^k\big([X,Y]_a\big)\in H_a^k$. By induction hypothesis we then have $(\tau X)_a=X_a$, $(\tau Y)_a=Y_a$ and hence $$\lambda\Kl v=\pi_a^k(\lambda\Kl v )=\pi_a^k\big((\tau[X,Y])_a\big)=\pi_a^k\big([\tau X, \tau Y]_a\big)=\pi_a^k\big([X,Y]_a\big)=v\,.\eqno{\qed}$$ The proof of Proposition shows that for every symmetry $\sigma$ of $M$ at $a$ the differential $\lambda=d_a\sigma$ satisfies $\lambda\Kl v=(-1)^kv$ for every $v\in H_a^k$ and every $k\ge-1$, i.e. $\lambda=\Sigma_k(-1)^k\pi_a^k$, or equivalently, $\,\Fix(\pm\lambda)=T_a^\pm$. A connected Hermitian CR-space $M$ is called [*symmetric*]{} (or an SCR-space for short) if every $a\in M$ is a symmetry point. The corresponding symmetry at $a$ is denoted by $s_a$. In the sequel we adopt the following notation: For a given SCR-space $M$ we denote as in section by $I=I_M$ the Lie group of all isometric CR-diffeomorphisms of $M$. Let $G=G_M$ be the closed subgroup of $I$ generated by all symmetries $s_a$, $a\in M$. Fix a base point $o\in M$ and denote by $K\colon=\{g\in G:g\Kl o=o\}$ the isotropy subgroup at $o$. $G$ is a Lie group acting transitively and properly on $M$. The connected identity component $G^0$ of $G$ has index $\le2$ in $G$ and coincides with the closed subgroup of $I_M$ generated by all transformations $\, s_a\circ s_b$ with $a,b\in M$. The isotropy subgroup $K$ is compact and $M$ can be canonically identified with the homogeneous manifold $G/K$ via $g\Kl o \mapsto gK$. $M$ is compact if and only if $G$ is a compact Lie group. There exists an open subset $U\ne\emptyset$ of $M$ such that for every $k\ge-1$ the dimension of $H_a^k$ does not depend on $a\in U$. Therefore, every tensor field $\pi^k$ is smooth over $U$, i.e. the orthogonal decomposition depends smoothly on $a$ as long as $a$ stays in $U$. We may assume without loss of generality that for a fixed $\epsilon>0$ and every $a\in U$, the exponential mapping $\Exp_a$ is defined on the open ball $B_a$ of radius $\epsilon$ about the origin in $T_a$ and that $\Exp_a$ is a diffeomorphism from $B_a$ onto a neighbourhood $N_a\subset M$ of $a$. Every isometric diffeomorphism $\phi\in I_M$ is linear in local normal coordinates, more precisely, for every $a\in U$ with $c\colon=\phi\Kl a \in U$, the diagram $$\diagram{B_a&\mapright{\displaystyle d_a\phi} &B_c\cr \mapdown{\displaystyle\hskip-29pt\Exp_a} & &\mapdown{\[4\displaystyle\Exp_c} \cr N_a&\mapright{\displaystyle\phi} &N_c\cr}$$ commutes. Since $d_a\[2s_a=\sum_k(-1)^k\pi_a^k$ depends smoothly on $a\in U$ and since $G$ consists of isometries this implies the smoothness of the mapping $U\times U\to M$ defined by $(a,b)\mapsto s_a(b)$. Now fix $u\in U$ and denote by $A\colon=G\Kl u $ the orbit of $u$ under the group $G$. Then $A$ is a closed smooth submanifold of $M$ since $G$ acts properly on $M$. Fix $a\in A$ and $v\in T_a^-M$ arbitrarily. Choose a smooth curve $\gamma\colon[0,1]\to U$ with $\gamma(0)=a$ and $\gamma'(0)=v$. Then $\alpha(t)=s_{\gamma(t)}\Kl a $ defines a smooth curve in $A$ with $\alpha(0)=a$ and $\alpha'(0)=2v$. This proves $T_a^-M\subset T_aA$ and hence $T_c^{-1}M\subset T_cA$ for all $c\in A$ since $a\in A$ was arbitrarily chosen. Now fix a vector $w\in T_a^+M$. Then there exist smooth vector fields $X^1,\dots,X^{2k}$ on $M$ such that $X_c^j\in T_c^-M$ for all $1\le j\le2k$, $c\in A$ and such that $w=\sum_{j=1}^k[X^j,X^{k+j}]_a$. But we know already that all $X^j$’s are tangent to the submanifold $A\subset M$, i.e. all their brackets are tangent to $A$ and therefore $w\in T_aA$. This implies $T_aA=T_aM$ and $A=M$ since $A$ is closed in $M$. Therefore $G$ acts transitively on $M$. The proof of Proposition shows that an Hermitian CR-space $M$ is already symmetric as soon as the set of symmetry points of $M$ has an interior point in $M$. The transitivity of the $G$-action has several consequences. Every SCR-space $M$ has a unique structure of a real-analytic CR-manifold in such a way that every isometric diffeomorphism of $M$ is real-analytic. All tensors $\pi^k$ are real-analytic on $M$ and also the mapping $a\mapsto s_a$ from $M$ to $G$ is real-analytic. In particular, the dimension of $H_a^k\subset T_a$ does not depend on $a\in M$ for every $k$. In the following we will always consider SCR-spaces as real-analytic manifolds according to . The number $\kappa=\kappa(M)\colon=\max\{k\ge-1:H_a^k\ne0\}$ does not depend on $a\in M$. Example will show that arbitrary values of $\kappa\ne0$ occur. $\CC^n\!\!\times\!\RR^m$ is a Levi flat CR-submanifold of $\CC^{n+m}$ and as another corollary of we have: Let $M$ be a symmetric CR-space with CR-dimension $n$ and CR-codimension $m$. Then for every $a\in M$ the following conditions are equivalent. $a$ is an isolated fixed point of the the symmetry $s_a$. $M$ is Levi flat. $M$ is locally CR-isomorphic to an open subset of $\CC^n\!\!\times\!\RR^m$. In particular, $M$ is a CR-manifold. 12. The differential $d_as_a$ is the identity on the subspace $H_a^2\subset T_a$ due to Remark . Therefore, if $a$ is isolated in $\Fix(s_a)$, we have $H_a^2=0$ and hence $L_a=0$. By homogeneity then the Levi form vanishes at every point of $M$, i.e. $M$ is Levi flat.23. Suppose that $M$ is Levi flat. Then the holomorphic tangent spaces form an involutive distribution on $M$ and define a foliation of $M$. Let $N$ be the leaf through $a$, i.e. the maximal connected immersed smooth submanifold $N$ of $M$ with $T_cN=H_cM$ for all $c\in N$. Then $N$ is an Hermitian almost complex manifold in the leaf topology invariant under every symmetry $s_c$, $c\in N$. Therefore, $N$ is an Hermitian symmetric space and in particular a complex manifold, see . But $M$ locally is CR-isomorphic to a direct product $U\times V$, where $U\subset N$ and $V\subset\RR^m$ are open subsets.31. Condition (iii) implies $H_a^{-1}\oplus H_a^1=T_a$ and hence $d_as_a=-\id$, i.e. $a$ is an isolated fixed point of $s_a$. Every SCR-space $M$ may be considered as a reflection space in the sense of , i.e. if a ‘multiplication’ on $M$ is defined by $x\cdot y\colon= s_xy$ for all $x,y\in M$, the following rules hold: $x\cdot x=x$, $\;x\cdot(x\cdot y)=y\;$ and $\;x\cdot(y\cdot z)=(x\cdot y)\cdot(x\cdot z)$ for all $x,y,z\in M$. The SCR-spaces form in various ways a category. We prefer here the following notion (ignoring the Riemannian metrics on $M$ and $N$): A CR-map $\phi\colon M\to N$ is called an SCR-map, if $\phi(x\cdot y)=\phi\Kl x \cdot\phi(y)$ for all $x,y\in M$. If $\phi$ in addition is contractive we also call it a [*metric SCR-map*]{}. In this sense it is clear what (metric, isometric) SCR-isomorphisms, SCR-automorphisms are. For instance, $G_M$ consists of isometric SCR-automorphisms of $M$. Also, the universal covering $\pi\colon\widetilde M\to M$ of an SCR-space $M$ has a unique structure of an SCR-space such that locally $\pi$ is an isometric CR-diffeomorphism. The following statement can be used to construct SCR-spaces. Let $M$ be a connected Hermitian CR-space with a base point $o$ and let $H\subset I_M$ be a subgroup acting transitively on $M$. Suppose that $\sigma\colon M\to M$ is a diffeomorphism with $\sigma\Kl o=o$ and $\sigma^2=\id$ such that the following conditions are satisfied: $\sigma\circ H=H\circ\sigma$, the differential $d_o\sigma\in\5L(T_oM)$ is a linear isometry with $(H_oM\oplus T_o^{rr}M)\;\subset\;\Fix(-d_o\sigma)$. Then $\sigma$ is a symmetry of $M$. Fix $a\in M$ and choose $g,h\in H$ with $g\Kl a=o$ and $\sigma=h\,\circ\,\sigma\,\circ\,g$. The claim follows from the identity $d_a\sigma=d_oh\,\circ\,d_o\sigma\,\circ\,d_a g$. Let $E,F$ be complex Hilbert spaces of finite dimension. Suppose that $w\mapsto w^*$ is a conjugation of $F$ (i.e. a conjugate linear, involutive isometry of $E$) and let $\Phi\colon E\times E\to F$ be an Hermitian mapping with respect to the conjugation $^*$ (i.e. $\Phi$ is sesqui-linear and $\Phi(v,u)=\Phi(u,v)^*$ for all $u,v\in E$). Set $$V\colon=\{w\in F:w+w^*=0\}\steil{and} M\colon=\{(z,w)\in E\oplus F:w+w^*=\Phi(z,z)\}\,.$$ Then $M$ is a CR-submanifold of $E\oplus F$ with $$\eqalign{T_aM=&\{(z,w)\in E\oplus F:w+w^*=\Phi(e,z)+\Phi(z,e)\}\cr H_aM=&\{(z,w)\in E\oplus F:w=\Phi(e,z)\}\cr}$$ for every $a=(e,c)\in M$. The group $$\Lambda\colon=\big\{(z,w)\mapsto(z+e,w+\Phi(e,z)+\Phi(e,e)/2 +v):e\in E,v\in V\big\}$$ acts transitively and freely on $M$ by affine CR-diffeomorphisms. Therefore $M$ has a group structure, a generalization of the Heisenberg group. For $o\colon=(0,0)\in M$, there exists a unique $\Lambda$-invariant Riemannian metric on $M$ such that $T_oM=E\oplus V$ is the orthogonal sum of $E$ and $V$. By Lemma , $M$ is a symmetric CR-manifold – for every $a=(e,c)\in M$, the corresponding symmetry $s_a$ is given by $(z,w)\mapsto\big(2e-z,w+\Phi(2e,e-z)\big)$ and $G_M=\Lambda\cup s_o\Lambda$, $G^0_M=\Lambda$. The full group $\AUT(M)$ does not act properly on $M$ since it contains all transformations of the form $(z,w)\mapsto(tz,t\overline t\[1w)$, $t\in\CC^*$. For every $\xi\in E$, the vector field $X$ on $E\oplus F$ defined by $X_{(z,w)}=(\xi,\Phi(z,\xi))$ satisfies $X_a\in H_aM$ for all $a\in M$. From this it is easily derived that the Levi form $L_o$ at $o\in M$ as defined in coincides with the Hermitian map $\Phi\colon E\times E\to F$ after the identification $H_oM=E$ and $(T_oM/H_oM)\otimes_\RR\CC\cong F$. The next statement will be used later. Let $E,V\subset F,M,\Phi,\Lambda$ be as in Example . Assume that $\Phi$ is non-degenerate in the following sense: For every $e\in E$ with $e\ne0$ there exists $c\in E$ with $\Phi(e,c)\ne0$. Then $$\Aff(M)=\Lambda\rtimes\GL(M)\;\;\4{and}$$ $$\GL(M)=\{(\eta\times\epsilon)\in\GL(E)\times\GL(V): \Phi(\eta z,\eta z)=\epsilon\[1\Phi(z,z)\steil{for all}z\in E\}\,.$$ Furthermore, the group $I_M$ of all isometric CR-diffeomorphisms of $M$ is given by $$I_M=\Lambda\rtimes\,\Gamma\subset\Aff(M)\,,\steil{where} \Gamma\colon=\{(\eta\times\epsilon)\in\GL(M):\eta\;\4{unitary},\, \epsilon\;\4{orthogonal}\}\,.$$ Let us start with an arbitrary real-analytic CR-diffeomorphism $\phi$ of $M$ satisfying $\phi(o)=o$. Then $\phi$ extends to a holomorphicühic map $\phi\colon U\to E\oplus F$ for a suitable open connected neighbourhood $U$ of $M$ in $E\oplus F$ (see e.g. , §1.7). The differential $g\colon=d_o\phi\in\GL(E\oplus F)$ leaves $H_oM=E$ invariant and hence can be written as operator matrix $g={\eta\,\alpha\choose0\,\epsilon}\in\GL(E\oplus F)$ with a linear operator $\alpha\colon F\to E$. Since also $T_oM=E\oplus V$ is invariant under $g$, the operator $\epsilon\in\GL(F)$ must leave invariant the subspace $V\subset F$, i.e. $\epsilon\in\GL(V)\subset\GL(F)$ and, in particular, $\epsilon(w^*)=(\epsilon w)^*$ for all $w\in F$. There exist holomorphic functions $h\colon U\to E$ , $f\colon U\to F$ vanishing of order $\ge2$ at $o$ such that $$\phi(z,w)\;=\;\big(\eta z+\alpha w+h(z,w),\epsilon w+f(z,w)\big)$$ for all $(z,w)\in M$. For every $z\in E$, $v\in V$ and $$w=w(z,v)\colon=v+\Phi(z,z)/2$$ we have $(z,w)\in M$ and hence $$\epsilon\,\Phi(z,z)+f(z,w)+f(z,w)^* =\Phi(\eta z+\alpha w+h(z,w),\eta z+\alpha w+h(z,w))\leqno{(*)}$$ for all $z\in E$ and $v\in V$. Comparing terms in $(*)$ we derive $\epsilon\,\Phi(z,z)=\Phi(\eta z,\eta z)$ for all $z\in E$. Now suppose that $\phi$ is affine, i.e. $f=0$ and $h=0$. Comparing terms in $(*)$ again we get $\Phi(\alpha v,\eta z)=0$ for all $z\in E$ and $v\in V$. But then the non-degeneracy of $\Phi$ implies $\alpha v=0$ for all $v\in V$, i.e. $\alpha=0$. This proves that the groups $\Aff(M)$ and $\GL(M)$ have the claimed forms.Now suppose that $\phi\in I_M$ is an isometry. Since there is a unique real-analytic structure on $M$ such that the Lie group $I_M$ acts as a real-analytic transformation group. Since the same holds for the Lie subgroup $\Lambda$, these two structures must coincide, i.e. $\phi$ is real-analytic. Furthermore, $g=d_o\phi$ is a linear isometry of $E\oplus V$, i.e. $g={\eta\,0\choose0\,\epsilon}$ with $\eta$ unitary and $\epsilon$ orthogonal. Together with $\epsilon\,\Phi(z,z)=\Phi(\eta z,\eta z)$ for all $z\in E$ this implies $g\in I_M$ and hence $\phi=g\in\GL(M)$ as a consequence of the Uniqueness Theorem . For all CR-manifolds $M$ in Example with $\Phi\ne0$ the group $G^0$ is nilpotent of nilpotency class $2=\kappa(M)$. For examples with nilpotent groups of higher class compare section . An explicit example of class $3$ with lowest possible dimension is the following. Set $$M\colon=\big\{(z,w,v)\in\CC^3:\Im\Kl w =z\overline z,\;\Im\Kl v=\Im(w\overline z)\big\}\,.$$ Then $M$ is a CR-submanifold of $\CC^3$ with CR-dimension 1 and CR-codimension 2 and every $(a,b,c)\in M$ induces an affine CR-automorphism of $M$ by $$\big(z,w,v\big)\;\longmapsto\;\big(z+a,\,w+2i\overline az+b,\, v+(2i\overline a^2-\overline b)z+(a+2\overline a)w+c\big)\,.\leqno{(*)}$$ Indeed, if we denote the right hand side of $(*)$ by ${\bf(z,w,v)}$ then $$\eqalign{\Im({\bf w\overline z})\;& =\;\Im(w\overline z+2iz\overline z\overline a+\overline zb +2i\overline a^2z+\overline aw+\overline ab)\cr &=\;\Im(v+(w-\overline w)\overline a-\overline bz+2i\overline a^2z+ \overline aw+c)\;=\;\Im({\bf v})\,.\cr}$$ An elementary calculation shows that the transformations $(*)$ form a nilpotent Lie group $G^0$ acting freely and transitively on $M$. In particular, $M$ has the structure of a group with the product $(a,b,c){\scriptstyle\,\odot\,}(z,w,v)\colon={\bf(z,w,v)}$ and the unit $o\colon=(0,0,0)$. There is a unique $G^0$-invariant Riemannian metric on $M$ whose restriction to the tangent space $T_oM=\CC\oplus\RR^2\subset\CC^3$ is the one inherited from $\CC^3$. The transformation $s_o:(z,w,v)\mapsto(-z,w,-v)$ is a symmetry at $o$ by Lemma . Hence $M$ is a symmetric CR-manifold. It can be verified that $I_M=G_M=G^0\cup s_oG^0$ is the group of all CR-isometries. The action of $\AUT(M)$ is not proper since this group contains all transformations of the form $(z,w,v)\mapsto(tz,t^2w,t^3v)$, $t\in\RR^*$.The group $Z$ of all translations $(z,w,v)\mapsto(z,w,v+c)$, $c\in\RR$, is in the center of $G^0$ and $M/Z$ is CR-isomorphic to the classical Heisenberg group $\{(z,w)\in\CC^2:\Im\Kl w=z\overline z\}$ that already occurred in Example in a slightly different form. We start with the inclusion of the complex manifolds $$\BB_n\subset\CC^n\subset\PP_n\,,\Leqno{SHSP}$$ where $\BB_n\colon=\BB\colon=\{z\in\CC^n:(z|z)<1\}$ is the euclidean ball and $\PP_n=\PP_n(\CC)$ is the complex projective space with homogeneous coordinates $[z_0,z_1,\dots,z_n]$. We identify every $z\in\CC^n$ with the point $[1,z]\in\PP_n$. It is known that the group $\Aut(\PP_n)$ of all biholomorphic automorphisms of $\PP_n$ coincides with the group of all projective linear transformations $\PSL(n+1,\CC)$. Furthermore, $$\Aut(\BB)\;=\;\{g\in\Aut(\PP_n):g(\BB)=\BB\}\;=\;\PSU(1,n)\,.$$ The group $\Aut(\BB)\subset\Aut(\PP_n)$ has three orbits in $\PP_n$ – the ball $\BB$, the unit sphere $S\colon=\partial\BB$ and the outer domain $\DD\colon=\PP_n\backslash\overline\BB$. Actually, it is known that again $$\Aut(\DD)\;=\;\{g\in\Aut(\PP_n):g(\DD)=\DD\}\;=\;\Aut(\BB)$$ holds. The spaces $\BB_n$, $\CC^n$ and $\PP_n$ are symmetric Hermitian manifolds with constant holomorphic sectional curvature $<0$, $=0$ and $>0$ respectively. Also, $\BB_n$ and $\PP_n$ are dual to each other in the sense of symmetric Hermitian manifolds. The unit sphere $S=\partial\BB=\{z\in\CC^n:(z|z)=1\}$ is a CR-submanifold of $\CC^n$, whose holomorphic tangent space at $a\in S$ is $H_a=\{v\in\CC^n:(a|v)=0\}$. To avoid the totally real case $n=1$ let us assume for the rest of the section that always $n>1$ holds. Then $S$ is a minimal CR-manifold. It is known (compare f.i. ) that for every pair $U,V$ of domains in $S$ and every CR-diffeomorphism $\phi\colon U\to V$ there exists a biholomorphic transformation $g\in\Aut(\PP_n)$ with $\phi=g|U$. In particular this implies $$\AUT(S)\;=\;\{g\in\Aut(\PP_n):g(S)=S\}\;=\;\Aut(\BB)$$ and the maximal compact subgroups of $\AUT(S)$ are in one-to-one correspondence to the points of $\BB$.By restricting the flat Hermitian metric of $\CC^n$, the sphere $S$ becomes an Hermitian CR-manifold. The unitary group $\U(n)$ coincides with $\{g\in\AUT(S):g(0)=0\}$, acts transitively on $S$ by isometric CR-diffeomorphisms and it is easy to see that actually $I_M=\U(n)$ holds. Moreover, $z\mapsto2(a|z)a-z$ defines a symmetry at $a\in S$ and $G_M=\{g\in\U(n):\det(g)=(\pm1)^{n-1}\}$. The group $G^0=\SU(n)$ is simple whereas $I_M=\U(n)$ has center $Z\cong\TT$. For every closed subgroup $A\subset Z$ also $S/A$ is an SCR-space with CR-dimension $n-1$ in a natural way - for instance for $A=\{\pm\id\}$ we get the real projective space $\PP_{2n-1}(\RR)$ and for $A=Z$ the complex projective space $\PP_{n-1}(\CC)$.  The space of (projective) hyperplanes $L\subset\PP_n$ is again a complex projective space of dimension $n$ on which the group $\AUT(S)\subset\Aut(\PP_n)$ acts with three orbits. These consist of all $L$ meeting $S$ in no, in precisely one and in more than one point respectively. Assume $L\cap S\ne\emptyset$ in the following and consider the domain $W\colon=S\backslash L$ in $S$. Let $o\in W$ be the point with the maximal distance from the hyperplane $L\cap\CC^n$. We will see that $W$ has the structure of a symmetric CR-manifold. We claim that there exists a CR-isomorphic model $Q\subset\CC^n$ of $W$ in such a way that $Q$ is closed in $\CC^n$ and such that every CR-diffeomorphism of $Q$ is the restriction of a complex affine transformation of $\CC^n$, that is $\AUT(Q)=\Aff(Q)$. Indeed, choose a transformation $g\in\Aut(\PP_n)$ with $g(L)\cap\CC^n=\emptyset$ and put $Q\colon=g(W)$. We call $Q$ an [*affine model*]{} of $W$. Let us consider the two cases $L\cap\BB=\emptyset$ and $L\cap\BB\ne\emptyset$ separately. Let $U\colon=S\backslash L$ for a hyperplane $L$ with $L\cap\BB\ne\emptyset$, say $U=\{z\in S:z_1\ne0\}$ and $o=(1,0,\dots,0)$. The group $\AUT(U)$ acts transitively on $U$ and has compact isotropy subgroup $\U(n\]4-\]51)$ at $o$. Therefore, $U$ is a symmetric CR-manifold with $$I_U=\AUT(U)\;\cong\;\U(1,n\]4-\]51)$$ and the symmetry $\rho=s_o$ at $o$ is given by $\rho(t,v)=(t,-v)$ for all $(t,v)\in\CC^{1+(n-1)}$. An affine model is $$R\colon=\{z\in\CC^n:(\rho z|z)=1\}\;=\;\{(t,v)\in \CC^{1+(n-1)}:t\overline t-(v|v)=1\}\Leqno{DUAL}$$ with $(t,v)\mapsto (1/t,v/t)$ a CR-diffeomorphism $U\to R$. The universal covering $\widetilde R$ of $R$ is again a symmetric CR-manifold and can be realized via $(s,v)\mapsto(\exp\Kl s ,v)$ as (see also Example ) $$\widetilde R=\{(s,v)\in\CC^{1+(n-1)}:\exp(s+\overline s)-(v|v)=1\}\,.\eqno{\qex}$$ Let $V\colon=S\backslash\{a\}$ for some point $a\in S$, say $a\colon=(0,\dots,0,1)$ and hence $o=-a$. Then $V$ is a cell in $S$ and the Cayley transform $(v,t)\mapsto\big(\sqrt2\,(v/(1-t),(1+t)/(1-t)\big)$ for all $(v,t)\in\CC^{(n-1)+1}$, $t\ne1$, defines a CR-diffeomorphism of $V$ onto the affine model $$N\colon=\big\{(v,t)\in\CC^{(n-1)+1}:t+\overline t=(v|v)\big\}\,.$$ This SCR-space occurs already in Example and as a consequence of we have $$I_N\;=\;G_{\]3N}^0\[3\rtimes\,\U(n\]4-\]51)\steil{and}\AUT(N)\;=\;\Aff(N)\; =\;G_{\]3N}^0\[3\rtimes\,\big(\RR^+\times\U(n\]4-\]51)\big)\,,$$ where the groups $\U(n\]4-\]51)$ and $\RR^+$ act on $N$ by $(v,t)\mapsto(\epsilon v,t)$ and $(v,t)\mapsto(sv,s^2t)$ respectively. Consider for every $s\in\RR^+$, the central subgroup $\Gamma_s\colon=\{(v,t)\mapsto(v,t+ins):n\in\ZZ\}$ of $G_N$. Then the quotient manifold $N_s\colon=N/\Gamma_s$ is an SCR-space diffeomorphic to $\CC^n\times\TT$. But since the groups $\Gamma_s$ and $\Gamma_{\tilde s}$ are not conjugate in $\AUT(N)$ for $s\ne\tilde s$ the manifolds $N_s$ and $N_{\tilde s}$ are not isomorphic as CR-manifolds. We obtain a continuous family of symmetric CR-manifolds that are pairwise non-isomorphic even in the category of CR-manifolds. In analogy to we have inclusions $$U\subset V\subset S\,.\Leqno{INCL}$$ But in contrast to $U$ is not ‘a bounded domain’ (meaning relatively compact) in the cell $V$. In all three cases $M=U,V,S$, the center $Z$ of $I_M$ is either $\TT$ or $\RR$ and the quotient CR-manifold $M/Z$ is $\BB_{n-1}$, $\CC^{n-1}$ and $\PP_{n-1}$ respectively. We call the symmetric CR-manifold $R\cong U$ the [*dual unit sphere*]{} in $\CC^n$ (compare also the discussion at the end of section ). We remark that the action of $\TT$ on $U$ given by scalar multiplication has as quotient the complex manifold $\CC^{n-1}$ which is not biholomorphically equivalent to the bounded domain $\BB_{n-1}=U/Z$. The group $G_M$ is simple in case $M=U,S$ and is nilpotent in case $M=V$. Also, the isotropy subgroup $K$ at a point $o\in M$ acts irreducibly on the tangent space $T_aM$ if $M=U,S$ and $K$ is a finite group in the third case. In all three cases the group $I_M$ acts transitively on the subbundle $\{v\in H_aM:a\in M,\,\|v\|=1\}$ of the tangent bundle $TM$. In particular, $M$ is an SCR-space of constant holomorphic sectional curvature. Let again $M$ be an SCR-space and let $o\in M$ be a fixed point, called base point in the following. Let $G=G_M$ and $K=\{g\in G:g\Kl o=o\}$ be as before. Then $s_o$ is in the center of $K$ and $\sigma(g)\colon= s_og s_o$ defines an involutive group automorphism $\sigma$ of $G$. Therefore $K$ is contained in the closed subgroup $\Fix(\sigma)\subset G$. Let $L$ be the smallest [*open*]{} subgroup of $\Fix(\sigma)$ containing $K$. Then $s_o$ is contained in the center of $L$ and $s_a= s_o$ for all $a\in L\Kl o $. Identify as before $M$ with the homogeneous space $G/K$ and put $N\colon=G/L$. Then we have canonical fibre bundles $$G\buildrel \mu\over\longrightarrow M\buildrel\nu\over\longrightarrow N$$ defined by $g\mapsto gK\mapsto gL$. The typical fibres are $K$ for $\mu$ and the connected homogeneous space $L/K$ for $\nu$. The following statement follows directly from the definition of $\sigma$. The fibration $\mu$ satisfies $\mu\circ\sigma= s_o\circ \mu$. Because of Lemma , $\sigma$ can be seen as a lifting of $s_o$ via $\mu$. On the other hand, $s_o$ can be pushed forward via $\nu$: For every $c\in N$, there exists a unique involutive diffeomorphism $\, s_c\colon N\to N$ such that $\nu\circ s_a= s_c\circ\nu$ for all $a\in M$ with $\nu\Kl a=c$. The differential $d_a\nu$ has kernel $T_a^+M$ in $T_aM$ and hence induces a linear isomorphism from $T_a^-M$ onto $T_cN$. Every $s_c$ has $c$ as an isolated fixed point. Furthermore, $N$ is simply connected if $M$ is simply connected. Since the fibration $\nu$ is $G$-equivariant we have to establish the map $s_c$ only for the base point $c:=\nu\Kl o $ of $N$. But then $s_c$ is given by $gL\mapsto\sigma(g)L$ since $s_o$ can be identified with the map $gK\mapsto\sigma(g)K$ of $M$. The tangent space $T_oF$ of the fiber $F\colon=\nu^{-1}(c)=L\Kl o $ at $o$ is $T_o^+M$ and hence is the kernel of the differential $d_o\nu$. Consequently, the differential $d_c s_c$ is the negative identity on $T_cN$ and hence $c$ is an isolated fixed point of $s_c$. Now suppose that $M$ is simply connected and denote by $\alpha\colon H\to G^0$ the universal covering group of $G^0$. Then the subgroup $\alpha^{-1}(K\cap G^0)$ of $H$ is connected and hence by the construction of $L$ also the subgroup $\alpha^{-1}(L\cap G^0)$ is connected, i.e. $N=G^0/(L\cap G^0)$ is simply connected. The manifold $N$ in Proposition together with all involutive diffeomorphisms $s_c$, $c\in N$, is a symmetric space in the sense of . An interesting case occurs when $N$ has the structure of a symmetric CR-manifold in such a way that $\nu$ is a CR-map and $M$, $N$ have the same CR-dimension. $\nu$ is a partial isometry, i.e. the restriction of $d_a\nu$ to $T_a^-M$ is an isometry for every $a\in M$. $s_c$ is a symmetry at $c$ for every $c\in N$. We say that the SCR-space $M$ has [*symmetric reduction*]{} $N$ if the properties (i) – (iii) are satisfied. For every $g\in L$ and $a=g\Kl o $ we have the commutative diagram $$\diagram{T_o^-M&\mapright{\displaystyle d_o\Phi_g} &T_a^-M\cr \mapdown{\displaystyle\hskip-23pt d_o\nu}&&\mapdown{\[4\displaystyle d_a\nu}\cr T_cN&\mapright{\displaystyle d_c\]1\Psi_g} &T_cN\cr}$$ where for better distinction we denote by $\Phi_g$ the diffeomorphism of $M$ given by $g$ and by $\Psi_g$ the corresponding diffeomorphism of $N$ (that is, $\Phi_g(hK)=ghK$ and $\Psi_g(hL)=ghL\;)$. Put $H_cN\colon=d_o\nu(H_oM)$ and give it the complex structure for which $d_o\nu\colon H_oM\to H_cN$ is a complex linear ismorphism. Furthermore, endow $T_cN$ with the Riemannian metric for which $d_o\nu\colon T_o^-M\to T_cN$ is an isometry. Then the existence of a $G$-invariant almost CR-structure on $N$ with property (i) is equivalent to the condition that all operators $d_c\]1\Psi_g$, $g\in L$, leave the subspace $H_cN$ invariant and are complex linear there. In the same way, a $G$-invariant Riemannian metric on $N$ with property (ii) exists if and only every $d_c\]1\Psi_g$, $g\in L$, is an isometry of $T_cN$. This happens for instance (after possibly changing the metric of $M$) if the group $L$ is compact, or more generally, if the linear group $\{d_c\]1\Psi_g:g\in L\}$ is compact. We notice that as a consequence of Lemma , condition (iii) is automatically satisfied if (i), (ii) hold. Also, in case that for the fibration $\nu\colon M\to N$ there exists a Riemannian metric on $N$ with the property (ii) and such that in addition every $\nu$-fibre is a symmetric Riemannian manifold, the space $M$ is a bisymmetric space in the sense of . The following sufficient condition for the existence of a symmetric reduction is easily seen, we leave the proof to the reader. Suppose that the subgroup $\{g\in I_M:\nu\circ g=\nu\}$ acts transitively on some $\nu$-fibre. Then this group acts transitively on every $\nu$-fibre and $M$ has a symmetric reduction. Not every symmetric CR-space has a symmetric reduction. Consider for instance Example . Then $L^0$ is the subgroup of all $(0,b,0)\in M$ with $b\in\RR$, and $N\colon=G/L$ can be identified with $\CC\times\RR$ in such a way that $\nu(z,w,v)=\big(z,\Re(v)-3\Re(w)\Re(z)\big)$. Furthermore, the action of $L^0$ on $\CC\times\RR$ is given by $(z,t)\mapsto\big(z,t-4\[1b\[1\Re(z)\big)$, $b\in\RR$. This implies that there cannot exist any $G$-invariant Riemannian metric on $N$. Also, there is no CR-structure on $N$ satisfying property (i). In this section we give a Lie theoretical construction of symmetric CR-spaces such that every SCR-space can be obtained is this way. We start with an arbitrary connected Lie group $G^0$ together with an involutive group automorphism $\sigma$ of $G^0$. Then there is a Lie group $G$ with connected identity component $G^0$ and an element $s\in G$ with $G=G^0\,\cup\,s\[2G^0$ and $\sigma(g)=sgs$ for all $g$. Let $\7g$ be the Lie algebra of $G$ and denote by $\tau\colon=\Ad(s)$ the Lie algebra automorphism of $\7g$ induced by $\sigma$ (here and in the following $\Ad$ always refers to the group $G$). Put $$\7l\colon=\Fix(\tau)\steil{and}\7m\colon=\Fix(-\tau)\Leqno{FIXT}$$ Then $\7l$ is a Lie subalgebra of $\7g$ and $\7m$ is a Lie triple system, see . For every $g\in\Fix(\sigma)$, the decomposition $\7g=\7l\oplus\7m$ is invariant under $\Ad(g)$. Now choose a compact subgroup $K\subset\Fix(\sigma)$, an $\Ad(K)$-invariant Riemannian metric on $\7g$ and a linear subspace $\7h\subset\7m$ together with a complex structure $J$ on $\7h$ satisfying the following properties: $\|Jx\|=\|x\|$ for all $x\in\7h$. $K$ contains the element $s$. $\Ad(g)$ leaves the subspace $\7h$ invariant and commutes there with $J$ for every $g\in K$. Notice that $K=\{s,e\}$ with arbitrary $\7h\subset\7m$ and arbitrary $J$ always is an admissible choice. Also, if the compact group $K$ has been chosen, every closed subgroup of $K$ containing $s$ is again an admissible choice. Since $K$ is compact there exists an $\Ad(K)$-invariant decomposition $\7l=\7k\oplus\7n$, where $\7k$ is the Lie algebra of $K$. With $\7p\colon=\7n\oplus\7m$ therefore we get the $\Ad(K)$-invariant decomposition $\7g=\7k\oplus\7p$. Consider the connected homogeneous $G$-manifold $M\colon=G/K$ and declare $o\colon=K\in M$ as base point. In the following we identify the tangent space $T_oM$ in the canonical way with the Hilbert space $\7p$. Denote by $\Phi_g$ the diffeomorphism of $M$ induced by $g$ , that is $$\Phi_g\colon M\to M,\quad hK\longmapsto ghK\,,$$ (we do not require here that the $G$-action is effective, this could be easily achieved by reducing out the kernel of ineffectivity from the beginning). Then for every $g\in K$, the differential $d_o\Phi_g$ is nothing but the restriction of $\Ad(g)$ to $\7p$ and hence there exists a unique $G$-invariant almost CR-structure with $H_oM=\7h$ and also a unique $G$-invariant Riemannian metric on $M$ extending the given Hilbert norm of $T_oM=\7p$. In particular, $s_o\colon=\Phi_s$ is an involutive isometric diffeomorphism of $M$ with fixed point $o$. Clearly, $s_o$ is a symmetry of $M$ at $o$ if and only if $H_o^{-1}M\subset\7m$, where $H_o^{-1}M\subset T_oM$ is the subspace defined in section . A more convenient condition for this is given by the following statement. Let $\7a$ and $\7b$ be the Lie subalgebras of $\7g$ generated by $\7m$ and $\7h$ respectively. Then $\7a=[\7m,\7m]\oplus\7m$ holds, $\7a$ is an $\Ad(K)$-invariant ideal of $\7g$ and $M$ is a minimal symmetric CR-manifold with symmetry $s_o$ at $o$ if and only if $\7g=\7k+\7b$. In case $s_o\colon=\Phi_s$ is a symmetry of $M$ at $o$, the weaker condition $\7g=\7k+\7a$ holds. First notice that $[\7m,\7m]\subset\7l$, $[\7l,\7m]\subset\7m$ by and hence that $\7a=[\7m,\7m]\oplus\7m$ holds. Obviously, $\7a$ is invariant under $\ad(\7l)$ as well as $\ad(\7m)$, i.e. $\7a$ is an ideal in $\7g$. Now suppose that $s_o$ is a symmetry of $M$ at $o$. Then $M$ is a symmetric CR-manifold by the transitivity of the group $G$ and $\7g=\7k+\7a$ follows as in the proof of Proposition . Now suppose that in addition that $M$ is minimal as an almost CR-manifold. Define inductively $\7h^k\colon=\7h^{k-1}+[\7h,\7h^{k-1}]$ and $\7h^0=0$. Then $\7h^k/\7h^{k-1}$ is isomorphic to $H_o^kM$ and $\7g=\7k+\7h^k$ for $k$ sufficiently large, i.e. $\7g=\7k+\7b$. Conversely, suppose that $\7g=\7k+\7b$ holds. Then $M$ is minimal and the differential of $s_0$ is the negative identity on $\7h=H_oM$, i.e. $s_o$ is a symmetry at $o$. As an illustration of the construction principle fix integers $p,q\ge0$ with $n\colon=p+q\ge2$ and set $G\colon=\SU(n)$. Then the corresponding Lie algebra $\7{g=su}(n)$ is a real Hilbert subspace of $\CC^{n\times n}$. Write every $g\in\CC^{n\times n}$ in the form ${a\[1b\choose c\[1d}$ with $a,b,c,d$ matrices of sizes $p\times q, p\times q, q\times p$, $q\times q$ respectively and denote by $\sigma$ the automorphism of $G$ defined by ${a\[1b\choose c\[1d} \mapsto{\;a\;-b\choose-c\;\,d}$. Fix a closed subgroup $K\subset F\colon=\Fix(\sigma)$ and put $M\colon=G/K$ with base point $o\colon=K\in M$. Identify $T_oM$ with the orthogonal complement $\7p$ of $\7k$ in $\7g$ and put $H_oM\colon=\7m\colon=\{{0\[1b\choose c\[20}\in\7p\}\approx\CC^{p\times q}$ with complex structure defined by ${0\[3b\choose c\[40}\mapsto {\,\[90\[{15}ib\choose-ic\[60}$. These data give a unique $G$-invariant Hermitian metric and a unique $G$-invariant almost CR-structure on $M$ with $gK\mapsto\sigma(g)K$ a symmetry at $o\in M$. It is easily seen that $\7m+[\7m,\7m]=\7g$ as well as the integrability condition hold, i.e. $M$ is a compact minimal symmetric CR-manifold with symmetric reduction $N\colon=G/L$. Here $N$ is the Grassmannian $\GG_{p,q}$ of all linear subspaces of dimension $p$ in $\CC^n$ and in particular is a symmetric Hermitian space. If we replace $G=\SU(n)$ by the group $G^d\colon=\SU(p,q)$ and define $\sigma$ by the same formula, then $L=\Fix(\sigma)$ remains unchanged and for every compact subgroup $K\subset L$ we get the two minimal symmetric CR-manifolds $M=G/K$ and $M^d\colon=G^d/K$ which we call dual to each other. In particular, $N^d\colon=G^d/L$ is a bounded symmetric domain and is the dual of the Grassmannian $\GG_{p,q}$ in the sense of symmetric Hermitian spaces. Assume that $M$ is an arbitrary CR-space with base point $o\in M$, not necessarily symmetric to begin with. Assume that $G$ is a Lie group acting smoothly and transitively on $M$ by CR-diffeomorphisms. Let $K\colon=\{g\in G:g\Kl o=o\}$ be the isotropy subgroup at $o$ and denote by $\7k\subset\7g$ the corresponding Lie algebras. Then the canonical map $\theta\colon\7g\to T_oM$ is surjective and has $\7k$ as kernel. Choose a linear subspace $\7h\subset\7g$ such that $\theta\colon\7h\to H_oM$ is a linear isomorphism. Then there is a unique complex structure $J$ on $\7h$ making $\theta|_{\7h}$ complex linear. It is clear that $\7k\oplus\7h=\theta^{-1}(H_oM)$ does not depend on the choice of $\7h$. Let $\8g\colon=\7g\oplus i\7g$ be the complexification of $\7g$ and denote for linear subspaces of $\7g$ its complex linear span by the corresponding [*boldface letter,*]{} that is for instance $\8a=\7a\oplus i\7a$ in case of $\7a$. The complex structure $J$ of $\7h$ extends in a unique way to a complex linear endomorphism $\3J$ of $\8h$. Denote by $\8h^\pm$ the eigenspaces of $\3J$ in $\8h$ to the eigenvalues $\pm i$: $\8h^\pm=\{Jx\pm ix:x\in\7h\}$. Put $\8l\colon=\8k\oplus\8h^-$ for the following. This space does not depend on the choice of $\7h$. The composition of the canonical maps $\7h\hookrightarrow\8h\to\8h/\8h^-$ induces a complex linear isomorphism $\7h\,\cong\,\8h/\8h^-\cong\,\8h^+$. As a consequence, $\7g/\7k\hookrightarrow\8g/\8l$ realizes $T_oM=\7g/\7k$ as a linear subspace of the complex vector space $\8g/\8l$ in such a way that there $H_oM=T_oM\cap iT_oM$ holds. This property will be the key in the proof of Proposition . The CR-structure of $M$ is integrable if and only if $\8l$ is a Lie algebra. Let $\3TM$ be the complexified tangent bundle of $M$ and denote by $\8V=\7V\oplus i\7V$ the complex Lie algebra of all smooth complexified vector fields on $M$, i.e. of all smooth sections $M\to\3TM$. Then $\8K\colon=\{X\in\8V:X_o=0\}$ is a complex Lie subalgebra of $\8V$. The almost CR-structure of $M$ gives a complex subbundle $\3H^{0,1}M\subset\3TM$. Denote by $\8H^{0,1}\subset\8V$ the linear subspace of all vector fields of type $(0,1)$, i.e. $X_a\in\3H^{0,1}_a\]2M$ for all $a\in M$. The integrability conditon for $M$ is equivalent to $\8H^{0,1}$ being a Lie subalgebra of $\8V$. The action of $G$ on $M$ induces a Lie homomorphism $\7g\to\7V$ that uniquely extends to a complex linear homomorphism $\phi\colon\8g\to\8V$. The assumption that $G$ acts by CR-diffeomorphisms implies $\big[\phi(\8g),\8H^{0,1}\big]\subset\8H^{0,1}$. For $\8L\colon=\8K+\8H^{0,1}$ we have $\8l=\phi^{-1}(\8L)$. Now suppose that $M$ is integrable. We claim that $\8l$ is a Lie algebra and consider arbitrary vector fields $X,Y\in\phi(\8l)$. It is enough to show for $Z\colon=[X,Y]$ that $Z_o\in\3H_o^{0,1}\]2M$ holds, i.e. that $Z$ is contained in $\8L$. Write $X=X'+X''$, $Y=Y'+Y''$ with $X',Y'\in\8K\,$ and $X'',Y''\in\8H^{0,1}$. Then we have $$[X',Y'']=[X-X'',Y'']\equiv[X,Y'']\steil{and hence}$$ $$Z\;\equiv\;[X',Y'']+[X'',Y']\;\equiv\;[X,Y'']+ [X'',Y]\;\equiv\;0\steil{mod}\8L\;.\leqno{(*)}$$ Conversely, suppose that $\8l$ is a Lie algebra. Since $G$ acts transitively on $M$ we have $\8L=\8K+\phi(\8l)$. We claim that $M$ is integrable. Consider arbitrary vector fields $X,Y\in\8H^{0,1}$ and write $X=X'+X''$, $Y=Y'+Y''$ with $X',Y'\in\8K\,$ and $X'',Y''\in\phi(\8l)$. We have to show that $Z\colon=[X,Y]$ is contained in $\8H^{0,1}$. Since $G$ acts transitively on $M$ and leaves $\8H^{0,1}$ invariant it is enough to show that $Z_o\in\3H_o^{0,1}\]2M$ holds, i.e. that $Z\in\8L$. But this follows as in $(*)$. Suppose that $M$ in is symmetric with $G\colon=G_M$. Then, if $\7h\subset\7g$ is chosen to be $\Ad(K)$-invariant, $M$ is integrable if and only if $[\8h^-,\8h^-]\subset\8k$. By the choice of $\7h$ we have $[\7k,\7h]\subset\7h$ and hence $[\8k,\8h^-]\subset\8h^-$. The involution $\Ad(s_o)$ of $\7g$ extends to a complex linear involution $\tau$ of $\8g$ with $\8h\subset\Fix(-\tau)$. Therefore, $\8l$ is a Lie algebra if and only if the inclusion $[\8h^-,\8h^-]\subset\8l$ holds, that is $[\8h^-,\8h^-]\;\subset\;\8l\cap\Fix(\tau)=\8k$. We remark that and remain valid for $\8h^+$ in place of $\8h^-$. Let $M=G/K$ be a homogeneous CR-manifold as in Proposition . Suppose, there exist complex Lie groups $\3L\subset\3G$ with Lie algebras $\8l\subset\8g$, where $\8l$ and $\8g=\7g\oplus i\7g$ are as before. Suppose furthermore that $G$ can be realized as real Lie subgroup $G\subset\3G$ in such a way that the corresponding injection $\7g\to\8g$ is the canonical one and such that $G\3L$ is locally closed in $\3G$. Then, if $\3L\cap G=K$ holds and if $\3L$ is closed in $\3G$, $gK\mapsto g\3L$ realizes $M$ as a locally closed generic CR-submanifold of the homogeneous complex manifold $\3M\colon=\3G/\3L$. The assumptions guarantee that $M$ is imbedded in $\3M$ as a locally closed real-analytic submanifold with $H_oM=T_oM\cap iT_oM$ in $T_o\3M$. The result follows since $M$ is a $G$-orbit in $\3M$. In general, the CR-submanifold $M$ is not closed in $\3M$. For instance, if $M$ is a bounded symmetric domain and $G=\Aut(M)$ is the biholomorphic automorphism group of $M$ then $\3M$ can be chosen to be the corresponding compact dual symmetric Hermitian manifold which contains $M$ as an open subset. For the sphere $M\colon=\partial\BB_n\subset\CC^n$ and $G=\AUT(M)$ we may chose $\3M=\PP_n$. On the other hand, for the same sphere $M=\partial\BB_n$ but $G=\U(n)$ we may obtain for $\3M$ the domain $\CC^n\backslash\{0\}$ in $\CC^n$ – but also the Hopf manifold $\Quot{\CC^n\backslash\{0\}}{\alpha^{\ZZ}}$ for some complex number $\alpha$ with $|\alpha|>1$. In the following we illustrate the statements – by various examples. Denote by $\sigma$ the inner automorphism of $\CC^{n\times n}$ given by $(a_{ij})\mapsto(\Kl{\]5-\]51}^{i+j}a_{ij})$. On the contrary to symmetric Hermitian spaces, the CR-structure of a symmetric CR-space does not need to be integrable. For $n\ge3$ let $M\subset\CC^{n\times n}$ be the nilpotent subgroup of all unipotent lower triangular matrices, i.e. of all $a=(a_{ij})$ with $a_{ii}=1$ and $a_{ij}=0$ if $i<j$. Then for the identity $e\in M$, the tangent space $T_eM$ will be identified with the nilpotent algebra $\7g$ of all strictly lower triangular matrices. Denote by $G$ the group generated by all left multiplications with elements from $M$ and denote the restriction of $\sigma$ to $M$ by the same symbol. Then $G=G^0\cup\sigma G^0$ acts transitively on $M$ and there exists a unique $G$-invariant Riemannian metric on $N$ which coincides on $T_eM$ with the one inherited from $\CC^{n\times n}$. Also there is a unique $G$-invariant almost CR-structure on $M$ with $$H_eM=\7h\colon=\{a\in\7g:a_{ij}=0\;\;\4{if}\;j\ne i+1\}$$ and complex structure on $\8h$ inherited from $\CC^{n\times n}$. With this structure $M$ is symmetric and minimal. Because of $[\8h^-,\8h^-]\ne0$ the CR-structure is not integrable. Let $n>d\ge1$ be fixed integers with $d\le n/2$ and denote by $\7g$ the space of all matrices in $\CC^{n\times n}$ having the form . Also, denote by $\7h\subset\7g$ the subspace of all matrices with $z_k=\alpha_j=0$ for all $k>d$ and all $j$. A simple calculation shows that $\7g$ is a real Lie subalgebra of $\CC^{n\times n}$ and that $\7h$ generates $\7g$ as Lie algebra. Identifying $z=(z_1,\dots,z_d)\in\CC^d$ in the obvious way with the corresponding matrix in $\7h$ we get a complex structure $J$ on $\7h\subset\Fix(-\sigma)$. For all $x,y\in\7h$ the identity $[Jx,y]+[x,Jy]=0$ is easily verified. $M\colon\,=\,\exp(\7g)$ is a closed nilpotent subgroup of $\GL(n,\CC)$ invariant under $\sigma$. Precisely as in Example $M$ becomes a symmetric minimal CR-manifold. But this time $[\8h^-,\8h^-]=0$ holds, that is, $M$ is integrable. Furthermore, $\kappa(M)=[(n-1)/(2d-1)]$ and $M$ has CR-dimension $d$. Proposition gives a prescription for a generic embedding of $M$. Let $\3G$ be the connected, simply connected complex Lie group with Lie algebra $\8g=\7g\oplus i\7g$. Then $\exp\colon\8g\to\3G$ is biholomorphic and in particular, $\3L\colon=\exp(\8h^-)$ is a closed abelian complex subgroup of $\3G$ – notice that we have $\8k=0$ in this case. Now, $M$ embeds in the canonical way into the complex manifold $\3M=\3G/\3L$ which is biholomorphic to the complex vector space $\8g/\8h^-$. Easy for explicit calculations is the case $n$ odd – then $\7g\cap i\7g=0$ holds in $\CC^{n\times n}$ and we can realize the complexification $\8g$ within the complex Lie algebra $\CC^{n\times n}$. The commutative subalgebra $\8h^-$ then consists of all matrices obtained from by keeping all $\overline z_1,\overline z_2,\dots,\overline z_d$ and replacing all other entries (including all $z_k$) by $0$. For $n=3$, $d=1$ we find the realization $$M\cong\{(z,w)\in\CC^2:w+\overline w=z\overline z\}$$ which is the classical Heisenberg group, compare Example . For $n=5$, $d=1$ one can show $$\eqalign{M\cong\Big\{(z,w,v_1,v_2,u)\in\CC^5:w+\overline w=z\overline z, \;\;v_1-\overline v_2&=z\overline z(z-\overline z)/6+\overline wz,\cr u+\overline u&=w\overline w+(z\overline v_1+\overline zv_1)+z\overline zz\overline z/4 \Big\}\cr}$$ which is a symmetric CR-manifold with CR-dimension 1, CR-codimension 4 and $\kappa(M)=4$, compare also Example . The symmetry at the origin is given by $(z,w,v_1,v_2,u)\mapsto(-z,w,-v_1,-v_2,u)$. Fix an integer $k>1$ and consider in $\CC^2$ the connected CR-submanifold $$M\colon=\big\{(s,v)\in\CC^2:|s|^{2k}-|v|^2=1\big\}$$ which is a $k$-fold cover of the symmetric CR-manifold $R$ in Example via the map $(s,v)\mapsto(s^k,v)$. Therefore also $M$ is a symmetric CR-manifold and $G^0_M$ is a $k$-fold covering group of $G^0_R=SU(1,1)\cong\SL(2,\RR)$. Denote by $\7g$ the Lie algebra of $G_M$. Then it is known that for $\8g=\7g\oplus i\7g$ there does not exist any complex Lie group $\3G$ into which $G$ admits an embedding induced by the canonical injection $\7g\hookrightarrow\8g$. Therefore the conclusion of Proposition cannot hold for this example.The group $G^0_M$ consists of all transformations $$(s,v)\mapsto\big((as^k+bv)^{1/k},\overline bs+\overline av\big)\,,$$ where $a,b\in\CC$ satisfy $a\overline a-b\overline b=1$. It follows that the action of $G_M$ does not extend to all of $\CC^2$. But it extends to the domain $$D\colon=\big\{(s,v)\in\CC^2:|v|<|s|^k\big\}\;=\;\RR^+M$$ on which the group $\RR^+\!\!\times\!G^0_M$ acts transitively and freely. Suppose that $E$ is a complex vector space of dimension $n$ and that $D\subset E$ is a bounded symmetric domain. Then $\Aut(D)$ is a semi-simple Lie group and at every point of $D$ the corresponding isotropy subgroup is a maximal compact subgroup (see f.i. ). It is known that there exists a complex norm $\|\!\cdot\!\|_\infty$ on $E$ such that $D$ can biholomorphically be realized as the open unit ball $$D=\{z\in E:\|z\|_\infty<1\}\Leqno{REAL}$$ with respect to this norm and that any two realizations of this type are linearly equivalent. In this realization the isotropy subgroup at the origin is linear, i.e. $$\{g\in\Aut(D):g(0)=0\}\;=\;\GL(D)\,.$$ Moreover, $\GL(D)$ is compact. Therefore there exists a $\GL(D)$-invariant complex Hilbert norm $\|\!\cdot\!\|$ on $E$ that we fix for the sequel and hence consider $E$ as a complex Hilbert space in the following. We will also always assume that $D$ is given in the form . For shorter notation we use for the whole section the abbreviation $$\Gamma\colon=\Aut(D)^0\steil{and}K\colon=\GL(D)^0\,.$$ As a generalization of there exists a compact complex manifold $Q$ with $$D\subset E\subset Q\steil{and}\Aut(D)=\{g\in\Aut(Q):g(D)=D\}\,.\Leqno{COMP}$$ $Q$ is the dual of $D$ in the sense of symmetric Hermitian manifolds and $\Aut(Q)$ is a complex Lie group acting holomorphically and transitively on $Q$. The domain $E$ is open and dense in $Q$ and the set $Q\backslash E$ of the ‘points at infinity’ is an analytic subset of $Q$, but not a complex submanifold in general. The boundary $\partial D$ of $D$ is smooth only in the very special case, where also $\|\!\cdot\!\|_\infty$ is a Hilbert norm. Nevertheless, $\partial D$ is a finite union of $\Gamma$-orbits, which are locally closed CR-submanifolds of $E$. Every $K$-orbit $M$ in $\partial D$ is an Hermitian CR-submanifold of $E$ with respect to the metric induced from $E$, where $K$ acts by CR-isometries. We start with an orbit of a special nature: Denote by $S=S(D)$ the set of all extreme points of the closed convex set $\overline D$. The following two statements are well known, but will also be obvious from our discussion below. $S$ is a connected generic CR-submanifold of $E$. Moreover, $S$ is the only compact $\Gamma$-orbit in $\overline D$, consists of all $e\in\overline D$ with $K(e)=\Gamma(e)$ and coincides with the Shilov boundary of $D$. The CR-submanifold $S$ is totally real if and only if $D$ is biholomorphically equivalent to a ‘tube domain’ $\{z\in\CC^n:\Re(z)\in\Omega\}$, where $\Omega\subset\RR^n$ is an open convex cone. In this case $D$ is said to be of tube type. A bounded symmetric domain $D$ is called [*irreducible*]{} if it is not biholomorphically equivalent to a direct product of complex manifolds of lower dimensions. This is known to be equivalent to $K\subset\GL(E)$ acting irreducibly on $E$. There exists (up to order) a unique representation of $D$ as direct product $D=D_1\times\cdots\times D_k$, where all $D_j$ are irreducible bounded symmetric domains and are of the form $D_j=E_j\cap D$ for linear subspaces $E_j\subset E$ with $E=E_1\oplus\cdots\oplus E_k$. Also, there exist direct product representations $S(D)=S(D_1)\times\cdots\times S(D_k)$ for the Shilov boundaries and $K=\GL(D_1)^0\times\cdots\times \GL(D_k)^0$. We call the $D_j$ the [*factors*]{} of $D$. We are now able to formulate the main result of this section. Let $D$ be a bounded symmetric domain. Then the Shilov boundary $S$ of $D$ is a symmetric CR-manifold and the following conditions are equivalent. The Levi cone of $S$ has non-empty interior at every point. $S$ is a minimal CR-manifold. Every smooth CR-function $f$ on $S$ has a unique holomorphic extension to $D$ that has the same smoothness degree on $\overline D$ as $f$. $\Aut(D)=\AUT(S)$. $D$ does not have a factor of tube type. For the proof we use the Jordan theoretic approach to bounded symmetric domains as originated by [Koecher]{} , for details in the following always compare : There exists a Jordan triple product $E^3\to E$, $(x,y,z)\mapsto \{xyz\}$, that contains the full structural information of $D$. This triple product is symmetric bilinear in the outer variables $(x,z)$, conjugate linear in the inner variable $y$ and satisfies certain algebraic and spectral properties. The group $\GL(D)$ of all linear $\|\!\cdot\!\|_\infty$-isometries of $E$ coincides with the group of all linear triple automorphisms, more precisely $$\GL(D)\;=\;\big\{g\in\GL(E):g\{xyz\}=\{gx\,gy\,gz\} \steil{for all}x,y,z\in E\big\}\,.$$ An element $e\in E$ is called a [*tripotent*]{} if $\{eee\}=e$ holds. The set $\Tri(E)$ of all tripotents in $E$ is a compact real-analytic submanifold of $E$ and the group $K$ acts transitively on every connected component of $\Tri(E)$. Except for $\{0\}$ every other connected component of $\Tri(E)$ has positive dimension and is contained in $\partial D$. Tripotents may also be characterized geometrically as ‘affine symmetry points’ of $\overline D$ in the following sense. The element $a\in\overline D$ is a tripotent if and only if there exists an operator $\sigma\in\GL(D)$ with $\sigma(a)=a$, $\sigma(v)=-v$ for all $v\in E$ with $\|a+tv\|\le1$ for all $t\in\TT$. We will postpone the proof of this criterion and fix a tripotent $e\in E$ for a moment. The triple multiplication operator $\mu=\mu_e\in\5L(E)$ defined by $z\mapsto \{eez\}$ is Hermitian and splits $E$ into an orthogonal sum $E=E_1\oplus E_{1/2}\oplus E_0$ of eigenspaces to the eigenvalues $1,1/2,0$, called the [*Peirce spaces*]{} of the tripotent $e$. The canonical projection $P_k\colon E\to E_k$ maps $D$ into itself and clearly is a polynomial in $\mu$, more precisely $$P_1=\mu(2\mu-1)\,,\qquad P_{1/2}=4\mu(1-\mu)\,, \qquad P_0=(1-\mu)(1-2\mu)\,.\Leqno{PROJ}$$ The ‘Peirce reflection’ $\rho\colon=\exp(2\pi i\mu)=P_1-P_{1/2}+P_0$ is contained in $K$, fixes $e$ and leaves $\Tri(E)$ invariant. In particular, also the projection $P_1+P_0$ maps $D$ into itself. The tripotent $e\ne0$ is called [*minimal*]{} if $E_1=\CC e$ holds and is called [*maximal*]{} if $E_0=0$ holds. For instance, the Shilov boundary $S$ of $D$ is just the set of all maximal tripotents. $E$ becomes a complex [*Jordan algebra*]{} (depending on the tripotent $e$) with respect to the commutative product $a\circ b\colon=\{aeb\}$, and $e^2\colon=e\circ e=e$ is an idempotent in $E$. The Peirce space $E_1$ is a unital complex Jordan subalgebra with identity element $e$ and conjugate linear algebra involution $z\mapsto z^*\colon=\{eze\}$. For every $a\in E_1$ and powers inductively defined by $a^{k+1}\colon=a^k\circ a$, $\;a^0\colon=e$, the linear subspace $\CC[a]\subset E_1$ is a commutative, associative subalgebra (notice that the Jordan algebra $E_1$ is not associative in general). The element $a\in E_1$ is called [*invertible*]{} if $a$ has an inverse $a^{-1}\in\CC[a]$. The selfadjoint part $A\colon=\{z\in E_1:z^*=z\}$ of $E_1$ is a [*formally real Jordan algebra*]{}, i.e. a real Jordan algebra such that $x^2+y^2=0$ implies $x=y=0$ for all $x,y\in A$. Clearly, $E_1=A\oplus iA$ holds since the involution is conjugate linear. For all $z\in E_1$ we denote by $\Re(z)\colon=\Quot{(z+z^*)}{2}\in A$ the real part of $z$. The set $Y\colon=\{a^2:a\in A\}$ of all squares in $A$ is a closed convex cone with $A=Y-Y$ and $Y\cap-Y=\{0\}$. The interior $$\Omega\colon=\hbox{\rm Interior of }Y$$ coincides with $\exp(A)\subset A$ and also with the set of all $a\in Y$ that are invertible in $A$. $\Omega$ is an open convex linearly-homogeneous cone in $A$. The sesqui-linear mapping $\Phi\colon E_{1/2}\oplus E_{1/2}\to E$ defined by $\Phi(u,v)=2\{euv\}$ takes values in $E_1$ and satisfies $\Phi(z,z)\in\overline\Omega$ for all $z\in E_{1/2}$, and $\Phi(u,u)=0$ if and only if $u=0$. To indicate the dependence on the tripotent $e\in E$ we also write $E_k(e)$, $k=1,1/2,0$, for the Peirce spaces as well as $A(e)$, $Y(e)$, $\Omega(e)$, $\rho_e$ and $\Phi_e$. Let us illustrate these objects by a typical example. Fix arbitrary integers $p\ge q\ge1$ and consider the complex Hilbert space $E\colon=\CC^{p\times q}$ of dimension $n=pq$. Then $D\colon=\{z\in E:\One-z^*\!z>0\}$ is a bounded symmetric domain in $E$, where $\One=\One_q$ is the $q\times q$-unit matrix. $\|z\|_\infty^2$ is the largest eigenvalue of the Hermitian matrix $z^*\!z$, i.e. $\|z\|_\infty$ may be considered as the operator norm of $z$ if considered as operator $\CC^q\to\CC^p$. The triple product is given by $\{xyz\}=(xy^*\!z+zy^*\!x)/2$ and $K\subset\GL(E)$ is the subgroup of all transformations $z\mapsto uzv$ with $u\in\U(p)$ and $v\in\U(q)$. The Hilbert norm on $E$ given by $\|z\|^2=\tr(z^*\!z)$ is $K$-invariant. $\Tri(E)$ is the disjoint union of the $K$-orbits $S_0,S_1,\dots,S_q$, where $S_k$ is the set of all tripotents $e\in E$ that have matrix rank $k$. In particular, if we write every $z\in E$ as block matrix ${ab\choose cd}$ with $a\in\CC^{k\times k}$ and matrices $b,c,d$ of suitable sizes, then $e={\one_{\]4q}0\choose 0\;0}$ is a tripotent in $S_k$. The corresponding Peirce spaces $E_1$, $E_{1/2}$ and $E_0$ consist of all matrices of the forms ${a0\choose00}$, ${0b\choose c0}$ and ${00\choose0d}$ respectively. Furthermore, $A$ is the real subspace of all Hermitian matrices in $E_1$ and $\Omega\subset A$ is the convex cone of all matrices ${a0\choose00}$ with $a\in\CC^{k\times k}$ positive definite Hermitian. For every $u={0b\choose c0}\in H_eS_k$ we have $\Phi_e(u,u)=2\{euu\}={a0\choose00}$ with $a=bb^*+c^*\!c$. Finally, $S=S_q$ is the Shilov boundary of $D$. $S$ consists of all matrices in $E$ whose column vectors are orthogonal in $\CC^p$, or equivalently, which represent isometries $\CC^q\to\CC^p$. The group $\Gamma$ is the set of all transformations $$z\longmapsto(\alpha z+\beta)(\gamma z+\delta)^{-1} \Steil{with}\pmatrix{\alpha&\beta\cr\gamma&\delta\cr}\in\SU(p,q)$$ and $\alpha,\beta,\gamma,\delta$ matrices of sizes $p\times p$, $p\times q$, $q\times p$ and $q\times q$ respectively. In case $p=q>1$ the groups $\GL(D)$ and $\Aut(D)$ have two connected components, in all other cases these groups are connected. – For the special case $q=1$ we get for $D$ the euclidean ball $\BB$ in $E=\CC^p$ with Shilov boundary the unit sphere $S=S_1=\partial D$ as studied in Example . For every $e\in S$ then $E_1(e)=\CC e$ holds and $E_{1/2}(e)$ is the orthogonal complement of $e$ in the Hilbert space $E$. Two tripotents $e,c\in E$ are called (triple) [*orthogonal*]{} if $c\in E_0(e)$ holds. Then also $e\in E_0(c)$ is true and $e\pm c$ are tripotents. An ordered tuple $(e_1,e_2,\dots,e_r)$ of pairwise orthogonal minimal tripotents in $E$ is called a [*frame*]{} in $E$ if there does not exist a minimal tripotent $e\in E$ that is orthogonal to all $e_j$ in the triple sense. All frames in $E$ have the same length $r$, which is called the [*rank*]{} of the bounded symmetric domain. Every element $a\in E$ has a representation $$a=\lambda_1e_1+\lambda_2e_2+\cdots+\lambda_re_r,\qquad\qquad \|a\|_\infty=\lambda_1\ge\lambda_2\ge\cdots\ge\lambda_r\ge0\,,\Leqno{SING}$$ where $(e_1,e_2,\dots,e_r)$ is a frame depending on $a$. The real numbers $\lambda_j=\lambda_j(a)$ are uniquely determined by $a$ and are called the [*singular values*]{} of $a$. In general, the frame $(e_1,e_2,\dots,e_r)$ is not uniquely determined by $a$. For every $a\in\overline D$ there is a unique representation $$a=e+u\Steil{with}e=\colon\epsilon(a)\in\Tri(E) \steil{and}u\in D\cap E_0(e)\,.\Leqno{TYPI}$$ The Shilov-boundary of $D$ is given by $$S=\{a\in E: \lambda_1(a)=\lambda_2(a)=\cdots=\lambda_r(a)=1\}\,.\Leqno{SHIL}$$ In case $D$ is irreducible, the compact group $K$ acts transitively on the set of all frames and hence any two elements $a,b\in E$ are in the same $K$-orbit if and only if $\lambda_j(a)=\lambda_j(b)$ holds for all $j$. These considerations can be used to prove the following property. If $S$ is totally real, it is rationally convex. For every $e\in S$, the Jordan algebra $E=E_1(e)$ has $e$ as the unit element. It is known (compare for instance or ) that there exists a unique homogeneous polynomial function $N\colon E\to\CC$ of degree $r$ such that the following is satisfied: $z\in E$ is invertible if and only if $N(z)\ne0$ and (ii) $N(e)=1$. $N$ is called the (generic) norm of the unital Jordan algebra $E$. It is known that there exists a character $\chi\colon K\to\TT$ such that $N(gz)=\chi(g)N(z)$ holds for all $g\in K$ and all $z\in E$. On the other hand, for every frame $(e_1,\dots,e_r)$ in $E$ with $e_1+\cdots+e_r=e$ and every complex linear combination $z=z_1e_1+\cdots+z_re_r$ we have $N(z)=z_1z_2\cdots z_r$. This implies the following characterization of the Shilov boundary in the tube type case. $$S=\{z\in\overline D:|N(z)|=1\}\,.$$ In particular, for every $a\in\overline D\backslash S$, the rational function $(N-N(a))^{-1}$ is holomorphic in a neighbourhood of $S$ and has no holomorphic extension to $a$, i.e. the rational convex hull of $S$ in $E$ coincides with $S$. The Shilov boundary $S$ of $D$ in Example is totally real if and only if $p=q$ holds, and then $S=\U(q)$ is the unitary group. For the unit matrix $e\in E=\CC^{q\times q}$ the Jordan product on $E$ is given by $a\circ b=(ab+ba)/2$ and invertibility in the Jordan sense is the same as in the associative sense. In particular, $N(z)=\det(z)$ is the norm of $E$. In case $a$ is a tripotent, every $v\in E$ with $\|a+tv\|\le1$ for all $t\in\TT$ is contained in $E_0(a)$ and we may take $\sigma\colon=-\exp(\pi i\mu_a)=P_1-iP_{1/2}-P_0\in K$, where $\mu_a$ is the triple multiplication operator $z\mapsto \{aaz\}$ on $E$. Conversely, suppose that $a$ satisfies .i-ii and write $a=e+u$ as in . Then $\sigma(u)=-u$ follows from the assumptions. For every $t>1$ with $tu\in\overline D$ we have $a-(1+t)u=e-tu\in\overline D$ and hence $\sigma(a-(1+t)u)=e+(2+t)u\in\overline D$, i.e $\;(t+2)u\in\overline D$ and hence $u=0$. Therefore, $a=e$ is a tripotent. Fix a frame $(e_1,e_2,\dots,e_r)$ in $E$ and consider for all integers $0\le i,j\le r$, the [*refined Peirce spaces*]{}: $$E_{ij}\colon=\big\{z\in E\;:\;2\{e_ke_kz\}=(\delta_{ik}+ \delta_{kj})z\steil{for}1\le k\le r\big\}\,.$$ Then, if we put $e_0\colon=0$, $$E=\bigoplus_{0\le i\le j\le r}\!\!E_{ij}\;,\qquad E_{ii}=\CC\[2e_i \Steil{and}\{E_{ij}E_{jk}E_{kl}\}\subset E_{il}$$ hold for all $0\le i,j,k,l\le r$. Also, $\{E_{ij}E_{kl}E\}=0$ if the index sets $\{i,j\}$ and $\{k,l\}$ are disjoint. Furthermore, $D$ has no tube type factor if and only if $E_{i0}\ne 0$ for $1\le i\le r$. To indicate the dependence of $E_{ij}$ on the given frame we also write $E_{ij}(e_1,e_2,\dots,e_r)$. Now consider a $\Gamma$-orbit $\Sigma\subset\overline D$. Then it is known that there is a tripotent $e$ in $E$ with $\Sigma=\Gamma(e)$ and that $T_e\Sigma=iA\oplus E_{1/2}\oplus E_0$ is the tangent space at $e\in \Sigma$, where the Peirce spaces refer to the tripotent $e$. This implies that $\Sigma$ is a homogeneous generic locally-closed CR-submanifold of $E$ with holomorphic tangent space $H_e\Sigma=E_{1/2}\oplus E_0$. The orbit $M\colon=K(e)$ is a compact submanifold of $\Sigma$ with tangent space $T_eM=iA\oplus E_{1/2}$ and holomorphic tangent space $H_eM=E_{1/2}$. Furthermore, $M=\Sigma\cap\Tri(E)$ and $\epsilon\colon\Sigma\to M$ (compare ) is a fibre bundle with typical fibre $D\cap E_0$. Every connected component $M$ of $\Tri(E)$ is a symmetric CR-manifold. Fix an arbitrary element $e\in M$. For the decomposition of $D$ into a direct product $D_1\times\cdots\times D_k$ of irreducible factors we get a decomposition $e=e_1+\cdots+e_k$ with tripotents $e_j\in E_j$ and a decomposition $M=M_1\times\cdots\times M_k$ with $M_j=\GL(D_j)^0(e_j)$, that is, we may assume without loss of generality that $D$ is irreducible. To begin with, suppose that $M$ is totally real, i.e. $E_{1/2}=0$. Then, by irreducibility, also $E_0=0$ holds and $M=\exp(iA)$ is the ‘generalized unit circle’ in $E_1=E$. Furthermore, $s_e(z)=z^*$ leaves $M$ invariant and hence is a symmetry of $M$ at $e$, i.e. $M$ is symmetric in this case. Now suppose, that $M$ is not totally real, i.e. $H_eM=E_{1/2}\ne0$. Then the Peirce reflection $\rho_e$ maps $M$ into itself and satisfies $H_eM\subset\Fix(-\rho_e)$. Therefore, as soon as we know that $M$ is a minimal CR-manifold we know that $\rho_e$ is a symmetry of $M$ at $e$ and hence that $M$ is symmetric. For the minimality of $M$ it is enough to show that $H_e^2M=iA$ holds, where $H_e^2M$ is as in section . But this is a consequence of the following Proposition . Let $e$ be a tripotent in $E$ and denote by $M$ the connected $e$-component of $\Tri(E)$. Then $\,H_e^2M\subset iA(e)$ and the Levi form $E_{1/2}(e)\times E_{1/2}(e)\to E_1(e)$ of $M$ at $e$ is given by $(u,v)\mapsto-2\{euv\}$, i.e. $L-e=-\Phi_e$. In case $D$ has no tube type factor, the convex hull of $\{\Phi_e(u,u):u\in E_{1/2}(e)\}$ in $A(e)$ has the cone $-\Omega(e)$ as interior and then, in particular, $H_e^2M=iA(e)$ holds. For every $u\in H_eM=E_{1/2}(e)$ define the vector field $X^u$ on $E$ by $X_a^u=4\{aau\}-4\{aa\{aau\}\}$ for all $a\in M$. Then $X_e^u=u$ and $X_a^u\in H_aM$ for all $a\in M$ by . A simple calculation gives $[X^u,X^v]_e=2\{evu\}-2\{euv\}\in iA(e)$. This shows that $-\Phi_e$ is the Levi form at $e\in M$. Let $C$ be the convex hull of $\{\Phi_e(u,u):u\in E_{1/2}(e)\}$. Then $C\subset\overline\Omega(e)$ is clear. For the proof of the opposite inclusion fix an arbitrary element $a\in\overline\Omega(e)$. Then there exists an integer $k\le r$ and a representation $a=\lambda_1e_1+\cdots+\lambda_ke_k$, where $(e_1,\dots,e_k)$ is a family of pairwise orthogonal minimal idempotents in the formally real Jordan algebra $A(e)$ summing up to $e$ and where all coefficients $\lambda_j$ are $\ge0$. This means that we only need to show that $e_j\in C$ for $1\le j\le k$. For this we extend $(e_1,\dots,e_k)$ to a frame $(e_1,\dots,e_r)$ of $E$ and fix $j\le k$. Since by assumption $D$ has no tube type factor we have $E_{j0}\ne0$. But then $\Phi_e(u,v)=2\{e_juu\}$ cannot vanish for all $u,v\in E_{j0}$ since otherwise there would exist a tripotent $c\ne0$ in $E_{j0}$ that is orthogonal to all $e_i$, $1\le i\le r$. This implies $e_j\in C$. $S$ is symmetric by Lemma since $S$ is a connected component of $\Tri(E)$. Fix an element $e\in S$. Then with $Q$ as in there exists an automorphism $\gamma\in\Aut(Q)$, called [*Cayley transformation,*]{} mapping $D$ biholomorphically onto the Siegel domain $$\HH\colon=\big\{(t,v)\in E_1\oplus E_{1/2}:t+\overline t-\Phi(v,v)\in\Omega\big\}$$ in $E=E_1\oplus E_{1/2}$, where the Peirce spaces $E_k$, the cone $\Omega\subset A$ and the Hermitian map $\Phi\colon E_{1/2}\times E_{1/2}\to E_1$ refer to the tripotent $e$. The transformation $\gamma$ satisfies $\gamma^4=\id$,$S\cap\Fix(\gamma)=\{\pm ie\}$, $\gamma(-e)=0$ and is given by $$\gamma(t,v)=\big((e-t)^{-1}\!\circ(e+t),\sqrt 2\,(e-t)^{-1}\!\circ v\big)\,,$$ where $(e-t)^{-1}$ is the inverse in the unital Jordan algebra $E_1$. The domain $$V\colon=\big\{(t,v)\in S:(e-t)\;\steil{is invertible in}E_1\big\}$$ is dense in $S$ and $\gamma$ defines a CR-diffeomorphism from $V$ onto the CR-submanifold $$N\colon=\big\{(t,v)\in E_1\oplus E_{1/2}:t+\overline t=\Phi(v,v)\big\}\;\subset\;\partial\HH$$ of $E$, compare also Example . 12 is an immediate consequence of the definitions (see ) and holds for every CR-manifold. 23. Suppose, $S$ is minimal. Since the Shilov boundary of a bounded symmetric domain of tube type is totally real, $D$ cannot have a factor of tube type. Then by Proposition , the interior of $\{\Phi(v,v):v\in E_{1/2}\}=\{L(v,v):v\in E_{1/2}\}$ coincides with the cone $\Omega$, where $L$ denotes the Levi form of $N$ at $0$ with respect to the obvious identification $(T_0N/H_0N)\otimes\CC \cong E_1$. Let $v\in\Omega$ be an arbitrary vector. By the extension result of , every CR-function $f$ on $N$ extends holomorphically to a small wedge in the direction $v$, in particular, to a neighbourhood of a subset of the type $(N+ \RR_+ v)\cap U$, where $U$ is a neighbourhood of $0$ in $E_1\oplus E_{1/2}$. Furthermore, the wedge extension is of the same smoothness degree as $f$ (due to , see also , Theorem 7.5.1, since $f$ is of slow growth by the Cauchy estimates). Using the transformations $(t,v)\mapsto(st,s^2v)$, $s>0$, we see that $f$ automatically extends to a neighbourhood of $(N+ \RR_+ v)$. Since $v\in\Omega$ is arbitrary, $f$ extends holomorphically to the whole of $\HH$. This implies via the Cayley transformation that every CR-function $f$ on $S$ has a smooth extension to $D\cap S$ which is holomorphic on $D$. It remains to prove that the extension of $f$ is of the same smoothness degree on the boundary $\partial D$. For every $0<r<1$, define the continuous function $f_r$ on $\overline D$ by $f_r(z):=f(rz)$. Then for $r\to1$, the functions $f_r$ converge uniformly on $S$ to $f$. Since $S$ is the Shilov boundary of $D$ the convergence is also uniform on $\overline D$, i.e. $f$ extends continuously to $\overline D$. The smoothness is obtained by the same argument applied to the partial derivatives. 34. $\Aut(D)\subset\AUT(S)$ follows from . Assume (ii) and consider a transformation $g\in\AUT(S)$. Then $g$ extends to a continuous mapping $g\colon\overline D\to E$ which is holomorphic on $D$. As a consequence of the maximum principle, $g(\overline D)$ is contained in the closed convex hull of $g(S)=S$, which is $\overline D$. By the same argument, $h\colon=g^{-1}$ extends to a continuous map $h\colon\overline D\to\overline D$ which is holomorphic on $D$. Then $h\circ g=g\circ h=\id$ shows $g\in\Aut(D)$. 45. Suppose $D$ has a factor of tube type. Then $S$ is a direct product of a CR-manifold with a totally real CR-manifold of positive dimension. In particular, $\AUT(S)$ cannot be a Lie group of finite dimension like $\Aut(D)$. 51 follows from Proposition . Theorem  together with Proposition  can be used to calculate both polynomial and rational convex hulls of $S$ explicitly. In particular, they are finite unions of disjoint connected real-analytic CR-submanifolds (forming a stratification in the sense of Whitney). We call a smooth function on such a union a CR-function if it is CR on each single CR-submanifold (this notion is independent of the partition into CR-submanifolds). Let $E=E_1\oplus E_2$ be the canonical splitting such that $D_1:=D\cap E_1$ is of tube type and $D_2:=D\cap E_2$ has no tube type factor. Denote by $S_1\subset \partial D_1$ and $S_2\subset \partial D_2$ the corresponding Shilov boundaries. Then the following holds. Both convex and polynomial convex hulls of $S$ coincide with $\overline D$. The rational convex hull $\hat S$ of $S$ is given by $\hat S = S_1\times\overline D_2$. Every smooth CR-function $f$ on $S$ extends uniquely to a CR-function on $\hat S$ of the same smoothness degree. Since $S$ is the Shilov boundary of $D$, $|P(z)|\le\|P\|_S$ holds for every holomorphic polynomial $P$ and every $z\in \overline D$. Hence $\overline D$ is contained in the polynomial convex hull of $S$. The latter is always contained in the convex hull of $S$, which is $\overline D$. This proves (i) (the statement about the convex hull also follows from the classical Krein-Milman theorem). For the rational convex hull, we obtain $\hat S_1 = S_1$ by Proposition . This shows $\hat S\subset S_1\times\overline D_2$. On the other hand, every rational function on $E_2$, holomorphic in a neighbourhood of $S_2$, is continuous on $\overline D_2$ by Theorem . This implies the opposite inclusion $\hat S\supset S_1\times\overline D_2$ and therefore (ii). Finally, let $f$ be a CR-function on $S$. Then, for every $z_1\in S_1$, Theorem  guarantees that $f$ has a unique smooth extension $\hat f$ to $\{z_1\}\times\overline D_2$ which is holomorphic on $\{z_1\}\times D_2$. By the smoothness, $\hat f$ is CR on each CR-submanifold of the boundary $\{z_1\}\times\partial D_2$. To prove the smoothness of $\hat f$ on $\hat S$, we fix a convergent sequence $z_1^m\to z_1^0$. Then $\hat f(z_1^m,\cdot)$ converges to $\hat f(z_1^0,\cdot)$ uniformly on $S_2$ and therefore on $\partial D_2$, because $S_2$ is the Shilov boundary. This shows that $\hat f$ is continuous. The same argument applied to the partial derivatives of $\hat f$ shows that $\hat f$ is of the same smoothness degree as $f$. Since $S_1$ is totally real, the holomorphic tangent spaces to every CR-submanifold of $\hat S$ are contained in $E_2$. This shows that $\hat f$ is CR and finishes the proof of (iii). From the classification of all irreducible bounded symmetric domains into the 6 types [**I, II,…,VI**]{} (compare f.i. 4.11) it follows that there are precisely the following 3 types of irreducible non-tube domains: with $p>q\ge1$ arbitrary integers. Then, as in Example , $E=\CC^{p\times q}$ and $D=\{z\in E:\One-z^*\!z>0\}$ is the bounded symmetric domain of rank $r=q$, where $\One$ is the $q\times q$-unit matrix. The Shilov boundary of $D$ is the set $S\colon=S_q$ of all matrices in $E$ whose column vectors are orthogonal in $\CC^p$, i.e. $$S=\{z\in\CC^{p\times q}:z^*\!z=\One\}\,.\Leqno{ZITA}$$ On $S$ the group $\SU(p)$ acts transitively by matrix multiplication from the left with isotropy subgroup $\One\times\SU(p\!-\!q)$ at $e\colon={\one\choose0}\in S$, i.e. $S=\SU(p)/(\One\times\SU(p\!-\!q))$ is simply connected, has CR-dimension $(p-q)q$ and CR-codimension $q^2$. Also, $\AUT(S)\approx\U(p,q)/\TT$ is connected. Every closed subgroup $L\subset\U(q)$ acts freely on $S$ by matrix multiplication from the right and $S/L$ again is a symmetric CR-manifold of the same CR-dimension in a natural way. For $L=U(q)$ we get the Grassmannian of all $q$-planes in $\CC^p$ which is the reduction of $S$ as defined in section . The typical fibre of the reduction map is the group $\U(q)$. with $p=2q+1$ an arbitrary odd integer $>3$. Let $E\colon=\{z\in\CC^{p\times p}:z'=-z\}$ and define the bounded domain $D\subset E$ as well as the Jordan triple product by the same formulae as for [**I**]{}$_{p,q}$. Then again $D$ is a bounded symmetric domain of rank $r=q$ and $\Gamma\subset\GL(E)$ is the group of all transformations $z\mapsto uzu'$ with $u\in\U(p)$. For $j\colon={\;0\hskip6pt\one\choose -\one\;0}\in\CC^{2q\times 2q}$ the matrix $e\colon={j\;0\choose0\;0}\in\CC^{p\times p}$ is in $S\colon=S_q$ and $\Gamma$ has isotropy subgroup $\Sp(q)\times\TT$ at $e$. Therefore $S=SU(p)/(\Sp(q)\times1)$ is simply connected. The holomorphic tangent space $H_eS$ is the space of all $z={0\;u\choose v\;0}\in E$ with $u\!=\!-v'\in\CC^{2q}$, i.e. $S$ has CR-dimension $2q$ and CR-codimension $q(2q-1)$. The reduction is the projective space $\PP_{2q}(\CC)=\SU(p)/S(\U(2q)\times\TT)$ and $\SU(2q)/\Sp(q)$ is the typical reduction fibre. The group $\AUT(S)\;\approx\;\SO^*(2p)/\{\pm1\}$ is connected (compare 451 and 518 for the non-compact type D III). Here $D$ is the exceptional bounded symmetric domain of dimension 16 (non-compact type E III on 518 of ). $D$ has rank 2 and the Shilov boundary $S\colon=S_2$ has CR-dimension 8 and CR-codimension 8. On $S$ the group $\Spin(10)$ acts transitively and the reduction $\widetilde S$ of $S$ is the symmetric Hermitian manifold $\SO(10)/(\SO(2)\times\SO(8))$, the complex nonsingular quadric of dimension 8. The group $\AUT(S)$ is a non-compact simple exceptional real Lie group of type $E_6$ and has dimension 78. As a generalization of Example also the dual of (compare and section ) can be described explicitely. Fix $e={\one\choose0}\in S$ and denote by $\rho=\rho_e$ the corresponding Peirce reflection of $E=\CC^{p\times q}$. Then $\Fix(\rho)=\CC^{q\times q}$ and $\Fix(-\rho)=\CC^{(p-q)\times q}$. On $$R=\{z\in\CC^{p\times q}:\rho\Kl z ^*\!z=\One\}$$ the group $\U(q,p\!-\!q)$ acts transitively from the left with compact isotropy subgroup $\One\times\U(p\!-\!q)$ at $e$. Therefore there is a unique $\U(q,p\!-\!q)$-invariant Riemannian metric on $R$ which coincides on $T_eR$ with the one induced from $E$. The restriction of $\rho$ to $R$ is a symmetry of $R$ at $e$, i.e. is a symmetric CR-manifold. Again, every closed subgroup $L\subset\U(q)$ acts freely on $R$ from the right and $S/L$ is a symmetric CR-manifold of the same CR-dimension. For $L=U(q)$ we get the bounded symmetric domain of type [**I**]{}$_{q,p-q}$, the reduction of $R$. 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--- abstract: 'Associated to any supermanifold is a filtration by spaces, referred to as *thickenings*. It is the objective of this article to study them up to a certain equivalence and then up to isomorphism in the complex-analytic setting. We study them from two points of view: (1) as structures embedded in supermanifolds and (2) abstractly. Throughout, we will be guided by the goal to clarify and address the question: *when does a given thickening embed in a supermanifold?* Such a question was, in principle, first studied by Eastwood and LeBrun. In this article we begin with a pedagogical account of their study, after which we further study thickenings in supergeometry and present a classification of thickenings of a given order. As a complement to our study, we comment on the moduli problem for complex supermanifolds and consider the analogous problem for thickenings. Finally, to illustrate the ideas in this article, we conclude by describing obstructed thickenings of the complex projective plane.' author: - Kowshik Bettadapura bibliography: - 'Bibliography.bib' title: ' Obstructed Thickenings and Supermanifolds\' --- Introduction ============ Supermanifolds are constructs which have arisen from considerations of supersymmetry in physics. In the context of superstring theory, super Riemann surfaces and their moduli are of particular interest, for it is on the moduli space of super Riemann surfaces where one calculates scattering amplitudes. The methods for undertaking these calculations are sensitive to the ‘obstruction theory’ of the moduli space and so, for at least that reason, one can find motivation for studying obstruction theory more generally for supermanifolds.\ In this article we study obstruction theory with the view to further understand the classification of supermanifolds. This problem, of classifying supermanifolds, was itself initiated by Batchelor in [@BAT] where a classification is obtained in the smooth setting. In the complex-analytic setting, the analogous problem becomes more subtle and inroads into the classification problem are made in [@GREEN; @YMAN; @ONISHCLASS]. The moduli variety of complex supermanifolds was constructed and studied by Onishchik in a particular instance in [@ONISHMOD] and then in more generality in [@ONISHCLASS]. We review these studies here and, in line with the overall theme of this article, propose that this variety will admit a filtration.\ The primary focus of this article however is on studying the obstruction theory of a more fundamental construct than that of a supermanifold, being that of a *thickening*. These thickenings are more fundamental in the sense that: *associated to any supermanifold will be a thickening, but not necessarily conversely*. In taking inspiration from the study of thickenings in complex geometry by Griffiths in [@GRIFF], analogous notions are developed by Eastwood and LeBrun in [@EASTBRU]. There, the rudiments of an obstruction theory incorporating thickenings is laid out and it is this obstruction theory that we investigate further in this article. We begin with a pedagogical review of the work in [@EASTBRU] by reference to methods similar to those of Kodaira-Spencer deformation theory, expounded in [@KS]. To justify this method, we re-derive one of the main results in [@EASTBRU] pertaining to the identification of the space of obstructions to finding thickenings, leading then to a classification.\ \ In relation to the classification problem for supermanifolds, the idea here behind studying thickenings is the following: in order to classify supermanifolds, one might instead try and classify these thickenings and throw out those which will not ‘give rise’ to a supermanifold. Along this vein, our first step is in observing that a thickening will come in two basic flavours: *unobstructed* and *obstructed*. The obstructed thickenings will never give rise to a supermanifold whereas the unobstructed thickenings *may* or *may not*. In this way we can relate the obstruction theory for supermanifolds with that for thickenings. These observations are formulated at the level of cohomology, leading to meaningful decompositions of certain cohomology groups which we describe here.\ \ The main new result in this article is in the construction of obstructed thickenings of the complex projective plane. We infer the existence of obstructed thickenings of ${\mathbb{C}}{\mathbb{P}}^2$ equipped with a rank 3, holomorphic vector bundle $E$ which is: (1) split; and (2) non-split but decomposable. Outline and Summary ------------------- This article is divided into eight sections, including this introduction and some concluding remarks. The contents of the other sections are briefly summarised below.\ \ In Section \[2\] the notation is set and some relevant, background theory is provided so as to be called upon as needed throughout this article. In Section \[3\] we discuss thickenings as developed by Eastwood and LeBrun in [@EASTBRU] and set the motivating theme, guiding considerations in the sections to come. Importantly, the notion of an *obstructed thickening* is defined and one of the central results in [@EASTBRU] is stated here in Theorem \[jnvjknknvkjnvk\], pertaining to the existence of obstructed thickenings. Section \[4\] is devoted to a proof of this result (Theorem \[jnvjknknvkjnvk\]) by methods similar to those found in Kodaira-Spencer deformation theory, detailed in [@KS]. This involves describing a construction of thickenings by means of gluing and working directly with this gluing data. As a prelude, to justify our argument, we provide a ‘direct’ proof of Theorem \[jnvjknknvkjnvk\] in a particular instance. To then give some further context in which this theorem can sit, we look at the problem of classifying thickenings of a given order, culminating in Theorem \[o4993uf0oijwoijfwojpw\]. In Section \[5\], we recall the second object in title of this article: *supermanifolds*. After a brief interlude on obstruction theory for supermanifolds, we look to integrate this obstruction theory with the more general theory for thickenings. This is expressed in the form of a decomoposition of certain 1-cohomology groups valued in appropriate sheaves of abelian groups. A precise formulation is given in . We have so far been studying thickenings up to a certain equivalence that is stronger than isomorphism. In Section \[6\] we explore moduli problems, which considers these objects—supermanifolds and thickenings, up to isomorphism. The moduli problem for complex supermanifolds was studied by Onishchik in [@ONISHCLASS] from an analytic point of view and by Vaintrob in [@VAIN] from an algebraic point of view. We propose here a relation between these viewpoints in Claim \[fkrpokvpokvp4k\], but will not attempt to prove this claim. We then consider the moduli variety constructed by Onishchik in [@ONISHCLASS] and propose that it admit a filtration in analogy with supermanifolds in Conjecture \[dnohf3fh839hf3h\]. We conclude by formulating the moduli problem for thickenings and argue that the previous-mentioned decomposition in will hold here, i.e., up to isomorphism. In the penultimate section, Section \[7\], we present the main new result in this article, being the construction of obstructed thickenings of the complex projective plane ${\mathbb{C}}{\mathbb{P}}^2$. We consider the case where ${\mathbb{C}}{\mathbb{P}}^2$ is equipped with a rank 3, holomorphic vector bundle which is either: (1) split; and (2) non-split but decomposable. Acknowledgements {#acknowledgements .unnumbered} ---------------- I would like to acknowledge the many helpful discussions and support I have had from Peter Bouwknegt and Bryan Wang. I would also like to acknowledge the Australian Postgraduate Award for providing financial support during the time when much of the work in this article was undertaken. Preliminaries {#2} ============= Supermanifolds -------------- The definition of a supermanifold may be quite succinctly given in the framework of algebraic geometry. We refer to [@QFAS] where this point of view is emphasised. We firstly have: \[mironufuir\] *A complex manifold $M$ of dimension $n$ is a locally ringed space $(|M|, {\mathcal{C}}_M)$, for $|M|$ a topological space, which is locally isomorphic to $({\mathbb{C}}^n, {\mathcal{C}}_{{\mathbb{C}}^n})$, where ${\mathcal{C}}_{{\mathbb{C}}^n}$ denotes the sheaf of (germs of) holomorphic functions on ${\mathbb{C}}^n$.* The isomorphism mentioned in the definition of a complex manifold in Definition \[mironufuir\] above is in the category of *locally ringed spaces*. Note that a (real) smooth manifold may be defined analogously. The definition of *supermanifold* is now somewhat natural: \[ncnrnv322rnk\] In what follows we make sense of describing a supermanifold as being ‘modelled’ on a manifold $M$ and vector bundle $E{\rightarrow}M$. The Split Model {#fjf9j490fj40fj40} --------------- A supermanifold ${\mathfrak{X}}$ of dimension $(p|q)$ is said to have *even* dimension $p$ and *odd* dimension $q$. For the purposes of this article it will be convenient to think about a supermanifold ${\mathfrak{X}}$ as being modelled on two bits of data: a manifold $M$, called the *reduced space*, and a vector bundle $E{\rightarrow}M$ called the *modelling bundle*. Let ${\mathcal{E}}$ denote the sheaf of sections of $E$. To see how to make sense of a supermanifold ${\mathfrak{X}}$ as being ‘modelled’ on $(M, E)$, recall from Definition \[ncnrnv322rnk\](ii) the epimorphism $\iota^\sharp : {\mathcal{O}}_M {\rightarrow}{\mathcal{C}}_M$ and set ${\mathcal{J}}= \ker\iota^\sharp$. It is called the *nilpotent ideal*. By Definition \[ncnrnv322rnk\](iii) we see that the quotient ${\mathcal{J}}/{\mathcal{J}}^2$ will be a sheaf of locally free ${\mathcal{C}}_M$-modules and hence correspond to the sheaf of sections of a vector bundle, say $E{\rightarrow}M$. The structure sheaf ${\mathcal{O}}_M$ of ${\mathfrak{X}}= (M, {\mathcal{O}}_M)$ will then be locally isomorphic to the sheaf of sections $\wedge^{\bullet}{\mathcal{E}}$ of the bundle of exterior algebras $\wedge^{\bullet}E$. If there exists an isomorphism ${\mathcal{J}}/{\mathcal{J}}^2\stackrel{\cong}{{\rightarrow}}{\mathcal{E}}$, then we say ${\mathfrak{X}}$ will be *modelled* on a pair $(M, E)$ and use the notation ${\mathfrak{X}}_{(M, E)}$.[^1] \[ifh4fgiuho48h04\] *For a supermanifold ${\mathfrak{X}}_{(M, E)}$, a choice of isomorphism ${\varphi}: {\mathcal{J}}/{\mathcal{J}}^2\stackrel{\cong}{{\rightarrow}}{\mathcal{E}}$ is referred to as a *coframe* for ${\mathfrak{X}}_{(M, E)}$.* We will justify the terminology ‘coframe’ in the above definition later on in this article. *The supermanifold ${\mathfrak{X}}_{(M, E)}$ is said to be *smooth* (resp. *holomorphic*) if the pair $(M, E)$ consists of a smooth (resp. complex-analytic) manifold and smooth (resp. holomorphic) vector bundle.* In this article we will be concerned with holomorphic supermanifolds.\ \ Now from any given pair $(M, E)$ we may construct a supermanifold by simply taking the structure sheaf ${\mathcal{O}}_M$ to be $\wedge^{\bullet}{\mathcal{E}}$. This gives what is termed the *split model* and is denoted here by $\Pi E$. As a locally ringed space $\Pi E = (M, \wedge^{\bullet}{\mathcal{E}})$. A supermanifold ${\mathfrak{X}}$ modelled on a pair $(M, E)$ is locally isomorphic to $\Pi E$. To see why, note by Definition \[ncnrnv322rnk\](iii) that ${\mathfrak{X}}$ will be locally isomorphic to $\wedge^{\bullet}({\mathcal{J}}/{\mathcal{J}}^2)$. If ${\mathfrak{X}}$ is modelled on $(M, E)$, then ${\mathcal{J}}/{\mathcal{J}}^2\cong {\mathcal{E}}$. Therefore $\wedge^{\bullet}({\mathcal{J}}/{\mathcal{J}}^2)\cong \wedge^{\bullet}{\mathcal{E}}$ and so ${\mathfrak{X}}$ and $\Pi E$ are locally isomorphic. *Since ${\mathcal{O}}_M$ is ${\mathbb{Z}}_2$-graded then, if ${\mathcal{O}}^{{\mathrm{ev}}/{\mathrm{odd}}}_M$ denote the even and odd components respectively, we have: ${\mathcal{O}}^{{\mathrm{ev}}/{\mathrm{odd}}}_M \cong_{\mathrm{loc.}}\wedge^{{\mathrm{ev}}/{\mathrm{odd}}}{\mathcal{E}}$ as sheaves of ${\mathcal{C}}_M$-modules.* Embedded Thickenings {#kd4k9fk30fk30kf3p} -------------------- It is our intent in this article to study thickenings independently of supermanifolds, in a suitable sense. Before giving their definition however, we will present the following construction associated to any supermanifold. It may be found in [@BER; @YMAN]. \[nruiciurhf94h9h\] Let ${\mathfrak{X}}= (M, {\mathcal{O}}_M)$ be a supermanifold and denote by ${\mathcal{J}}$ the nilpotent ideal. Recall that it is identified with the kernel of the morphism ${\mathcal{O}}_M \twoheadrightarrow {\mathcal{C}}_M$. That is, we have an exact sequence of sheaves of ${\mathcal{C}}_M$-modules: $$0 {\rightarrow}{\mathcal{J}}\hookrightarrow {\mathcal{O}}_M \twoheadrightarrow {\mathcal{C}}_M{\rightarrow}0.$$ Consider now the ${\mathcal{J}}$-adic filtration on ${\mathcal{O}}_M$, i.e., ${\mathcal{O}}_M\supset {\mathcal{J}}\supset {\mathcal{J}}^2\supset\cdots$. If ${\mathfrak{X}}$ has odd-dimension $q$, then the ${\mathcal{J}}$-adic filtration will have length $q$, i.e., ${\mathcal{J}}^{q+1} = 0$. Now set ${\mathcal{O}}_M^{(k)} := {\mathcal{O}}_M/{\mathcal{J}}^{k+1}$. Then we will obtain, in this way, a locally ringed space ${\mathfrak{X}}^{(k)} = (M, {\mathcal{O}}^{(k)}_M)$, with: ${\mathfrak{X}}^{(k)}\subset {\mathfrak{X}}$. We now have the following remarks: if $k =0$, then we recover the underlying, complex manifold since ${\mathcal{O}}_M^{(0)} = {\mathcal{O}}_M/{\mathcal{J}}= {\mathcal{C}}_M$. If we fix a pair $(M, E)$, then ${\mathfrak{X}}^{(1)}_{(M, E)}$ and $\Pi E^{(1)}$ coincide. Finally, if $k \geq q$ then clearly ${\mathcal{O}}_M^{(k)} = {\mathcal{O}}_M$ and so ${\mathfrak{X}}^{(k)} = {\mathfrak{X}}$. Note that the sheaf ${\mathcal{O}}_M^{(k)}$, for $0 < k < q$, will not be locally isomorphic to a sheaf of exterior algebras, and so the thickenings ${\mathfrak{X}}^{(k)}$ will not be supermanifolds in the sense of Definition \[ncnrnv322rnk\]. Regarding ${\mathfrak{X}}_{(M,E)}$, we have by construction a filtration: $$\begin{aligned} M \subset \Pi E^{(1)} \subset {\mathfrak{X}}_{(M, E)}^{(2)}\subset \cdots\subset {\mathfrak{X}}^{(q-1)}_{(M, E)} \subset {\mathfrak{X}}_{(M, E)}. \label{bcbfjhbvrbjh}\end{aligned}$$ In light of we have the following definition, motivated by [@EASTBRU]: \[nkcdkjkfoieoe\] *The locally ringed space ${\mathfrak{X}}_{(M, E)}^{(k)}$ in is referred to as the *$k$-th order thickening of $M$ in ${\mathfrak{X}}_{(M, E)}$*.* Another way to think of a $k$-th order thickening ${\mathfrak{X}}^{(k)}_{(M, E)}$ in ${\mathfrak{X}}_{(M, E)}$ is as a thickening of $M$ equipped with an embedding $\iota: {\mathfrak{X}}^{(k)}_{(M, E)} \subset{\mathfrak{X}}_{(M, E)}$. Hence we may refer to such a thickening as an *embedded* thickening. As mentioned at the start of this section, we are interested in thickenings ‘independently’ of supermanifolds and by this we mean thickenings that need not be embedded in some higher structure such as a supermanifold, i.e., an *abstract* thickening. The focus of this article is then: *given an abstract thickening, will there exist an embedding of it into some higher structure, e.g., such as a thickening of higher order, or a supermanifold?*\ \ In the next section we will look at the tangent sheaf of a supermanifold and the split model. The Tangent Sheaf ----------------- Let ${\mathcal{O}}_U$ be a sheaf of (super-)commutative rings on a topological space $U$. Then the sheaf of derivations ${\mathrm{Der}}~{\mathcal{O}}_U$ is locally free, as shown in [@LEI; @QFAS]. The *tangent sheaf* of a supermanifold ${\mathfrak{X}}= (M, {\mathcal{O}}_M)$ is then defined to be ${\mathfrak{T}}_{\mathfrak{X}}:={\mathrm{Der}}~{\mathcal{O}}_M$. It is locally free and identified with the sheaf of sections of the *tangent bundle* of ${\mathfrak{X}}$. We will work directly with the sheaf ${\mathfrak{T}}_{\mathfrak{X}}$ rather than the bundle in this article. Our first observation here is: *since the split model $\Pi E$ is ${\mathbb{Z}}$-graded then so is the tangent sheaf ${\mathfrak{T}}_{\Pi E}$*. As a supermanifold ${\mathfrak{X}}$ is more generally only ${\mathbb{Z}}_2$-graded, it follows that the tangent sheaf ${\mathfrak{T}}_{\mathfrak{X}}$ is ${\mathbb{Z}}_2$-graded. It will admit more structure than simply a ${\mathbb{Z}}_2$-grading however. As described in [@ONISHCLASS], it is in fact a filtered ${\mathcal{O}}_M$-module. Let ${\mathfrak{X}}$ be a $(p|q)$-dimensional supermanifold and set: $${\mathfrak{T}}_{\mathfrak{X}}[k] := \left\{ \nu\in {\mathrm{Der}}~{\mathcal{O}}_M \mid \nu({\mathcal{O}}_M)\subset {\mathcal{J}}^k~~\mbox{and}~~\nu({\mathcal{J}}^l)\subset {\mathcal{J}}^{k+l}~\mbox{for all $l>0$}\right\}.$$ Then since ${\mathcal{J}}\supset {\mathcal{J}}^2\supset\cdots$, it follows that ${\mathfrak{T}}_{\mathfrak{X}}[k]\supset {\mathfrak{T}}_{\mathfrak{X}}[k+1]$ for all $k$. Set ${\mathfrak{T}}_{\mathfrak{X}}[-1] := {\mathfrak{T}}_{\mathfrak{X}}$. We now have: $${\mathfrak{T}}_{\mathfrak{X}}= {\mathfrak{T}}_{\mathfrak{X}}[-1] \supset {\mathfrak{T}}_{\mathfrak{X}}[0] \supset {\mathfrak{T}}_{\mathfrak{X}}[1]\supset\cdots\supset {\mathfrak{T}}_{\mathfrak{X}}[q]\supset {\mathfrak{T}}_{\mathfrak{X}}[q+1]=0.$$ In this way ${\mathfrak{T}}_{\mathfrak{X}}$ is a filtered ${\mathcal{O}}_M$-module. Now let ${\mathfrak{X}}$ be a supermanifold modelled on $(M, E)$. Then it is locally isomorphic to its split model $\Pi E$. Denote by ${\mathfrak{T}}_{(M, E)}$ its tangent sheaf. We have the following result, an argument for which we refer to [@ONISHCLASS p. 311]: \[jd90d098jdoijp3\] There exists a short-exact sequence of sheaves of ${\mathcal{C}}_M$-modules: $$0 {\rightarrow}{\mathfrak{T}}_{(M, E)}[k+1] \hookrightarrow {\mathfrak{T}}_{(M, E)}[k] \twoheadrightarrow {\mathfrak{T}}_{\Pi E}[k]{\rightarrow}0$$ for all $k\geq-1$. As for the tangent sheaf of the split model itself, we have the following: \[hjbbjhhveyjj\] There exists a short-exact sequence of sheaves of ${\mathcal{C}}_M$-modules: $$\begin{aligned} 0 {\rightarrow}\wedge^{k+1}{\mathcal{E}}\otimes{\mathcal{E}}^\vee \hookrightarrow {\mathfrak{T}}_{\Pi E}[k] \twoheadrightarrow {\mathfrak{T}}_M\otimes\wedge^k{\mathcal{E}}{\rightarrow}0 \label{jkkbkbkbk}\end{aligned}$$ for all $k\geq-1$. Thickenings in Supergeometry {#3} ============================ In Construction \[nruiciurhf94h9h\] we obtained locally ringed spaces from a given supermanifold ${\mathfrak{X}}$. To indicate this dependence on ${\mathfrak{X}}$, we referred to these spaces as *thickenings in ${\mathfrak{X}}$* in Definition \[nkcdkjkfoieoe\]. In the present section we will give a treatment of thickenings in the spirit of [@EASTBRU]. Preliminaries: Thickenings of Complex Manifolds ----------------------------------------------- We refer to [@EASTBRU; @GRIFF] for a more complete treatment on the theory of thickenings of complex manifolds. Here only the basic notions will be described with the goal to motivate similar considerations in the context of supergeometry.\ \ Firstly fix a complex manifold $M = (|M|, {\mathcal{C}}_M)$, thought of here as a locally ringed space (see Definition \[mironufuir\]). \[djcnknckjncjkrccr\] An instructive example of a thickening is provided by holomorphically embedded sub-manifolds: \[jcnkrncjkrnckjr\] Suppose $N$ is a complex manifold and $M\subset N$ is a holomorphically embedded, co-dimension one submanifold. If ${\mathcal{C}}_N$ denotes the structure sheaf of $N$, consider the ideal ${\mathcal{J}}_M\subset {\mathcal{C}}_N$ comprising those functions on $N$ which vanish on $M$. Then an $m$-th order thickening of $M$ is given by the locally ringed space $M^{(m)} = (M, {\mathcal{O}}_M^{(m)})$, where ${\mathcal{O}}_M^{(m)} := \iota^*({\mathcal{C}}_N/{\mathcal{J}}_M^{m+1})$, for $\iota : M\hookrightarrow N$ the embedding. Indeed, given a thickening of order $m$, it is possible to construct another thickening of order $l<m$ in the same manner as in Construction \[nruiciurhf94h9h\]. \[djcnknjcknjcknrkc\] Suppose we are given an $m$-th order thickening $M^{(m)}$ of $M$. Then we obtain an ideal ${\mathcal{J}}_{(m)}$ as the following kernel, $$\begin{aligned} {\mathcal{J}}_{(m)} := \ker\{ {\mathcal{O}}_M^{(m)} \twoheadrightarrow {\mathcal{C}}_M\}. \label{jcnkrnckrnckjrcrcr}\end{aligned}$$ Now consider the ${\mathcal{J}}_{(m)}$-adic filtration of ${\mathcal{O}}_M^{(m)}$ and define ${\mathcal{O}}_M^{(l)}$ by, $$\begin{aligned} {\mathcal{J}}^{l+1}_{(m)} \hookrightarrow {\mathcal{O}}_M^{(m)} \twoheadrightarrow {\mathcal{O}}_M^{(l)} \label{jcnkjrnckrcnkrnc}\end{aligned}$$ for $0\leq l< m$. Then $(|M|, {\mathcal{O}}_M^{(l)})$ will be a thickening of $M$ of order $l$. Now by definition of ${\mathcal{J}}_{(m)}$ in we see that $M = M^{(0)}$. Moreover, since ${\mathcal{J}}^{l}_{(m)}\supset {\mathcal{J}}^{l+1}_{(m)}$, we are led to the following commuting diagram of exact sequences: $$\xymatrix{ 0 \ar[r] & \ar@{^{(}->}[d] {\mathcal{J}}^{l+1}_{(m)} \ar[r] & \ar@{=}[d]{\mathcal{O}}_M^{(m)} \ar[r] & \ar[d]{\mathcal{O}}_M^{(l)} \ar[r] & 0\\ 0 \ar[r] & {\mathcal{J}}^{l}_{(m)} \ar[r] & {\mathcal{O}}_M^{(m)} \ar[r] & {\mathcal{O}}_M^{(l-1)} \ar[r] & 0 }$$ The morphism ${\mathcal{O}}_M^{(l)} {\rightarrow}{\mathcal{O}}_M^{(l-1)}$ will be surjective. As these are the structure sheaves of the ringed spaces $M^{(l)}$ and $M^{(l-1)}$, this morphism will therefore correspond to an embedding $M^{(l-1)}\subset M^{(l)}$. In general we have a filtration $$\begin{aligned} M = M^{(0)} \subset M^{(1)} \subset M^{(2)} \subset \cdots\subset M^{(m)}. \label{idcmlmclrmclr}\end{aligned}$$ In this way we see that associated to any $m$-th order thickening of $M$ are thickenings of orders $l$, for $l = 0, \ldots, m-1$, filtered as in . We now have the following natural question: \[thickeningsquestion\] Given a thickening $M^{(m)}$ of $M$ does there exist a thickening $M^{(m+1)}$ of $M$ containing $M^{(m)}$? One of the main results in [@EASTBRU] is in identifying the space of obstructions to the existence of such a thickening. We will not state this result here but rather investigate the analogous question in the context of supergeometry. Thickenings in Supergeometry {#thickenings-in-supergeometry} ---------------------------- We firstly note the similarities between Definition \[djcnknckjncjkrccr\] of a thickening of a complex manifold and Definition \[ncnrnv322rnk\] of a (complex) supermanifold. This motivates the following definition of an abstract, order-$m$ ‘super-geometric’ thickening of a complex manifold: \[xomcomocoocoko\] \[rfi4hf984hoj3jp3\]*The definition of an abstract, super-geometric thickening given in Definition \[xomcomocoocoko\] above is taken from [@EASTBRU] where the terminology ‘*supersymmetric* thickening’ is employed. The word supersymmetry has some connotation in physics so we have opted here to use the term ‘super-geometric’ rather than supersymmetric. However, we will later drop the prefix super-geometric where confusion is unlikely to arise.* \[deduieuidh4hd487h\] *Let ${\mathfrak{X}}^{(m)}$ be an abstract, super-geometric thickening of $M$ of order-$m$. We say it is *trivial* if its structure sheaf ${\mathcal{O}}_M^{(m)}$ is globally isomorphic to the quotient algebra ${\mathcal{C}}_M[\xi_1, \ldots, \xi_q]/J^{m+1}$.* As with supermanifolds, if ${\mathcal{J}}= \ker\iota^\sharp$ then ${\mathcal{J}}/{\mathcal{J}}^2$ is identified with the sheaf of sections of a vector bundle ${\mathcal{E}}$ of $E{\rightarrow}M$. So in analogy with the notion of an abstract supermanifold modelled on a pair $(M, E)$, we have the notion of an abstract, *super-geometric thickening* of the pair $(M, E)$ of order $m$, denoted ${\mathfrak{X}}^{(m)}_{(M, E)}$. It is defined just as in Definition \[xomcomocoocoko\] with Condition (iii) replaced by: $$\begin{aligned} {\mathcal{O}}_M^{(m)}\cong_{\mathrm{loc.}} \wedge^{{\bullet}}{\mathcal{E}}/J^{m+1}, \label{nfo3hof3hf8o3hf03}\end{aligned}$$ where $J= \ker\{\wedge^{\bullet}{\mathcal{E}}\twoheadrightarrow {\mathcal{C}}_M\}$. \[fiugf78g79h380fj09\] For any thickening ${\mathfrak{X}}^{(m)}_{(M, E)}$ we have ${\mathfrak{X}}^{(1)}_{(M, E)} \cong \Pi E^{(1)}$. This follows almost immediately from Definition \[xomcomocoocoko\](i). To see this, recall that ${\mathfrak{X}}^{(m)}_{(M, E)}$ is the thickening ${\mathfrak{X}}^{(m)} = (M, {\mathcal{O}}^{(m)}_M)$. Moreover, we have an isomorphism ${\mathcal{J}}/{\mathcal{J}}^2\stackrel{\cong}{{\rightarrow}}{\mathcal{E}}$. Now since ${\mathcal{O}}^{(m)}_M$ is ${\mathbb{Z}}_2$-graded we can write ${\mathcal{O}}^{(m)}_M = {\mathcal{O}}_M^{(m);{\mathrm{ev}}}\oplus {\mathcal{O}}_M^{(m);{\mathrm{odd}}}$. The ideal ${\mathcal{J}}$ is generated by odd sections and so $$\begin{aligned} {\mathcal{O}}_M^{(m);{\mathrm{ev}}}/{\mathcal{J}}^{2k} = {\mathcal{O}}_M^{(m);{\mathrm{ev}}}/{\mathcal{J}}^{2k+1} && \mbox{and} && {\mathcal{O}}_M^{(m);{\mathrm{odd}}}/{\mathcal{J}}^{2k+1} = {\mathcal{O}}_M^{(m);{\mathrm{odd}}}/{\mathcal{J}}^{2k+2}. \label{3hf8gf83gf793hf39}\end{aligned}$$ In particular, for ${\mathfrak{X}}^{(1)}_{(M, E)}$ we find: $$\begin{aligned} {\mathcal{O}}_M^{(1)} := {\mathcal{O}}^{(m)}_M/{\mathcal{J}}^2 &= ({\mathcal{O}}_M^{(m);{\mathrm{ev}}}/{\mathcal{J}}^2)\oplus({\mathcal{O}}_M^{(m);{\mathrm{odd}}}/{\mathcal{J}}^2) \\ &= ({\mathcal{O}}^{(m)}_M/{\mathcal{J}})\oplus({\mathcal{J}}/{\mathcal{J}}^2) &&\mbox{from \eqref{3hf8gf83gf793hf39}} \\ &\cong {\mathcal{C}}_M\oplus {\mathcal{E}}\\ &= \wedge^{\bullet}{\mathcal{E}}^{(1)}.\end{aligned}$$ Hence ${\mathfrak{X}}^{(1)}_{(M, E)}\cong \Pi E^{(1)}$. \[dnclncjkdddjdjd\] *For $E{\rightarrow}M$ a vector bundle of rank $q$, a thickening of $(M, E)$ of order $q$ or higher will define a $(p|q)$-dimensional supermanifold ${\mathfrak{X}}$ modelled on $(M, E)$. In this way, a supermanifold ${\mathfrak{X}}_{(M, E)}$ may be thought of as a ‘maximal’, super-geometric thickening of the pair $(M, E)$.* *To avoid cumbersome terminology, we will typically refer to a ‘$(p|q)$-dimensional, order $m$, abstract, super-geometric thickening’ as a *$(p|q)$-dimensional thickening*. Moreover, if $E{\rightarrow}M$ is a fixed, rank-$q$ vector bundle then the terminology ‘$(p|q)$-dimensional’ is redundant and so will be dropped.* Now similarly to and , we have a filtration of locally ringed spaces: $$\begin{aligned} M \subset \Pi E^{(1)} \subset {\mathfrak{X}}_{(M, E)}^{(2)}\subset\cdots\subset{\mathfrak{X}}_{(M, E)}^{(m)}. \label{rhcbjrbchjrchjrnc}\end{aligned}$$ Then just as in the case of thickenings of complex manifolds, we want to know what the obstructions are to extending a given, abstract thickening. That is, we want a definitive answer to Question \[thickeningsquestion\], adapted to thickenings in the present context: \[SUSYthickeningsquestion\] Given an abstract (super-geometric) thickening ${\mathfrak{X}}_{(M, E)}^{(m)}$ of $M$, does there exist a (super-geometric) thickening ${\mathfrak{X}}_{(M, E)}^{(m+1)}$ of $M$ which contains ${\mathfrak{X}}_{(M, E)}^{(m)}$? In order to gain traction on the Question \[SUSYthickeningsquestion\] we submit the following definition. \[smeioiof894f89jcoi\] *A given, abstract thickening ${\mathfrak{X}}_{(M, E)}^{(m)}$ is said to be *obstructed* if there does not exist any $(m+1)$-th order thickening ${\mathfrak{X}}_{(M, E)}^{(m+1)}$ containing it. Otherwise, it is said to be *unobstructed*.* Hence if a given, abstract thickening ${\mathfrak{X}}_{(M, E)}^{(m)}$ is unobstructed, there will exist a thickening ${\mathfrak{X}}_{(M, E)}^{(m+1)}$ containing it and so an embedding ${\mathfrak{X}}_{(M, E)}^{(m)}\subset {\mathfrak{X}}_{(M, E)}^{(m+1)}$. In this article one of the goals is to give an example of an obstructed thickening, i.e., an abstract thickening for which no embedding into a higher structure will exist. In the sections to follow we will look to give a classification of thickenings and a first step toward doing so is a notion of equivalence for thickenings, defined below. \[equivextensusythick\] *Two thickenings ${\mathfrak{X}}_{(M, E)}^{(m+1)}$ and $\tilde{\mathfrak{X}}_{(M, E)}^{(m+1)}$ of ${\mathfrak{X}}^{(m)}$ are said to be *equivalent* if there exists an isomorphism ${\mathfrak{X}}_{(M, E)}^{(m+1)} \stackrel{\cong}{{\rightarrow}}\tilde {\mathfrak{X}}_{(M, E)}^{(m+1)}$, as locally ringed spaces, which restricts to the identity on ${\mathfrak{X}}_{(M, E)}^{(m)}$, i.e., a commutative diagram of locally ringed spaces: $$\xymatrix{ \ar@{^{(}->}[d] {\mathfrak{X}}_{(M, E)}^{(m)} \ar@{^{(}->}[r] & {\mathfrak{X}}_{(M, E)}^{(m+1)}\ar[dl]_\cong\\ \tilde {\mathfrak{X}}_{(M, E)}^{(m+1)}. & }$$ We denote by $\mathrm{T}^1({\mathfrak{X}}_{(M, E)}^{(m)})$ the set of equivalence classes of $(m+1)$-th order thickenings containing ${\mathfrak{X}}_{(M, E)}^{(m)}$.* A first step toward an answer to Question \[SUSYthickeningsquestion\] is provided by the following result: \[jnvjknknvkjnvk\] Suppose ${\mathfrak{X}}_{(M, E)}^{(m)}$ is a thickening of $(M,E)$ of order $m$, where $m\geq2$. Then the space of obstructions to finding a thickening of ${\mathfrak{X}}_{(M, E)}^{(m)}$ is identified with either: $$\begin{aligned} \mbox{$H^2(M, {\mathfrak{T}}_M\otimes\wedge^{m+1}{\mathcal{E}})$ if $m$ is odd} &&\mbox{or}&& \mbox{$H^2(M, \wedge^{m+1}{\mathcal{E}}\otimes {\mathcal{E}}^\vee)$ if $m$ is even.} \end{aligned}$$ A statement of Theorem \[jnvjknknvkjnvk\] may be found in [@EASTBRU p. 1188], where it is deduced from the more general theory of extensions of holomorphic vector bundles, detailed in [@GRIFF]. We will give a more elementary proof of Theorem \[jnvjknknvkjnvk\] in the section to follow. Obstructions to Existence {#4} ========================= In Remark \[dnclncjkdddjdjd\] it was observed that a supermanifold modelled on $(M, E)$ may be thought of as a ‘maximal’ super-geometric thickening of $(M, E)$. Question \[SUSYthickeningsquestion\] then motivates asking: *given a thickening ${\mathfrak{X}}^{(m)}_{(M, E)}$, does there exist a supermanifold ${\mathfrak{X}}_{(M, E)}$ which contains it?* We will ask this question again later. For now, we concentrate on Theorem \[jnvjknknvkjnvk\] where the space of obstructions to the existence of such a supermanifold is identified. We will present here firstly a ‘direct’ proof of Theorem \[jnvjknknvkjnvk\] for $m = 2$, from which a useful criterion for checking whether a given thickening is obstructed or not will become apparent. A full (and more indirect) proof of Theorem \[jnvjknknvkjnvk\] will then be given. To give these proofs, it will be convenient to digress and describe thickenings by means of gluing. Gluing Data for Thickenings --------------------------- Associated to any manifold (smooth or complex) is the data of an atlas, charts and transition functions. Conversely, the *manifold gluing construction* allows one to construct a manifold from a given collection of such data. This is described in [@LEE] for smooth manifolds and in [@KS] for complex manifolds. Importantly, a similar description exists for supermanifolds and thickenings. \[knkniucbruibckj\] (for supermanifolds) If $U$ is an open subset of ${\mathbb{C}}^p$, then ${\mathcal{U}}:= U\times{\mathbb{C}}^{0|q}$ will be an open subset of ${\mathbb{C}}^{p|q}$ with ${\mathcal{U}}_{\mathrm{red}}= U$. Now let ${\mathfrak{U}}= \{U, V, W, \ldots\}$ be a collection of open subsets of ${\mathbb{C}}^p$ and $\{{\mathcal{U}}, {\mathcal{V}},{\mathcal{W}}, \ldots\}$ a corresponding collection of subsets of ${\mathbb{C}}^{p|q}$. Furthermore, let $\{{\mathcal{U}}_{\mathcal{V}}, {\mathcal{U}}_{\mathcal{W}}, \ldots\}$ be a collection of subsets where ${\mathcal{U}}_{\bullet}\subset {\mathcal{U}}; {\mathcal{V}}_{\bullet}\subset {\mathcal{V}}$ and so on. The transition data $\rho = \{\rho_{{\mathcal{U}}{\mathcal{V}}}\}$ then consists of a collection of isomorphisms $\rho_{{\mathcal{U}}{\mathcal{V}}} : {\mathcal{U}}_{\mathcal{V}}\stackrel{\cong}{{\rightarrow}}{\mathcal{V}}_{\mathcal{U}}$ satisfying the cocycle condition, represented here by commutativity of the following diagram: $$\begin{aligned} \xymatrix{ \ar[dr]_{\rho_{{\mathcal{U}}{\mathcal{W}}}} {\mathcal{U}}_{\mathcal{V}}\cap {\mathcal{U}}_{\mathcal{W}}\ar[rr]^{\rho_{{\mathcal{U}}{\mathcal{V}}}} & & {\mathcal{V}}_{\mathcal{U}}\cap {\mathcal{V}}_{\mathcal{W}}\ar[dl]^{\rho_{{\mathcal{V}}{\mathcal{W}}}}\\ & {\mathcal{W}}_{\mathcal{U}}\cap{\mathcal{W}}_{\mathcal{V}}& } \label{mciornoihroi}\end{aligned}$$ The isomorphisms $\{\rho_{{\mathcal{U}}{\mathcal{V}}}\}$ are required to preserve the ${\mathbb{Z}}_2$-grading, in accordance with Definition $\ref{ncnrnv322rnk}$. If $(x^\mu, {\theta}_a)$ (resp. $(y^\mu, \eta_a)$) denote coordinates on ${\mathcal{U}}$ (resp. ${\mathcal{V}}$), then on the intersection ${\mathcal{U}}\cap{\mathcal{V}}$ we can write: $$\begin{aligned} y^\mu &= \rho_{{\mathcal{U}}{\mathcal{V}}}^\mu(x, {\theta}) = f_{UV}^\mu(x) + \sum_{|I|>0} f_{UV}^{\mu|2I}{\theta}_{2I} \label{uicuibviuv78v3} \\ \eta_a &= \rho_{{\mathcal{U}}{\mathcal{V}}, a}(x, {\theta}) = \zeta_{UV, a}^b(x)~{\theta}_b + \sum_{|I|>0} \zeta_{UV, a}^{2I+1}(x)~{\theta}_{2I+1} \label{porviorhg894h89}\end{aligned}$$ where $I$ is a multi-index and $|I|$ its length; for $I = (i_1, \ldots, i_n)$ that ${\theta}_I = {\theta}_{i_1}\wedge\cdots\wedge{\theta}_{i_n}$; by $2I$ (resp. $2I+1$) it is meant the multi-indices of even (resp. odd) length. The coefficient functions $\{(f_{UV}^{\mu|2I})\}$ (resp. $\{ (\zeta_{UV, a}^{2I+1})\}$) are holomorphic and defined on the intersection ${\mathcal{U}}\cap {\mathcal{V}}$. Then just as for manifolds, a supermanifold ${\mathfrak{X}}_{(M, E)}$ is constructed by setting:[^2] $${\mathfrak{X}}_{(M, E)} = \frac{\bigsqcup_{\{{\mathcal{U}},{\mathcal{V}},\ldots\}} {\mathcal{U}}}{\{{\mathcal{U}}_{\mathcal{V}}\sim_{\rho_{{\mathcal{U}}{\mathcal{V}}}}{\mathcal{V}}_{\mathcal{U}}\}}.$$ The transition data $\rho = \{\rho_{{\mathcal{U}}{\mathcal{V}}}\}$ is graded by the degree of the monomials ${\theta}_I$. In this way, observe that the transition data for the manifold $M$ (resp. vector bundle $E$) is encoded in the degree-zero (resp. degree-one) components of $\rho$. For more details on this construction see [@ROG p. 92]. \[k49fk90fk3p\] *Gluing data for a supermanifold ${\mathfrak{X}}$ described in Construction \[knkniucbruibckj\] will be referred to as a *trivialisation* and denoted $({\mathfrak{U}}, \rho)$, or simply $\rho$.* Following Construction \[nruiciurhf94h9h\] and \[knkniucbruibckj\] above, we describe gluing data for thickenings in what follows. Central to the construction is the dichotomy between trivial and non-trivial thickenings (see Definition \[deduieuidh4hd487h\]). \[opnciornvruo4\] Fix subsets $\{U, V, W, \ldots\}$ of ${\mathbb{C}}^p$. For the open set $U$, let ${\mathcal{C}}_U$ denote its structure sheaf. Then associated to each $U$ is the locally ringed space ${\mathcal{U}}^{(m)} = (U, {\mathcal{O}}_U^{(m)})$, where ${\mathcal{O}}_U^{(m)} = {\mathcal{C}}_U\otimes\wedge^{k\leq m}{\mathbb{C}}^q$.[^3] By Definition *\[deduieuidh4hd487h\]* we see that ${\mathcal{U}}^{(m)}$ will be a trivial, $m$-th order thickening of $U$. In this way, we see that associated to $\{U, V, W, \ldots\}$ will be a collection of trivial, $m$-th order thickenings ${\mathfrak{U}}^{(m)} = \{{\mathcal{U}}^{(m)}, {\mathcal{V}}^{(m)},{\mathcal{W}}^{(m)}, \ldots\}$. Now just as in Construction *\[knkniucbruibckj\]* we fix a collection of subsets $\{{\mathcal{U}}_{\mathcal{V}}^{(m)}, {\mathcal{U}}_{\mathcal{W}}^{(m)}, \ldots\}$, where ${\mathcal{U}}_{\bullet}^{(m)}\subset {\mathcal{U}}^{(m)}$, and a collection of isomorphisms $\rho^{(m)} = \{\rho^{(m)}_{{\mathcal{U}}{\mathcal{V}}}\}$, where $\rho^{(m)}_{{\mathcal{U}}{\mathcal{V}}} : {\mathcal{U}}_{\mathcal{V}}^{(m)} \stackrel{\cong}{{\rightarrow}}{\mathcal{V}}_{\mathcal{U}}^{(m)}$ satisfies the cocycle condition as in $:$ $$\begin{aligned} \xymatrix{ \ar[dr]_{\rho^{(m)}_{{\mathcal{U}}{\mathcal{W}}}} {\mathcal{U}}^{(m)}_{\mathcal{V}}\cap {\mathcal{U}}^{(m)}_{\mathcal{W}}\ar[rr]^{\rho^{(m)}_{{\mathcal{U}}{\mathcal{V}}}} & & {\mathcal{V}}^{(m)}_{\mathcal{U}}\cap {\mathcal{V}}^{(m)}_{\mathcal{W}}\ar[dl]^{\rho^{(m)}_{{\mathcal{V}}{\mathcal{W}}}}\\ & {\mathcal{W}}^{(m)}_{\mathcal{U}}\cap{\mathcal{W}}^{(m)}_{\mathcal{V}}& } \label{090038083j}\end{aligned}$$ Just as in and we have here on ${\mathcal{U}}\cap{\mathcal{V}}$ $$\begin{aligned} y^\mu &= \rho_{{\mathcal{U}}{\mathcal{V}}}^\mu(x, {\theta}) = f_{UV}^\mu(x) + \sum_{0<|I|\leq m} f_{UV}^{\mu|2I}{\theta}_{2I} \\ \eta_a &= \rho_{{\mathcal{U}}{\mathcal{V}}, a}(x, {\theta}) = \zeta_{UV, a}^b(x)~{\theta}_b + \sum_{0<|I|\leq m} \zeta_{UV, a}^{2I+1}(x)~{\theta}_{2I+1}.\end{aligned}$$ With this data a super-geometric thickening ${\mathfrak{X}}_{(M, E)}^{(l)}$ is given by: $${\mathfrak{X}}^{(m)}_{(M, E)} = \frac{\bigsqcup_{{\mathcal{U}}^{(m)}\in {\mathfrak{U}}^{(m)}} {\mathcal{U}}^{(m)}}{\{{\mathcal{U}}^{(m)}_{\mathcal{V}}\sim_{\rho^{(m)}_{{\mathcal{U}}{\mathcal{V}}}}{\mathcal{V}}^{(m)}_{\mathcal{U}}\}}.$$ Note, this is consistent with the construction of thickenings associated to supermanifolds described in Construction *\[nruiciurhf94h9h\]*. Of course, the thickening described here does not presume the existence of some supermanifold containing it and so is abstract. *Gluing data for an $m$-th order thickening ${\mathfrak{X}}_{(M, E)}^{(m)}$ as described in Construction \[opnciornvruo4\] will be referred to as a *trivialisation* and denoted $({\mathfrak{U}}^{(m)}, \rho^{(m)})$.* From Construction \[opnciornvruo4\] it is evident that Question \[SUSYthickeningsquestion\] may be reformulated at the level of trivialisations as follows: *given a trivialisation $({\mathfrak{U}}^{(m)}, \rho^{(m)})$ for ${\mathfrak{X}}^{(m)}$, does there exist a trivialisation $({\mathfrak{U}}^{(m+1)}, \rho^{(m+1)})$ for a thickening ${\mathfrak{X}}^{(m+1)}$ such that each ${\mathcal{U}}^{(m+1)}$ is a thickening of ${\mathcal{U}}^{(m)}$ and $\rho^{(m)} \equiv \rho^{(m+1)}\mod {\mathcal{J}}^{m+1}$?* We address this question in what follows for $m = 2$ and will thereby arrive at a proof of Theorem \[jnvjknknvkjnvk\] in this particular case. Proof of Theorem $\ref{jnvjknknvkjnvk}_{m=2}$ --------------------------------------------- We begin with the following result, whose proof can be found, in principle, in [@GREEN; @YMAN]. \[pjiroich4h894ho\] There exists a bijection $\mathrm{T}^1(\Pi E^{(1)})\cong H^1(M, {\mathfrak{T}}_M\otimes\wedge^2{\mathcal{E}})$. Let ${\mathfrak{X}}^{(2)}_{(M, E)}$ be a second order thickening with trivialisation $({\mathfrak{U}}^{(2)}, \rho^{(2)})$. Here $\rho^{(2)} = \{\rho^{(2)}_{{\mathcal{U}}{\mathcal{V}}}\}$ and, following Construction \[opnciornvruo4\], we write: $$\begin{aligned} \rho_{{\mathcal{U}}{\mathcal{V}}}^{\mu;(2)} = f_{UV}^\mu + f_{UV}^{\mu|ij}~{\theta}_{ij} && \mbox{and} && \rho_{{\mathcal{U}}{\mathcal{V}}, a}^{(2)} = \zeta_{UV, a}^b~{\theta}_b. \label{piorf84hf4j0f43}\end{aligned}$$ The summation in over the indices $i, j$ is implicit. Now as a result of the cocycle condition we will find that $f^{(2)} = \{(f^{\mu|ij}_{UV})\}$ is a cocycle representative of a class ${\omega}_{(M, E)}\in H^1(M, \wedge^2{\mathcal{E}}\otimes{\mathfrak{T}}_M)$. Furthermore, it is a straightforward check that the notion of equivalence in Definition \[equivextensusythick\] is precisely captured by that notion of equivalence at the level of cohomology. That is, if ${\mathfrak{X}}^{(2){\prime}}_{(M, E)}$ is another first order thickening of $\Pi E^{(1)}$ and if the corresponding cocycle representative $f^{(2){\prime}}$ is cohomologous to $f^{(2)}$, then it is straightforward to construct an equivalence between the thickenings ${\mathfrak{X}}^{(2)}_{(M, E)}$ and ${\mathfrak{X}}^{(2){\prime}}_{(M, E)}$ and vice-versa. For the calculations justifying our assertions, we refer to [@BETTPHD Chapter 2]. Now if we are given a second order thickening ${\mathfrak{X}}^{(2)}_{(M, E)}$, then by construction it will be a thickening of $\Pi E^{(1)}$ by Lemma \[fiugf78g79h380fj09\]. A such, by Lemma \[pjiroich4h894ho\], it will define a class in $H^1(M, {\mathfrak{T}}_M\otimes\wedge^2{\mathcal{E}})$. We denote this class by ${\omega}_{(M, E)}$ and refer to it as the *obstruction class associated to ${\mathfrak{X}}^{(2)}_{(M, E)}$*. The central result of the present section, which we will prove en route to Theorem \[jnvjknknvkjnvk\], is the following: \[popiioh4o8fh84o\] Consider the exact sequence associated to the degree-two component of tangent sheaf ${\mathfrak{T}}_{\Pi E}$ in Lemma *\[hjbbjhhveyjj\]*, i.e., the sequence in at $k = 2$. It induces a long-exact sequence on sheaf cohomology containing the piece: $$\begin{aligned} \xymatrix{ \cdots \ar[r] & H^1(M, {\mathfrak{T}}_M\otimes\wedge^2{\mathcal{E}}) \ar[r]^{{\partial}_*} & H^2(M, \wedge^3{\mathcal{E}}\otimes{\mathcal{E}}^\vee) \ar[r] & \cdots } \label{poijirjciojj49j4}\end{aligned}$$ Then a given, second-order thickening ${\mathfrak{X}}^{(2)}_{(M, E)}$ is unobstructed if and only if its obstruction class satisfies: ${\partial}_*({\omega}_{(M, E)})=0$. We will now give a proof of Theorem \[jnvjknknvkjnvk\] for $m = 2$ by appealing to the description of a thickening by means of trivialisations, as in Construction \[opnciornvruo4\].\ \ *Proof of Theorem $\emph{\ref{jnvjknknvkjnvk}}_{m=2}$*. Following on from the proof of Lemma \[pjiroich4h894ho\], we consider here the analogous picture for any thickening ${\mathfrak{X}}^{(3)}_{(M, E)}$ of ${\mathfrak{X}}^{(2)}_{(M, E)}$. If $({\mathfrak{U}}^{(3)}, \rho^{(3)})$ is a trivialisation for ${\mathfrak{X}}^{(3)}_{(M, E)}$, then $\rho^{(3)}$ will be as in , however we will have an additional term: $$\begin{aligned} \rho^{(3);\mu}_{{\mathcal{U}}{\mathcal{V}}} = \rho^{(2);\mu}_{{\mathcal{U}}{\mathcal{V}}} && \mbox{and} && \rho^{(3)}_{{\mathcal{U}}{\mathcal{V}}, a} = \rho^{(2)}_{{\mathcal{U}}{\mathcal{V}}, a} + \zeta_{UV, a}^{ijk}~{\theta}_{ijk}. \label{piorf83f33f34hf4j0f43}\end{aligned}$$ The summation in over the indices $i, j, k$ is implicit. In assuming that a thickening ${\mathfrak{X}}^{(3)}_{(M, E)}$ of ${\mathfrak{X}}^{(2)}_{(M, E)}$ exists, we will be faced with a non-trivial condition that needs to hold. We will write down this condition explicitly here as an equation which, we argue, will lie in the second cohomology group. This is the assertion of the present theorem. To begin note that $\zeta^{(3)} = \{(\zeta_{UV, a}^{ijk})\}$ here is a $1$-cochain valued in $\wedge^3{\mathcal{E}}\otimes{\mathcal{E}}^\vee$. In imposing the cocycle condition on $\rho^{(3)}$ we obtain: $$\begin{aligned} \zeta_{UW, a}^{lmn}~{\theta}_{lmn}\frac{{\partial}}{{\partial}\xi_a} - \zeta_{UV, b}^{lmn}~{\theta}_{lmn}\frac{{\partial}}{{\partial}\eta_b} ~-~ &\zeta_{VW, a}^{ijk}~\eta_{ijk}\frac{{\partial}}{{\partial}\xi_a} \label{ioiohfojfoifjof} \\ &= \notag \\ \frac{{\partial}\zeta_{VW, a}^b}{{\partial}y^\mu}&f_{UV}^{\mu|lm}\zeta_{UV, b}^n~{\theta}_{lmn}\frac{{\partial}}{{\partial}\xi_a}. \label{ejckencjkencjkenc}\end{aligned}$$ Now note that $\eqref{ejckencjkencjkenc}$ is a 2-cochain. The proof of Theorem $\ref{opnciornvruo4}_{m=2}$ will follow if we can successfully argue that $\eqref{ejckencjkencjkenc}$ is in fact a 2-cocycle, for then the equality $\eqref{ioiohfojfoifjof} = \eqref{ejckencjkencjkenc}$ will make sense as a statement in the second cohomology group, which is precisely what is asserted in Theorem $\ref{jnvjknknvkjnvk}_{m=2}$. We will now show that is a 2-cocycle in a way that will also prove Proposition \[popiioh4o8fh84o\] with the following: \[lkliioolkss\] $\eqref{ejckencjkencjkenc}$ is a $2$-cocycle representative of ${\partial}_*({\omega}_{(M, E)})$. The proof follows essentially by construction. Indeed, firstly note that the map ${\mathfrak{T}}_{\Pi E}[2]{\rightarrow}{\mathfrak{T}}_M\otimes\wedge^2{\mathcal{E}}$ is given by sending a vector field $v$ in ${\mathfrak{T}}_{\Pi E}[2]$ to $v\mod {\mathcal{J}}^3$. At the level of $1$-cocycles: let $f^{(2)} =\{f^{(2)}_{{\mathcal{U}}{\mathcal{V}}}\} = \{(f_{{\mathcal{U}}{\mathcal{V}}}^{\mu|ij})\}$ be a cocycle representative for ${\omega}_{(M, E)}$. Then we have the cocycle condition: $$\begin{aligned} f_{UW}^{{\sigma}|kl}~{\theta}_{kl}\frac{{\partial}}{{\partial}z^{\sigma}} = f^{\tau|kl}_{UV}~{\theta}_{kl} \frac{{\partial}}{{\partial}y^\tau}+ f^{{\sigma}|ij}_{VW}~\eta_{ij}\frac{{\partial}}{{\partial}z^{\sigma}} \label{4h9f8h5hiurhh89}\end{aligned}$$ Now let $\{Y_{{\mathcal{U}}{\mathcal{V}}}\}$ be such that $Y_{{\mathcal{U}}{\mathcal{V}}}\equiv f^{(2)}_{{\mathcal{U}}{\mathcal{V}}}$ modulo ${\mathcal{J}}^3$. Then we have the 2-cocycle $Y_{{\mathcal{U}}{\mathcal{V}}{\mathcal{W}}} = \{Y_{{\mathcal{U}}{\mathcal{W}}} - Y_{{\mathcal{U}}{\mathcal{V}}} - Y_{{\mathcal{V}}{\mathcal{W}}}\}$ which, by , vanishes modulo ${\mathcal{J}}^3$. Now by exactness of the induced, long-exact sequence on cohomology in we have, at the level of 2-cocycles, a 2-cocycle $Z = \{Z_{{\mathcal{U}}{\mathcal{V}}{\mathcal{W}}}\}$ in $\mathcal Z^2({\mathfrak{U}}, \wedge^3{\mathcal{E}}\otimes{\mathcal{E}}^\vee)$ mapping to $Y = \{Y_{{\mathcal{U}}{\mathcal{V}}{\mathcal{W}}}\}$. That is $Z = Y|_{\wedge^3{\mathcal{E}}\otimes{\mathcal{E}}^\vee}$. Its cohomology class $[Z]$ is defined to be ${\partial}_*({\omega}_{(M, E)})$. To get an expression for its representative we will make use of the transition data $\rho^{(2)}$. Firstly recall that $f^{(2)}_{{\mathcal{U}}{\mathcal{V}}} = (f^{\mu|ij}_{{\mathcal{U}}{\mathcal{V}}})$ defines a vector field on ${\mathcal{V}}_{\mathcal{U}}$. With $\rho^{(2)}_{{\mathcal{V}}{\mathcal{W}}} : {\mathcal{V}}^{(2)}_{\mathcal{W}}\stackrel{\cong}{{\rightarrow}}{\mathcal{W}}^{(2)}_{\mathcal{U}}$, we can send $f^{(2)}_{{\mathcal{U}}{\mathcal{V}}}$ to a vector field $(\rho^{(2)}_{{\mathcal{V}}{\mathcal{W}}})_*f^{(2)}_{{\mathcal{U}}{\mathcal{V}}}$ on ${\mathcal{W}}_{\mathcal{U}}$. Now note that $\rho^{(2)} = \rho^{(3)}\mod {\mathcal{J}}^3$. In writing $\rho^{(2)}$ in terms of its even and odd components $\rho^{(2)} = (\rho^{(2);+}, \rho^{(2);-})$, the condition in asserts, on ${\mathcal{W}}_{\mathcal{U}}$, that: $f_{{\mathcal{U}}{\mathcal{W}}}^{(2)} = (\rho_{{\mathcal{U}}{\mathcal{V}}}^{(2);+})_*f^{(2)}_{{\mathcal{U}}{\mathcal{V}}} + f^{(2)}_{{\mathcal{V}}{\mathcal{W}}}$. Hence, as a vector field on ${\mathcal{W}}_{\mathcal{U}}$, we have: $$\begin{aligned} Y_{{\mathcal{U}}{\mathcal{V}}{\mathcal{W}}} &= (\rho^{(2);-}_{{\mathcal{V}}{\mathcal{W}}})_*f^{(2)}_{{\mathcal{U}}{\mathcal{V}}} \label{9g859gh95hfFD} \\ \notag &=f^{\mu|ij}_{UV}{\theta}_{ij} ~(\rho^{(2)}_-)_*\left( \frac{{\partial}}{{\partial}y^\mu}\right) \\ \notag &= f^{\mu|ij}_{UV}{\theta}_{ij} \cdot \left( \frac{{\partial}\zeta_{VW, a}^b}{{\partial}y^\mu}\eta_b\frac{{\partial}}{{\partial}\xi_a}\right) \\ \notag &=f^{\mu|ij}_{UV}\frac{{\partial}\zeta_{VW, a}^b}{{\partial}y^\mu}\zeta_{UV, b}^c~ {\theta}_{ijc} \frac{{\partial}}{{\partial}\xi_a}.\end{aligned}$$ The lemma now follows by comparing with . An alternative and more direct check that satisfies the conditions to be a 2-cocycle can be found in [@BETTPHD Appendix A.1]. The proof of Theorem $\ref{jnvjknknvkjnvk}_{m=2}$ and Proposition \[popiioh4o8fh84o\] now follows.\ A much simpler proof of Theorem $\ref{jnvjknknvkjnvk}_{m=2}$ may be given more indirectly and independently of the direct verification in Lemma \[lkliioolkss\]. This uses the assumption that ${\mathfrak{X}}^{(2)}_{(M, E)}$ is a second order thickening, meaning $\rho^{(2)}$ must satisfy the cocycle condition. The proof is inductive and can in fact be applied to give a full proof of Theorem \[jnvjknknvkjnvk\], which we do so in what follows. An Elementary Proof of Theorem $\ref{jnvjknknvkjnvk}$ ----------------------------------------------------- We will consider the case where we are given a super-geometric thickening of odd order ${\mathfrak{X}}^{(2m-1)}$. The argument remains essentially unchanged in the even case so we omit it here. Firstly, to briefly recap: if ${\mathfrak{X}}^{(2m)}$ denotes a $(2m)$-th order, super-geometric thickening containing ${\mathfrak{X}}^{(2m-1)}$, then it will admit a trivialisation $({\mathfrak{U}}^{(2m)}, \rho^{(2m)})$, with $\rho^{(2m)} = \{\rho^{(2m)}_{{\mathcal{U}}{\mathcal{V}}}\}$ subject to the cocycle condition in , i.e., that: $$\begin{aligned} \rho^{(2m);\mu}_{{\mathcal{U}}{\mathcal{W}}} = \rho^{(2m);\mu}_{{\mathcal{V}}{\mathcal{W}}}\circ \rho^{(2m)}_{{\mathcal{U}}{\mathcal{V}}}. \label{dkncrcurcrknckmcr87h3}\end{aligned}$$ Note that $\rho_{{\mathcal{U}}{\mathcal{V}}, a}^{(2m)} = \rho_{{\mathcal{U}}{\mathcal{V}}, a}^{(2m-1)}$ and that, by assumption, $\rho^{(2m-1)}$ will satisfy the cocycle condition. Now write, $$\begin{aligned} \rho_{{\mathcal{U}}{\mathcal{V}}}^{(2m);\mu} = \rho_{{\mathcal{U}}{\mathcal{V}}}^{(2m-2);\mu} + \phi_{{\mathcal{U}}{\mathcal{V}}}^\mu \label{pprirjiovrio}\end{aligned}$$ where $\phi = \{(\phi_{{\mathcal{U}}{\mathcal{V}}}^\mu)\}$ comprises homogeneous quantities of degree-$(2m)$. In applying the decomposition in to we arrive at the following expression: $$\begin{aligned} \rho_{{\mathcal{U}}{\mathcal{W}}}^{(2m-2);\mu} - \rho_{{\mathcal{V}}{\mathcal{W}}}^{(2m-2);\mu}\circ \rho_{{\mathcal{U}}{\mathcal{V}}}^{(2m-2)} = \phi^\mu_{{\mathcal{V}}{\mathcal{W}}}\circ \rho_{{\mathcal{U}}{\mathcal{V}}} + \frac{{\partial}f^\mu_{VW}}{{\partial}y^\nu}\phi_{{\mathcal{U}}{\mathcal{V}}}^\nu - \phi^\mu_{{\mathcal{U}}{\mathcal{W}}}. \label{dcnn8984jid}\end{aligned}$$ We see that the right-hand side of consists only of those quantities which regulate the extension of $\rho^{(2m-1)}$ to $\rho^{(2m)}$. As for the left-hand side of , we set: $$\begin{aligned} {\Gamma}_{{\mathcal{U}}{\mathcal{V}}{\mathcal{W}}}^\mu := \rho_{{\mathcal{U}}{\mathcal{W}}}^{(2m-2);\mu} - \rho_{{\mathcal{V}}{\mathcal{W}}}^{(2m-2);\mu}\circ \rho_{{\mathcal{U}}{\mathcal{V}}}^{(2m-2)}. \label{mchrehehruei38deu383j}\end{aligned}$$ Since $\rho^{(2m-2)}$ satisfies the cocycle condition (modulo ${\mathcal{J}}^{2m}$), it follows that ${\Gamma}_{{\mathcal{U}}{\mathcal{V}}{\mathcal{W}}}^\mu$ will be homogeneous and of degree-$(2m)$.[^4] \[bipalabophhhh89\] Let $${\Gamma}_{{\mathcal{U}}{\mathcal{V}}{\mathcal{W}}} = {\Gamma}_{{\mathcal{U}}{\mathcal{V}}{\mathcal{W}}}^\mu\frac{{\partial}}{{\partial}z^\mu}.$$ Then ${\Gamma}= \{{\Gamma}_{{\mathcal{U}}{\mathcal{V}}{\mathcal{W}}}\}$ will be a $2$-cocycle valued in ${\mathfrak{T}}_M\otimes\wedge^{2m}{\mathcal{E}}$. The proof follows along essentially the same lines as that given in [@KS p. 254-255], so we omit it here. Finally, from we obtain the following equation: $$\begin{aligned} \phi^\mu_{{\mathcal{V}}{\mathcal{W}}} \frac{{\partial}}{{\partial}z^\mu} + \phi_{{\mathcal{U}}{\mathcal{V}}}^\nu \frac{{\partial}}{{\partial}y^\nu} - \phi^\mu_{{\mathcal{U}}{\mathcal{W}}} \frac{{\partial}}{{\partial}z^\mu} = {\Gamma}_{{\mathcal{U}}{\mathcal{V}}{\mathcal{W}}}^\mu\frac{{\partial}}{{\partial}z^\mu}. \label{dmodj833jso}\end{aligned}$$ The left-hand side of is clearly the coboundary of a 1-cochain valued in ${\mathfrak{T}}_M\otimes\wedge^{2m}{\mathcal{E}}$, whereas the right-hand side is a 2-cocycle valued in ${\mathfrak{T}}_M\otimes\wedge^{2m}{\mathcal{E}}$ by Lemma \[bipalabophhhh89\]. The theorem now follows. *Our usage of the word ‘elementary’ to term the proof of Theorem \[jnvjknknvkjnvk\] given here is motivated by a similar usage of this word in [@KS]. There it is used in reference to a proof of the existence of (formal) deformations of complex structures on compact manifolds.* *Up to sign, the equation in is explicitly written down as the equality $\eqref{ioiohfojfoifjof} = \eqref{ejckencjkencjkenc}$ in the case where we are given an even thickening ${\mathfrak{X}}^{(2)}_{(M, E)}$.* A Classification of Thickenings ------------------------------- Consequences of Theorem \[jnvjknknvkjnvk\] and its proof provided in the previous (sub)section are the results to be given here regarding classifications of thickenings. It will be convenient to introduce the following notation: $$\begin{aligned} \mathcal Q^{(k);+}_E := {\mathfrak{T}}_M\otimes \wedge^k{\mathcal{E}}&& \mbox{and} && \mathcal Q^{(k);-}_E := \wedge^k{\mathcal{E}}\otimes{\mathcal{E}}^\vee. \label{dafoprmijo4jo4}\end{aligned}$$ For further convenience we will write: $$\mathcal Q_E^{(i);\pm} := \left\{\begin{array}{ll} \mathcal Q_E^{(i);+} & \mbox{if $i$ is even} \\ \mathcal Q_E^{(i);-} &\mbox{if $i$ is odd}. \end{array} \right.$$ With this notation the exact sequence in Lemma \[hjbbjhhveyjj\] becomes: $$\begin{aligned} 0 {\rightarrow}\mathcal Q_E^{(k+1);-} \hookrightarrow {\mathfrak{T}}_{\Pi E}[k] \twoheadrightarrow \mathcal Q_E^{(k);+}{\rightarrow}0. \label{oinriobuoioie}\end{aligned}$$ In particular, we see that $\mathcal Q_E^{(i);\pm}$ will always be affiliated with sections of the tangent sheaf of even-degree tangent vectors.\ \ We have so far been guided by Question \[SUSYthickeningsquestion\], which was addressed in part in Theorem \[jnvjknknvkjnvk\]. In what follows we present a result which can be seen as a generalisation of Proposition \[popiioh4o8fh84o\] or as a slight furthering of Theorem \[jnvjknknvkjnvk\]. In particular, it will address Question \[SUSYthickeningsquestion\] more directly. \[o3jdioj3ijp3\] Let ${\mathfrak{X}}^{(l)}_{(M, E)}$ be a thickening of order $l$ and suppose $\mathrm{T}^1({\mathfrak{X}}_{(M, E)}^{(l)})$ is non-empty. Then there exists a map $${\partial}_* : \mathrm{T}^1({\mathfrak{X}}^{(l)}_{(M, E)}) {\longrightarrow}H^2(M, \mathcal Q^{(l+2);\pm}_E)$$ which measures the failure for a thickening of ${\mathfrak{X}}^{(l)}_{(M, E)}$ to be unobstructed. The map ${\partial}_*$ can be described explicitly by appealing to a trivialisation. To do so, let ${\mathfrak{X}}^{(l)}_{(M, E)}$ be a thickening of order $l$ and ${\mathfrak{X}}^{(l+1)}_{(M, E)}$ a thickening of ${\mathfrak{X}}^{(l)}_{(M, E)}$. We think of ${\mathfrak{X}}_{(M,E)}^{(l+1)}$ as a representative of a class $[{\mathfrak{X}}_{(M, E)}^{(l+1)}]\in \mathrm{T}^1({\mathfrak{X}}_{(M, E)}^{(l)})$. Let $({\mathfrak{U}}^{(l+1)}, \rho^{(l+1)})$ be a trivialisation for ${\mathfrak{X}}_{(M, E)}^{(l+1)}$. Then associated to $\rho^{(l+1)}$ will, by Lemma \[bipalabophhhh89\], be the 2-cocycle $\{{\Gamma}^{(l+2)}_{{\mathcal{U}}{\mathcal{V}}{\mathcal{W}}}\}\in \mathcal Z^2({\mathfrak{U}}, \mathcal Q^{(l+2);\pm}_E)$. We set: $$\begin{aligned} {\partial}_*\left( [{\mathfrak{X}}^{(l+1)}_{(M, E)}]\right) := \left[ \left\{{\Gamma}^{(l+2)}_{{\mathcal{U}}{\mathcal{V}}{\mathcal{W}}}\right\}\right]. \label{mdio3nfuih3fh3o}\end{aligned}$$ The goal now is to argue that is well-defined. To that end, let $\tilde{\mathfrak{X}}^{(l+1)}_{(M, E)}$ be another thickening of ${\mathfrak{X}}^{(l)}_{(M, E)}$ and suppose ${\mathfrak{X}}^{(l+1)}_{(M, E)}\sim \tilde{\mathfrak{X}}^{(l+1)}_{(M, E)}$ as extensions. This means there exists an isomorphism ${\lambda}: {\mathfrak{X}}^{(l+1)}_{(M, E)}\cong \tilde{\mathfrak{X}}^{(l+1)}_{(M, E)}$ which is the identity when restricted to ${\mathfrak{X}}^{(l)}_{(M, E)}$. Let $({\mathfrak{U}}^{(l+1)}, \tilde\rho^{(l+1)})$ be a trivialisation for $\tilde{\mathfrak{X}}^{(l+1)}_{(M, E)}$ and denote by $\{\tilde{\Gamma}^{(l+2)}_{{\mathcal{U}}{\mathcal{V}}{\mathcal{W}}}\}$ the 2-cocycle constructed from the trivialisation $\tilde\rho^{(l+1)}$. In order to show that the map ${\partial}_*$ in is well-defined, we need to show that $[\{{\Gamma}^{(l+2)}_{{\mathcal{U}}{\mathcal{V}}{\mathcal{W}}}\}]= [\{\tilde{\Gamma}^{(l+2)}_{{\mathcal{U}}{\mathcal{V}}{\mathcal{W}}}\}]$ in $H^2(M, \mathcal Q^{(l+2);\pm}_E)$. In fact, under the assumptions here we will find that the cocycle representatives are themselves equal, i.e., that ${\Gamma}^{(l+2)}_{{\mathcal{U}}{\mathcal{V}}{\mathcal{W}}} = \tilde{\Gamma}^{(l+2)}_{{\mathcal{U}}{\mathcal{V}}{\mathcal{W}}}$. So firstly, as we assume ${\mathfrak{X}}^{(l+1)}_{(M, E)}\sim \tilde{\mathfrak{X}}^{(l+1)}_{(M, E)}$, we have on the intersection ${\mathcal{U}}\cap{\mathcal{V}}$ that $$\begin{aligned} \rho_{{\mathcal{U}}{\mathcal{V}}}^{(l+1)}\circ {\lambda}_{\mathcal{U}}= {\lambda}_{\mathcal{V}}\circ \tilde\rho^{(l+1)}_{{\mathcal{U}}{\mathcal{V}}} \label{4903jf903jf093}\end{aligned}$$ where $\{{\lambda}_{\mathcal{U}}\}$ is a 0-cochain representative of ${\lambda}: {\mathfrak{X}}^{(l+1)}_{(M, E)}\cong \tilde{\mathfrak{X}}^{(l+1)}_{(M, E)}$ with respect to the cover ${\mathfrak{U}}^{(l+1)}$. Now since ${\lambda}_{\mathcal{U}}: {\mathcal{U}}^{(l+1)}\cong {\mathcal{U}}^{(l+1)}$ is trivial modulo ${\mathcal{J}}^l$ (i.e., that ${\lambda}_{\mathcal{U}}|_{{\mathcal{U}}^{(l)}}$ is the identity), we can construct from $\{{\lambda}_{\mathcal{U}}\}$ a $\mathcal Q_E^{(l+1);\pm}$-valued, 0-cochain $\{\nu_{\mathcal{U}}\}$.[^5] Expanding will now reveal that: $$\begin{aligned} \rho^{(l+1)}_{{\mathcal{U}}{\mathcal{V}}} - \tilde\rho^{(l+1)}_{{\mathcal{U}}{\mathcal{V}}} = ({\delta}\nu)_{{\mathcal{U}}{\mathcal{V}}} = \nu_{\mathcal{V}}-\nu_{\mathcal{U}}\label{fj4j49jf94kf4kp}\end{aligned}$$ and as a result: $$\begin{aligned} \tilde {\Gamma}^{(l+2)}_{{\mathcal{U}}{\mathcal{V}}{\mathcal{W}}} &= \tilde \rho^{(l+1)}_{{\mathcal{U}}{\mathcal{W}}} - \tilde\rho^{(l+1)}_{{\mathcal{W}}{\mathcal{V}}}\circ \tilde\rho^{(l+1)}_{{\mathcal{U}}{\mathcal{V}}} \notag \\ &= {\lambda}_{\mathcal{W}}^{-1}\circ \rho^{(l+1)}_{{\mathcal{U}}{\mathcal{W}}}\circ {\lambda}_{\mathcal{U}}- \left({\lambda}_{\mathcal{W}}^{-1}\circ\rho^{(l+1)}_{{\mathcal{V}}{\mathcal{W}}}\circ{\lambda}_{\mathcal{V}}\right)\circ \left({\lambda}_{\mathcal{V}}^{-1}\circ \rho^{(l+1)}_{{\mathcal{U}}{\mathcal{V}}}\circ {\lambda}_{\mathcal{U}}\right) \notag \\ &= {\Gamma}^{(l+2)}_{{\mathcal{U}}{\mathcal{V}}{\mathcal{W}}} - ({\delta}\nu)_{{\mathcal{U}}{\mathcal{W}}} + ({\delta}\nu)_{{\mathcal{V}}{\mathcal{W}}} + ({\delta}\nu)_{{\mathcal{U}}{\mathcal{V}}} \notag \\ &= {\Gamma}^{(l+2)}_{{\mathcal{U}}{\mathcal{V}}{\mathcal{W}}}. \label{39j03jf30jfp3}\end{aligned}$$ Hence we see that if ${\mathfrak{X}}^{(l+1)}\sim \tilde {\mathfrak{X}}^{(l+1)}$, then the 2-cocycles ${\Gamma}_{{\mathcal{U}}{\mathcal{V}}{\mathcal{W}}}$ and $\tilde {\Gamma}_{{\mathcal{U}}{\mathcal{V}}{\mathcal{W}}}$ associated to trivialisations $\rho^{(l+1)}$ and $\tilde\rho^{(l+1)}$ of ${\mathfrak{X}}^{(l+1)}$ and $\tilde {\mathfrak{X}}^{(l+1)}$ respectively will coincide. In particular, the map ${\partial}_*$ in is well-defined. Now, from the proof of Theorem \[jnvjknknvkjnvk\] given earlier, we see that ${\mathfrak{X}}^{(l+1)}_{(M, E)}$ will be an obstructed thickening if and only if ${\partial}_*( [{\mathfrak{X}}^{(l+1)}_{(M, E)}])\neq0$. It is in this sense that ${\partial}_*$ measures the failure for a given first-order extension of ${\mathfrak{X}}^{(l)}$ to be unobstructed. The proposition now follows. Recall that ${\mathfrak{X}}_{(M, E)}$ will admit a trivialisation by Construction \[opnciornvruo4\] which is unique up to common refinement. Should we consider trivialisations of ${\mathfrak{X}}_{(M, E)}$ directly, then from the proof of Proposition \[o3jdioj3ijp3\] above—and in particular from , we obtain a stronger result: \[eiooihf839hf3ho\] Let ${\mathfrak{X}}_{(M, E)}^{(l)}$ be an unobstructed thickening and suppose $({\mathfrak{U}}^{(l)}, \rho^{(l)})$ is a trivialisation for ${\mathfrak{X}}_{(M, E)}^{(l)}$. Denote by $\mathrm{T}^1({\mathfrak{U}}^{(l)}, \rho^{(l)})$ the set of thickenings of ${\mathfrak{X}}_{(M, E)}^{(l)}$ equipped with this choice of trivialisation $({\mathfrak{U}}^{(l)}, \rho^{(l)})$ and suppose it is non-empty. Then the following diagram commutes: $$\begin{aligned} \xymatrix{ \ar[dr]_{{\partial}_*}\mathrm{T}^1({\mathfrak{U}}^{(l)}, \rho^{(l)}) \ar[r] & \mathcal Z^2({\mathfrak{U}}, \mathcal Q_E^{(l+2);\pm})\ar[d] \\ & \mbox{\emph{ \v H}}^2({\mathfrak{U}}, Q_E^{(l+2);\pm}) } \label{prfpejpj3pj390}\end{aligned}$$ where in : the horizontal map is given by sending a trivialisation $({\mathfrak{U}}^{(l+1)}, \rho^{(l+1)})$ to its obstruction class ${\Gamma}^{(l+2)}$; the vertical arrow is the natural map on Čech cohomology of the covering ${\mathfrak{U}}$; and ${\partial}_*$ is the map from Proposition $\ref{o3jdioj3ijp3}$. We turn our attention now to the set of thickenings itself with the intent to show that, in general, it will admit the structure of a pseudo-torsor over a certain group. The notion of a torsor itself comes up in many areas of mathematics and the definition we present below is taken from [@HARTDEF p. 49]. Following on from Proposition \[o3jdioj3ijp3\] we have: \[o4993uf0oijwoijfwojpw\] Given an $l$-th order thickening ${\mathfrak{X}}_{(M, E)}^{(l)}$, the set of thickenings $\mathrm{T}^1({\mathfrak{X}}_{(M, E)}^{(l)})$ is a pseudo-torsor for the group $H^1(M, \mathcal Q_E^{(l+1);\pm})$. Assume $\mathrm{T}^1({\mathfrak{X}}_{(M, E)}^{(l)})$ is non-empty. This means ${\mathfrak{X}}_{(M, E)}^{(l)}$ is unobstructed. Then associated to any thickening ${\mathfrak{X}}_{(M, E)}^{(l+1)}$ will be a trivialisation $({\mathfrak{U}}^{(l+1)}, \rho^{(l+1)})$ and associated to $\rho^{(l+1)}$ will be 1-cochains $c^{(j)}(\rho)=\{c_{{\mathcal{U}}{\mathcal{V}}}^{(j)}\}\in C^1({\mathfrak{U}}^{(l+1)}, \mathcal Q_E^{(j);\pm})$ for $j = 2,\cdots l+1$. On the intersection ${\mathcal{U}}\cap{\mathcal{V}}$ the term $c_{{\mathcal{U}}{\mathcal{V}}}^{(j)}$ is precisely the homogeneous, degree-$j$ component of $\rho_{{\mathcal{U}}{\mathcal{V}}}^{(l+1)}$. Now recall that ${\mathfrak{X}}_{(M, E)}^{(l)}$ is unobstructed from our assumption. As such we know from Proposition \[o3jdioj3ijp3\] that the 2-cocycle ${\Gamma}^{(l+1)}$ corresponding to $\rho^{(l+1)}$ will be cohomologically trivial. Now let ${\alpha}^{(l+1)}\in \mathcal Z^1({\mathfrak{U}}, \mathcal Q_E^{(l+1);\pm})$. Then ${\delta}{\alpha}^{(l+1)} = 0$. Define $\tilde\rho^{(l+1)} = \rho^{(l+1)} + {\alpha}^{(l+1)}$. Evidently the 2-cocycle $\tilde{\Gamma}$ associated to $\tilde\rho^{(l+1)}$ is $$\tilde{\Gamma}_{{\mathcal{U}}{\mathcal{V}}{\mathcal{W}}} = {\Gamma}_{{\mathcal{U}}{\mathcal{V}}{\mathcal{W}}} +({\delta}{\alpha})_{{\mathcal{U}}{\mathcal{V}}{\mathcal{W}}} = {\Gamma}_{{\mathcal{U}}{\mathcal{V}}{\mathcal{W}}}$$ and since ${\Gamma}_{{\mathcal{U}}{\mathcal{V}}{\mathcal{W}}}\sim 0$, we find $\tilde {\Gamma}_{{\mathcal{U}}{\mathcal{V}}{\mathcal{W}}}\sim 0$ also. Hence $({\mathfrak{U}}^{(l+1)}, \tilde\rho^{(l+1)})$ defines a trivialisation from which we can construct another, unobstructed thickening $\tilde {\mathfrak{X}}_{(M, E)}^{(l+1)}$ of ${\mathfrak{X}}^{(l)}$ as in Construction \[opnciornvruo4\]. From here, it is not too hard to deduce that ${\mathfrak{X}}_{(M, E)}^{(l+1)}$ and $\tilde {\mathfrak{X}}_{(M, E)}^{(l+1)}$ will be equivalent as thickenings of ${\mathfrak{X}}_{(M, E)}^{(l)}$ if and only if ${\alpha}^{(l+1)}\sim 0$. If ${\mathfrak{X}}^{(l)}_{(M, E)}$ is obstructed then by definition there will not exist any thickening containing it, which means $\mathrm{T}^1({\mathfrak{X}}_{(M, E)}^{(l)})=\eset$. The present theorem now follows. For more details see [@BETTPHD Chapter 2]. \[dopmropvopejpv\] $\mathrm{T}^1(\Pi E^{(l)})\cong H^1(M, \mathcal Q_E^{(l+1);\pm})$. *Note that Corollary \[dopmropvopejpv\] is a generalisation of Lemma \[pjiroich4h894ho\].* A Decomposition of the Obstruction Spaces {#5} ========================================= It was mentioned in the previous section that the discussion so far had and results presented were guided by Question \[SUSYthickeningsquestion\], which was answered in Theorem \[jnvjknknvkjnvk\] and Proposition \[o3jdioj3ijp3\]. In the present section we consider the following variant incorporating supermanifolds: \[kd93f3jf3kf9p4\] Suppose ${\mathfrak{X}}^{(k)}$ is a given, unobstructed thickening. Then will there exist some supermanifold ${\mathfrak{X}}$ containing ${\mathfrak{X}}^{(k)}$? In general the answer to Question \[kd93f3jf3kf9p4\] above will be in the negative and, just as in Proposition \[o3jdioj3ijp3\], we can identify conditions under which an unobstructed thickening will be associated to a supermanifold. Before doing so it will be necessary to briefly deliberate on obstruction theory for supermanifolds. Preliminaries: Obstruction Theory for Supermanifolds ---------------------------------------------------- Let ${\mathfrak{X}}_{(M, E)}$ be a complex supermanifold with split model $\Pi E$ (see Section \[fjf9j490fj40fj40\]). Then by Definition \[ncnrnv322rnk\] we know that ${\mathfrak{X}}_{(M, E)}$ and $\Pi E$ will be *locally* isomorphic. *The supermanifold ${\mathfrak{X}}_{(M, E)}$ is said to be *split* if it is globally isomorphic to its split model.* With this definition of splitness we can ask the following natural question guiding considerations in obstruction theory for supermanifolds: \[kf93j890jf0j\] Let ${\mathfrak{X}}$ be a supermanifold. Is it split? As a smooth supermanifold ${\mathfrak{X}}_{(M, E)}$ is split, a result which is well known and originally due to Batchelor in [@BAT]. In the holomorphic setting, a given complex supermanifold certainly need not be split and the study of Question \[kf93j890jf0j\] in this setting is referred to as ‘obstruction theory’. In this section we present some well known results in obstruction theory following [@BER; @GREEN; @YMAN; @ONISHCLASS]. Our starting point is in the construction of the sheaf of groups ${\mathcal{G}}^{(i)}_E$ for any $i\geq2$ as follows: on a complex manifold $M$ let $E{\rightarrow}M$ be a rank $q$, holomorphic vector bundle and ${\mathcal{E}}$ its sheaf of holomorphic sections. Define, $${\mathcal{G}}^{(i)}_E := \left\{ {\alpha}\in \mathscr Aut\wedge^{\bullet}{\mathcal{E}}\mid {\alpha}(u) - u\in {\mathcal{J}}^i\right\}.$$ Then by construction, for $i = 2$, we have the following short-exact sequence of sheaves of groups on $M$ which is split, as observed by Onishchik in [@ONISHCLASS]: $$1 {\rightarrow}{\mathcal{G}}^{(2)}_E \hookrightarrow \mathscr Aut\wedge^{\bullet}{\mathcal{E}}\twoheadrightarrow \mathscr Aut~{\mathcal{E}}{\rightarrow}1.$$ For $i>2$ the relation of these sheaves ${\mathcal{G}}^{(i)}_E$ to $\mathcal Q_E^{(i);\pm}$ from is captured in the following result by Green in [@GREEN]: \[dm30j8fjf3o\] For each $i$ the sheaf ${\mathcal{G}}^{(i)}_E$ contains ${\mathcal{G}}^{(i+1)}_E$ as a (sheaf of) normal subgroups. Moreover, for $i> 2$, there exists an isomorphism $$\frac{{\mathcal{G}}_E^{(i)}}{{\mathcal{G}}_E^{(i+1)}} \cong \mathcal Q_E^{(i);\pm}.$$ If $i> q$, then ${\mathcal{G}}^{(i)}_E = \{1\}$. *As a result of the last sentence in the statement of Proposition \[dm30j8fjf3o\] we have ${\mathcal{G}}^{(q)}_E \cong \mathcal Q^{(q);\pm}_E$ and is therefore abelian.* By the general theory of non-abelian sheaf cohomology developed in [@GROTHNONAB] and [@BRY p. 158-62], the results in Proposition \[dm30j8fjf3o\] may be recast into a short-exact sequence of sheaves of groups from which we obtain a long exact sequence on Čech cohomology (as pointed sets). A piece of this sequence is given below: $$\begin{aligned} \cdots {\longrightarrow}H^1({\mathcal{G}}^{(i+1)}_E) {\longrightarrow}H^1({\mathcal{G}}^{(i)}_E) \stackrel{{\omega}_*}{{\longrightarrow}} H^1(\mathcal Q_E^{(i);\pm}) \label{8393hf93j3fojf83}\end{aligned}$$ where $H^i(-) = H^i(M, -)$ above. Hence, in this way, we find the obstruction spaces $H^1(\mathcal Q_E^{(i);\pm})$ appearing here. Following [@DW1] we submit: \[rfijf903jf903kf\]*Elements in $H^1(M, {\mathcal{G}}^{(2)}_E)$ will be referred to as *trivialisations*. Elements in $H^1(M, {\mathcal{G}}^{(i)}_E)$, for $i>2$, will be referred to as *lifts of trivialisations* or *level-$(i+1)$ trivialisations** \[kd93jf03j0j3j3p\]*Elements in the image of ${\omega}_*$ are referred to as *obstruction classes to splitting*.* By construction, the obstruction classes to splitting depend on a choice of trivialisation or lifts thereof. To see the relevance of these constructs to supermanifold theory, firstly consider the following set: $$\begin{aligned} {\mathfrak{M}}_{(M, E)} = \left\{\mbox{supermanifolds modelled on $(M, E)$}\right\}/\cong \label{rkfj490fj4fp3okp3ofk3p}\end{aligned}$$ We think of ${\mathfrak{M}}_{(M, E)}$ here as a pointed set, with base-point represented by the split model $\Pi E$. We now have from [@GREEN]: \[dn8j83jp3k9k922\] Any supermanifold ${\mathfrak{X}}_{(M, E)}$ will define a trivialisation in $H^1(M, {\mathcal{G}}^{(2)}_E)$. Furthermore, there exists a one-to-one correspondence as pointed sets: $${\mathfrak{M}}_{(M, E)} \cong \frac{H^1(M, {\mathcal{G}}^{(2)}_E)}{H^0(M, \mathscr Aut~{\mathcal{E}})}.$$ As a result of Theorem \[dn8j83jp3k9k922\] above we deduce the following statements pertaining to Question \[kf93j890jf0j\]: (i) any supermanifold ${\mathfrak{X}}_{(M, E)}$ will define an element in $H^1(M, \mathcal Q^{(i);\pm}_E)$ for *some $i$* by which, by Definition \[kd93jf03j0j3j3p\], is called an obstruction class; (ii) if ${\mathfrak{X}}$ is non-split, then there will not exist a level-$(q+1)$ trivialisation for ${\mathfrak{X}}$, and; (iii) if ${\mathfrak{X}}$ is split, then there will exist a level-$(q+1)$ trivialisation for ${\mathfrak{X}}$. As remarked in [@DW1 p. 15], the obstruction classes themselves do not generally constitute a system of invariants of a given supermanifold. To elaborate further on this point we provide the following illustration. \[kopdj389hf883jfpo3\] Consider the split model $\Pi E$. By Theorem $\ref{dn8j83jp3k9k922}$ it will define a trivialisation $\rho$ in $H^1(M, {\mathcal{G}}^{(2)}_E)$, and by construction this trivialisation $\rho$ corresponds to the choice of base-point in the pointed set $H^1(M, {\mathcal{G}}^{(2)}_E)$. Furthermore, we also have that ${\omega}_*(\rho) = 0$ in $H^1(M, \mathcal Q^{(2);+}_E)$ since ${\omega}_*$ preserves the base-point. Then by exactness of the sequence in at $i = 2$, we will obtain a lift of $\rho$ to some level-$3$ trivialisation ${\varphi}_3\in H^1(M, {\mathcal{G}}^{(3)}_E)$. The map $H^1(M, {\mathcal{G}}^{(3)}_E) {\rightarrow}H^1(M, {\mathcal{G}}^{(2)}_E)$ is, in general, not injective however so there may exist many lifts ${\varphi}_3$ of $\rho$ and there is no reason to suppose, in general, that ${\omega}_*({\varphi}_3) = 0$ in $H^1(M, \mathcal Q^{(3);-}_E)$. In accordance with *(iii)* above, if ${\mathfrak{X}}$ is split then associated to $\rho$ will be lifts ${\varphi}_i\in H^1(M, {\mathcal{G}}^{(i)}_E)$ such that ${\omega}_*({\varphi}_i) = 0$ for each $i$. Lifts ${\varphi}_i$ of the split supermanifold $\rho$ for which ${\omega}_*({\varphi}_i)\neq0$ are referred to in [@DW1] as ‘exotic lifts’. Supermanifolds and Thickenings ------------------------------ We revisit thickenings now by considering the following variant of Question \[kd93f3jf3kf9p4\]: \[dmopeififp3p33\] Let $\eta\in H^1(M,\mathcal Q_E^{(i);\pm})$. Then does there exist a supermanifold ${\mathfrak{X}}_{(M, E)}$ which admits a trivialisation whose obstruction class is realised by this given element $\eta$? From Corollary \[dopmropvopejpv\] note that the space $H^1(M,\mathcal Q_E^{(i);\pm})$ can be identified with $i$-th order thickenings ${\mathfrak{X}}_{(M, E)}^{(i)}$ of $\Pi E^{(i-1)}$. In particular, a partial answer to Question \[dmopeififp3p33\] is readily given by Proposition \[o3jdioj3ijp3\]: *if ${\partial}_*(\eta) \neq0$, then there will not exist any supermanifold realising $\eta$ as an obstruction class*. Hence we find in Proposition \[o3jdioj3ijp3\] a necessary condition for $\eta$ to be realised by some supermanifold. Naturally, we can then ask: *is this sufficient?* This question is addressed in the following result: \[ppoppieepeee\] ${\mathrm{im}}~{\omega}_*\subset \ker{\partial}_*$ but not necessarily conversely. As mentioned, by Corollary \[dopmropvopejpv\] we know that $H^1(M,\mathcal Q_E^{(i);\pm})\cong \mathrm{Ext}^1(\Pi E^{(i-1)})$. Let ${\mathfrak{X}}^{(i)}_{(M, E)}\supset \Pi E^{(i-1)}$ be a thickening and suppose that ${\partial}_*([{\mathfrak{X}}^{(i)}_{(M, E)}]) = 0$. Then by Proposition \[o3jdioj3ijp3\] it will be unobstructed. Hence we can find a thickening ${\mathfrak{X}}^{(i+1)}_{(M, E)}\supset {\mathfrak{X}}^{(i)}_{(M, E)}$. Now, by Theorem \[dn8j83jp3k9k922\] and the succeeding discussion, we know that any supermanifold ${\mathfrak{X}}_{(M, E)}$ will define an obstruction class in $H^1(M, \mathcal Q_E^{(i);\pm})$ for *some $i$*, not necessarily unique, but in the image of the obstruction map ${\omega}_*$. Suppose it admits a level-$i$ trivialisation with $\eta = {\omega}_*({\varphi}_i)$. Then we can write: $$\begin{aligned} M \subset \Pi E^{(i)} \subset \cdots\subset \Pi E^{(i-1)}\subset {\mathfrak{X}}^{(i)}_{(M, E)}\subset {\mathfrak{X}}^{(i+1)}_{(M, E)}\subset\cdots \subset {\mathfrak{X}}_{(M, E)} \label{dkjfjf03j93322}\end{aligned}$$ and evidently ${\omega}_*({\varphi}_i) = [{\mathfrak{X}}_{(M, E)}^{(i)}]$. Importantly, we see that the thickening ${\mathfrak{X}}^{(i+1)}_{(M, E)}\supset {\mathfrak{X}}^{(i)}_{(M, E)}$ must be unobstructed and so must thickenings of it, and so on (c.f., Remark \[dnclncjkdddjdjd\]). More generally however there is of course no reason to expect, just because the thickening ${\mathfrak{X}}^{(i)}_{(M, E)}\supset \Pi E^{(i-1)}$ is unobstructed, that the thickening ${\mathfrak{X}}^{(i+1)}_{(M, E)}\supset {\mathfrak{X}}^{(i)}_{(M, E)}$ will be unobstructed also. In this case we will not be able to construct a supermanifold as in and so we *cannot* write $ [{\mathfrak{X}}_{(M, E)}^{(i)}] = {\omega}_*({\varphi}_i)$ for any level-$i$ trivialisation ${\varphi}_i$. This shows that while we certainly have ${\mathrm{im}}~{\omega}_*\subset \ker{\partial}_*$, the reverse containment need not hold. The above result motivates the following definition. \[f904fj04fjj33\]*A thickening that defines a non-trivial class in $\ker{\partial}_*/{\mathrm{im}}~{\omega}_*$ is referred to as a *pseudo-supermanifold*.* From the argument presented in the proof of Proposition \[ppoppieepeee\], it is clear that pseudo-supermanifolds can be characterised as those thickenings ${\mathfrak{X}}^{(l)}_{(M, E)}$ given by a filtration: $$\begin{aligned} M \subset \Pi E^{(1)} \subset \cdots\subset \Pi E^{(l-1)} \subset {\mathfrak{X}}^{(l)}_{(M, E)} \subset \cdots \subset {\mathfrak{X}}^{(k)}_{(M, E)} \label{4j903f03jf03j9}\end{aligned}$$ for *some* $k$, where: (1) $l<k< q$, for $q$ the rank of the vector bundle $E$; (2) that $[{\mathfrak{X}}^{(l)}_{(M, E)}]\in \ker{\partial}_*/{\mathrm{im}}~{\omega}_*$ is non-trivial and; (3) ${\partial}_*([{\mathfrak{X}}^{(k)}_{(M, E)}])\neq0$. Hence, for any $i$, we have: $$\begin{aligned} H^1(M, \mathcal Q^{(i);\pm}_E) = {\mathrm{im}}~{\omega}_*\oplus \left(\frac{\ker {\partial}_*}{{\mathrm{im}}~{\omega}_*}\right)\oplus (\ker{\partial}_*)^\perp. \label{49d93j8fj3fj3p}\end{aligned}$$ In words, we can decompose the cohomology groups $H^1(M, \mathcal Q_E^{(i);\pm})$ into: (i) supermanifolds; (ii) pseudo-supermanifolds and; (iii) obstructed thickenings. This addresses Question \[dmopeififp3p33\]. In what follows we will illustrate some of the results so far presented in a more invariant manner for thickenings of even order. Thickenings of Even Order ------------------------- We have appealed to trivialisations to prove all the results so far presented. In the present section we will give simple illustrations of Proposition \[o3jdioj3ijp3\] and \[ppoppieepeee\] for thickenings ${\mathfrak{X}}^{(2j)}_{(M, E)}\supset \Pi E^{(2j-1)}$. The new ingredient here is the following morphism of exact sequences concerning the groups ${\mathcal{G}}^{(i)}_E$. This result extends Proposition \[dm30j8fjf3o\] and can be found in [@ONISHCLASS p. 55]. \[dmopifoijfojfo3\] There exists a morphism of short-exact sequences of sheaves of groups on $M$: $$\xymatrix{ 1\ar[r] & \ar@{^{(}->}[d] {\mathcal{G}}^{(2j+2)}_E \ar@{^{(}->}[r] &\ar@{=}[d] {\mathcal{G}}^{(2j)}_E\ar[r]^{{\lambda}~~} &\ar[d]^p {\mathfrak{T}}_{\Pi E}[2j]\ar[r] & 1 \\ 1 \ar[r] & {\mathcal{G}}^{(2j+1)}_E \ar@{^{(}->}[r] & {\mathcal{G}}^{(2j)}_E \ar[r]^{{\omega}~~} & \mathcal Q_E^{(2j);+}\ar[r] & 1 }$$ By naturality of the sheaf cohomology functor $H^i$ we will obtain the following diagram on cohomology from the diagram in Proposition \[dmopifoijfojfo3\] (horizontal) and the short-exact sequence in Lemma \[hjbbjhhveyjj\] (vertical): $$\begin{aligned} \xymatrix{ & &\vdots\ar[d] \\ & & H^1(M, \mathcal Q^{(2j+1);-}_E)\ar[d] \\ \cdots \ar[r] & \ar@{=}[d]H^1(M, {\mathcal{G}}^{(2j)}_E) \ar[r]^{{\lambda}_*~~} & H^1(M, {\mathfrak{T}}_{\Pi E}[2j])\ar[d]^{p_*} \\ \cdots\ar[r] & H^1(M, {\mathcal{G}}^{(2j)}_E)\ar[r]^{{\omega}_*} & H^1(M, \mathcal Q_E^{(2j);+})\ar[d]^{{\partial}_*} \\ & & H^2(M, \mathcal Q_E^{(2j+1);-})\ar[d] \\ & & \vdots } \label{miojfioejfoijf039}\end{aligned}$$ From the above diagram we have the following: (1) the map ${\partial}_*$ in Proposition \[o3jdioj3ijp3\] coincides with ${\partial}_* : H^1(\mathcal Q_E^{(2j);+}) {\rightarrow}H^2(\mathcal Q_E^{(2j+1);-})$ in in the case where ${\mathfrak{X}}^{(l)}_{(M, E)} = \Pi E^{(2j-1)}$ (c.f., Corollary \[dopmropvopejpv\]) and; (2) commutativity of and long-exactness on cohomology allows us to deduce: $${\mathrm{im}}~{\omega}_* = {\mathrm{im}}~(p_*\circ {\lambda}_*) \subset {\mathrm{im}}~p_* = \ker{\partial}_*.$$ Hence we have ${\mathrm{im}}~{\omega}_*\subset \ker{\partial}_*$, which is an instance of Proposition \[ppoppieepeee\]. We conclude now our foray into thickenings and supermanifolds. In the remaining sections we comment on the corresponding moduli problems and provide illustrations of obstructed thickenings of the complex projective plane. Before doing so however, we will compare the results obtained here with what one might find in the literature in what follows. Comparisons with Known Results ------------------------------ The results so far presented and discussed in this article—mainly Proposition \[o3jdioj3ijp3\] and Theorem \[o4993uf0oijwoijfwojpw\], are reminiscent of similar results one finds in studies of deformation theory in both complex-analytic and algebraic geometry. In the complex-analytic setting, we can compare Theorem \[jnvjknknvkjnvk\] and Proposition \[o3jdioj3ijp3\] with [@KS Theorem 5.1, p. 214]. In the algebraic setting, we compare Proposition \[o3jdioj3ijp3\] with [@STACKPROJ Tag 08L8] and Theorem \[o4993uf0oijwoijfwojpw\] with [@STACKPROJ Tag 08UC]. Where complex supermanifolds are concerned, the development of a deformation theory for them can be found in works such as [@ROTH] in the complex-analytic setting and [@VAIN] in the more algebro-geometric setting.\ \ In [@ROTH], the main idea is to regard a complex supermanifold as a deformation of its split model. A guiding question is then: *what are the obstructions to deforming the split model?* Sufficient conditions, being the vanishing of the second cohomology group, are identified in [@ROTH Theorem 3, p. 259]. In this way, the central result in [@ROTH] mirrors Theorem \[jnvjknknvkjnvk\] and thereby also the results in [@EASTBRU]. In addition to these sufficient conditions, we consider necessary conditions in this article. Indeed the subject of the illustrations on the projective plane, provided at the end of this article, focus on these necessary conditions.\ \ The developments in [@VAIN] follow the more algebraic route to deformation theory, in the spirit of [@HARTDEF; @STACKPROJ]. These developments are however concerned more with the deformation theory of a given supermanifold, as opposed to the problem of existence of a supermanifold as a deformation. In this subtle way the articles [@ROTH] and [@VAIN] differ[^6] and the material presented so far in this article is more closely related to [@ROTH]. In the following section we will consider the moduli problem for supermanifolds and thickenings. Moduli Problems: Supermanifolds and Thickenings {#6} =============================================== Moduli problems in considerable generality are discussed in [@HARTDEF p. 150]. Loosely speaking, one aspect of the moduli problem concerns itself with giving sets such as the structure of an algebraic variety or scheme or some other such geometric object. Such problems admit nice, categorical reformulations where they can be reduced to the problem of representability of an appropriately constructed functor.\ \ In [@HARTDEF], sets such as are themselves referred to as moduli problems. In this section we will briefly comment on the moduli problem for complex supermanifolds, following studies by Onishchik in [@ONISHMOD; @ONISHCLASS]. We aim to then formulate a variant of this problem in order to investigate the structure of the corresponding moduli variety. We will submit a claim regarding this structure but will not attempt to study it further here. Finally, in taking motivation from the decomposition in , we conclude by arguing that this decomposition will hold at the level of moduli. Framed Supermanifolds --------------------- The moduli problem for complex supermanifolds, modelled on a given complex manifold $M$ and holomorphic vector bundle $E$, is written down in and given further meaning in Theorem \[dn8j83jp3k9k922\]. We consider here a variant of the moduli problem in .\ \ In Definition \[ifh4fgiuho48h04\] we defined a coframe for a supermanifold ${\mathfrak{X}}_{(M, E)}$. Dual to a coframe is a frame and we define it in what follows. Starting with a supermanifold modelled on $(M, E)$, recall the exact sequences in Lemma \[jd90d098jdoijp3\] and \[hjbbjhhveyjj\]: $$\begin{aligned} 0{\rightarrow}{\mathfrak{T}}_{(M, E)}[0] \hookrightarrow {\mathfrak{T}}_{(M, E)}[-1] \twoheadrightarrow {\mathfrak{T}}_{\Pi E}[-1] {\rightarrow}0. \label{rofopjofj3o9fj930jf3}\end{aligned}$$ Since ${\mathfrak{T}}_{(M, E)}[-1] = {\mathfrak{T}}_{(M, E)}$ and ${\mathfrak{T}}_{\Pi E}[-1] \cong {\mathcal{E}}^\vee$ we obtain from a surjective morphism of sheaves ${\alpha}: {\mathfrak{T}}_{(M, E)} \twoheadrightarrow {\mathcal{E}}^\vee$ which is an isomorphism modulo ${\mathfrak{T}}_{(M, E)}[0]$. \[fmojfiojf4fjp3\] *To any supermanifold ${\mathfrak{X}}_{(M, E)}$, an epimorphism ${\alpha}:{\mathfrak{T}}_{(M, E)} {\rightarrow}{\mathcal{E}}^\vee$ with $\ker {\alpha}= {\mathfrak{T}}_{(M, E)}[0]$ will be referred to as a *framing*. A supermanifold ${\mathfrak{X}}_{(M, E)}$ equipped with a choice of framing ${\alpha}$ will be called *framed**. To justify the terminology ‘coframe’ we have: \[fh389fh98hfoij83\] Let ${\alpha}$ denote a framing on ${\mathfrak{X}}_{(M,E)}$. Up to a scale factor, we have ${\alpha}= {\varphi}^\vee$ modulo ${\mathfrak{T}}_{(M, E)}[0]$. This follows from the identity $ {\mathfrak{T}}_{(M, E)}[-1] = ({\mathcal{J}}/{\mathcal{J}}^2)^\vee$ modulo ${\mathfrak{T}}_{(M, E)}[0]$. The reason for introducing a framing in this section is so we can make sense of the following notion of equivalence.[^7] \[rjf930j09j3fjp3fjp3w\] Similarly to , the moduli problem here is: $$\begin{aligned} {\mathfrak{M}}_{(M, E)}^{\mathrm{framed}} = \left\{\mbox{framed supermanifolds $({\mathfrak{X}}_{(M, E)}, {\alpha})$}\right\}/\sim \label{rkfj490fj4fp3okp3ofk3p9d039jd}\end{aligned}$$ where the equivalence ‘$\sim$’ in above is in the sense of Definition \[rjf930j09j3fjp3fjp3w\]. As a set ${\mathfrak{M}}_{(M, E)}^{\mathrm{framed}}$ consists of equivalence classes of pairs $[({\mathfrak{X}}_{(M, E)}, {\alpha})]$ where ${\alpha}: {\mathfrak{T}}_{(M, E)} \twoheadrightarrow {\mathcal{E}}^\vee$ is a framing. *The moduli problem ${\mathfrak{M}}_{(M, E)}^{\mathrm{framed}}$ is considered also in [@VAIN p. 2151]. In [@VAIN Chapter 3] it is argued that the corresponding moduli functor is quasi-representable.* Evidently we have a map ${\mathfrak{M}}_{(M, E)}^{\mathrm{framed}} {\rightarrow}{\mathfrak{M}}_{(M, E)}$ given by forgetting this choice of framing ${\alpha}$. The moduli problems and are related in this way. Indeed, it is shown in the proof of Theorem \[dn8j83jp3k9k922\] in [@GREEN] and remarked also in [@DW1 p. 13] the following, which we state as a proposition and for which we provide a sketch of a proof: \[jf983f3jfoj3ojg93jgp\] The moduli problem for framed, complex supermanifolds modelled on a given pair $(M, E)$ is in one-to-one correspondence with $H^1(M, {\mathcal{G}}^{(2)}_E)$. *Proof Sketch*. A framed supermanifold is a supermanifold ${\mathfrak{X}}_{(M, E)}$ equipped with a framing ${\alpha}$. We are free to change ${\alpha}$ by a global automorphism of ${\mathcal{E}}$, i.e., an element of $H^0(M, \mathscr Aut~{\mathcal{E}})$ and so ${\alpha}$ is certainly non-canonical. Now suppose ${\mathfrak{X}}_{(M, E)}$ and ${\mathfrak{X}}_{(M, E)}^{\prime}$ are framed with respective framings ${\alpha}$ and ${\alpha}^{\prime}$ and moreover suppose $({\mathfrak{X}}_{(M, E)}, {\alpha})\sim ({\mathfrak{X}}^{\prime}_{(M, E)}, {\alpha}^{\prime})$. Then there exists an isomorphism ${\lambda}: {\mathfrak{X}}_{(M, E)}\stackrel{\cong}{{\rightarrow}}{\mathfrak{X}}^{\prime}_{(M,E)}$ and by Definition \[rjf930j09j3fjp3fjp3w\](i) we have ${\mathcal{O}}_M/{\mathcal{J}}^2 = {\mathcal{O}}_M^{\prime}/{\mathcal{J}}^{{\prime}2}$. The isomorphism ${\lambda}$ preserves the ${\mathbb{Z}}_2$-grading which means ${\mathcal{J}}/{\mathcal{J}}^2 = {\mathcal{J}}^{\prime}/{\mathcal{J}}^{{\prime}2}$.[^8] That the coframes ${\varphi}$ and ${\varphi}^{\prime}$ for ${\mathfrak{X}}_{(M, E)}$ resp. ${\mathfrak{X}}_{(M, E)}^{\prime}$ coincide follows from Definition \[rjf930j09j3fjp3fjp3w\](ii) and Lemma \[fh389fh98hfoij83\]. Hence the frames ${\alpha}$ and ${\alpha}^{\prime}$ coincide, modulo ${\mathfrak{T}}_{(M, E)}[0]$. Thus $H^0(M, \mathscr Aut~{\mathcal{E}})$ separates equivalence classes of framed supermanifolds. Using this and the fact that any supermanifold modelled on $(M, E)$ defines an element in $H^1(M, {\mathcal{G}}^{(2)}_E)$, we deduce ${\mathfrak{M}}_{(M, E)}^{\mathrm{framed}}\subseteq H^1(M, {\mathcal{G}}^{(2)}_E)$. We do not deliberate on the reverse inclusion here. *In comparing Proposition \[jf983f3jfoj3ojg93jgp\] with Theorem \[dn8j83jp3k9k922\] we see that the map ${\mathfrak{M}}_{(M, E)}^{\mathrm{framed}} {\rightarrow}{\mathfrak{M}}_{(M, E)}$ is given by the quotient map $H^1({\mathcal{G}}^{(2)}_E) {\rightarrow}H^1({\mathcal{G}}^{(2)}_E)/H^0(\mathscr Aut~{\mathcal{E}})$. In this way, we can think of the action of $H^0(M, \mathscr Aut~{\mathcal{E}})$ on $H^1(M, {\mathcal{G}}^{(2)}_E)$ as changing the frame.* We have so far been embroiled in set-theoretic aspects of the moduli problem. Onishchik in [@ONISHCLASS p. 68] addresses the moduli problem in by constructing an algebraic variety parametrising framed supermanifolds. Paraphrasing this result we have: \[4jf93jf903jf3jfp3j\] Let $${\mathfrak{T}}_{\Pi E}^{\geq2} := \bigoplus_{k\geq 2} {\mathfrak{T}}_{\Pi E}[k].$$ Then there exists a connected, complex-analytic subvariety $\mathbb V_{(M, E)}\subseteq H^1(M, {\mathfrak{T}}_{\Pi E}^{\geq2})$ whose set of points lie in one-to-one correspondence with ${\mathfrak{M}}^{\mathrm{framed}}_{(M, E)}$. As briefly mentioned at the outset of this section, a more complete formulation of the moduli problem consists of finding a geometric object parametrising the given set of isomorphism classes. Where supermanifolds are concerned, the results in [@VAIN] suggest that we might expect the corresponding moduli space to be a space in the sense of supergeometry, i.e., some sort of superspace. We will refer to such a space as a *supermoduli space*.[^9] For the super-geometric analogue of varieties and schemes see [@RABGLOB pp. 129-33] and [@LEITSPEC]. For framed supermanifolds ${\mathfrak{M}}_{(M, E)}^{\mathrm{framed}}$ it is tempting to conclude from Theorem \[4jf93jf903jf3jfp3j\] that the corresponding supermoduli space is the variety $\mathbb V_{(M, E)}$. However, it is not clear here as to how this variety can be interpreted as an object in supergeometry. That is, as a ‘super’-variety or ‘super’-scheme. One possible reason for this might be due to the distinction between *classification* and *deformation*, mentioned briefly in footnote . It is remarked in the introductory section in [@ONISHCLASS] that $\mathbb V_{(M, E)}$ is related to classifications, not deformations. Supermoduli spaces however should be related to deformation theory. As such, using the language in [@HARTDEF], we suspect that a statement along the following lines might be true, but we do not venture to study it further here: \[fkrpokvpokvp4k\] There exists a coarse supermoduli space $\mathscr V_{(M, E)}$ for the moduli problem ${\mathfrak{M}}_{(M, E)}^{\mathrm{framed}}$ with $(\mathscr V_{(M, E)})_{\mathrm{red}} = \mathbb V_{(M, E)}$. In what follows we turn to another aspect of the moduli problem $\mathfrak M_{(M, E)}^{\mathrm{framed}}$, motivated by Proposition \[jf983f3jfoj3ojg93jgp\]. The Moduli Problem with Level Structures ---------------------------------------- We wish to formulate here a variant of the moduli problem in and submit a claim regarding the structure of the moduli variety $\mathbb V_{(M, E)}$. The variant we have in mind is motivated in some sense by the entire theme underlying this article—that of thickenings and filtrations. Recall that any complex supermanifold comes equipped with thickenings that fit together to define a filtration, depicted in . We suspect that a similar structure will manifest itself on the moduli variety $\mathbb V_{(M, E)}$ and in this section we will clarify our suspicions further. We begin with the following definitions: \[nfiuh78fg47fh93hf038\] *A supermanifold is said to be be *$j$-trivialisable* if it admits a level-$(j+1)$ trivialisation ${\varphi}_j$. A supermanifold equipped with a level-$(j+1)$ trivialisation ${\varphi}_j$ is said to be *$j$-trivialised* and is denoted by the pair $({\mathfrak{X}}_{(M, E)}, {\varphi}_j)$.[^10]* Note that a $j$-trivialised supermanifold ${\mathfrak{X}}_{(M, E)}$ is filtered as follows: $$\begin{aligned} M \subset \Pi E^{(1)} \subset \Pi E^{(2)}\subset \cdots\subset \Pi E^{(j)}\subset {\mathfrak{X}}_{(M, E)}^{(j+1)}\subset \cdots\subset {\mathfrak{X}}_{(M, E)}. \label{fiorjfiofojfp3}\end{aligned}$$ Given a $j$-trivialised supermanifold $({\mathfrak{X}}_{(M, E)}, {\varphi}_j)$, an isomorphism ${\mathfrak{X}}_{(M, E)}\cong {\mathfrak{X}}_{(M, E)}^{\prime}$ certainly need not preserve the level of the trivialisation ${\varphi}_j$, as Illustration \[kopdj389hf883jfpo3\] shows. As such we consider the following notion of equivalence: \[fiojfoj90fj93jpf3\] We present now the following variant of the moduli problem in : $$\begin{aligned} {\mathfrak{M}}_{(M, E)}^{(j);\mathrm{triv.}} = \left\{\mbox{level-$i$ trivialised supermanifolds with $i\geq j$}\right\}/\sim \label{rkfj490fj3r3f3ff334fp3okp3ofk3p9d039jd}\end{aligned}$$ the above equivalence being that in the sense of Definition \[fiojfoj90fj93jpf3\]. \[fgdfvbhjbvevy3r4\] ${\mathfrak{M}}_{(M, E)}^{(1);\mathrm{triv.}} = {\mathfrak{M}}_{(M, E)}^{\mathrm{framed}}$. A 1-trivialisation for a supermanifold ${\mathfrak{X}}_{(M, E)}$ is a trivialisation $({\mathfrak{U}}, \rho)$ described in Construction \[knkniucbruibckj\], where $\rho$ is as in and . If $(x^\mu, {\theta}_a)$ denote coordinates on a patch ${\mathcal{U}}$ a vector field $X$ can be written $$X = f^\mu\frac{{\partial}}{{\partial}x^\mu} + g_a\frac{{\partial}}{{\partial}{\theta}_a}$$ where the indices are implicitly summed. Now consider a map ${\alpha}$ sending $X$ to $(g_a \mod {\mathcal{J}})~{\partial}/{\partial}{\theta}_a$ and observe that on the intersection ${\mathcal{U}}\cap{\mathcal{V}}$, $$\begin{aligned} g_a\frac{{\partial}}{{\partial}{\theta}_a} &= g_a \circ f_{UV} \left(\frac{{\partial}\rho^\nu_{{\mathcal{U}}{\mathcal{V}}}}{{\partial}{\theta}_a}\frac{{\partial}}{{\partial}y^\nu} + \frac{{\partial}\rho_{{\mathcal{U}}{\mathcal{V}};b}}{{\partial}{\theta}_a}\frac{{\partial}}{{\partial}\eta_b} \right) \stackrel{{\alpha}}{\longmapsto} \left(g_a \circ f_{UV} \mod {\mathcal{J}}\right) \zeta_{UV, b}^a\frac{{\partial}}{{\partial}\eta_b}\end{aligned}$$ where $(y^\nu,\eta_b)$ denote coordinates on ${\mathcal{V}}$. Hence $$(g_a\mod {\mathcal{J}})~\frac{{\partial}}{{\partial}{\theta}_a} = \left(g_a \circ f_{UV} \mod {\mathcal{J}}\right) \zeta_{UV, b}^a\frac{{\partial}}{{\partial}\eta_b}$$ and so ${\alpha}(X)$ is a section of ${\mathcal{E}}^\vee$. By construction ${\alpha}: {\mathfrak{T}}_{(M, E)}{\rightarrow}{\mathcal{E}}^\vee$ will be a framing. Note that the framing ${\alpha}$ depends on the trivialisation $({\mathfrak{U}}, \rho)$ up to common refinement. Now by Definition \[fiojfoj90fj93jpf3\], two 1-trivialised supermanifolds ${\mathfrak{X}}_{(M, E)}$ and ${\mathfrak{X}}^{\prime}_{(M, E)}$ will be equivalent if and only if there exists an isomorphism ${\lambda}: {\mathfrak{X}}_{(M, E)}\stackrel{\cong}{{\rightarrow}} {\mathfrak{X}}^{\prime}_{(M, E)}$ that restricts to the identity along $\Pi E^{(1)}$. Comparing with Definition \[rjf930j09j3fjp3fjp3w\](i) it is not too hard to see that ${\lambda}$ will also preserve the framing and thereby be an equivalence of framed supermanifolds. The converse assertion is straightforward. Following on from Lemma \[fgdfvbhjbvevy3r4\] above we have the following generalisation of Proposition \[jf983f3jfoj3ojg93jgp\]. \[dop3oiehoijip\] There exists a bijection: ${\mathfrak{M}}_{(M, E)}^{(j);\mathrm{triv.}}\cong H^1(M, {\mathcal{G}}^{(j+1)}_E)$. The existence of map ${\mathfrak{M}}_{(M, E)}^{(j);\mathrm{triv.}}{\rightarrow}H^1(M, {\mathcal{G}}^{(j+1)}_E)$ follows immediately from Definition \[nfiuh78fg47fh93hf038\]. It is given by sending $({\mathfrak{X}}_{(M, E)}, {\varphi}_i) \mapsto {\varphi}_i$, where $i\geq j$. That this map is a bijection is precisely what is captured by Definition \[fiojfoj90fj93jpf3\]. We tautologically have maps ${\mathfrak{M}}_{(M, E)}^{(j);\mathrm{triv.}}{\rightarrow}{\mathfrak{M}}_{(M, E)}^{(j-1);\mathrm{triv.}}$ and forgetting the trivialisation corresponds to the map ${\mathfrak{M}}_{(M, E)}^{(j)}{\rightarrow}{\mathfrak{M}}_{(M, E)}$. Evidently we have a commutative diagram: $$\begin{aligned} \xymatrix{ \ar[drrrr]{\mathfrak{M}}_{(M, E)}^{(j);\mathrm{triv.}} \ar[rr]& & \ar[drr]{\mathfrak{M}}_{(M, E)}^{(j-1);\mathrm{triv.}}\ar[r] & \cdots \ar[r] & {\mathfrak{M}}_{(M, E)}^{\mathrm{framed}}\ar[d] \\ &&&&{\mathfrak{M}}_{(M, E)}. } \label{4di3j8j30j3pkp3}\end{aligned}$$ By Proposition \[dop3oiehoijip\] we can compare the horizontal maps in with the maps on 1-cohomology induced by the normal filtration of the sheaf of groups ${\mathcal{G}}_E^{(2)}$. Now as observed in Illustration \[kopdj389hf883jfpo3\], the horizontal maps in need not be injective and so the moduli variety parametrising them, should it exist, need not fit together to define a filtration of $\mathbb V_{(M, E)}$. With this observation and Theorem \[4jf93jf903jf3jfp3j\] we conjecture: \[dnohf3fh839hf3h\] Fix a complex manifold $M$ and holomorphic vector bundle $E{\rightarrow}M$ of rank $q$ and set $${\mathfrak{T}}_{\Pi E}^{\geq j} := \bigoplus_{k\geq j} {\mathfrak{T}}_{\Pi E}[k].$$ For each $j$ there exists a moduli problem $\widetilde {\mathfrak{M}}_{(M, E)}^{(j)}\subseteq {\mathfrak{M}}_{(M, E)}^{(j);\mathrm{triv.}}$ and a connected, algebraic subvariety $\mathbb V^{(j)}_{(M, E)}\subset H^1(M,{\mathfrak{T}}_{\Pi E}^{\geq j})$ such that: (i) $\widetilde {\mathfrak{M}}_{(M, E)}^{(1)} = {\mathfrak{M}}_{(M, E)}^{\mathrm{framed}}$ and $\mathbb V^{(1)}_{(M, E)} = \mathbb V_{(M, E)}$; (ii) the set of points of $\mathbb V^{(j)}_{(M, E)}$ are in bijective correspondence with $\widetilde {\mathfrak{M}}_{(M, E)}^{(j)}$ and; (iii) there exists a descending filtration $$\mathbb V_{(M, E)} = \mathbb V_{(M, E)}^{(1)} \supset \mathbb V_{(M, E)}^{(2)}\supset \mathbb V_{(M, E)}^{(3)}\supset\cdots\supset \mathbb V_{(M, E)}^{(q)}\supset 0.$$ Hence, in the same way that a given supermanifold will admit a filtration, we suspect that the moduli variety of complex supermanifolds also will. In what follows we consider the moduli problem for thickenings of a given order. The Moduli Problem for Thickenings ---------------------------------- In Theorem \[4jf93jf903jf3jfp3j\] it is the first-cohomology group of the sheaf ${\mathfrak{T}}_{\Pi E}^{\geq2}$ which is relevant. Note however that the obstruction spaces in incorporate the sheaves $\mathcal Q_E^{(j);\pm}$ and, while they are related to ${\mathfrak{T}}_{\Pi E}[j]$ via the short-exact sequences in , the sheaves $\mathcal Q_E^{(j);\pm}$ themselves contain more information (see e.g., (2) following ). The first cohomology group of $\mathcal Q_E^{(j);\pm}$ comprises supermanifolds in addition to pseudo-supermanifolds and obstructed thickenings and so we suspect that it will house a variety representing, in a sense, a larger moduli problem than framed supermanifolds in . We will concentrate here on the moduli problem for thickenings directly and refrain from making any statements about the moduli variety itself.\ \ Our intent in this section is to show that the decomposition in holds at the level of moduli, i.e., up to isomorphism. This leads to the following notion of a morphism of thickenings, generalising Definition \[equivextensusythick\]. \[303hfjeipjej3fj\] The following construct, which we term the *moduli problem for order-$k$ thickenings*, now makes sense: $$\begin{aligned} \mathcal M^{(k)}_{(M, E)} = \left\{ \mbox{thickenings of $(M, E)$ of order $k$} \right\}/\cong. \label{dk3j903jf093jf903j}\end{aligned}$$ As mentioned above, we wish to argue that the decomposition in classifying thickenings into obstructed thickenings, pseudo-supermanifolds and supermanifolds, holds at the level of moduli. Our stating point is the following: \[4f4f8j0fj0\] Let ${\mathfrak{X}}^{(k)}_{(M, E)}$ be a thickening of order $k$ and suppose that ${\mathfrak{X}}^{(k)}_{(M, E)}\cong {\mathfrak{X}}^{(k){\prime}}_{(M, E)}$, for another order-$k$ thickening ${\mathfrak{X}}^{(k){\prime}}_{(M, E)}$. Then there exists a bijection: $$\mathrm{T}^1({\mathfrak{X}}_{(M, E)}^{(k)})\cong\mathrm{T}^1({\mathfrak{X}}_{(M, E)}^{(k){\prime}}).$$ It suffices to show that if ${\mathfrak{X}}^{(k+1)}_{(M, E)}\supset {\mathfrak{X}}_{(M, E)}^{(k)}$ is a thickening, then we can use the given isomorphism ${\mathfrak{X}}^{(k)}_{(M, E)}\cong {\mathfrak{X}}^{(k){\prime}}_{(M, E)}$ to deduce that ${\mathfrak{X}}^{(k){\prime}}_{(M, E)}$ must be unobstructed. The method of proof is similar to that for Proposition \[o3jdioj3ijp3\]. Firstly let ${\mathfrak{X}}^{(k+1)}_{(M, E)}\supset {\mathfrak{X}}_{(M, E)}^{(k)}$ be a thickening and denote by $({\mathfrak{U}}^{(k+1)}, \rho^{(k+1)})$ a trivialisation for it. From $\rho^{(k+1)}$ we obtain the trivialisation $({\mathfrak{U}}^{(k)}, \rho^{(k)})$ for ${\mathfrak{X}}_{(M, E)}^{(k)}$ by setting $\rho^{(k)} := \rho^{(k+1)}\mod {\mathcal{J}}^{k+1}$. Now, by assumption there exists an isomorphism ${\lambda}: {\mathfrak{X}}^{(k)}_{(M, E)}\cong {\mathfrak{X}}^{(k){\prime}}_{(M, E)}$. If $({\mathfrak{U}}^{(k)}, \rho^{(k){\prime}})$ denotes a trivialisation for ${\mathfrak{X}}_{(M, E)}^{(k){\prime}}$, then on the intersection ${\mathcal{U}}\cap{\mathcal{V}}$ we have: $$\begin{aligned} {\lambda}_{\mathcal{V}}\circ \rho^{(k)}_{{\mathcal{U}}{\mathcal{V}}} = \rho^{(k){\prime}}_{{\mathcal{U}}{\mathcal{V}}}\circ {\lambda}_{\mathcal{U}}. \label{fiofio3jfio3jf3}\end{aligned}$$ Before we proceed, the following construct will be convenient: $$\begin{aligned} \mathcal Q_E^{\geq2;\pm} := \bigoplus_{j\geq 2}\mathcal Q_E^{(j);\pm}. \label{39f0j30fj3pjfp3vv}\end{aligned}$$ Now just as in we will find that, more generally: $$\begin{aligned} \rho_{{\mathcal{U}}{\mathcal{V}}}^{(k)} - \rho^{(k){\prime}}_{{\mathcal{U}}{\mathcal{V}}} = ({\delta}\nu)_{{\mathcal{U}}{\mathcal{V}}} + w_{{\mathcal{U}}{\mathcal{V}}} \label{4f4jf4jg4ogj49}\end{aligned}$$ for $\nu = \{\nu_{\mathcal{U}}\}\in C^0(\mathfrak U, \mathcal Q_E^{(k);\pm})$ and $w = \{w_{{\mathcal{U}}{\mathcal{V}}}\}\in C^1({\mathfrak{U}}, \mathcal Q_E^{\geq2;\pm})$.[^11] The reason $w\neq 0$ here, in contrast to , is for the reason that the isomorphism ${\lambda}$ need not be trivial modulo ${\mathcal{J}}^k$. However, it must be trivial modulo ${\mathcal{J}}^2$ by Definition \[303hfjeipjej3fj\](ii), thereby justifying the summation in . As a consequence of we find: $$\begin{aligned} {\Gamma}^{(k+1){\prime}}_{{\mathcal{U}}{\mathcal{V}}{\mathcal{W}}} &= \rho^{(k){\prime}}_{{\mathcal{U}}{\mathcal{W}}} - \rho^{(k){\prime}}_{{\mathcal{V}}{\mathcal{W}}}\circ \rho^{(k){\prime}}_{{\mathcal{U}}{\mathcal{V}}} \notag\\ &= \rho^{(k)}_{{\mathcal{U}}{\mathcal{W}}} - ({\delta}\nu)_{{\mathcal{U}}{\mathcal{W}}} + w_{{\mathcal{U}}{\mathcal{W}}} - \rho^{(k)}_{{\mathcal{V}}{\mathcal{W}}}\circ \rho^{(k)}_{{\mathcal{U}}{\mathcal{V}}} - ({\delta}\nu)_{{\mathcal{V}}{\mathcal{W}}} - w_{{\mathcal{V}}{\mathcal{W}}} -({\delta}\nu)_{{\mathcal{U}}{\mathcal{V}}} - w_{{\mathcal{U}}{\mathcal{V}}} \notag\\ &= {\Gamma}_{{\mathcal{U}}{\mathcal{V}}{\mathcal{W}}}^{(k+1)} + ({\delta}w)_{{\mathcal{U}}{\mathcal{V}}{\mathcal{W}}}. \label{38893hf83hf83h}\end{aligned}$$ Note that while $w = \{w_{{\mathcal{U}}{\mathcal{V}}}\}\in C^1({\mathfrak{U}}, \mathcal Q_E^{\geq2;\pm})$, the quantity $({\delta}w)_{{\mathcal{U}}{\mathcal{V}}{\mathcal{W}}}$ in above will be homogeneous and of degree-$(k+1)$. This is because, by assumption, both ${\Gamma}^{(k+1){\prime}}_{{\mathcal{U}}{\mathcal{V}}{\mathcal{W}}}$ and ${\Gamma}_{{\mathcal{U}}{\mathcal{V}}{\mathcal{W}}}^{(k+1)}$ will vanish identically modulo ${\mathcal{J}}^{k+1}$ implying therefore that so must $({\delta}w)_{{\mathcal{U}}{\mathcal{V}}{\mathcal{W}}}$. Hence, we can make sense of as a statement about $\mathcal Q_E^{(k+1);\pm}$-valued 2-cocycles. In particular that $\{{\Gamma}^{(k+1){\prime}}_{{\mathcal{U}}{\mathcal{V}}{\mathcal{W}}}\} = {\Gamma}^{(k+1){\prime}} \sim {\Gamma}^{(k+1)} = \{{\Gamma}_{{\mathcal{U}}{\mathcal{V}}{\mathcal{W}}}^{(k+1)}\}$. Since we assume that ${\mathfrak{X}}^{(k)}_{(M, E)}$ is unobstructed we know that ${\Gamma}^{(k+1)}\sim 0$ and therefore ${\Gamma}^{(k+1){\prime}}\sim 0$ also which means ${\mathfrak{X}}^{(k){\prime}}$ will be unobstructed. The present result now follows from Theorem \[o4993uf0oijwoijfwojpw\]. \[38f03hfoinio30\] An obstructed thickening cannot be isomorphic to an unobstructed thickening. If ${\mathfrak{X}}^{(k)}$ is an obstructed thickening then $\mathrm T^1({\mathfrak{X}}^{(k)}) = \eset$ whereas for ${\mathfrak{X}}^{(k){\prime}}$ unobstructed, we have $\mathrm T^1({\mathfrak{X}}^{(k){\prime}})\neq\eset$ by Theorem \[o4993uf0oijwoijfwojpw\]. The present corollary now follows from Proposition \[4f4f8j0fj0\]. Continuing on with the theme set by Corollary \[38f03hfoinio30\], we now consider pseudo-supermanifolds. \[fffggfefgjee\] Let ${\mathfrak{X}}^{(k+1)}\supset {\mathfrak{X}}^{(k)}$ and suppose ${\mathfrak{X}}^{(k)}\cong{\mathfrak{X}}^{(k){\prime}}$ for some other thickening ${\mathfrak{X}}^{(k)}$. Then there exists a thickening ${\mathfrak{X}}^{(k+1){\prime}}\supset {\mathfrak{X}}^{(k){\prime}}$ such that ${\mathfrak{X}}^{(k+1){\prime}}\cong {\mathfrak{X}}^{(k+1){\prime}}$ as locally ringed spaces. We are given the data of a thickening ${\mathfrak{X}}^{(k+1)}\supset {\mathfrak{X}}^{(k)}$ and an isomorphism ${\mathfrak{X}}^{(k)}\cong {\mathfrak{X}}^{(k){\prime}}$. With respect to a cover ${\mathfrak{U}}$, denote by $\rho^{(k+1)}$ and $\rho^{(k){\prime}}$ trivialisations for the thickenings ${\mathfrak{X}}^{(k+1)}$ and ${\mathfrak{X}}^{(k){\prime}}$ respectively. Note that $\rho^{(k)} := \rho^{(k+1)}\mod {\mathcal{J}}^{k+1}$ will be a trivialisation for ${\mathfrak{X}}^{(k)}$. Now just as in , the assumption that ${\mathfrak{X}}^{(k)}\cong {\mathfrak{X}}^{(k){\prime}}$ means there exists an isomorphism ${\lambda}= \{{\lambda}_{\mathcal{U}}\}$ such that on all intersections ${\mathcal{U}}\cap{\mathcal{V}}$, $$\begin{aligned} {\lambda}_{\mathcal{V}}\circ \rho^{(k)}_{{\mathcal{U}}{\mathcal{V}}} = \rho^{(k){\prime}}_{{\mathcal{U}}{\mathcal{V}}}\circ {\lambda}_{\mathcal{U}}. \label{orooievio3o33}\end{aligned}$$ Consider now the following object, defined on intersections: $$\begin{aligned} \rho^{(k+1){\prime}}_{{\mathcal{U}}{\mathcal{V}}} := {\lambda}_{\mathcal{V}}\circ \rho^{(k+1)}_{{\mathcal{U}}{\mathcal{V}}}\circ {\lambda}_{\mathcal{U}}^{-1}. \label{39f03foeij40j3}\end{aligned}$$ Since $\rho^{(k+1)}$ satisfies the cocycle condition, then so will $\rho^{(k+1){\prime}}$. Hence $({\mathfrak{U}}, \rho^{(k+1){\prime}})$ will trivialise *something*. We claim that it will define a trivialisation for a thickening ${\mathfrak{X}}^{(k+1){\prime}}\supset{\mathfrak{X}}^{(k){\prime}}$. The thickening ${\mathfrak{X}}^{(k+1){\prime}}$ will, by construction, be isomorphic to ${\mathfrak{X}}^{(k+1)}$ and the lemma will then follow. So to argue that $({\mathfrak{U}}, \rho^{(k+1){\prime}})$ will trivialise ${\mathfrak{X}}^{(k+1){\prime}}\supset {\mathfrak{X}}^{(k){\prime}}$, it suffices to show that $\rho^{(k+1){\prime}} \equiv \rho^{(k){\prime}}$ modulo ${\mathcal{J}}^{k+1}$. To show this, write: $\rho^{(k+1)} = \rho^{(k)} + \phi^{(k+1)}$, for $\phi^{(k+1)}$ a $\mathcal Q_E^{(k+1);\pm}$-valued 1-cochain. This will vanish modulo ${\mathcal{J}}^{k+1}$. Evaluating gives: $$\begin{aligned} \rho^{(k+1){\prime}}_{{\mathcal{U}}{\mathcal{V}}} = {\lambda}_{\mathcal{V}}\circ \rho^{(k+1)}_{{\mathcal{U}}{\mathcal{V}}}\circ {\lambda}_{\mathcal{U}}^{-1} &= {\lambda}_{\mathcal{V}}\circ \left(\rho_{{\mathcal{U}}{\mathcal{V}}}^{(k)} + \phi_{{\mathcal{U}}{\mathcal{V}}}^{(k+1)}\right) \circ {\lambda}_{\mathcal{U}}^{-1} \\ &= {\lambda}_{\mathcal{V}}\circ \rho^{(k)}_{{\mathcal{U}}{\mathcal{V}}}\circ {\lambda}_{\mathcal{U}}^{-1} +\ldots \\ &= \rho_{{\mathcal{U}}{\mathcal{V}}}^{(k){\prime}} + \ldots &&\mbox{by \eqref{orooievio3o33}}\end{aligned}$$ where the ellipses ‘$\ldots$’ denote terms proportional to $\phi^{(k+1)}$. In particular, such terms lie in ${\mathcal{J}}^{k+1}$ which means, by , that $\rho^{(k+1){\prime}} \equiv \rho^{(k){\prime}}$ modulo ${\mathcal{J}}^{k+1}$. The lemma now follows. As an illustration, we can use Proposition \[4f4f8j0fj0\] and Lemma \[fffggfefgjee\] to address Question \[kd93f3jf3kf9p4\]: \[39f3hfoi33j\] Let ${\mathfrak{X}}^{(k)}$ be a thickening and suppose it is isomorphic to some thickening $\tilde{\mathfrak{X}}^{(k)}$ which embeds in some supermanifold $\tilde{\mathfrak{X}}$. Then there exists a supermanifold ${\mathfrak{X}}$ containing this given thickening ${\mathfrak{X}}^{(k)}$. We are given ${\mathfrak{X}}^{(k)}$. Since we assume ${\mathfrak{X}}^{(k)}\cong \tilde{\mathfrak{X}}^{(k)}$ and since $\tilde{\mathfrak{X}}^{(k)}$ is assumed to be embed in some supermanifold $\tilde{\mathfrak{X}}$, it follows that there will be a thickening $\tilde{\mathfrak{X}}^{(k+1)}\supset \tilde{\mathfrak{X}}^{(k)}$. Then by Lemma \[fffggfefgjee\] we see that there must then exist a thickening ${\mathfrak{X}}^{(k+1)}\supset {\mathfrak{X}}^{(k)}$ with ${\mathfrak{X}}^{(k+1)}\cong \tilde{\mathfrak{X}}^{(k+1)}$. Now observe that we are in the same situation as we were at the start of this proof, i.e., that we have a thickening ${\mathfrak{X}}^{(k+1)}$ and an isomorphism between this given thickening and some thickening $\tilde{\mathfrak{X}}^{(k+1)}$ embedded in a supermanifold $\tilde{\mathfrak{X}}$. Continuing on inductively we can deduce (in finitely many steps) the existence of a supermanifold ${\mathfrak{X}}$, which is isomorphic to $\tilde{\mathfrak{X}}$, and contains ${\mathfrak{X}}^{(k)}$ as an embedded, $k$-th order thickening. Now just like Corollary \[38f03hfoinio30\] we have: \[38hfofjoijfoijo\] A pseudo-supermanifold will not be isomorphic to any thickening that embeds in some supermanifold. To recap now, from Corollary \[38f03hfoinio30\] an obstructed thickening of order $k$ cannot be isomorphic to a pseudo-supermanifold of order-$k$; and from Corollary \[38hfofjoijfoijo\] a pseudo-supermanifold can never be isomorphic to a thickening embedded in some supermanifold. Hence, a straightforward consequence is the following decomposition of the moduli problem, mirroring : $$\begin{aligned} \mathcal M^{(k)}_{(M, E)} = {\mathcal{M}}^{(k);\mathrm{smfld.}}_{(M, E)}\cup\mathcal M^{(k);\mathrm{pseudo.}}_{(M, E)} \cup \mathcal M^{(k);\mathrm{obs.}}_{(M, E)} \label{jifjefiojoif3j9f03j}\end{aligned}$$ where ${\mathcal{M}}^{(k);\mathrm{smfld.}}_{(M, E)}$; resp. $\mathcal M^{(k);\mathrm{pseudo.}}_{(M, E)}$; resp. $\mathcal M^{(k);\mathrm{obs.}}_{(M, E)}$ are moduli problems for order-$k$ thickenings embedded in supermanifolds; resp. pseudo-supermanifolds of order $k$; resp. obstructed, $k$-th order thickenings.\ \ In what follows we will infer the existence of obstructed thickenings of the complex projective plane. Illustrations on the Complex Projective Plane {#7} ============================================= A corollary of Theorem \[jnvjknknvkjnvk\] is that *any* thickening ${\mathfrak{X}}^{(l)}$ of a Riemann surface $C$ will be unobstructed, in the sense of Definition \[smeioiof894f89jcoi\]. We wish to describe here an example of an *obstructed*, second order thickening so we will therefore have to look at thickenings ${\mathfrak{X}}^{(l)}_{(M, E)}$ of $M$ where $M$ a complex manifold with $\dim_{\mathbb{C}}M\geq 2$ and $E{\rightarrow}M$ is a holomorphic vector bundle of rank at least $3$. We will consider here the complex projective plane $M = {\mathbb{C}}{\mathbb{P}}^2$ and appeal to Proposition \[popiioh4o8fh84o\]. Integral to our considerations will be the *Bott formula*—a formula for computing the complex dimensions of certain cohomology groups on projective spaces; and *Serre duality*. Regarding the Bott formula, we state this from [@OSS p. 4] below: $$\begin{aligned} h^q\left({\Omega}_{{\mathbb{C}}{\mathbb{P}}^n}^p(k)\right) = \left\{ \begin{array}{cl} \binom{k+n-p}{k}\binom{k-1}{p} & \mbox{for $q = 0$ and $0\leq p\leq n$ and $k> p$}\\ 1 & \mbox{for $0\leq p = q\leq n$ and $k = 0$}\\ \binom{-k+p}{-k}\binom{-k-1}{n-p} & \mbox{for $q = n$ and $0\leq p \leq n$ and $k<p-n$}\\ 0 & \mbox{otherwise}. \end{array} \right. \label{BOTT}\end{aligned}$$ Here ${\Omega}^p_{{\mathbb{C}}{\mathbb{P}}^n}(k) = {\Omega}^p_{{\mathbb{C}}{\mathbb{P}}^n}\otimes{\mathcal{O}}_{{\mathbb{C}}{\mathbb{P}}^n}(k)$, where ${\mathcal{O}}_{{\mathbb{C}}{\mathbb{P}}^n}(k)$ is the $|k|$-th tensor power of the hyperplane divisor if $k> 0$, or its dual for $k< 0$; and $h^q( -) = \dim_{\mathbb{C}}H^q({\mathbb{C}}{\mathbb{P}}^n, -)$. The celebrated Serre duality theorem for vector bundles on projective space is: $$h^i({\mathcal{E}}) = h^{n-i}({\mathcal{E}}^\vee(-n-1))$$ for $E{\rightarrow}{\mathbb{C}}{\mathbb{P}}^n$ a holomorphic vector bundle and $i = 0, \ldots, n$. As mentioned, we are interested in constructing an example of an obstructed, second order thickening. Our starting point is the exact sequence in for $k = 2$ which we give below for convenience: $$0 {\rightarrow}\wedge^3{\mathcal{E}}\otimes{\mathcal{E}}^\vee\hookrightarrow {\mathfrak{T}}_{\Pi E}[2] \twoheadrightarrow \wedge^2{\mathcal{E}}\otimes{\mathfrak{T}}_M{\rightarrow}0.$$ This sequence induces a long-exact sequence on sheaf cohomology containing the following piece: $$\begin{aligned} \xymatrix{ \ldots\ar[r] & H^1(M,\wedge^3{\mathcal{E}}\otimes{\mathcal{E}}^\vee)\ar[r] & H^1(M,{\mathfrak{T}}_{\Pi E}[2]) \ar[r]^{r_*~~~} & H^1(M,\wedge^2{\mathcal{E}}\otimes{\mathfrak{T}}_M) \ar[d]^{{\partial}_*} \\ & &\ldots & \ar[l]H^2(M,\wedge^3{\mathcal{E}}\otimes{\mathcal{E}}^\vee) } \label{dmiomdiorjf94j09}\end{aligned}$$ In the case where $\dim_{\mathbb{C}}M=2$ we have: \[dmoeiciojciojc904jc409dmoeiciojciojc904jc409\] Let $M$ be $2$-dimensional and $E{\rightarrow}M$ a rank $3$ vector bundle with $\deg E = k$. Suppose $$\begin{aligned} h^1( {\mathfrak{T}}_{M}\otimes\wedge^2{\mathcal{E}})\neq0; && h^2({\mathfrak{T}}_{M}\otimes\wedge^2{\mathcal{E}})\neq0, &&\mbox{and}&& h^2( {\mathcal{E}}^\vee(k))\neq0, \label{dmoeiciojciojc904jc409}\end{aligned}$$ where $h^i(-) = h^i(M, -)$. Then there will exist an obstructed, second order thickening ${\mathfrak{X}}_{(M, E)}^{(2)}$ of $\Pi E^{(1)}$. As $M$ is a two-dimensional, complex manifold it follows that $H^i(M, {\mathcal{F}}) = 0$ for all $i> 2$ and any sheaf of abelian groups ${\mathcal{F}}$. This allows us to conclude that the sequence in for $M$ continues as follows: $$\begin{aligned} \xymatrix{ \ldots\ar[r] & H^1({\mathcal{E}}^\vee(k))\ar[r]^{s^1_*} & H^1({\mathfrak{T}}_{\Pi E}[2]) \ar[r]^{r^1_*~~~} & H^1(\wedge^2{\mathcal{E}}\otimes{\mathfrak{T}}_M) \ar[d]^{{\partial}_*} \\ 0&\ar[l]H^2(\wedge^2{\mathcal{E}}\otimes{\mathfrak{T}}_M) &\ar[l]_{~~~~~~r^2_*}H^2({\mathfrak{T}}_{\Pi E}[2]) & \ar[l]_{~~s^2_*}H^2({\mathcal{E}}^\vee(k)) } \label{dmiomdiorjf94j043433d9}\end{aligned}$$ where $H^i(-) = H^i(M, -)$. Our objective is to definitively conclude that the boundary map ${\partial}_*$ is non-trivial under the hypotheses in , for then we can conclude that the complement of ${\mathrm{im}}~r^1_*$ in $H^1(\wedge^2{\mathcal{E}}\otimes{\mathfrak{T}}_M)$, denoted $({\mathrm{im}}~r^1_*)^\perp$, will be non-empty. This is important since, by exactness of , Lemma \[lkliioolkss\] and Theorem \[jnvjknknvkjnvk\], the set $({\mathrm{im}}~r^1_*)^\perp$ will contain trivialisations for obstructed, second order thickenings. Now, the former two conditions in are clearly necessary conditions. To see that all three conditions in are sufficient to deduce $({\mathrm{im}}~r^1_*)^c\neq\eset$, firstly note by exactness of that: ${\mathrm{im}}~ {\partial}_* = \ker~s_*^2$. Hence it suffices to characterise the map $s^2_*$. Now, by exactness again we know that $r^2_*$ will be surjective and non-trivial. If $r^2_*$ is *bijective*, then: $$\begin{aligned} \{0\} = \ker r^2_* = {\mathrm{im}}~s^2_* {\Longrightarrow}\ker~s_*^2\neq0 {\Longrightarrow}{\mathrm{im}}~{\partial}_*\neq0. \label{ijoijoij494kk39kpk}\end{aligned}$$ Hence, if $r^2_*$ is bijective, we see that ${\partial}_*$ will be non-trivial. Suppose however $r_*^2$ is *not* bijective. Then, if $s^2_*$ is not bijective either, we may use the reasoning in again to deduce that ${\partial}_*$ is non-trivial. However, if $s_*^2$ *is* bijective, then ${\partial}_*$ must necessarily be trivial, but note that bijectivity of $s^2_*$ will contradict the surjectivity of $r^2_*$. Hence $s^2_*$ cannot be bijective, which means ${\partial}_*$ will be non-trivial. The lemma now follows. \[8jf48f4unovu4no4oi\] (for split, rank-$3$ vector bundles on ${\mathbb{C}}{\mathbb{P}}^2$) From we have: $$\begin{aligned} h^0({\mathfrak{T}}_{{\mathbb{C}}{\mathbb{P}}^2}(l))&\neq 0 \iff l>2 \label{splitvdcp3condn1} \\ h^1({\mathfrak{T}}_{{\mathbb{C}}{\mathbb{P}}^2}(l)) &\neq 0\iff l=-3 \label{splitvdcp3condn2} \\ h^2({\mathcal{O}}_{{\mathbb{C}}{\mathbb{P}}^2}(l))&\neq 0\iff l<-3. \label{splitvdcp3condn3}\end{aligned}$$ Now let ${\mathcal{E}}= \bigoplus_{a=1}^3{\mathcal{O}}_{{\mathbb{C}}{\mathbb{P}}^2}(k_a)$ be a split, rank-$3$, holomorphic vector bundle on ${\mathbb{C}}{\mathbb{P}}^2$. The degree of ${\mathcal{E}}$ is $k = k_1 + k_2+k_3$ and the second exterior power is given by: $$\wedge^2{\mathcal{E}}= {\mathcal{O}}_{{\mathbb{C}}{\mathbb{P}}^2}(k_1+k_2) \oplus {\mathcal{O}}_{{\mathbb{C}}{\mathbb{P}}^2}(k_1+k_3)\oplus {\mathcal{O}}_{{\mathbb{C}}{\mathbb{P}}^2}(k_2+k_3).$$ Now, the sheaf cohomology functor $H^i(-)$ on projective space commutes with (countably-many) direct sums, as discussed in [@HARTALG p. 209], and so $h^i({\mathcal{E}}) = \sum_{a=1}^3h^i({\mathcal{O}}_{{\mathbb{C}}{\mathbb{P}}^2}(k_a))$. Then, in order to ensure , it suffices to choose the triple $(k_1, k_2, k_3)$ such that , and hold. This leads to the following constraints: $$\begin{aligned} k_1 + k_2>2 && k_1 + k_3 = -3 && \mbox{and} && k_2 + k_3 < -3. \label{jfuf484h84frifoe}\end{aligned}$$ Evidently, a solution to exists for any distinct pair of integers $(k_1, k_2)$ which satisfy $k_1 + k_2>2$. Then for such an ${\mathcal{E}}$ the conditions in will hold and so, by Lemma $\ref{dmoeiciojciojc904jc409dmoeiciojciojc904jc409}$, there will exist obstructed, second order thickenings ${\mathfrak{X}}_{({\mathbb{C}}{\mathbb{P}}^2, E)}^{(2)}$ of $\Pi E^{(1)}$. With regards to non-split bundles, it is not so straightforward to deduce the existence of obstructed thickenings as in the split case in Example \[8jf48f4unovu4no4oi\]. In this article we will consider the next logical step after rank-3, split bundles being: rank-3, non-split, decomposable bundles. Non-split, Decomoposable Vector Bundles --------------------------------------- Non-split, decomposable vector bundles of any rank (greater than $3$) exist on the projective plane by virtue of the construction of indecomposable bundles (of any rank) by Schwarzenberger in [@SCH]. In rank-3 any non-split, decomposable bundle must necessarily be a direct sum of a rank-2, indecomposable bundle with a line bundle. Now suppose $E{\rightarrow}{\mathbb{C}}{\mathbb{P}}^2$ has rank-3. Then $\wedge^3{\mathcal{E}}$ will be a line bundle and so $\wedge^3{\mathcal{E}}= {\mathcal{O}}_{{\mathbb{C}}{\mathbb{P}}^2}(k)$, for $k = \deg(E)$. We set ${\mathcal{E}}^\vee(k) := {\mathcal{E}}^\vee\otimes {\mathcal{O}}_{{\mathbb{C}}{\mathbb{P}}^2}(k)$. Our method of inferring the existence of obstructed thickenings here will follow that in Example \[8jf48f4unovu4no4oi\]. That is, we will appeal to Lemma \[dmoeiciojciojc904jc409dmoeiciojciojc904jc409\]. To this extent we present the following construction of a rank-2, indecomposable, holomorphic vector bundle from [@OSS]. \[dnnriufj4jf904j094c\] Let $Y\subset {\mathbb{C}}{\mathbb{P}}^2$ comprise a collection of $m$ points, $m>0$, and let ${\mathcal{J}}_Y\subset {\mathcal{C}}_{{\mathbb{C}}{\mathbb{P}}^2}$ be the sub-sheaf of holomorphic functions on ${\mathbb{C}}{\mathbb{P}}^2$ which vanish on $Y$, i.e., an ideal sheaf. Then in *[@OSS p. 53]* it is constructed a rank-$2$, holomorphic vector bundle $F{\rightarrow}{\mathbb{C}}{\mathbb{P}}^2$ and a global section $s\in H^0({\mathbb{C}}{\mathbb{P}}^2, {\mathcal{F}})$ such that firstly, for a fixed integer $k^{\prime}<3$, $$\begin{aligned} c_1(F) = k^{\prime};&&c_2(F) = m &&\mbox{and}&& Y = \{\mbox{zeroes of $s$}\}; \label{dnieief894jf9j44j}\end{aligned}$$ and secondly that the sheaf of holomorphic sections ${\mathcal{F}}$ of $F$ fits into the exact sequence: $$\begin{aligned} 0 {\rightarrow}{\mathcal{C}}_{{\mathbb{C}}{\mathbb{P}}^2}\stackrel{\cdot s}{\hookrightarrow} {\mathcal{F}}\twoheadrightarrow {\mathcal{J}}_Y(k^{\prime}){\rightarrow}0 \label{deioieojf4jj43djd}\end{aligned}$$ where ${\mathcal{J}}_Y(k^{\prime}) = {\mathcal{J}}_Y\otimes{\mathcal{O}}_{{\mathbb{C}}{\mathbb{P}}^2}(k^{\prime})$ and ${\mathcal{F}}$ the sheaf of sections of $F$. We will be interested in the degree of ${\mathcal{F}}$ which, from is $k^{\prime}$ and so, henceforth, we will denote the bundle described here by ${\mathcal{F}}_{k^{\prime}}$. More generally, a construction of rank-2, indecomposable bundles of a similar nature to that of ${\mathcal{F}}_{k^{\prime}}$ in Construction \[dnnriufj4jf904j094c\] on complex surfaces other than ${\mathbb{C}}{\mathbb{P}}^2$ is given in [@PAGGH p. 726]. We limit our considerations here to the projective plane. Integral to inferring the existence of some $E{\rightarrow}{\mathbb{C}}{\mathbb{P}}^2$ such that holds is another famous theorem of Serre, which we state from [@OSS]: Let ${\mathcal{F}}$ be a coherent, analytic sheaf on ${\mathbb{C}}{\mathbb{P}}^n$. Then there exists a $k_0\in {\mathbb{Z}}$ such that, for any $l\geq k_0$, the sheaf ${\mathcal{F}}(l)$ is generated by its global sections. To elaborate on Serre’s Theorem A, a sheaf ${\mathcal{F}}$ is said to be generated by its global sections if the evaluation map $H^0({\mathcal{F}}) \otimes {\mathcal{O}}{\rightarrow}{\mathcal{F}}$ is surjective, where ${\mathcal{O}}$ denotes the structure sheaf. In particular, if ${\mathcal{F}}$ is a holomorphic vector bundle on ${\mathbb{C}}{\mathbb{P}}^n$, we see that $h^0({\mathcal{F}}(l))\neq 0$ for all $l\geq k_0$ and some $k_0\in {\mathbb{Z}}$. Now consider the bundle $${\mathcal{E}}_{(k^{\prime}, l)} := {\mathcal{F}}_{k^{\prime}}\oplus {\mathcal{O}}_{{\mathbb{C}}{\mathbb{P}}^2}(l),$$ for some $l$. Then ${\mathcal{E}}_{(k^{\prime}, l)}$ will be the sheaf of holomorphic sections of a non-split, decomposable, rank-3 vector bundle $E_{(k^{\prime}, l)}$ over ${\mathbb{C}}{\mathbb{P}}^2$. By construction we have that $\deg E_{(k^{\prime}, l)} = k^{\prime}+ l$. \[hghgghdvvy3vdy3hdhhgwhfgabababbybwu\] For sufficiently small $l$ the bundle ${\mathcal{E}}_{(k^{\prime}, l)}$ will be such that will be satisfied. We firstly have: $$\begin{aligned} \wedge^2{\mathcal{E}}_{(k^{\prime}, l)} &\cong \left(\wedge^2{\mathcal{F}}_{k^{\prime}}\otimes \wedge^0{\mathcal{O}}_{{\mathbb{C}}{\mathbb{P}}^2}(l)\right)\oplus \left(\wedge^1{\mathcal{F}}_{k^{\prime}}\otimes \wedge^1{\mathcal{O}}_{{\mathbb{C}}{\mathbb{P}}^2}(l)\right) \notag \\ &= {\mathcal{O}}_{{\mathbb{C}}{\mathbb{P}}^2}(k^{\prime}) \oplus {\mathcal{F}}_{k^{\prime}}(l). \label{040f903joi3mpwmvmep}\end{aligned}$$ Now, the sheaf cohomology functor $H^i({\mathbb{C}}{\mathbb{P}}^n, -)$ will commute with countably-many direct sums so we may deduce that $h^i({\mathcal{F}}\oplus {\mathcal{G}}) = h^i({\mathcal{F}}) + h^i({\mathcal{G}})$. Using this, Serre duality and , it will then follow from that $$\begin{aligned} h^1({\mathfrak{T}}_{{\mathbb{C}}{\mathbb{P}}^2}\otimes\wedge^2{\mathcal{E}}_{(k^{\prime}, l)}) \geq h^1({\Omega}^1_{{\mathbb{C}}{\mathbb{P}}^2}(-k^{\prime}-3)) = 1~~\mbox{iff}~k^{\prime}= -3. \label{fuihhfi4hf4h89h94}\end{aligned}$$ Now, recall that we must have $k^{\prime}< 3$ by construction of ${\mathcal{F}}_{k^{\prime}}$ in Construction \[dnnriufj4jf904j094c\]. In setting $k^{\prime}= -3$, we will be assured in $h^1({\mathfrak{T}}_{{\mathbb{C}}{\mathbb{P}}^2}\otimes\wedge^2{\mathcal{E}}_{(k^{\prime}, l)})\neq0$ from . Now, as a result of setting $k^{\prime}= -3$, note from that: $$\begin{aligned} h^2({\mathfrak{T}}_{{\mathbb{C}}{\mathbb{P}}^2}\otimes\wedge^2{\mathcal{E}}_{(-3, l)}) &= h^0({\Omega}^1_{{\mathbb{C}}{\mathbb{P}}^2} (0)) + h^0({\Omega}^1_{{\mathbb{C}}{\mathbb{P}}^2}\otimes {\mathcal{F}}^\vee_{-3}(-l-3)) \notag \\ &= h^0({\Omega}^1_{{\mathbb{C}}{\mathbb{P}}^2}\otimes {\mathcal{F}}^\vee_{-3}(-l-3)). \label{neiofio3f9830j303}\end{aligned}$$ We are yet to impose any constraints on $l$ here. In appealing to Serre’s Theorem A, we may choose $l$ sufficiently small (i.e., sufficiently negative) so that $-l-3 \gg 0$. This will ensure that $h^0({\Omega}^1_{{\mathbb{C}}{\mathbb{P}}^2}\otimes {\mathcal{F}}^\vee_{-3}(-l-3))\neq0$. Similarly, regarding the latter-most cohomology group in , we have: $$\begin{aligned} h^2({\mathcal{E}}^\vee(-3+l)) &= h^2({\mathcal{F}}^\vee_{-3}(-3+l)) + h^2({\mathcal{O}}_{{\mathbb{C}}{\mathbb{P}}^2}(-3)) = h^0({\mathcal{F}}_{-3}(-l+3)) \end{aligned}$$ and just as in we see, for sufficiently small $l$, that $h^0({\mathcal{F}}_{-3}(-l+3))>0$ by Serre’s Theorem A. The proposition now follows. From Lemma \[dmoeiciojciojc904jc409dmoeiciojciojc904jc409\] and Proposition \[hghgghdvvy3vdy3hdhhgwhfgabababbybwu\] we conclude: Let $E_{(-3, l)} {\rightarrow}{\mathbb{C}}{\mathbb{P}}^2$ be the vector bundle whose sheaf of sections is ${\mathcal{E}}_{(-3, l)}$. Then for sufficiently small $l$, there will exist an obstructed, second order thickening of $\Pi E^{(1)}_{(-3, l)}$. Concluding Remarks ================== A common technique in studies of supersymmetric sigma-models and field theories is to move from the superspace formulation to the component formulation, which is achieved by carrying out a Berezin integral at some stage. See [@FREEDSUSY; @DELFREEDSUSY] for a more in depth treatment of this general procedure for superspace Lagrangians. Importantly, employing this technique allows for well studied methods from geometry to become readily applicable to calculate quantities of interest. There are however instances in which this technique will not be well-defined and this issue, of well-definedness, can be directly related to the subtleties of obstruction theory. A motivating example highlighting precisely this issue is superstring theory—the quantities of interest here being scattering amplitudes for the superstring. For more on this topic see [@HOKERPHONG1; @HOKER1].\ \ In light of these observations we might view obstruction theory as being that theory dictating the interplay between geometry and supergeometry—at least in the complex-analytic setting. It is a hope of the author that obstruction theory can provide insights into supersymmetric theories more generally. For instance, as a means to study certain supersymmetric field theories within the superspace formulation itself. At the very least, it seems clear that to circumvent some of the issues plaguing superstring theory, further developments in complex supergeometry will be desirable and these will certainly require a greater understanding of obstruction theory. \ <span style="font-variant:small-caps;">Kowshik Bettadapura, Mathematical Sciences Institute, Australian National University, Canberra, ACT 2601, Australia</span>\ *E-mail address:* <kowshik.bettadapura@anu.edu.au> [^1]: The distinction between ${\mathfrak{X}}$ and ${\mathfrak{X}}_{(M,E)}$ is superficial. We make it only to emphasise that, in this article, we adopt a ‘bottom-up’ point of view on supermanifolds. That is, rather than starting with some supermanifold, we start with a manifold $M$, a vector bundle $E{\rightarrow}M$ and study supermanifold structures associated to $(M, E)$. [^2]: note, this is well defined up to common refinement of the open covering. In this way ${\mathfrak{X}}_{(M, E)}$ will not depend on this choice of open covering. [^3]: Here we define: $\wedge^{k\leq l}{\mathbb{C}}^q :=\wedge^{\bullet}{\mathbb{C}}^q /J^{l+1}$ where $J = \ker\{\wedge^{\bullet}{\mathbb{C}}^q \twoheadrightarrow {\mathbb{C}}\}$. As ${\mathbb{C}}$-modules (complex vector spaces) we have $\wedge^{k\leq l}{\mathbb{C}}^q\cong \bigoplus_{k=0}^l\wedge^k{\mathbb{C}}^q$. [^4]: Recall that we are implicitly considering all quantities here modulo ${\mathcal{J}}^{2m+1}$. [^5]: a simpler result justifying our assertions here is a certain exact sequence of sheaves of groups which we discuss in the section to come (Proposition \[dm30j8fjf3o\]). [^6]: \[ijfoeho8f4ho\]this is explicitly mentioned by Vaintrob in [@VAIN] in the introductory paragraph. The difference is attributed to the difference between the terms *classification* and *deformation*. [^7]: We could also consider an equivalence by appealing directly to the coframe ${\varphi}: {\mathcal{J}}/{\mathcal{J}}^2\stackrel{\cong}{{\rightarrow}}{\mathcal{E}}$. However, it seems more natural to use framings. Moreover, framings are more closely related to similar discussions in [@DW1]. [^8]: c.f., Lemma \[fiugf78g79h380fj09\] [^9]: At the beginning of this article, by way of motivation, the moduli space of super Riemann surfaces was mentioned. This object is an example of a supermoduli space. [^10]: c.f., Definition \[rfijf903jf903kf\]. [^11]: note of course that need not vanish modulo ${\mathcal{J}}^{k+2}$. An apt comparison is the construction of the obstruction element ${\Gamma}$ in .
--- abstract: 'We show that symmetric and positive profiles of ground-state standing-wave of the non-linear Schrödinger equation are non-degenerate and unique up to a translation of the argument and multiplication by complex numbers in the unit sphere. The non-linear term is a combination of two or three pure-powers. The class of non-linearities satisfying the mentioned properties can be extended beyond two or three power combinations. Specifically, it is sufficient that an Euler differential inequality is satisfied and that a certain auxiliary function is such that the first local maximum is also an absolute maximum.' address: - - | Vladimir Georgiev\ Department of Mathematics, University of Pisa, Largo Bruno Pontecorvo 5 I - 56127 Pisa, Italy\ and\ Faculty of Science and Engineering, Waseda University, 3-4-1, Okubo, Shinjuku-ku, Tokyo 169-8555 Japan author: - Daniele Garrisi - Vladimir Georgiev date: '2017, July 26' title: 'Uniqueness of standing-waves for a non-linear Schrödinger equation with three pure-power combinations in dimension one' --- [^1] [^2] The role of the uniqueness and non-degeneracy in the stability ============================================================== A standing-wave is a function defined as $ \phi(t,x) := e^{i\omega t} u(x) $, where $ \omega $ is a real number, $ u $ is a complex-valued function in $ H^1 (\mathbb{R};\mathbb{C}) $ and $ \phi $ is a solution to the non-linear Schrödinger equation $$\label{eq.2017-07-27.10:05} i\partial_t \phi(t,x) + \partial_{xx}^2 \phi(t,x) - F'(\phi(t,x)) = 0,$$ The profile of a standing-wave is just $ R(x) := |u(x)| $. The literature is concerned with the existence and the stability of standing-waves whose profiles obey prescribed variational characterizations. The profiles we are interested in are minima of the energy functional $$E(u) := \frac{1}{2}{\int_{-\infty}^{+\infty}}|u'(x)|^2 dx + {\int_{-\infty}^{+\infty}}F(u(x)) dx$$ on the constrained defined as $ S(\lambda) := \{u\in H^1 (\mathbb{R})\mid \|u\|_{L^2}^2 = \lambda\} $ where $ \lambda > 0 $. As one can easily check, if $ u $ is a minimum of the energy functional, then $ v(x) := zu(x + y) $ belongs to the same constraint and has the same energy. Therefore, it is a new minimum, for every choice of $ z $ in $ S^1 $ (complex numbers in the unit sphere) and $ y $ in $ \mathbb{R} $. Then, $ u $ clearly a degenerate critical point of $ E $ on the constraint $ S(\lambda) $, as the transformations defined above show that $ u $ is the limit of a sequence of critical points. Therefore, both uniqueness and non-degeneracy need to be defined. We introduce the notation $$\mathcal{G}_\lambda := \{u\in S(\lambda)\mid E(u) = \inf_{S(\lambda)} E\}.$$ The set we defined is sometimes called *ground state*, as in [@BBBM10], even if the literature occasionally adopts this term to address more generally positive solutions to semi-linear elliptic equations, [@DdPG13]. We denote by $ H^1 _r (\mathbb{R}) $ the set of real-valued $ H^1 $ functions which are radially symmetric with respect to the origin. A pair $ (F,\lambda) $ satisfies the uniqueness property if given $ u $ and $ v $ in $ \mathcal{G}_\lambda $, there exists $ (z,y) $ in $ S^1\times\mathbb{R} $ such that $ u(x) = zv(x + y) $ for every $ x $ in $ \mathbb{R} $. It satisfies the non-degeneracy property if the function $ E_r $ obtained as a restriction of $ E $ on $ S(\lambda)\cap H^1 _r (\mathbb{R}) $ has non-degenerate minima. Uniqueness and non-degeneracy are not interesting features of the energy functional, but also play a role in the orbital stability of standing-wave solutions to . We say that is globally well-posed in $ H^1 (\mathbb{R};\mathbb{C}) $ if, given $ u_0 $ in $ H^1 (\mathbb{R};\mathbb{C}) $, there exists a solution $$\phi\colon [0,+\infty)\times \mathbb{R}\to\mathbb{C}$$ such that $ \phi(0,x) = u_0 (x) $ and the map $$U\colon [0,+\infty)\to H^1 (\mathbb{R};\mathbb{C})\to H^1 (\mathbb{R};\mathbb{C}),\quad U_t (u_0) := \phi(t,\cdot)$$ is of class $$C\sp 1 \big([0,+\infty);H^{-1}(\mathbb{R};\mathbb{C})\big)\cap C\big([0,+\infty);H\sp 1 (\mathbb{R};\mathbb{C})\big).$$ On the set $ H^1 (\mathbb{R};\mathbb{C}) $ we consider the metric induced by the scalar product $$(u,w)_{H^1 (\mathbb{R};\mathbb{C})} := \text{Re}\int_{\mathbb{R}} u(x)\overline{w}(x) dx + \text{Re}\int_{\mathbb{R}} u'(x)\cdot \overline{w'(x)} dx$$ and denote it by $ d $. A subset $ \mathcal{G} $ of $ H\sp 1 (\mathbb{R};\mathbb{C}) $ is said stable if for every $ \delta > 0 $ there exists $ \varepsilon > 0 $ such that $ d(u_0,\mathcal{G}) < \varepsilon\implies d(U_t (u_0),\mathcal{G}) < \delta $ for every $ t\geq 0 $. Given $ u $ in $ \mathcal{G}_\lambda $, we define $$\label{eq.2017-07-27.10:49} \mathcal{G}_\lambda (u) := \{zu(\cdot + y)\mid (z,y)\in S^1\times\mathbb{R}\}.$$ In general, if $ u $ is a minimum of $ E $, then $ \mathcal{G}_\lambda (u) $ is a subset of the ground state $ \mathcal{G}_\lambda $. The stability of these two sets is object of interest of the literature since the work of T. Cazenave and P. L. Lions, [@CL82], where pure-powers are considered. Results of stability of the ground-state have been extended to more general non-linearities, as in [@BBGM07; @Shi14]. We also mention other references which target the stability of the ground-state in other evolutionary equation, as multi-constraint non-linear Schrödinger systems, [@Iko14; @Bha15; @LNW16], coupled non-linear Schrödinger systems (NLS + NLS), [@Oht96; @NW11; @GJ16], coupled non-linear Schrödinger and Kortweg-de Vries equation (NLS + KdV), [@AA03], non-linear Klein-Gordon equation (NLKG), [@BBBM10], (NLKG + NLKG), [@Gar12]. In most cases, the stability of the ground-state is a consequence of the Concentration-Compactness Lemma, [@Lio84a; @Lio84b]. Coupled equations present some additional difficulties (rescalings do not work) but they can be worked around with *ad hoc* rescalings, as in [@AA03] or with inequalities obtained through symmetric rearrangements for more general non-linearities, as [@Gar12 Lemma 3.1] and [@Bye00 Proposition 1.4], or through the *coupled rearrangement* defined in [@Shi13 §2.2]. .5em The stability of $ \mathcal{G}_\lambda (u) $ is more challenging than the stability of $ \mathcal{G}_\lambda $: there might be solutions to with initial values close to $ \mathcal{G}_\lambda (u) $, but intermediate values far from it. Another application of the Concentration-Compactness Lemma and the stability of the ground state implies that these intermediate values are close to another set $ \mathcal{G}_\lambda (v) $ (as shown in <span style="font-variant:small-caps;">Figure</span> \[fig.2017-07-27.15:17\]). A simple way to rule out the existence of these trajectories is to prove that there is only one $ \mathcal{G}_\lambda (u) $, as $ u $ varies in $ \mathcal{G}_\lambda $. This is the approach followed in [@CL82] with the help of a uniqueness result, [@MS87], which specifically applies to pure-powers. Therefore, $ \mathcal{G}_\lambda = \mathcal{G}_\lambda (u) $, and the second set is stable because the first one is stable. Another way is to show that there are only finitely many of these sets $ \mathcal{G}_\lambda (u) $. In this case (see <span style="font-variant:small-caps;">Figure</span> \[fig.2017-07-27.15:17\]), trajectories bridging two different sets need to achieve a minimum amount of energy, which is too high if the initial value is too close to $ \mathcal{G}_\lambda (u) $, as it follows from [@GG17 §4]. Now, from the work of L. Jeanjean and J. Byeon, [@BJM09], in every set $ \mathcal{G}_\lambda (u) $ there exists a unique positive $ R $ in $ H^1 _r $. Therefore, the problem of the stability of the set reduces to showing that $ \mathcal{G}_{\lambda,r} := \mathcal{G}_\lambda\cap H^1 _r $ is finite, [@GG17 Proposition 5]. And this follows straightforwardly from the non-degeneracy of minima of $ E_r $. From [@GG17 Corollary 2], the uniqueness holds if $ \mathcal{G}_{\lambda,r} ^+ := \mathcal{G}_{\lambda}\cap H^1 _{r,+} $ is a singleton. (0,-1)(4,1.5) (0,0)[2]{} (4,0)[2]{} (0,0)(4,0) (0,0)(0,0.7) (-0.2,0.3) (0.2,-0.3) (4.2,-0.3) (0,0.9) (0,0)(0.5,0) (0.5,0)(3.5,0) (3.5,0)(4,0) Assumptions on $ F $ and non-degeneracy ======================================= The non-linearity $ F $ is a $ C^2 $ real valued function defined on $ \mathbb{C} $; $ F(s) = G(|s|) $ for every $ s $ in $ \mathbb{C} $. We list our assumptions trying to keep the notation consistent with [@GG17]: $$\begin{gathered} \label{G1} \tag{G1} \exists s_0 > 0\text{ such that } G(s_0) < 0\\ \tag{G2b} \label{G2b} -C|s|^{p^*}\leq G(s),\quad s\geq s_*,\quad 2 < p_* < 6\\ \tag{G4'} \label{G4} G(0) = G'(0),\quad |G''(s)|\leq C(|s|^{p - 2} + |s|^{q - 2}),\quad 2 < p < q\end{gathered}$$ are satisfied. makes sure that the energy functional $ E $ is $ C^2(H^1(\mathbb{R};\mathbb{C}),\mathbb{R}) $, which is a consequence of regularity theorems on Nemytski operators proved in [@AP93]; makes $ E $ a coercive functional on $ S(\lambda) $ and provides a-priori estimates, and thus the global well-posedness of . For we refer to [@BBGM07]; together with , it ensures that a minimum of $ E $ on $ S(\lambda) $ exists, if $ \lambda $ is large enough, see [@BBGM07 Theorem 2]. In the quoted reference, a stronger condition than is actually used, by setting $ s_* $ is to zero. However, their proof applies under the weaked assumption as well. We illustrate how the non-degeneracy of minima of $ E $ on $ S(\lambda) $ are obtained in [@GG17]. We set $ S_r (\lambda) := S(\lambda)\cap H^1 _r (\mathbb{R};\mathbb{R}) $. Suppose that $ R_0 $ is a minimum of $ E_r $ on $ S_r (\lambda) $. Then, there exists $ \omega_0 $ such that $$\label{eq.5} R_0 '' - G'(R_0) - \omega_0 R_0 = 0.$$ By looking at conserved quantities, as in [@GG17 Proposition 3], there holds $ \omega_0 > 0 $. In order to prove that $ R_0 $ is a non-degenerate minimum of $ E_r $, it is enough to consider a function $ v $ in $ H^1 _r $ such that $ (v,R_0)_2 = 0 $ and show that there exists $ C $ not depending on $ v $ such that $$\label{eq.2017-07-28.13:58} D^2 E(R_0)[v,v]\geq C\|v\|^2 _{H^1}.$$ In order to evaluate the Hessian, it only takes to define a smooth curve $ \alpha $ from $ (-\varepsilon,\varepsilon) $ to $ S_r (\lambda) $ such that $ \alpha(0) = R_0 $, $ \alpha'(0) = v $. Then $$(E\circ\alpha)''(0) = D^2 E(R_0)[v,v] = {\int_{-\infty}^{+\infty}}\Big(|v'(x)|^2 + (G''(R_0(x)) + \omega_0) v(x)^2\Big) dx =: \xi(v).$$ Since the functional above is homogeneous, we only need to to show that the infimum of $ \xi $ is positive when restricted to the unit sphere $ S_r (1) $ in $ L^2 $. Then will follow from the Banach-Steinhaus theorem. The functional $ \xi $ is certainly non-negative, because the fact that $ R_0 $ is a minimum is part of our assumptions. It is convenient to prove that $ \xi $ does actually achieve its infimum. We use as a reference the proof of [@Wei85 Proposition 2.9]. In fact, although that deals specifically with pure-powers, it can be applied to more general non-linearities, provided $ G''(0) = 0 $. Let $ v_0 $ be a minimum of $ \xi $. If $ \xi(v) = 0 $, then there exists $ \beta $ in $ \mathbb{R} $ such that $$L_+ (v) := -v'' + G''(R_0)v + \omega_0 v = \beta R_0.$$ From $ R_0 $ we construct a one-parameter family of solutions starting from $ R_0 $. We premise a few remarks. Firstly, $ R_0(0) $ is a solution to the equation $$\label{eq.2017-07-28.12:07} V(R_0 (0)) = \omega_0,\quad V(s) := -\frac{2G(s)}{s^2}$$ which is the auxiliary function mentioned in the abstract of this paper. Secondly, from [@BL83a Theorem 5], $ R_0 (0) $ is the least positive solution to . Moreover, $ R_0 $ is also an even function decreasing on $ [0,+\infty) $. Therefore, $ R_0 '' (0) < 0 $. Then $$R_0 (0) = \inf\{s > 0\mid V(s) = \omega_0\},\quad V'(R_0 (0)) > 0.$$ The second inequality is obtained by combining two equalities which in turn can be obtained by multiplying by $ R_0 $ and $ R_0 ' $, as in [@GG17 Proposition 4]. The construction of the one-parameter family is made as follows: the function $$R_* (\omega) := \inf\{s > 0\mid V(s) = \omega\}$$ is smooth in a neighborhood of $ \omega_0 $, because $ V'(R_0 (0)) > 0 $. We define the function $ R_\omega $ as solution to the initial value problem $$\label{eq.2017-07-27.13:21} R_\omega '' (x) - G'(R_\omega (x)) - \omega R_\omega(x) = 0 \quad R_\omega '(0) = 0,\quad R_\omega(0) = R_* (\omega).$$ We set $ S(\omega,x) = \frac{\partial R}{\partial \omega} (\omega,x) $ and define $ S(\omega_0,x) := S_0 (x) $. Therefore, taking the derivative with respect to $ \omega $ in , and evaluating at $ \omega = \omega_0 $, we obtain $$L_+ (S_0) = R_0.$$ Taking the $ L^2 $ scalar product with $ R_0 $, we obtain $$\frac{1}{2}\frac{d}{d\omega}\|R(\omega_0,\cdot)\|_2 ^2 = (L^+ (S_0),R_0)_2.$$ Up to a sign-change, the quantity appearing in the left term is the one denoted by $ Q(\omega) $ in the paper of N. G. Vakhitov and A. A. Kolokolov [@VK73]. Therefore, it is worth to investigate the behavior of the derivative of the function $ \lambda(\omega) := \|R(\omega,\cdot)\|_2 ^2 $ at the point $ \omega = \omega_0 $. The calculations made in [@GG17 §4] can be summarized as follows: there exists a positive function $ \Psi $ such that $$\label{eq.2017-07-31.14:47} \frac{d\lambda}{d\omega}(\omega_0) = -\frac{2R_* '(\omega_0)}{R_*(\omega_0)^5}\int_0\sp 1 \frac{\theta^2 (K(R_* (\omega_0)) - K(\theta R_* (\omega_0)))} {(\Psi(\theta,R_*(\omega_0),\omega_0))^{3/2}}d\theta\geq 0,$$ where $ K(s) = \frac{1}{s^2}(-6G(s) + sG'(s)) $. At this point, provided $ K $ is a strictly non-decreasing function, we have $ \lambda'(\omega_0) > 0 $. Since $$K'(s) = \frac{12sG(s) - 7s^2G'(s) + s^3 G''(s)}{s^4}$$ this computation suggests to require that $ 12G(s) - 7sG'(s) + s^2 G''(s) > 0 $ for every $ s $ in the interval $ (0,R_* (\omega_0)) $. In fact, there is no need to have a strict inequality here: since the integrand in is non-negative, if $ \lambda'(\omega_0) $ vanishes, then $ 12G(s) - 7sG'(s) + s^2 G''(s) = 0 $ on $ (0,R_* (\omega_0)) $, which means that on this interval $ G $ is a linear combination of $ s^2 $ and $ s^6 $. However, the coefficient of $ s^2 $ is zero, by , while the coefficient of $ s^6 $ is equal to zero because it is the pure-power critical case where minima of $ E $ over $ S(\lambda) $ do not exist, see [@GG17 Proof of Lemma 3.1]. Finally, since we wish to address all the minima, regardless of the constraint, the set where the requirement holds should apply to the images of all the minima. We define $$\Omega := \bigcup_{\lambda > 0}\bigcup_{R\in\mathcal{G}_\lambda} \text{Img}(R).$$ By [@GG17 Proposition 4], $ \Omega = (0,+\infty) $ if $ V $ is not bounded or $ V $ is bounded but $ \sup(V) $ is not achieved. Otherwise, $ \Omega = (0,R_* (\max(V)) $. Therefore, in [@GG17] we required $$\label{G3} \tag{G3} L(s) := 12G(s) - 7sG'(s) + s^2 G''(s)\geq 0 \text{ on } \Omega.$$ There are several non-linearities satisfying the condition above, starting from pure-powers $ G(s) = -as^p $ with $ a > 0 $ and $ 2 < p\leq 6 $. Another example is the combined pure-power $ G(s) = -as^p + bs^q $ with $ a,b > 0 $ and $ p < q $; clearly, in the latter case, $$L(s) = a(p - 2)(6 - p)s^p - b(q - 2)(6 - q) s^q$$ might changes sign. However, the function is non-negative on $ \Omega $ which is a bounded interval for this choice of $ G $. In fact, is satisfied, [@GG17 Corollary 2]. Uniqueness of standing-waves {#s.uniqueness} ============================ The idea of how we obtain the uniqueness of standing-waves is the following: if there are two minima $ R_0 $ and $ R_1 $ belonging to the same constraint $ S(\lambda) $, we consider the corresponding Lagrange multipliers $ -\omega_0 $ and $ -\omega_1 $. From , the function $ \lambda $ is injective on $ [\omega_0,\omega_1] $, which implies that $ \lambda $ is constant, because achieves the same values at the endpoints. Then $ L \equiv 0 $ on $ (0,R_*(\omega_1)) $ which implies that $ G $ is a linear combination of $ s^2 $ and $ s^6 $ and gives a contradiction with the sub-critical assumptions. The only thing we need to take care of is the definition of $ \lambda $, which is smooth as long as $ R_* $ is smooth. In turn $ R_* $ is smooth on $ \omega $ if $ V'(R_* (\omega))\neq 0 $. Therefore, critical points of $ V $ represent potential discontinuities of the function $ R_* $. However, $ R_* $ is continuous everywhere if, for instance, $ V $ does not have local maxima or the first local maximum is an absolute maximum. Therefore, we set $$A := \{s > 0\mid s\text{ is a local maximum of } V\}.$$ The assumption introduced in [@GG17] reads $$\label{G5} \tag{G5} A = \emptyset\text{ or } ( A\neq\emptyset,\text{ } V\text{ is bounded and } V(\inf(A)) = \sup(V) < +\infty). $$ To summarize, condition allows to state that the set $ \mathcal{G}_{\lambda,r}^+ $ is finite. If holds as well, then $ \mathcal{G}_{\lambda,r}^+ $ is a singleton, [@GG17 Theorem 1.4]. If the conditions , , , and hold, then $ \mathcal{G}_\lambda\cap H^1 _r $ consists of exactly two functions, $ R_+ $ and $ R_- $. The first is positive while $ R_- = -R_+ $. The assumption (G2a) in [@GG17] has been omitted here, as it can be replaced by . This explains the slight difference with the referenced theorem. We consider $$G(s) = {\varepsilon}_a a s^p + {\varepsilon}_b b s^q + {\varepsilon}_c c s^r,\ \{{\varepsilon}_a,{\varepsilon}_b,{\varepsilon}_c\}\subseteq\{-1,0,1\},\ a,b,c > 0,\ 2 < p < q < r$$ and discuss the assumptions mentioned above. In the remainder of the paper we will describe the behavior of the two properties for pure-powers, combined pure-power, and three pure-power combinations. Some cases have already been illustrated in [@GG17 §5], but we included them for the sake of completeness. We will leave out the cases $ \{{\varepsilon}_a,{\varepsilon}_b,{\varepsilon}_c\}\subseteq\{0,1\} $ as is not fulfilled. follows from the fact that all the exponents are bigger than two. When the coefficient of highest order term at infinity is positive is satisfied. Pure-powers {#ss.pp} ----------- If $ G(s) = -as^p $, then is satisfied because $ G < 0 $ and holds if $ p < 6 $. Then the function $ L(s) = a(p - 2)(6 - p)s^p $ is non-negative, while $ V = 2as^{p - 2} $ does not have local maxima, implying that is satisfied. Combined pure-powers {#ss.cp} -------------------- Firstly, we consider the case $ G(s) = -as^p + bs^q $ which clearly achieves negative values. The function $ V $ is bounded and has a single local maximum. Therefore, is satisfied and is satisfied if the (unique) zero of $ L $ occurs *before* the local maximum of $ V $, which is the unique zero of $ V' $. We will show that $ V'(s_0) = 0 $ implies $ L(s_0) > 0 $. In fact, $ V'(s_0) = 0 $ gives $$2a(p - 2)s_0 ^{p - 3} - 2b(q - 2)s_0 ^{q - 3} = 0.$$ If we multiply it by $ s_0 ^3 $, and substitute $ 2b(q - 2) s_0 ^q $ with $ 2a(p - 2)s_0 ^p $ in $ L $, we obtain $ L(s_0) = a(p - 2)(q - p)s_0^p > 0 $ which implies . In fact, no sub-critical assumption (which was required in [@GG17 §5]) is needed. If $ G = -as^p - bs^q $ then $ G < 0 $ which implies . For to hold, we need $ q < 6 $. Then $ L > 0 $ on $ (0,+\infty) $. is satisfied because $ V' > 0 $ on $ (0,+\infty) $, so $ A = \emptyset $. Finally, if $ G(s) = as^p - bs^q $, $ q < 6 $. Since $ V' $ goes to $ +\infty $, $ \Omega = (0,+\infty) $. However, $ L $ clearly changes sign in a neighborhood of the origin. Therefore is not satisfied. Three pure-power combinations ----------------------------- The cases with three negative coefficients are ruled out as in \[ss.cp\]. Then all the assumptions are fulfilled. can be easily checked except for the case $ ({\varepsilon}_a,{\varepsilon}_b,{\varepsilon}_c) = (1,-1,1) $ in §\[sss.pmp\]. The following remark will be useful in §\[sss.pmp\] and §\[sss.mpm\]: given the function $$k(s) := A - Bs^{q - p} + Cs^{r - q}$$ with $ p < q < r $ and $ A,B,C $ positive real numbers there holds $$\label{eq.2} \inf(k)\geq 0\iff A\geq B^{\frac{r - p}{r - q}} C^{\frac{p - q}{r - q}} d_*$$ where $$\label{eq.6} d_* := \left[\left(\frac{q - p}{r - p}\right)^{\frac{q - p}{r - p}} - \left(\frac{q - p}{r - p}\right)^{\frac{r - p}{r - q}}\right] > 0.$$ It is obtained by evaluating $ k $ on its unique minimum, obtained by solving explicitly $ k' = 0 $. ### $ G(s) = -as^p - bs^q + cs^r $ {#sss.mmp} Clearly holds, because the set $ A $ is a singleton. If $ p\geq 6 $, then $ L < 0 $ in a neighborhood of the origin. Therefore does not hold because $ \Omega $ contains small neighborhoods of the origin, as shown in Figure \[fig.mmp\]. For the case $ p < 6 $ it is convenient to divide $ V' $ and $ L $ by the leading coefficient, $ s^{q - p} $, and use the substitution $ t = s^{q - p} $. As in §\[ss.cp\], we need to know the behavior of $ L $ at the unique zero of $ V' $. We set $$\begin{aligned} g(t) &:= 1 + \frac{b(q - 2)(6 - q)}{a(p - 2)(6 - p)} t - \frac{c(r - 2)(6 - r)}{a(p - 2)(6 - p)}t^{\frac{r - p}{q - p}}\\ h(t) &:= 1 + \frac{b(q - 2)}{a(p - 2)} t - \frac{c(r - 2)}{a(p - 2)} t^{\frac{r - p}{q - p}}.\end{aligned}$$ Let $ t_0 $ be the unique zero of $ h $. From $ h(t_0) = 0 $ we obtain $$\frac{c(r - 2)}{a(p - 2)} t_0 ^{\frac{r - p}{q - p}} = \left(\frac{b(q - 2)}{a(p - 2)} t_0 + 1\right).$$ Then $$\begin{split} g(t_0) &= 1 + \frac{b(q - 2)(6 - q)}{a(p - 2)(6 - p)} t_0 - \frac{6 - r}{6 - p}\left( \frac{b(q - 2)}{a(p - 2)} t_0 + 1\right) \\ &= \frac{b(q - 2)(r - q)}{a(p - 2)(6 - p)} t_0 + 1 - \frac{6 - r}{6 - p} > 0. \end{split}$$ Then holds. The behavior of $ g $ and $ h $ is represented in Figure \[fig.mmp.2\]. (0,-2)(7.2,2.5) (-0.2,-2.1)(7.4,-2.1)(7.4,2.1)(-0.2,2.1)(-0.2,-2.1) (6.5,-0.3)[$ s_V $]{} (6.5,0) [0.01\*(-40\*x + 2\*x\^2 + x\^3)]{} [0.1\*(x\^2 - 0.1\*x\^3)]{} (0,0)(7.2,0) (0,-1.5)(0,1.8) (6.5,0)(6.5,1.5) (7.2,-0.2)[$ s $]{} (3,1)[$ V $]{} (2,-1)[$ L $]{} ### $ G(s) = -as^p + bs^q + cs^r $ {#sss.mpp} always holds as $ V $ has a single local maximum. If $ p = 6 $, then $ L > 0 $ everywhere. For $ p\neq 6 $ we can define the functions $ g $ and $ h $ in the same fashion as in §\[sss.mmp\] $$\begin{aligned} g(t) &:= 1 - \frac{b(q - 2)(6 - q)}{a(p - 2)(6 - p)} t - \frac{c(r - 2)(6 - r)}{a(p - 2)(6 - p)}t^{\frac{r - p}{q - p}}\\ h(t) &:= 1 - \frac{b(q - 2)}{a(p - 2)} t - \frac{c(r - 2)}{a(p - 2)} t^{\frac{r - p}{q - p}}.\end{aligned}$$ If $ p < 6 $ then $ g\geq h $ because each coefficient of $ g $ is larger than the corresponding coefficient of $ h $. Therefore, the first zero of $ L $ occurs after the first zero of $ V' $, and holds, Figure \[fig.mmp\]. If $ p > 6 $, then $ L $ is negative in a neighborhood of the origin, therefore does not hold. (0,-2)(7.2,2.5) (-0.2,-2.1)(7.4,-2.1)(7.4,2.1)(-0.2,2.1)(-0.2,-2.1) (2,-0.3)[$ s_V $]{} (1.95,0) [0.1\*(10 - 0.5\*x\^2)]{} [0.05\*(30 - 8\*x\^2)]{} (0,0)(7.2,0) (0,-1.5)(0,1.8) (7.2,-0.2)[$ t $]{} (1.9,1)[$ g $]{} (2.6,-0.8)[$ h $]{} (1,-1.7) ### $ G(s) = as^p - bs^q - cs^r $ {#sss.pmm} For to hold, $ r < 6 $ must be satisfied. $ \Omega = (0,+\infty) $, while $ \inf(L) < 0 $. Then holds, but does not. ### $ G(s) = as^p - bs^q + cs^r $ {#sss.pmp} For to hold we need $ \inf(G) < 0 $. If we set $ k := [as^p]^{-1} G $, the equivalence gives $$\label{eq.2017-08-26.10:59} a < b^{\frac{r - p}{r - q}} c^{\frac{p - q}{r - q}} d_*.$$ If $ p \leq 6 $, then does not hold, because $ L $ is negative in a neighborhood of the origin, as in Figure \[fig.mmp\]. Before looking at the case $ p > 6 $ it is useful to observe that from we have $ \sup(V') > 0 $. On the contrary, $ \inf(-V')\geq 0 $. We apply to $ k := [-a(p - 2)s^{p - 3}]^{-1} V' $ and obtain $$\label{eq.2017-08-26.10:58} a(p - 2)\geq [b(q - 2)]^{\frac{r - p}{r - q}} [c(r - 2)]^{\frac{p - q}{r - q}} d_*.$$ Dividing term-wise by , we obtain $$\label{eq.2017-08-26.11:04} p - 2 > (q - 2)^{\frac{r - p}{r - q}} (r - 2)^{\frac{p - q}{r - q}}.$$ By exponentiating both terms to $ r - q $, dividing by $ (p - 2)^{r - q} $ and applying the variable changes $ x = p - 2 $, $ y = q - 2 $ and $ z = r - 2 $, reads $ M(x,y,z) > 1 $ which contradicts Lemma \[lem.1\]. Then $ \sup(V') > 0 $ and holds too. When $ p > 6 $ we need to compare $ L $ and $ V' $. Since $ \sup(V') > 0 $, it has two distinct zeroes. We will show that $ L $ is negative in the first zero of $ V' $, as in Figure \[fig.pmp\]. We set $$\begin{aligned} g(t) &:= 1 - \frac{b(q - 2)(6 - q)}{a(p - 2)(6 - p)} t + \frac{c(r - 2)(6 - r)}{a(p - 2)(6 - p)}t^{\frac{r - p}{q - p}}\\ h(t) &:= 1 - \frac{b(q - 2)}{a(p - 2)} t + \frac{c(r - 2)}{a(p - 2)} t^{\frac{r - p}{q - p}}.\end{aligned}$$ We call $ t_1 $ the first zero of $ h $. Since $ hV' < 0 $ on $ (0,+\infty) $, $ h(t_1) = 0 $ and $ h'(t_1) < 0 $. From $ h(t_1) = 0 $, we obtain $$\label{eq.2017-08-26.09:48} \frac{c(r - 2)}{a(p - 2)} t_1 ^{\frac{r - p}{q - p}} = \frac{b(q - 2)}{a(p - 2)} t_1 - 1$$ which we can substitute into the inequality $ t_1 h'(t_1) < 0 $ and obtain $$- \frac{b(q - 2)}{a(p - 2)} t_1 + \frac{r - p}{q - p} \left(\frac{b(q - 2)}{a(p - 2)} t_1 - 1\right) < 0$$ which gives $$\label{eq.2017-08-25.17:49} t_1 < \frac{a(p - 2)(r - p)}{b(q - 2)(r - q)}.$$ Therefore, from and $$\begin{split} g(t_1)&= 1 - \frac{b(q - 2)(6 - q)}{a(p - 2)(6 - p)} t_1 + \frac{6 - r}{6 - p} \left(\frac{b(q - 2)}{a(p - 2)} t_1 - 1\right)\\ &= \frac{b(q - 2)(r - q)}{a(p - 2)(p - 6)} t_1 - \frac{r - p}{p - 6}\\ &< \frac{b(q - 2)(r - q)}{a(p - 2)(p - 6)}\cdot \frac{a(p - 2)(r - p)}{b(q - 2)(r - q)} - \frac{r - p}{p - 6} = 0. \end{split}$$ Then, regardless of the values of the exponents, is never met, while holds. (0,-2)(7.2,2.5) (-0.2,-2.1)(7.4,-2.1)(7.4,2.1)(-0.2,2.1)(-0.2,-2.1) (2.51,0.3)[$ t_1 $]{} (2.51,0) (5.4,-0.3)[$ s_V $]{} (5.35,0) [0.01\*(20 - 10\*x\^2 + 1.8\*x\^3)]{} [0.05\*(30 - 8\*x\^2 + 1.3\*x\^3)]{} (0,0)(7.2,0) (0,-1.5)(0,1.8) (7.2,-0.2)[$ t $]{} (1.8,1)[$ h $]{} (2,-0.3)[$ g $]{} (1,-1.7) ### $ G(s) = as^p + bs^q - cs^r $ {#sss.ppm} implies $ r < 6 $. Therefore, $ \inf(L) < 0 $ on $ (0,{\varepsilon}) $ for $ {\varepsilon}$ suitably small and does not hold, as in Figure \[fig.mmp\]. ### $ G(s) = -as^p + bs^q - cs^r $ {#sss.mpm} The conclusions we reached so far depend only on the exponents and not on the coefficients, as long as their signs are prescribed. This case is an exception. Firstly, forces $ r < 6 $. Since $ V $ is not bounded, $ \Omega = (0,+\infty) $. If we apply with $ k := [a(p - 2)(6 - p)s^p]^{-1} L $, we need $$\label{eq.1} a(p - 2)(6 - p)\geq b^{\frac{r - p}{r - q}} (q - 2)^{\frac{r - p}{r - q}} (6 - q)^{\frac{r - p}{r - q}} \cdot c^{\frac{p - q}{r - q}} (r - 2)^{\frac{p - q}{r - q}} (6 - r)^{\frac{p - q}{r - q}} d_*$$ where $ d_* $ has been defined in . can be true or false depending on whether $ a $ is large or small, respectively. Therefore, in this section we just show that whenever is satisfied, is satisfied too. In fact, if does not hold, then $ \inf(V') < 0 $. By applying with $ k := [a(p - 2)s^{p - 3}]V' $, we obtain $$\label{eq.3} a(p - 2) < b^{\frac{r - p}{r - q}} (q - 2)^{\frac{r - p}{r - q}} c^{\frac{p - q}{r - q}} (r - 2)^{\frac{p - q}{r - q}} d_*.$$ We divide term-wise by and obtain $$\label{eq.4} (6 - p) > (6 - q)^{\frac{r - p}{r - q}} (6 - r)^{\frac{p - q}{r - q}}.$$ By exponentiating both terms to $ r - q $, dividing by $ (6 - p)^{r - q} $ and applying the variable changes $ x = 6 - p $, $ y = 6 - q $ and $ z = 6 - r $, reads $ M(x,y,z) > 1 $ which contradicts Lemma \[lem.1\]. .5em We give a proof of the lemma we referred to in §\[sss.pmp\] and §\[sss.mpm\]. \[lem.1\] Let $ M $ and $ D $ be the function and domain defined as $$M(x,y,z) = y^{z - x} z^{x - y} x^{y - z},\quad D := \{0 < z \leq y \leq x\}.$$ Then $ \sup_D (M) = 1 $ and $ M < 1 $ in the interior of $ D $. Moreover, for every $ (x,y,z) $ in $ D $, $ M(x,y,z) = 1 $ if and only if $ x = y $ or $ y = z $. We have $$\begin{split} M(x,y,z) = \frac{z^{x - y} x^{y - z}}{y^{x - z}} = \left(\frac{z}{y}\right)^{x - y} \left(\frac{x}{y}\right)^{y - z} = \left[\left(\frac{z}{y}\right)^{\frac{x}{y} - 1} \left(\frac{x}{y}\right)^{1 - \frac{z}{y}}\right]^y. \end{split}$$ In order to show that $ M < 1 $ it is enough to prove that $ M^{1/y} < 1 $. We substitute $ \frac{x}{y} $ with $ a $ and $ \frac{z}{x} $ with $ b $. Then $$M^{1/y} = a^{a(1 - b)} b^{a - 1}.$$ We define $$H(a,b) = \ln(M^{1/y}) = a(1 - b)\ln(a) + (a - 1)\ln(b).$$ We fix $ 0 < b < 1 $ and consider the function $ H(a,b) $ on the interval $ b\leq a\leq \frac{1}{b} $. Clearly, $ H(1,b) = H(\frac{1}{b},b) = 0 $. Moreover, $$\frac{\partial H}{\partial a} (a,b) = (1 - b)\ln(a) + (1 - b) + \ln(b).$$ We have $ \frac{\partial H}{\partial a} (1,b) = (1 - b) + \ln(b) $ which is negative on the interval $ (0,1) $ and has $ \partial_a H $ is monotonically increasing on the interval $ [1,\frac{1}{b}] $. Therefore, $ \partial_a H $ has at most one zero on this interval. In conclusion, $ H(\cdot,b) < 0 $ on the interval $ (1,\frac{1}{b}) $ which implies $ M < 1 $ in the interior of $ D $, while the fact that $ H(\cdot,b) = 0 $ on the boundary gives the second part of the statement. [|\*[10]{}[c|]{}]{} $ {\varepsilon}_a $ & $ {\varepsilon}_b $ & $ {\varepsilon}_c $ & $ 6 - p $ & $ 6 - q $ & $ 6 - r $ & $ \#A $ & $ \Omega $ & Assumptions & Section\ [$ - $]{}& 0 & 0 & [$ + $]{}& & & 0 & $ (0,+\infty) $ & (G3) $ \wedge $ (G5) & \[ss.pp\]\ [$ - $]{}& [$ + $]{}& 0 & & & & 1 & bounded & (G3) $ \wedge $ (G5) & \[ss.cp\]\ [$ + $]{}& [$ - $]{}& 0 & [$ + $]{}& [$ + $]{}& & 0 & $ (0,+\infty) $ & $ \neg $(G3) $ \wedge $ (G5) & \[ss.cp\]\ [$ - $]{}& [$ - $]{}& 0 & [$ + $]{}& [$ + $]{}& & 0 & $ (0,+\infty) $ & (G3) $ \wedge $ (G5) & \[ss.cp\]\ [$ + $]{}& [$ + $]{}& [$ - $]{}& [$ + $]{}& [$ + $]{}& [$ + $]{}& 0 & $ (0,+\infty) $ & $ \neg $(G3) $ \wedge $ (G5) & \[sss.ppm\]\ [$ + $]{}& [$ - $]{}& [$ + $]{}& & & & 0 & $ (0,+\infty) $ & $ \neg $(G3) $ \wedge $ (G5) & \[sss.pmp\]\ [$ + $]{}& [$ - $]{}& [$ - $]{}& [$ + $]{}& [$ + $]{}& [$ + $]{}& 0 & $ (0,+\infty) $ & $ \neg $(G3) $ \wedge $ (G5) & \[sss.pmm\]\ [$ - $]{}& [$ + $]{}& [$ + $]{}& [$ + $]{}& & & 1 & bounded & (G3) $ \wedge $ (G5) & \[sss.mpp\]\ [$ - $]{}& [$ + $]{}& [$ + $]{}& 0 & [$ - $]{}& [$ - $]{}& 1 & bounded & (G3) $ \wedge $ (G5) & \[sss.mpp\]\ [$ - $]{}& [$ + $]{}& [$ + $]{}& [$ - $]{}& [$ - $]{}& [$ - $]{}& 1 & bounded & $ \neg $(G3) $ \wedge $ (G5) & \[sss.mpp\]\ [$ - $]{}& [$ + $]{}& [$ - $]{}& [$ + $]{}& [$ + $]{}& [$ + $]{}& 0 & $ (0,+\infty) $ & (G3) $ \Rightarrow $ (G5) & \[sss.mpm\]\ [$ - $]{}& [$ - $]{}& [$ + $]{}& [$ - $]{}& [$ - $]{}& [$ - $]{}& 1 & bounded & $ \neg $(G3) $ \wedge $ (G5) & \[sss.mmp\]\ [$ - $]{}& [$ - $]{}& [$ + $]{}& 0 & [$ - $]{}& [$ - $]{}& 1 & bounded & $ \neg $(G3) $ \wedge $ (G5) & \[sss.mmp\]\ [$ - $]{}& [$ - $]{}& [$ + $]{}& [$ + $]{}& & & 1 & bounded & (G3) $ \wedge $ (G5) & \[sss.mmp\]\ For every $ \lambda > 0 $ if the set $ \mathcal{G}_\lambda\cap H^1 _{r,+} $ is non-empty then it is a singleton, provided $ G $ (i) is a pure-power with $ {\varepsilon}_a < 0 $ (ii) is a combined pure-power with $ {\varepsilon}_a < 0 $, or $ {\varepsilon}_a > 0 $ in sub-critical regime (iii) is a three pure-power combination with $ ({\varepsilon}_a,{\varepsilon}_b,{\varepsilon}_c) = (-1,1,1) $ and $ p\leq 6 $ or (iv) $ ({\varepsilon}_a,{\varepsilon}_b,{\varepsilon}_c) = (-1,-1,1) $ with $ p < 6 $ or (v) $ ({\varepsilon}_a,{\varepsilon}_b,{\varepsilon}_c) = (-1,1,-1) $ provided $ r < 6 $ and inequality holds. 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[^2]: The second author was supported in part by INDAM, GNAMPA - Gruppo Nazionale per l’Analisi Matematica, la “Probabilità e le loro Applicazioni” and by Institute of Mathematics and Informatics, Bulgarian Academy of Sciences and Top Global University Project, Waseda University.
--- abstract: 'We recently showed (Class. Quantum Grav. 23, 3353; gr-qc/0602091) that aligned Petrov type D purely magnetic perfect fluids are necessarily locally rotationally symmetric (LRS) and hence are all explicitly known. We provide here a more transparent proof.' --- [**Note on aligned Petrov type D purely magnetic perfect fluids**]{} Lode Wylleman[^1] and Norbert Van den Bergh[^2] [Faculty of Applied Sciences TW16, Gent University, Galglaan 2, 9000 Gent, Belgium]{} Aligned purely magnetic perfect fluids are defined to be non-conformally flat solutions of Einstein’s field equations $$G_{ab}\equiv R_{ab}-\frac{1}{2}R g_{ab}= (w+p) u_a u_b -p g_{ab},$$ for which the electric part of the Weyl tensor w.r.t. to the unique normalized timelike eigenvector field $u^a$ of the Ricci tensor vanishes: $$\label{E_ab} E_{ac}\equiv C_{abcd} u^b u^d=0.$$ All locally rotationally symmetric solutions of this kind were recently found [@Lozanovski2; @Lozanovski3]. They are of class III or I in the classification by Stewart and Ellis [@StewartEllis], wherefor the metrics can then be explicitly constructed or determined up to a single third-order differential equation, respectively. It was shown in [@VdBWyll], within the Newman-Penrose formalism, that these LRS classes actually exhaust Petrov type D. To come to the result, use was made of a further tetrad fixation, which is unnecessary and hides somewhat the exact freedom which is left in the NP and Bianchi-equations at a certain level of the reasoning. The streamlined proof below is designed to bring more clarity, and also demonstrates in a straightforward way that the LRS classes are I or III. We will still make use of the Newman-Penrose formalism and follow the notation of [@Kramer], whereby the Newman-Penrose equations (7.21a – 7.21r) and Bianchi identities (7.32i – 7.32h) will be indicated as ($np1$ – $np18$) and ($b9$ – $b11$). Regarding (7.32a – 7.32h), appropriate combinations with (7.32i – 7.32h) are made to remove derivatives of the Ricci scalar, and the resulting equations are ($b1$ – $b11$). In the philosophy of *not* further fixing ${\cal B}$, the GHP ‘edth’ operator $\dh$ will also come into play. We use a canonical type D tetrad ${\cal B}$, with $$\label{e:psis} \Psi_0=\Psi_1=\Psi_3=\Psi_4=0 ,$$ with the condition (\[E\_ab\]) being expressed as $$\label{e:psi2} \overline{\Psi_2}=-\Psi_2\neq 0$$ Choosing a boost in the $(k,\ell)$ plane such that the fluid velocity $u=(k+\ell)/ \sqrt{2}$ and introducing $S=w+p$ as a new variable, one has $$\label{e:phis} \Phi_{00}=\Phi_{22}=2 \Phi_{11}=\frac{S}{4}\ \textrm{and}\ R=4 w-3 S.$$ [**Theorem.**]{} A PMpf of Petrov type D is LRS class I or III. More precisely, the spin coefficients of a canonical Weyl null-tetrad obey $\l=\s=\b+\ac=\k=\tau=\pi=\nu=0$, with moreover $$\label{extra} \mu=q\,\rho,\quad g=q\,e, \Psi_2=\frac{q}{2}(\rc-\r)(2e+\r+\rc),$$ where $g\equiv \g+\gc$ and $e\equiv \e+\ec$, and $q$ is 1 for class I and -1 for class III.\ <span style="font-variant:small-caps;">Proof.</span> From $(bi5)$ and $(bi6)$ one gets $$\begin{aligned} (D-\D)S &=& -(g+e)S-9(\r+\mu-(\rc+\mc))\Psi_2,\label{d:DminDeltaS}\\ (D-\D)\Psi_2 &=& \frac{3}{2}(\r+\mu+\rc+\mc)\Psi_2-\frac{1}{4}(\r+\mu-(\rc+\mc))S, \label{d:DminDeltaPsi2}\\ (D+\D)\Psi_2 &=& -\frac{3}{2}(\mu-\rc+(\mc-\r))\Psi_2, \label{d:DplusDeltaPsi}\end{aligned}$$ together with an algebraic relation $$\label{e:Rextra} 18(\rho-\rc+\mc-\mu)\Psi_2- (\rho+\rc-\mu-\mc+2(e-g))S=0.$$ Next, $\l=\s=0$ immediately follows from $(bi2)$ and $(bi3)$. From $(bi1)+\bar{(bi4)}+\bar{(bi7)}+(bi8)$ one gets $$\kappa+3(\overline{\pi}+\tau) +\overline{\nu} = 0. \label{e:lsnu}$$ The first key observation is that herewith, the combination $(np2)+3\overline{(np4)} +3 (np10) + \overline{(np14)}$ is algebraic and factorises as follows: $$\label{e:main} (\b+\ac)(2\k+3 \tau+3 \pc)=0.$$ If $\b+\ac$ were not identically zero, then from (\[e:lsnu\]) and (\[e:main\]) one would obtain $\kappa = \nc = -\frac{3}{2} (\tau+\pc)$ after which $(bi1)-\bar{(bi4)}$ and $(np2)-\bar{(np10)}$ respectively yield $$\b+\ac = \tau+\pc,\quad (\b+\ac)(\tau+\pc)=0,$$ such that one would come to the contradiction $\b+\ac=0$. With $\b+\ac=0$, we respectively get from $(np11),\bar{(np13)},\bar{(np4)}+(np5)$ and $(np15)+\bar{(np18)}$ that $$\begin{aligned} \d\r&=&\t(\r-\rc), \label{d:rho} \\ \d\mc&=&\pc(\mu-\mc)-(\k+3(\t+\pc))(\r-\rc), \\ \d e&=&-\pc(\rc+\e)+\k(\mu+g), \\ \d g&=&\t(\mu+g)+(\k+3(\t+\pc))(\rc+e) \label{d:g},\end{aligned}$$ whereas from $(bi1)$, $(bi8)-\bar{(bi7)}$ and $(np2),\bar{(np7)},(np16)$ we get an autonomous dynamical system in $\Psi[2],S,\k,\tau,\pc$ with $\dh$ as differential operator, for which (a) $\dh\Psi_2$ and $\dh S$ are linear and homogeneous in $\Psi_2$ and $S$, and (b) $\k,\tau,\pc$ form a subsystem: $$\begin{aligned} \dh\Psi_2 &=& \frac{3}{2}(\tau-\pc)\Psi_2 -\frac{1}{2}(\tau+\pc)S=\d\Psi_2,\label{d:edthPsi2}\\ \dh S &=& -12\k\Psi_2+(\k-\pc)S=\d S,\label{d:edthS}\\ \dh\k &=& (\tau-\pc)\k, \label{d:edthkc}\\ \dh\pc &=& -(3\k+\pc)\pc-(\k+3\tau)\k, \label{d:edthpi}\\ \dh\tau &=& (3\k+\tau)\tau+(\k+3\pc)\k.\label{d:edthtc}\end{aligned}$$ However, from $\bar{(bi4)}-(bi1)$ we get that these variables are constrained by $$\label{e:edthconstraint1} 3(2\k+3\tau+3\pc)\Psi_2+(\tau+\pc)S=0.$$ Hence, on applying $\dh$ to (\[e:edthconstraint1\]), the homogeneous and linear character in $\Psi_2$ and $S$ is preserved, and as $(\Psi_2,S)\neq (0,0)$ we must have $$\label{e:edthconstraint2} 2\k^2+9\tau^2+6\pc^2+7\k\tau+5\k\pc+15\tau\pc=0.$$ Applying $\dh$ to \[e:edthconstraint2\] one further finds $$\label{e:edthconstraint3} 2\k^3+18\tau^3-12\pc^3+13\tau\k^2+23\k\tau^2+5\pc\k^2-\k\pc^2 +15\pc\tau^2-15\tau\pc^2+16\tau\k\pc=0.$$ Eliminating $\pc$, resp. $\t$, from (\[e:edthconstraint2\]) and (\[e:edthconstraint3\]) by calculating resultants one obtains $$\label{e:edthresultant} (9\k^2-2\k\tau+9\tau^2)\k^2(\k+\tau)^2=0, \quad (3\k^2+4\k\pc+3\pc^2)\k^2(\k-\pc)^2=0.$$ Suppose $\k\neq 0$. By substituting into (\[e:edthconstraint2\]) and (\[e:edthconstraint3\]) the relations $\k=-\tau$ and $\k=\pc$ one finds that these are equivalent, hence $\tau+\pc=0$, in contradiction with (\[e:edthconstraint1\]) and $\Psi_2\neq 0\neq \k$. Hence possible ratio’s $\t/\k$ and $\pc/\k$ are roots of the first factors in (\[e:edthconstraint3\]) divided by $\k^2$. Checking (\[e:edthconstraint2\]) for the four different possibilities, one finds that only two couples are allowed: $$\label{e:soltaupc} \left(\tau,\pc\right)=\left(\frac{1}{9}(1+z 4i\sqrt{5})\k, -\frac{1}{3}(2+z i\sqrt{5})\k\right),\quad z=\pm 1.$$ Inserting this in (\[e:edthconstraint1\]) and dividing by $\k$ we correspondingly find $S$ and $\Psi_2$ to be constantly proportional: $$\label{e:solS} S=z \frac{9i}{\sqrt{5}}\Psi_2.$$ We immediately get two additional (real) algebraic equations in $e,g,\r,\rc,\mu,\mc$ from (\[e:solS\]), by substituting it into (\[e:Rextra\]) on one hand, and applying $D-\D$ to it and use (\[d:DminDeltaS\])-(\[d:DminDeltaPsi2\]). With $\Psi_2\neq 0$, this respectively yields $$\begin{aligned} z\sqrt{5}(\mu+\mc-(\r+\rc)+2(g-e))+10i(\mu-\mc-(\r-\rc)) &=& 0, \label{e:eq1} \\ z2\sqrt{5}(3(\mu+\mc+\r+\rc)+2(g+e))-11i(\mu-\mc+\r-\rc) &=& 0. \label{e:eq2}\end{aligned}$$ To come to a quick contradiction we would like to have four more real (or two more complex) homogeneous and linear equations in $e,g,\r,\rc,\mu,\mc$. The amazing fact is that *just* two such complex equations (and not more!) become easily available, as follows. Combining (\[e:soltaupc\]) and (\[e:lsnu\]), one finds that the ratio’s $\nc/tau$ and $\pc/\k$ are equal $$\label{e:equalratios} \frac{\nc}{\tau}=\frac{\pc}{\k}=-\frac{1}{3}(2+z i\sqrt{5}):=a(z).$$ The second key observation is that herewith, $\bar{(np9)}-a(z)(np3)$ becomes algebraic and of the wanted form: $$\label{e:eq3} z\sqrt{5}(\r-\mc+6(g-e))-i(5(\r+\mc)+12(g+e))=0.$$ By means of (\[d:rho\])-(\[d:g\]) we can get another algebraic equation on applying $\d$ on (\[e:eq3\]), but this turns out to give an identity when inserting (\[e:soltaupc\]). However, we can build one other homogeneous and linear complex equation in $\r,\mc,e,g$ by eliminating $\rc$ and $\mu$ from (\[e:eq1\]),(\[e:eq2\]) and the complex conjugate of (\[e:equalratiosnp\]). This yields $$\label{e:kraak} z\sqrt{5}(25(\r-\mc)+31(g-e))+2i(65(\r+\mc)+63(g+e))=0.$$ Applying $\d$ on this, inserting (\[e:soltaupc\]) and dividing by $\k$ now gives $$\begin{aligned} \label{e:eq5} && z(255\rho+873\rc-13\mu+765\mc+752g+1128e)\nonumber\\ &&-i\sqrt{5}(357(\r-\rc)+529\mu-153\mc+376g)=0.\end{aligned}$$ Thus $\{(\ref{e:eq1}),(\ref{e:eq2}),(\ref{e:eq3}),\bar{(\ref{e:eq3})}, (\ref{e:eq5}),\bar{(\ref{e:eq5})}\}$ forms a homogeneous and linear system in $\r,\rc,\mu,\mc,e,g$, the determinant of which is computed to be non-zero (and independent of $z$). Thus the only solution is the zero-solution. However, inserting $\r=\mu=0$ in the imaginary part of $(np12)$ then yields $\Psi_2=0$: contradiction. Hence $\k=0$. Substituting this into (\[e:edthconstraint2\]) and (\[e:edthconstraint3\]) one finds $$\label{plop} (3\t+2\pc)(\tau+\pc)=(-6\tau^2+\pc\t+4\pc^2)(\t+\pc)=0$$ such that $\t+\pc=0$, and $\nu=0$ by (\[e:lsnu\]). With the so far obtained specifications, $(bi10)$ reduces to $\d w=\dc w=0$, while from $(bi9)$ and $(bi11)$ we respectively get: $$\begin{aligned} Dw &=& \frac{9}{2}(\r+\mu-(\rc+\mc))\Psi_2+\frac{1}{2}(g-e+\r+\rc-\mu-\mc)S, \\ \D w &=& -\frac{9}{2}(\r+\mu-(\rc+\mc))\Psi_2+\frac{1}{2}(g-e+\r+\rc-\mu-\mc)S.\end{aligned}$$ Herewith, the $[\d,\dc]$ commutator applied on $w$ is algebraic, and elimination of $\Psi_2$ from it by combination with (\[e:Rextra\]) yields a factorisation $$\label{e:LRSIorIII} S(\mu-\rc-(\mc-\r))(\mu-\rc+(\mc-\r))=0.$$ Hence $\mu-\rc$ is either real or imaginary, where the real case corresponds to vanishing vorticity of the $u^a$-congruence, see the appendix. Application of the $[\d,D+\D]$ commutator to $w$ results for both cases in $$\label{e:taufactor} \tau(\mu-\rc+g-e)=0.$$ In the case where $\mu-\rc$ is real, the imaginary part of $(np12)$ precisely gives $$\label{e:Psi2} \Psi_2=\frac{1}{2}(\r-\rc)(\mu-\rc+g-e)$$ such that the second factor of \[e:taufactor\] cannot vanish. It can vanish neither when $\mu-\rc$ is imaginary, as then $\mu-\rc$ would be zero as $g-e$ is real. Therefore $\tau,\k,\nu,\pi,\l$ and $\s$ are all zero, and hence the solutions are LRS according to corollary 1 in [@GoodeWainwright]. Further, the Ricci equations $(np1),(np8),(np17)$ and $(np14)$ respectively yield $$\begin{aligned} D\r &=& \r(\r+e)+\frac{S}{4} \\ D\mu &=& \mu(\rc-e)+\Psi_2+\frac{w}{3}-\frac{S}{4}\\ \D\r &=& -\r(\mc-g) -\Psi_2-\frac{w}{3}+\frac{S}{4}\\ \D\mu &=& -\mu(\mu+g)-\frac{S}{4}.\end{aligned}$$ The non-rotating case ($\mu-\rc$ real) was fully treated in [@Lozanovski2]. Application of the commutator $[\d,\dc]$ to $\Psi_2$ and the evolution operator $D+\D$ to $\mu+\r-\mc-\rc=0$ respectively yield $$\label{e:plok} \Psi_2(\r-\rc)(\r+\mu)=0,\quad (\r-\rc)(e+g)=0,$$ such that $\r+\mu=g+e=0$ by (\[e:Psi2\]), where $g+e=0$ is equivalent to the perfect fluid $u^a$-congruence being geodesic, see the appendix. As $\rho=\mu=0$ is not allowed by (\[e:Psi2\]), one has $\rho/\mu=-1 <0$, and the corresponding space-times are LRS class III, see [@GoodeWainwright]. For imaginary $\mu-\rc\neq 0$, application of $D-\D$ to $\r+\rc-\mu-\mc=0$ yields $$\label{e:jiha1} (\mu+\mc)(e-g)+(\mu-\r)(\mu-\rc)=0,$$ whereas from the application of the commutator $[\d,\dc]$ to $\Psi_2$ one now obtains $$\label{e:aha1} (\mu-\rho)[(\r-\mc)S+6(\mu+\mc)\Psi_2]=0.$$ If the second factor in (\[e:aha1\]) vanishes, then eliminating $S$ from it in combination with (\[e:Rextra\]) yields $$\label{e:jiha2} (\mu+\mc)(e-g)+3(\mu-\r)(\mu-\rc)=0.$$ Hence, consistency with (\[e:jiha1\]) requires $\r=\mu$ anyway. Substitution of this into (\[e:aha1\]) finally gives $g=e$, and the $u^a$-congruence is shear-free and non-expanding, see the appendix. The imaginary part of $(np12)$ gives the expression (\[extra\]) for $\Psi_2$ with $q=1$. Again, $\rho=\mu=0$ is not allowed, hence $\rho/\mu=1>0$ and the corresponding space-times are LRS class I [@GoodeWainwright]. [$\Box$]{}\ [**Remark.**]{} All LRS class I, resp. class III, solutions are stationary, resp. orthogonally spatially homogeneous, and hence exhibit an equation of state, which can be most explicitly be determined for LRS class III PMpf’s, see [@Lozanovski2]. LRS class II only contains purely electric solutions. A beautiful overview of LRS space-times in a 1+3 covariant approach, summarizing many features of the different classes and discussing their practical meaning, can be found in [@vanElst1]. Appendix {#appendix .unnumbered} ======== Choosing an orthonormal tetrad such that $\delta\equiv(e_1 - i e_2)/\sqrt{2}$, $D \equiv(e_3+e_4)/\sqrt{2}$ and $\Delta \equiv (e_4-e_3)/\sqrt{2}$ ($e_4=u$ being the fluid velocity), the components of the fluid kinematical quantities are given by the following expressions: (expansion tensor) $$\begin{aligned} \theta_{12} = 0 \\ \theta_{13}+i \theta_{23} = (\alpha+\bc+2\pi+2\tc)/\sqrt{2} \\ \theta_{11} = \theta_{22} = (\mu+\mc-\rho-\rc)/(2\sqrt{2})\\ \theta_{33}= (\epsilon+\ec-\gamma-\gc)/\sqrt{2}\end{aligned}$$ (acceleration vector) $$\begin{aligned} \dot{u}_1+ i \dot{u}_2 = -\sqrt{2}(\pi+\kc+2\tc)\\ \dot{u}_3 = (\epsilon+\ec+\gamma+\gc)/\sqrt{2}\end{aligned}$$ (vorticity vector) $$\begin{aligned} \omega_1+i \omega_2 = \frac{i}{2} (\alpha+\bc - 2 \tc -2 \pi)/\sqrt{2}\\ \omega_3 = \frac{i}{2}(\rho-\rc+\mu-\mc)/\sqrt{2}\end{aligned}$$ [99]{} Lozanovski C Lozanovski C and Carminati J J.M. Stewart and G.F.R. Ellis Van den Bergh N and Wylleman L Stephani H, Kramer D, MacCallum M A H, Hoenselaers C and Herlt E 2003, *Exact Solutions to Einstein’s Field Equations (Second Edition)*, Cambridge: University Press Goode S W and Wainwright J van Elst H and Ellis G F R [^1]: Research assistant supported by the Fund for Scientific Research Flanders(F.W.O.), e-mail: lwyllema@cage.ugent.be [^2]: e-mail: norbert.vandenbergh@ugent.be
--- abstract: 'A general framework of quantum state amplification using the language of quantum state transformation is given systematically for the first time. The concept of amplification of quantum states is defined specifically and the amplification of a set of quantum states is formulated generally as the transformation of quantum states. Three different kinds of important quantum amplifications, i.e., deterministic noisy quantum amplification, probabilistic noiseless quantum amplification, and deterministic noiseless quantum amplification are identified and discussed. For deterministic quantum amplification, the linearity of amplification is proven to be incompatible with the noiseless amplification while it is not true for probabilistic quantum amplification. However, deterministic noiseless quantum amplification is shown physically attainable if the linearity of amplification is given up. The connection between the gain of amplification and the successful probability is discussed for probabilistic quantum amplification. Assuming that successful probability is the same for all quantum states to be amplified, we obtain a generally valid relation between the gain of amplification and the successful probability. Particular interest is given to phase-preserving quantum amplification of Gaussian states which has been shown of theoretical interest and of practical importance in quantum information and quantum communication recently. Our results of quantum state amplification not only enrich the research of quantum amplification but also can be helpful for further practical applications.' author: - 'Meng-Jun Hu' - 'Yong-Sheng Zhang' title: Unified view of quantum amplification based on quantum transformation --- **Introduction** ================ Quantum amplification (QA) is at the heart of quantum measurement and quantum metrology. Restricted by fundamental laws of physics, it is difficult for detectors to measure sufficiently weak signal, especially for quantum signal. Amplifying signal first and then detect it with proper detector thus becomes a general way for signal detection. Unfortunately, the intrinsic noise accompanying the signal is also amplified during the process of amplification and extra noises may be introduced to make the signal-to-noise ratio (SNR) worse. For a linear phase-preserving quantum amplifier, due to the constraint of bosonic commutation relation $[a,a^{\dagger}]=1$, it has been shown that there is at least $(g^{2}-1)\hbar\omega$ total noise power per unit bandwidth out of its output-port, where $g^{2}$ is the power gain [@1; @2]. As quantum signal amplification is equivalent to quantum state amplification, it seems that a universal linear phase-preserving quantum amplifier that can amplify any coherent state determinately and noiselessly is impossible [@3]. For the special set of coherent states, however, the probabilistic noiseless amplification of coherent states is physically realizable [@4; @5; @6] and there have been some experimental reports on the realization of noiseless amplification of quantum light states [@7; @8; @9]. Moreover, regarding quantum state amplification as quantum state transformation, our recent research shows that for a certain set of coherent states, which satisfies particular conditions, there always exists a specific quantum amplifier that can amplify coherent state in that set determinately and noiselessly [@10]. Although various investigations are given to quantum amplification, there is still lack of a general framework to discuss it. In this paper, a general framework of quantum state amplification using the language of quantum state transformation is given systematically for the first time. The paper is organized as follows. In Sec. II, the concept of amplification of quantum state is definitely defined. The general discussion of amplification of quantum states using the language of quantum state transformation is given and three different important amplifications, i.e., deterministic noisy QA, probabilistic noiseless QA and deterministic noiseless QA are identified and discussed. In Sec. III, phase-preserving QA is considered and the particular interest is given to the deterministic noiseless Gaussian state amplification. An example of application in phase measurement using deterministic noiseless QA is shown and the situation of amplified states in noisy environment is discussed. In Sec. IV, we give a summary of these results. **Amplification of quantum states** =================================== Before the detail discussion of quantum state amplification, an explicit definition of amplification of quantum state is necessary. The quantum state amplification defined as quantum state transformation is then illustrated and three different kinds of important QA are identified and discussed. **The definition of amplification of quantum state** ---------------------------------------------------- Suppose a system in which we have interest is in quantum state $|\psi\rangle$. When we measure the observable $\hat{A}$ in this system, the physical quantity obtained in the measurement is the expectation of $\hat{A}$ in state $|\psi\rangle$, that is $\langle\psi|\hat{A}|\psi\rangle $. The expectation value $\langle\psi|\hat{A}|\psi\rangle$ of observable $\hat{A}$ in state $|\psi\rangle$ is usually called quantum signal. Amplification of the quantum signal means the amplification of the expectation value $\langle\psi|\hat{A}|\psi\rangle$, which is equivalent to the amplification of state $|\psi\rangle$. Now suppose that the system evolves under an operation that transforms the state $|\psi\rangle$ into the another state $|\phi\rangle$. If the expectation value of observable $\hat{A}$ in the state $|\phi\rangle$ is larger than the expectation value in state $|\psi\rangle$, that is $\langle\phi|\hat{A}|\phi\rangle>\langle\psi|\hat{A}|\psi\rangle$, then we say the quantum signal is amplified, or equivalently, the state is amplified with respect to observable $\hat{A}$. The process of quantum signal amplification is thus actually a process of quantum state transformation. Note that QA is related with the measurement of observable. We thus give the following definition of amplification of quantum state. [**Definition 1:**]{} If there exists an operation that transforms the state $|\psi\rangle$ of quantum system into the another state $|\phi\rangle$ such that $\langle\phi|\hat{A}|\phi\rangle>\langle\psi|\hat{A}|\psi\rangle$, then this process is called the amplification of the state $|\psi\rangle$ with respect to observable $\hat{A}$. The state $|\phi\rangle$ is called the amplified state of the state $|\psi\rangle$ with respect to observable $\hat{A}$. If $\langle\phi|\hat{A}|\phi\rangle=g\langle\psi|\hat{A}|\psi\rangle$ with $g>1$, we say that the state $|\psi\rangle$ is amplified with gain of $g$. If there exists a setup that can amplify state $|\psi\rangle$ into the amplified state $|\phi\rangle$ with the gain of $g$ , then we call that setup a quantum amplifier with gain of $g$ of the amplification. A quantum amplifier is thus a physical system whose function is to transform the input state $|\psi\rangle$ to the amplified state $|\phi\rangle$. Before the amplification, the fluctuation of observable $\hat{A}$ in state $|\psi\rangle$ is $\Delta\hat{A}_{|\psi\rangle}=\sqrt{\langle\psi|\hat{A}^{2}|\psi\rangle-\langle\psi|\hat{A}|\psi\rangle^{2}}$. After the amplification, the fluctuation of observable $\hat{A}$ in the amplified state $|\phi\rangle$ becomes $\Delta\hat{A}_{|\phi\rangle}=\sqrt{\langle\phi|\hat{A}^{2}|\phi\rangle-\langle\phi|\hat{A}|\phi\rangle^{2}}$. The amplification is called noiseless if $\Delta\hat{A}_{|\phi\rangle}=\Delta\hat{A}_{|\psi\rangle}$, otherwise, the amplification is noisy. The definition of noiseless amplification of quantum state is thus defined as below. [**Definition 2:**]{} The quantum amplification is called noiseless if the fluctuation of observable $\hat{A}$ is the same in the state $|\psi\rangle$ and the amplified state $|\phi\rangle$, otherwise the quantum amplification is noisy. The above definition can be easily extended to mixed state of a quantum system. Suppose that the state of the quantum system is $\rho$, if we send this state to a quantum amplifier with gain of $g$ and the amplified state is $\sigma$, then we have $\langle\hat{A}\rangle_{\sigma}=g\langle\hat{A}\rangle_{\rho}$, where $\langle\hat{A}\rangle_{\sigma}=\mathrm{Tr}(\hat{A}\sigma)$ and $\langle\hat{A}\rangle_{\rho}=\mathrm{Tr}(\hat{A}\rho)$ are the expectation values of observable $\hat{A}$ in state $\sigma$ and $\rho$ respectively. The amplification is noiseless if $\Delta\hat{A}_{\sigma}=\Delta\hat{A}_{\rho}$, where $\Delta\hat{A}_{\sigma}=\sqrt{\mathrm{Tr}(\hat{A}^{2}\sigma)-\mathrm{Tr}(\hat{A}\sigma)^{2}}$ and $\Delta\hat{A}_{\rho}=\sqrt{\mathrm{Tr}(\hat{A}^{2}\rho)-\mathrm{Tr}(\hat{A}\rho)^{2}}$ are fluctuations of observable $\hat{A}$ in state $\sigma$ and $\rho$ respectively. The definition of amplification of quantum state with respect to two observables or more than two observables is also valid. For example, in the case of two observables $\hat{A}$ and $\hat{B}$, we have $\langle\phi|\hat{A}|\psi\rangle=g_{1}\langle\psi|\hat{A}|\psi\rangle$ and $\langle\phi|\hat{B}|\phi\rangle=g_{2}\langle\psi|\hat{B}|\psi\rangle$, where $g_{1}$ and $g_{2}$ are the gains of amplification with respect to observables $\hat{A}$ and $\hat{B}$ respectively. The gains of $g_{1}$ and $g_{2}$ can be equal to each other that $g_{1}=g_{2}$ or unequal to each other that $g_{1}\neq g_{2}$. The amplification is noiseless if and only if $\Delta\hat{A}_{|\phi\rangle}=\Delta\hat{A}_{|\psi\rangle}$ and $\Delta\hat{B}_{|\phi\rangle}=\Delta\hat{B}_{|\psi\rangle}$. If $\Delta\hat{A}_{|\phi\rangle}=\Delta\hat{A}_{|\psi\rangle}$ but $\Delta\hat{B}_{|\phi\rangle}\neq\Delta\hat{B}_{|\psi\rangle}$, then the amplification is only noiseless with respect to observable $\hat{A}$. Conversely, if $\Delta\hat{A}_{|\phi\rangle}\neq\Delta\hat{A}_{|\psi\rangle}$ but $\Delta\hat{B}_{|\phi\rangle}=\Delta\hat{B}_{|\psi\rangle}$, then the amplification is only noiseless with respect to observable $\hat{B}$. In all other cases, the amplification is not noiseless. **Quantum state amplification viewed as quantum state transformation** ---------------------------------------------------------------------- According to the definition of amplification of quantum state, the quantum state amplification is actually a quantum state transformation, thus it is natural to discuss the amplification of the quantum state using the language of quantum state transformation. In fact, the language of quantum state transformation provides a unified framework describing the quantum state cloning, the unambiguous discrimination of states, and the quantum state amplification [@11; @12; @13]. In the quantum operation theory, any physically permissible state transformation of a quantum system can be determined by a completely positive (CP), linear, trace non-increasing map: $\Lambda: \rho\rightarrow\Lambda(\rho)$ [@14; @15]. The transformation is physically realizable in principle if such a map $\Lambda$ mathematically exists. The first representation theorem gives the general representation of the CP, linear, trace nonincreasing map $\Lambda$ [@16]. It states that any CP, linear, trace non-increasing map $\Lambda$ can be represented by an operator-sum form $\Lambda(\rho)=\sum_{k}\hat{M}_{k}\rho\hat{M}_{k}^{\dagger}$, where $\hat{M}_{k}$ is the Kraus operator that satisfies $\sum_{k}\hat{M}_{k}^{\dagger}\hat{M}_{k}\leq\hat{I}$. For deterministic transformation $\sum_{k}\hat{M}_{k}^{\dagger}\hat{M}_{k}=\hat{I}$, while for probabilistic transformation $\sum_{k}\hat{M}_{k}^{\dagger}\hat{M}_{k}<\hat{I}$ [@17; @18]. The unitary evolution of a quantum system that $\Lambda(\rho)=\hat{U}\rho\hat{U}^{\dagger}$ is a special case of this representation. Another more physical way to represent the map $\Lambda$ is considering the unitary evolution on an enlarged Hilbert space. If an ancillary system is introduced to a quantum system, and a unitary transformation is applied onto the composite system, the map $\Lambda$ is realized when we make projective measurement on the ancillary system. Mathematically, it can be expressed as $\Lambda(\rho)=\mathrm{Tr}_{E^{\prime}}\lbrace \hat{U}\rho\bigotimes\rho_{E}\hat{U}^{\dagger}\hat{I}\bigotimes P_{E^{\prime}}\rbrace $, where $\rho_{E}$ is the initial state of the ancillary system and $P_{E^{\prime}}$ is a projector in the transformed ancillary Hilbert space [@15]. For convenience, we will use both of them optionally. In practice, the significant case is the amplification of a set of quantum states. The amplification of a specific state is trivial since we can always prepare the system in the expected quantum state. For simplicity, the amplification of a set of $N$ linear-independent pure states $\Pi\equiv\lbrace|\psi_{i}\rangle\rbrace$ with $i=1,2...N$ is preferred to be considered. Suppose that the set of quantum states $\Xi\equiv\lbrace|\phi_{i}\rangle\rbrace$ is the corresponding amplified set of quantum states with respect to observable $\hat{A}$. The operator $\hat{A}$ can be a single observable, or represent a set of specific observables, which will not be mentioned below except for the case of necessary. For any quantum state $|\psi_{i}\rangle$ in the set $\Pi$, there should exist a unitary operation $\hat{U}$ that transforms the state $|\psi_{i}\rangle$ into the corresponding amplified state $|\phi_{i}\rangle$ with the assistance of an ancillary system. In general, the transformation is probabilistic and the successful probability $p$ may be varied for different states in the set $\Pi$. For an arbitrary state $|\psi_{i}\rangle$ in set $\Pi$, we thus have the following formula of quantum state transformation: $$\hat{U}|\psi_{i}\rangle|0\rangle=\sqrt{p_{i}}|\phi_{i}\rangle|\mu_{i}\rangle|+\rangle+\sqrt{1-p_{i}}|Fail\rangle|\nu_{i}\rangle|-\rangle.$$ Here $|+\rangle$, $|-\rangle$ are the pointer states that indicate the amplification is successful or failure respectively. The initial state of the ancillary system is simply denoted by $|0\rangle$ and it is assumed that there is no correlation between the system and the ancillary initially. $|\mu_{i}\rangle$ and $|\nu_{i}\rangle$ are states of the ancillary system corresponding to successful and failure amplification respectively. The complex conjugate of Eq. (1) can be written as $$\langle0|\langle\psi_{j}|\hat{U}^{\dagger}=\sqrt{p_{j}}\langle\phi_{j}|\langle\mu_{j}|\langle+|+\sqrt{1-p_{j}}\langle Fail|\langle\nu_{j}|\langle-|.$$ Taking the inner product of Eq. (1) and Eq. (2) gives $$\langle\psi_{j}|\psi_{i}\rangle=\sqrt{p_{i}p_{j}}\langle\phi_{j}|\phi_{i}\rangle\langle\mu_{j}|\mu_{i}\rangle+\sqrt{(1-p_{i})(1-p_{j})}\langle\nu_{j}|\nu_{i}\rangle.$$ Equation (3) is valid for any pair of states $\lbrace|\psi_{i}\rangle,|\psi_{j}\rangle\rbrace$ in the set $\Pi$, and it can be recast as $$G_{\Pi}=G_{\Xi}\circ \Omega+K.$$ Here $G_{\Pi}$ and $G_{\Xi}$ are Gram matrices of set $\Pi$ and $\Xi$ respectively. The Gram matrix $\Omega$ is defined as $\Omega_{ij}=\sqrt{p_{i}p_{j}}\langle\mu_{j}|\mu_{i}\rangle$ and the Gram matrix $K$ is defined as $K_{ij}=\sqrt{(1-p_{i})(1-p_{j})}\langle\nu_{j}|\nu_{i}\rangle$. It is obvious that the matrix $\Omega$ satisfies the following conditions: $\Omega\geq0$; $\mathrm{diag}(\Omega)=\vec{p}=(p_{1},p_{2},...,p_{N})$; and $G_{\Pi}-G_{\Xi}\circ\Omega\geq0$. According to quantum transformation theorem of sets of pure states [@18], the matrix $\Omega$ can be factorized as $C^{\dagger}C$ where $C=[c_{ki}]$ is a $M\times N$ matrix. The amplification Kraus operators can be constructed as $$\hat{M}_{ks}=\sum_{i}\dfrac{c_{ki}}{\langle\tilde{\psi_{i}}|\psi_{i}\rangle}|\phi_{i}\rangle\langle\tilde{\psi_{i}}|,$$ where $\hat{M}_{ks}(k=1,2,...,M)$ are the Kraus operators for successful amplification. State $|\tilde{\psi_{i}}\rangle$ is orthogonal to any state in set $\Omega$ except for state $|\psi_{i}\rangle$ and $\langle\tilde{\psi_{i}}|\psi_{j}\rangle=\gamma_{i}\delta_{ij}$, where $\gamma_{i}\neq 0$ is a constant. For the successful amplification, it can be seen that for any state $|\psi_{i}\rangle$ $$\hat{M}_{ks}|\psi_{i}\rangle=c_{ki}|\phi_{i}\rangle.$$ Though only pure state amplification is considered above, the extension to the case of general state amplification is not difficult. Suppose that the set of $N$ linear-independent general quantum states $\Pi=\lbrace\rho_{1},\rho_{2},...,\rho_{N}\rbrace$ is to be amplified, and the set $\Xi=\lbrace\sigma_{1},\sigma_{2},...,\sigma_{N}\rbrace$ is the corresponding amplified set of quantum states. For any state $\rho_{i}$ in set $\Pi$, the transformation of amplification can be formulated as: $$\hat{U}\rho_{i}\otimes\rho_{E}\hat{U}^{\dagger}=p_{i}\sigma_{i}\otimes\tau_{i}\otimes|+\rangle\langle+|+(1-p_{i})\varrho\otimes\upsilon_{i}\otimes|-\rangle\langle-|,$$ where $|+\rangle\langle+|,|-\rangle\langle-|$ are pointer states, $\tau_{i}$ and $\upsilon_{i}$ are states of the ancillary system corresponding to successful and failure amplification respectively, and $\varrho$ is the failure state of the quantum system. Taking the trace of the product of Eq. (7) and its conjugate gives $$\mathrm{Tr}(\rho_{i}\rho_{j})=p_{i}p_{j}\mathrm{Tr}(\sigma_{i}\sigma_{j})\mathrm{Tr}(\tau_{i}\tau_{j})+(1-p_{i})(1-p_{j})\mathrm{Tr}(\upsilon_{i}\upsilon_{j}),$$ which is valid for any pair of states $\lbrace\rho_{i},\rho_{j}\rbrace$ in set $\Pi$. In the above formula, we have assumed that $\rho_{E}$ is a pure state and $\mathrm{Tr}(\varrho^{2})$ is a constant that involved in term $\mathrm{Tr}(\upsilon_{i}\upsilon_{j})$. The Kraus operator representation of successful amplification now becomes $$\hat{M}_{ks}\rho_{i}\hat{M}_{ks}^{\dagger}=|c_{ki}|^{2}\sigma_{i}.$$ Since a mixed state $\rho$ can be considered as an ensemble of pure quantum states, the amplification of a set of general quantum states is equivalent to the amplification of a specific set of pure quantum states. In the following discussion, the QA of a set of states refers to the amplification of pure states without pointing out it explicitly. **The classification of quantum amplification** ----------------------------------------------- There are mainly two different kinds of quantum amplification, i.e., deterministic QA and probabilistic QA that can be obviously seen from Eq. (8). We first consider the deterministic amplification of the set of quantum states $\Pi=\lbrace\rho_{1},\rho_{2},...,\rho_{N}\rbrace$. The successful probability of amplification is unity for any state $\rho_{i}$ in set $\Pi$, that is $p_{i}=1,\forall\rho_{i}\subseteq\Pi$. The second term in the right-hand side (RHS) of Eq. (7) vanishes for this case and Eq. (8) becomes $$\mathrm{Tr}(\rho_{i}\rho_{j})=\mathrm{Tr}(\sigma_{i}\sigma_{j})\mathrm{Tr}(\tau_{i}\tau_{j}).$$ Since the overlap of any two states is less than unity, we have $\mathrm{Tr}(\rho_{i}\rho_{j})=\mathrm{Tr}(\sigma_{i}\sigma_{j})\mathrm{Tr}(\tau_{i}\tau_{j})\leq \mathrm{Tr}(\sigma_{i}\sigma_{j})$ for any states $\rho_{i},\rho_{j}\subseteq\Pi$, which implies that the overlap between any two states to be amplified is no more than the overlap between two corresponding amplified states for deterministic amplification. From the point of view of information, the distinguishability of any two states to be amplified does not increase after the deterministic QA. The underlying physics of this consequence is that a quantum system will dissipate some of its information to the environment because of the interaction between them. In fact, for any quantum operation $L$ on the set of states $\Pi$, the distinguishability of any two states $\rho_{i},\rho_{j}$ in the set does not increase after operation, which means $$D(\rho_{i},\rho_{j})\geq D(L\rho_{i},L\rho_{j}),$$ where $D(\rho_{i},\rho_{j})$ represents the distinguishability of states $\rho_{i},\rho_{j}$. The distinguishability does not increase after amplification provides us an essential tool to investigate deterministic QA further. Suppose that the amplification is noiseless, which means that the fluctuation of observable $\hat{A}$ in any state $\rho_{i}\subseteq\Pi$ and its corresponding amplified state $\sigma_{i}\subseteq\Xi$ stay unchanged, i.e., $\Delta\hat{A}_{\rho_{i}}=\Delta\hat{A}_{\sigma_{i}}$. For a linear quantum amplifier with fixed gain $g$ of amplification, which means $\mathrm{Tr}(\hat{A}\sigma_{i})=g\mathrm{Tr}(\hat{A}\rho_{i})$ for any state $\rho_{i}\subseteq\Pi$, we will show that the linearity of quantum amplifier is incompatible with the noiseless amplification. ![image](new1.eps) Suppose that the eigenvalues of observable $\hat{A}$ are $\lbrace a_{f}\rbrace$ and the corresponding eigenstates are $\lbrace|a_{f}\rangle\rbrace$ so that $\hat{A}=\sum_{f}a_{f}|a_{f}\rangle\langle a_{f}|$. The expectation of observable $\hat{A}$ in any quantum state $\rho$ now can be expressed as $$\mathrm{Tr}(\hat{A}\rho)=\sum_{f}a_{f}\langle a_{f}|\rho|a_{f}\rangle.$$ For a linear quantum amplifier with gain $g$, $\mathrm{Tr}(\hat{A}\rho_{i})=g\mathrm{Tr}(\hat{A}\sigma_{i})$ holds for any state $\rho_{i}\subseteq\Pi$ and the corresponding amplified state $\sigma_{i}\subseteq\Xi$. In the representation of observable $\hat{A}$, the fluctuation of $\hat{A}$ in quantum state $\rho_{i}$ reads $$\begin{split} \Delta\hat{A}_{\rho_{i}}^{2}&=\mathrm{Tr}(\hat{A}^{2}\rho_{i})-\mathrm{Tr}(\hat{A}\rho_{i})^{2} \\ &=\sum_{f}a_{f}^{2}\langle a_{f}|\rho_{i}|a_{f}\rangle-(\sum_{f}a_{f}\langle a_{f}|\rho_{i}|a_{f}\rangle)^{2}. \end{split}$$ In the amplified state $\sigma_{i}$, the fluctuation of observable $\hat{A}$ is $$\begin{split} \Delta\hat{A}_{\sigma_{i}}^{2}&=\sum_{f}a_{f}^{2}\langle a_{f}|\sigma_{i}|a_{f}\rangle-(\sum_{f}a_{f}\langle a_{f}|\sigma_{i}|a_{f}\rangle)^{2}\\ &=\sum_{f}a_{f}^{2}\langle a_{f}|\sigma_{i}|a_{f}\rangle-g^{2}(\sum_{f}a_{f}\langle a_{f}|\rho_{i}|a_{f}\rangle)^{2}. \end{split}$$ If the amplification is noiseless, then $\Delta\hat{A}_{\rho_{i}}^{2}=\Delta\hat{A}_{\sigma_{i}}^{2}$ gives $$(g^{2}-1)\mathrm{Tr}(\hat{A}\rho_{i})^{2}=\mathrm{Tr}\lbrace\hat{A}^{2}(\sigma_{i}-\rho_{i})\rbrace.$$ Considering the term in the RHS of Eq.(15), we have $$\begin{split} \mathrm{Tr}\lbrace\hat{A}^{2}(\sigma_{i}-\rho_{i})\rbrace=\sum_{f}a_{f}^{2}\langle a_{f}|(\sigma_{i}-\rho_{i})|a_{f}\rangle \\ <a_{max}(g-1)\mathrm{Tr}(\hat{A}\rho_{i}), \end{split}$$ where $a_{max}$ is the maximum eigenvalue of observable $\hat{A}$. Substituting Eq.(16) into Eq. (15) gives $$(g^{2}-1)\mathrm{Tr}(\hat{A}\rho_{i})^{2}<a_{max}(g-1)\mathrm{Tr}(\hat{A}\rho_{i}),$$ which is obviously not hold in general. It is obvious from Eq. (14) that $\Delta\hat{A}_{\sigma_{i}}\neq \Delta\hat{A}_{\rho_{i}}$ except for $g=1$, which means there is no amplification at all. We thus conclude that a deterministic linear quantum amplifier with fixed gain $g$ can not be noiseless. $\mathrm{Tr}(\rho_{i}\rho_{j})$ is equal to the overlap of two distribution of $\rho_{i}$ and $\rho_{j}$ in the representation of $\hat{A}$, since $$\mathrm{Tr}(\rho_{i}\rho_{j})=\sum_{f,g}\langle a_{f}|\rho_{i}|a_{g}\rangle\langle a_{g}|\rho_{j}|a_{f}\rangle.$$ The sum should be replaced by an integral if $\hat{A}$ is a continuous observable $$\mathrm{Tr}(\rho_{i}\rho_{j})=\int dxdy\langle x|\rho_{i}|y\rangle\langle y|\rho_{j}|x\rangle.$$ The added noise to the amplified states must ensure that the overlap between amplified states larger than corresponding initial states. It is easy to understand just with this added noise the distinguishability of any two amplified quantum states does not increase. Although the deterministic amplification can not be noiseless for a linear quantum amplifier, the deterministic noiseless amplification is attainable if the gain of amplification can be state-dependent, i.e., the quantum amplifier is nonlinear [@10]. Here, we assume that the gain of amplification of state $\rho_{i}$ is $g_{i}$, i.e., $\mathrm{Tr}(\hat{A}\sigma_{i})=g_{i}\mathrm{Tr}(\hat{A}\rho_{i})$. For noiseless amplification, the overlap between any two states $\rho_{i},\rho_{j}\subseteq\Pi$ can be less than the overlap between two corresponding amplified states, that means $\mathrm{Tr}(\rho_{i}\rho_{j})\leqslant\mathrm{Tr}(\sigma_{i}\sigma_{j})$, which is impossible for a linear quantum amplifier. The distinguishability of any two states $\rho_{i},\rho_{j}\subseteq\Pi$ does not increase for a deterministic noiseless amplification means $D(\rho_{i},\rho_{j})\geqslant D(\sigma_{i},\sigma_{j})$ is the only physical requirement. For the set of states $\Pi$ to be amplified and the noiseless amplified set of states $\Xi$, if $D(\rho_{i},\rho_{j})\geqslant D(\sigma_{i},\sigma_{j})$ for any two states $\rho_{i},\rho_{j}$ and their corresponding noiseless amplified states $\sigma_{i},\sigma_{j}$ is satisfied, then such deterministic noiseless quantum amplifier always can be constructed physically. The linear quantum amplifier with gain $g$ can be noiseless if we do not demand the amplification is deterministic, and in this case the only physical requirement is that Eq. (8) must be hold. The probabilistic noiseless linear quantum amplifier is thus physically allowed. The relation between the gain $g$ and the successful probability $p_{i}$ can also be obtained from Eq. (8) since for noiseless amplification $\mathrm{Tr}(\rho_{i}\rho_{j})=f_{ij}(g)\mathrm{Tr}(\sigma_{i}\sigma_{j})$, where $f(g)$ is proportional to $g$ and depends on the states $\rho_{i}$ and $\rho_{j}$ to be amplified. The form of $f_{ij}(g)$ relies on the specific case we consider, for instance, in the noiseless amplification of coherent states, $f_{ij}(g)=e^{(g^{2}-1)|\alpha_{i}-\alpha_{j}|^{2}}$, where $|\alpha_{i}\rangle$ and $|\alpha_{j}\rangle$ are coherent states to be amplified. The Eq. (8) thus becomes $$f_{ij}(g)=p_{i}p_{j}\mathrm{Tr}(\tau_{i}\tau_{j})+(1-p_{i})(1-p_{j})\dfrac{\mathrm{Tr}(\upsilon_{i}\upsilon_{j})}{\mathrm{Tr}(\sigma_{i}\sigma_{j})}.$$ Assuming that the successful probability is the same for all states $\rho_{i}\subseteq\Pi$ , then we have $$f_{ij}(g)=C_{i,j}p^{2}+V_{i,j}(1-p)^{2},$$ where $C_{i,j}\equiv\mathrm{Tr}(\tau_{i}\tau_{j})$ and $V_{i,j}\equiv \dfrac{\mathrm{Tr}(\upsilon_{i}\upsilon_{j})}{\mathrm{Tr}(\sigma_{i}\sigma_{j})}$. Eq. (21) is valid for any pair of states $\rho_{i},\rho_{j}\subseteq\Pi$. We emphasize that this kind of probabilistic noiseless QA is totally different from the classic-like noiseless QA that analogous to the classical amplifier works based on the unambiguous identification of input states and the preparation of desired amplified states [@19; @20]. Though the successful probability is limited by identification of input states, the gain of classic-like quantum amplifier can be arbitrary high which is not the case for our probabilistic quantum amplifier. The probabilistic linear amplification of course can be noisy or noiseless when the probabilistic amplification is nonlinear, but we usually do not consider them since they are practically trivial. As a summary, three different kinds of important QA, i.e., linear deterministic noisy QA, nonlinear deterministic noiseless QA and linear probabilistic noiseless QA are identified and discussed in detail. For deterministic QA, the linearity of the amplifier is incompatible with the noiseless amplification, while it is not a problem for probabilistic QA. The clear relation between this three different kinds of quantum amplification is shown in Fig. 1. **Phase-preserving amplification of quantum states** ==================================================== Phase-preserving QA is of theoretical interest and of practical importance [@1; @2; @4; @6; @7; @21; @22; @23], especially for the phase-preserving quantum amplification of Gaussian states in phase space. **Definition of phase-preserving quantum amplification** -------------------------------------------------------- The phase-preserving QA is the amplification of states with respect to two canonical observables $\hat{X}_{1}$ and $\hat{X}_{2}$, e.g., position $\hat{Q}$ and momentum $\hat{P}$. For any state $\rho_{i}\subseteq\Pi$, the phase-preserving quantum amplifier amplifies the state $\rho_{i}$ into the corresponding amplified state $\sigma_{i}\subseteq\Xi$ with $\mathrm{Tr}(\hat{X}_{1}\sigma_{i})=g_{i}\mathrm{Tr}(\hat{X}_{1}\rho_{i})$ and $\mathrm{Tr}(\hat{X}_{2}\sigma_{i})=g_{i}\mathrm{Tr}(\hat{X}_{2}\rho_{i})$. If $g_{i}$ is the same for any state $\rho_{i}$, then it is a linear phase-preserving quantum amplifier, otherwise it is non-linear. ![(Color Online) [*Illustration of the distribution of quantum state $\rho$ in $(\hat{X}_{1},\hat{X}_{2})$ space and phase-preserving quantum amplification*]{}. (a) The distribution of quantum state $\rho$ in $(\hat{X}_{1},\hat{X}_{2})$ space. For the simplicity of figure, we assume here that the distribution is Gaussian. The position of distribution totally determined by amplitude $r$ and phase $\theta$. (b) Phase-preserving amplification of quantum state $\rho$. The amplitude is amplified by $g$-fold while the phase stays unchanged.](hu.eps) We can construct an operator $\hat{a}=\hat{X}_{1}+i\hat{X}_{2}$ which is not Hermitian. Though $\hat{a}$ is not an observable, the phase-preserving amplification with respect to observables $\hat{X}_{1}$ and $\hat{X}_{2}$ can be regarded as amplification with respect to $\hat{a}$ since $\mathrm{Tr}(\hat{a}\sigma_{i})=g_{i}\mathrm{Tr}(\hat{a}\rho_{i})$. Analogous to phase space based on position $\hat{Q}$ and momentum $\hat{P}$, the general phase space based on canonical observables $\hat{X}_{1}$ and $\hat{X}_{2}$ can be defined. Any quantum state $\rho$ can be represented by its distribution in $(\hat{X}_{1},\hat{X}_{2})$ space that denoted by $G_{\rho}(x_{1},x_{2})$. The marginal distribution of $G_{\rho}(x_{1},x_{2})$ can be given in the form $$\begin{split} \int dx_{2}G_{\rho}(x_{1},x_{2})&=G_{\rho}(x_{1}),\\ \int dx_{1}G_{\rho}(x_{1},x_{2})&=G_{\rho}(x_{2}), \end{split}$$ where we have assumed that $\hat{X}_{1}$ and $\hat{X}_{2}$ are continuous observables and $G_{\rho}(x_{1})\equiv\langle x_{1}|\rho|x_{1}\rangle$ , $G_{\rho}(x_{2})\equiv\langle x_{2}|\rho|x_{2}\rangle$ are distribution of $\rho$ in the representation of $\hat{X}_{1}$, $\hat{X}_{2}$ respectively. When $\hat{X}_{1}=\hat{Q}$ and $\hat{X}_{2}=\hat{P}$, the distribution $G_{\rho}(x_{1},x_{2})$ is just the Wigner function $W_{\rho}(q,p)$ in phase space. The phase-preserving amplification of quantum state $\rho$ can be restated as the amplitude of $\rho$ in $(\hat{X}_{1},\hat{X}_{2})$ space being amplified while the phase being unchanged. The phase stays unchanged after the amplification is the reason that we call it phase-preserving amplification. As it can be seen from Fig. 2, the amplitude of $\rho$ in $(\hat{X}_{1},\hat{X}_{2})$ space is defined as $r=\sqrt{\mathrm{Tr}(\hat{X}_{1}\rho)^{2}+\mathrm{Tr}(\hat{X}_{2}\rho)^{2}}$ and the phase is determined by $\theta=\mathrm{arctan}[\dfrac{\mathrm{Tr}(\hat{X}_{2}\rho)}{\mathrm{Tr}(\hat{X}_{1}\rho)}]$. In practice, the particular focus is given to the spacial case of $\hat{X}_{1}=\hat{Q}$ and $\hat{X}_{2}=\hat{P}$. The distribution of quantum state $\rho$ is described by the Wigner function defined as [@24] $$W_{\rho}(q,p)=\dfrac{1}{2\pi\hbar}\int_{-\infty}^{+\infty}d\xi \mathrm{exp}(-\dfrac{i}{\hbar}p\xi)\langle q+\dfrac{1}{2}\xi|\rho|x-\dfrac{1}{2}\xi\rangle.$$ For any two states $\lbrace\rho_{i},\rho_{j}\rbrace$, their overlap is equal to the overlap of their corresponding distributions in phase space [@25] $$\mathrm{Tr}(\rho_{i}\rho_{j})=2\pi\hbar\int_{-\infty}^{+\infty}dq\int_{-\infty}^{+\infty}dp W_{\rho_{i}}(q,p)W_{\rho_{j}}(q,p).$$ **Phase-preserving quantum amplification of Gaussian states** ------------------------------------------------------------- Gaussian states play an important role in quantum information and quantum communication [@26; @27; @28]. The phase-preserving QA of Gaussian states is not only of theoretical interest [@1; @10] but also of practical importance [@29; @30; @31; @32; @33]. A Gaussian state is defined as such a state that its characteristic function, or equivalently its Wigner function, is Gaussian. The vacuum state, coherent states, squeezed states, and thermal states are typical Gaussian states. The Gaussian state is fully characterized by its first moment $\vec{d}$ and second moment $\vec{\gamma}$ [@27; @28]. For a single-mode Gaussian state $\rho$, the first moment $\vec{d}=(d_{1},d_{2})$ is defined as $$d_{i}=\mathrm{Tr}(\hat{X}_{i}\rho),$$ where $\hat{X}_{i}$ represent quadrature operators $\hat{Q}$ and $\hat{P}$. The second moment $\vec{\gamma}$, which forms the so-called covariance matrix $\vec{\gamma}=(\gamma_{ij})(i,j=1,2)$, is given by $$\gamma_{ij}=\mathrm{Tr}(\lbrace\hat{X}_{i},\hat{X}_{j}\rbrace\rho)-2d_{i}d_{j},$$ where $\lbrace\hat{X}_{i},\hat{X}_{j}\rbrace\equiv\hat{X}_{i}\hat{X}_{j}+\hat{X}_{j}\hat{X}_{i}$ is the anti-commutator of $\hat{X}_{i}$ and $\hat{X}_{j}$. For simplicity and convenience, only single-mode Gaussian states are considered here. Results about phase-preserving QA of single-mode Gaussian states can be easily extended into multi-mode case. We consider the phase-preserving QA of a set of $N$ single-mode Gaussian states $\Pi=\lbrace\rho_{1},\rho_{2},...,\rho_{N}\rbrace$ here. Note that all Gaussian states in set $\Pi$ should be in the same single-mode, otherwise different Gaussian states can be identified because of mode identification. The corresponding set of $N$ amplified Gaussian states is denoted by $\Xi=\lbrace\sigma_{1},\sigma_{2},...,\sigma_{N}\rbrace$. For any state $\rho_{i}\subseteq\Pi$, the phase-preserving QA means the amplification of the first moment $$\vec{d}_{\sigma_{i}}=g_{i}\vec{d}_{\rho_{i}},$$ where $\vec{d}_{\rho}$ represents the first moment of Gaussian state $\rho$. If the gain $g_{i}$ is the same for all states $\rho_{i}\subseteq\Pi$, then the phase-preserving QA of Gaussian states is linear, otherwise it is nonlinear. For noiseless amplification, the second moments should stay unchanged $$\vec{\gamma}_{\sigma_{i}}=\vec{\gamma}_{\rho_{i}}, \forall\rho_{i}\subseteq\Pi,$$ where $\vec{\gamma}_{\rho}$ represents the second moment of state $\rho$. The distinguishability of any two Gaussian states can be measured by their distance in phase space. The distance between any two Gaussian states $\rho$ and $\sigma$ in phase space can be written as $$D(\rho,\sigma)=(\vec{d}_{\rho}-\vec{d}_{\sigma})^{2}.$$ The distinguishability of two states $\rho$ and $\sigma$ that is inversely proportional to overlap $\mathrm{Tr}(\rho\sigma)$ is also proportional to their distance $D(\rho,\sigma)$ in phase space. The larger the distance between two Gaussian states, the more the distinguishability of these two states and vice versa. For a deterministic quantum amplifier, it has been shown above that the linearity of amplification is incompatible with noiseless amplification. The linear deterministic phase-preserving QA of Gaussian states can not be noiseless. This can be seen by calculating the distance of two noiseless amplified Gaussian states. Suppose that the amplification is noiseless, the distance between any two amplified Gaussian states $\sigma_{i},\sigma_{j}\subseteq\Xi$ is $$D(\sigma_{i},\sigma_{j})=(\vec{d}_{\sigma_{i}}-\vec{d}_{\sigma_{j}})^{2}=g^{2}D(\rho_{i},\rho_{j}),$$ where $D(\rho_{i},\rho_{j})=(\vec{d}_{\rho_{i}}-\vec{d}_{\rho_{j}})^{2}$ and Eq. (27) is used in the last step above. Obviously, for $g>1$, $D(\sigma_{i},\sigma_{j})>D(\rho_{i},\rho_{j})$, which implies that Gaussian states are more distinguishable after deterministic QA, which is directly conflicted with the physical fact that the distinguishability of any two states in the set $\Pi$ does not increase after deterministic QA. Of course, the linear probabilistic phase-preserving QA of Gaussian states can be noiseless. The gain of amplification $g$ and the successful probability $p$ is limited by Eq. (20). Furthermore, Eq. (21) is satisfied if the successful probability $p$ is the same for all states $\rho_{i}\subseteq\Pi$. Consider the linear probabilistic QA of set of two Gaussian states $\lbrace\rho_{1},\rho_{2}\rbrace$ and its corresponding amplified set of coherent states is $\lbrace\sigma_{1},\sigma_{2}\rbrace$ with gain $g$ of QA. If the QA is noiseless then Eq.(21) becomes $$f(g)=Cp^{2}+V(1-p)^{2},$$ where $C\equiv\mathrm{Tr}(\tau_{1}\tau_{2})$ and $V\equiv\dfrac{\mathrm{Tr}(\upsilon_{1}\upsilon_{2})}{\mathrm{Tr}(\sigma_{1}\sigma_{2})}$ are constants in this case and $f(g)$ is proportional to $g$. The minimum gain of QA $g_{min}$ exists since $\dfrac{d^{2}f(g)}{dp^{2}}=2(C+V)>0$. We obtain the minimum point $p_{0}=\dfrac{V}{V+C}$ by setting $\dfrac{df(g)}{dp}=0$. The $g_{min}$ is thus determined by $$f(g_{min})=\sqrt{Cp_{0}^{2}+V(1-p_{0})^{2}}.$$ For the case of QA of coherent states, $f(g)=e^{(g^{2}-1)D(\rho_{1},\rho_{2})}$ and we have $$g_{min}=\sqrt{\dfrac{1}{2}\dfrac{\mathrm{ln}[Cp_{0}^{2}lnV(1-p_{0})^{2}]}{D(\rho_{1},\rho_{2})}+1}.$$ The existence of minimum gain of amplification $g_{min}$ can be easily understood since the amplified state $\sigma_{i}$ should be distinguished from the initial state $\rho_{i}$. To distinguish the two Gaussian states requires the distinguishability or the distance of two Gaussian states is larger or equal to a threshold value $\epsilon$ below which the two Gaussian states can not be distinguished from each other. The threshold value $\epsilon$ determines the minimum gain of amplification $g_{min}$. The distance between state $\rho_{i}$ and the corresponding amplified state $\sigma_{i}$ depend on the first moment of state $\rho_{i}$ and the gain $g$ of amplification since $$D(\sigma_{i},\rho_{i})=[(g-1)\vec{d}_{\rho_{i}}]^{2}.$$ The threshold value $\epsilon$ should be proportional to the minimum distance that defined as $D_{min}\equiv\mathrm{min}\lbrace D(\sigma_{i},\rho_{i}),\forall\rho_{i}\subseteq\Pi\rbrace$, that is $$\epsilon=\kappa D_{min},$$ where $\kappa$ is a constant factor for a definite set of Gaussian states $\Pi$. The minimum gain of amplification can be calculated as $$g_{min}=\sqrt{\dfrac{\epsilon}{\kappa(\vec{d}_{\rho,min})^{2}}}+1,$$ where $\vec{d}_{\rho_{i},min}\equiv\mathrm{min}\lbrace \vec{d}_{\rho_{i}},\forall\rho_{i}\subseteq\Pi\rbrace$ is the minimum first moment of states $\rho_{i}$ in set $\Pi$. In general, the situation is more complicated so that we have to consider the set of equations in the form of Eq.(21). The deterministic phase-preserving QA of Gaussian states can be noiseless if the requirement of linearity of amplification is relaxed. Suppose that the gain of amplification is dependent on different Gaussian states $\rho_{i}\subseteq\Pi$, that is $\mathrm{Tr}(\hat{a}\sigma_{i})=g_{i}\mathrm{Tr}(\hat{a}\rho_{i})$ with $\hat{a}=\hat{Q}+i\hat{P}$. According to Eq. (10), the distinguishability of any two Gaussian states does not increase after deterministic QA. In other words, the distance does not increase after deterministic noiseless QA, i.e., $$D(\rho_{i},\rho_{j})\geq D(\sigma_{i},\sigma_{j}).$$ Using the definition of distance, the simple calculation gives $$\mathrm{cos}\vartheta_{ij}\geq \dfrac{\sqrt{(g_{i}^{2}-1)(g_{j}^{2}-1)}}{g_{i}g_{j}-1},$$ where $\vartheta_{ij}$ is the relative phase of states $\rho_{i}$ and $\rho_{j}$ in phase space. For a deterministic noiseless QA, the set of Gaussian states $\Pi$ and gain of amplification $\lbrace g_{i}\rbrace$ must satisfy Eq. (38). On the other hand, if Eq. (38) is satisfied, we can always construct a deterministic noiseless phase-preserving quantum amplifier that amplifies any Gaussian state $\rho_{i}\subseteq\Pi$ to the corresponding amplified Gaussian state $\sigma_{i}\subseteq\Xi$ physically. We thus obtain the following theorem about deterministic noiseless phase-preserving QA of Gaussian states. [**Theorem:**]{} Suppose the set of $N$ Gaussian states $\Pi=\lbrace\rho_{1},\rho_{2},...,\rho_{N}\rbrace$ is to be amplified and its corresponding phase-preserving amplified set is $\Xi=\lbrace\sigma_{1},\sigma_{2},...,\sigma_{N}\rbrace$ with $\vec{d}_{\rho_{i}}=g_{i}\vec{d}_{\sigma_{i}}$ and $\vec{\gamma}_{\sigma_{i}}=\vec{\gamma}_{\rho_{i}}$ for any Gaussian state $\rho_{i}\subseteq\Pi$. The deterministic noiseless phase-preserving quantum amplifier that amplifies the state $\rho_{i}$ randomly chosen from set $\Pi$ into the corresponding amplified state $\sigma_{i}$ in the set $\Xi$ exists if and only if the states in set $\Pi$ and the gain of amplification satisfy the condition $\mathrm{cos}\vartheta_{ij}\geq \dfrac{\sqrt{(g_{i}^{2}-1)(g_{j}^{2}-1)}}{g_{i}g_{j}-1}$ for any two states $\rho_{i},\rho_{j}\subseteq\Pi$, where $\vartheta_{ij}$ is the relative phase between states $\rho_{i}$ and $\rho_{j}$ in phase space. There exists a special case that all the final amplified states have the same amplitude in phase space. In this case, $\vec{d}_{\sigma_{i}}^{2}=\vec{d}_{\sigma_{j}}^{2}$ for any two states $\sigma_{i},\sigma_{j}\subseteq\Xi$, which implies that $g_{i}^{2}\vec{d}_{\rho_{i}}^{2}=g_{j}^{2}\vec{d}_{\rho_{j}}^{2}$. Combining this requirement with Eq. (38) finally gives $$\mathrm{cos}\vartheta_{ij}\geq \sqrt{\dfrac{g_{i}^{2}\vec{d}_{\rho_{i}}^{2}-\vec{d}_{\rho_{j}}^{2}}{(g_{i}^{2}-1)\vec{d}_{\rho_{i}}^{2}\vec{d}_{\rho_{j}}^{2}}}.$$ [**Corollary:**]{} The deterministic noiseless phase-preserving quantum amplifier that amplifies the Gaussian state $\rho_{i}$ randomly chosen from set $\Pi$ to corresponding amplified Gaussian state $\sigma_{i}$ in the set $\Xi$ with all the same amplitude in phase space exists if and only if the Gaussian states in set $\Pi$ and the gain of amplification satisfy the condition $\mathrm{cos}\vartheta_{ij}\geq \sqrt{\dfrac{g_{i}^{2}\vec{d}_{\rho_{i}}^{2}-\vec{d}_{\rho_{j}}^{2}}{(g_{i}^{2}-1)\vec{d}_{\rho_{i}}^{2}\vec{d}_{\rho_{j}}^{2}}}$ for any two states $\rho_{i},\rho_{j}\subseteq\Pi$, where $\vartheta_{ij}$ is the relative phase between states $\rho_{i}$ and $\rho_{j}$ in phase space. **The phase measurement with phase-preserving amplification** ------------------------------------------------------------- After phase-preserving QA, the phase of Gaussian states stays unchanged. This property can be exploited to improve the precision of phase measurement, for instance, the phase measurement of coherent states using a balanced homodyne detector. The difference of two photo-detector measurements is obtained in homodyne detection and the output signal is determined by $$\hat{n}_{d}=-i(\hat{b}^{\dagger}\hat{c}-\hat{c}^{\dagger}\hat{b}),$$ where $\hat{n}_{d}$ is the number difference operator and $\hat{b},\hat{c}$ are annihilation operators [@34]. A large amplitude coherent state $|c\rangle$ with fixed phase is used in one input port as reference. When a coherent state $|b\rangle$ is sent into another input port, the mean measured signal at the output is $$\langle\hat{n}_{d}\rangle=2|b|\cdot|c|\mathrm{sin}\delta,$$ where $\delta$ is the relative phase between coherent states $|b\rangle$ and $|c\rangle$. The variance of signal is calculated as $$\Delta\hat{n}_{d}=\sqrt{|b|^{2}+|c|^{2}}.$$ The sensitivity of the measured phase, according to error transfer formula, becomes $$\Delta\delta=\dfrac{\Delta\hat{n}_{d}}{|\partial\langle\hat{n}_{d}\rangle/\partial\delta|}=\dfrac{\sqrt{1+(|c|/|b|)^{2}}}{2|c|\mathrm{cos}\delta}.$$ Suppose that the input coherent state $|b\rangle$ is randomly chosen from a definite set of coherent states. A probabilistic noiseless quantum amplifier can be applied to amplify input coherent state $|b\rangle$ before detection. According to Eq. (41) and Eq. (43), not only the mean output signal is enhanced but also the precision of measured phase is improved in this case. Moreover, if this set of coherent states satisfy the condition of deterministic noiseless QA, a deterministic noiseless quantum amplifier can be designed to amplify the input coherent state noiselessly and determinately. **Deterministic amplification in noisy circumstance** ----------------------------------------------------- The distinguishabilty of two quantum states decreases in general after deterministic QA. However, the situation may be different in noisy circumstance. The distinguishability of two amplified states through the noisy channel may be larger than the two states through the same noisy channel without amplification. In order to see it explicitly, suppose that the two states $\rho_{1},\rho_{2}$ are to be amplified and $\sigma_{1},\sigma_{2}$ are the two corresponding amplified states. The noisy channel can be described by superoperator $\hat{V}(t)$ so that the state evolution of quantum system in noisy channel is $\rho(t)=\hat{V}(t)[\rho(0)]$ [@35]. In general, the distance of two states decreases monotonously in noisy channel because of the information dissipation of quantum system $D(\rho_{1}(t),\rho_{2}(t))\leq D(\rho_{1}(0),\rho_{2}(0))$. The decay rate of distance can de defined as [@36] $$\chi(\rho_{1}(t),\rho_{2}(t))=\dfrac{d}{dt}D(\rho_{1}(t),\rho_{2}(t)).$$ $\chi(\rho_{1}(t),\rho_{2}(t))\leq 0$ implies the distinguishability of two states decreases with time in noisy channel while in non-Markovian channel $\chi(\rho_{1}(t),\rho_{2}(t))>0$ [@37; @38]. The decay rate of two amplified quantum states in the same noisy channel can be defined similarly. The decay rate of distance depends on the initial states which means the possible different decay rate for states and amplified states. It is thus possible that the distinguishability of two amplified states may be larger than the states without amplification through the same noisy channel. Besides, the detector is not truly ideal in practice. The unavoidable noises in detector will lower our precision of distinguish quantum states. For an imperfect detector, the amplified states may be more distinguishable than states without amplification. **Summary** =========== In this paper, using the language of quantum state transformation, we give a general framework of quantum state amplification systematically for the first time. Based on explicit definition of amplification of quantum states, we formulate the amplification of set of quantum states as transformation of quantum states. Three different kinds of important QA, i.e., deterministic noisy QA, probabilistic noiseless QA, and deterministic noiseless QA, are identified and discussed in detail. For deterministic QA, we show that the linearity of amplification is incompatible with the noiseless amplification while it is not a problem for probabilistic QA. For probabilistic noiseless QA, we discussed the connection between the gain of amplification and the successful probability. We obtain a generally valid relation between the gain of amplification and the successful probability for the case in which successful probability is the same for all quantum states to be amplified. Particular interest is focused on phase-preserving QA of Gaussian states that has been shown of theoretical interest and of practical importance in quantum information and quantum communication. Using the concept of distance between Gaussian states, the theorem of deterministic noiseless QA of Gaussian states with explicit formula is obtained for the first time. Also, the application of noiseless QA in phase measurement is given and deterministic QA in noisy circumstance is discussed. Though the distinguishability of two quantum states decreases after deterministic amplification in general, the situation may be very different in noisy circumstance. 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--- abstract: 'Using operator algebra, we extend the series for the activity density in a one-dimensional stochastic sandpile with fixed particle density $p$, the first terms of which were obtained via perturbation theory \[R. Dickman and R. Vidigal, J. Phys. A [**35**]{}, 7269 (2002)\]. The expansion is in powers of the time; the coefficients are polynomials in $p$. We devise an algorithm for evaluating expectations of operator products and extend the series to ${\cal O}(t^{16})$. Constructing Padé approximants to a suitably transformed series, we obtain predictions for the activity that compare well against simulations, in the supercritical regime.' address: | $^1$Departamento de Física, Universidade Federal Fluminense, Av. Litorânea s/n,\ 24210-340\ Niterói - Rio de Janeiro, Brasil\ $^2$Departamento de Física, ICEx, Universidade Federal de Minas Gerais,\ 30123-970 Belo Horizonte - Minas Gerais, Brasil\ author: - 'Jürgen F. Stilck$^{1,*}$, Ronald Dickman$^{2,\dagger}$ and Ronaldo R. Vidigal$^{2,\ddagger}$' title: Series expansion for a stochastic sandpile --- 1em PACS: 05.70.Ln, 02.50.Ga, 05.10.Gg, 05.40.-a [$^*$electronic address: jstilck@if.uff.br]{}\ [$^\dagger$electronic address: dickman@fisica.ufmg.br]{}\ [$^\ddagger$electronic address: rvidigal@dedalus.lcc.ufmg.br]{} Introduction ============ Sandpiles with a strictly conserved particle density (so-called [*fixed-energy sandpiles*]{} or FES [@dvz]), exhibit absorbing-state phase transitions [@marro; @hinrichsen; @bjp00], and have recently attracted much interest. Until now, most quantitative results for FES have been based on simulations [@fes2d; @rossi; @manna1d; @mnrst], an important exception being the solution by Priezzhev [*et al.*]{} [@priezzhev] of a directed, fixed-energy version of the Maslov-Zhang model [@mz], via the Bethe ansatz. It is therefore of great interest to develop theoretical approaches for FES. Series analysis has proved to be one of the most accurate and reliable approaches to critical phenomena, in both equilibrium and nonequilibrium contexts [@baker; @rdjsp; @jsp2; @iwandp]. Series expansion typically functions best in low-dimensional systems (because longer series can be derived), that is, for just those systems in which the renormalization group and expansion about an upper critical dimension $d_c$ are less reliable. In the case of sandpiles, systematic epsilon expansions are as yet unavailable, and the value of $d_c$ in fact remains controversial [@pmb; @socprl; @wijland]. Simulation results suggest novel critical behavior in the one-dimensional FES, although conflicting critical exponent values have been reported [@manna1d; @mnrst; @lubeck; @chate], which may reflect finite-size effects. Series expansions, on the other hand, implicity treat the infinite-size limit, and so provide important information, complementary to that afforded by simulations. In light of these observations, we believe it highly desirable to apply series methods to sandpile models. This paper is one of a series analyzing a stochastic sandpile using operator methods. In an earlier work [@manft] a path-integral representation was developed and an expansion derived for the order parameter (activity density) in powers of time. While the path-integral formalism reveals interesting features of the model, and may be applied in any number of dimensions, the complexity of the diagrammatic expansion limits the number of terms that can be obtained. (In Ref. [@manft] terms up to ${\cal O}(t^5)$ are reported.) In this paper we employ a different approach, which permits us to extend the series for the one-dimensional case considerably. After casting the master equation for the sandpile in terms of an operator formalism, we analyze the direct expansion of its (formal) solution, leading to an algorithm for generating the series coefficients. We consider Manna’s stochastic sandpile in its fixed-energy (particle-conserving) version [@manna1d; @manft; @manna; @manna2]. The configuration is specified by the occupation number $n$ at each site; sites with $n \geq 2$ are said to be [*active*]{}, and have a positive rate of [*toppling*]{}. When a site topples, it loses exactly two particles (“grains of sand"), which move randomly and independently to nearest-neighbor (NN) sites. (Any configuration devoid of active sites is [*absorbing*]{}, i.e., no futher evolution of the system is possible once such a configuration is reached.) In this work, as in Ref. [@manft], we adopt a toppling rate of $n(n\!-\!1)$ at a site having $n$ particles, which leads us to define the order parameter as $\rho = \langle n(n-1) \rangle$. While this choice of rate represents a slight departure from the usual definition (in which all active sites have the same toppling rate), it leads to a much simpler evolution operator, and should yield the same scaling properties [@manft]. Preliminary simulation results [@rdunp] indicate that in one dimension the model exhibits a continuous phase transition at $p_c \!=\! 0.9493$. In the following section we define the model and review the operator formalism introduced in Ref. [@manft]. This is followed in Sec. III by an analysis leading to an expansion in terms of so-called reduced commutators. Implementation of the expansion in a computational algorithm is described in Sec. IV. Then in Sec. V we report numerical results of the series analysis. A summary and discussion is provided in Sec. VI. Model ===== As discussed in Ref. [@manft], the master equation for this model may be written in the form $$\frac{d|\Psi\rangle}{dt} = L |\Psi\rangle \;, \label{me}$$ where $$|\Psi \rangle = \sum_{\{n_i\}} p(\{n_i\},t) |\{n_i\} \rangle,$$ is the probability distribution, and the evolution operator takes the form, $$L = \sum_i \left[ \frac{1}{4} (\pi_{i-1} + \pi_{i+1})^2 - \pi_i^2 \right] a_i^2 \equiv \sum_i L_i. \label{evop}$$ Here $a_i$ and $\pi_i$ are, respectively, annihilation and creation operators associated with site $i$, defined via $$a_i |n_i\rangle = n_i |n_i\!-\!1\rangle$$ and $$\pi_i |n_i\rangle = |n_i\!+\!1\rangle .$$ The formal solution of the master equation is $|\Psi (t)\rangle = e^{tL}|\Psi (0)\rangle $; that for the activity density is: $$\rho(t) = \langle \;|a_0^2 e^{tL} |\Psi (0)\rangle. \label{act}$$ Here we have introduced the notation: $$\langle \;| \equiv \sum_{\{n\}} \langle \{n\} |$$ for the projection onto all possible states; thus normalization reads: $\langle \;| \Psi \rangle = 1$. We consider a uniform Poisson-product initial distribution. Letting $p_n = e^{-p} p^n/n!$, and using $|P \rangle_i = \sum_{n_i} p_{n_i} |n \rangle_i$ to denote a Poisson distribution at site $i$, we have, $$|\Psi (0)\rangle = \prod_j |P \rangle _j \;. \label{psi0}$$ We shall expand equation (\[act\]) for the activity density in powers of $t$. Operator Algebra ================ To begin we note some basic properties of operators $a_j$, $\pi_j$ and $L_j$: $$a_j^n |P\rangle_j = p^n |P\rangle_j \label{aP}$$ $$\langle \;| \pi_j = \langle \;| \label{normpi}$$ $$\langle \;| L_i = 0 \label{Lcons}$$ $$[a_i,\pi_j] = \delta_{i,j} \label{comm}$$ $$[L_i,L_j] = [a_i,L_j] = 0 \;\;\; \mbox{for} \; |i-j| > 1 \;. \label{comm0}$$ The second relation expresses the fact that the creation operator conserves the normalization of any state, while the third shows that $L_i$ conserves probability, as it must. The coefficient of $t^n/n!$ in the expansion of the activity is: $$\sum_{\cal S} \langle \;|a_0^2 L_{s_1} L_{s_2}...L_{s_n} |P\rangle \;, \label{coefn}$$ where the sum is over all sequences ${\cal S}$ of sites $s_0 \equiv 0,s_1,,,s_n$ with $|s_1| \leq 1$, and $s_{j+1} \in \{s_{j,min}-1,...,s_{j,max}+1\}$, for $j \geq 1$, where $s_{j,min} = \min \{s_0,...s_j\}$, and $s_{j,max}$ is the maximum of this set. The restriction on sequences follows from equations (\[Lcons\]) and (\[comm0\]); if the condition were violated, it would be possible to move one of the $L_j$ to the left of all other operators, yielding a result of zero. Our strategy for evaluating $\rho_{\cal S}$ is to commute each $L_j$ to the left of $a_0^2$. The first step replaces $a_0^2 L_{s_1}$ by its commutator, due to equation (\[Lcons\]). If we write this commutator in [*normal order*]{}, that is, with all creation operators $\pi_j$ to the left of all annihilation operators, then the $\pi$’s may be replaced by 1, by equation (\[normpi\]). Thus, $$\langle \;| a_0^2 L_j = \langle \;| [a_0^2,L_j]_R \;, \label{troca}$$ where the subscript $R$ denotes a [*reduced commutator*]{}, that is, the commutator in normal order, with all $\pi$’s replaced by unity. Evidently $[a_0^2,L_j]_R$ involves only annihilation operators. The two nontrivial expressions of this kind are: $$[a_0^2,L_0]_R = -2a_0^2 - 4a_0^3 \;, \label{cr00}$$ and $$[a_0^2,L_1]_R = \frac{1}{2}a_1^2 + 2 a_0 a_1^2 \;. \label{cr01}$$ In the computational algorithm, discussed in some detail below, it is not necessary to generate the tree structure of sequences explicitly, since each monomial is processed separately and both translation and reflection symmetries may be used in the calculations of the contributions $\rho_{\cal S}$. Evaluating the expectation of each term in $\rho_{\cal S}$ is trivial, because $$\langle \;| a_{s_1}^{m_1}...a_{s_n}^{m_n}|P\rangle = p^M, \label{exppa}$$ where $M = \sum_j m_j$ is the number of annihilation operators, irrespective of which sites are involved. It remains to find a general expression for the reduced commutator $[F^{(j)},L_k]_R$. Since $F^{(j)}$ is a linear combination of products of annihilation operators, and recalling that $a_i$ and $L_k$ commute if $|i \!-\! k| > 1$, we see that the problem reduces to evaluating $$C(p,q,r) \equiv [a_{-1}^p a_0^q a_1^r, L_0]_R \;.$$ (Commutators involving $L_j$ with $j \!\neq \! 0$ are obtained using translation invariance.) It is straightforward to evaluate $C(p,q,r)$ using the following identities. First we note that $$[a_j^p, \pi_j] = pa_j^{p-1} \;, \label{id1}$$ as is readily shown by induction. Using this it is simple to show $$[a_j^p, \pi_j^2]_R = p(p\!-\!1) a_j^{p-2} +2pa_j^{p-1} \;, \label{id2}$$ and $$[a_j^p, \pi_j^2 a_j^2]_R = p(p\!-\!1) a_j^p +2pa_j^{p+1} \;. \label{id3}$$ Finally, we may use equation (\[id1\]) to show that for $i \! \neq \! j$, $$[a_i^p a_j^r, \pi_i \pi_j]_R = pa_i^{p-1} a_j^r + r a_i^p a_j^{r-1} + pr a_i^{p-1} a_j^{r-1} \;. \label{id4}$$ Applying these relations one readily finds: $$\begin{aligned} \nonumber C(p,q,r) &=& a_0^{q+2} \left[ \frac{1}{4}p(p\!-\!1)a_{-1}^{p-2} a_1^r + \frac{1}{4}r(r\!-\!1)a_{-1}^p a_1^{r-2} \right. \\ \nonumber &+& \left. \frac{1}{2}pr a_{-1}^{p-1} a_1^{r-1} + p a_{-1}^{p-1} a_1^r + r a_{-1}^p a_1^{r-1} \right] \\ &-& q a_{-1}^p a_1^r \left[ 2 a_0^{q+1} + (q\!-\!1) a_0^q \right] \;. \label{cpqr}\end{aligned}$$ Using this result, we can evaluate the reduced commutators in a computer algorithm. Computational Algorithm ======================= Let us discuss some details of the computer algorithm used to generate the series for the activity. We employ a recursive procedure to generate the contributions of order $n+1$ on the basis of those of order $n$. From equation (\[act\]) and with a Poisson-product initial distribution defined in equation (\[psi0\]) we notice that $$\frac{d \rho}{d t}=\langle \, |a_0^2 \, L \, e^{tL}|P\rangle= \langle \, | [a_0^2,L]_R \, e^{tL}|P\rangle=-\rho+\langle \, |(4a_0a_1^2-4a_0^3) \, e^{tL}|P\rangle. \label{drho}$$ The last equality above may be understood using the reduced commutators (\[cr00\]) and (\[cr01\]). Using reflection symmetry, we have $\langle \, | [a_0^2,L]_R \, e^{tL}|P\rangle = \langle \, | \{[a_0^2,L_0]_R+2[a_0^2,L_1]_R \} e^{tL}|P\rangle$ and further simplification is provided by translation symmetry. The coefficient of $t^n/n!$ in the expansion of the activity may be identified with $C_n$, where $$C_n=\left. \frac{d^n \rho}{d t^n} \right|_{t=0}. \label{cdef}$$ Using the procedure described above we obtain a recursion relation $$C_{n+1}=-C_n+\langle |F_{n+1}|P\rangle, \label{rr}$$ where $F_{n+1}=[F_n,L]$ and $F_1=4a_0a_1^2-4a_0^3$. To complete the algorithm, we have that $C_0(0)=\rho(0)=\langle \, |a_0^2| \rangle=p^2$, where relation (\[exppa\]) is used. An immediate consequence of these recursion relations is that the coefficient of $p^2$ in the term of order $n$ in $t$ is given by $(-1)^n$, since each monomial in the functions $F$ has at least three annihilation operators[@note1]. This was already shown in Ref. [@manft]. The calculations were done in two steps. Initially, the functions $F$ were calculated up to order 12. Each monomial was represented by three integer variables: an eight-byte integer for the numerator of the coefficient, a four-byte integer for the denominator (which is always a power of 2), and another eight-byte integer to store the number of factors of each annihilation operator. Since all calculations are done in integer arithmetic, there are no roundoff errors. Using translation invariance, each monomial was put in a form such that the the annihilation operator of lowest spatial index is $a_0$. The power $m_i$ associated with each annihilation operator $a_{s_i}$ (as in equation (\[exppa\])) is stored in four bits of the eight-byte integer variable mentioned above. As each new monomial is generated, a search is performed for any existing term with the same set of powers; storing all powers in a single integer facilitates the search. As a consequence of equation (\[comm0\]), when the reduced commutator of a monomial with $L$ is calculated, nonzero contributions may arise only from the commutator of the monomial with $L_{-1}$, $L_0$, $L_1, \ldots, L_{i+1}$, where $i$ is the largest index in the monomial. As is clear from equation (\[cpqr\]), each of these commutators can give rise to up to seven new monomials. Thus, it is apparent that the number of monomials grows very rapidly as the order is increased; the function $F_{12}$ involves $519 \; 115$ monomials. To go beyond order 12 it is necessary to handle integers larger than can be represented using eight bytes and to process monomials with more than 16 exponents, which can no longer be stored in a single 8-bite integer variable. In fact, at order 12 most of the processing time is used in the search procedure. Therefore, our results from order 13 to 16 were obtained processing the monomials in $F_{12}$ one-by-one, generating all contributions from it at orders 13-16. In these calculations the numerators were represented by two eight-byte integers. The limiting order (16) is determined by the large number of new monomials generated; a single monomial in $F_{15}$ may generate on the order of 40 monomials in $F_{16}$. The results presented here required about 170 hours of cpu time on an Athlon K7 1800 MHz computer. It is convenient to write the expansion in the form: $$\overline{\rho}(t) \equiv \frac{\rho(t)}{p^2} = \sum_n \frac{(-t)^n}{n!} \sum_{m=0}^{n-1} b_{n,m} p^m \;. \label{bnm}$$ The series coefficients $b_{n,m}$ are listed in Table I. In Ref. [@manft] it was shown that $m \leq n\!-\!1$, with $$b_{n,n-1} = 2^{4n-1} \frac{(2n-1)!!}{(2n+2)!!}. \label{last}$$ The coefficients reported in Table I satisfy this relation at each order, and agree with those derived (for $n \leq 5$) using the path-integral formalism [@manft]. Analysis of Series ================== The coefficients $b_{n,m}/n!$ in the time series, equation (\[bnm\]), grow rapidly with $n$; the rate of growth appears to be faster than exponential. This is evident from an analysis of $$h_{n,m} \equiv \ln \left( \frac{b_{n,m}}{n!} \right) \label{deff}$$ To see if the $h_{n,m}$ follow a systematic trend we analyze these quantities for a given $q \equiv (m-1)/(n-2)$. (For a fixed value of $n$, the $h_{n,m}$ appear to trace out a smooth curve, so that $h_n(q)$ for intermediate values of $q$ can be estimated via interpolation.) As shown in figure 1, $h_n(q)$ appears to grow faster than exponentially with $n$, away from the limits $q=0$ and $q=1$. (Observe that for $m = 0$, $h_{m,n} \to - \infty$ as $n \to \infty$ since $b_{n,0} = 1$, and similarly for $m=n-1$, since equation (\[last\]) implies that $b_{n,n-1}$ grows more slowly than $n!$.) A reasonable description of the dominant growth in the series coefficientes is $h_n(q) \sim n^\alpha(q)$, with the exponent $\alpha$ (see the inset of figure 1) taking its maximum value of about 1.2 for $q \simeq 0.4$. This of course implies faster-than-exponential growth for the coefficients $b_{n,m}/n!$. Next we examine the behavior of the coefficients in the time series for specific values of the particle density $p$. Let $$\overline{\rho}(t) = \sum_n c_n (-t)^n \label{rhot}$$ where $$c_n \equiv \frac{1}{n!} \sum_{m=0}^{n-1} b_{n,m} p^m \;. \label{cn}$$ For $p=1$ (slightly above the critical value of 0.9493), $c_n$ is simply the sum of all coefficients at order $n$, divided by $n!$. The coefficients $c_n$ again grow faster than exponentially, with $\ln c_n \sim n^{1.15} $ for $p=1$ and $\sim n^{1.10}$ for $p=2$. (Given the limited number of coefficients, we cannot make very precise estimates of the exponent. The key point is that the growth appears to be faster than exponential.) These results imply that equation (\[rhot\]) is a [*divergent series*]{} with zero radius of convergence. We turn now to an analysis of the series for $\overline{\rho}(t)$. As is well known, it is often possible to obtain useful results from divergent series by means of a resummation technique. In the present case we construct Padé approximants to the time series or to the series obtained via a transformation of variable [@jsp2; @baker75; @orszag; @guttmann]. We have examined many transformations, for example $$y = \frac{1 - e^{-bt}}{b} \;, \label{exp}$$ $$x = \frac{t}{1+bt} \;, \label{euler}$$ $$z = 1 - \frac{1}{(1+bt)^\gamma} \;, \label{gamma}$$ $$w = 1 - \frac{1}{1+ \ln(1+bt)} \;, \label{ln}$$ and $$v = 1 - \exp \left\{b\left[1-(1+t)^\gamma \right] \right\} \;. \label{ste}$$ Each transformation maps the interval $t \geq 0$ to a finite interval, and can be expanded as a power series in $t$ about $t=0$, with the lowest-order term $\propto t$. Each is readily inverted permitting one to express the time $t$ in powers of the new variable. The slow convergence associated with the power-law or logarithmic forms in the last four expressions is motivated by the numerical finding of slow relaxation in the sandpile model, even far from the critical point [@mannrel]. We analyze the transformed series for $\overline{\rho}$ (or for $\ln \overline{\rho}$) using Padé approximants. The degree of success depends greatly on the range of $p$ under consideration. (Each transformation includes a free parameter $b$, which can be adjusted to optimize the regularity of the result, or to obtain consistency between different approximants. Except where noted, the results do not exhibit much sensitivity to the choice of this parameter.) For small values of $p$, the best results are obtained via Padé approximants to the $t$-series [*without*]{} any transformation of variable. Figure 2 compares series predictions for $p=0.5$ (obtained using the \[6,7\], \[7,8\] and \[7,9\] approximants to the series for $\overline{\rho}(t)$) against the result of a Monte Carlo simulation for a system of 500 sites (for $p=0.5$ finite-size effects are negligible at this system size). The \[7,8\] approximant is reliable for $t \leq 10$. (Various other approximants, such as \[8,8\] and \[7,7\], are ill-behaved and provide reasonable predictions only for quite short times, typically $t \leq 2$.) We have not been able to improve the series prediction for longer times, either by a change of variable or through analysis of $\ln \overline{\rho}(t)$ or its time derivative. Although some improvement could be expected with longer series, it appears unlikely that the asymptotic decay of $\overline{\rho}(t)$ in the subcritical regime will be accessible through analysis of an expansion in powers of time. For larger values of $p$ the transformation defined in equation (\[gamma\]), using $\gamma = 1/2$, is the most useful of those studied. In figure 3 we compare the \[8,8\] approximant (obtained using $b = 0.57$) with simulation data for $p=1$. The situation is markedly better than for $p=1/2$: the series prediction accompanies the simulation result up to around $t=1000$. It must be noted, however, that the good agreement seen here depends on the choice of the transformation parameter $b$. For other values the agreement with simulation is not as good. (A more suitable criterion for choosing $b$ would be by seeking agreement among various approximants [@jsp2]. In the present case this is not possible because the off-diagonal approximants to the $z$ series are ill-behaved, while the \[7,7\] approximant behaves very similarly to the \[8,8\] used here.) Despite the good agreement up to times on the order of 1000, the present series seems incapable of capturing the asymptotic long-time relaxation of $\overline{\rho}(t)$, which is non-monotonic, as shown in figure 3. Remarkably, the reliability of the series improves dramatically at larger values of the particle density $p$. Series and simulation results for $\overline{\rho}(t)$ at $p=2$ are compared in figure 4; the maximum relative error is about 0.1%. (The series prediction is generated as for $p=1$, but using $b=1.5$ in this case.) The good agreement, moreover, persists at long times, motivating a study of $\overline{\rho}_\infty \equiv lim_{t \to \infty} \overline{\rho}(t)$, corresponding to the transformed series with $z=1$ in equation (\[gamma\]). (We again use the \[8,8\] approximant to the $z$-series.) The series prediction for $\overline{\rho}_\infty$ is compared with simulation in figure 5, using parameters $b=0.57$ and $b=5$. Excellent agreement is found for $p \geq 2$, the relative error being $\leq 0.2\%$. The smaller $b$ value yields better results for $p \simeq 1$, whereas slightly better results are obtained for larger $p$, using the larger $b$ value. For $p \geq 2$ we may claim quantitative accuracy for the series prediction. Nearer the critical point, the agreement appears reasonable (at least on the scale of figure 5), but it is clear that the 16-term series cannot be used to study critical properties. For example, the prediction using $b=0.57$ yields a critical value of about 0.906, that is, the extrapolated activity density goes to zero at this $p$ value. The critical value found in simulations is 0.9493. In summary, the present series seems quite reliable in the supercritical regime, both at short and at asymptotically long times, whereas its utility in the critical and subcritical regimes is restricted to rather short times. Just above the critical point, rather good predictions are possible for short and intermediate times, but this depends on a judicious choice of the transformation parameter $b$. Discussion ========== We develop an algebraic method leading to a time series for the activity density of the stochastic sandpile model introduced in [@manft]. Determination of the series coefficients depends on evaluation of certain commutators, an algebraic task readily codified in a computational algorithm. We extend the series for the one-dimensional case to sixteen terms. Analysis of the series yields disappointing results for the subcritical and critical regimes, but very good predictions in the supercritical region, as judged by comparison with Monte Carlo simulation. At first glance this is surprising, since in the subcritical regime the stationary state is inactive and might be regarded as trivial. Relaxation to this inactive state (and to the active state at or near the critical point $p_c$) is however nontrivial, characterized by stretched exponential, power-law, or other slowly-converging forms [@mannrel]. It appears to be very difficult to capture such behavior in the kind of temporal series developed here, which employs a Poissonian initial distribution. The reason is that for smaller values of the particle density ($p < 2$, say), the one-site stationary occupation distribution $P(n)$ is far from Poissonian. As $p$ increases, the second factorial moment $\langle n(n-1) \rangle = \rho $ approaches $p^2$, as expected for a Poisson distribution. Figure 7 shows that the stationary one-site distribution observed in simulations approaches the corresponding Poisson distribution with the same density $p$. (Even for $p=8$ there are significant differences between the distributions; but analysis of the third and fourth factorial moments suggests convergence to a Poisson distribution as $p \to \infty$.) An important open question is whether simply increasing the number of terms would permit one to analyze the small-$p$ regime. The present results suggest that even with 20 or 30 terms this region would remain inaccessible. It appears to be more promising to approach the critical region from above, since for larger particle densities we find good agreement with numerical results. In this context it is interesting that the quality of predictions near the critical point improves greatly on going from 12 to 16 terms. This suggests that further extension of the series, to 20 or more terms, would yield quantitative results for critical properties, through study of relaxational properties at $p_c$, or or of stationary properties (as in figure 5) as $p_c$ is approached from above. It would also be of great interest to develop an expansion for a stationary property such as $\overline{\rho}_\infty$ directly in powers of the particle density $p$, but this appears to be much more difficult than deriving an expansion in powers of the time. We leave such investigations, using modified or extended series, as subjects for future work. [**Acknowledgments**]{} We thank the Referee for a suggestion that allowed us to derive an extended series. This work was supported by CNPq, CAPES, and FAPERJ, Brazil. [99]{} R. Dickman, A. Vespignani and S. Zapperi, Phys. Rev. E [**57**]{}, 5095 (1998). J. Marro and R. 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Bender and S. A. Orszag, [*Advanced Mathematical Methods for Scientists and Engineers*]{} (McGraw-Hill, New York, 1978). A. J. Guttmann, in [*Phase Transitions and Critical Phenomena*]{}, Vol. 13, C. Domb and J. L. Lebowitz, eds. (Academic Press, New York, 1989). ----------------------------------------------------------------------------- $n$ $m$ $b_{n,m}$ ----- ----- ----------------------------------------------------------------- 0 0 1 1 0 1 2 0 1 1 8 3 0 1 1 66 2 80 4 0 1 1 442 2 $2 \; 076$ 3 896 5 0 1 1 $2 \; 842$ 2 $35 \; 396$ 3 $52 \; 240$ 4 $10 \; 752$ 6 0 1 1 $18 \; 118$ 2 $516 \; 880$ 3 $1 \; 737 \; 952$ 4 $1 \; 187 \; 968$ 5 $135 \; 168$ 7 0 1 1 $\frac{\small 922 \; 265}{\small 8}$ 2 $7 \; 040 \; 282$ 3 $45 \; 847 \; 512$ 4 $67 \; 368 \; 480$ 5 $25 \; 614 \; 368$ 6 $1 \; 757 \; 184$ 8 0 1 1 $\frac{\small 5 \; 865 \; 473}{\small 8}$ 2 $\frac{\small 370 \; 752 \; 137}{\small 4}$ 3 $1 \; 078 \; 168 \; 434$ 4 $2 \; 871 \; 388 \; 040$ 5 $2 \; 283 \; 464 \; 832$ 6 $536 \; 472 \; 640$ 7 $23 \; 429 \; 120$ 9 0 1 1 $\frac{\small 74 \; 596 \; 747}{\small 16}$ 2 $\frac{\small 4 \; 797 \; 745 \; 191}{\small 4}$ 3 $23 \; 841 \; 662 \; 132$ 4 $105 \; 679 \; 404 \; 154$ 5 $147 \; 137 \; 780 \; 760$ 6 $71 \; 353 \; 965 \; 088$ 7 $11 \; 072 \; 770 \; 560$ 8 $318 \; 636 \; 032$ 10 0 1 1 $\frac{\small 474 \; 336 \; 627}{\small 16}$ 2 $\frac{\small 123 \; 077 \; 063 \; 429}{\small 8}$ 3 $\frac{\small 1 \; 018 \; 938 \; 641 \; 745}{\small 2}$ 4 $3 \; 584 \; 915 \; 570 \; 625$ 5 $7 \; 999 \; 349 \; 570 \; 432$ 6 $6 \; 656 \; 488 \; 808 \; 368$ 7 $2 \; 121 \; 777 \; 710 \; 528$ 8 $227 \; 436 \; 059 \; 136$ 9 $4 \; 402 \; 970 \; 624$ 11 0 1 1 $\frac{\small 12 \; 064 \; 410 \; 263}{\small 64}$ 2 $\frac{\small 3 \; 142 \; 928 \; 518 \; 289}{\small 16}$ 3 $\frac{\small 85 \; 419 \; 503 \; 179 \; 415}{\small 8}$ 4 $116 \; 020 \; 128 \; 091 \; 449$ 5 $394 \; 806 \; 480 \; 115 \; 048$ 6 $514 \; 548 \; 057 \; 479 \; 072$ 7 $278 \; 154 \; 455 \; 793 \; 952$ 8 $61 \;313 \; 513 \; 593 \; 600$ 9 $4 \; 683 \; 285 \; 856 \; 256$ 10 $61 \; 641 \; 588 \; 736$ 12 0 1 1 $\frac{\small 76 \; 711 \; 895 \; 439}{\small 64}$ 2 $\frac{\small 80 \; 070 \; 040 \; 225 \; 479}{\small 32}$ 3 $\frac{\small 1 \; 770 \; 456 \; 755 \; 814 \; 995}{\small 8}$ 4 $\frac{\small 14 \; 610 \; 068 \; 149 \; 248 \; 089}{\small 4}$ 5 $18 \; 396 \; 700 \; 126 \; 638 \; 476$ 6 $35 \; 650 \; 110 \; 284 \; 461 \; 928$ 7 $29 \; 745 \; 976 \; 515 \; 005 \; 712$ 8 $11 \; 054 \; 665 \; 928 \; 232 \; 448$ 9 $1 \; 747 \; 506 \; 609 \; 502 \; 464$ 10 $97 \; 252 \; 577 \; 107 \; 968$ 11 $872 \; 465 \; 563 \; 648$ 13 0 1 1 $\frac{\small 1 \; 951 \; 093 \; 993 \; 893}{\small 256}$ 2 $\frac{\small 2 \; 037 \; 418 \; 656 \; 354 \; 491}{\small 64}$ 3 $\frac{\small 72 \; 926 \; 486 \; 692 \; 093 \; 419}{\small 16}$ 4 $\frac{\small 905 \; 058 \; 014 \; 398 \; 112 \; 835}{\small 8}$ 5 $\frac{\small 1 \; 655 \; 391 \; 460 \; 208 \; 555 \; 433}{\small 2}$ 6 $2 \; 309 \; 179 \; 626 \; 832 \; 648 \; 726$ 7 $2 \; 816 \; 714 \; 002 \; 502 \; 804 \; 952$ 8 $1 \; 601 \; 275 \; 099 \; 838 \; 022 \; 656$ 9 $426 \; 223 \; 203 \; 786 \; 122 \; 496$ 10 $49 \; 655 \; 626 \; 778 \; 919 \; 936$ 11 $2 \; 046 \; 635 \; 410 \; 882 \; 560$ 12 $12 \; 463 \; 793 \; 766 \; 400$ 14 0 1 1 $\frac{\small 2 \; 613 \; 736 \; 799 \; 297}{\small 64}$ 2 $\frac{\small 11 \; 934 \; 019 \; 637 \; 184 \; 639}{\small 32}$ 3 $\frac{\small 711 \; 799 \; 150 \; 376 \; 749 \; 517}{\small 8}$ 4 $\frac{\small 53 \; 725 \; 847 \; 102 \; 644 \; 113 \; 051}{\small 16}$ 5 $\frac{\small 142 \; 451 \; 880 \; 202 \; 934 \; 178 \; 839}{\small 4}$ 6 $140 \; 971 \; 191 \; 603 \; 315 \; 396 \; 510$ 7 $243 \; 993 \; 717 \; 303 \; 561 \; 711 \; 492$ 8 $201 \; 186 \; 302 \; 850 \; 192 \; 322 \; 944$ 9 $81 \; 746 \; 928 \; 038 \; 823 \; 569 \; 408$ 10 $16 \; 113 \; 035 \; 142 \; 846 \; 829 \; 824$ 11 $ 1 \; 415 \; 730 \; 263 \; 534 \; 155 \; 776$ 12 $ 43 \; 811 \; 063 \; 460 \; 921 \; 344$ 13 $179 \; 478 \; 630 \; 236 \; 160$ 15 0 1 1 $\frac{\small 531 \; 822 \; 407 \; 449 \; 409}{\small 2048}$ 2 $\frac{\small 606 \; 748 \; 047 \; 325 \; 325 \; 193}{\small 128}$ 3 $\frac{\small 116 \; 603 \; 968 \; 592 \; 784 \; 196 \; 927}{\small 64}$ 4 $\frac{\small 819 \; 559 \; 563 \; 865 \; 455 \; 675 \; 379}{\small 8}$ 5 $\frac{\small 12 \; 371 \; 190 \; 775 \; 573 \; 187 \; 034 \; 899}{\small 8}$ 6 $8 \; 499 \; 377 \; 166 \; 784 \; 889 \; 887 \; 638$ 7 $20 \; 286 \; 352 \; 375 \; 993 \; 324 \; 998 \; 496$ 8 $23 \; 298 \; 464 \; 567 \; 721 \; 533 \; 566 \; 328$ 9 $13 \; 579 \; 307 \; 980 \; 015 \; 469 \; 945 \; 184$ 10 $4 \; 068 \; 154 \; 005 \; 082 \; 098 \; 401 \; 408$ 11 $607 \; 110 \; 051 \; 667 \; 479 \; 652 \; 352$ 12 $40 \; 856 \; 912 \; 571 \; 394 \; 580 \; 480$ 13 $957 \; 525 \; 442 \; 027 \; 462 \; 656$ 14 $2 \; 602 \; 440 \; 138 \; 424 \; 320$ 16 0 1 1 $\frac{\small 422 \; 699 \; 161 \; 810 \; 361}{\small 256}$ 2 $\frac{\small 61 \; 687 \; 281 \; 835 \; 997 \; 869 \; 817}{\small 1024}$ 3 $\frac{\small 1 \; 192 \; 545 \; 870 \; 991 \; 479 \; 699 \; 681}{\small 32}$ 4 $\frac{\small 12 \; 465 \; 559 \; 773 \; 385 \; 531 \; 628 \; 949}{\small 4}$ 5 $\frac{\small 133 \; 184 \; 916 \; 649 \; 036 \; 025 \; 384 \; 179}{\small 2}$ 6 $502 \; 866 \; 857 \; 598 \; 243 \; 404 \; 511 \; 546$ 7 $ 1 \; 627 \; 359 \; 034 \; 988 \; 855 \; 536 \; 002 \; 199$ 8 $2 \; 537 \; 319 \; 446 \; 210 \; 036 \; 202 \; 445 \; 148$ 9 $2 \; 040 \; 132 \; 355 \; 769 \; 944 \; 077 \; 918 \; 400$ 10 $872 \; 811 \; 268 \; 389 \; 569 \; 306 \; 302 \; 976$ 11 $198 \; 169 \; 235 \; 678 \; 101 \; 485 \; 620 \; 992$ 12 $22 \; 853 \; 161 \; 358 \; 227 \; 259 \; 040 \; 768$ 13 $1 \; 195 \; 547 \; 596 \; 367 \; 062 \; 589 \; 440$ 14 $21 \; 401 \; 594 \; 847 \; 260 \; 721 \; 152$ 15 $37 \; 965 \; 009 \; 078 \; 190 \; 080$ ----------------------------------------------------------------------------- \[tab1\] [Table I. Series coefficients in the expansion of the activity.]{} FIGURE CAPTIONS Figure 1. Function $h_n(q)$ as defined in text, for $q=0.2$ ($\blacksquare$); $q=0.4$ ($\bullet$); $q=0.6$ ($\circ$); $q=0.8$ ($\Box$). Observe that $h$ grows faster than linearly for $q=0.2$, 0.4. Inset: growth exponent $\alpha (q)$ defined via $h_n(q) \sim n^{\alpha(q)}$. Figure 2. Normalized activity $\overline{\rho} \equiv \rho(t)/p^2$ versus time for $p=1/2$ from simulation and various Padé approximants to the times series as indicated. Figure 3. Normalized activity for $p=1$. Symbols: simulation result; curve: series prediction as described in text. Figure 4. As in figure 3 for but for $p=2$. Figure 5. Main graph: limiting activity $\overline{\rho}_\infty$ versus particle density $p$. Points: simulation; solid curve: series prediction using transformation (\[gamma\]) with $b=0.57$, \[8,8\] Padé approximant; dashed line: same approximant and transformation but using $b=5$. Inset: difference $\Delta = \overline{\rho}_{\infty,series} - \overline{\rho}_{\infty,sim}$ for $b=0.57$ ($\blacksquare$) and $b=5$ ($\Box$). Figure 6. Single-site occupancy distributions $P(n)$ obtained in simulation ($\blacksquare$) compared with the corresponding Poisson distribution ($\Box$). Upper panel: $p=1.2$; middle: $p=3$; lower: $p=8$.
--- abstract: 'Intuitionistic first-order logic extended with a restricted form of Markov’s principle is constructive and admits a Curry-Howard correspondence, as shown by Herbelin. We provide a simpler proof of that result and then we study intuitionistic first-order logic extended with unrestricted Markov’s principle. Starting from classical natural deduction, we restrict the excluded middle and we obtain a natural deduction system and a parallel Curry-Howard isomorphism for the logic. We show that proof terms for existentially quantified formulas reduce to a list of individual terms representing all possible witnesses. As corollary, we derive that the logic is Herbrand constructive: whenever it proves any existential formula, it proves also an Herbrand disjunction for the formula. Finally, using the techniques just introduced, we also provide a new computational interpretation of Arithmetic with Markov’s principle.' author: - 'Federico Aschieri[^1]' - 'Matteo Manighetti[^2]' title: 'On Natural Deduction for Herbrand Constructive Logics II: Curry-Howard Correspondence for Markov’s Principle in First-Order Logic and Arithmetic' --- Introduction ============ Markov’s Principle was introduced by Markov in the context of his theory of Constructive Recursive Mathematics (see [@Troelstra2]). Its original formulation is tied to Arithmetic: it states that given a recursive function $f: \mathbb{N} \rightarrow \mathbb{N}$, if it is impossible that for every natural number $n$, $f(n)\neq 0$, then there exists a $n$ such that $f(n)=0$. Markov’s original argument for justifying it was simply the following: if it is not possible that for all $n$, $f(n)\neq 0$, then by computing in sequence $f(0), f(1), f(2), \ldots$, one will eventually hit a number $n$ such that $f(n)=0$ and will *effectively* recognize it as a witness. Markov’s principle is readily formalized in Heyting Arithmetic as the axiom scheme $$\lnot\lnot \exists \alpha^{{ {\tt N} }} {P}\rightarrow \exists \alpha^{{ {\tt N} }} {P}$$ where ${P}$ is a primitive recursive predicate [@Troelstra]. When added to Heyting Arithmetic, Markov’s principle gives rise to a *constructive* system, that is, one enjoying the disjunction and the existential witness property [@Troelstra] (if a disjunction is derivable, one of the disjoints is derivable too, and if an existential statement is derivable, so it is one instance of it). Furthermore, witnesses for any provable existential formula can be effectively computed using either Markov’s unbounded search and Kleene’s realizability [@Kleene] or much more efficient functional interpretations [@Godeldialectica; @AZMarkov]. Markov’s Principle in First-Order Logic --------------------------------------- The very shape of Markov’s principle makes it also a purely logical principle, namely an instance of the double negation elimination axiom. But in pure logic, what exactly should Markov’s principle correspond to? In particular, what class of formulas should $P$ be restricted to? Since Markov’s principle was originally understood as a constructive principle, it is natural to restrict $P$ as little as possible, while maintaining the logical system as constructive as possible. As proven by Herbelin [@Herbelin], it turns out that asking that $P$ is propositional and with no implication $\rightarrow$ symbols guarantees that intuitionistic logic extended with such a version of Markov’s principle is constructive. The proof of this result employs a Curry-Howard isomorphism based on a mechanism for raising and catching exceptions. As opposed to the aforementioned functional interpretations of Markov’s principle, Herbelin’s calculus is fully isomorphic to an intuitionistic logic: there is a perfect match between reduction steps at the level of programs and detour eliminations at the level of proofs. Moreover, witnesses for provable existential statements are computed by the associated proof terms. Nevertheless, as we shall later show, the mechanism of throwing exceptions plays no role during these computations: intuitionistic reductions are entirely enough for computing witnesses. A question is now naturally raised: as no special mechanism is required for witness computation using Herbelin’s restriction of Markov’s principle, can the first be further relaxed so that the second becomes stronger as well as computationally *and* constructively meaningful? Allowing the propositional matrix $P$ to contain implication destroys the constructivity of the logic. It turns out, however, that *Herbrand constructivity* is preserved. An intermediate logic is called [Herbrand constructive]{} if it enjoys a strong form of Herbrand’s Theorem [@Buss; @AschieriZH]: for *every* provable formula $\exists \alpha\, A$, the logic proves as well an Herbrand disjunction $$A[m_{1}/\alpha]\lor \ldots \lor A[m_{k}/\alpha]$$ So the Markov principle we shall interpret in this paper is $${\mathsf{MP}}: \lnot\lnot \exists \alpha\, {P}\rightarrow \exists \alpha\, {P} \mbox{\qquad($P$ propositional formula)}$$ and show that when added to intuitionistic first-order logic, the resulting system is Herbrand constructive. This is the most general form of Markov’s principle that allows a significant constructive interpretation: we shall show how to non-trivially compute lists of witnesses for provable existential formulas thanks to an exception raising construct and a parallel computation operator. ${\mathsf{MP}}$ can also be used in conjunction with negative translations to compute Herbrand disjunctions in classical logic, something which is not possible with Herbelin’s form of Markov’s principle. Restricted Excluded Middle -------------------------- The Curry-Howard correspondence we present here is by no means an ad hoc construction, only tailored for Markov’s principle. It is a simple restriction of the Curry-Howard correspondence for classical first-order logic introduced in [@AschieriZH], where classical reasoning is formalized by the excluded middle inference rule: It is enough to restrict the conclusion $C$ of this rule to be a simply existential statement and the ${{\mathsf{Q}}}$ in the premises $\forall x \, {{\mathsf{Q}}}, \exists x\, \neg {{\mathsf{Q}}}$ to be propositional. We shall show that the rule is intuitionistically equivalent to ${\mathsf{MP}}$. With our approach, strong normalization is just inherited and the transition from classical logic to intuitionistic logic with ${\mathsf{MP}}$ is smooth and natural. Markov’s Principle in Arithmetic -------------------------------- We shall also provide a computational interpretation of Heyting Arithmetic with ${\mathsf{MP}}$. The system is constructive and witnesses for provable existential statements can be computed. This time, we shall restrict the excluded middle as formalized in $\cite{ABB}$ and we shall directly obtain the desired Curry-Howard correspondence. As a matter of fact, the interpretation of ${\mathsf{MP}}$ in Arithmetic ends up to be a simplification of the methods we use in first-order logic, because the decidability of atomic formulas greatly reduces parallelism and eliminates case distinction on the truth of atomic formulas. Plan of the Paper ----------------- In , we provide a simple computational interpretation of first-order intuitionistic logic extended with Herbelin’s restriction of Markov’s principle. We also show that the full Markov principle ${\mathsf{MP}}$ cannot be proved in that system. In , we provide a Curry-Howard correspondence for intuitionistic logic with ${\mathsf{MP}}$, by restricting the excluded middle, and show that the system is Herbrand constructive. In , we extend the Curry-Howard to Arithmetic with ${\mathsf{MP}}$ and show that the system becomes again constructive. Herbelin’s Restriction of Markov’s Principle {#sec:herb} ============================================ In [@Herbelin] Herbelin introduced a Curry-Howard isomorphism for an extended intuitionistic logic. By employing exception raising operators and new reduction rules, he proved that the logic is constructive and can derive the axiom scheme $${\mathsf{HMP}}: \neg \neg \exists \alpha \, P \to \exists \alpha \, P \text{\qquad($P$ propositional and $\rightarrow$ not occurring in $P$)}$$ Actually, Herbelin allowed $P$ also to contain existential quantifiers, but in that case the axiom scheme is intuitionistically equivalent to $ \neg \neg \exists \alpha_{1}\ldots \exists \alpha_{n} \, P \to \exists\alpha_{1}\ldots \exists \alpha_{n} \, P$, again with $P$ propositional and $\rightarrow$ not occurring in $P$. All of the methods of our paper apply to this case as well, but for avoiding trivial details, we keep the present ${\mathsf{HMP}}$. Our first goal is to show that ${\mathsf{HMP}}$ has a simpler computational interpretation and to provide a straightforward proof that, when added on top of first-order intuitionistic logic, ${\mathsf{HMP}}$ gives rise to a constructive system. In particular, we show that the ordinary Prawitz reduction rules for intuitionistic logic and thus the standard Curry-Howard isomorphism [@Sorensen] are enough for extracting witnesses for provable existential formulas. The crucial insight, as we shall see, is that ${\mathsf{HMP}}$ can never actually appear in the head of a closed proof term having existential type. It thus plays no computational role in computing witnesses; it plays rather a logical role, in that it may be used to prove the correctness of the witnesses. To achieve our goal, we consider the usual natural deduction system for intuitionistic first-order logic [@Prawitz; @Sorensen], to which we add ${\mathsf{HMP}}$. Accordingly, we add to the associated lambda calculus the constants ${\mathcal{M}_{P}}: \neg \neg \exists \alpha \, P \to \exists \alpha \, P$. The resulting Curry-Howard system is called ${\mathsf{IL}}+{\mathsf{HMP}}$ and is presented in \[fig:system\]. The reduction rules for ${\mathsf{IL}}+{\mathsf{HMP}}$ presented in \[fig:red-ilmp\] are just the ordinary ones of lambda calculus. On the other hand, ${\mathcal{M}_{P}}$ has no computational content and thus no associated reduction rule. Of course, the strong normalization of ${\mathsf{IL}}+{\mathsf{HMP}}$ holds by virtue of the result for standard intuitionistic Curry-Howard. The system ${\mathsf{IL}}+{\mathsf{HMP}}$ is strongly normalizing Reduction Rules for ${\mathsf{IL}}$ : $$(\lambda x. u)t\mapsto u[t/x]$$ $$(\lambda \alpha. u)m\mapsto u[m/\alpha]$$ $${{\langle u_0, u_1 \rangle} \pi_{i}}\mapsto u_i, \mbox{ for i=0,1}$$ $${{\upiota_{i}(u)}[x_{1}.t_{1}, x_{2}.t_{2}]}\mapsto t_{i}[u/x_{i}], \mbox{ for i=0,1}$$ $${{(m, u)}[(\alpha,x).v]} \mapsto v[m/\alpha][u/x], \mbox{ for each term $m$ of $\mathcal{L}$}$$ As we shall see in , the reason why ${\mathsf{HMP}}$ cannot be appear in the head of a closed proof term having existential type is that its premise $\lnot\lnot\exists \alpha\, P$ is never classically valid, let alone provable in intuitionistic logic. \[theorem:no-sigma-taut\] Assume that the symbol $\rightarrow$ does not occur in the propositional formula $P$. Then $\lnot\lnot\exists \alpha\, P$ is not classically provable. We provide a semantical argument. $\lnot\lnot\exists \alpha\, P$ is classically provable if and only if it is classically valid and thus if and only if $\exists \alpha\, P$ is classically valid. For every such a formula, we shall exhibit a model falsifying it. Consider the model $\mathfrak{M}$ where every $n$-ary predicate is interpreted as the empty $n$-ary relation. We show by induction on the complexity of the formula $P$ that $P^\mathfrak{M} = \bot$ for every assignment of individuals to the free variables of $P$, and therefore $(\exists \alpha \, P)^\mathfrak{M} = \bot$. - If $P$ is atomic, then by definition of $\mathfrak{M}$, we have $P^\mathfrak{M} = \bot$ for every assignment of the variables. - If $P=P_1 \land P_2$, then since by induction $P_1^\mathfrak{M} = \bot$, $(P_1\land P_2)^\mathfrak{M} = \bot$ - If $P=P_1 \lor P_2$, then since by induction $P_1^\mathfrak{M} = \bot$ and $P_2^\mathfrak{M} = \bot$, $(P_1\lor P_2)^\mathfrak{M} = \bot$ In order to derive constructivity of ${\mathsf{IL}}+{\mathsf{HMP}}$, we shall just have to inspect the normal forms of proof terms. Our main argument, in particular, will use the following well-known syntactic characterization of the shape of proof terms. \[theorem:head-form\] Every proof-term of ${\mathsf{IL}}+{\mathsf{HMP}}$ is of the form $$\lambda z_1 \dots \lambda z_n.\, r u_1 \dots u_k$$ where - $r$ is either a variable or a constant or a term corresponding to an introduction rule: $\lambda x . t$, $\lambda \alpha . t$, ${\langle t_1, t_2 \rangle}$, ${\upiota_{i}(t)}$, ${(m, t)}$ - $u_1, \dots u_k$ are either proof terms, first order terms, or one of the following expressions corresponding to elimination rules: ${ \pi_{i}}$, ${[x.w_1, y.w_2]}$, ${[(\alpha,x).t]}$. Standard. We are now able to prove that ${\mathsf{IL}}+{\mathsf{HMP}}$ is constructive. \[thm:construct-il-mp\] 1. If ${\mathsf{IL}}+{\mathsf{HMP}}\vdash t: \exists \alpha \, A$, and $t$ is in normal form, then $t={(m, u)}$ and ${\mathsf{IL}}+{\mathsf{HMP}}\vdash u: A[m/\alpha]$. 2. If ${\mathsf{IL}}+{\mathsf{HMP}}\vdash t: A \lor B$ and $t$ is in normal form, then either $t={\upiota_{0}(u)}$ and ${\mathsf{IL}}+{\mathsf{HMP}}\vdash u: A$ or $t={\upiota_{1}(u)}$ and ${\mathsf{IL}}+{\mathsf{HMP}}\vdash u: B$. <!-- --> 1. By , $t$ must be of the form $r u_1\dots u_k$. Let us consider the possible forms of $r$. - Since $t$ is closed, $r$ cannot be a variable. - We show that $r$ cannot be ${\mathcal{M}_{P}}$. If $r$ were ${\mathcal{M}_{P}} : \neg \neg \exists x \, P \to \exists \alpha \, P$ for some $P$, then ${\mathsf{IL}}+\mathsf{MP} \vdash u_1 : \neg \neg \exists \alpha \, P$. Since ${\mathsf{IL}}+{\mathsf{HMP}}$ is contained in classical logic, we have that $\neg \neg \exists \alpha \, P$ is classically provable. However we know from that this cannot be the case, which is a contradiction. - We also show that $r$ cannot be ${{\mathtt{H}^{{{\mathsf{\bot\rightarrow P}}}}}}$. Indeed, if $r$ were ${{\mathtt{H}^{{{\mathsf{\bot\rightarrow P}}}}}}$ for some $P$, then ${\mathsf{IL}}+\mathsf{MP} \vdash u_1 : \bot$, which is a contradiction. - The only possibility is thus that $r$ is one among $\lambda x . t$, $\lambda \alpha . t$, ${\langle t_1, t_2 \rangle}$, ${\upiota_{i}(t)}$, ${(m, t)}$. In this case, $k$ must be 0 as otherwise we would have a redex. This means that $t=r$ and thus $t={(m, u)}$ with ${\mathsf{IL}}+{\mathsf{HMP}}\vdash u : A(m)$. 2. The proof goes along the same lines of case 1. Finally, we prove that ${\mathsf{IL}}+{\mathsf{HMP}}$ is not powerful enough to express full Markov’s principle ${\mathsf{MP}}$. Intuitively, the reason is that ${\mathsf{IL}}+{\mathsf{HMP}}$ is a constructive system and thus cannot be strong enough to interpret classical reasoning. This would indeed be the case if ${\mathsf{IL}}+{\mathsf{HMP}}$ proved ${\mathsf{MP}}$, an axiom which complements very well negative translations. ${\mathsf{IL}}+{\mathsf{HMP}}\nvdash {\mathsf{MP}}$. Suppose for the sake of contradiction that ${\mathsf{IL}}+{\mathsf{HMP}}\vdash {\mathsf{MP}}$. Consider any proof in classical first-order logic of a simply existential statement $\exists \alpha\,{{\mathsf{P}}}$. By the Gödel-Gentzen negative translation (see [@Troelstra]), we can then obtain an intuitionistic proof of $\neg \neg \exists \alpha \, {{\mathsf{P}}}^{N}$, where ${{\mathsf{P}}}^{N}$ is the negative translation of ${{\mathsf{P}}}$, and thus ${\mathsf{IL}}+{\mathsf{HMP}}\vdash\exists \alpha\, {{\mathsf{P}}}^{N}$. By , there is a first-order term $m$ such that ${\mathsf{IL}}+{\mathsf{HMP}}\vdash {{\mathsf{P}}}^{N}[m/\alpha]$. Since ${{\mathsf{P}}}^{N}[m/\alpha]$ is classically equivalent to ${{\mathsf{P}}}[m/\alpha]$, we would have a single witness for every classically valid simply existential statement. But this is not possible: consider for example the first-order language $\mathcal{L}=\{P,a,b\}$ and the formula $F = (P(a) \lor P(b)) \to P(\alpha)$ where $P$ is an atomic predicate. Then the formula $\exists \alpha \, F$ is classically provable, but there is no term $m$ such that $F[m/\alpha]$ is valid, let alone provable: - it cannot be $m=a$, as it is shown by picking a model where $P$ is interpreted as the set $\{a\}$ - it cannot be $m=b$, because we can interpret $P$ as the set $\{b\}$. Full Markov Principle and Restricted Excluded Middle in First-Order Logic {#sec:ilemeno} ========================================================================= In this section we describe the natural deduction system and Curry-Howard correspondence ${\mathsf{IL}}+{\mathsf{EM}}_{1}^{-}$, which arise by restricting the excluded-middle in classical natural deduction [@AschieriZH]. This computational system is based on delimited exceptions and a parallel operator. We will show that on one hand full Markov principle ${\mathsf{MP}}$ is provable in ${\mathsf{IL}}+{\mathsf{EM}}_{1}^{-}$ and, on the other hand, that ${\mathsf{IL}}+{\mathsf{MP}}$ derives all of the restricted classical reasoning that can be expressed in ${\mathsf{IL}}+{\mathsf{EM}}_{1}^{-}$, so that the two systems are actually equivalent. Finally, we show that the system ${\mathsf{IL}}+{\mathsf{EM}}_{1}^{-}$ is Herbrand constructive and that witnesses can effectively be computed.\ All of the classical reasoning in ${\mathsf{IL}}+{\mathsf{EM}}_{1}^{-}$ is formally restricted to negative formulas. We denote propositional formulas as ${{\mathsf{P_1}}},\dots {{\mathsf{P_n}}},{{\mathsf{Q}}},{{\mathsf{R}}},\dots$. We say that a propositional formula is *negative* whenever $\lor$ does not occur in it. Formulas of the form $\forall \alpha_{1}\ldots \forall \alpha_{n}\, {{\mathsf{P}}}$, with ${{\mathsf{P}}}$ negative, will be called *simply universal*. In order to computationally interpret Markov’s principle, we consider the rule ${\mathsf{EM}}_{1}^{-}$, which is obtained by restricting the conclusion of the excluded middle ${\mathsf{EM}}_{1}$ [@AschieriZH; @ABB] to be a simply existential formula, where both ${{\mathsf{P}}}$ and ${{\mathsf{Q}}}$ are negative formulas. This inference rule is complemented by the axioms: $$\Gamma, a:{\forall \alpha {{\mathsf{P}}}}\vdash {{\mathtt{H}_{a}^{\forall {\alpha} {{{\mathsf{P}}}}}}}: \forall\alpha {{\mathsf{P}}}$$ $$\Gamma, a:{\exists \alpha \lnot {{\mathsf{P}}}}\vdash {\mathtt{W}_{a}^{\exists {\alpha} {\neg {{\mathsf{P}}}} }}: \exists\alpha \lnot {{\mathsf{P}}}$$ These last two rules correspond respectively to a term making an *Hypothesis* and a term waiting for a *Witness* and these terms are put in communication via ${\mathsf{EM}_{1}^-}$. A term of the form ${{\mathtt{H}_{a}^{\forall {\alpha} {{{\mathsf{P}}}}}}} m$, with $m$ first-order term, is said to be *active*, if its only free variable is $a$: it represents a raise operator which has been turned on. The term ${{ u \parallel_{a} v}}$ supports an exception mechanism: $u$ is the ordinary computation, $v$ is the exceptional one and $a$ is the communication channel. Raising exceptions is the task of the term ${{\mathtt{H}_{a}^{\forall {\alpha} {{{\mathsf{P}}}}}}}$, when it encounters a counterexample $m$ to $\forall\alpha\, {{\mathsf{P}}}$; catching exceptions is performed by the term ${\mathtt{W}_{a}^{\exists {\alpha} {\neg {{\mathsf{P}}}} }}$. In first-order logic, however, there is an issue: when should an exception be thrown? Since the truth of atomic predicates depends on models, one cannot know. Therefore, each time ${{\mathtt{H}_{a}^{\forall {\alpha} {{{\mathsf{P}}}}}}}$ is applied to a term $m$, a new pair of parallel independent computational paths is created, according as to whether ${{\mathsf{P}}}[m/\alpha]$ is false or true. In one path the exception is thrown, in the other not, and the two computations will never join again. To render this computational behaviour, we add the rule ${\mathsf{EM}_{0}}$ of propositional excluded middle over negative formulas even if in principle it is derivable from ${\mathsf{EM}}_{1}^{-}$; we also add the axiom $$\Gamma, a:{{\mathsf{P}}} \vdash {{\mathtt{H}^{{{\mathsf{P}}}}}}: {{\mathsf{P}}}$$ We call the resulting system ${\mathsf{IL}}+{\mathsf{EM}}_1^-$ (\[fig:system-ilem\]) and present its reduction rules in \[fig:red\]; they just form a restriction of the system ${\mathsf{IL}}+{\mathsf{EM}}$ described in [@AschieriZH]. The reduction rules are in \[fig:red\] and are based on the following definition, which formalizes the raise and catch mechanism. \[def:witsub\] \[definition-witsub\] Suppose $v$ is any proof term and $m$ is a term of $\mathcal{L}$. Then: 1. If every free occurrence of $a$ in $v$ is of the form ${\mathtt{W}_{a}^{\exists {\alpha} { {{\mathsf{P}}}} }}$, we define $$v[a:=m]$$ as the term obtained from $v$ by replacing each subterm ${\mathtt{W}_{a}^{\exists {\alpha} {{{\mathsf{P}}}} }}$ corresponding to a free occurrence of $a$ in $v$ by $(m, {{\mathtt{H}^{{{\mathsf{P}}}[m/\alpha]}}})$. 2. If every free occurrence of $a$ in $v$ is of the form ${{\mathtt{H}_{a}^{\forall {\alpha} {{{\mathsf{P}}}}}}}$, we define $$v[a:=m]$$ as the term obtained from $v$ by replacing each subterm ${{\mathtt{H}_{a}^{\forall {\alpha} {{{\mathsf{P}}}}}}}m$ corresponding to a free occurrence of $a$ in $v$ by ${{\mathtt{H}^{\mathsf{P}[m/\alpha]}}}$. Reduction Rules for ${\mathsf{IL}}$ : $$(\lambda x. u)t\mapsto u[t/x]\qquad (\lambda \alpha. u)m\mapsto u[m/\alpha]$$ $${{\langle u_0, u_1 \rangle} \pi_{i}}\mapsto u_i, \mbox{ for i=0,1}$$ $${{\upiota_{i}(u)}[x_{1}.t_{1}, x_{2}.t_{2}]}\mapsto t_{i}[u/x_{i}], \mbox{ for i=0,1}$$ $${{(m, u)}[(\alpha,x).v]} \mapsto v[m/\alpha][u/x], \mbox{ for each term $m$ of $\mathcal{L}$}$$ Permutation Rules for ${\mathsf{EM}_{0}}$ : $$({{ u\, |_{}\, v}}) w \mapsto {{ uw\, |_{}\, vw}}$$ $${({{ u\, |_{}\, v}}) \pi_{i}} \mapsto {{ {u \pi_{i}}\, |_{}\, {v \pi_{i}}}}$$ $${({{ u\, |_{}\, v}})[x.w_{1}, y.w_{2}]} \mapsto {{ {u[x.w_{1}, y]}{w_{2}}\, |_{}\, {v[x.w_{1}, y]}{w_{2}}}}$$ $${({{ u\, |_{}\, v}})[(\alpha,x).w]} \mapsto {{ {u[(\alpha,x).w]}\, |_{}\, {v[(\alpha,x).w]}}}$$ Reduction Rules for ${\mathsf{EM}_{1}^-}$ : $${{ u \parallel_{a} v}}\mapsto u,\, \mbox{ if $a$ does not occur free in $u$ }$$ $${{ u \parallel_{a} v}}\mapsto {{ v [a:=m]\, |_{}\, ({{ u [a:=m] \parallel_{a} v}})}},\mbox{ whenever $u$ has some \emph{active} subterm ${{\mathtt{H}_{a}^{\forall {\alpha} {{{\mathsf{P}}}}}}} m$}$$ As we anticipated, our system is capable of proving the full Markov Principle ${\mathsf{MP}}$ and thus its particular case ${\mathsf{HMP}}$. ${\mathsf{IL}}+ {\mathsf{EM}}_1^- \vdash {\mathsf{MP}}$ First note that with the use of ${\mathsf{EM}_{0}}$, we obtain that ${\mathsf{IL}}+{\mathsf{EM}}_{1}^{-}\vdash P \lor \neg P$ for any atomic formula $P$. Therefore ${\mathsf{IL}}+{\mathsf{EM}}_{1}^{-}$ can prove any propositional tautology, and in particular ${\mathsf{IL}}+{\mathsf{EM}}_{1}^{-}\vdash{{\mathsf{P}}} \lor {{\mathsf{Q}}} \leftrightarrow \neg (\neg {{\mathsf{P}}} \land \neg {{\mathsf{Q}}})$ for any propositional formulas ${{\mathsf{P}}}, {{\mathsf{Q}}}$, thus proving that each propositional formula is equivalent to a negative one. Consider now any instance $\neg \neg \exists \alpha\, {{\mathsf{Q}}} \to \exists \alpha\, {{\mathsf{Q}}}$ of ${\mathsf{MP}}$. Thanks to the previous observation, we obtain $${\mathsf{IL}}+{\mathsf{EM}}_{1}^{-}\vdash \big(\neg \neg \exists \alpha\, {{\mathsf{Q}}} \to \exists \alpha\, {{\mathsf{Q}}}\big) \leftrightarrow \big(\neg \neg \exists \alpha\, {{\mathsf{P}}} \to \exists \alpha\, {{\mathsf{P}}}\big)$$ for some negative formula ${{\mathsf{P}}}$ logically equivalent to ${{\mathsf{Q}}}$. The following formal proof shows that ${\mathsf{IL}}+{\mathsf{EM}_{1}^-} \vdash\neg \neg \exists \alpha\, {{\mathsf{P}}} \to \exists \alpha\, {{\mathsf{P}}}$. Finally, this implies ${\mathsf{IL}}+{\mathsf{EM}_{1}^-} \vdash \neg \neg \exists \alpha\, {{\mathsf{Q}}} \to \exists \alpha\, {{\mathsf{Q}}}$. Conversely, everything which is provable within our system can be proven by means of first-order logic with full Markov principle. If ${\mathsf{IL}}+{\mathsf{EM}_{1}^-} \vdash F$, then ${\mathsf{IL}}+ {\mathsf{MP}}\vdash F$. We just need to show that ${\mathsf{IL}}+ {\mathsf{MP}}$ can prove the rules ${\mathsf{EM}_{1}^-}$ and ${\mathsf{EM}_{0}}$. For the case of ${\mathsf{EM}_{0}}$, note that ${\mathsf{IL}}+{\mathsf{MP}}\vdash\neg \neg {{\mathsf{P}}} \to {{\mathsf{P}}}$ for all propositional formulas ${{\mathsf{P}}}$, thanks to ${\mathsf{MP}}$. Since for every propositional ${{\mathsf{Q}}}$ we have ${\mathsf{IL}}+{\mathsf{MP}}\vdash \lnot\lnot ({{\mathsf{Q}}}\lor \lnot {{\mathsf{Q}}})$, we obtain ${\mathsf{IL}}+{\mathsf{MP}}\vdash{{\mathsf{Q}}} \lor \neg {{\mathsf{Q}}}$, and therefore ${\mathsf{IL}}+{\mathsf{MP}}$ can prove ${\mathsf{EM}_{0}}$ by mean of an ordinary disjunction elimination. In the case of ${\mathsf{EM}_{1}^-}$, if we are given the proofs of and in ${\mathsf{IL}}+{\mathsf{MP}}$, the following derivation shows a proof of $\exists \alpha\, {{\mathsf{C}}}$ in ${\mathsf{IL}}+{\mathsf{MP}}$. As in [@AschieriZH], all of our main results about witness extraction are valid not only for closed terms, but also for quasi-closed ones, which are those containing only pure universal assumptions. \[def:quasi\] An untyped proof term $t$ is said to be *quasi-closed*, if it contains as free variables only hypothesis variables $a_{1}, \ldots, a_{n}$, such that each occurrence of them is of the form ${{\mathtt{H}_{a_{i}}^{\forall {\vec{\alpha}} {{{\mathsf{P}}}_i}}}}$, where $\forall \vec{\alpha}\, {{\mathsf{P}}}_{i}$ is simply universal. ${\mathsf{IL}}+{\mathsf{EM}_{1}^-}$ with the reduction rules in figure \[fig:red\] enjoys the Subject Reduction Theorem, as a particular case of the Subject Reduction for ${\mathsf{IL}}+{\mathsf{EM}}$ presented in [@AschieriZH]. \[subjectred\] If $\Gamma \vdash t : C$ and $t \mapsto u$, then $\Gamma \vdash u : C$. No term of ${\mathsf{IL}}+{\mathsf{EM}_{1}^-}$ gives rise to an infinite reduction sequence [@AschieriZH]. Every term typable in ${\mathsf{IL}}+{\mathsf{EM}_{1}^-}$ is strongly normalizing. We now update the characterization of proof-terms heads given in to the case of ${\mathsf{IL}}+{\mathsf{EM}_{1}^-}$. \[theorem:head-form-em\] Every proof term of ${\mathsf{IL}}+{\mathsf{EM}_{1}^-}$ is of the form: $$\lambda z_1 \dots \lambda z_n . r u_1 \dots u_k$$ where - $r$ is either a variable $x$, a constant ${{\mathtt{H}^{P}}}$ or ${{\mathtt{H}_{a}^{\forall {\alpha} {A}}}}$ or ${\mathtt{W}_{a}^{\exists {\alpha} {{{\mathsf{P}}}} }}$ or an excluded middle term ${{ u \parallel_{a} v}}$ or ${{ u\, |_{}\, v}}$, or a term corresponding to an introduction rule $\lambda x . t$, $\lambda \alpha . t$, ${\langle t_1, t_2 \rangle}$, ${\upiota_{i}(t)}$, ${(m, t)}$ - $u_1, \dots u_k$ are either lambda terms, first order terms, or one of the following expressions corresponding to elimination rules: ${ \pi_{i}}$, ${[x.w_1, y.w_2]}$, ${[(\alpha,x).t]}$ Standard. We now study the shape of the normal terms with the most simple types. \[prop:pnf\] Let ${{\mathsf{P}}},{{\mathsf{P}}}_1,\dots {{\mathsf{P}}}_n$ be negative propositional formulas, $A_1, \dots, A_m$ simply universal formulas. Suppose that $$\Gamma = z_1: {{\mathsf{P}}}_1, \dots z_n: {{\mathsf{P}}}_n, a_1 : \forall \alpha_1 A_1, \dots a_m : \forall \alpha_m A_m$$ and $\Gamma \vdash t:\exists {\alpha}\, {{\mathsf{P}}}$ or $\Gamma \vdash t: {{\mathsf{P}}}$, with $t$ in normal form and having all its free variables among $z_1, \dots z_n, a_1, \dots a_m $. Then: 1. Every occurrence in $t$ of every term ${{\mathtt{H}_{a_{i}}^{\forall {\alpha_{i}} {A_i}}}}$ is of the active form ${{\mathtt{H}_{a_{i}}^{\forall {\alpha_{i}} {A_i}}}}m$, where $m$ is a term of $\mathcal{L}$. 2. $t$ cannot be of the form $u\parallel_{a} v$. We prove 1. and 2. simultaneously and by induction on $t$. There are several cases, according to the shape of $t$:\ - $t=(m, u)$, $\Gamma\vdash t:\exists {\alpha}\, {{\mathsf{P}}}$ and $\Gamma \vdash u: {{\mathsf{P}}}[m/\alpha]$. We immediately get 1. by induction hypothesis applied to $u$, while 2. is obviously verified. - $t=\lambda x\, u$, $\Gamma \vdash t: {{\mathsf{P}}}={{\mathsf{Q}}}\rightarrow {{\mathsf{R}}}$ and $\Gamma, x: {{\mathsf{Q}}} \vdash u: {{\mathsf{R}}}$. We immediately get 1. by induction hypothesis applied to $u$, while 2. is obviously verified. - $t=\langle u, v\rangle$, $\Gamma \vdash t: {{\mathsf{P}}}={{\mathsf{Q}}}\land {{\mathsf{R}}}$, $\Gamma \vdash u: {{\mathsf{Q}}}$ and $\Gamma \vdash v: {{\mathsf{R}}}$. We immediately get 1. by induction hypothesis applied to $u$, while 2. is obviously verified. - $t={{ u\, |_{}\, v}}$, $\Gamma, a: \neg {{\mathsf{Q}}} \vdash u: \exists {\alpha}\, {{\mathsf{P}}}$ (resp. $u: {{\mathsf{P}}}$) and $\Gamma, a: {{\mathsf{Q}}} \vdash v: \exists {\alpha}\,{{\mathsf{P}}}$ (resp. $v: {{\mathsf{P}}}$). We immediately get the thesis by induction hypothesis applied to $u$ and $v$, while 2. is obviously verified. - $t={{ u \parallel_{a} v}}$. We show that this is not possible. Note that $a$ must occur free in $u$, otherwise $t$ is not in normal form. Since $\Gamma, a: \forall \beta\, A \vdash u: \exists {\alpha}\, {{\mathsf{P}}}$, we can apply the induction hypothesis to $u$, and obtain that all occurrences of hypothetical terms must be active; in particular, this must be the case for the occurrences of ${{\mathtt{H}_{a}^{\forall {\beta} {A}}}}$, but this is not possible since $t$ is in normal form. - $t={{\mathtt{H}_{a_{i}}^{\forall {\alpha} {A_{i}}}}}$. This case is not possible, for $\Gamma\vdash t:\exists {\alpha}\, {{\mathsf{P}}}$ or $\Gamma\vdash t: {{\mathsf{P}}}$. - $t={{\mathtt{H}^{{{\mathsf{P}}}}}}$. In this case, 1. and 2. are trivially true. - $t$ is obtained by an elimination rule and by we can write it as $r\, t_{1}\,t_{2}\ldots t_n$. Notice that in this case $r$ cannot correspond to an introduction rule neither be a term of the form ${{ u \parallel_{a} v}}$, because of the induction hypothesis, nor ${{ u\, |_{}\, v}}$, because of the permutation rules and $t$ being in normal form; moreover, $r$ cannot be ${\mathtt{W}_{b}^{\exists {\alpha} {P} }}$, otherwise $b$ would be free in $t$ and $b\neq a_{1}, \ldots, a_{n}$. We have now two remaining cases: 1. $r=x_{i}$ (resp. $r={{\mathtt{H}^{{{\mathsf{P}}}}}}$). Then, since $\Gamma \vdash x_{i}: {{\mathsf{P}}}_{i}$ (resp. $\Gamma \vdash {{\mathtt{H}^{{{\mathsf{P}}}}}}: {{\mathsf{P}}}$), we have that for each $i$, either $t_{i}$ is $\pi_{j}$ or $\Gamma\vdash t_{i}: {{\mathsf{Q}}}$, where ${{\mathsf{Q}}}$ is a negative propositional formula. By induction hypothesis, each $t_{i}$ satisfies 1. and also $t$. 2. is obviously verified. 2. $r={{\mathtt{H}_{a_{i}}^{\forall {\alpha_i} {{A}_{i}}}}}$. Then, $t_{1}$ is $m$, for some closed term of $\mathcal{L}$. Let $A_{i}=\forall\gamma_1\ldots\forall\gamma_l \, {{\mathsf{Q}}}$, with ${{\mathsf{Q}}}$ propositional, we have that for each $i$, either $t_i$ is a closed term $m_i$ of $\mathcal{L}$ or $t_{i}$ is $\pi_{j}$ or $\Gamma\vdash t_{i}: {{\mathsf{R}}}$, where ${{\mathsf{R}}}$ is a negative propositional formula. By induction hypothesis, each $t_{i}$ satisfies 1. and thus also $t$, while 2. is obviously verified. If we omit the parentheses, we will show that every normal proof-term having as type an existential formula can be written as ${{ {{ {{ v_0\, |_{}\, v_{1}}}\, |_{}\, }}\ldots\, |_{}\, v_{n}}}$, where each $v_{i}$ is not of the form ${{ u\, |_{}\, v}}$; if for every $i$, $v_i$ is of the form $(m_i,u_i)$, then we call the whole term an *Herbrand normal form*, because it is essentially a list of the witnesses appearing in an Herbrand disjunction. Formally: \[definition-hnf\] We define by induction a set of proof terms, called *Herbrand normal forms*, as follows: - Every normal proof-term $(m,u)$ is an Herbrand normal form; - if $u$ and $v$ are Herbrand normal forms, ${{ u\, |_{}\, v}}$ is an Herbrand normal form. Our last task is to prove that all quasi-closed proofs of any existential statement $\exists \alpha\, A$ include an exhaustive sequence $m_{1}, m_{2}, \ldots, m_{k}$ of possible witnesses. This theorem is stronger than the usual Herbrand theorem for classical logic [@AschieriZH], since we are stating it for any existential formula and not just for formulas with a single and existential quantifier. \[theorem-extraction\] Let $\exists\alpha\,A$ be any closed formula. Suppose $\Gamma \vdash t: \exists \alpha\, A$ in ${\mathsf{IL}}+{\mathsf{EM}_{1}^-}$ for a quasi closed term $t$, and $t\mapsto^{*} t'$ with $t^\prime$ in normal form. Then $\Gamma \vdash t': \exists \alpha\, A$ and $t'$ is an Herbrand normal form $${{ {{ {{ (m_{0}, u_{0})\, |_{}\, (m_{1}, u_{1})}}\, |_{}\, }}\ldots\, |_{}\, (m_{k}, u_{k})}}$$ Moreover, $\Gamma \vdash A[m_{1}/\alpha]\lor \dots \lor A[m_{k}/\alpha]$. By the Subject Reduction Theorem \[subjectred\], $\Gamma\vdash t': \exists \alpha\, A$. We proceed by induction on the structure of $t^\prime$. According to , we can write $t^\prime$ as $r u_1\dots u_n$. Note that since $t'$ is quasi closed, $r$ cannot be a variable $x$; moreover, $r$ cannot be a term ${{\mathtt{H}^{{{\mathsf{P}}}}}}$ or ${{\mathtt{H}_{b}^{\forall {\alpha} {B}}}}$, otherwise $t'$ would not have type $\exists \alpha\, A$, nor a term ${\mathtt{W}_{b}^{\exists {\alpha} {{{\mathsf{P}}}} }}$, otherwise $t'$ would not be quasi closed. $r$ also cannot be of the shape ${{ u \parallel_{a} v}}$, otherwise $\Gamma \vdash {{ u \parallel_{a} v}} : \exists \alpha\, {{\mathsf{Q}}}$, for some negative propositional ${{\mathsf{Q}}}$, but from we know that this is not possible. By , we are now left with only two possibilities. 1. $r$ is obtained by an introduction rule. Then $n=0$, otherwise there is a redex, and thus the only possibility is $t^\prime = r = {(n, u)}$ which is an Herbrand Normal Form. 2. $r = {{ u\, |_{}\, v}}$. Again $n=0$, otherwise we could apply a permutation rule; then $t^\prime = r = {{ u\, |_{}\, v}}$, and the thesis follows by applying the induction hypothesis on $u$ and $v$. We have thus shown that $t^\prime$ is an Herbrand normal form $${{ {{ {{ (m_{0}, u_{0})\, |_{}\, (m_{1}, u_{1})}}\, |_{}\, }}\ldots\, |_{}\, (m_{k}, u_{k})}}$$ Finally, we have that for each $i$, $\Gamma_{i}\vdash u_{i}: A[m_{i}/\alpha]$, for the very same $\Gamma_{i}$ that types $(m_{i}, u_{i})$ of type $\exists \alpha\, A$ in $t'$. Therefore, for each $i$, $\Gamma_{i}\vdash u_{i}^{+}: A[m_{1}/\alpha]\lor \dots \lor A[m_{k}/\alpha]$, where $u_{i}^{+}$ is of the form ${\upiota_{i_{1}}(\ldots {\upiota_{i_{k}}(u_{i})}\ldots )}$. We conclude that $$\Gamma \vdash {{ {{ {{ u_{0}^{+}\, |_{}\, u_{1}^{+}}}\, |_{}\, }}\ldots\, |_{}\, u_{k}^{+}}}: A[m_{1}/\alpha]\lor \dots \lor A[m_{k}/\alpha]$$ Markov’s Principle in Arithmetic {#sec:arithmetic} ================================ The original statement of Markov’s principle refers to Arithmetic and can be formulated in the system of Heyting Arithmetic ${\mathsf{HA}}$ as $$\neg \neg \exists \alpha \, {{\mathsf{P}}} \to \exists \alpha \, {{\mathsf{P}}} \text{, for } {{\mathsf{P}}} \text{ atomic}$$ By adapting ${\mathsf{IL}}+{\mathsf{EM}_{1}^-}$ to Arithmetic, following [@ABB], we will now provide a new computational interpretation of Markov’s principle. Note first of all that propositional formulas are decidable in intuitionistic Arithmetic ${\mathsf{HA}}$: therefore we will not need the rule ${\mathsf{EM}_{0}^-}$ and the parallelism operator. For the very same reason, we can expect the system ${\mathsf{HA}}+{\mathsf{EM}_{1}^-}$ to be constructive and the proof to be similar to the one of Herbrand constructivity for ${\mathsf{IL}}+{\mathsf{EM}_{1}^-}$. In this section indeed we will give such a syntactic proof. We could also have used the realizability interpretation for ${\mathsf{HA}}+{\mathsf{EM}_{1}}$ introduced in [@ABB] (see [@Manighetti]). The system ${\mathsf{HA}}+{\mathsf{EM}_{1}^-}$ ---------------------------------------------- We will now introduce the system ${\mathsf{HA}}+{\mathsf{EM}_{1}^-}$. We start by defining the language: \[definition-languagear\] The language $\mathcal{L}$ of ${\mathsf{HA}}+ {\mathsf{EM}}_1$ is defined as follows. 1. The terms of $\mathcal{L}$ are inductively defined as either variables $\alpha, \beta,\ldots$ or $0$ or ${\mathsf{S}}(t)$ with $t\in\mathcal{L}$. A numeral is a term of the form ${\mathsf{S}}\ldots {\mathsf{S}}0$.\ 2. There is one symbol $\mathcal{P}$ for every primitive recursive relation over $\mathbb{N}$; with $\mathcal{P}^{\bot}$ we denote the symbol for the complement of the relation denoted by $\mathcal{P}$. The atomic formulas of $\mathcal{L}$ are all the expressions of the form $\mathcal{P}(t_{1}, \ldots, t_{n})$ such that $t_{1}, \ldots, t_{n}$ are terms of $\mathcal{L}$ and $n$ is the arity of $\mathcal{P}$. Atomic formulas will also be denoted as ${{\mathsf{P}}}, \mathsf{Q}, {{\mathsf{P_i}}}, \ldots$ and $\mathcal{P}(t_{1}, \ldots, t_{n})^{\bot}:=\mathcal{P}^{\bot}(t_{1}, \ldots, t_{n})$.\ 3. The formulas of $\mathcal{L}$ are built from atomic formulas of $\mathcal{L}$ by the connectives $\lor,\land,\rightarrow, \forall,\exists$ as usual, with quantifiers ranging over numeric variables $\alpha^{{ {\tt N} }}, \beta^{{ {\tt N} }}, \ldots$.\ The system ${\mathsf{HA}}+{\mathsf{EM}_{1}^-}$ in \[fig:haemeno\] extends the usual Curry-Howard correspondence for ${\mathsf{HA}}$ with our rule ${\mathsf{EM}_{1}^-}$ and is a restriction of the system introduced in [@ABB]. The purely universal arithmetical axioms are introduced by means of Post rules, as in Prawitz [@Prawitz]. As we anticipated, there is no need for a parallelism operator. Therefore ${\mathsf{EM}_{1}^-}$ introduces a pure delimited exception mechanism, explained by the reduction rules in \[fig:F\]: whenever we have a term ${{ u \parallel_{a} v}}$ and ${{\mathtt{H}_{a}^{\forall {\alpha} {{{\mathsf{P}}}}}}}m$ appears inside $u$, we can recursively *check* whether ${{\mathsf{P}}}[m/\alpha]$ holds, and switch to the exceptional path if it doesn’t; alternatively, if it does hold we can remove the instance of the assumption. When there are no free assumptions relative to $a$ left in $u$, we can forget about the exceptional path. Grammar of Untyped Terms : $$t,u, v::=\ x\ |\ tu\ |\ tm\ |\ \lambda x\, u\ |\ \lambda \alpha\, u\ |\ {\langle t, u \rangle}\ |\ {u \pi_{0}}\ |\ {u \pi_{1}} \ |\ {\upiota_{0}(u)}\ |\ {\upiota_{1}(u)}\ |\ {t[x.u, y.v]}\ |\ (m,t)\ |\ t[(\alpha, x). u]$$ $$|\ ({{ u \parallel_{a} v}})\ |\ {{\mathtt{H}_{a}^{\forall {\alpha} {{{\mathsf{P}}}}}}}\ |\ {\mathtt{W}_{a}^{\exists {\alpha} {{{\mathsf{P}}}} }}\ |\ { {\texttt{True}} }\ |\ {\mathsf{R}}u v m \ |\ \mathsf{r}t_{1}\ldots t_{n}$$ where $m$ ranges over terms of $\mathcal{L}$, $x$ over variables of the lambda calculus and $a$ over ${\mathsf{EM}}_1$ hypothesis variables. Moreover, in terms of the form ${{ u \parallel_{a} v}}$ there is a ${{\mathsf{P}}}$ such that all the free occurrences of $a$ in $u$ are of the form ${{\mathtt{H}_{a}^{\forall {\alpha} {{{\mathsf{P}}}}}}}$ and those in $v$ are of the form ${\mathtt{W}_{a}^{\exists {\alpha} {{{\mathsf{P}}}^{\bot}} }}$. Contexts : With $\Gamma$ we denote contexts of the form $e_1:A_1, \ldots, e_n:A_n$, where $e_{i}$ is either a proof-term variable $x, y, z\ldots$ or a ${\mathsf{EM}}_{1}$ hypothesis variable $a, b, \ldots$ Axioms : $\begin{array}{c} \Gamma, x:{A}\vdash x: A \end{array}$ $\begin{array}{c} \Gamma, a:{\forall \alpha^{{ {\tt N} }} {{\mathsf{P}}}}\vdash {{\mathtt{H}_{a}^{\forall {\alpha} {{{\mathsf{P}}}}}}}: \forall\alpha^{{ {\tt N} }} {{\mathsf{P}}} \end{array}$ $\begin{array}{c} \Gamma, a:{\exists \alpha^{{ {\tt N} }} {{\mathsf{P}}}^\bot}\vdash {\mathtt{W}_{a}^{\exists {\alpha} {P} }}: \exists\alpha^{{ {\tt N} }} {{\mathsf{P}}}^\bot \end{array}$ Conjunction : $\begin{array}{c} \Gamma \vdash u: A\ \ \ \Gamma\vdash t: B\\ \hline \Gamma\vdash {\langle u, t \rangle}: A\wedge B \end{array}\ \ \ \ $ $\begin{array}{c} \Gamma \vdash u: A\wedge B\\ \hline\Gamma \vdash {u \pi_{0}}: A \end{array}\ \ \ \ $ $\begin{array}{c} \Gamma \vdash u: A\wedge B\\ \hline \Gamma\vdash {u \pi_{1}} : B \end{array}$\ \ Implication : $\begin{array}{c} \Gamma\vdash t: A\rightarrow B\ \ \ \Gamma\vdash u:A \\ \hline \Gamma\vdash tu:B \end{array}\ \ \ \ $ $\begin{array}{c} \Gamma, x:A \vdash u: B\\ \hline \Gamma\vdash \lambda x\, u: A\rightarrow B \end{array}$\ \ Disjunction Introduction : $\begin{array}{c} \Gamma \vdash u: A\\ \hline \Gamma\vdash {\upiota_{0}(u)}: A\vee B \end{array}\ \ \ \ $ $\begin{array}{c} \Gamma \vdash u: B\\ \hline \Gamma\vdash{\upiota_{1}(u)}: A\vee B \end{array}$\ \ Disjunction Elimination : $\begin{array}{c} \Gamma\vdash u: A\vee B\ \ \ \Gamma, x: A \vdash w_1: C\ \ \ \Gamma, y:B\vdash w_2: C\\ \hline \Gamma\vdash u\, [x.w_{1}, y.w_{2}]: C \end{array}$\ \ Universal Quantification : $\begin{array}{c} \Gamma \vdash u:\forall \alpha^{ {\tt N} }A\\ \hline \Gamma\vdash um: A[m/\alpha] \end{array} $ $\begin{array}{c} \Gamma \vdash u: A\\ \hline \Gamma\vdash \lambda \alpha\, u: \forall \alpha^{ {\tt N} }A \end{array}$\ where $m$ is any term of the language $\mathcal{L}$ and $\alpha$ does not occur free in any formula $B$ occurring in $\Gamma$.\ Existential Quantification : $\begin{array}{c}\Gamma\vdash u: A[m/\alpha]\\ \hline \Gamma\vdash ( m,u): \exists \alpha^{ {\tt N} }A \end{array}$    $\begin{array}{c} \Gamma\vdash u: \exists \alpha^{ {\tt N} }A\ \ \ \Gamma, x: A \vdash t:C\\ \hline \Gamma\vdash u\, [(\alpha, x). t]: C \end{array} $\ where $\alpha$ is not free in $C$ nor in any formula $B$ occurring in $\Gamma$.\ Induction : $\begin{array}{c} \Gamma\vdash u: A(0)\ \ \ \Gamma\vdash v:\forall \alpha^{{ {\tt N} }}. A(\alpha)\rightarrow A({\mathsf{S}}(\alpha))\\ \hline \Gamma\vdash {\mathsf{R}}uvm : A[m/\alpha] \end{array}$, where $m$ is a term of $\mathcal{L}$ Post Rules : $\begin{array}{c} \Gamma\vdash u_1: A_1\ \Gamma\vdash u_2: A_2\ \cdots \ \Gamma\vdash u_n: A_n\\ \hline\Gamma\vdash u: A \end{array}$ where $A_1,A_2,\ldots,A_n,A$ are atomic formulas of ${\mathsf{HA}}$ and the rule is a Post rule for equality, for a Peano axiom or for a classical propositional tautology or for booleans and if $n>0$, $u=\mathsf{r} u_{1}\ldots u_{n}$, otherwise $u={ {\texttt{True}} }$. ${\mathsf{EM}}_1^-$ : $\begin{array}{c} \Gamma, a: \forall \alpha \, {{\mathsf{P}}} \vdash u: \exists \beta \ {{\mathsf{Q}}} \ \ \ \ \Gamma, a: \exists \alpha\, \neg {{\mathsf{P}}} \vdash v: \exists \beta \ {{\mathsf{Q}}} \\ \hline \Gamma\vdash {{ u \parallel_{a} v}} : \exists \beta \ {{\mathsf{Q}}} \end{array} \ \text{(} {{\mathsf{P}}} \text{ atomic},{{\mathsf{Q}}} \text{ negative propositional) }$ <!-- --> Reduction Rules for ${\mathsf{HA}}$ : $$(\lambda x. u)t\mapsto u[t/x]\qquad (\lambda \alpha. u)m\mapsto u[m/\alpha]$$ $${{\langle u_0, u_1 \rangle} \pi_{i}}\mapsto u_i, \mbox{ for i=0,1}$$ $${{\upiota_{i}(u)}[x_{1}.t_{1}, x_{2}.t_{2}]}\mapsto t_{i}[u/x_{i}], \mbox{ for i=0,1}$$ $${{(m, u)}[(\alpha,x).v]} \mapsto v[m/\alpha][u/x], \mbox{ for each term $m$ of $\mathcal{L}$}$$ $${\mathsf{R}}u v 0 \mapsto u$$ $${\mathsf{R}}u v ({\mathsf{S}}n) \mapsto v n ({\mathsf{R}}u v n), \mbox{ for each numeral $n$}$$ Reduction Rules for ${\mathsf{EM}_{1}^-}$ : $${{ u \parallel_{a} v}}\mapsto u,\, \mbox{ if $a$ does not occur free in $u$ }$$ $${{ u \parallel_{a} v}}\mapsto v[a:=n],\mbox{ if ${{\mathtt{H}_{a}^{\forall {\alpha} {{{\mathsf{P}}}}}}}n$ occurs in $u$ and ${{\mathsf{P}}}[n/\alpha]$ is \emph{closed} and ${{\mathsf{P}}}[n/\alpha] = { {\texttt{False}} }$}$$ $$({{\mathtt{H}_{a}^{\forall {\alpha} {{{\mathsf{P}}}}}}})n \mapsto { {\texttt{True}} }\mbox{ if ${{\mathsf{P}}}[n/\alpha]$ is \emph{closed} and ${{\mathsf{P}}}[n/\alpha] \equiv { {\texttt{True}} }$}$$ Similarly to the previous sections, we extend the characterization of the proof-term heads to take into account the new constructs. \[theorem:head-form-ha\] Every proof term of ${\mathsf{HA}}+{\mathsf{EM}_{1}^-}$ is of the form: $$\lambda z_1 \dots \lambda z_n . r u_1 \dots u_k$$ where - $r$ is either a variable $x$, a constant ${{\mathtt{H}_{a}^{\forall {\alpha} {P}}}}$, ${\mathtt{W}_{a}^{\exists {\alpha} {P} }}$, $\mathsf{r}$ or ${\mathsf{R}}$, an excluded middle term ${{ u \parallel_{a} v}}$, or a term corresponding to an introduction rule $\lambda x . t$, $\lambda \alpha . t$, ${\langle t_1, t_2 \rangle}$, ${\upiota_{i}(t)}$, ${(m, t)}$ - $u_1, \dots u_k$ are either lambda terms, first order terms, or one of the following expressions corresponding to elimination rules: ${ \pi_{i}}$, ${[x.w_1, y.w_2]}$, ${[(\alpha,x).t]}$ The new system proves exactly the same formulas that can be proven by making use of Markov’s principle in Heyting Arithmetic. For any formula $F$ in the language $\mathcal{L}$, ${\mathsf{HA}}+{\mathsf{MP}}\vdash F$ if and only if ${\mathsf{HA}}+ {\mathsf{EM}}_1^- \vdash F$ The proof is identical as the one in the previous section. ${\mathsf{HA}}+{\mathsf{EM}_{1}^-}$ with the reduction rules in figure \[fig:red\] enjoys the Subject Reduction Theorem [@ABB; @Manighetti]. \[subjectred3\] If $\Gamma \vdash t : C$ and $t \mapsto u$, then $\Gamma \vdash u : C$. No term of ${\mathsf{HA}}+{\mathsf{EM}_{1}^-}$ gives rise to an infinite reduction sequence [@AschieriCOS]. Every term typable in ${\mathsf{HA}}+{\mathsf{EM}_{1}^-}$ is strongly normalizing. ${\mathsf{HA}}+{\mathsf{EM}_{1}^-}$ is Constructive --------------------------------------------------- We can now proceed to prove the constructivity of the system, that is the disjunction and existential properties. We will do this again by inspecting the normal forms of the proof terms; the first thing to do is adapting to ${\mathsf{HA}}+{\mathsf{EM}_{1}^-}$. \[prop:pnf2\] Let ${{\mathsf{P}}},{{\mathsf{P}}}_1,\dots {{\mathsf{P}}}_n$ be negative propositional formulas, $A_1, \dots A_m$ simply universal formulas. Suppose that $$\Gamma = z_1: {{\mathsf{P}}}_1, \dots z_n: {{\mathsf{P}}}_n, a_1 : \forall \alpha_1 A_1, \dots a_m : \forall \alpha_m A_m$$ and $\Gamma \vdash t:\exists {\alpha}\, {{\mathsf{P}}}$ or $\Gamma \vdash t: {{\mathsf{P}}}$, with $t$ in normal form and having all its free variables among $z_1, \dots z_n, a_1, \dots a_m $. Then: 1. Every occurrence in $t$ of every term ${{\mathtt{H}_{a_{i}}^{\forall {\alpha_{i}} {A_i}}}}$ is of the active form ${{\mathtt{H}_{a_{i}}^{\forall {\alpha_{i}} {A_i}}}}m$, where $m$ is a term of $\mathcal{L}$ 2. $t$ cannot be of the form $u\parallel_{a} v$. The proof is identical to the proof of . We just need to consider the following additional cases: - $t=\mathsf{r} t_1 t_2 \dots t_n$. Then $\Gamma \vdash t_i : {{\mathsf{Q}}}_i$ for some atomic ${{\mathsf{Q}}}_i$ and for $i=1 \dots n$; 1. holds by applying the inductive hypothesis to the $t_i$, while 2. is obviously verified. - $t = {\mathsf{R}}t_1 \dots t_n$. This case is not possible, otherwise since $t_{3}$ is a numeral and thus $t$ would not be in normal form. Thanks to this, we can now state the main theorem. The proof of the existential property is the same as the one for : we just need to observe that since we don’t have a parallelism operator in ${\mathsf{HA}}+{\mathsf{EM}_{1}^-}$, every Herbrand disjunction will consist of a single term. The disjunction property will follow similarly. - If ${\mathsf{HA}}+{\mathsf{EM}_{1}^-} \vdash t : \exists \alpha A$, then there exists a term $t' = (n,u)$ such that $t\mapsto^{*}t'$ and ${\mathsf{HA}}+{\mathsf{EM}_{1}^-} \vdash u : A[n/\alpha]$ - If ${\mathsf{HA}}+{\mathsf{EM}_{1}^-} \vdash t : A \lor B$, then there exists a term $t'$ such that $t\mapsto^{*}t'$ and either $t'={\upiota_{0}(u)}$ and ${\mathsf{HA}}+{\mathsf{EM}_{1}^-} \vdash u : A$, or $t'={\upiota_{1}(u)}$ and ${\mathsf{HA}}+{\mathsf{EM}_{1}^-} \vdash u: B$ For both cases, we start by considering a term $t'$ such that $t \mapsto^* t'$ and $t'$ is in normal form. By the Subject Reduction we have that ${\mathsf{HA}}+{\mathsf{EM}_{1}^-} \vdash t' : \exists \alpha A$ (resp. ${\mathsf{HA}}+{\mathsf{EM}_{1}^-} \vdash t': A \lor B$). By we can write $t'$ as $r t_1 \dots t_n$. Since $t'$ is closed, $r$ cannot be a variable $x$ or a term ${{\mathtt{H}_{a}^{\forall {\alpha} {{{\mathsf{P}}}}}}}$ or ${\mathtt{W}_{a}^{\exists {\alpha} {{{\mathsf{P}}}} }}$; moreover it cannot be $\mathsf{r}$, otherwise the type of $t'$ would have to be atomic, and it cannot be ${\mathsf{R}}$, otherwise the term would not be in normal form. $r$ also cannot have been obtained by ${\mathsf{EM}_{1}^-}$, otherwise ${\mathsf{HA}}+{\mathsf{EM}_{1}^-} \vdash r : \exists \alpha{{\mathsf{P}}}$, for ${{\mathsf{P}}}$ atomic and $r={{ t_1 \parallel_{a} t_2}}$; but this is not possible due to . Therefore, $r$ must be obtained by an introduction rule. We distinguish now the two cases: - ${\mathsf{HA}}+{\mathsf{EM}_{1}^-} \vdash t' : \exists \alpha B$. Since the term is in normal form, $n$ has to be 0, that is $t'=r$ and $r={(n, u)}$; hence also ${\mathsf{HA}}+{\mathsf{EM}_{1}^-} \vdash u : A(n)$. - ${\mathsf{HA}}+{\mathsf{EM}_{1}^-} \vdash t' : A \lor B$. 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Univ., Dipl.-Arb., 2016, [arXiv:1611.03714](https://arxiv.org/abs/1611.03714). D. Prawitz, *Ideas and Results in Proof Theory*, Proceedings of the Second Scandinavian Logic Symposium, pp. 235–306,1971. M. H. Sorensen, P. Urzyczyn, *Lectures on the Curry-Howard isomorphism*, Studies in Logic and the Foundations of Mathematics, vol. 149, Elsevier, 2006. A. Troelstra, *Metamathematical Investigations of Intuitionistic Arithmetic and Analysis*, Lectures Notes in Mathematics, [344]{}, Springer-Verlag, 1973. A. Troelstra, D. van Dalen: *Constructivism in Mathematics Volume I*, North-Holland, 1988. [^1]: Funded by the Austrian Science Fund FWF Lise Meitner grant M 1930–N35 [^2]: Funded by the Vienna Science Fund WWTF project VRG12-004
--- abstract: 'The quantum and classical dynamics of particles kicked by a gaussian attractive potential are studied. Classically, it is an open mixed system (the motion in some parts of the phase space is chaotic, and in some parts it is regular). The fidelity (Loschmidt echo) is found to exhibit oscillations that can be determined from classical considerations but are sensitive to phase space structures that are smaller than Planck’s constant. Families of quasi-energies are determined from classical phase space structures. Substantial differences between the classical and quantum dynamics are found for time dependent scattering. It is argued that the system can be experimentally realized by cold atoms kicked by a gaussian light beam.' author: - Yevgeny Krivolapov - Shmuel Fishman - Edward Ott - 'Thomas M. Antonsen' title: 'Quantum chaos of a mixed, open system of kicked cold atoms' --- Introduction ============ The quantum behavior of classically chaotic systems has been extensively studied both with time dependent and time independent Hamiltonians [@Tabor1989; @OzoriodeAlmeida1988; @Varenna1991; @Gutzwiller1990; @Oppo1994; @Haake2001; @Ott2002]. The main issue is that of determining fingerprints of classical chaos in the quantum mechanical behavior. For example, the spectral statistics of closed classically integrable [@Berry1976; @Berry1977a; @Berry1977b] and classically chaotic [@Bohigas1984; @Sieber2001; @MullerHaake2004] quantum systems have been predicted to have clearly distinct properties. Many of the systems that are of physical interest are mixed, where some parts of the phase space are chaotic and some parts are regular. Spectral properties of mixed systems with time independent Hamiltonians were studied by Berry and Robnik [@Berry1984]. In the present paper we study the classical/quantum correspondence properties of a mixed, open, time dependent system. (Here by “open” we mean that both position and momentum are unbounded.) The system we study consists of a particle kicked by a Gaussian potential defined by the Hamiltonian,$$H=\frac{p^{2}}{2m}-K'Te^{-\frac{x^{2}}{2\Delta^{2}}}\sum_{n=-\infty}^{\infty}\delta\left(t-Tn\right).\label{eq:Hamiltonian_with_units}$$ Models of this form were studied by Jensen who used it to investigate quantum effects on scattering in classically chaotic [@Jensen1994] and mixed [@Jensen1992] systems. This system can be experimentally approximated by a Gaussian laser beam acting on a cloud of cold atoms, somewhat similar to the realization of the kicked rotor by Raizen and coworkers [@MooreRaizen1995]. As we will show, the study of Hamiltonian is particularly suited to the investigation of generic behavior of kicked, open, mixed-phase-space systems. In particular, we will focus on issues of fidelity [@Peres1984; @Jalabert2001; @Jacquod2002; @Gorin2006], decoherence [@HansonOtt1984a; @Cohen1991] and scattering [@Blummel1988; @Jalabert1990; @Doron1991; @Lai1992]. Our main motivation in studying Hamiltonian is that, with likely future technological advances (see Sec. \[sec:Discussion\] for discussion), the phenomena we consider may soon become accessible to experimental investigation. Quantum mechanically, it is expected that classical phase space details on the scale of Planck’s constant are washed out [@Berry1972; @Berry1977c]. In contrast, one of our results will be that quantum dynamics can be sensitive to extremely fine structures in phase space, and this sensitivity is stable in the presence of noise [@HansonOtt1984a; @Cohen1991]. Phase space tunneling has been studied extensively [@Hanson1984; @Tomsovic1994; @Sheinman2006; @Lock2010]. For systems with many phase space structures complications arise due to transport between these structures. For our Hamiltonian the motion is unbounded (i.e., the system is “open”), and therefore this system is ideal for the exploration of tunneling out of phase space structures and, in particular, for study of resonance assisted tunneling, a current active field of research [@Tomsovic1994; @Sheinman2006; @Lock2010]. The outline of our paper is as follows. Section \[sec:The-model\] presents and discusses our model system. Section \[sec:Quasi-energies-of-an\] considers the quasi-energies of quantum states localized to island chains. Section \[sec:Fidelity\] introduces the fidelity concept and applies it to study different regions of the phase space including the main, central KAM island (Sec. \[sub:Fidelity\_central\]), island chains (Sec. \[sub:Fidelity\_chain\]), and chaotic regions (Sec. \[sub:Fidelity\_chaos\]). Experimentally there is always some noise present in such systems. Also, noise can be intentionally introduced. Section \[sec:Dephasing\] considers this issue. Section \[sec:Scattering\] presents a study of the scattering properties of the system. Conclusions and further discussion are given in Sec. \[sec:Discussion\]. \[sec:The-model\]The model ========================== Formulation ----------- A particle kicked by a Gaussian beam is modeled by the Hamiltonian, Eq. , with the classical equations of motion,$$\begin{aligned} \dot{p} & = & -\frac{\partial H}{\partial x}=-K'T\frac{x}{\Delta^{2}}e^{-\frac{x^{2}}{2\Delta^{2}}}\sum_{n=-\infty}^{\infty}\delta\left(t-Tn\right),\label{eq:Ham_eq_motion}\\ \dot{x} & = & \frac{\partial H}{\partial p}=\frac{p}{m}.\nonumber \end{aligned}$$ We rewrite the equations of motion in dimensionless form by defining $\bar{x}=x/\Delta$, $\bar{t}=t/T$. The dimensionless momentum is correspondingly defined as $\bar{p}=pT/\left(m\Delta\right)$. Thus we obtain the dimensionless equations of motion,$$\begin{aligned} \dot{\bar{p}} & = & -K\bar{x}e^{-\frac{\bar{x}^{2}}{2}}\sum_{n=-\infty}^{\infty}\delta\left(\bar{t}-n\right),\\ \dot{\bar{x}} & = & \bar{p},\nonumber \end{aligned}$$ where$$K=\frac{K'T^{2}}{m\Delta^{2}}.$$ Since in what follows we deal with the rescaled position, momentum and time we will drop the bar notation for convenience. By integrating and defining $p_{n}=p\left(t=n_{-}\right)$, $x_{n}=x\left(t=n_{-}\right)$, where $t=n_{-}$ is a time just before the *n-*th kick, we can rewrite the differential equations of the motion as a mapping, $M$,$$M:\qquad\begin{cases} p_{n+1} & =p_{n}-Kx_{n}e^{-\frac{x_{n}^{2}}{2}},\\ x_{n+1} & =x_{n}+p_{n+1}.\end{cases}\label{eq:classical_map}$$ The corresponding quantum dynamics in rescaled units is given by the Hamiltonian,$$H=\frac{p^{2}}{2}-Ke^{-\frac{x^{2}}{2}}\sum_{n=-\infty}^{\infty}\delta\left(t-n\right),\label{eq:Hamiltonian}$$ where $p=-i\tau\partial_{x}$, and $\tau=\hbar T/\left(m\Delta^{2}\right)$ is the rescaled $\hbar$, namely, $\left[x,p\right]=i\tau$. The quantum evolution is given by$$i\tau\partial_{t}\psi=-\frac{\tau^{2}}{2}\partial_{xx}\psi-Ke^{-\frac{x^{2}}{2}}\sum_{n=-\infty}^{\infty}\delta\left(t-n\right)\psi,$$ or by the one kick propagator$$U_{1}=e^{-i\frac{p^{2}}{2\tau}}\exp\left(i\frac{K}{\tau}e^{-\frac{x^{2}}{2}}\right).$$ Properties of the Classical Map and the phase portrait ------------------------------------------------------ In this subsection the classical properties of the map Eq. will be presented. The first property is reflection symmetry, $\left(x,p\right)\rightarrow\left(-x,-p\right)$. Phase portraits such as these presented in Fig. \[fig:phase\_space\_K1\] and Fig. \[fig:phase\_space\_K4.5\] ![\[fig:phase\_space\_K1\](Color online) The phase space for $K=1$. Colors (shades) distinguish different orbits.](phase_space_K1){width="8cm"} ![\[fig:phase\_space\_K4.5\](Color online) The phase space for $K=4.5$. Colors (shades) distinguish different orbits.](phase_space_K4\lyxdot 5){width="8cm"} are clearly seen to satisfy this property. Like the standard map, Eq. can be written as a product of two involutions, $M=J_{2}J_{1}$, where$$\begin{aligned} J_{1}:\qquad\left(p,x\right) & \rightarrow & \left(-p,x+p\right)\label{eq:involutions}\\ J_{2}:\qquad\left(p,x\right) & \rightarrow & \left(-p-Kxe^{-\frac{x^{2}}{2}},x\right).\nonumber \end{aligned}$$ We will use this property for the calculation of the periodic orbits. From we see that the only fixed point is $\left(x=0,p=0\right)$. Linearizing around this point, we find that the trace of the tangent map is $2-K$. Therefore, this point is elliptic for $0<K<4$, and, for $K=1$, the phase portrait of Fig. \[fig:phase\_space\_K1\] is found, while for $K>4$ this point is hyperbolic, leading to phase portraits like that of Fig. \[fig:phase\_space\_K4.5\]. Since the kicking as a function of $x$ is bounded by $K$, for large initial momentum the particle is nearly not affected by the kicks, and continues to move in its initial direction. For $0<K<4$ we find a large island around the elliptic point $\left(x,p\right)=\mbox{\ensuremath{\left(0,0\right)}}$, and, for nearly all initial conditions near $x=p=0$, the motion is regular (i.e., lies on KAM surfaces). Further away from this fixed point, one finds island chains embedded in a chaotic strip. And even further away, the motion is unbounded. \[sec:Quasi-energies-of-an\]Quasi-energies of an island chain ============================================================= In the semiclassical regime quasi-energies are related to classical structures. In this section we assume the existence of quasi-energy eigenfunctions $u_{n}\left(x\right)$,$$U_{1}u_{n}\left(x\right)=e^{-iE_{n}}u_{n}\left(x\right),$$ such that $u_{n}\left(x\right)$ is strongly localized to an island chain of order $r$, and we attempt to calculate the quasi-energy $E_{n}$. For this purpose we use the one-kick propagator $U_{1}$ to generate successive jumps in the island chain,$$U_{1}\psi_{i}=\psi_{i+1},$$ where $\psi_{i}$ is a wavefunction which is localized in island number $i$ within the island chain. Further, we assume that this wavefunction can be expanded using the quasi-eigenstates of the island chain,$$\psi_{i}=\sum_{n}c_{in}u_{n}\left(x\right).\label{eq:initial_wavepacket}$$ Using this expansion we obtain a system of equations,$$U_{1}\psi_{i}=\sum_{n}c_{in}u_{n}\left(x\right)e^{-iE_{n}},$$ and$$U_{1}^{r}\psi_{i}=\sum_{n}c_{in}u_{n}\left(x\right)e^{-iE_{n}r}.$$ Classically the $i-$th island is transformed to itself by $r$ successive applications of the map $M$. In particular, an elliptic fixed point of the map $M^{r}$ is located in the center of the island. In the semiclassical limit the eigenstates of $U_{1}^{r}$ are determined by $M^{r}$ and are close to the eigenstates of a harmonic oscillator centered on the fixed point of $M^{r}$. The frequency of the oscillator, $\nu_{i}$, is such that the eigenvalues of the tangent map of $M^{r}$, which transforms the $i-$th island to itself, are $e^{\pm i\nu_{i}}$. This tangent map can be written in terms of the product of the tangent maps of $M\left(i\rightarrow i+1\right),$ which transform the $i-$th island to the $\left(i+1\right)-$th island. Consequently, since the eigenvalues are determined by the trace of the product of the tangent maps, they are independent of $i$ (due to the invariance of the trace to cyclic permutations). In what follows we therefore drop the index $i$ from $\nu_{i}$. Choosing $\psi_{i}$ as the eigenstate of $U_{1}^{r}$, means that$$U_{1}^{r}\psi_{i}=e^{i\bar{\beta}}\psi_{i},$$ where $\bar{\beta}=\tau\nu/2$ and we have taken $\psi_{i}$ to be the ground state of the harmonic oscillator. Therefore,$$\psi_{i}=\sum_{n}c_{in}u_{n}\left(x\right)e^{-i\left(E_{n}r+\bar{\beta}\right)}.\label{eq:16}$$ Using the orthogonality of the $u_{n}\left(x\right)$, Eqs. and yield$$e^{-i\left(E_{n}r+\bar{\beta}\right)}=1.\label{eq:quasien_cond}$$ The quasi-energies, obtained from are, therefore,$$E_{n}=\frac{2\pi}{r}n+\beta,\qquad0\leq n\leq r,\label{eq:quasi_en_formula}$$ where $\beta=-\bar{\beta}/r$. Approximations to the quasi-energies can be calculated numerically by launching a wavepacket into one island in the island chain and propagating it in time, which gives$$\psi\left(x,N\right)=\sum_{n}c_{in}u_{n}\left(x\right)e^{-iE_{n}N}.$$ Taking a Fourier transform with respect to $N$ gives the quasi-energies. We have found that for $K=1$ the chains with $r=8$ and $r=16$ accurately satisfy . \[sec:Fidelity\]Fidelity ======================== The concept of quantum fidelity was introduced by Peres [@Peres1984] as a fingerprint of classical chaos in quantum dynamics. It has subsequently been extensively utilized in theoretical [@Jalabert2001; @Jacquod2002; @Cerruti2002; @Wimberger2006] and experimental studies [@Andersen2004; @Kaplan2005; @Wimberger2006; @Andersen2006], for a review see [@Gorin2006]. Most of this research has focused on the difference between chaotic and regular systems. Here we discuss fidelity for a mixed system. We have calculated the fidelity,$$S\left(t\right)=\left|\left\langle \phi_{0}\right|e^{iH_{1}t/\tau}e^{-iH_{2}t/\tau}\left|\phi_{0}\right\rangle \right|^{2},\label{eq:Fidelity}$$ where $H_{1,2}$ are Hamiltonians of the form with with slightly different kicking strengths, $K_{1,2}$, and $\phi_{0}$ is the initial wavefunction. We note that the fidelity $S\left(t\right)$ can be experimentally measured by the Ramsey method, as used in Ref. [@Kaplan2005]. The fidelity is related to an integral over Wigner functions,$$S\left(t\right)=2\pi\tau\int dxdp\, P_{\phi_{1}}\left(x,p\right)P_{\phi_{2}}\left(x,p\right),\label{eq:Fidelity_Wigner}$$ where $P_{\phi_{1,2}}$ are the Wigner functions of $\phi_{1,2}=e^{-iH_{1,2}t/\tau}\phi_{0}$, respectively. We study separately the fidelity in the central island, in the island chain, and in the chaotic region (i.e., Eq. with $\phi_{0}$ localized to these regions). \[sub:Fidelity\_central\]Fidelity of a wavepacket in the central island ----------------------------------------------------------------------- First we prepare the initial wavefunction $\phi_{0}$ as a Gaussian wavepacket with a minimal uncertainty, namely, $\Delta x=\Delta p=\left(\tau/2\right)^{\nicefrac{1}{2}}$, $$\phi_{0}\left(x\right)=\frac{1}{\left(2\pi\left(\Delta x\right)^{2}\right)^{\nicefrac{1}{4}}}e^{-ip_{0}x/\tau}\exp\left[-\frac{\left(x-x_{0}\right)^{2}}{4\left(\Delta x\right)^{2}}\right].\label{eq:Gaussian_quantum}$$ We place $\phi_{0}$ in the center of the island, namely, $x_{0}=p_{0}=0$. Since the center of the wavepacket is initially at the fixed point, for $\Delta x$ and $\Delta p$ classically small, its dynamics are approximately determined by the tangent map of the fixed point. For this purpose we linearize the classical map around the point $x=p=0$. This gives the equation for the deviations,$$\left(\begin{array}{c} \delta x_{n+1}\\ \delta p_{n+1}\end{array}\right)=\left(\begin{array}{cc} \left(1-K\right) & 1\\ -K & 1\end{array}\right)\left(\begin{array}{c} \delta x_{n}\\ \delta p_{n}\end{array}\right).$$ The eigenvalues of this equation are,$$\alpha_{1,2}=\frac{\left(2-K\right)}{2}\pm\frac{i\sqrt{K\left(4-K\right)}}{2}\equiv e^{\pm i\omega},$$ with$$\omega=\arctan\frac{\sqrt{K\left(4-K\right)}}{\left(2-K\right)},$$ which is the angular velocity of the points around the origin. In the vicinity of the fixed point, the system behaves like a harmonic oscillator with a frequency $\omega$. Classically, the motion of the trajectories, starting near the elliptic fixed point, $x=p=0$, stays there because the region is bounded by KAM curves that surround this point. For small effective Planck’s constant, $\tau$, the quantum behavior is expected to mimic the classical behavior for a long time. Inspired by the relation between the fidelity and the Wigner function (see ), we have defined a classical fidelity, $S_{c}\left(t\right)$, as the overlap between coarse-grained Liouville densities of $H_{1}$ and $H_{2}$ (this is similar to the classical fidelity defined in [@Nielsen2000]). To do this we first randomly generate a large number of initial classical positions using the initial distribution function,$$f_{0}\left(x,p\right)=\frac{1}{2\pi\Delta x\Delta p}\exp\left\{ -\frac{1}{2}\left[\left(\frac{x-x_{0}}{\Delta x}\right)^{2}+\left(\frac{p-p_{0}}{\Delta p}\right)^{2}\right]\right\} ,\label{eq:Gaussian_classical}$$ corresponding to our initial $\phi_{0}$ given by . The coarse grained densities for $H_{1}$ and $H_{2}$ are then computed by first integrating these initial conditions and then coarse graining to a grid of squares in phase space of area $\tau$ [@Berry1979]. The motivation for this procedure is to check if structures in phase space of size smaller than $\tau$ are of importance to the fidelity. A comparison between $S\left(t\right)$ and $S_{c}\left(t\right)$ for $x_{0}=-0.25$, $p_{0}=0$, and $\tau=0.01$ is presented in Fig. \[fig:Quantum\_classical\_fidelity\]. ![\[fig:Quantum\_classical\_fidelity\](Color online) Quantum fidelity, $S\left(t\right)$, (dashed red) and classical fidelity, $S_{c}\left(t\right)$ (solid blue). $K_{1}=1$, $K_{2}=1.01$, $\tau=0.01$, $x_{0}=-0.25$ and $p_{0}=0$.](Fidelity_quant_class_tau0\lyxdot 01_K1_x0-0\lyxdot 25_kicks1e+5){width="8cm"} The initial wavepacket is smeared on a ring in the phase space due to the twist property of the map. Since the probability density is preserved, the “whorl” which is formed contains very dense and thin tendrils. In Fig. \[fig:Whorls\] ![\[fig:Whorls\](Color online) Classical density, which was initially placed at $x_{0}=-0.25$ and $p_{0}=0$ after $500$ kicks. Blue (dark) dots are for $K_{1}=1$ and green (light) dots are for $K_{2}=1.01$.](Whorls_K1_1_K2_1\lyxdot 01_tau_0\lyxdot 01_x0_-0\lyxdot 25_num_kicks_500){width="8cm"} two such “whorls” are presented for $H_{1}$ with $K_{1}=1$ and $H_{2}$ with $K_{2}=1.01$. When the two “whorls” coincide a fidelity revival is formed. Coarse graining the densities to a boxes of size $\tau$ averages the differences between the two “whorls”, obtained by $H_{1}$ and $H_{2}$. This explains why the classical fidelity approaches $1$ as the number of kicks becomes large. On the other hand, the quantum fidelity shows strong revivals which suggests that it feels the difference in trajectories between the two Hamiltonians. To understand the period of the revivals, we calculate, $\delta\omega$, the frequency difference between the two Hamiltonians, $H_{1,2}$. Expanding $\omega$ around $K_{1}$ gives$$\omega\left(K\right)=\omega\left(K_{1}\right)+\frac{K-K_{1}}{\sqrt{K\left(4-K\right)}}+O\left(\left(K-K_{1}\right)^{2}\right).$$ Therefore, the difference in angular velocity between two orbits of Hamiltonians, $H_{1}$ and $H_{2}$ is given by$$\delta\omega=\omega\left(K_{2}\right)-\omega\left(K_{1}\right)=\frac{\delta K}{\sqrt{K_{2}\left(4-K_{2}\right)}},$$ for $\delta K=K_{2}-K_{1}$. This suggests that the fidelity, $S\left(t\right)$ will be periodic, with the period $T=\pi/\delta\omega$. Note that we predict $T=\pi/\delta\omega$, rather than $T=2\pi/\delta\omega$. This is because of the symmetry of the initial condition. Each point of $H_{1}$ is chasing a point of $H_{2}$ which is its reflection through the origin of the phase space and, therefore, is found first at an angle of $\pi$ and not $2\pi$. To check this, we have calculated the period of the revivals numerically for $0<K<4$. First, fidelity was computed and Fourier transformed, then the second most significant value was taken as the period. In Fig. \[fig:Fidelity\_center\] we present a comparison of the analytic calculation of the period of the fidelity and the numerical computation. The correspondence is good through the whole range of the stochasticity parameter $K$ but degrades near $K=4$, where the elliptic point at the origin becomes unstable. Also, near $K=2$, resonance chains appear near the fixed point $x=p=0$, which results in poor agreement with the theoretical prediction, see Fig. \[fig:phase\_space\_K2.1\]. ![\[fig:Fidelity\_center\](Color online) A numerical (solid blue) and an analytical (dashed green) computation of the period of the fidelity revival as a function of $K$, $\delta K=0.1$, $x_{0}=p_{0}=0$, $\tau=0.01$.](Fidelity_period_vs_K_x0_p0){width="8cm"} Very often it is assumed that quantum mechanical behavior is insensitive to phase space structures with areas smaller than Planck’s constant, which results in an effective averaging on this scale [@Berry1972; @Berry1977c]. While this assumption is often correct [@Sheinman2006], sometimes it is not [@HensingerPhillips2001; @SteckRaizen2001; @SteckRaizen2002; @AverbukhMoiseyev1995; @AverbukhMoiseyev2002; @OsovskiMoiseyev2005]. The difference between $S\left(t\right)$ and $S_{c}\left(t\right)$ demonstrated in Fig. \[fig:Quantum\_classical\_fidelity\] shows that fidelity may be sensitive to extremely small details in the classical phase space. In particular, a “whorl” [@Berry1972; @Berry1977c] affects the quantum dynamics. The small decay of the quantum fidelity seen in Fig. \[fig:Quantum\_classical\_fidelity\] is a result of tunneling. We stress that to observe the oscillations which appear on Fig. \[fig:Quantum\_classical\_fidelity\] requires sensitivity to the structure of the “whorl” of Fig. \[fig:Whorls\]. In our quantum calculation the effective Planck’s constant is $\tau=0.01$ and it is obvious that the “whorl” of Fig. \[fig:Whorls\] exhibits structures on smaller scale, for example, in a square with sides of length $0.1$ in phase space (of Fig. \[fig:Whorls\]) one finds several stripes of the “whorl”. Indeed, averaging over such a square leads to the classical fidelity that does not exhibit oscillations as the quantum fidelity does. We conclude that the structures on the scale smaller than the effective Planck’s constant, $\tau$, are crucial for the oscillations in the quantum fidelity. Hence, structures of scales smaller than Planck’s constant may dominate fidelity, which is a quantum quantity. For a wavepacket started around an initial point $\left(x_{0},p_{0}\right)\neq\left(0,0\right)$ the behavior is similar, but with a slightly different period due to a decrease in the angular velocity for points far from the fixed point. Similarly to the case of $\left(x_{0},p_{0}\right)=\left(0,0\right)$, we have calculated numerically the revival period for different values of $K$; this is shown in Fig. \[fig:Fidelity\_near\_center\] ![\[fig:Fidelity\_near\_center\]A numerical (blue circles) and an analytical (solid blue line) computation of the period of the fidelity revival as a function of $K$, $\delta K=0.01$, $x_{0}=-0.25$, $p_{0}=0$, $\tau=2\times10^{-4}$.](Fidelity_period_vs_K_x0-0\lyxdot 25_tau2e-4_dK_0\lyxdot 01){width="8cm"} . For $K>1.5$ resonances appear near the launching point which introduce additional periods into the fidelity, making the analysis more complicated. \[sub:Fidelity\_chain\]Fidelity for a wavepacket in an island chain ------------------------------------------------------------------- We consider two different island chains occurring for different values of $K$. For $K=2.1$ we have examined a chain of order $r=4$ (see Fig. \[fig:phase\_space\_K2.1\]) ![\[fig:phase\_space\_K2.1\](Color online) The phase space for $K=2.1$. Colors (shades) distinguish different orbits.](phase_space_K2\lyxdot 1){width="8cm"} , and for $K=1$ we have studied a chain of order $r=8$ (see Fig. \[fig:phase\_space\_K1\]). The initial wavepacket was launched inside one of the islands of the chain, and the we numerically computed the fidelity. In Figs. \[fig:Fidelity-chain4\] ($K_{1}=2.10$, $K_{2}=2.11$) and \[fig:Fidelity-chain8\] ($K_{1}=1.00$, $K_{2}=1.01$) ![\[fig:Fidelity-chain4\]Fidelity of packet started inside an island chain of order $4$. $K_{1}=2.10$, $K_{2}=2.11$, $\tau=2\times10^{-4}$, and the center of the packet is started at $x=0.3198$, $p=0$, in the center of one of the islands of the chain.](Fidelity_K1_2\lyxdot 1_K2_2\lyxdot 11_tau_0\lyxdot 0002_x0_0\lyxdot 3198.pdf){width="8cm"} ![\[fig:Fidelity-chain8\]Fidelity of packet started inside an island chain of order $8$. $K_{1}=1$, $K_{2}=1.01$, $\tau=2\times10^{-4}$, and the center of the packet is started at $x=1.1312$, $p=0$, in the center of one of the islands of the chain. The inset is a zoom on the graph.](Fidelity_K1_1_K2_1\lyxdot 01_tau_0\lyxdot 0002_x0_1\lyxdot 1313){width="8cm"} we show the results of these computations. It is notable that there are three timescales in the graph of the fidelity. The shortest timescale is visible only in the inset of Fig. \[fig:Fidelity-chain8\] and may be understood taking into account the symmetry of the equations of motion, $x\rightarrow-x,$ $p\rightarrow-p$. This symmetry implies that each island has a “twin” which is found by reflection through the origin, $x=p=0$. Therefore, the overlap between the islands of $H_{1}$ and $H_{2}$ is a periodic function with a period of $\nicefrac{r}{2}$, where $r$ is the number of islands in the chain. Consequently, for the island chains used to obtain Fig. \[fig:Fidelity-chain4\] and \[fig:Fidelity-chain8\], the fidelity has periods of $2$ and $4$, respectively, on its shortest timescale. The intermediate timescale is due to a rotation of the wavepacket around the elliptic points of the island where it is initially launched. The central point in the island is a fixed point of $M^{r}$. In $r$ iterations, points in the island rotate with an angular velocity $\omega_{1}$ and $\omega_{2}$ for $H_{1}$ and $H_{2}$, respectively. The angular velocities can be calculated numerically by linearization of the tangent map of $M^{r}$ around the fixed point of the map $M^{r}$. We find the fixed point by reducing $M$ to a product of involutions , which allows us to reduce the search for the fixed points to the line $p=0$ in the phase space since any point on this line is a fixed point of $J_{1}$ [@Greene1979; @Hihinashvili2007]. For $K_{1}=1$ and $K_{2}=1.01$, the angular velocities are found to be $\omega_{1}=1.10$ and $\omega_{2}=1.147$. For $K_{1}=2.10$ and $K_{2}=2.11$, the angular velocities are found to be $\omega_{1}=0.391$ and $\omega_{2}=0.429$. Therefore, the time it takes for a packet to accomplish a full revolution around the fixed points of $M^{r}$ is $2\pi r/\bar{\omega}$, where $\bar{\omega}=\frac{1}{2}\left(\omega_{1}+\omega_{2}\right)\approx\omega_{1}\approx\omega_{2}$ (see Table \[tab:chains\_periods\]). The longest timescale of the fidelity is the timescale when the difference between the angular velocities is resolved $T=2\pi r/\delta\omega$. In Table \[tab:chains\_periods\] we compare those periods deduced directly from Fig. \[fig:Fidelity-chain4\] and Fig. \[fig:Fidelity-chain8\] and the periods calculated by finding $\omega_{1,2}$ from the tangent map. We see that the agreement is excellent. ----------------- ----------------------------- ------------- ----------------------------- ------------- Fig.\[fig:Fidelity-chain4\] Tangent map Fig.\[fig:Fidelity-chain8\] Tangent map shortest period 2 2 4 4 medium period 62 $61.3$ 44 $44.7$ longest period 651 $657.8$ 1061 $1077.4$ ----------------- ----------------------------- ------------- ----------------------------- ------------- : \[tab:chains\_periods\]This table compares two ways of calculating the periods of revivals for the resonance chains. In one way we have deduced them from the Figures \[fig:Fidelity-chain4\],\[fig:Fidelity-chain8\], and in the other way we have calculated them using the tangent map. This is done for two different resonances: $r=4$, for $K_{1}=2.1$, $K_{2}=2.11$; and $r=8$ for $K_{1}=1$, $K_{2}=1.01$. For both cases $\tau=2\times10^{-4}$. \[sub:Fidelity\_chaos\]Fidelity of the wavepacket in the chaotic strip ---------------------------------------------------------------------- For the fidelity of a packet started inside the chaotic strip (see Fig. \[fig:Fidelity-chaos\]), we notice a strong revival after 6 kicks which is dependent on $K_{1}$. This is half a period in this chain/strip. After this revival the fidelity decays to zero, which is a characteristic of chaotic regions. ![\[fig:Fidelity-chaos\]Fidelity of packet started inside a chaotic layer. $K_{1}=1$, $K_{2}=1.01$, $\tau=2\times10^{-4}$ and the center of the packet is started at $x=-2$, $p=0$.](Fidelity_chaos_tau2e-4_K1_x0-2){width="8cm"} Detailed exploration of this region is left for further studies. \[sec:Dephasing\]Dephasing ========================== We now investigate the effect of dephasing by adding temporal noise to the time between the kicks. The classical equations of motion with the dephasing are given by$$\begin{aligned} p_{n+1} & = & p_{n}-Kx_{n}e^{-\frac{x_{n}^{2}}{2}},\\ x_{n+1} & = & x_{n}+\left(1+\delta t_{n}\right)\cdot p_{n+1},\nonumber \end{aligned}$$ and the quantum one kick propagator is$$U_{1}=e^{-i\frac{p^{2}}{2\tau}\left(1+\delta t_{n}\right)}\exp\left(i\frac{K}{\tau}e^{-\frac{x^{2}}{2}}\right),$$ where $\delta t_{n}$ is a random variable which is normally distributed with zero mean and a standard deviation $\sigma_{t}$. The standard deviation of the $\delta t_{n}$, corresponds to the strength of the noise. We find that the noise results in an escape outside of the island, which yields additional decay in the fidelity. Since we are interested in the difference between the two wavefunctions only inside the main island, for each kick we normalize the wavefunctions of $H_{1}$ and $H_{2}$ such that their norm is equal to $1$ inside a region of $\left|x\right|\leq x_{b}=3$. This gives the following expression for the fidelity$$S\left(t\right)=\frac{\int_{-x_{b}}^{x_{b}}\left(e^{-iH_{1}t/\tau}\phi_{0}\left(x'\right)\right)\left(e^{-iH_{2}t/\tau}\phi_{0}\left(x'\right)\right)\, dx'}{\left(\int_{-x_{b}}^{x_{b}}\left|e^{-iH_{1}t/\tau}\phi_{0}\left(x'\right)\right|^{2}dx'\right)^{\nicefrac{1}{2}}\left(\int_{-x_{b}}^{x_{b}}\left|e^{-iH_{2}t/\tau}\phi_{0}\left(x'\right)\right|^{2}dx'\right)^{\nicefrac{1}{2}}},$$ with he classical fidelity $S_{c}\left(t\right)$ defined in a similar way. We have numerically calculated the fidelity for the same situation as in Fig. \[fig:Quantum\_classical\_fidelity\] with added relative noise of $\sigma_{t}=0.01$ (Fig. \[fig:Quantum\_classical\_fidelity\_noise0.01\]) and $\sigma_{t}=0.001$ (Fig. \[fig:Quantum\_classical\_fidelity\_noise0.001\]). ![\[fig:Quantum\_classical\_fidelity\_noise0.01\](Color online) Quantum fidelity, $S\left(t\right)$, (dashed red) and classical fidelity, $S_{c}\left(t\right)$ (solid blue) for a dephasing noise of strength $\sigma_{t}=0.01$, $K_{1}=1$, $K_{2}=1.01$, $\tau=0.01$, $x_{0}=-0.25$ and $p_{0}=0$.](Fidelity_K1_1_K2_1\lyxdot 01_tau_0\lyxdot 01_x0_-0\lyxdot 25_2e+4_noise_1e-2){width="8cm"} ![\[fig:Quantum\_classical\_fidelity\_noise0.001\](Color online) Quantum fidelity, $S\left(t\right)$, (dashed red) and classical fidelity, $S_{c}\left(t\right)$ (solid blue) for a dephasing noise of strength $\sigma_{t}=0.001$, $K_{1}=1$, $K_{2}=1.01$, $\tau=0.01$, $x_{0}=-0.25$ and $p_{0}=0$.](Fidelity_K1_1_K2_1\lyxdot 01_tau_0\lyxdot 01_x0_-0\lyxdot 25_2e+4_noise_1e-3){width="8cm"} We notice that the noise introduces additional decay in the quantum fidelity. To isolate the effect of noise from the decay in the fidelity due to the difference between $K_{1}$ and $K_{2}$ we set $K_{1}=K_{2}$ and use two different noise realizations with the same strength $\sigma_{t}$. From Fig. \[fig:Quantum\_classical\_fidelity\_noise0.01\_sameK\] ![\[fig:Quantum\_classical\_fidelity\_noise0.01\_sameK\](Color online) Quantum fidelity, $S\left(t\right)$, (dashed light red) and classical fidelity, $S_{c}\left(t\right)$ (solid dark blue) for two different realizations of a dephasing noise of strength $\sigma_{t}=0.001$, $K_{1}=K_{2}=1$, $\tau=0.01$, $x_{0}=-0.25$ and $p_{0}=0$.](Fidelity_K1_1_K2_1_tau_0\lyxdot 01_x0_-0\lyxdot 25_5e+4_noise_1e-2){width="8cm"} we notice that classical fidelity initially decays very fast due to the noise and than slowly recovers approaching a value of $0.8$. This is due to the coarse graining to the scale of $\tau$. To illustrate this we plot in Fig. \[fig:Whorls-sameK\] ![\[fig:Whorls-sameK\](Color online) Classical density, which was initially placed at $x_{0}=-0.25$ and $p_{0}=0$ after $5\times10^{4}$ kicks, $K_{1}=K_{2}=1$. Colors (shades) correspond to two different realizations of a dephasing noise of strength $\sigma_{t}=0.01$.](Whorls_K1_1_K2_1_tau_0\lyxdot 01_x0_-0\lyxdot 25_num_kicks_5e+4_noise_1e-2){width="8cm"} the classical densities after $5\times10^{4}$ kicks for a packet initially launched at $x_{0}=-0.25$. We notice that the densities for the two Hamiltonians highly overlap, which explains the high fidelity. In Fig. \[fig:Psi-sameK\] ![\[fig:Psi-sameK\](Color online) Wavepackets, which were initially placed at $x_{0}=-0.25$ and $p_{0}=0$ after $5\times10^{4}$ kicks, $K_{1}=K_{2}=1$. Colors (shades) correspond to two different realizations of a dephasing noise of strength $\sigma_{t}=0.01$.](Psi_K1_1_K2_1_tau_0\lyxdot 01_x0_-0\lyxdot 25_5e+4_noise_1e-2){width="8cm"} we observe the corresponding quantum wavepackets. Contrary to the classical fidelity, the quantum fidelity decays rather slowly with the noise, suggesting that it is more robust to noise than the classical fidelity. \[sec:Scattering\]Scattering ============================ We now investigate the difference between quantum and classical scattering behavior by studying the evolution of a wavepacket initialized outside of the main island of the phase space, Eq. with $x_{0}=-2$, $p_{0}=0$, $\Delta x=\Delta p=\left(\tau/2\right)^{\nicefrac{1}{2}}$. In the classical case both the classical chaos, as well as the numerous small island structures introduce, an erratic behavior for the transmission and reflection coefficients as a function of the initial launching position and energy [@Jensen1992; @Jensen1994]. Due to effective phase space smoothing of areas much smaller than our effective Planck’s constant, $\tau$, we expect that fine scale fractal-like features in the classically erratic scattering dependence will be averaged out. To quantify this behavior, we measure the transmission and reflection coefficients for a wavepacket defined as the transfered or reflected probability mass, either quantum or classical. Classically, it is the fraction of initial trajectories (generated using ) reflected or transmitted by the main island for a given time, while quantum mechanically, we measure the total escaped probability up to time $t$ from the island area, $\left|x\right|\leq x_{b}$, $$\begin{aligned} L\left(t\right) & = & \int_{0}^{t}dt'\int_{-\infty}^{-x_{b}}\left|\psi\left(x,t'\right)\right|^{2}dx\\ R\left(t\right) & = & \int_{0}^{t}dt'\int_{x_{b}}^{\infty}\left|\psi\left(x,t'\right)\right|^{2}dx,\nonumber \end{aligned}$$ where $x_{b}$ is the margin of the main island (we choose $x_{b}=4$), $L\left(t\right)$ and $R\left(t\right)$ are probabilities to be scattered to the left or the right of the island till time $t$, correspondingly. To determine those probabilities, we use the continuity equation for the probability,$$\partial_{t}\left(\int_{a}^{b}\left|\psi\right|^{2}dx\right)=\tau\operatorname{Im}\left[\left(\psi\partial_{x}\psi^{*}\right)|_{x=b}-\left(\psi\partial_{x}\psi^{*}\right)|_{x=a}\right],$$ so that,$$\begin{aligned} L\left(t\right) & = & \frac{\tau}{2i}\int_{0}^{t}dt'\int_{0}^{t'}dt''\left(\psi\partial_{x}\psi^{*}-\psi^{*}\partial_{x}\psi\right)|_{x=-x_{b}},\\ R\left(t\right) & = & -\frac{\tau}{2i}\int_{0}^{t}dt'\left(\psi\partial_{x}\psi^{*}-\psi^{*}\partial_{x}\psi\right)|_{x=x_{b}}.\nonumber \end{aligned}$$ In Figs. \[fig:scatter\_L\]-\[fig:scatter\_C-3\] ![\[fig:scatter\_L\](Color online) Total quantum (solid blue line) and classical (blue dots) probabilities for scattering to the left of the island $\left(x<-x_{b}\right)$ as a function of the number of kicks. $K=1$, $\tau=0.01$, $x_{0}=-2$, $p_{0}=0$.](Scattering_L_K_1_tau_0\lyxdot 01_x0_-2_p0_0num_kicks_1000){width="8cm"} ![\[fig:scatter\_R\](Color online) Total quantum (solid blue line) and classical (blue dots) probabilities for scattering to the right of the island $\left(x>x_{b}\right)$ as a function of the number of kicks. $K=1$, $\tau=0.01$, $x_{0}=-2$, $p_{0}=0$.](Scattering_R_K_1_tau_0\lyxdot 01_x0_-2_p0_0num_kicks_1000){width="8cm"} ![\[fig:scatter\_C\]Total quantum (solid blue line) and classical (blue dots) probabilities to stay in the island $\left(\left|x\right|\leq x_{b}\right)$ as a function of the number of kicks. $K=1$, $\tau=0.01$, $x_{0}=-2$, $p_{0}=0$.](Scattering_C_K_1_tau_0\lyxdot 01_x0_-2_p0_0num_kicks_1000){width="8cm"} ![\[fig:scatter\_L-3\]Total quantum (solid blue line) and classical (blue dots) probabilities for scattering to the left of the island $\left(x<-x_{b}\right)$ as a function of the number of kicks. $K=1$, $\tau=0.01$, $x_{0}=-3$, $p_{0}=0$.](Scattering_L_K_1_tau_0\lyxdot 01_x0_-3_p0_0num_kicks_1000){width="8cm"} ![\[fig:scatter\_R-3\]Total quantum (solid blue line) and classical (blue dots) probabilities for scattering to the right of the island $\left(x>x_{b}\right)$ as a function of the number of kicks. $K=1$, $\tau=0.01$, $x_{0}=-3$, $p_{0}=0$.](Scattering_R_K_1_tau_0\lyxdot 01_x0_-3_p0_0num_kicks_1000){width="8cm"} ![\[fig:scatter\_C-3\]Total quantum (solid blue line) and classical (blue dots) probabilities to stay in the island $\left(\left|x\right|\leq x_{b}\right)$ as a function of the number of kicks. $K=1$, $\tau=0.01$, $x_{0}=-3$, $p_{0}=0$.](Scattering_C_K_1_tau_0\lyxdot 01_x0_-3_p0_0num_kicks_1000){width="8cm"} we compare the quantum and classical scattering of a wavepacket launched from the left of the main island. We notice that there is a substantial difference, which decreases when we decrease the effective Planck’s constant, $\tau$. Figures \[fig:scatter\_L\]-\[fig:scatter\_C\] and Figs. \[fig:scatter\_L-3\]-\[fig:scatter\_C-3\] differ in the initial launching position of the wavepacket ($x_{0}=-2$, for Figs. \[fig:scatter\_L\]-\[fig:scatter\_C\] and $x_{0}=-3$ for Figs. \[fig:scatter\_L-3\]-\[fig:scatter\_C-3\]). We notice that the scattering is sensitive to $x_{0}$. Different aspects of chaotic scattering for this problem were explored in [@Jensen1992], and in particular, the effect of small $\hbar$ on washing out rainbow singularities of the classical scattering function. \[sec:Discussion\]Discussion and Conclusions ============================================ Discussion of experimental realizability ---------------------------------------- In the present work the classical and quantum dynamics of a system with a mixed phase space were studied. It is proposed to realize this system by injecting cold atoms into a coherent, pulsed, gaussian light beam. The phase space structures, which can be seen on Figs. \[fig:phase\_space\_K1\],\[fig:phase\_space\_K4.5\] and \[fig:phase\_space\_K2.1\] are controlled by the parameters of the beam via the parameter $K$. Since it is relatively straightforward to control the parameters of gaussian beams, the proposed system is ideal for the exploration of dynamics of mixed systems. In what follows limitations on experimental realizations are discussed. First we consider the realizability of an approximately one dimensional situation necessary for the validity of our theoretical results. Let us assume that the gaussian beam propagates in the $z$ direction. Its profile in the $xy$ plane is $$e^{-\frac{x^{2}}{2\Delta^{2}}-\frac{y^{2}}{2\Delta_{y}^{2}}}.\label{eq:*1}$$ Assuming that the extent of the light beam is much smaller than the Rayleigh length, $z_{R}=\pi\Delta^{2}/\lambda$ where $\lambda$ is the wavelength, and the $z$ dependence of the potential can be ignored. The potential of Eq. can be well approximated by $\exp\left(-x^{2}/\left(2\Delta^{2}\right)\right)$ in , for sufficiently small values of $y^{2}/\Delta_{y}^{2}$, and, to facilitate this, it is appropriate to consider $\Delta_{y}\gg\Delta$, i.e., a quasi-sheet-like beam. Such beams are experimentally realizable via routine methods. To analyze this situation, the normalized map $M$ of should be replaced by one with $\exp\left(-x^{2}/2\right)$ replaced by $\exp\left(-\left[x^{2}/2+y^{2}\left(\Delta/\Delta_{y}\right)^{2}\right]\right)$. In addition, there are equations for $y_{n}$ and its conjugate momentum $p_{y,n}$, which in dimensionless units with $y$ and $p_{y,n}$ rescaled by $\Delta$ and $T/\left(m\Delta\right)$, respectively, take the form,$$\begin{aligned} p_{y,n+1} & = & p_{y,n}-K_{y}y_{n}e^{-\frac{x_{n}^{2}}{2}-\frac{1}{2}\left(\frac{\Delta}{\Delta_{y}}\right)^{2}y_{n}^{2}},\nonumber \\ y_{n+1} & = & y_{n}+p_{y,n+1},\label{eq:*2}\end{aligned}$$ where $$K_{y}=K\left(\frac{\Delta}{\Delta_{y}}\right)^{2}.\label{eq:*3}$$ Since $K\approx1$ and $\Delta/\Delta_{y}\ll1$, it can be assumed that $K_{y}\ll1$. Therefore, the motion in the $y$ direction is slow relative to the motion in the $x$ direction. Thus $\exp\left(-x_{n}^{2}/2\right)$ can be approximated by its time average $\left\langle \exp\left(-x^{2}/\left(2\Delta^{2}\right)\right)\right\rangle \equiv\rho$ (which is of order unity), and, for sufficiently small $y$ the $y-$motion can be described by a Harmonic oscillator with a force constant $2K_{y}\rho\ll1$. Conservation of energy $E_{y}$ implies that the maximal value of $y$ satisfies$$E_{y}=2K_{y}\rho y_{\mathrm{max}}^{2}.\label{eq:*5}$$ The energy $E_{y}$ is determined by the initial preparation. Let us assume that initially the atoms form a Bose-Einstein Condensate (BEC) and are in a harmonic trap that is anisotropic where the frequency in the $y$ direction is $\nu_{y}^{\prime}$ in experimental units, and $\nu_{y}=T\nu_{y}^{\prime}$ in our rescaled units. We assume that the center of this trap $y_{0}$ satisfies $y_{0}\ll y_{max}$. The experiment starts when the trap is turned off. Assuming the atoms are in the ground state, their energy in our rescales units is$$\frac{1}{2}\hbar\nu_{y}^{\prime}\left(\frac{T^{2}}{m\Delta^{2}}\right)=\frac{1}{2}\nu_{y}\tau\leq E_{y}.\label{eq:*6}$$ We desire the effect of the motion in the $y$ direction on the motion in the $x$ direction (Eq. with $\exp\left(-x^{2}/\left(2\Delta^{2}\right)\right)$ replaced by $V$ of ) to be negligible. Thus it is required that$$\eta\equiv y_{\mathrm{max}}^{2}\left(\frac{\Delta}{\Delta_{y}}\right)^{2}\ll1.\label{eq:*7}$$ In this case the $y$ motion corresponds to a variation in $K$ of the order $\Delta K\sim K\eta$. Using and , condition reduces to$$\frac{1}{4}\frac{\nu_{y}\tau}{K}\leq\frac{E_{y}}{2K}=\eta\ll1,\label{eq:*8}$$ where, since we are interested only in crude estimates, we have replaced $\rho$ by one. The initial spread in $y$ is given by the ground state of the harmonic oscillator, where $\left\langle y^{2}\right\rangle =\tau/\left(2\nu_{y}\right)$, and we require that the expectation value of $y^{2}$ satisfies $\left\langle y^{2}\right\rangle \ll y_{\mathrm{max}}^{2}$, resulting in $$\left(\frac{\Delta}{\Delta_{y}}\right)^{2}\frac{\tau}{2\nu_{y}}\ll\eta.\label{eq:*9}$$ For both inequalities and to be satisfied it is required that$$\left(\frac{\Delta}{\Delta_{y}}\right)^{2}\frac{\tau}{2\eta}\ll\nu_{y}\leq\frac{4K}{\tau}\eta.\label{eq:*10}$$ The resulting fundamental lower bound on $\eta$ is $$\left(\frac{\Delta}{\Delta_{y}}\right)^{2}\frac{\tau^{2}}{8K}\ll\eta.\label{eq:*11}$$ Reasonable experimental values are $\Delta/\Delta_{y}\approx10^{-2}$ and $\nu_{y}\approx0.1$. For $\tau=10^{-2}$ and $K\approx1$ the lower bound on $\eta$ is $10^{-5}$ leaving a wide range for ‘engineering’ of BEC traps so that the $\nu_{y}$ is in the range . For $\nu_{y}\approx0.1$ and $\tau=10^{-2}$ and $K\approx1$ we can make $\eta\lesssim10^{-3}$. Since this value of $\eta$ is small compared to the value of $\Delta K=K_{2}-K_{1}$, used in our fidelity calculations (Figs \[fig:Quantum\_classical\_fidelity\], \[fig:Fidelity\_center\], \[fig:Quantum\_classical\_fidelity\_noise0.01\], \[fig:Quantum\_classical\_fidelity\_noise0.001\]), those calculations are expected to be uneffected by $y$ motion for our assumed parameters. It is also encouraging to see that noise of a higher level does not destroy fidelity oscillations (see Fig. \[fig:Quantum\_classical\_fidelity\_noise0.001\]). One should note, however, that the variation of the effective $K$ of the motion in the $x$ direction is slow, with effective frequency $\Delta/\Delta_{y}$ that for $\Delta/\Delta_{y}\approx10^{-2}$ is of order $10^{-2}$. For these reasons, we expect that, the model that we have explored theoretically in the present work should be realizable for a wide range of experimental parameters. Conclusions ----------- The main result of this paper is that the quantum fidelity is sensitive to the phase space details that are finer than Planck’s constant, contrary to expectations of Refs. [@Berry1972; @Berry1977c]. In particular, the fidelity was studied and predicted to oscillate with frequencies that can be predicted from classical considerations. This behavior is characteristic of regular regions. Fidelity exhibits a periodic sequence of peaks. For wavepackets in the main island, it was checked that the peak structure is stable in the presence of external noise but the amplitude decays with time. For wavepackets initialized in a chain of regular islands, it was found that the fidelity exhibits several time scales that can be predicted from classical considerations. 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--- abstract: 'Through the development of neural machine translation, the quality of machine translation systems has been improved significantly. By exploiting advancements in deep learning, systems are now able to better approximate the complex mapping from source sentences to target sentences. But with this ability, new challenges also arise. An example is the translation of partial sentences in low-latency speech translation. Since the model has only seen complete sentences in training, it will always try to generate a complete sentence, though the input may only be a partial sentence. We show that NMT systems can be adapted to scenarios where no task-specific training data is available. Furthermore, this is possible without losing performance on the original training data. We achieve this by creating artificial data and by using multi-task learning. After adaptation, we are able to reduce the number of corrections displayed during incremental output construction by 45%, without a decrease in translation quality.' address: | Institute for Anthropomatics and Robotics\ KIT - Karlsruhe Institute of Technology, Germany bibliography: - 'mybib.bib' title: 'Low-Latency Neural Speech Translation' --- **Index Terms**: speech translation, low-latency Introduction ============ Neural machine translation (NMT) is currently the state-of-the-art in machine translation, significantly improving translation quality in text translation [@WMT] as well as in speech translation [@IWSLT], where the translation input is the output from a speech recognizer. The main strength of neural machine translation is improved output fluency compared to traditional approaches, such as rule-based or statistical machine translation. However, while the model is able to capture more complex dependencies between the source and target languages, it relies heavily on training data examples to do so. As a consequence, the model lacks the robustness at test time to handle data that is fundamentally different from what was seen in training. There are several scenarios where this can be observed. For example, if the input is incorrectly cased, or if a different dialect is presented at test time which has different spelling or phrasings. In this work, we will concentrate on a speech translation use case in which the translation system is required to provide an initial translation in real time, before the complete sentence has been spoken. To this end, [@Niehues2016LT] presented an approach where partial sentences are translated and later replaced if necessary with the translations of the complete sentences. While we focus on this use case, the results of this work can be easily adapted to other use cases where there are differences between the training and testing scenarios. When applying partial sentence translation to neural machine translation systems, we encounter the problem that the MT system has only been trained on complete sentences, and thus the decoder is biased to generate complete target sentences. When receiving inputs which are partial sentences, the translation outputs are not guaranteed to exactly match with the input content, which can be seen in Example \[motivationExample\]. We observe that the translation is often “fantasized” by the model to be a full sentence, as would have occurred in the training data. In the example below, although the English input ends with *‘all of’*, the system generates the translation *‘todo el mundo’* (Engl. *‘all of the world’*). In other cases, the decoder can fall into an over-generation state and repeat the last word several times (eg. ‘debería, debería, debería’). \[motivationExample\] Examples of challenges in using NMT to translate spoken utterances. ---------- ------------------------------------- English: I encourage all of Spanish: yo animo a todo el mundo . English: now , I should Spanish: ahora debería , debería , debería . ---------- ------------------------------------- In this work, we aim to remedy the problem of partial sentence translation in NMT. Ideally, we want a model that is able to generate appropriate translations for incomplete sentences, without any compromise during other translation use cases. Our approach involves using multi-task learning and the automatic generation of parallel corpora in which both the source and the target sentences are incomplete sentences. Related Work ============ The main topic of our work is adapting to different types of inputs for neural machine translation. Previous works have focused on domain mismatch between the training data and test data [@Kobus2017]. In the case of speech translation, the model may only be exposed to specific issues arising from speech recognition outputs during test time. Since speech input can carry over errors from the ASR system to translation, it is necessary to adapt the model to noisier circumstances. To handle this scenario, previous work has proposed introducing artificially corrupted inputs at training time [@Sperber2017a] or direct training on lattices produced by the speech recognition system [@Sperber2017]. Multi-task learning has commonly been used in various NLP problems to jointly train a single model for several well-established NLP tasks, reducing overhead and improving performance. Such implementations can be seen using the encoder-decoder model with attention mechanism [@Luong2015], in which a single model is trained for part-of-speech tagging, named-entity tagging and machine translation simultaneously [@Niehues2017]. Regarding low-latency speech translation, various approaches to translating small text segments exist using statistical phrase-based models [@sridhar2013segmentation; @oda2014optimizing; @shavarani2015learning] or neural networks [@Gu2017]. Due to the fact that the whole input sentence is not available, it is necessary to find a compromise between translation quality and latency. The decoding process of the neural models also needs to be changed to deal with a stream of inputs, which is non-negligible. It is also possible to use the revision strategy to update the partial translations, which has been implemented in practical systems [@Niehues2016LT]. Low-Latency Speech Translation ============================== In the practice of simultaneous speech translation, translation quality is not the only criterion; it is also important to produce a translation for a spoken utterance in real-time and at low latency. Since a speaker’s utterance can be arbitrarily long, it is necessary for the translation system to start operating before the speaker stops, in which case the system input will consist of incomplete spoken segments instead of full sentences. We explore the translation revision method of [@Niehues2016LT], which has been successfully applied to statistical translation systems. The key idea of this method is that the system iteratively revises translations by re-translating new messages sent by the speech recognition component. These newly sent messages are either replacements of or concatenations to previous ones. As a result, the user sees the translation continually updated in the interface. For example, for the sentence *‘I encourage all of you’*, the system first receives only the beginning of the sentence *‘I’*, with the intermediate translation being *‘yo’*. Afterwards, it receives an update which is the continuation of the previous one: *‘I encourage all of’*. The resulting translation from a typical neural model would be *‘yo animo a todo el mundo’*, hypothesizes a final word. Finally, the whole source sentence is available, and the MT system will update the translation of the sentence to *‘yo animo a todos ustedes’*. As can be seen from the above example, in the last translation step the interface has to update the words *‘todo el mundo’* for *‘todos ustedes’*, which was generated only when the full sentence was available. As a result, we experience a delay which comes from the second to last translation step, which is longer than necessary. The interface also suffers from the update, since nearly half of the sentence needs to be replaced. Despite the fact that the final translation quality in the end does not depend on the processing of each segment, the intermediate translation outputs may change drastically due to source sequence updates. The problem is exacerbated by the fact that a neural machine translation model trained with normal parallel corpora is not able to flexibly generate translation for partial input segments, which were not available during training. The aim of our work is to build an online machine translation models which minimizes the number of words which need to be corrected until the full sentence has been seen. We aim to minimizing such criteria while maintaining translation quality for the complete utterance. Partial Translation =================== As motivated in the introduction, an out-of-the-box NMT system struggles with partial input sentences. In order to improve the flexibility of the model, we investigate generating parallel corpora in which the input and output are also partial sentences. Subsequently, we adjust the training process to make use of the data in order to build a single system that is proficient at translating partial as well as full sentences. Generating Partial Parallel Corpora {#task} ----------------------------------- In order to build a system that is good at translating partial sentences, we need to build partial sentence training data. Since such data is not available, we investigated methods to build an artificial training set from standard parallel data. This has the advantage that the methods can be easily applied to any language pair and domain and no new data has to be collected. Creating the source data is straightforward. Given a source sentence $S=s_1 \ldots s_I$, we can generate $I$ input samples $S^{(i)}=s_1 \ldots s_{i}$ by selecting the first $i$ words. The challenge arises from defining the correct translation for this source string. Since we are using this data in a low-latency speech translation system, the translation of the partial sentence should meet several conditions. First, it should be as long as possible in order to minimize the latency of the system. If we always used only the first word, we would not improve latency over a system that waits until the sentence is finished. Furthermore, to minimize the number of corrections, the translation of $S^{(i)}$ should be a substring of $S^{(i')}$ for all $i' > i$. Thus the translation of $S^{(i)}$ should be a substring of the reference translations. One possible solution is to take the reference translation of the whole sentence. But, it is unrealistic to be able to generate the whole target sentence from only a single word in the source string. Therefore, we investigated two methods to select a reasonable substrings from the reference translation. The first method is motivated by the idea that the translation should constantly generate longer target sequences when receiving longer source segments. Furthermore, word reordering may exist between two languages, for many languages sentence structure is similar. Consequently, a first approximation is to use the same proportion of words from the reference translation as we have from the source sentence. One problem in this case is that we introduce additional noise. If the word order is different, we force the system to guess the words coming next in the source sentence. To avoid this problem, we first generate a word alignment using Giza++ [@Och2003] between the source and target sentences. Then, we select the longest prefix of the reference so that no target words in the prefix are aligned to source words that are not in the partial sentences: $$T^{(i)}=\operatorname*{arg\,max}_{j \in J}\{t_1\ldots t_j | \vee j' \le j: a(j') \le i\}$$ Training Process ---------------- #### Multi-task training {#multi-task-training .unnumbered} Given the artificially produced training data, a first step is to train a model on the newly created partial sentence data and use it for speech translation only. Since both tasks are very similar, we first pre-train a standard NMT system and then fine-tune the system to translate partial sentences. The disadvantage of this approach is that the performance on complete sentences might drop, since NMT models tend to rapidly forget what they have learned before. In order to have a system that is able to generate high quality translations of both complete sentences and partial sentences, we opt to use multi-task learning, treating these as two separate tasks. In our approach, we randomly subsample the partial sentence training data to make it the same size as the original training data, so that the model can put equal emphasis on both tasks. The mixed training data then has twice as many sentences as the baseline system, but significantly less than the system using all partial sentences. Then, we fine-tune the NMT system on both tasks: translating complete sentences as well as the partial counterparts. #### Sequence level optimization {#sequence-level-optimization .unnumbered} Beside multi-task learning, we can also guide the search operation of the model so that the generated output is better matched to the source input. We use reinforcement learning with policy gradient methods [@williams1992simple; @ranzato2015sequence] to train the model to maximize the GLEU score [@wu2016google], which is the combination of $n$-gram precision and recall. This reward function restricts the model from generating sentences that are too long. Since this method is known to have high variance gradients, we follow the method in [@rennie2016self], which estimates a baseline using greedy search to reduce the variance. Experimental Results ==================== ------------- ----------------- ---------------- ------- ----------------- ------- --------- ----------- System Valid (tst2011) Test (tst2012) BLEU BLEU BLEU length (tokens) BLEU Word Up Mssg. Up. Baseline 36.86 31.33 26.66 509K 25.97 182K 15.0K Partial 35.45 30.29 29.48 375K 25.54 98K 11.8K Multi-task 37.05 31.27 30.09 376K 26.00 101K 12.0K Align. ref. 37.13 31.06 30.29 371K 26.30 98K 11.5K RL 37.21 31.25 30.08 540K 26.61 179K 15.1K RL + Multi 37.50 31.21 30.31 377K 26.77 82K 11.5K ------------- ----------------- ---------------- ------- ----------------- ------- --------- ----------- We evaluate the method on three different languages pairs: English-Spanish, English-French and German-English. System description ------------------ For all experiments, we trained systems on the Europarl [@Koehn2005] and the WIT-TED corpora [@Cettolo2012WIT] and tested on test sets from the IWSLT evaluation campaign. All systems were adapted to the TED domain by fine-tuning on the in-domain TED data. For the English$\rightarrow$Spanish and English$\rightarrow$French directions, we also optimize the models towards GLEU scores [@wu2016google] using reinforcement learning (RL). For partial sentence translation (both data generation and training), we utilize only the TED corpus. We used the OpenNMT-py toolkit [@opennmt] to train the systems. For each language pair, we jointly trained BPE [@Sennrich2016] for the source and target languages. ----------- ------- ------- ------- --------- ----------- System Final Mix BLEU Word Up Mssg. Up. Baseline 34.11 31.18 23.84 216K 16.3K Multi 34.40 34.71 23.83 128K 13.5K RL 35.08 34.09 24.31 140K 15.0K RL +Multi 34.84 42.51 24.23 99K 12.1K ----------- ------- ------- ------- --------- ----------- : \[result:enfr\] Results for English to French Evaluation metrics ------------------ We evaluated the translation quality using the BLEU score [@Papineni2002]. Since the ASR output uses automatic sentence segmentation, we need to re-segment the translation to fit the reference translations. Therefore, we used the method described in [@Matusov2005], where the automatic translation is re-segmented in a way that minimizes the word error rate to the reference. In addition, we also need to measure the extent to which we are able to reduce the number of corrections in the spoken language translation (SLT) system. To do so, we roll out all updates from the ASR system and translate each. For each updated translation, we measure the overlap between pairs of consecutive updates $s_t$ and $s_{t+1}$ and calculate the amount of re-writing necessary to produce $s_{t+1}$ after $s_t$. Specifically, the number of corrected words is calculated by the length of the translation of $s_t$, minus the length of the common prefix both translations. As illustrated in the example in Section 3, the final update would lead to $3$ corrected words (Word Up). Since an intermediate word change will force the user to reread all following words, our metric also counts all words following the first corrected word as corrected. We also report the number of messages where at least one word is corrected (Messg. Up.). Experiments ----------- #### Initial results {#initial-results .unnumbered} Our initial results on the English$\rightarrow$Spanish translation task are shown in Table \[result:enes\]. We report results in BLEU on the test and validation set with full sentence translation. Next, we report results on the test data with all possible prefixes, and finally, results on the ASR output. In the initial experiments, we use length ratio to determine the length of the reference for the partial translations as described in Section \[task\]. The baseline system is only trained on complete sentences. All other systems use the baseline system, and continue training using different strategies. The system “Partial” is fine-tuned on all partial sentences. As shown in the first two lines of Table \[result:enes\], the final translation quality drops significantly by ${\sim}1$ BLEU point. On the other hand, the BLEU score calculated on only partial sentences improves by ${\sim}3$ BLEU points. As shown by the number of tokens, the length of translations is reduced by 25%. So, a major problem of the baseline system is that it generates translations that are too long for the partial sentences. When testing on the ASR output in the last two columns, we see that the translation quality of the final hypothesis also drops, but the number of words which are updated is reduced by 45%. Also, the number of messages where at least one word changed is reduced by 20%. The system in the third line uses multi-task learning. In this case, the system is trained to perform both tasks: translation of partial and full sentences. Using this technique, we can combine the advantages of both models and maximize the translation quality of the final hypothesis, while minimizing the number of updates. The system has the same translation quality as the baseline system, with the same reduction in updates as the partial system. #### Performance w.r.t the artificial data {#performance-w.r.t-the-artificial-data .unnumbered} In the second set of experiments, we analyzed the use of the artificial data. We used the alignment-based method to generate the references for the partial sentences. In this case, we again fine-tuned used multi-task learning. As shown in the results in Table \[result:enes\], there is no clear performance difference between the two approaches. When translating text input, the system using length-ratio references is better, while the system using alignment-based methods is better on partial sentence and speech translation. Since the length-ratio based method is simpler, we used this approach for the remainder of this work. #### Sequence-level optimization {#sequence-level-optimization-1 .unnumbered} Finally, we also used reinforcement learning (RL) to optimize the performance of the system directly towards BLEU. These systems are first trained using cross-entropy and continue training using reinforcement learning. Here again, we have a baseline system trained only on the full sentences, and a multi-task system trained on both the full and partial sentences (final two lines of Table \[result:enes\]). As above, we observe that with multi-task learning, we do not lose performance on full sentences, while we can significantly reduce the number of updates. In this case, the number of words updated is further reduced, reaching more than 50% less than the baseline. #### English$\rightarrow$French {#englishrightarrowfrench .unnumbered} We also performed experiments using two English to French systems, which are summarized in Table \[result:enfr\]. We again have a baseline system trained with cross-entropy, and a baseline system where training is continued with RL. The models were evaluated on the full sentences and the mixed set of complete and partial sentences, as well as on the ASR output. Similar to the other translation directions, we improved translation performance on partial sentences and reduced the number of rewritten tokens for the SLT output by using multi-task learning. Interestingly, using reinforcement learning also helped us improve the performance on partial sentences. The RL criteria evaluates the n-gram precision as well as the recall of the translation, which is punished when the generated output is too long. Both methods can be combined to achieve the overall best performance, which reduced the rewritten tokens by up to 50% without compromising translation performance. #### German to English {#german-to-english .unnumbered} Finally, we also performed experiments on English to German as shown in Table \[result:deen\]. Again, we can improve the number of rewrites needed to produce the final output. For this language pair, however, the number of updates messages is only slightly reduced. The reason for this might be the larger reordering needed between the two language pairs. ------------ ------- ----------- ----------- System BLEU Words Up. Messg. Up Baseline 15.52 246K 23.6K Multi-task 15.64 172K 23.1K ------------ ------- ----------- ----------- : \[result:deen\] Results for German to English Examples -------- In addition to the evaluation in the last section, improvements using our approach can be seen through examples for English-Spanish and German-English, shown in Table \[example:deen\]. In both cases, we see that the baseline system is not able to generate translations for very short sequences. In this case, the last word is repeated several times. In addition, since the NMT system is tested on input it is not accustomed to, we see that the NMT decoder relies more heavily on language modeling information and completes the sentence in a way that is typical in the target language, regardless of the source input. For example, we see the added *and so on* in the first message for the German to English system. The multi-task system, however, has been trained to handle partial sentences and is therefore able to generate a correct translation. [ll]{}\ Input: & now,\ Baseline: & ahora ,\ Multi-task: & ahora ,\ Input : & now, I should\ Baseline: & ahora debería , debería , debería .\ Multi-task: & ahora debería\ Input : & now, I should men\ Baseline: & ahora debería hombres hombres .\ Multi-task: & ahora debería\ Input : & now, I should mention that this\ Baseline: & ahora debería mencionar esto .\ Multi-task: & ahora , debo mencionarlo .\ \ Input: & Und\ Baseline: & And and and and and so on.\ Multi-task: & And\ Input : & Und ich habe\ Baseline: & And I have\ Multi-task: & And I have\ Input : & Und ich empfehle Ihnen\ Baseline: & And I recommend you to you\ Multi-task: & And I recommend you\ Input : & Und ich empfehle Ihnen .\ Baseline: & And I recommend you .\ Multi-task: & And I recommend you .\ Finally, another interesting point is how the systems handle punctuation. The baseline model for German to English is only able to generate the correct translation if the sentences ends with a punctuation mark. This can be seen in the two last examples, which contain the same words, but only the second has punctuation. The multi-task system, in contrast, is able to generate the correct translation before the input is correctly punctuated. While most errors happen in very short (one or two words) partial sentences, longer partial sentences can also be problematic because of issues like punctuation, which suggests that ignoring short sentences is not a proper solution. Conclusion ========== Low latency translation is important for real-time speech translation systems. To address this challenge, we improve upon a mechanism to translate partial speech input and make updates in real-time. Our main contribution is to propose a simple method to deal with scenarios where data at inference time is different from the training data, which can be resolved with adaptation. We first showed that using simple techniques to generate artificial data are effective to get more fluent output with less correction. We also illustrated that multi-task learning can help adapt the model to the new inference condition, without losing the original capability to translate full sentences. Combining these two ideas, we are able to maintain high quality translation at low latency, minimizing the number of corrected words by 45%, which significantly improves user experience for practical applications. Acknowledgements ================ This work was supported by the Carl-Zeiss-Stiftung.
--- abstract: 'We explore the bifurcations and dynamics of a scalar differential equation with a single constant delay which models the population of human hematopoietic stem cells in the bone marrow. One parameter continuation reveals that with a delay of just a few days, stable periodic dynamics can be generated of all periods from about one week up to one decade! The long period orbits seem to be generated by several mechanisms, one of which is a canard explosion, for which we approximate the dynamics near the slow manifold. Two-parameter continuation reveals parameter regions with even more exotic dynamics including quasi-periodic and phase-locked tori, and chaotic solutions. The panoply of dynamics that we find in the model demonstrates that instability in the stem cell dynamics could be sufficient to generate the rich behaviour seen in dynamic hematological diseases.' author: - 'Daniel C. De Souza' - 'Antony R. Humphries' title: 'Dynamics of a Mathematical Hematopoietic Stem-Cell Population Model' --- Introduction {#sec.introduction} ============ We study the dynamics and bifurcations of the delay-differential equation (DDE) $$\label{Qprime} Q^{\prime}(t) = -(\kappa+\beta(Q(t)))Q(t)+A\beta(Q(t-\tau))Q(t-\tau),$$ where $Q(t){\geqslant}0$ represents the concentration of hematopoietic stem cells (HSCs) in the bone marrow, $A\in(1,2)$ is the amplification factor for cells undergoing division, $\tau>0$ is the division time, the rate at which cells enter division, $\beta(Q)$, is a monotonically decreasing function of $Q$ with $\lim_{Q\to\infty}\beta(Q)=0$, and $\kappa>0$ is the rate that the stem cells differentiate to the progenitors of circulating blood cells. The DDE  represents the $G_0$ cell proliferation model of Burns and Tannock [@Burns_1970]. A full derivation of the DDE can be found in Mackey and Rudnicki [@Mackey_1994], though it was first stated in the form  in Mackey [@Mackey_1978]. In Bernard *et al* [@Bernard_2003] equation  was used to describe the stem cell dynamics as one component of a larger model describing the regulation of circulating neutrophil concentrations. Since then many mathematical models have appeared which contain , or a variant, within larger hematopoiesis models for the production and regulation of neutrophils, erythrocytes and platelets [@Adimy_2006a; @Colijn_2005a; @Colijn_2005b; @Craig_2016; @Langlois2017]. Multiple versions of  have also been coupled together to model discrete levels of stem cell maturity [@Adimy_2006c; @Qu_2010]. The human hematopoietic system produces about $10^{11}$ blood cells of various types per day [@Kaushansky_2016], of which erythrocytes (red blood cells), neutrophils (a type of white blood cell) and platelets are the most common, in a production process which is tightly regulated by a myriad of feedback loops. In dynamical diseases including cyclic neutropenia (CN), cyclic thrombocytopenia (CT) and periodic chronic myelogenous leukemia (PCML), oscillations are observed in the circulating concentrations of one or more of the cell lines [@Foley_2009a]. In mathematical models of hematopoiesis these oscillations arise through Hopf bifurcations as one or more parameters are varied in the model. Two principle mechanisms have been proposed to drive the oscillations in different dynamical diseases [@Bernard_2003]. There can be an instability in the production of the HSCs themselves, with the oscillating HSC numbers then leading to oscillations in the concentrations of peripheral cells. This occurs in PCML where leukemic HSCs typically present a chromosome abnormality [@Pujo_Menjouet_2005]. An alternative mechanism is that oscillations in one cell line can be created through an abnormality in the production of precursor cells in that cell lineage, with the feedback loops from that lineage causing oscillations in the numbers of HSCs differentiating into other cell lines creating concomitant oscillations in the other cell lineages. This occurs in CN, for which a mutation in the ELANE gene that encodes neutrophil elastase leads to increased apoptosis in the neutrophil progenitor cells during mitosis [@Dale_2002]. In other dynamical diseases it remains an open question whether the oscillations are driven by an inherent instability in the HSCs, or whether an instability in production of one of the blood cell lineages is creating the oscillations seen in circulating concentrations. The second possibility is difficult to investigate directly, due to the complexity of the hematopoietic models, with for example the granulopoiesis model of Craig *et al* [@Craig_2016] having five equations, over twenty parameters, and state-dependent delays. In the current work, motivated by the first possibility, we investigate the dynamics of the simple HSC model  as parameters are varied and explore the dynamics that arise. Such an approach alone will not definitively answer the question of whether or not the oscillations in specific dynamical diseases are driven by inherent instability in the HSC dynamics. However, equation  is often incorporated in more complicated hematopoietic models, and if the HSCs can oscillate in the decoupled equation , these oscillations could drive oscillations in the production rates of the mature circulating blood cells which are all produced from the HSCs. Thus, studying the dynamics of equation  allows us to determine when instabilities in the HSC dynamics may arise, and hence if it is feasible for these to drive oscillations in the circulating blood cells concentrations. Although  has been used and studied in numerous models, the codimension-one bifurcation analysis is incomplete, and little is known about codimension-two bifurcations. In Section \[sec.decoup.hsc\] we review the model  and its basic dynamical properties including existence and positivity of solutions, non-dimensionalised formulation, homeostasis (the stable state of an organism maintained by physiological processes) parameter values and existence and stability of steady states. In Section \[sec.stab.bound\] we discuss the stability boundary of the steady state with respect to the delay $\tau$. In Section \[sec.bifurc.hsc\] we carry out a numerical bifurcation analysis of Eq.  by performing parameter continuation on solutions as three of the parameters that control the dynamics are varied individually or pairwise. The one parameter continuations reveal sub- and supercritical Hopf bifurcations, fold bifurcations of periodic orbits and period-doubling bifurcations. This results in bistability between a stable periodic orbit and a steady state, and bistability of two periodic orbits. We also find limit cycles of periods ranging from a week to over 9 years, and an apparent canard explosion [@Benoit1981]. The two-parameter continuations allow us to map out the curves of Hopf, period-doubling and fold bifurcations to determine regions of parameter space for which interesting dynamics occur, and also reveal torus (or Neimark-Sacker) bifurcations. In the following sections we explore some of the more interesting dynamics in more detail. In Section \[sec.longp.hsc\] we study a canard explosion for which the period of solutions increases from about 50 days to over 700 days over an exponentially small parameter interval. We show how to approximate the slow manifold associated with these solutions and show that this manifold has both stable and unstable components. In Section \[sec.chaos.hsc\] we consider non-periodic and chaotic solutions. First in Section \[sec.torus\] we investigate the torus bifurcations found in Section \[sec.bifurc.hsc\], and find a stable invariant torus in the dynamics. We compute Lyapunov exponents and a Poincaré section to show that the dynamics do indeed correspond to a quasi-periodic orbit which envelopes the unstable-periodic orbit from which the torus bifurcated. We also find parameter values for which there is phase-locking on the torus and present the resulting stable periodic orbits. In Section \[sec.chaos\] we study the dynamics between the period-doublings and find period doubling cascades leading to chaos. The chaotic nature of the dynamics is verified numerically by showing that the leading Lyapunov exponent is positive, and visualisations of the attractor which reveal some of its fractal structure. Parameter continuation in opposite directions reveals hysteresis with parameter intervals for which stable chaotic dynamics can co-exist with a stable periodic orbit, or even co-exist with a second chaotic attractor. We also find parameter values for which there appears to be transient chaos. In Section \[sec.snaking\] we present an example of a branch of periodic orbits which snakes in parameter space resulting in a small parameter region in which over 50 limit cycles co-exist. Period-doubling cascades either side of this region lead to additional parameter regions of chaotic dynamics. In Section \[sec.physiol\] we discuss the physiological plausibility and implications of the results with regards to periodic hematological disorders, and Section \[sec.conc\] includes further discussion and conclusions. Hematopoietic Stem Cell Equation {#sec.decoup.hsc} ================================ HSC dynamics can be described by the classic $G_{0}$ cell-cycle model of Burns and Tannock [@Burns_1970]. The HSCs are distinguished between two phases, the proliferating phase and the resting or $G_0$ phase. We denote the concentration of HSCs in the resting phase by $Q$. From the resting phase HSCs may enter the proliferating phase at a rate $\beta(Q)$, or differentiate at a constant rate $\kappa$, or remain in the resting phase. Once HSCs enter the proliferating phase they are lost by apoptosis with a constant rate $\gamma$ or undergo mitosis. The time to complete the cell cycle is $\tau$. After mitosis cells return to the $G_{0}$ resting phase, from whence the cycle may begin again. In the resting phase HSCs are quiescent, while in the proliferating phase they are active and distinguished between four subphases: $G_{1}$, $S$, $G_{2}$ and $M$. Cells at gap $G_{1}$ increase in size and are committed to go through the cell cycle and undergo mitosis. At $S$ phase DNA synthesis occurs, at gap $G_{2}$ cells continue to grow, while in the mitotic phase $M$ cells stop growing and undergo cell division. [fig\_SchematicHSC]{} (6.5,18.9)[$\gamma$]{} (2.1,15.4) (5.2,12.6) (17.2,23.8)[$G_{2}$]{} (26.9,30.5)[$M$]{} (24.5,19) (24,16.5) (25.9,14.0)[$\tau$]{} (17.5,11.2)[$S$]{} (26.0,3.8)[$G_1$]{} (43.4,33.2) (43.6,29.3) (44.0,26.8) (46.8,24.5) (46.5,5.5)[$\beta(Q)$]{} (41.5,1.1) (65.8,17.9) (66.6,15.4) (76.1,16.5)[$G_{0}$]{} (88.5,18.4)[$\kappa$]{} (81.0,15.4) (83.0,13) A schematic of the model is presented in Figure \[fig\_SchematicHSC\]. Following Mackey [@Mackey_1978; @Mackey_1994] this model can be stated as the DDE  where $\beta(Q)$ is a Hill function defined by $$\label{beta2} \beta(Q) = f\frac{\theta^{s}}{\theta^{s}+Q^{s}},$$ and the HSC amplification factor $A$ is given by $$\label{AQ2} A = 2\mathnormal{e}^{-\gamma\tau}.$$ The parameters $\kappa$, $\gamma$, $\tau$, $\theta$, $f$ and $s$ are all strictly positive, and we are interested in non-negative solutions $Q(t){\geqslant}0$, since $Q(t)$ represents a blood cell population. Aspects of the dynamics of the HSC model  have been studied by a number of authors, mainly concentrating on the existence and stability of the steady states, and one parameter continuation of some of the periodic orbits that arise [@Andersen_2001; @Bernard_2004; @Mackey_1994]. A special case of  with $\beta(Q)$ replaced by a step function, corresponding to the limit as $s\to\infty$ in , allows explicit stable periodic solutions to be constructed [@Pujo_Menjouet_2005; @Pujo_Menjouet_2004]. The existence of stable periodic solutions for $s$ large was subsequently established [@Pujo_Menjouet_2006], as a perturbation of the $s=\infty$ solutions. In contrast, we will study the dynamics of  with $\beta(Q)$ a Hill function, with $s$ small, as is usually considered to be the case in hematopoiesis models. In order to solve the DDE  for $t{\geqslant}0$ it is necessary to define an initial function $Q(t)=\varphi(t)$ for $t\in[-\tau,0]$. For $\varphi\in C\coloneqq C([-\tau,0],[0,\infty))$, that is continuous, bounded and non-negative, from the following theorem the solution of  is also bounded and non-negative. It follows that the DDE  can be considered as an infinite dimensional dynamical system with phase space $C$ [@Smith_2010]. \[theorem.bound\] If $Q(t)=\varphi(t)$ for $t\in[-\tau,0]$ where $\varphi\in C([-\tau,0],[0,\infty))$, then equation  has a unique solution $Q(t)$ defined for all $t{\geqslant}0$ and which satisfies $Q(t)\in[0,M]$ for all $t{\geqslant}0$ for some $M<\infty$. As already noted by other authors, uniqueness and local existence of solutions follows from the method of steps. It suffices to show that solutions are bounded to obtain global existence and complete the proof [@Pujo_Menjouet_2006; @Pujo_Menjouet_2005; @Pujo_Menjouet_2004]. To show positivity of the solution, let $t^*{\geqslant}0$ such that $Q(t){\geqslant}0$ for $t\in[-\tau,t^*]$. It follows easily from that for all $t\in(t^*,t^*+\tau)$ for which $Q(t)\in(-\theta,0)$ that $Q'(t)>0$. This leads to a contradiction unless $Q(t){\geqslant}0$ for all $t\in(t^*,t^*+\tau)$. Hence $Q(t){\geqslant}0$ for all $t{\geqslant}0$. The existence of an upper bound $Q(t){\leqslant}M$ follows from $\limsup_{t\to\infty}Q(t)$ being bounded, which was shown by Mackey and Rudnicki [@Mackey_1994] for non-negative solutions with a general class of monotonic functions $\beta(Q)$ that includes . In - there are six parameters $\{\kappa,\gamma,\tau,\theta,f,s\}$, but we can reduce these to four by non-dimensionalising the equations. Let $\hat{t}\coloneqq t/\tau$, $\hat{f}\coloneqq\tau f$$, \hat{\kappa}\coloneqq \kappa/f$, $\hat{\gamma}\coloneqq \gamma\tau$, and $\hat{Q}(\hat t)\coloneqq Q(t)/\theta$, then $\hat{Q}(\hat t-1)= Q(t-\tau)/\theta$. It is also convenient to define $\hat{A}\coloneqq 2\mathnormal{e}^{-\hat{\gamma}}=2\mathnormal{e}^{-\gamma\tau}=A$, and notice that $\beta(\hat{Q})=f/(1+\hat{Q}^{s}).$ Then becomes $$\label{dQdt_hat} \frac{1}{\hat{f}}\frac{d\hat{Q}}{d\hat{t}}(\hat t)= -\hat{\kappa}\hat{Q}(\hat t) -\frac{\hat{Q}(\hat t)}{1+\hat{Q}(\hat t)^{s}}+\hat{A}\frac{\hat{Q}(\hat t-1)}{1+\hat{Q}(\hat t-1)^{s}},$$ which depends on the four parameters $\hat{f}$, $\hat{\kappa}$, $s$ and $\hat{A}$ (or $\hat\gamma$). Although many mathematicians would prefer to study the non-dimensionalised DDE  instead of , we chose to work with  so that the solutions and bifurcations that we find have direct physiological interpretations. But, as suggested by the non-dimensionalisation, we need only vary four parameters in . It is easily seen that varying the four parameters $\gamma$, $\kappa$, $\tau$ and $s$ in , with the other parameters held constant, we can reproduce all possible values of $\hat{f}$, $\hat{\kappa}$, $s$ and $\hat{A}$ in , and hence we will only need to consider the variation of parameters from amongst these four. In Table \[tab.model.par\] we state homeostatic values of the parameters for the model -. The values in Table \[tab.model.par\] are all taken from Craig *et al* [@Craig_2016]. The first two parameters in the table, $Q^{h}$ the homeostatic concentration of HSCs, and $\beta(Q^h)$ the homeostatic rate that cells enter the cell cycle, do not appear explicitly in the model -, but are used to calculate the last two parameters. To ensure that $Q^{h}$ is exactly the homeostatic concentration of HSCs in the model -, the last three parameters are computed in double precision. While $A$ is given directly by , rearranging   and  at homeostasis implies that $$\label{homeoparas} \theta=Q^h\left(\frac{f}{\beta(Q^h)} - 1\right)^{-1/s}, \qquad \kappa=(A-1)\beta(Q^h).$$ We will use the parameters of Table \[tab.model.par\] as a starting point for our bifurcation studies. Name Interpretation Value Units -------------- --------------------------------------- ------------- ----------------------------- -- $Q^{h}$ HSC homeostasis concentration $1.1$ ${10}^{6}\mathrm{cells/kg}$ $\beta(Q^h)$ homeostasis cell cycle entry rate $0.043$ $\mathrm{days}^{-1}$ $\gamma$ HSC apoptosis rate $0.1$ $\mathrm{days}^{-1}$ $\tau$ Time for HSC re-entry $2.8$ $\mathrm{days}$ $f$ Maximal HSC re-entry rate $8$ $\mathrm{days}^{-1}$ $s$ HSC re-entry Hill coefficient $2$ $-$ $A$ HSC Amplification Factor $1.512^*$ $-$ $\theta$ Half-effect HSC concentration $0.08086^*$ ${10}^{6}\mathrm{cells/kg}$ $\kappa$ HSC differentiation rate to all lines $0.022^*$ $\mathrm{days}^{-1}$ : Homeostasis values of the parameters for the mathematical granulopoiesis model -, which are all taken from Craig *et al* [@Craig_2016]. The last three parameters with values denoted by $^*$, are only stated to 4 significant figures here, but actual values are computed to full double precision in MATLAB [@Matlab] using Eqs.  and .[]{data-label="tab.model.par"} For general parameter values, from , steady states satisfy $$\label{dQdt.star} 0 = [(A-1)\beta(Q)-\kappa]Q.$$ Hence the DDE  has the trivial steady state $Q(t)=0$ for all values of the parameters. The trivial steady state has been shown to be globally asymptotically stable if $\kappa>f(A-1)$ [@Mackey_1994]. Eq.  has another solution $Q^*$ which satisfies $$\label{betaQ.star} \beta(Q^{*}) = \frac{\kappa}{(A-1)}.$$ Since $\beta(Q)$ is monotonic, this defines a unique nontrivial steady state $Q^*$. Using  we have $$\label{Q.star} Q^{*} = \theta\left[f\frac{(A-1)}{\kappa} - 1\right]^{1/s}.$$ From , along with the relation  we obtain that $Q^*>0$ if and only if the upper bounds $$\label{kgt} \kappa < f(A-1),\qquad \gamma\tau < \ln\left(\frac{2f}{\kappa+f}\right),$$ on the parameters $\kappa$, $\gamma$ and $\tau$ are satisfied. In the rest of this work, we will consider the case where  holds and there are two steady states, $Q=0$ and $Q^*>0$. From  and  we require $A\in(1+\kappa/f,2)$ for $Q^*>0$ to exist. Regarding the steady state $Q^*>0$ as a function of the differentiation rate $\kappa$, the death rate $\gamma$, or the cell cycle duration $\tau$, it is easy to see from  that $Q^*$ is a monotonically decreasing function with respect to each of these parameters, as would be expected from a physiological point of view. Furthermore $Q^*\to0$ as equality is approached in . We denote the non-zero steady state by $Q^h$ only when the parameters take their homeostasis values from Table \[tab.model.par\], and by $Q^*$ otherwise. Unless otherwise stated, all the values in this paper are given in the same units as in Table \[tab.model.par\]. Stability Boundary {#sec.stab.bound} ------------------ To determine the stability of the steady state $Q^*$, linearise the DDE  around $Q^{*}$, with $z(t)\coloneqq Q(t)-Q^{*}$ to get $$\label{dQdtLin} z^{\prime}(t)=az(t)+bz(t-\tau).$$ It is convenient to define $$\label{hx} h(x)\coloneqq x\beta(x),$$ then the parameters $a$ and $b$ can be written as $$\label{ab} a =-\kappa-\beta(Q^{*})-Q^{*}\beta^{\prime}(Q^{*})=-\kappa-h'(Q^*),\qquad b = A\left(\beta(Q^{*})+Q^{*}\beta^{\prime}(Q^{*})\right)=Ah'(Q^*).$$ Seeking a nontrivial solution $z(t)=z_{0}\mathnormal{e}^{\lambda t}$ for equation , we get the characteristic equation $$\label{hayes} p(\lambda)\coloneqq \lambda - a - b\mathnormal{e}^{-\lambda\tau} = 0,$$ first studied by Hayes [@Hayes_1950]. According to the *Principle of Linearised Stability* [@Smith_2010] the stability analysis of the steady states for the nonlinear DDE  is reduced to the stability analysis of the steady state of the linearised equation . Stability analysis of equation  is a standard example in DDEs, and can be found in [@Breda_2015; @Hale_1993; @Insperger_2011; @Smith_2010]. The steady state is unstable if $b>-a$ with a characteristic value $\lambda>0$, and it is asymptotically stable if $a<-|b|<0$, which is sometimes called the delay-independent stability region [@Kolmanovskii_1999]. The interesting parameter region is for $b{\leqslant}-|a|<0$, where the steady state is asymptotically stable for $$\label{tau1} \tau<\tau_1(a,b)\coloneqq\cos^{-1}(-a/b)/\sqrt{b^2-a^2}$$ and unstable if $\tau>\tau_1(a,b)$. The curve $C_0$ described by $\tau_1(a,b)$ is contained in the region $a{\leqslant}1/\tau$ and $b{\leqslant}-1/\tau$ and can be parameterised [@Smith_2010] as $$\label{C0} C_0=\bigl\{(a,b)=(\omega\cot(\omega)/\tau,-\omega\csc(\omega)/\tau), \quad \omega\in[0,\pi)\bigr\}.$$ On this curve the characteristic equation  has an imaginary solution $\lambda=\pm i\omega$. The parameter region $(a\tau,b\tau)\in\mathbb{R}^2$ for which $Q^*$ is stable is illustrated in Figure \[fig:Hayesabstab\]. In the context of the DDE  the trivial steady state $Q=0$ is unstable when $Q^*>0$ [@Qu_2010]. For the stability of $Q^*$, from  and  we have $a+b=(A-1)Q^*\beta'(Q^*),$ and hence using  and  we find that $-s\kappa<-s\kappa(1-\kappa/[f(A-1)])=a+b<0.$ Thus when $Q^*>0$ we have $a+b<0$, and $Q^*$ can only lose stability if $(a,b)$ crosses the curve $C_0$ as parameters are varied. The steady state $Q^*>0$ is stable for all $\tau$ sufficiently small, since $\lambda=-(a+b)<0$ when $\tau=0$. It is also stable for all $\tau$ sufficiently large. This follows from noting that $b>0$ when $h'(Q^*)>0$, which holds whenever $\tau>\tau_2$ where $\tau_2$ is defined by below. Then, for $\tau>\tau_2$ the parameters $(a,b)$ are in the upper half of the delay-independent stability region $a<-|b|<0$. The steady state $Q^*$ may lose stability in a Hopf bifurcation if the parameters cross the curve $C_0$ in the $(a,b)$ parameter space, but since the steady state is stable for $\tau$ small and for $\tau$ large, such Hopf bifurcations will occur in pairs, corresponding to crossings of $C_0$ in opposite directions. For many DDEs the steady state gains ever more characteristic values with positive real part as the delay $\tau$ is increased [@Longtin1998]. The Burns-Tannock DDE  does not behave like that because of the exponential term in $A=2e^{-\gamma\tau}$ representing mortality during the cell cycle. For the homeostasis parameter values in Table \[tab.model.par\] we have $f(A-1)=4.093>\kappa=0.022$ so is satisfied and there exists a unique positive steady state, $Q^h=1.1>0$. Furthermore, at homeostasis $a=0.020540$ and $b=-0.064298$ and $\tau=2.8$ which is inside the stability region, as illustrated in Figure \[fig:Hayesabstab\], so the homeostasis steady state $Q^h$ is asymptotically stable. If $\tau$ is varied and all the other parameters are at their homeostasis values from Table \[tab.model.par\], we find that $Q^*>0$ is stable for $\tau\in[0,\tau_1^-)$ and for $\tau\in(\tau_1^+,\tau_{\max})$ where $$\label{hstabtau} 0 < \tau^{h} = 2.8 < \tau_1^- = 5.74851 < \tau_1^+ = 6.87437 < \tau_2 = 6.87662 < \tau_{\max} = 6.90401.$$ Here, $\tau_{\max}$ is given by , while $$\label{tau12} \tau_1^\pm=\frac{\cos^{-1}((\kappa+h'(Q^*))/A h'(Q^*))}{\sqrt{(A h'(Q^*))^2-(\kappa+h'(Q^*))^2}}, \qquad \tau_2=\frac{1}{\gamma}\ln\left(\frac12\left[1+\frac{\kappa s}{f(s-1)}\right]\right)^{-1}.$$ The formula for $\tau_2$ is found by using the expressions for $h(Q)$ and $Q^*$ and solving for $b=0$. This formula was already stated in Pujo Menjouet *et al* [@Pujo_Menjouet_2005; @Pujo_Menjouet_2004], where it was erroneously claimed that $\tau=\tau_2$ was a stability boundary. As noted above already, and as shown in Figure \[fig:Hayesabstab\], the parameters corresponding to $\tau=\tau_2$ are in the interior of the stability region. The locus of the parameters $(a,b)$ and the corresponding $\tau$ values from  are illustrated in Figure \[fig:Hayesabstab\]. ![The parameter regions for which the steady state of equation  is stable (white) and unstable (shaded). Also shown (red) is the locus in $(a,b)$ of the parameters defined by as $\tau$ is varied with the other parameters all at their values from Table \[tab.model.par\]. The values of $\tau$ from  are indicated on this curve.[]{data-label="fig:Hayesabstab"}](HayesabstabShaded "fig:"){width="55.00000%"} (-10,0)[$a\tau$]{} (-275,135)[$b\tau$]{} (-65,41)[$\tau_1^-$]{} (-38,48.5)[$\tau_1^+$]{} (-217,117)[$\tau_2$]{} (-189,102)[$\tau^h$]{} (-115,25)[$C_{0}$]{} If parameters are varied so that $\tau$ crosses the boundary $\tau_1$, either by varying $\tau$ itself, or by varying parameters to change the values of $\tau_1$, then the stability of $Q^*$ changes and a periodic orbit with period $2\pi/\omega$ is created in a Hopf bifurcation. The periodic orbits thus created have been explored to some extent [@Fowler_Mackey_2002; @Pujo_Menjouet_2006; @Pujo_Menjouet_2005; @Pujo_Menjouet_2004], but the details of the Hopf bifurcation and its normal form have only been studied more recently [@Qu_2010]. Because  has infinitely many roots, it is possible for additional pairs of complex conjugate characteristic values to cross the imaginary axis resulting in additional Hopf bifurcations. These will occur on curves $C_{2n}$, which are defined by but for $\omega\in[2n\pi,(2n+1)\pi)$ [@Smith_2010]. None of these curves intersect, so there are no double-Hopf bifurcations in which two pairs of characteristic values cross the imaginary axis at the same time. Bifurcations of the HSC Equation {#sec.bifurc.hsc} ================================ In this section we vary parameters from their homeostasis values so that the perturbed steady state $Q^*$ becomes unstable, and survey the bifurcations and dynamics that arise. We begin our numerical bifurcation analysis of the DDE , by studying codimension-one bifurcations performing parameter continuation on $\tau$, $\gamma$ and $\kappa$, one at a time. We will also study codimension-two bifurcations carrying out a two-parameter continuation for each pair of these three parameters. Our numerical bifurcation diagrams are constructed using the well-established DDEBiftool package [@Engelborghs_2002; @DDEBiftool15], which runs under MATLAB [@Matlab]. This software finds periodic orbits using a boundary value approach, and is able to find stable and unstable solutions, and continue the solutions as parameter(s) are varied and detect stability and bifurcations. As noted after , the dynamics of the HSC DDE  only depend on the four parameters $\tau$, $\gamma$, $\kappa$ and $s$. The dependence of the dynamics on the parameter $s$ has previously been studied for small integer values of $s$ (through numerical simulation) and analytically in the limit as $s\to\infty$ (in which case the Hill function  simplifies to a Heaviside function) [@Pujo_Menjouet_2006; @Pujo_Menjouet_2005; @Pujo_Menjouet_2004]. In the current work we will keep $s=2$ fixed and equal to its value in Craig *et al* [@Craig_2016], and we will only vary the parameters $\tau$, $\gamma$, and $\kappa$. In the following, unless mentioned otherwise, all parameters take the values stated in Table \[tab.model.par\]. Recall that the homeostasis steady state $Q^h$ is stable. One Parameter Continuation {#sec.1p} -------------------------- ![(i) Bifurcation diagram showing stability of the steady state $Q^*$, along with the branch of periodic orbits which bifurcates from $Q^*$ at the Hopf bifurcation points. All parameters except $\kappa$ take values from Table \[tab.model.par\]. In all bifurcation diagrams stable (unstable) steady states are represented by solid (dotted) black lines while the stable (unstable) limit cycles are represented by solid (dotted) blue lines. For periodic orbits we plot both the maximum and minimum value of $Q$ over the period, so the upper and lower curve both represent the same periodic orbit. Hopf bifurcation points , saddle-node bifurcation of limit cycles points , and period-doubling bifurcation points , are indicated, and also highlighted in insets. The saddle-node bifurcation in the right inset creates an interval for $\kappa\in(0.16872,0.17632)$ of bistability between the periodic orbit and the steady state $Q^*$, while the pair of steady fold bifurcations seen in the left inset create an interval for $\kappa\in(0.97254,1.0247)$ of bistability between two different periodic orbits. (ii) Shows the period of the periodic orbits seen in (i). Two period doubling bifurcations and an interval for $\kappa\in(1.173,1.2506)$ of period-doubled solutions are also shown. These can also be seen in the left inset in (i).[]{data-label="fig:KappaCont"}](fig2_HSC_Biftool1D_KappaA "fig:"){width="49.00000%" height="44mm"} (-245,2.5) ![(i) Bifurcation diagram showing stability of the steady state $Q^*$, along with the branch of periodic orbits which bifurcates from $Q^*$ at the Hopf bifurcation points. All parameters except $\kappa$ take values from Table \[tab.model.par\]. In all bifurcation diagrams stable (unstable) steady states are represented by solid (dotted) black lines while the stable (unstable) limit cycles are represented by solid (dotted) blue lines. For periodic orbits we plot both the maximum and minimum value of $Q$ over the period, so the upper and lower curve both represent the same periodic orbit. Hopf bifurcation points , saddle-node bifurcation of limit cycles points , and period-doubling bifurcation points , are indicated, and also highlighted in insets. The saddle-node bifurcation in the right inset creates an interval for $\kappa\in(0.16872,0.17632)$ of bistability between the periodic orbit and the steady state $Q^*$, while the pair of steady fold bifurcations seen in the left inset create an interval for $\kappa\in(0.97254,1.0247)$ of bistability between two different periodic orbits. (ii) Shows the period of the periodic orbits seen in (i). Two period doubling bifurcations and an interval for $\kappa\in(1.173,1.2506)$ of period-doubled solutions are also shown. These can also be seen in the left inset in (i).[]{data-label="fig:KappaCont"}](fig3_HSC_Biftool1D_KappaA "fig:"){width="49.00000%" height="44mm"} (-245,2.5) We begin by varying the differentiation rate $\kappa$ (with all the other parameters held constant), with the bifurcation diagram presented in Figure \[fig:KappaCont\]. The non-trivial steady state $Q^*$ (given by ) is seen to be unstable for an interval of $\kappa$ values between two Hopf bifurcation points, and stable for $\kappa$ outside this interval. The Hopf bifurcation point at $\kappa\approx0.17632$ is subcritical leading to a branch of unstable periodic orbits which becomes stable at $\kappa\approx0.16872$ in a fold bifurcation of periodic orbits. This creates a small interval of bistability between the stable steady state $Q^*$ and the stable periodic orbit of amplitude close to $1$. Similar bistability has been observed before in hematopoiesis models. Bernard *et al* [@Bernard_2003; @Bernard_2004] studied a model for white blood cell (WBC) production which incorporated  to model the stem cell dynamics, and found bistability for WBCs between a stable steady state and a stable periodic orbit. But in that model the bistability was seen as an amplification parameter in the WBC proliferation was varied, and was not associated with the variation of any parameter in the HSC equation. We are not aware of bistability having been observed previously in a stand-alone model for HSC dynamics. There is a pair of fold bifurcations of periodic orbits which creates an interval for $\kappa\in(0.97254,1.0247)$ of bistability of periodic orbits. Bistability is interesting, as it allows the possibility for a short term perturbation of the system (such as during treatment) to cause the solution to switch between one solution and another, and for the new stable dynamics to persist indefinitely. The two instances of bistability observed here occur for relatively small parameter intervals far from the homeostasis parameters and so are unlikely to be of great direct physiological relevance for healthy subjects. However the existence of bistability is interesting in the context of dynamical diseases which are related to bifurcations that occur when parameters in the system are varied. There is also a pair of period-doubling bifurcations on the branch of periodic orbits illustrated in Figure \[fig:KappaCont\]. This leads to an interval for $\kappa\in(1.173,1.2506)$ where a period-doubled orbit is stable. The amplitude of the periodic orbits on the main branch tends to decrease along the branch, and the periodic orbits disappear in a super-critical Hopf bifurcation at $\kappa\approx1.5317$. For $\kappa$ between this value and its bound given by , the steady-state $Q^*$ is stable. The periods of the orbits shown in Figure \[fig:KappaCont\](ii) strongly correlate with the amplitude of the orbits. The periods are all larger than a week, much larger than the delay $\tau=2.8$ days. ![Periodic Orbits from the branch seen in Figure \[fig:KappaCont\]. (i) For $\kappa=0.17$ the stable (solid line) and unstable (dotted line) periodic orbits which coexist with the stable steady state $Q^*$ just before the subcritical Hopf bifurcation. (ii) The same periodic orbits shown in the two-dimensional $(Q(t),Q(t-\tau))$ projection of phase space. (iii-iv) For $\kappa=1$ the two coexisting stable periodic orbits along with the unstable periodic orbit and unstable steady state. (v-vi) For $\kappa=1.2$ the stable period-doubled orbit (solid line), along with the unstable periodic orbit from which it bifurcates (in panel (v) we show two periods of this orbit). []{data-label="KappaCont_Orbits_Bistab_kappa_plots"}](KappaCont_Orbits_Bistab_kappa_017 "fig:"){width="49.00000%" height="44mm"} (-244,2.5) ![Periodic Orbits from the branch seen in Figure \[fig:KappaCont\]. (i) For $\kappa=0.17$ the stable (solid line) and unstable (dotted line) periodic orbits which coexist with the stable steady state $Q^*$ just before the subcritical Hopf bifurcation. (ii) The same periodic orbits shown in the two-dimensional $(Q(t),Q(t-\tau))$ projection of phase space. (iii-iv) For $\kappa=1$ the two coexisting stable periodic orbits along with the unstable periodic orbit and unstable steady state. (v-vi) For $\kappa=1.2$ the stable period-doubled orbit (solid line), along with the unstable periodic orbit from which it bifurcates (in panel (v) we show two periods of this orbit). []{data-label="KappaCont_Orbits_Bistab_kappa_plots"}](KappaCont_Orbits_PS_Bistab_kappa_017 "fig:"){width="49.00000%" height="44mm"} (-244,2.5) ![Periodic Orbits from the branch seen in Figure \[fig:KappaCont\]. (i) For $\kappa=0.17$ the stable (solid line) and unstable (dotted line) periodic orbits which coexist with the stable steady state $Q^*$ just before the subcritical Hopf bifurcation. (ii) The same periodic orbits shown in the two-dimensional $(Q(t),Q(t-\tau))$ projection of phase space. (iii-iv) For $\kappa=1$ the two coexisting stable periodic orbits along with the unstable periodic orbit and unstable steady state. (v-vi) For $\kappa=1.2$ the stable period-doubled orbit (solid line), along with the unstable periodic orbit from which it bifurcates (in panel (v) we show two periods of this orbit). []{data-label="KappaCont_Orbits_Bistab_kappa_plots"}](KappaCont_Orbits_Bistab_kappa_1 "fig:"){width="49.00000%" height="44mm"} (-244,2.5) ![Periodic Orbits from the branch seen in Figure \[fig:KappaCont\]. (i) For $\kappa=0.17$ the stable (solid line) and unstable (dotted line) periodic orbits which coexist with the stable steady state $Q^*$ just before the subcritical Hopf bifurcation. (ii) The same periodic orbits shown in the two-dimensional $(Q(t),Q(t-\tau))$ projection of phase space. (iii-iv) For $\kappa=1$ the two coexisting stable periodic orbits along with the unstable periodic orbit and unstable steady state. (v-vi) For $\kappa=1.2$ the stable period-doubled orbit (solid line), along with the unstable periodic orbit from which it bifurcates (in panel (v) we show two periods of this orbit). []{data-label="KappaCont_Orbits_Bistab_kappa_plots"}](KappaCont_Orbits_PS_Bistab_kappa_1 "fig:"){width="49.00000%" height="44mm"} (-244,2.5)\ ![Periodic Orbits from the branch seen in Figure \[fig:KappaCont\]. (i) For $\kappa=0.17$ the stable (solid line) and unstable (dotted line) periodic orbits which coexist with the stable steady state $Q^*$ just before the subcritical Hopf bifurcation. (ii) The same periodic orbits shown in the two-dimensional $(Q(t),Q(t-\tau))$ projection of phase space. (iii-iv) For $\kappa=1$ the two coexisting stable periodic orbits along with the unstable periodic orbit and unstable steady state. (v-vi) For $\kappa=1.2$ the stable period-doubled orbit (solid line), along with the unstable periodic orbit from which it bifurcates (in panel (v) we show two periods of this orbit). []{data-label="KappaCont_Orbits_Bistab_kappa_plots"}](KappaCont_Orbits_PD_kappa_121 "fig:"){width="49.00000%" height="44mm"} (-244,2.5) ![Periodic Orbits from the branch seen in Figure \[fig:KappaCont\]. (i) For $\kappa=0.17$ the stable (solid line) and unstable (dotted line) periodic orbits which coexist with the stable steady state $Q^*$ just before the subcritical Hopf bifurcation. (ii) The same periodic orbits shown in the two-dimensional $(Q(t),Q(t-\tau))$ projection of phase space. (iii-iv) For $\kappa=1$ the two coexisting stable periodic orbits along with the unstable periodic orbit and unstable steady state. (v-vi) For $\kappa=1.2$ the stable period-doubled orbit (solid line), along with the unstable periodic orbit from which it bifurcates (in panel (v) we show two periods of this orbit). []{data-label="KappaCont_Orbits_Bistab_kappa_plots"}](KappaCont_Orbits_PS_PD_kappa_121 "fig:"){width="49.00000%" height="44mm"} (-244,2.5) Figure \[KappaCont\_Orbits\_Bistab\_kappa\_plots\] illustrates some of the more interesting periodic orbits found during the $\kappa$ continuation, including examples of bistability and period doubled orbits. The left panels show periodic solution profiles over one period, while the right panels display the time-delay embedding of the same solutions. Recalling that $Q(t)$ is a scalar, but that the DDE  defines an infinite dimensional dynamical system, $(Q(t),Q(t-\tau))$ gives a useful two-dimensional projection of the infinite dimensional solutions, which has been used widely since it was introduced by Glass and Mackey [@Glass_Mackey_1979]. Since it is only a projection of phase space, orbits may appear to cross each other, but because it incorporates both the terms $Q(t)$ and $Q(t-\tau)$ that appear in  this projection is often very revealing. ![Continuation in $\gamma$ with other parameters taking values from Table \[tab.model.par\]. (i) Bifurcation diagram showing stability of the steady state $Q^*$, along with the branch of periodic orbits which bifurcates from the steady state at the Hopf bifurcation points at $\gamma=0.227918$ and $\gamma=0.245375$. A saddle-node bifurcation seen in the top inset creates an interval for $\kappa\in(0.227766,0.227918)$ of bistability between the periodic orbit and the steady state $Q^*$. (ii) Shows the period of the periodic orbits seen in (i). []{data-label="fig:GammaCont"}](fig2_HSC_Biftool2D_Gamma_Kappa_TonyParams_Insets "fig:"){width="49.00000%" height="44mm"} (-245,2.5) ![Continuation in $\gamma$ with other parameters taking values from Table \[tab.model.par\]. (i) Bifurcation diagram showing stability of the steady state $Q^*$, along with the branch of periodic orbits which bifurcates from the steady state at the Hopf bifurcation points at $\gamma=0.227918$ and $\gamma=0.245375$. A saddle-node bifurcation seen in the top inset creates an interval for $\kappa\in(0.227766,0.227918)$ of bistability between the periodic orbit and the steady state $Q^*$. (ii) Shows the period of the periodic orbits seen in (i). []{data-label="fig:GammaCont"}](fig3_HSC_Biftool2D_Gamma_Kappa_TonyParamsC_Insets "fig:"){width="49.00000%" height="44mm"} (-245,2.5) ![Three example periodic orbits from the branch shown in Figure \[fig:GammaCont\]. (i) Solution profiles and (ii) the corresponding $(Q(t),Q(t-\tau))$ delay embeddings, with the location of the orbits on the bifurcation branch indicated on the insets.[]{data-label="fig:GammaOrb"}](figGammaOrbitA "fig:"){width="49.00000%" height="44mm"} (-245,2.5) ![Three example periodic orbits from the branch shown in Figure \[fig:GammaCont\]. (i) Solution profiles and (ii) the corresponding $(Q(t),Q(t-\tau))$ delay embeddings, with the location of the orbits on the bifurcation branch indicated on the insets.[]{data-label="fig:GammaOrb"}](figGammaEmbbA "fig:"){width="49.00000%" height="44mm"} (-245,2.5) Figure \[fig:GammaCont\] shows the results of applying one parameter continuation in the apoptosis rate $\gamma$, with the other parameters all held at their values in Table \[tab.model.par\]. The steady state $Q^*$ is stable unless $\gamma$ is close to its upper bound defined by . There is again a pair of Hopf bifurcations with the left bifurcation at $\gamma\approx0.227918$ subcritical leading to an unstable periodic orbit with period $\approx36.7$ days at the bifurcation point. The period grows to about $82$ days at a fold bifurcation of periodic orbits with $\gamma\approx0.227766$. The periodic orbit becomes stable at the fold bifurcation leading to a very short interval of bistability between the stable periodic orbit and the stable steady state. As $\gamma$ is increased from the fold bifurcation the stable periodic orbits gradually decrease in amplitude but increase in period reaching a maximum period of about $714$ days when $\gamma\approx0.2453692$. Some of these stable periodic orbits are illustrated in Figure \[fig:GammaOrb\]. These orbits all have a single peak above $Q^*$ which is only achieved once per period. After this peak the value of $Q$ quickly drops to below $0.1$, and there is then a very low amplitude oscillation in $Q$ with a period of about $3$ days (slightly larger than the delay $\tau=2.8$) which decays in amplitude before the next spike in the number of HSCs. At $\gamma=0.2453692$ equation  gives that $Q^*=0.089673$ and the longest period orbit shown in Figure \[fig:GammaOrb\] is close to homoclinic to the steady state $Q^*$ ($Q'(t)\approx0.00025$ when $Q(t)=Q^*$). The long-period orbits appear to be relaxation oscillations; these have been observed and studied previously for   [@Colijn2006; @Fowler_Mackey_2002]. Visually, these solution profiles are more reminiscent of a spiking neuron [@Izhikevich_2007] than what one would naively expect to see in blood cell concentrations. After the maximum period is achieved at $\gamma\approx0.2453692$ the period declines precipitously to approximately $48$ days at the Hopf bifurcation when $\gamma\approx0.245375$ in an apparent canard explosion. We are not aware of a canard being observed in a scalar DDE before, and we will investigate this phenomenon in Section \[sec.longp.hsc\]. ![Continuation in $\tau$ with other parameters taking values from Table \[tab.model.par\]. (i) Bifurcation diagram showing stability of the steady state $Q^*$, along with the branch of periodic orbits which bifurcates from the steady state at the Hopf bifurcation points at $\tau=5.74851$ and $\tau=6.87437$. A saddle-node bifurcation seen in the top inset creates an interval for $\tau\in(5.72939,5.74851)$ of bistability between the periodic orbit and the steady state $Q^*$. (ii) Shows the period of the periodic orbits seen in (i).[]{data-label="fig:TauCont"}](fig2_HSC_Biftool1D_TauMerged "fig:"){width="49.00000%" height="44mm"} (-245,2.5) ![Continuation in $\tau$ with other parameters taking values from Table \[tab.model.par\]. (i) Bifurcation diagram showing stability of the steady state $Q^*$, along with the branch of periodic orbits which bifurcates from the steady state at the Hopf bifurcation points at $\tau=5.74851$ and $\tau=6.87437$. A saddle-node bifurcation seen in the top inset creates an interval for $\tau\in(5.72939,5.74851)$ of bistability between the periodic orbit and the steady state $Q^*$. (ii) Shows the period of the periodic orbits seen in (i).[]{data-label="fig:TauCont"}](fig3_HSC_Biftool1D_TauMerged "fig:"){width="49.00000%" height="44mm"} (-245,2.5) ![Four stable periodic orbits from neighbouring peaks and troughs of the ripples illustrate how they arise. As seen in the inset of the time-embedding plot, the large period and amplitude orbits from the peaks of the ripples perform one more small amplitude oscillation before escaping to $Q(t)\gg Q^*$ compared to the smaller period and amplitude solution from the preceding trough in the ripples. []{data-label="fig:TauOrb"}](fig_tau_bi_orb3B "fig:"){width="49.00000%" height="44mm"} (-245,2.5) ![Four stable periodic orbits from neighbouring peaks and troughs of the ripples illustrate how they arise. As seen in the inset of the time-embedding plot, the large period and amplitude orbits from the peaks of the ripples perform one more small amplitude oscillation before escaping to $Q(t)\gg Q^*$ compared to the smaller period and amplitude solution from the preceding trough in the ripples. []{data-label="fig:TauOrb"}](fig_tau_bi_embbB "fig:"){width="49.00000%" height="44mm"} (-245,2.5) In Figure \[fig:TauCont\] we present the dynamics observed from applying one parameter continuation in the delay $\tau$, with the other parameters held at their values in Table \[tab.model.par\]. The steady state $Q^*$ is again stable unless the delay $\tau$ is close to its upper bound, and only unstable for $\tau$ between the pair of Hopf bifurcation points, which occur at $\tau=\tau_1^-$ and $\tau=\tau_1^+$, where $\tau^\pm$ are defined in . The left Hopf point is subcritical, leading to unstable orbits of period about $52$ days, growing to a period of about $124$ days at a saddle-node bifurcation of periodic orbits with $\tau\approx5.72940$ days, where the periodic orbits become stable, creating an interval of bistability between the steady state and the stable periodic orbits. Ripples are visible in the amplitude of the branch of stable periodic orbits, with the magnitude of these undulations decreasing to zero as $\tau$ approaches the right Hopf bifurcation point, as shown in the bottom right inset of Figure \[fig:TauCont\](i). There are corresponding ripples in the period of the orbits, visible in the first inset of Figure \[fig:TauCont\](ii). The other insets show details of the branch of periodic orbits near the Hopf bifurcation points. Figure \[fig:TauOrb\] illustrates stable periodic orbits from the left part of this branch. Although these orbits superficially resemble those of Figure \[fig:GammaOrb\], with a single peak above $Q^*$ and a small amplitude oscillation close to $Q=0$, the periodic orbits seen in Figure \[fig:TauOrb\] have a quite different character to those seen in Figure \[fig:GammaOrb\]. Specifically the orbits have a growing oscillation close to $Q=0$ with a period close to the delay $\tau$. In contrast, the solutions seen in Figure \[fig:GammaOrb\] have a decaying oscillation near their minimum value, which is not particularly close to $Q=0$. Figure \[fig:TauOrb\](ii) shows the delay embedding of the solutions, from which we see that the ripples in the amplitude and period along the branch are related to the number of low amplitude oscillations in the solution, with the smaller amplitude and period solutions at the bottom of the ripples performing one less oscillation in the $(Q(t),Q(t-\tau))$ projection before escaping to $Q(t)\gg Q^*$. The dynamics of the oscillations close to $Q(t)=0$ are easy to describe but harder to explain. The trivial steady state $Q=0$ has one positive real characteristic value ($\lambda=0.017605$ when $\tau=6$) and infinitely many complex conjugate characteristic values, three pairs of which have positive real part. Thus the steady state $Q=0$ of the DDE has a seven-dimensional unstable manifold and an infinite-dimensional stable manifold in the full infinite dimensional phase space of the functional differential equation. However, complex characteristic values would give rise to oscillatory solutions about $Q=0$ which change sign. Since, by Theorem \[theorem.bound\], solutions with positive initial conditions remain positive, oscillations about $Q=0$ will not arise with physiological initial history functions. (Remark: This implies that for any complex characteristic value $\lambda=\alpha\pm i\omega$ that $|\omega|{\geqslant}\pi/\tau$, since otherwise $\tau$ would be less than half the period of the oscillation, and the positive half of the oscillation could be used to define an initial function $\varphi=-{\varepsilon}e^{\alpha t}\sin(\omega t)$ for $t\in[-\tau,0]$ so the DDE that would have a solution close to $Q(t)=-{\varepsilon}e^{\alpha t}\sin(\omega t)$ for $0<t\ll 1$ which would violate the positivity of solutions). Consequently, in the restricted phase space of positive solutions that we consider $Q=0$ has a one-dimensional unstable manifold and a trivial stable manifold. \[positive\] In the inset of Figure \[fig:TauOrb\](ii) we see that in the delay embedding $(Q(t),Q(t-\tau))$ that $Q(t)$ takes its minimum value on the periodic orbit when $Q(t-\tau)$ is close to $0.5$ but decreasing. $Q(t)$ then increases slightly before decreasing again to its next minimum which occurs very close to the minimum of $Q(t-\tau)$ on the solution. This sets in train a clockwise oscillation in the $(Q(t),Q(t-\tau))$ projection close to $0$. Along this oscillation the local minima of $Q(t)$ and $Q(t-\tau)$ occur very close to each other in time because the period of this low amplitude oscillation (as seen in Figure \[fig:TauOrb\](i)) is very close to the delay $\tau$. After the double local minima the solution grows close to the local unstable manifold of $Q=0$ with $Q(t-\tau)\approx Q(t)e^{-\lambda\tau}$ for a time, until $Q(t-\tau)$ starts to decrease again (towards the previous local minima of $Q(t)$), after which $Q'(t)$ becomes negative and $Q(t)$ decreases to its next local minima, completing one cycle. The amplitude of this oscillation grows slightly with each subsequent cycle, until eventually the oscillation escapes to $Q(t)\gg Q^*$. As $\tau$ increases across the branch of stable orbits the character of the periodic orbits changes (not illustrated), with the growth rate of the small amplitude oscillations progressively decreasing and the period of the orbit increasing. For $\tau$ sufficiently large the small amplitude oscillations decay instead of grow, and thereafter the periodic orbits resemble the longest period orbit shown in Figure \[fig:GammaOrb\](i). The period of the orbit, but not the amplitude, continues to grow until the amplitude and period of the solutions decreases abruptly just before the right bifurcation point, apparently in a canard explosion. The period reaches its maximum value of $3281$ days for $\tau\approx 6.874295373$ and decreases dramatically to $182$ days while the value of $\tau$ remains constant to 10 significant digits (see right inset of Figure \[fig:TauCont\](ii)). Although we do not find a homoclinic bifurcation, the longest period orbit is about 9 years, nearly $500$ times larger than the delay in the system, with the orbit close to homoclinic to the non-trivial steady state $Q^*$. The solutions in this region are similar to the long-period orbits displayed in Figure \[fig:GammaOrb\]. Two-parameter continuation {#sec.2p} -------------------------- ![Two parameter $(\kappa,\tau)$-bifurcation diagram for the DDE  with other parameter values given by Table \[tab.model.par\]. Each solid curve represents the locus of bifurcation of periodic orbits, as indicated in the key. The brown square denotes a Bautin bifurcation; see text. The black cross marks the homeostasis values of the parameters. Within the inset, the black square indicates the parameter values of the torus investigated in Section \[sec.torus\], and the red and blue triangles represent chaotic attractors seen in Section \[sec.chaos\].[]{data-label="fig:KappaTau_2D_Cont"}](fig5_HSC_Biftool2D_Kappa_TauB "fig:"){width="79.00000%"} (-80,102) (-63,86) (-63,71) (-63,55) (-63,42) (-63,27) Recent versions of DDEBiftool [@Engelborghs_2002; @DDEBiftool15] have the facility to perform two-parameter continuation of bifurcations of periodic orbits, and we used this to study the bifurcations of the DDE  as the parameters $\kappa$, $\gamma$ and $\tau$ are varied pairwise. In Figure \[fig:KappaTau\_2D\_Cont\] we present the two-parameter bifurcation diagram as $\kappa$ and $\tau$ are varied, with all other parameters at their values in Table \[tab.model.par\], which reveals the curves of Hopf, period-doubling, saddle-node, torus and steady state bifurcation as this pair of parameters are varied. Taking a straight line through Figure \[fig:KappaTau\_2D\_Cont\] with $\tau=2.8$ or with $\kappa=0.022$ reveals the bifurcations found in Figures \[fig:KappaCont\] and \[fig:TauCont\] respectively. From Figure \[fig:KappaTau\_2D\_Cont\] we see that the homeostasis parameters $(\kappa,\tau)=(0.022,2.8)$ are not particularly close to any bifurcations, with the Hopf curve and an associated curve of saddle-node of limit-cycle bifurcations being the only other bifurcations near to that part of parameter space. The Hopf bifurcations are subcritical to the left of the Bautin or generalised Hopf bifurcation at $(\kappa,\tau)=(0.4960,1.525)$ and supercritical otherwise. We already saw instances of the subcritical Hopf bifurcations in Figures \[fig:KappaCont\] and \[fig:TauCont\]; Bernard *et al* [@Bernard_2004] previously presented an example with both Hopf bifurcations supercritical. If the delay is small ($\tau<1$) there are no bifurcations at all, while the bifurcation structures become more complicated as $\tau$ is increased with two curves of fold bifurcations of periodic orbits created in a cusp bifurcation at $(\kappa,\tau)\approx(0.81600,2.3956)$, and a further cusp bifurcation and torus bifurcation curves only occurring for $\tau>3$. Figure \[fig:KappaTau\_2D\_Cont\] suggests that for one parameter continuation in $\kappa$, taking $\tau$ close to $4$ will lead to more complicated dynamics than was seen in Figure \[fig:KappaCont\] for $\tau=2.8$. Indeed, the curves seen in Figure \[fig:KappaTau\_2D\_Cont\] were seeded by performing a one parameter continuation in $\kappa$ with $\tau=3.9$ (see Figure \[fig2\_HSC\_Biftool1D\_KappaDA\]) and consequently Figure \[fig:KappaTau\_2D\_Cont\] shows all the bifurcation curves that cross $\tau=3.9$. There may be other bifurcation curves that remain above $\tau=3.9$, but it appears from Figure \[fig:KappaTau\_2D\_Cont\] that they would be constrained to be near the right Hopf bifurcation. ![Continuation in $(\kappa,\gamma)$ with other parameters at homeostasis (see Table \[tab.model.par\]). Each solid curve represents the locus of a certain type of bifurcation of periodic orbits as specified in the key. []{data-label="fig:GammaKappa_2D_Cont"}](fig5_HSC_Biftool2D_Kappa_GammaC "fig:"){width="79.00000%"} (-80,83) (-61,70) (-61,55) (-61,42) (-61,27) Figure \[fig:GammaKappa\_2D\_Cont\] shows the bifurcation curves found for two-parameter continuation in $(\kappa,\gamma)$ with all the other parameters taking their values from Table \[tab.model.par\], revealing an alternating sequence of curves of period-doubling and fold bifurcations (of limit cycles), and associated cusp bifurcation of limit cycles. The closed curves of bifurcations become shorter and narrower as parameters approach the upper bound on $\gamma$ for periodic orbits to exist, and also progressively more delicate to compute numerically. There may be additional curves of bifurcations for $\gamma>0.2$ which we were not able to compute. A straight line through Figure \[fig:GammaKappa\_2D\_Cont\] with $\gamma=0.1$ or with $\kappa=0.022$ reveals the bifurcations found in Figures \[fig:KappaCont\] and \[fig:GammaCont\], respectively. The inset reveals that the period-doubling and saddle-node loci do not overlap. ![Continuation in $(\gamma,\tau)$ with other parameters at homeostasis values. Each solid curve represents the locus of a bifurcation of periodic orbits, as specified in the key to Figure \[fig:GammaKappa\_2D\_Cont\]. The brown square denotes a Bautin bifurcation. The brown dots and arc of brown curve emanating from the Bautin bifurcation form part of a curve of fold bifurcations of limit cycles, which DDEBiftool is only partially able to compute (see text).[]{data-label="fig:GammaTau_2D_Cont"}](fig5_HSC_Biftool2D_Gamma_Tau2B){width="79.00000%"} The results of two-parameter continuation in $(\gamma,\tau)$ are shown in Figure \[fig:GammaTau\_2D\_Cont\]. This reveals the locus of the Hopf bifurcations already observed in Figures \[fig:GammaCont\] and \[fig:TauCont\]. There is also a Bautin bifurcation at $(\gamma,\tau)=(0.4020,1.651)$ and a branch of saddle-node bifurcations of limit cycles which emerges this point. This branch represents the two-parameter continuation of the fold bifurcation seen in Figures \[fig:GammaCont\] and \[fig:TauCont\]. This bifurcation is very delicate for DDE-Biftool to compute and continue numerically, and we were not able to compute the full branch. For $\tau=2.8$, $4$, $6$ and $8$ we performed one parameter continuation in $\gamma$ to confirm that the fold bifurcation persists for larger $\tau$ values and also to verify that there are not other bifurcation curves missing from the diagram. For the larger values of $\tau$, DDEBiftool is not able to identify the fold bifurcation. While we are able to find the fold from a one-parameter continuation by simply looking for the minimum value of $\gamma$ along the branch (and we added these points to Figure \[fig:GammaTau\_2D\_Cont\]), DDEBiftool computes and continues fold bifurcations of limit-cycles in two parameters by solving the defining equations for a fold bifurcation of periodic orbits [@DDEBiftool15], which is a considerably more complicated computation. The two-parameter continuations in Figures \[fig:KappaTau\_2D\_Cont\], \[fig:GammaKappa\_2D\_Cont\] and \[fig:GammaTau\_2D\_Cont\] reveal that the non-trivial steady-state solution $Q^*>0$ remains stable for all reasonably small perturbations from the homeostasis parameter values of Table \[tab.model.par\]. We also see from Figures \[fig:KappaTau\_2D\_Cont\] and \[fig:GammaTau\_2D\_Cont\] that $Q^*$ is stable for all small delays $\tau<1$ (at least when the other parameters are varied one at a time). This suggests that an ODE model would not capture the instabilities driven by the delays. Since the cell-cycle time for stem cells is estimated to be much larger than one day ($2.8$ days in Craig [@Craig_2016]) it is essential to include the delay in the DDE model  to properly capture the possible dynamics of the system. That the two-parameter continuations in $(\kappa,\gamma)$ and $(\gamma,\tau)$ reveal less interesting bifurcation diagrams than for continuation in $(\kappa,\tau)$, is probably not intrinsic to the properties of the parameters in the model, but rather determined by the homeostasis value of the third parameter from Table \[tab.model.par\] when we perform two-parameter continuation. More complicated bifurcation diagrams can be generated by taking $(\kappa,\tau)$ close to $(0.86,4)$, in the interesting part of Figure \[fig:KappaTau\_2D\_Cont\], and then doing two-parameter continuation in any pair of these three parameters. For example, with $\kappa=0.68$, two-parameter continuation in $(\gamma,\tau)$ (not shown) reveals torus and period-doubling curves, quite unlike anything seen in Figure \[fig:GammaTau\_2D\_Cont\]. However, here we have based our continuation on using the homeostasis values of the parameters from Table \[tab.model.par\] to start one and two-parameter continuations. If we allowed all the parameters to vary its likely that we could find more exotic dynamics, but what the relationship, if any, that dynamics would have to Burns-Tannock HSC model is not clear. Long Period Orbits and Canard Explosion {#sec.longp.hsc} ======================================= A canard explosion is a dynamical phenomenon seen in fast-slow or singularly perturbed systems whereby over an exponentially small range of the continuation parameter a periodic orbit is transformed into a long period relaxation oscillation. For ODEs this requires at least two space dimensions, with classical examples being the van der Pol oscillator and Fitzhugh-Nagumo equations [@Benoit1981; @Wech13]. Canard explosions have already been explored in DDEs [@CSE09; @KrupaTouboul2016], but only in systems with at least two spatial dimensions that incorporate a delay. However, since DDEs are inherently infinite dimensional there is no reason why a canard explosion should not be seen in a scalar DDE such as . ![(i) Solution profiles and (ii) time-delay embedding $(Q(t),Q(t-\tau))$ of periodic solutions from the same $\gamma$ continuation shown in Figure \[fig:GammaCont\], show an apparent canard explosion at $\gamma\approx0.2453692$. The insets indicate where the illustrated orbits lie on the branch. (iii) and (iv) shows parts of (ii) at a much larger scale. (iii) reveals the oscillatory convergence of the solutions onto a slow manifold for $Q<Q^*$. (iv) shows the dynamics near to $Q^*$. The nullcline $Q'(t)=0$ is indicated by the red curves in panels (ii)-(iv).[]{data-label="fig:GammaCont_Canard"}](critmanorb "fig:"){width="49.00000%" height="44mm"} (-245,2.5) ![(i) Solution profiles and (ii) time-delay embedding $(Q(t),Q(t-\tau))$ of periodic solutions from the same $\gamma$ continuation shown in Figure \[fig:GammaCont\], show an apparent canard explosion at $\gamma\approx0.2453692$. The insets indicate where the illustrated orbits lie on the branch. (iii) and (iv) shows parts of (ii) at a much larger scale. (iii) reveals the oscillatory convergence of the solutions onto a slow manifold for $Q<Q^*$. (iv) shows the dynamics near to $Q^*$. The nullcline $Q'(t)=0$ is indicated by the red curves in panels (ii)-(iv).[]{data-label="fig:GammaCont_Canard"}](critman "fig:"){width="49.00000%" height="44mm"} (-245,2.5) ![(i) Solution profiles and (ii) time-delay embedding $(Q(t),Q(t-\tau))$ of periodic solutions from the same $\gamma$ continuation shown in Figure \[fig:GammaCont\], show an apparent canard explosion at $\gamma\approx0.2453692$. The insets indicate where the illustrated orbits lie on the branch. (iii) and (iv) shows parts of (ii) at a much larger scale. (iii) reveals the oscillatory convergence of the solutions onto a slow manifold for $Q<Q^*$. (iv) shows the dynamics near to $Q^*$. The nullcline $Q'(t)=0$ is indicated by the red curves in panels (ii)-(iv).[]{data-label="fig:GammaCont_Canard"}](critmanoscil "fig:"){width="49.00000%" height="44mm"} (-245,2.5) ![(i) Solution profiles and (ii) time-delay embedding $(Q(t),Q(t-\tau))$ of periodic solutions from the same $\gamma$ continuation shown in Figure \[fig:GammaCont\], show an apparent canard explosion at $\gamma\approx0.2453692$. The insets indicate where the illustrated orbits lie on the branch. (iii) and (iv) shows parts of (ii) at a much larger scale. (iii) reveals the oscillatory convergence of the solutions onto a slow manifold for $Q<Q^*$. (iv) shows the dynamics near to $Q^*$. The nullcline $Q'(t)=0$ is indicated by the red curves in panels (ii)-(iv).[]{data-label="fig:GammaCont_Canard"}](critmanbif "fig:"){width="49.00000%" height="44mm"} (-245,2.5) Recalling the continuation in $\gamma$ shown in Figure \[fig:GammaCont\], at the right Hopf bifurcation point $\gamma\approx0.2453746$ a periodic orbit is born with period approximately $48$ days, but at $\gamma\approx0.2453692$ the period increases dramatically to about $700$ days while the value of $\gamma$ remains constant to 7 significant figures. This would appear to be a canard explosion. Figure \[fig:GammaCont\_Canard\] illustrates orbits from this part of the branch as the period increases. Comparing the time plot with the $(Q(t),Q(t-\tau))$ phase space projection, the slow manifold appears to be close to $Q(t)=Q(t-\tau)$, with $Q'(t)$ gradually increasing along this curve, followed by a fast transition layer as $Q(t)$ decreases to close to its minimum value while $Q(t-\tau)$ remains close to its maximum. Then $Q(t-\tau)$ passes through the transition layer to also be close to its minimum value after which there is a slowly decaying oscillation with a period of about $3$ days as the solution converges back to the slow manifold. The largest period orbit illustrated has a period of about $701.3$ days, with $Q(t)$ crossing the steady state $Q^*$ once in each direction, with $Q(t)>Q^*$ for approximately $191.3$ days and $Q(t)<Q^*$ for the remaining $510.0$ days. Here the delay $\tau=2.8$ days, so this is an example of a (very) slowly oscillating periodic solution. Fast-slow systems in ODEs often have separate fast and slow variables which can be considered separately in the fast and slow subsystems. That separation of variables does not occur in the DDE , for which we have only one variable. Nevertheless, relaxation oscillators with both fast and slow segments within the solution can arise and have been studied in DDEs, by tackling the fast and slow segments of the solution separately. In particular a relaxation oscillator for the HSC DDE  has been studied using singular perturbation analysis [@Colijn2006; @Fowler_Mackey_2002]. Of particular note is the extensive work of Mallet-Paret and Nussbaum studying slowly oscillating periodic solutions in singularly perturbed constant and state-dependent DDEs [@JMPRN86; @JMPRNIII; @JMPRN11]. A complete analysis of how the canard explosion arises in  will be beyond the scope of this work, but we will show that equation  can be considered as singular perturbation problem. We identify the critical manifold, and also investigate its persistence by approximating the resulting slow manifold and studying its stability. We will show that a segment of this manifold for $Q<Q^*$ is stable with oscillatory convergence of nearby trajectories onto the manifold (see Figures \[fig:GammaCont\_Canard\](ii) and (iii)) while a segment for $Q>Q^*$ is unstable, leading to the divergence of trajectories from the manifold. In the current work, we will not study the fast dynamics in the transition layer. For $\gamma=0.2453692$ and all other parameters taking their values from Table \[tab.model.par\], we notice that $\kappa\approx0$ and $A\approx1$, so we introduce the perturbation parameter ${\varepsilon}{\geqslant}0$ and let $$\label{eq:epspars} {\varepsilon}=A-1=2e^{-\gamma\tau}-1, \qquad \kappa=\frac{{\varepsilon}f}{C}.$$ For the non-zero steady state $Q^*$ to exist the inequality must hold; equivalently the constant $C$ must satisfy $C>1$. Then $Q^*>0$ is given from by $$\label{Qstarsing} Q^*=\theta(C-1)^{1/s},$$ independent of the value of ${\varepsilon}>0$. For the parameters used in Figure \[fig:GammaCont\_Canard\] we have ${\varepsilon}=6.132\times10^{-3}$ and $C=2.23$ with $Q^*=0.0896868$ when $\gamma=0.2453692$. The parameter definitions in could be applied to the non-dimensionalised equation  with $\hat{A}={\varepsilon}$ and $\hat{\kappa}={\varepsilon}/C$, but we prefer to continue to study  directly. Letting $h(Q)=Q\beta(Q)$, as in , and using we re-write as $$\label{eq:epszeroman} Q^{\prime}(t) = -\frac{{\varepsilon}f}{C}Q(t)-h(Q(t))+(1+{\varepsilon})h(Q(t-\tau)).$$ When ${\varepsilon}=0$ this reduces to $$\label{eq:epszero} Q^{\prime}(t) =-h(Q(t))+h(Q(t-\tau))= -\frac{fQ(t)}{1+(Q(t)/\theta)^s}+\frac{fQ(t-\tau)}{1+(Q(t-\tau)/\theta)^s}.$$ While equation has the unique positive steady state $Q^*$ given by , when ${\varepsilon}=0$ equation has a line of equilibria with $Q$ being an arbitrary constant, which is the critical manifold. The linearisation of is given by with $a=-h'(Q)$ and $b=-a$ and so the characteristic function becomes $$p(\lambda)= \lambda - a + a\mathnormal{e}^{-\lambda\tau}.$$ This satisfies $p(0)=0$ and $p'(0)=1-a\tau$, and has $\lambda=0$ as a solution for any value of $a$. There is an additional real negative root if $p'(0)>0$, i.e. $a\tau<1$. This root crosses zero when $p'(0)=0$ and becomes positive for $a\tau>1$ when $p'(0)<0$. Thus the steady state stability changes when $h'(Q)=-1/\tau$. Using and and the non-dimensionalised variables of the identity $h'(Q)=-1/\tau$ reduces to a quadratic equation for $\hat Q^s$: $$\hat{Q}^{2s}+(2-(s-1)\hat f)\hat{Q}^{s}+1+\hat{f}=0.$$ Solving this with parameters corresponding to Figure \[fig:GammaCont\_Canard\] we find that the stability on the critical manifold changes when $Q=\theta\hat Q=0.0893174$, very close to the value $Q^*$. The critical manifold should persist where it is transversally hyperbolic as a slow manifold following the theory of Fenichel [@Fenichel_1979]. However, that theory was developed for ODEs in multiple space dimensions with at least one fast and one slow variable. Likewise, the previous examples of canards in DDEs considered systems with two spatial dimensions with one fast and one slow variable [@CSE09; @KrupaTouboul2016]. For the scalar DDE there is not an obvious separation into fast and slow variables, and it is not apparent how to proceed rigourously. Nevertheless, it is apparent from Figure \[fig:GammaCont\_Canard\] that there is a slow manifold, and in the remainder of this section we will show how to approximate the slow manifold and determine its stability. The slow manifold on which $Q'\approx0$ should be close to the nullcline $Q'(t)=0$, which from  is given by $$\label{eq:critman} 0 = -\frac{\kappa}{f}Q(t)-\frac{Q(t)}{1+(Q(t)/\theta)^s}+2e^{-\gamma\tau}\frac{Q(t-\tau)}{1+(Q(t-\tau)/\theta)^s}.$$ If one of $Q(t)$ or $Q(t-\tau)$ is fixed, then for $Q'(t)=0$ with $s=2$ from  the value of the other one is defined by a cubic equation. The resulting nullcline is displayed as the two red curves in Figure \[fig:GammaCont\_Canard\](ii), which are seen to be disjoint in Figure \[fig:GammaCont\_Canard\](iv) which shows an expanded view near to $(Q^*,Q^*)$. Typically, canard explosions are seen close to a bifurcation of the intersections of the nullclines of the slow and fast variables. Here we do not have separate fast and slow variables, but we see that we are close to a bifurcation of the $Q'(t)=0$ nullcline itself, with the two disjoint parts coming very close to each other in Figure \[fig:GammaCont\_Canard\](iv). Figure \[fig:GammaCont\_Canard\](iv) also shows that the periodic orbits that form the canard explosion appear to lie on a slow manifold between the branches of the nullcline and switch from following the lower branch to following the upper branch at the point close to $Q^*$ where the two curves are closest. It is thus likely essential for the canard explosion that the parameter set is close to this bifurcation of the nullcline structure. To obtain a simple approximation to the slow manifold, let $h(Q)=Q\beta(Q)$, as in , so becomes$$\label{dQdt3} Q^{\prime}(t) = -\kappa Q(t)- h(Q(t))+Ah(Q(t-\tau)).$$ Then use the approximation $$\label{hQtauapprox} h(Q(t-\tau))\approx h(Q(t))-\tau \frac{d}{dt}h(Q(t))=h(Q(t))-\tau Q'(t) h'(Q(t)),$$ to remove the delay from . Note that $\tau\gg0$ so  is only useful if $\tau^2\frac{d^2}{dt^2}h(Q(t))\ll1$. Substituting  into  and rearranging we obtain $$Q'(t)=\frac{-\kappa Q(t)+(A-1)h(Q(t))}{1+A\tau h'(Q(t))}.$$ On the slow manifold $Q(t-\tau)\approx Q(t)-\tau Q'(t)$ so we can approximate the manifold in the delay embedding $(Q(t),Q(t-\tau))$ by the curve $(Q,Q_\tau)$ where $$\label{eq:Qtaucanardslowman} Q_\tau = \frac{(1+\kappa\tau+A\tau h'(Q))Q-\tau(A-1)h(Q))}{1+A\tau h'(Q)}.$$ This gives a good approximation to the slow manifold for $Q\in(0,0.23)$, except for a very small interval $|Q-Q^*|<0.003$ about the non-trivial steady state. This is shown in Figure \[fig:linearslowman\](i). The curve $(Q,Q_\tau)$ is also an approximation to the unstable manifold of $Q=0$; the trivial steady state has a single positive characteristic value, and infinitely many pairs of complex conjugate characteristic values with negative real part. The canard explosion is *not* associated with a solution homoclinic to $Q=0$ though; no such homoclinic solution can exist in the space of non-negative solutions, as all solutions in the stable manifold of $Q=0$ will be oscillatory and violate the positivity of solutions (as already noted on page ). ![Approximations $(Q,Q_\tau)$ to the slow manifold of the canard solution where: (i) $Q_\tau$ is given by  and shown in violet; (ii) $Q_\tau$ is given by  with $Q_{r}$ replaced by $Q$ and shown in green. Both approximations are shown in the insets.[]{data-label="fig:linearslowman"}](fig_linearslowman1a "fig:"){width="49.00000%" height="44mm"} (-245,2.5) ![Approximations $(Q,Q_\tau)$ to the slow manifold of the canard solution where: (i) $Q_\tau$ is given by  and shown in violet; (ii) $Q_\tau$ is given by  with $Q_{r}$ replaced by $Q$ and shown in green. Both approximations are shown in the insets.[]{data-label="fig:linearslowman"}](fig_linearslowman1b "fig:"){width="49.00000%" height="44mm"} (-245,2.5) To determine the dynamics close to the slow manifold, it is necessary to take proper account of the delayed term, and so we will derive another slow manifold approximation. For this, linearise $h(Q)$ about $Q=Q_r$ for general $Q_r$ as $$\label{hlin} h(Q)\approx h(Q_r)+h'(Q_r)(Q-Q_r).$$ Then let $Q_s(t)$ be a solution on the slow manifold, and let $Q(t)$ be some other solution in a neighbourhood of this manifold, and let $w(t)=Q(t)-Q_s(t)$ be the difference between these two solutions, then using  and  we find that $$\label{eq:linslddenonaut} w'(t) = Q'(t)-Q_s'(t) \approx -(\kappa+h'(Q_s(t)))w(t)+Ah'(Q_s(t-\tau))w(t-\tau).$$ The linearisation  is actually valid as an approximation of the dynamics about any solution $Q_s(t)$ of the DDE . However, it is problematical to use, even about solutions on the slow manifold, since although  is linear it is non-autonomous and the time-dependent terms depend on the as yet unknown slow-manifold solution $Q_s(t)$. As an alternative, rather than linearise about a particular solution, close to the slow manifold we can use to linearise $h$ about $Q=Q_r$, for some reference value of $Q$ and convert  into a linear DDE for the dynamics near to $Q=Q_r$, with $$\label{DDEhlinderiv} Q^{\prime}(t) \approx (A-1)[h(Q_r)-h^{\prime}(Q_r)Q_r] -(\kappa+h^{\prime}(Q_r))Q(t)+Ah^{\prime}(Q_r)Q(t-\tau).$$ We rewrite  as $$\label{DDEhlinnon} Q^{\prime}(t)= aQ(t)+bQ(t-\tau)+c,$$ where letting $$\label{fQsQ} G(Q) := (A-1)\beta(Q)Q-\kappa Q=(A-1)h(Q)-\kappa Q,$$ we see that the constants $a$, $b$ and $c$ satisfy $$\begin{gathered} a=-(\kappa+h^{\prime}(Q_r)),\qquad b=Ah^{\prime}(Q_r),\\ c=(A-1)[h(Q_r)-h^{\prime}(Q_r)Q_r]=G(Q_r)+[\kappa-(A-1)h^{\prime}(Q_r)]Q_r=G(Q_r)-(a+b)Q_r.\end{gathered}$$ Noting that $$G'(Q_r)=(A-1)h'(Q_r)-\kappa=a+b,$$ we can rewrite $c$ as $c=G(Q_r)-G'(Q_r)Q_r.$ The function $G$ is unimodal, and recalling  it follows that $G(0)=G(Q^*)=0$, and $G$ has a unique maximum for $Q=Q_f$ for some $Q_f\in(0,Q^*)$. Thus $G'(Q_r)>0$ (resp. $G'(Q_r)<0$) for $Q<Q_f$ (resp. $Q>Q_f$). Another value of $Q$ which will be relevant below is $Q_h$, the value of $Q$ such that $h'(Q_h)=0$. It follows easily that $Q_h>Q_f$. With $\gamma = 0.2453692$ and the other parameters from Table \[tab.model.par\] we have $$Q_f=0.042263 \quad < \quad Q_h=(s-1)^{-\frac1s}\theta=\theta \quad < \quad Q^*=0.089686.$$ Solutions to the nonhomogeneous linear DDE  consist of a particular solution of  and any linear combination of solutions of the homogeneous linear DDE $$\label{DDEhlinaut} Q^{\prime}(t)= aQ(t)+bQ(t-\tau).$$ The DDE  is of the same form as equation  and has the same characteristic equation . This has infinitely many complex roots, which would lead to oscillatory solutions of . However, from above the slow manifold appears to be monotonic, hence we will seek a monotonic solution of , for which we require real roots of . Since $p'(\lambda)=1+b\tau e^{-\lambda\tau}$, for $Q{\leqslant}Q_h$ we have $p'(\lambda){\geqslant}1$ and equation  has a unique real root. Real roots of  can be found using the Lambert-$W$ function [@Corless1996]. Rearranging  we see that any root $\lambda$ satisfies $b\tau e^{-a\tau}=(\lambda-a)\tau e^{(\lambda-a)\tau}.$ Hence $W(b\tau e^{-a\tau})=(\lambda-a)\tau$, and so $$\label{lambdaW} \lambda=a+\frac{1}{\tau}W(b\tau e^{-a\tau}).$$ Consider first the case where $Q_r<Q_f$. Then $G'(Q_r)=a+b>0$ and $b>0>a$. Since $p(0)=-(a+b)$ and $p'(\lambda){\geqslant}1$ the characteristic equation  has a unique real root $\lambda_+=\lambda_+(Q_r)\in(0,G'(Q_r))$ given by . Equation  then admits a constant solution, $Q(t)=k$, where $$k=\frac{-c}{a+b}=\frac{-G(Q_r)+G'(Q_r)Q_r}{G'(Q_r)}=Q_r-\frac{G(Q_r)}{G'(Q_r)}.$$ Hence a monotonic solution of  passing through $Q(0)=Q_r$ is $$\label{QsollessQf} Q(t)=Q_r+(e^{\lambda_+ t}-1)G(Q_r)/G'(Q_r).$$ The general solution of  which describes the behaviour of solutions in a neighbourhood of the monotonic solution is obtained by adding an arbitrary linear combination of the solutions of , defined by the roots of . But since $\lambda_+>0$, even without solving for the complex characteristic values, we already know that the monotonic solution is not stable for $Q<Q_f$. Nevertheless, we can use  to approximate the slow manifold for $Q<Q_f$. From , when $Q=Q_r$ we have $Q'=\lambda_+G(Q_r)/G'(Q_r)<G(Q_r)$, and hence assuming that $Q(t-\tau)\approx Q(t)-\tau Q'(t)$, we can approximate the slow manifold $Q_s(t)$ in the delay embedding $(Q(t),Q(t-\tau))$ for $Q<Q_f$ by the curve $(Q_r,Q_\tau)$ where $$\label{eq:Qtauslowlin} Q_\tau= Q_r - \tau\lambda G(Q_r)/G'(Q_r)$$ and $\lambda=\lambda_+(Q_r)$. Next consider the case for which $Q_r\in(Q_f,Q_h)$. Then $b>0>a+b=G'(Q_r)$. Now, $p(0)=-(a+b)>0$ and $p'(\lambda){\geqslant}1$, so the characteristic equation  has a unique real root which is negative, $\lambda_-=\lambda_-(Q_r)\in(G'(Q_r),0)$ given by . Similarly to above, equation  then has a monotonic solution $$\label{QsolgtQf} Q(t)=Q_r+(e^{\lambda_- t}-1)G(Q_r)/G'(Q_r),$$ passing through $Q(0)=Q_r\in(Q_f,Q_h)$. The behaviour of nearby solutions is determined by the general solution of which is $$\label{QgensolmoreQf} Q(t)=Q_r+(e^{\lambda_- t}-1)\frac{G(Q_r)}{G'(Q_r)}+\beta_0e^{\lambda_-t} +\sum_{j{\geqslant}1}e^{\alpha_jt}(\beta_j\cos(\omega_jt)+\gamma_j\sin(\omega_jt)),$$ for constants $\beta_j$, $\gamma_j$ where $\lambda_j=\alpha_j\pm i\omega_j$ are the complex roots of . For this solution, not only is $\lambda_-<0$, but we can also show that all the complex characteristic values that solve  also have strictly negative real part. Taking real and imaginary parts of  we find that $\lambda_j=\alpha_j\pm i\omega_j$ satisfies $0=\alpha_j-a-be^{-\alpha_j\tau}\cos(\omega_j\tau)=\omega_j+be^{-\alpha_j\tau}\sin(\omega_j\tau),$ which implies that $\omega_j^2=b^2e^{-2\alpha_j\tau}-(\alpha_j-a)^2.$ But for $Q_r\in(Q_f,Q_h)$ we have $a<0$ and $-a>b>0$, hence for a characteristic root with $\alpha_j{\geqslant}0$ we have $\omega_j^2{\leqslant}b^2-a^2<0$, a contradiction, and so all characteristic values have $Re(\lambda_j)=\alpha_j<0$. For $Q\in(Q_f,Q_h)$ we can again approximate the slow manifold $Q_s(t)$ in the delay embedding $(Q(t),Q(t-\tau))$ by the curve $(Q_r,Q_\tau)$ where $Q_\tau$ is defined by  with $\lambda=\lambda_-(Q_r)$. Since $\alpha_j<0$ for all $j$ and $\lambda_-<0$ all the additional solution elements included in are decaying, and the solution defined by  and the resulting slow manifold are attracting in this region of phase space. $Q_r$ $\lambda_\pm$ $Q'$ $Q_\tau$ $\lambda_1$ -------- ---------------------- --------------------- --------------------- --------------------- $0.01$ $1.10\times10^{-5}$ $1.17\times10^{-5}$ $9.96\times10^{-3}$ $-0.0118 \pm 2.15i$ $0.02$ $9.56\times10^{-4}$ $2.45\times10^{-5}$ $1.99\times10^{-2}$ $-0.0155 \pm 2.13i$ $0.03$ $6.68\times10^{-4}$ $3.96\times10^{-5}$ $2.98\times10^{-2}$ $-0.0237 \pm 2.11i$ $0.04$ $1.60\times10^{-4}$ $5.81\times10^{-5}$ $3.98\times10^{-2}$ $-0.0408 \pm 2.07i$ $0.05$ $-7.40\times10^{-4}$ $8.13\times10^{-5}$ $4.97\times10^{-2}$ $-0.077 \pm 2.00i$ $0.06$ $-2.45\times10^{-3}$ $1.10\times10^{-4}$ $5.96\times10^{-2}$ $-0.157 \pm 1.90i$ $0.07$ $-6.28\times10^{-3}$ $1.47\times10^{-4}$ $6.95\times10^{-2}$ $-0.354 \pm 1.75i$ $0.08$ $-1.93\times10^{-2}$ $1.98\times10^{-4}$ $7.94\times10^{-2}$ $-1.35 \pm 1.42i$ : Table showing the real characteristic value $\lambda_-$ or $\lambda_+$ of  along with the first pair of complex characteristic values $\lambda_1$, and the derivative of the solution $Q'(t)$ given by  or , and the resulting approximation of $Q_\tau$ given by  for a range of values of $Q_r$.[]{data-label="tab:lambdas"} The convergence onto the slow manifold is oscillatory, as seen in Figure \[fig:GammaCont\_Canard\]. This is governed by the dominant complex characteristic value of , the value of which is stated as $\lambda_1$ in Table \[tab:lambdas\]. We see from the table that $Re(\lambda_1)$ becomes more negative as $Q$ increases, implying that the slow manifold becomes more attractive as $Q$ increases towards $Q_h$. This is clearly visible in Figure \[fig:GammaCont\_Canard\](iii) with progressively fewer oscillations visible for the orbits converging onto the slow manifold for larger values of $Q$. The period of these oscillations $2\pi/\omega_1$ also increases with $Q$ but not greatly, and is close to $3$ in the range of $Q$ values where the oscillations are most visible. Figure \[fig:canard\_lindyn\] illustrates how well our approximations perform in the region $Q\in(Q_f,Q_h)$ where the slow manifold is attracting. The blue curves in Figure \[fig:canard\_lindyn\] show part of the limit cycle of the nonlinear DDE  with period $297$ days when $\gamma= 0.2453692$, which occurs in the canard explosion and was previously shown in Figure \[fig:GammaCont\_Canard\]. Taking $Q_r=0.063224$ we find that the rightmost characteristic value is $\lambda_{-}= -3.33\times10^{-3}$, then  defines an approximation to the slow manifold which is shown as the black curve in Figure \[fig:canard\_lindyn\]. ![(i) Profile and (ii) time-delay embedding, for a periodic solution of the nonlinear DDE , and several of its approximations. For $\gamma = 0.2453692$ the DDE  has a limit cycle with period $297$ days, part of which is shown here (blue curve). For the approximations we take $Q_{r}=0.063224$, then the black curve shows the approximation  to the slow manifold. The red curve shows a solution to  computed numerically using a segment of the solution of the nonlinear DDE  for $t\in[-2.8,0]$ to define the initial function. The green curve shows a solution of  defined by  with $\beta_j=\gamma_j=0$ for all $j$, except $\gamma_1=0.003935$.[]{data-label="fig:canard_lindyn"}](fig3_DdeLinearABC297F "fig:"){width="49.00000%" height="44mm"} (-245,2.5) ![(i) Profile and (ii) time-delay embedding, for a periodic solution of the nonlinear DDE , and several of its approximations. For $\gamma = 0.2453692$ the DDE  has a limit cycle with period $297$ days, part of which is shown here (blue curve). For the approximations we take $Q_{r}=0.063224$, then the black curve shows the approximation  to the slow manifold. The red curve shows a solution to  computed numerically using a segment of the solution of the nonlinear DDE  for $t\in[-2.8,0]$ to define the initial function. The green curve shows a solution of  defined by  with $\beta_j=\gamma_j=0$ for all $j$, except $\gamma_1=0.003935$.[]{data-label="fig:canard_lindyn"}](fig4_DdeLinearABC297F "fig:"){width="49.00000%" height="44mm"} (-245,2.5) The second-rightmost characteristic value $\lambda_{1} = \alpha_1+i\omega_1=-0.202 + 1.86i$ yields the approximate oscillation time of 3.37 days. To show that this characteristic value governs the convergence of solutions onto the slow manifold in Figure \[fig:canard\_lindyn\] we show as the green curve the solution  of the linearised DDE  with $\beta_j=\gamma_j=0$ for all $j$, except for $\gamma_1\ne0$, so that the only oscillatory mode included in the solution is defined by $\lambda_{1}$. Additionally, the red curve shows the solution of  incorporating all modes, computed by solving numerically using part of the solution of  as the initial function. Both approximations have oscillations about the slow manifold with very similar period and decay rate as for the solution of the full nonlinear DDE , demonstrating the validity of our approximations. In the current work we will not describe the passage of the slow manifold past the steady state $Q^*$, but note that the behaviour of the solutions of  changes when $Q_r$ approaches $Q^*$. For $Q>Q_h$ we have $b<0$ and the different branches of $W(x)$ in  for $x<0$ can lead to zero or two real solutions for $\lambda$. There are two values of $Q$, $Q_{h'}^-<Q^*<Q_{h'}^+$, such that when $Q=Q_{h'}^\pm$, we have $b\tau e^{-a\tau}=-e^{-1}$ and the two branches of the Lambert-$W$ function coalesce. These points can be computed from the solution on each branch of $h'(Q_{h'}^\pm)=W(-e^{-1-\kappa\tau}/A)/\tau$, which leads to $Q_{h'}^-=0.08626$ and $Q_{h'}^+=0.09389$. For $Q\in(Q_{h'}^-,Q_{h'}^+)$ equation  has no real roots. At the boundaries, $Q_{h'}^\pm$, of this interval a pair of complex conjugate characteristic roots coalesce, and for $Q<Q_{h'}^-$ or $Q>Q_{h'}^+$ there are two real characteristic roots. At the steady state $Q^*\in(Q_{h'}^-,Q_{h'}^+)$, there is a single pair of characteristic roots with positive real part and leading characteristic roots of are $\lambda_1=0.0070 \pm 0.1303i$ and $\lambda_2=-0.73\pm2.67i$. For $Q>Q_{h'}^+$, the function $p(\lambda)$ in  is convex with $p(0)=-(a+b)>0$, and $p(\lambda)>0$ for $\lambda{\geqslant}a$. With the other parameters as stated,  has two positive solutions provided $Q_r{\leqslant}0.28577$. Using the smaller of these two roots, we obtain a monotonic solution of the same form as  and , which can be similarly used to construct an approximation to the slow manifold for $Q>Q_{h'}^+$, as shown in Figure \[fig:linearslowman\](ii). Because of the two positive characteristic roots, this part of the slow manifold is unstable, as seen in the dynamics where the periodic orbits of different amplitudes and periods are seen in Figure \[fig:GammaCont\_Canard\](ii) to peel away from each other sooner or later depending on their amplitude and period. Thus we have approximated the attracting and repelling parts of the slow manifold either side of the steady state $Q^*$. A complete analysis of the canard explosion would require the dynamics that join these segments of the slow manifold, both near to the steady state, and also the fast dynamics when the solution is far from the slow manifold. Non-periodic and Chaotic Dynamics {#sec.chaos.hsc} ================================= Quasi-Periodic Dynamics {#sec.torus} ----------------------- It is somewhat surprising to find torus bifurcations for the HSC model , because as we already noted in Section \[sec.decoup.hsc\], its linearisation about a steady state is equation  which does not admit any double Hopf bifurcations. Double Hopf or Hopf-Hopf bifurcations are a standard mechanism for generating tori and curves of torus bifurcations [@Kuznetsov_2004], and arise frequently in systems with coupled oscillators and systems with multiple delays [@Calleja_2017]. However, the HSC model  is scalar, with a single delay, but nevertheless torus bifurcations do occur, as seen in Section \[sec.2p\]. So here we will investigate the existence of invariant tori for . ![(i) Bifurcation diagram for one-parameter continuation in $\kappa$ with $\tau=3.9$ and other parameters taking homeostasis values from Table \[tab.model.par\]. Hopf bifurcation points , saddle-node bifurcation of limit cycles points , period-doubling bifurcation points , and torus bifurcation points are indicated and highlighted in insets. (ii) Period of the limit cycles displayed in (i).[]{data-label="fig2_HSC_Biftool1D_KappaDA"}](fig2_HSC_Biftool1D_KappaDA "fig:"){width="49.00000%" height="44mm"} (-245,2.5) ![(i) Bifurcation diagram for one-parameter continuation in $\kappa$ with $\tau=3.9$ and other parameters taking homeostasis values from Table \[tab.model.par\]. Hopf bifurcation points , saddle-node bifurcation of limit cycles points , period-doubling bifurcation points , and torus bifurcation points are indicated and highlighted in insets. (ii) Period of the limit cycles displayed in (i).[]{data-label="fig2_HSC_Biftool1D_KappaDA"}](fig3_HSC_Biftool1D_KappaDA "fig:"){width="49.00000%" height="44mm"} (-247,2) The two-parameter continuation in $(\kappa,\tau)$ shown in Figure \[fig:KappaTau\_2D\_Cont\] reveals an isola of torus bifurcations for $\kappa\in(0.91859,1.0174)$ and $\tau\in(3.4857,4.1342)$. In a one parameter continuation, as $\kappa$ is varied with $\tau=3.9$ fixed, Figure \[fig2\_HSC\_Biftool1D\_KappaDA\] reveals that the main branch of periodic orbits loses stability for $\kappa\in(0.95004,0.97309)$ at a pair of torus or Neimark-Sacker bifurcations, corresponding to the points where the continuation crosses the isola found in the two-parameter continuation. The simplest explanation is that there should be a stable torus at $\kappa$ values between the two Neimark-Sacker bifurcations. Although DDEBiftool [@DDEBiftool15] cannot be used to find tori directly, a stable torus can be found by direct numerical simulation if a suitable initial function is chosen in the basin of attraction of the torus. Some care needs to be taken, because as the top right inset in Figure \[fig2\_HSC\_Biftool1D\_KappaDA\](i) reveals, folds on the main branch of periodic solutions result in a stable (as well as two unstable) periodic orbits existing for $\kappa\in(0.088007,1.1556)$, so if a stable torus exists it will co-exist with a stable periodic orbit. ![For $\tau=3.9$ and $\kappa=0.961$. (i) Time series and (ii) projected solution space $(Q(t),Q(t-\tau))$ for a quasi-periodic orbit (red curve) and the three periodic solutions (blue curves) seen in the top right inset of Figure \[fig2\_HSC\_Biftool1D\_KappaDA\](i) for $\kappa=0.961$. (iii) Projected Poincaré section of the quasi-periodic orbit (red) and the unstable orbit (blue) that it envelops onto the plane $(Q(t-\tau),Q(t-\tau/2))$ for crossing of the Poincaré section $Q(t)=c=0.14$ with $Q'(t)>0$. (iv) Initial convergence of the first three Lyapunov exponents.[]{data-label="fig1_DdeStemKappaQTori01"}](fig3_DdeStemKappaQTori01 "fig:"){width="49.00000%" height="44mm"} (-245,2.5) ![For $\tau=3.9$ and $\kappa=0.961$. (i) Time series and (ii) projected solution space $(Q(t),Q(t-\tau))$ for a quasi-periodic orbit (red curve) and the three periodic solutions (blue curves) seen in the top right inset of Figure \[fig2\_HSC\_Biftool1D\_KappaDA\](i) for $\kappa=0.961$. (iii) Projected Poincaré section of the quasi-periodic orbit (red) and the unstable orbit (blue) that it envelops onto the plane $(Q(t-\tau),Q(t-\tau/2))$ for crossing of the Poincaré section $Q(t)=c=0.14$ with $Q'(t)>0$. (iv) Initial convergence of the first three Lyapunov exponents.[]{data-label="fig1_DdeStemKappaQTori01"}](fig4_DdeStemKappaQTori01 "fig:"){width="49.00000%" height="44mm"} (-245,2.5) ![For $\tau=3.9$ and $\kappa=0.961$. (i) Time series and (ii) projected solution space $(Q(t),Q(t-\tau))$ for a quasi-periodic orbit (red curve) and the three periodic solutions (blue curves) seen in the top right inset of Figure \[fig2\_HSC\_Biftool1D\_KappaDA\](i) for $\kappa=0.961$. (iii) Projected Poincaré section of the quasi-periodic orbit (red) and the unstable orbit (blue) that it envelops onto the plane $(Q(t-\tau),Q(t-\tau/2))$ for crossing of the Poincaré section $Q(t)=c=0.14$ with $Q'(t)>0$. (iv) Initial convergence of the first three Lyapunov exponents.[]{data-label="fig1_DdeStemKappaQTori01"}](fig1_DdeStemKappaQTori01_Poincare "fig:"){width="49.00000%" height="44mm"} (-245,2.5) ![For $\tau=3.9$ and $\kappa=0.961$. (i) Time series and (ii) projected solution space $(Q(t),Q(t-\tau))$ for a quasi-periodic orbit (red curve) and the three periodic solutions (blue curves) seen in the top right inset of Figure \[fig2\_HSC\_Biftool1D\_KappaDA\](i) for $\kappa=0.961$. (iii) Projected Poincaré section of the quasi-periodic orbit (red) and the unstable orbit (blue) that it envelops onto the plane $(Q(t-\tau),Q(t-\tau/2))$ for crossing of the Poincaré section $Q(t)=c=0.14$ with $Q'(t)>0$. (iv) Initial convergence of the first three Lyapunov exponents.[]{data-label="fig1_DdeStemKappaQTori01"}](fig_DdeStemKappaQTori01_LE "fig:"){width="49.00000%" height="44mm"} (-245,2.5) To confirm the existence of a stable torus we performed a long time integration of the DDE using the MATLAB `dde23` routine [@Matlab] with initial history function very close to the unstable periodic orbit on the main branch of solutions. For $\kappa=0.961$ with $\tau=3.9$ and all other parameters taking their values from Table \[tab.model.par\] (this parameter combination is indicated by the black square in the inset within Figure \[fig:KappaTau\_2D\_Cont\]) we found a quasi-periodic torus which envelopes the unstable periodic orbit, as illustrated in Figure \[fig1\_DdeStemKappaQTori01\]. The existence of the quasi periodic torus was confirmed numerically both by plotting the Poincaré section and by computing the Lyapunov exponents. Recall from Section \[sec.decoup.hsc\] that the DDE  has the infinite dimensional phase space $C=C([-\tau,0],\mathbb{R})$, consequently a hyperplane defined by a Poincaré section is also infinite dimensional. For $\alpha\in[0,\tau]$ and some constant $c\in\mathbb{R}$ we define the Poincaré section ${\mathcal{P}}_\alpha:=\{u_t\in C: u_t(-\alpha)=c, \; u_t'(-\alpha)>0\}$. For $\alpha=0$ this is equivalent to looking for the points $\hat t$ along the solution trajectory such that $u(\hat t)=c$ and $u'(\hat t)>0$ and taking as the corresponding element of the Poincaré section the function segment $u(t)$ for $t\in[\hat t-\tau,\hat t]$, so that $u$ is equal to $c$ at the right-hand end of the function segment. Other choices of $\alpha$ are also possible, so for example with $\alpha=\tau$ the function $u$ will be equal to $c$ at the left-hand end of the interval. For the Poincaré section ${\mathcal{P}}_0$ to be useful we need to project it into finite dimensions. The simplest way to do this is to take the value of the solution $u(t)$ at a finite set of points in $[t-\tau,t]$. Since the choice $\alpha=0$ fixes $u(t)=c$, we choose the time points $t-\tau/2$ and $t-\tau$ and project the Poincaré section into $\mathbb{R}^2$ by plotting $u(t-\tau/2)$ against $u(t-\tau)$ for values of $t$ such that $u(t)=c$. This is equivalent to the projection $P:{\mathcal{P}}_0\to\mathbb{R}^2$ defined by $P(u_t)=(u(t-\tau),u(t-\tau/2))$. Figure \[fig1\_DdeStemKappaQTori01\](iii) reveals the results of doing this with $c=0.14$ for both the putative torus and the unstable periodic orbit that gave rise to it. This reveals the expected torus structure with the points representing the function segments in ${\mathcal{P}}_0$ lying on a closed curve that encloses the point representing the periodic orbit in the two-dimensional projection. Since each of the red points represents separate intersections of the same orbit with the Poincaré section, the orbit is either quasi-periodic or of period longer than a human adult lifespan (the time integration was $30000$ days, which is longer than $82$ years). We computed the Lyapunov exponents of the quasi-periodic orbit on the torus using the method of Breda and Van Vleck [@Breda_2014]. Figure \[fig1\_DdeStemKappaQTori01\](iv) shows the initial convergence of the numerical estimates for the three largest Lyapunov exponents. After $3\times10^4$ days, the six largest Lyapunov exponents are estimated to be $0.00052$, $-0.00066$, $-0.0093$, $-0.18$, $-0.29$ and $-0.29$. This reveals that up to the numerical accuracy the first two exponents are both zero, and the rest are negative, as is characteristic for a quasi-periodic two-torus. ![Time series over one period and the corresponding time-delay embedding $(Q(t),Q(t-\tau))$ for stable periodic orbits on the phase-locked torus with $\tau=3.9$ for (i-ii) $\kappa=0.965$, and, (iii-iv) $\kappa=0.957$, which intersect the Poincaré section ${\mathcal{P}}_0$ three and seven times respectively.[]{data-label="fig_phase_locking"}](fig6_DdeStemKappaQToriC01C "fig:"){width="49.00000%" height="44mm"} (-245,2.5) ![Time series over one period and the corresponding time-delay embedding $(Q(t),Q(t-\tau))$ for stable periodic orbits on the phase-locked torus with $\tau=3.9$ for (i-ii) $\kappa=0.965$, and, (iii-iv) $\kappa=0.957$, which intersect the Poincaré section ${\mathcal{P}}_0$ three and seven times respectively.[]{data-label="fig_phase_locking"}](fig4_DdeStemKappaQToriC01C "fig:"){width="49.00000%" height="44mm"} (-245,2.5) ![Time series over one period and the corresponding time-delay embedding $(Q(t),Q(t-\tau))$ for stable periodic orbits on the phase-locked torus with $\tau=3.9$ for (i-ii) $\kappa=0.965$, and, (iii-iv) $\kappa=0.957$, which intersect the Poincaré section ${\mathcal{P}}_0$ three and seven times respectively.[]{data-label="fig_phase_locking"}](fig6_DdeStemKappaQToriC01B "fig:"){width="49.00000%" height="44mm"} (-245,2.5) ![Time series over one period and the corresponding time-delay embedding $(Q(t),Q(t-\tau))$ for stable periodic orbits on the phase-locked torus with $\tau=3.9$ for (i-ii) $\kappa=0.965$, and, (iii-iv) $\kappa=0.957$, which intersect the Poincaré section ${\mathcal{P}}_0$ three and seven times respectively.[]{data-label="fig_phase_locking"}](fig4_DdeStemKappaQToriC01B "fig:"){width="49.00000%" height="44mm"} (-245,2.5) As is well known in torus dynamics [@Broer_1996], perturbing parameters in the system will change the dynamics on the torus, with parameter regions of phase locking, where there is a stable periodic orbit on the torus, interspersed with parameter sets for which the dynamics are truly quasi-periodic. So, although we cannot prove that there exists a quasi-periodic torus for exactly the parameters illustrated in Figure \[fig1\_DdeStemKappaQTori01\], there will be for nearby parameter values. Equally, there will be parameter sets for which phase locking occurs on the torus, leading to stable periodic orbits of large period. In Figure \[fig\_phase\_locking\] we show examples with $\kappa=0.957$ and $\kappa=0.965$ and all the other parameters at their values for the example of Figure \[fig1\_DdeStemKappaQTori01\] (in the $(\kappa,\tau)$ space of Figure \[fig:KappaTau\_2D\_Cont\] both these parameter sets are inside the torus curve close to the black square). These show stable periodic orbits which close after going around the torus $7$ and $3$ times, leading to periodic orbits of periods approximately $90.5$ and $38.3$ days. These orbits intersect the Poincaré section ${\mathcal{P}}_0$ and its projection into $\mathbb{R}^2$ seven and three times respectively. The phase locked orbits exist over a parameter region called an Arnold tongue. DDEBiftool can be used to find the edges of these Arnold tongues (which are bounded by a fold bifurcation of periodic orbits between the interleaved stable and unstable orbits that lie on the torus inside the parameter region of the Arnold tongue). These Arnold tongues will lie in the small parameter region indicated in the inset of Figure \[fig:KappaTau\_2D\_Cont\] where torus bifurcations occur. We will not pursue the Arnold tongue structure in this work; Arnold tongues have previously been computed for DDEs, even in the state-dependent delay case [@Calleja_2017]. Another curve of torus bifurcations is visible in Figure \[fig:KappaTau\_2D\_Cont\] close to $(\kappa,\tau)=(0.7,4)$. The corresponding torus bifurcation can be seen in Figure \[fig2\_HSC\_Biftool1D\_KappaDA\] at $\kappa\approx0.65046$ (with $\tau=3.9$), where the periodic orbit loses stability in a torus bifurcation. In this case there is not a second corresponding torus bifurcation where the periodic orbit regains stability. Instead there is a period-doubling bifurcation (which is a resonant torus bifurcation) near $\kappa=0.767$, but the periodic orbit on the principal branch does not regain stability at this point. That this period-doubling bifurcation is associated with the neighbouring torus bifurcation can be surmised from Figure \[fig:KappaTau\_2D\_Cont\] where we see that the endpoints of the curve of torus bifurcations lie on the period-doubling bifurcation curve. The torus dynamics are likely to be more complicated in this case, but we did not explore them. Chaotic Dynamics {#sec.chaos} ---------------- Having found long period and quasi-periodic orbits, it is natural to also ask whether  admits chaotic solutions. Kaplan and Yorke [@KaplanYorke1979] defined an attractor dimension, now known as the Lyapunov dimension, to be $$\label{lyapdim} d=k-\frac{1}{\lambda_{k+1}}\sum_{j=1}^k\lambda_j$$ where the Lyapunov exponents are ordered so $\lambda_1{\geqslant}\lambda_2{\geqslant}\ldots$, and $k$ is the largest integer so that the sum of the first $k$ exponents is non-negative, thus necessarily $\lambda_{k+1}<0$, and $d\in[k,k+1)$. For the torus seen in the previous section with $\lambda_1=\lambda_2=0>\lambda_3$ equation  gives a dimension of $d=2$, as expected for a torus. One generally accepted indication of chaos is the presence of a positive Lyapunov exponent, in which case the Lyapunov dimension will be larger than two. We will investigate the existence of chaotic solutions for  by numerically computing the Lyapunov exponents, again using the method of Breda and Van Vleck [@Breda_2014]. ![Orbit diagram showing local maxima and minima of solutions of as a function of the delay $\tau$, with $\kappa=0.865$ and other parameters taking their values from Table \[tab.model.par\]. The red and green dots denote respectively the local maxima and minima computed along a mesh with 30400 points for increasing $\tau$, while the black and blue dots denote the local maxima and minima computed by decreasing $\tau$. The bifurcation points of primary branch are identified using the same symbols as in Figures \[fig:KappaCont\] and \[fig2\_HSC\_Biftool1D\_KappaDA\]. The upper side panel shows a detail from the main panel, while the other two side panels show just the decreasing and increasing parameter scans, illustrating bistability and hysteresis in the system.[]{data-label="fig.chaos0"}](fig_DdeStemBifTauBoth3F_RF){width="\columnwidth"} ![Orbit diagram showing local maxima and minima of solutions of as a function of the delay $\tau$, with $\kappa=0.865$ and other parameters taking their values from Table \[tab.model.par\]. The red and green dots denote respectively the local maxima and minima computed along a mesh with 30400 points for increasing $\tau$, while the black and blue dots denote the local maxima and minima computed by decreasing $\tau$. The bifurcation points of primary branch are identified using the same symbols as in Figures \[fig:KappaCont\] and \[fig2\_HSC\_Biftool1D\_KappaDA\]. The upper side panel shows a detail from the main panel, while the other two side panels show just the decreasing and increasing parameter scans, illustrating bistability and hysteresis in the system.[]{data-label="fig.chaos0"}](fig_DdeStemBifTauBoth2Z_RF "fig:"){width="\columnwidth"}\ ![Orbit diagram showing local maxima and minima of solutions of as a function of the delay $\tau$, with $\kappa=0.865$ and other parameters taking their values from Table \[tab.model.par\]. The red and green dots denote respectively the local maxima and minima computed along a mesh with 30400 points for increasing $\tau$, while the black and blue dots denote the local maxima and minima computed by decreasing $\tau$. The bifurcation points of primary branch are identified using the same symbols as in Figures \[fig:KappaCont\] and \[fig2\_HSC\_Biftool1D\_KappaDA\]. The upper side panel shows a detail from the main panel, while the other two side panels show just the decreasing and increasing parameter scans, illustrating bistability and hysteresis in the system.[]{data-label="fig.chaos0"}](fig_DdeStemBifTauBoth2ZRL_RF "fig:"){width="\columnwidth"}\ ![Orbit diagram showing local maxima and minima of solutions of as a function of the delay $\tau$, with $\kappa=0.865$ and other parameters taking their values from Table \[tab.model.par\]. The red and green dots denote respectively the local maxima and minima computed along a mesh with 30400 points for increasing $\tau$, while the black and blue dots denote the local maxima and minima computed by decreasing $\tau$. The bifurcation points of primary branch are identified using the same symbols as in Figures \[fig:KappaCont\] and \[fig2\_HSC\_Biftool1D\_KappaDA\]. The upper side panel shows a detail from the main panel, while the other two side panels show just the decreasing and increasing parameter scans, illustrating bistability and hysteresis in the system.[]{data-label="fig.chaos0"}](fig_DdeStemBifTauBoth2ZLR_RF "fig:"){width="\columnwidth"} ![A sequence of windows of periodic dynamics with parameter intervals of apparent chaotic dynamics. Panel (i) shows a zoom of part of the Figure \[fig.chaos0\], while panel (ii) shows a zoom of part of panel (i). []{data-label="fig.chaosB"}](fig_DdeStemBifTauBothM_RF "fig:"){width="49.00000%" height="44mm"} (-245,2.5) ![A sequence of windows of periodic dynamics with parameter intervals of apparent chaotic dynamics. Panel (i) shows a zoom of part of the Figure \[fig.chaos0\], while panel (ii) shows a zoom of part of panel (i). []{data-label="fig.chaosB"}](fig_DdeStemBifTauBothN_RF "fig:"){width="49.00000%" height="44mm"} (-245,2.5) In Figure \[fig2\_HSC\_Biftool1D\_KappaDA\] we see that for $\kappa\in(0.795,0.929)$ both steady states and the periodic orbits on the main branch and on the period doubled branch are all unstable. However from Theorem \[theorem.bound\] we know that the dynamics must remain bounded, and so there must be a global attractor for these parameters. This parameter interval of unstable solutions for the $\kappa$ continuation lies between two period doubling bifurcations in Figure \[fig2\_HSC\_Biftool1D\_KappaDA\], which is also inside the lobe of period-doubling bifurcations depicted in the two-parameter continuation in $\kappa$ and $\tau$ in Figure \[fig:KappaTau\_2D\_Cont\], and we investigate the dynamics within this region. In Figure \[fig.chaos0\] we present an orbit diagram for  as $\tau$ is varied across this region with $\kappa=0.865$. Orbit diagrams are usually produced for maps, and we reduce the solution of  to a map by considering the crossings of a Poincaré section. Previously, we considered Poincaré sections with $Q(t)$ constant, which would not work so well in this case because the value of $Q^*$ changes as $\tau$ is varied, and we would need to vary the constant to ensure that the orbits cross the Poincaré section. Instead, we consider the local maxima and minima of $Q(t)$ along the solution, or equivalently the points where $Q'(t)=0$ with $Q''(t)<0$ or $Q''(t)>0$ (respectively). For each value of $\tau$ using the MATLAB `dde23` routine [@Matlab] we integrate through a time interval of $50\tau$ days, then plot the value of $Q$ at its last local maxima and minima. Since the dynamics are more interesting for some $\tau$ values then others, we defined a $\tau$ mesh with $9121$ points from $1$ to $3.4$, $19000$ points from $3.4$ to $4.4$, and $2281$ points from $4.4$ to $5$. These three meshes were combined to form a mesh of $30400$ points from 1 to 4 for increasing $\tau$. A second mesh with 30399 points interleaved with the previous mesh was used for decreasing $\tau$. For each mesh point the last $\tau$ time units of the solution was used as the initial function to compute the solution at the next mesh point. The results displayed in Figure \[fig.chaos0\] clearly reveal the bifurcations already shown in Figure \[fig:KappaTau\_2D\_Cont\] including the Hopf bifurcations at $\tau=1.1364$ and $4.6841$, the fold bifurcations near $\tau= 2.4379$, $2.4451$, $4.3281$ and $4.4364$ and the period doubling bifurcations at $\tau= 3.1303$, $4.3909$, $4.4215$ and $4.5575$. Between those period doubling bifurcations, Figure \[fig.chaos0\] reveals numerous period doubling cascades and several parameter intervals of apparent chaotic dynamics with windows of periodic dynamics. For some intervals of parameter values the results of sweeping left to right and right to left are significantly different, revealing the bistability of attracting states and hysteresis between them. The side panels to Figure \[fig.chaos0\] illustrate this for $\tau\approx3.85$, where increasing $\tau$ sequentially appears to reveal chaotic dynamics, but decreasing $\tau$ reveals a stable periodic-orbit which appears to undergo a period doubling cascade leading to a small interval of parameter values for $\tau\approx3.9$ for which there are apparently co-existing chaotic attractors. Figure \[fig.chaosB\] shows two successive magnifications from a small region of Figure \[fig.chaos0\]. To reveal the finer structure, we recomputed the orbit diagram for each of these $\tau$ intervals for $30000$ equally spaced increasing $\tau$ values, and a second interleaved mesh with one fewer point with decreasing $\tau$ values. Figures \[fig.chaos0\] and \[fig.chaosB\](i)-(ii) together suggest a self-similarity of the structure with sequences of windows of periodic dynamics separated by intervals of apparent chaotic dynamics, on ever smaller parameter intervals. While it would be interesting to study the scaling in the period doubling cascades, this is very difficult to do because the mapping is only implicitly defined, and requires that we numerically solve the DDE  between each extremum of $Q(t)$. Instead here, we will investigate the nature of the chaotic solutions. ![Chaotic orbit for $\tau=3.9$ and $\kappa=0.865$. (This parameter set is indicated by the red triangle in Figure \[fig:KappaTau\_2D\_Cont\] inset) (i) Orbit segment. (ii) Time-delay embedding $(Q(t),Q(t-\tau))$ and (iii) solution space $(Q(t),Q(t-\tau/2),Q(t-\tau))$ of the solution for $t\in[3000,6000]$. The black line and black dots in (i)-(ii) represent the unstable steady state $Q^*$. (iv) Convergence of the first three Lyapunov exponents. []{data-label="fig.chaos1"}](fig6_DdeStemKappaQ0865PSTau39C "fig:"){width="49.00000%" height="44mm"} (-245,2.5) ![Chaotic orbit for $\tau=3.9$ and $\kappa=0.865$. (This parameter set is indicated by the red triangle in Figure \[fig:KappaTau\_2D\_Cont\] inset) (i) Orbit segment. (ii) Time-delay embedding $(Q(t),Q(t-\tau))$ and (iii) solution space $(Q(t),Q(t-\tau/2),Q(t-\tau))$ of the solution for $t\in[3000,6000]$. The black line and black dots in (i)-(ii) represent the unstable steady state $Q^*$. (iv) Convergence of the first three Lyapunov exponents. []{data-label="fig.chaos1"}](fig4_DdeStemKappaQ0865PSTau39C "fig:"){width="49.00000%" height="44mm"} (-245,2.5) ![Chaotic orbit for $\tau=3.9$ and $\kappa=0.865$. (This parameter set is indicated by the red triangle in Figure \[fig:KappaTau\_2D\_Cont\] inset) (i) Orbit segment. (ii) Time-delay embedding $(Q(t),Q(t-\tau))$ and (iii) solution space $(Q(t),Q(t-\tau/2),Q(t-\tau))$ of the solution for $t\in[3000,6000]$. The black line and black dots in (i)-(ii) represent the unstable steady state $Q^*$. (iv) Convergence of the first three Lyapunov exponents. []{data-label="fig.chaos1"}](fig13_DdeStemKappaQ0865PSTau39C "fig:"){width="49.00000%" height="44mm"} (-245,2.5) ![Chaotic orbit for $\tau=3.9$ and $\kappa=0.865$. (This parameter set is indicated by the red triangle in Figure \[fig:KappaTau\_2D\_Cont\] inset) (i) Orbit segment. (ii) Time-delay embedding $(Q(t),Q(t-\tau))$ and (iii) solution space $(Q(t),Q(t-\tau/2),Q(t-\tau))$ of the solution for $t\in[3000,6000]$. The black line and black dots in (i)-(ii) represent the unstable steady state $Q^*$. (iv) Convergence of the first three Lyapunov exponents. []{data-label="fig.chaos1"}](fig_DdeStemKappaQ0865PSTau39E_LE "fig:"){width="49.00000%" height="44mm"} (-245,2.5) With $\kappa=0.865$ and $\tau=3.9$ the orbit diagram suggests that the dynamics should be chaotic, and this case is illustrated in Figure \[fig.chaos1\]. At first glance the time series in panel (i) resembles a period-doubled solution, but the maxima close to $Q(t)=0.2$ actually alternate in height, so the solution is closer to a period-quadrupled solution. However, the time-delay embeddings in panel (ii) and (iii) appear to show that the orbit is not periodic but that there is a very structured low-dimensional attractor. The solutions were computed by taking a constant initial history function close to $Q^*$ and integrating with `dde23` through the transient dynamics until the orbit converges to the attractor. The segment of the solution trajectory that is displayed in Figure \[fig.chaos1\](ii) and (iii) spans 3000 days. The initial convergence of the first three Lyapunov exponents is illustrated in Figure \[fig.chaos1\](iv), but the full computation of the exponents, using the method of Breda and Van Vleck [@Breda_2014], is over a time interval of $30000$ days, or 82 years. The dynamics are not periodic over this time interval and the leading Lyapunov exponents are computed numerically to be $0.0107$, $-0.0002$, $-0.0966$ and $-0.1577$. The second Lyapunov exponent here is $0$ to numerical accuracy, and the presence of a positive Lyapunov exponent indicates chaos. The appearance of the orbit being close to a period-quadrupled orbit is most likely just due to the provenance of the chaotic orbit being created through a period-doubling cascade. Had we only looked at the time-series we could have been wrongly led to conclude that the dynamics was not chaotic; the time-series of the solution alone is very rarely sufficient to determine the nature of the dynamics in the interesting cases. For the attractor shown in Figure \[fig.chaos1\] the Lyapunov dimension is computed from  to be $d=2.11$. ![Chaotic orbit for $(\kappa,\tau)=(0.865,4.07)$. (i) Segment of solution time series on the attractor. Time-Delay embeddings (ii) $(Q(t),Q(t-\tau))$, and, (iii) $(Q(t),Q(t-\tau/2),Q(t-\tau))$ for $t\in[3000,6000]$. The black line and black dots in (i)-(iii) represent the unstable steady state. (iv) Projection $P(u_t)=(u_t(-\tau/2),u_t(-\tau))$ of function elements $u_t$ in the Poincaré section ${\mathcal{P}}_\tau=\{u_t: u_t(0)=Q^{*}, u_t'(0)<0\}$. []{data-label="fig5_DdeStemKappaQ0865Torus05"}](fig6_DdeStemKappaQ0865Torus05 "fig:"){width="49.00000%" height="44mm"} (-245,2.5) ![Chaotic orbit for $(\kappa,\tau)=(0.865,4.07)$. (i) Segment of solution time series on the attractor. Time-Delay embeddings (ii) $(Q(t),Q(t-\tau))$, and, (iii) $(Q(t),Q(t-\tau/2),Q(t-\tau))$ for $t\in[3000,6000]$. The black line and black dots in (i)-(iii) represent the unstable steady state. (iv) Projection $P(u_t)=(u_t(-\tau/2),u_t(-\tau))$ of function elements $u_t$ in the Poincaré section ${\mathcal{P}}_\tau=\{u_t: u_t(0)=Q^{*}, u_t'(0)<0\}$. []{data-label="fig5_DdeStemKappaQ0865Torus05"}](fig4_DdeStemKappaQ0865Torus01Cont10B "fig:"){width="49.00000%" height="44mm"} (-242,2.5) ![Chaotic orbit for $(\kappa,\tau)=(0.865,4.07)$. (i) Segment of solution time series on the attractor. Time-Delay embeddings (ii) $(Q(t),Q(t-\tau))$, and, (iii) $(Q(t),Q(t-\tau/2),Q(t-\tau))$ for $t\in[3000,6000]$. The black line and black dots in (i)-(iii) represent the unstable steady state. (iv) Projection $P(u_t)=(u_t(-\tau/2),u_t(-\tau))$ of function elements $u_t$ in the Poincaré section ${\mathcal{P}}_\tau=\{u_t: u_t(0)=Q^{*}, u_t'(0)<0\}$. []{data-label="fig5_DdeStemKappaQ0865Torus05"}](fig13_DdeStemKappaQ0865Torus01Cont10 "fig:"){width="49.00000%" height="44mm"} (-240,2.5) ![Chaotic orbit for $(\kappa,\tau)=(0.865,4.07)$. (i) Segment of solution time series on the attractor. Time-Delay embeddings (ii) $(Q(t),Q(t-\tau))$, and, (iii) $(Q(t),Q(t-\tau/2),Q(t-\tau))$ for $t\in[3000,6000]$. The black line and black dots in (i)-(iii) represent the unstable steady state. (iv) Projection $P(u_t)=(u_t(-\tau/2),u_t(-\tau))$ of function elements $u_t$ in the Poincaré section ${\mathcal{P}}_\tau=\{u_t: u_t(0)=Q^{*}, u_t'(0)<0\}$. []{data-label="fig5_DdeStemKappaQ0865Torus05"}](fig11PS180000 "fig:"){width="49.00000%" height="44mm"} (-240,2.5) As can be seen from Figure \[fig.chaos0\] the character of the chaotic dynamics is very sensitive to changes in the parameter values. Changing $\tau$ from $3.9$ to $4.07$ while keeping $\kappa$ and all the other parameters at their values in Figure \[fig.chaos1\] (see the blue triangle in Figure \[fig:KappaTau\_2D\_Cont\] inset) the dynamics becomes as shown in Figure \[fig5\_DdeStemKappaQ0865Torus05\]. Now the time series in Figure \[fig.chaos1\](i) is visually non-periodic, and the time-delay embedding in Figure \[fig.chaos1\](ii) and (iii) appear to fill more of phase space. This is reflected in the Lyapunov dimension. Computing out to $1.8\times10^5$ days (about $500$ years) the leading Lyapunov exponents are estimated to be $0.03027$, $-0.00009$, $-0.11271$ and $-0.153236$. Using  the Lyapunov dimension is computed to be $d=2.268$, larger than in the previous example. ![For $\kappa=0.68$, $\gamma=0.0354608$ and $\tau=9.88888$ an orbit which appears to display transient chaos. (i) The transition from non-periodic to periodic motion. (ii)-(iii) Time series $Q(t)$ and delay embedding $(Q(t-\tau),Q(t))$ for the non-periodic part of the orbit, and (iv)-(v) the periodic orbit. []{data-label="fig_Transition"}](fig_TransitionQ "fig:"){height="36mm" width="100.00000%"} (-503,2.5) ![For $\kappa=0.68$, $\gamma=0.0354608$ and $\tau=9.88888$ an orbit which appears to display transient chaos. (i) The transition from non-periodic to periodic motion. (ii)-(iii) Time series $Q(t)$ and delay embedding $(Q(t-\tau),Q(t))$ for the non-periodic part of the orbit, and (iv)-(v) the periodic orbit. []{data-label="fig_Transition"}](fig_TransitionChaosQ "fig:"){width="49.00000%" height="44mm"} (-251,2.5) ![For $\kappa=0.68$, $\gamma=0.0354608$ and $\tau=9.88888$ an orbit which appears to display transient chaos. (i) The transition from non-periodic to periodic motion. (ii)-(iii) Time series $Q(t)$ and delay embedding $(Q(t-\tau),Q(t))$ for the non-periodic part of the orbit, and (iv)-(v) the periodic orbit. []{data-label="fig_Transition"}](fig_TransitionChaosQQtau "fig:"){width="49.00000%" height="44mm"} (-247,2.5) ![For $\kappa=0.68$, $\gamma=0.0354608$ and $\tau=9.88888$ an orbit which appears to display transient chaos. (i) The transition from non-periodic to periodic motion. (ii)-(iii) Time series $Q(t)$ and delay embedding $(Q(t-\tau),Q(t))$ for the non-periodic part of the orbit, and (iv)-(v) the periodic orbit. []{data-label="fig_Transition"}](fig_TransitionPeriodicQ "fig:"){width="49.00000%" height="44mm"} (-251,2.5) ![For $\kappa=0.68$, $\gamma=0.0354608$ and $\tau=9.88888$ an orbit which appears to display transient chaos. (i) The transition from non-periodic to periodic motion. (ii)-(iii) Time series $Q(t)$ and delay embedding $(Q(t-\tau),Q(t))$ for the non-periodic part of the orbit, and (iv)-(v) the periodic orbit. []{data-label="fig_Transition"}](fig_TransitionPeriodicQQtau "fig:"){width="49.00000%" height="44mm"} (-245,2.5) The Lyapunov exponents could have been obtained with a shorter integration interval; the reason to integrate out to $500$ years was to obtain many crossings of the Poincaré section $Q(t)=Q^*$ in order to try to reveal the fractal structure of the attractor. This is difficult to achieve because the mapping between the intersections with the Poincaré section is only implicitly defined by the solution of the DDE  which has to be solved numerically. Nevertheless Figure \[fig5\_DdeStemKappaQ0865Torus05\](iv) shows a projection of the crossing of the Poincaré section, with insets which reveal some of the fractal structure of the attractor. If we vary all three of $\kappa$, $\gamma$ and $\tau$, while still holding all the other parameters at their homeostasis values from Table \[tab.model.par\], further interesting chaotic solutions can be found. Figure \[fig\_Transition\] shows an orbit that appears to display transient chaos. We interpret this as co-existence of a chaotic invariant set which is not asymptotically stable along with a periodic orbit which is stable. The orbit initially appears to be chaotic with a high-dimensional attractor (see panels (ii) and (iii)) but after about 2850 days transitions to the stable period-doubled periodic orbit which has a period of about $87.75$ days. This orbit was found by taking parameters close to a point where two period-doubling bifurcation branches cross each other in a bifurcation diagram on parameter space $(\gamma,\tau)$ (not shown), similar to the diagram from Figure \[fig:GammaTau\_2D\_Cont\] but with $\kappa$ not at its homeostasis value. If the value of $\kappa$ is changed to $\kappa=0.662$, but all the other parameters are held at their values from Figure \[fig\_Transition\], then the chaos becomes persistent. The attractor (not shown) looks very similar to Figure \[fig\_Transition\](iii), but for $\kappa=0.662$ the chaos persists through at least $3\times10^4$ days. That the attractor is of higher dimension than the previous examples can be inferred by comparing how disordered the two-dimensional projection seen in Figure \[fig\_Transition\](ii) looks compared to the previous examples. The first six Lyapunov exponents are computed numerically to be $+0.02444$, $+0.008055$, $-0.00004119$, $-0.006071$, $-0.01771$, and $-0.02882$. So for this example there are two positive Lyapunov exponents, the sum of the first five exponents is positive, and the Lyapunov dimension of the attractor is $d=5.3$. This dimension is relatively high compared to our previous examples and many of the classical examples of chaotic attractors in ODEs, such as the Lorenz attractor [@Lorenz_1963], for which the dimension is often between $2$ and $3$. However, DDEs define infinite-dimensional dynamical systems, and it is well-known that they can generate high-dimensional chaotic attractors [@Longtin1998]. Snaking branch {#sec.snaking} -------------- Continuation in $\tau$ with all the other parameters at their values from Table \[tab.model.par\] was illustrated in Section \[sec.1p\] (see Figures \[fig:TauCont\]-\[fig:TauOrb\]), and appears to show a canard explosion, similar to the canard explosion for $\gamma$ continuation, described in Section \[sec.longp.hsc\]. Different behaviour is observed if we vary all three parameters $\gamma$, $\kappa$ and $\tau$. ![Parameter continuation in $\tau$ for periodic orbits with $\gamma=0.15$, $\kappa=0.2$. (i) Bifurcation diagram showing Hopf bifurcations at $\tau=2.21327$ and $4.26817$, fold bifurcation of periodic orbits at $\tau=2.19228$ and $4.262041$, and period-doubling bifurcation of periodic orbits at $4.261983$, $4.262037$, $4.262054$ and $4.262183$. The inset shows the period of the orbits. (ii) Details of the snaking-branch region of the bifurcation diagram from panel (i). (iii) Examples of co-existing periodic orbits for $\tau=4.26203$ and $\tau=4.2620395$. (iv) Delay embeddings of co-existing periodic orbits for $\tau=4.26203$ and inset showing that the orbits are out of phase. (v) Three solutions of Eq.  for $\tau=4.2620395$ computed using the MATLAB `dde23` routine [@Matlab] with initial functions given by DDEBiftool solutions for the corresponding coloured dots shown in panel (iii). All three orbits converge to the large amplitude stable limit cycle seen in the bifurcation diagram in panel (iii). (vi) Delay embeddings of the three orbits shown in panel (v) along with the stable limit cycle to which they converge show in blue, also denoted by the blue square in panel (iii).[]{data-label="figSnakCont"}](figSnakingAmpPer "fig:"){width="49.00000%" height="44mm"} (-245,2.5) ![Parameter continuation in $\tau$ for periodic orbits with $\gamma=0.15$, $\kappa=0.2$. (i) Bifurcation diagram showing Hopf bifurcations at $\tau=2.21327$ and $4.26817$, fold bifurcation of periodic orbits at $\tau=2.19228$ and $4.262041$, and period-doubling bifurcation of periodic orbits at $4.261983$, $4.262037$, $4.262054$ and $4.262183$. The inset shows the period of the orbits. (ii) Details of the snaking-branch region of the bifurcation diagram from panel (i). (iii) Examples of co-existing periodic orbits for $\tau=4.26203$ and $\tau=4.2620395$. (iv) Delay embeddings of co-existing periodic orbits for $\tau=4.26203$ and inset showing that the orbits are out of phase. (v) Three solutions of Eq.  for $\tau=4.2620395$ computed using the MATLAB `dde23` routine [@Matlab] with initial functions given by DDEBiftool solutions for the corresponding coloured dots shown in panel (iii). All three orbits converge to the large amplitude stable limit cycle seen in the bifurcation diagram in panel (iii). (vi) Delay embeddings of the three orbits shown in panel (v) along with the stable limit cycle to which they converge show in blue, also denoted by the blue square in panel (iii).[]{data-label="figSnakCont"}](fig2_HSC_Biftool1D_Tau_SnakingCMerged3InCInset "fig:"){width="49.00000%" height="44mm"} (-245,2.5) ![Parameter continuation in $\tau$ for periodic orbits with $\gamma=0.15$, $\kappa=0.2$. (i) Bifurcation diagram showing Hopf bifurcations at $\tau=2.21327$ and $4.26817$, fold bifurcation of periodic orbits at $\tau=2.19228$ and $4.262041$, and period-doubling bifurcation of periodic orbits at $4.261983$, $4.262037$, $4.262054$ and $4.262183$. The inset shows the period of the orbits. (ii) Details of the snaking-branch region of the bifurcation diagram from panel (i). (iii) Examples of co-existing periodic orbits for $\tau=4.26203$ and $\tau=4.2620395$. (iv) Delay embeddings of co-existing periodic orbits for $\tau=4.26203$ and inset showing that the orbits are out of phase. (v) Three solutions of Eq.  for $\tau=4.2620395$ computed using the MATLAB `dde23` routine [@Matlab] with initial functions given by DDEBiftool solutions for the corresponding coloured dots shown in panel (iii). All three orbits converge to the large amplitude stable limit cycle seen in the bifurcation diagram in panel (iii). (vi) Delay embeddings of the three orbits shown in panel (v) along with the stable limit cycle to which they converge show in blue, also denoted by the blue square in panel (iii).[]{data-label="figSnakCont"}](figSnakAmplitudeF "fig:"){width="49.00000%" height="44mm"} (-245,2.5) ![Parameter continuation in $\tau$ for periodic orbits with $\gamma=0.15$, $\kappa=0.2$. (i) Bifurcation diagram showing Hopf bifurcations at $\tau=2.21327$ and $4.26817$, fold bifurcation of periodic orbits at $\tau=2.19228$ and $4.262041$, and period-doubling bifurcation of periodic orbits at $4.261983$, $4.262037$, $4.262054$ and $4.262183$. The inset shows the period of the orbits. (ii) Details of the snaking-branch region of the bifurcation diagram from panel (i). (iii) Examples of co-existing periodic orbits for $\tau=4.26203$ and $\tau=4.2620395$. (iv) Delay embeddings of co-existing periodic orbits for $\tau=4.26203$ and inset showing that the orbits are out of phase. (v) Three solutions of Eq.  for $\tau=4.2620395$ computed using the MATLAB `dde23` routine [@Matlab] with initial functions given by DDEBiftool solutions for the corresponding coloured dots shown in panel (iii). All three orbits converge to the large amplitude stable limit cycle seen in the bifurcation diagram in panel (iii). (vi) Delay embeddings of the three orbits shown in panel (v) along with the stable limit cycle to which they converge show in blue, also denoted by the blue square in panel (iii).[]{data-label="figSnakCont"}](figSnakEmbeddingF_PNG "fig:"){width="49.00000%" height="44mm"} (-245,2.5) ![Parameter continuation in $\tau$ for periodic orbits with $\gamma=0.15$, $\kappa=0.2$. (i) Bifurcation diagram showing Hopf bifurcations at $\tau=2.21327$ and $4.26817$, fold bifurcation of periodic orbits at $\tau=2.19228$ and $4.262041$, and period-doubling bifurcation of periodic orbits at $4.261983$, $4.262037$, $4.262054$ and $4.262183$. The inset shows the period of the orbits. (ii) Details of the snaking-branch region of the bifurcation diagram from panel (i). (iii) Examples of co-existing periodic orbits for $\tau=4.26203$ and $\tau=4.2620395$. (iv) Delay embeddings of co-existing periodic orbits for $\tau=4.26203$ and inset showing that the orbits are out of phase. (v) Three solutions of Eq.  for $\tau=4.2620395$ computed using the MATLAB `dde23` routine [@Matlab] with initial functions given by DDEBiftool solutions for the corresponding coloured dots shown in panel (iii). All three orbits converge to the large amplitude stable limit cycle seen in the bifurcation diagram in panel (iii). (vi) Delay embeddings of the three orbits shown in panel (v) along with the stable limit cycle to which they converge show in blue, also denoted by the blue square in panel (iii).[]{data-label="figSnakCont"}](fig3_DdeStemPSSnakingAlignedBig "fig:"){width="49.00000%" height="44mm"} (-245,2.5) ![Parameter continuation in $\tau$ for periodic orbits with $\gamma=0.15$, $\kappa=0.2$. (i) Bifurcation diagram showing Hopf bifurcations at $\tau=2.21327$ and $4.26817$, fold bifurcation of periodic orbits at $\tau=2.19228$ and $4.262041$, and period-doubling bifurcation of periodic orbits at $4.261983$, $4.262037$, $4.262054$ and $4.262183$. The inset shows the period of the orbits. (ii) Details of the snaking-branch region of the bifurcation diagram from panel (i). (iii) Examples of co-existing periodic orbits for $\tau=4.26203$ and $\tau=4.2620395$. (iv) Delay embeddings of co-existing periodic orbits for $\tau=4.26203$ and inset showing that the orbits are out of phase. (v) Three solutions of Eq.  for $\tau=4.2620395$ computed using the MATLAB `dde23` routine [@Matlab] with initial functions given by DDEBiftool solutions for the corresponding coloured dots shown in panel (iii). All three orbits converge to the large amplitude stable limit cycle seen in the bifurcation diagram in panel (iii). (vi) Delay embeddings of the three orbits shown in panel (v) along with the stable limit cycle to which they converge show in blue, also denoted by the blue square in panel (iii).[]{data-label="figSnakCont"}](fig4_DdeStemPSSnakingAlignedBig "fig:"){width="49.00000%" height="44mm"} (-245,2.5) In Figure \[figSnakCont\] we present the results of one-parameter continuation in $\tau$ with $\gamma=0.15$, $\kappa=0.2$, and the other parameters at their values from Table \[tab.model.par\]. The bifurcation diagram in Figure \[figSnakCont\](i) appears to show similar behaviour to the earlier $\tau$ continuation, with the steady state stable except between a pair of Hopf bifurcations. There is again a subcritical Hopf bifurcation leading to an interval of bistability between the steady state and a stable limit cycle, and there are again ripples in the amplitude and period of solutions along the branch of stable periodic orbits. The period of the orbits but not the amplitude increases significantly to reach $180$ days just before the period collapses to $21.2$ days at the Hopf bifurcation. As illustrated in Figure \[figSnakCont\](ii) there is *not* a canard this time. Instead the bifurcation branch snakes about $28$ times across $\tau=4.262041$ creating a small interval of $\tau$ values for which there are 57 co-existing periodic orbits. If the periodic orbits had been computed just by simulating to only find the stable solutions, it would appear that the amplitude and period both suddenly increase as $\tau$ is decreased through $4.262041$, suggesting the possibility of a canard explosion. But the DDEBiftool computations, which allow us to compute unstable periodic orbits just as well as stable ones, show this not to be the case. At the top and bottom of the snake there is a pair of fold bifurcations of periodic orbits both at $\tau\approx4.262041$, with the $\tau$ values of the bifurcations points agreeing to at least $7$ significant figures. The large amplitude orbit at the top of the snaking branch is stable for very small interval of $\tau$ values ($\tau\in(4.262037,4.262041)$), before losing stability in a period doubling bifurcation at $\tau\approx4.262037$. The small amplitude orbit at the bottom of the snake is stable for $\tau\in(4.262037,4.262054)$, before also losing stability in a period doubling bifurcation at $\tau\approx4.262054$. We will come back to the dynamics resulting from these period doublings at the end of this section. On the snaking branch between the two fold bifurcations at $\tau\approx4.262041$ all the periodic orbits are unstable. Figure \[figSnakCont\](iv) shows the delay embeddings for the 57 unstable limit cycles that co-exist when $\tau=4.26203$. The positions of these orbits on the snaking branch are indicated on Figure \[figSnakCont\](iii), where we use shades of pink to red to indicate orbits which are on the legs of the snake for which the amplitude increases as $\tau$ increases, and shades of cyan to blue for orbits on the legs of the branch where the amplitude decreases as $\tau$ increases. Although Figure \[figSnakCont\](iv) is very reminiscent of Figure \[fig:linearslowman\], there are crucial differences between the dynamics. In particular the orbits shown in Figure \[figSnakCont\](iv) are all unstable and all co-exist, whereas those of Figure \[fig:linearslowman\] are stable and exist over an exponentially small parameter interval, with a unique orbit existing for each of the parameter value. Nevertheless, there are significant similarities between the dynamics in the two cases with Figure \[figSnakCont\](iv) also appearing to indicate the presence of a slow manifold which is stable for a certain range of $Q$ values, with the orbits appearing to spiral onto the slow manifold. The inset in Figure \[figSnakCont\](iv) shows that the phase of this convergence is different on the two legs of the snaking branch. In Figure \[figSnakCont\](v)-(vi) we illustrate the dynamics with $\tau= 4.2620395$ when the large amplitude orbit (indicated by the blue dot on Figure \[figSnakCont\](iii)) is stable. For three different initial functions corresponding to unstable periodic orbits on the snaking branch (also indicated by coloured dots on panel (iii)) we take a part of the periodic orbit generated by DDEBiftool as the initial function, then use the MATLAB `dde23` routine [@Matlab] to simulate the solution. All three orbits are seen to converge to the stable large amplitude limit cycle, with period about $180$ days, but the nature of that convergence is not simple to explain. All of the periodic orbits along with their unstable manifolds are squeezed very close together when the orbits follow the slow manifold before diverging from each other again when the slow manifold becomes unstable, and probably as a consequence of this the connecting orbits between the limit cycles do not appear to have a simple structure. In the figure we see that the orbit shown in orange passes close to the slow manifold many times before approaching the stable limit cycle, around $t=2000$ days, while the other two initial functions lead to solutions which converge to the stable periodic orbit relatively quickly. ![Example periodic orbits and their delay embedding from the snaking branch shown in Figure \[figSnakCont\]. The delay embedding shows that oscillations in the $(Q(t),Q(t-\tau))$ projection appear to be in anti-phase between the left and right sides of the snaking branch. The orbit profiles show the same behaviour when they are plotted with final time points and $Q$ values equal.[]{data-label="figSnakExtr"}](figSnakExtrProfile3 "fig:"){width="49.00000%" height="44mm"} (-245,2.5) ![Example periodic orbits and their delay embedding from the snaking branch shown in Figure \[figSnakCont\]. The delay embedding shows that oscillations in the $(Q(t),Q(t-\tau))$ projection appear to be in anti-phase between the left and right sides of the snaking branch. The orbit profiles show the same behaviour when they are plotted with final time points and $Q$ values equal.[]{data-label="figSnakExtr"}](figSnakExtrEmbedd "fig:"){width="49.00000%" height="44mm"} (-245,2.5) In Figure \[figSnakExtr\] we show 4 orbits located at adjacent local extrema of $\tau$ on the snaking branch, as shown in the inset of panel (i). The profiles in panel (i) and delay embeddings in panel (ii) illustrate how the periodic orbit changes along the snaking curve of solutions as the amplitude increases. In Figure \[figSnakExtr\](i) the phase of the orbits is aligned so that they all have the global maximum and minimum aligned (close to $t=150$). Looking back one period to the previous occurrence of the global maxima and minima, we see that the position on the snaking branch of solutions is related to the number of short period oscillations seen as the solution converges onto the slow manifold. Crossing each leg of the snaking branch with increasing amplitude corresponds to adding half a short period oscillation to the whole periodic orbit. So the points on the snaking branch at minima of $\tau$ display one less/more short oscillation than seen at the next minima of $\tau$ directly above/below them on the branch, and half a short period oscillation less/more than seen at the adjacent maxima of $\tau$ with larger/smaller amplitude. The time-delay embeddings in Figure \[figSnakExtr\](ii) and its inset reveal that orbits located in the same extrema of the snaking branch converge to the slow manifold in phase with each other, and in antiphase to orbits located in opposite extrema. Although we have seen how the solution changes along the snaking branch, this does not explain why the branch itself snakes; similar solution behaviour but without branch snaking was observed in Section \[sec.longp.hsc\] for the canard explosion. As noted near the beginning of the section, either side of the fold points at $\tau=4.262041$ there are period doubling bifurcations. These bifurcations actually come in pairs, resulting in two separate intervals, one each side of the snaking part of the branch, for which the periodic solutions on the principal branch are unstable. For large amplitude solutions this occurs for $\tau\in(4.261983,4.262037)$ with a period doubling bifurcation at each end of this interval. For small amplitude solutions, the unstable part of the branch between the period-doubling bifurcations is for $\tau\in(4.262054,4.262183)$. To explore the dynamics as $\tau$ is varied over these parameter intervals in Figure \[figSnakBif\] we present an orbit diagram showing the local maxima and minima of $Q(t)$ along the solutions of  as $\tau$ is varied across this region with $\gamma=0.15$, $\kappa=0.2$. This is computed similarly to Figure \[fig.chaos0\], but this time integrating through a transient of $1530\tau$ days, then plotting all the maxima and minima that occur over the next $170\tau$ days. A mesh of two thousand equally spaced points for $\tau\in[4.26197,4.26219]$ was used for decreasing $\tau$. In this case we did not observe any noticeable hysteresis effects. For each mesh the solution over the last $\tau$ days was used as the initial history to start the transient computation for the next adjacent $\tau$ value. The results displayed in Figure \[figSnakBif\] clearly reveal the bifurcations already shown in Figure \[figSnakCont\](ii) including the fold bifurcation near $\tau= 4.262041$ and the period doubling bifurcations at $\tau= 4.261984$, $4.262038$, $4.262054$ and $4.262184$. Between the pairs of period-doublings, much richer dynamics are displayed than we had expected. Several period-doubling cascades are clearly visible (in the inset to the figure), leading to several intervals of apparently stable chaotic dynamics. There are also windows of stable periodic dynamics, including a period-3 window, which suggests the possibility of unstable chaotic dynamics (period-3 implies chaos only for one-dimensional maps). Dynamical Diseases {#sec.physiol} ================== In dynamic hematological diseases oscillations are observed in the circulating concentrations of one or more of the cell lines [@Foley_2009a]. Mathematical interest has often focused on what have been termed periodic hematological disorders, including cyclic neutropenia (CN), cyclic thrombocytopenia (CT) and periodic chronic myelogenous leukemia (PCML). CN is one of the most studied of these periodic diseases, with the concentration of circulating neutrophils varying from very low to normal or high levels with a period of about 19 to 21 days [@Bernard_2003; @Colijn_2005b]. Patients experience a bout of neutropenia (abnormally low neutrophil concentrations) each period, during which time the immune system is impaired and they are more susceptible to infection [@Dale_2016]. For patients with CT, oscillations in platelet counts from normal to very low values are observed with periods between 20 to 40 days [@Haurie_1998]. For patients with PCML, cycling in white blood cells from normal to high levels with periods from approximately 30 to 100 days [@Haurie_1998] and 40 to 80 days [@Pujo_Menjouet_2005] is reported. Many mathematical models of hematopoiesis have been developed in an effort to understand these diseases and the origins of the oscillatory dynamics [@Bernard_2003; @Pujo_Menjouet_2004; @ZhugeMackeyLeiJTB2019]. Efforts have often focused on deriving mathematical models and associated parameter sets for which the model has a stable limit cycle with a period commensurate with a particular disease under consideration [@Colijn_2005a; @Colijn_2005b; @Langlois2017]. Clinical efforts focus in entirely different directions, typically concentrating on alleviating the cytopenia (dangerously low blood cell concentrations) either by raising the concentration nadir or by decreasing the time interval that concentrations are below the recognised cytopenia threshold. Periodic oscillations in the strict mathematical sense are of limited clinical interest, and in the clinical literature the terms periodic and cyclic are often used as synonyms for episodic, and it is not implied that the time intervals between episodes are fixed. Consequently, there are many other hematological disorders which at least for some subjects display dynamics with a periodic signature, but for which there are only scattered case reports of the periodicity. Examples include cyclic 100-day pancytopenia [@Birgens_BJH93], cyclic (approximately 60 day) bicytopenia with Shapiro syndrome [@Roeker_CyclicBicyt_CRH2013], and Polycyth[æ]{}mia Vera [@Morley_AAM69] with approximately 28 day cycling. CT typically involves oscillations of just the platelets [@Foley_2009a], though one case of multi-lineage CT has recently been reported [@Langlois_CCR2018], while for CN and PCML oscillations of all of the major blood cell groups are observed [@Foley_2009a]. This suggests that for CN and PCML the cycling in all cell lineages may be due to a dynamic destabilization at the stem cell level [@Foley_2009a]. This destabilization occurs through different mechanisms in these two diseases with leukemic HSCs typically presenting a chromosome abnormality in PCML [@Pujo_Menjouet_2005]. In CN a mutation in the ELANE gene leads to increased apoptosis in the neutrophil progenitor cells during mitosis [@Dale_2002], and the destabilization of the HSCs appears to be caused by a feedback mechanism from the neutrophil lineage. Considerable variation in the oscillatory periods is observed within and between these disorders. A Lomb periodogram [@Langlois2017] is typically used to extract a periodic signature from the data, but the data itself is never truly periodic. There can be many reasons for this including data sampling, measurement error, intrinsic stochasticity of cell proliferation and differentiation, environmental variation, adaptation of the model parameters, or simply that the actual dynamics are not periodic. In Table \[tab.model.par\] we gave specific values of the model parameters from which we start our bifurcation analysis. Other authors use somewhat different values, or more correctly report ranges for the parameter values [@Bernard_2003; @Colijn_2005a]. Through inter-individual variability we should expect that a single parameter set will not be suitable for all subjects. However, as seen in Section \[sec.bifurc.hsc\] there are no bifurcations near to the stated homeostasis parameters. Hence, using other similar parameter values in the model will also lead to an asymptotically stable steady state. To provoke a qualitative change in the dynamics of requires a large change in the parameters. This situation was already envisioned by Glass and Mackey [@Glass_Mackey_1979] who coined the term dynamical disease to describe physiological systems where the control system itself is intact, but operating in a parameter range leading to abnormal dynamics. With significant changes to one or more parameters we do observe non-trivial dynamics. These dynamics only become of physiological, rather than mathematical, interest when they produce oscillations with characteristics similar to the reported diseases, and we do observe behaviour reminiscent of CN, PCML and CT. An increased apoptosis rate $\gamma$ during the cell cycle, as illustrated in Figures \[fig:GammaCont\] and \[fig:GammaOrb\] results in stable oscillations in the HSCs of period between about 75 and 100 days for $\gamma\in[0.2278,0.24]$. The shortest period orbit illustrated in Figure \[fig:GammaOrb\](i) is of interest. This has a maximal value of $Q(t)$ greater than 70% of $Q^h$, and hence maximum differentiation rate $\kappa Q(t)$ to peripheral blood cell precursors above 70% of the homeostatic rate, while the interval of severely reduced HSC numbers is relatively short (below 4 weeks). Such cycling in the HSCs would naturally result in pancytopenia in a full model of the hematopoietic system. If the apoptosis rate $\gamma$ is increased slightly above $0.24$ longer periodic orbits result, but as seen from Figure \[fig:GammaOrb\](ii) these have severely reduced HSC numbers for intervals of hundreds of days, which is much longer than the lifespan of circulating erythrocytes and which would induce a fatal anemia. Still higher values of $\gamma$ result in complete depletion of the HSCs with $Q=0$ becoming the globally attracting stable steady state. In Figures \[fig:KappaCont\] and \[KappaCont\_Orbits\_Bistab\_kappa\_plots\] we illustrate periodic dynamics of the HSCs for increased values of the differentiation rate $\kappa$. The stable periodic orbits in Figure \[fig:KappaCont\] exist when the rate constant $\kappa$ is $6$ or more times its homeostatic value, meaning that in this scenario the rate $\kappa Q(t)$ at which HSCs differentiate to precursors of circulating hematopoietic cells can be elevated compared to the homeostasis value, even when the number of HSCs $Q(t)$ is less than $Q^h$. While the periods observed in Figure \[fig:KappaCont\] are too short for PCML, longer periods of 30-100 days consistent with PCML can be obtained by also increasing the cell cycle time $\tau$ as seen in Figure \[fig2\_HSC\_Biftool1D\_KappaDA\](i) and Figure \[figSnakCont\](i)-inset. The largest periods seen in Figure \[fig:KappaCont\](ii), corresponding to the largest amplitude orbits on the main branch, and also the period-doubled orbits have period about 17 days, which is close to but a little shorter than typical periods for CN. Varying three parameters in it is possible to find periodic orbits with periods typical of CN, for example $(\kappa,\gamma,\tau)\approx(0.09,0.28,3)$ results in stable limit cycles with period between 19 and 21 days. Stable large amplitude limit cycles are observed in Figure \[fig2\_HSC\_Biftool1D\_KappaDA\] with periods in the 20 to 40 day range typical of CT. We observed numerous instances of bistability, which allows for the possibility that a therapeutic intervention or some other outside affect on the hematopoietic system could cause it to flip between different stable states. This has been observed in practice, where for example G-CSF can induce neutrophil oscillations with a period of about 7 to 15 days for neutropenic individuals [@Haurie_1998]. In Section \[sec.longp.hsc\] we explored a canard explosion. The very long period orbits that we found are likely not physiologically relevant, as they include long time intervals during which the HSCs are severely depleted. During these intervals the production of peripheral blood cells would be so severely compromised, that a fatal cytopenia would likely result. Although we do not rule out the possibility that a canard explosion with other parameters might lead to physiologically feasible long period orbits, the singular parameter ${\varepsilon}$ suggests this is unlikely. We see from and that $\kappa Q^*\sim{\varepsilon}$, so in the parameter regime $0<{\varepsilon}\ll 1$ where we might expect the canard to exist the differentiation of HSCs towards mature blood cell lines will be severely comprised. The quasi-periodic and chaotic solutions observed in Section \[sec.chaos.hsc\] may be of more physiological relevance for two reasons. Firstly, although these solutions all have significantly reduced HSC concentrations compared to homeostasis, they are found in parameter regions where the differentiation rate $\kappa$ is significantly increased, so that the differentiation $\kappa Q(t)$ out of the HSC compartment is at or above the homeostatic rate when $Q(t)$ is close to a local maxima. These HSC dynamics would likely lead to episodic pancytopenia in a full model of the hematopoietic system, which could be an interesting topic for follow up study. A second reason why these dynamics are of physiological relevance is that they show the system generating non-constant non-periodic dynamics which is more akin to what seen in real data than the purely periodic solutions that we investigated earlier. In this section we highlighted some of the solutions that we observed with periods in ranges characteristic of dynamical diseases. The two-parameter continuations of Section \[sec.2p\] could be used as a starting point for an extended study to find additional parameter regions with periodic solutions commensurate with dynamical diseases. Although it would be tractable to do that for the HSC model , such a study would be more interesting in a model of the hematopoietic system that incorporates multiple mature cell lines. We have clearly shown that our HSC model can demonstrate the oscillatory dynamics characteristic of dynamical diseases, without the need for any feedback loops from more mature cell lines. However, many of the solutions with interesting dynamics are associated with an increased differentiation rate $\kappa$. It remains an open question in particular dynamical diseases whether the differentiation rate is actually raised, and if so whether this is intrinsic to the disease-state HSC dynamics, or caused by feedback from the peripheral blood cell dynamics. Discussion and Conclusions {#sec.conc} ========================== We set out to show that the HSC model  could generate limit cycles of periods typical in dynamical diseases, simply by changing some of the parameter values in the model. Long period orbits had previously been observed by varying $s$ [@Pujo_Menjouet_2006; @Pujo_Menjouet_2005; @Pujo_Menjouet_2004]. We varied the parameters $\gamma$, $\kappa$ and $\tau$ and found periodic orbits of periods from about one week up to 9 years, encompassing the 19-21 days typical of CN, 20-40 days of CT and the 30-100 days of PCML. Whereas the model  treats the HSCs as a single homogeneous population, more recent mathematical models couple multiple copies of  together [@Adimy_2006a; @Qu_2010] to represent the different maturity levels of HSCs, and should be able to generate similar dynamics. We also observed a plethora of more exotic dynamics including mixed mode oscillations, period-doubling cascades and chaotic solutions. In Section \[sec.torus\] we showed that the DDE  admits stable torus solutions. Elsewhere, in Section \[sec.longp.hsc\] we studied a putative canard explosion, identified the singular variable, and constructed an approximation to the slow manifold and nearby dynamics. We showed that the local stability of the slow manifold changes very close to the point where the stability of the critical manifold changes. Our analysis of the canard explosion is incomplete. Established analysis and constructions rely on separating the slow and fast variables [@Wech13]. In contrast, equation  is scalar, and does not have a simple natural separation into fast and slow subsystems. We believe this to be the first demonstration of canard-like behaviour in a scalar system, and a full analysis will require an extension to current theory. In the current work we present a detailed numerical investigation of the phenomenon, with the hope that it will intrigue the theoreticians to complete the analysis. Equation  clearly displays mixed mode oscillations (see the time plots in Figures \[fig:GammaOrb\], \[fig:TauOrb\], \[fig:GammaCont\_Canard\]). Such dynamics are usually associated with slow-fast systems and coupled oscillators, and it is rather curious to see these phenomena in the scalar DDE . It is well-known that such dynamics are possible when there are multiple delays, and in the case of two state-dependent delays no other nonlinearity is required other than the state-dependency of the delays [@Calleja_2017]. In that case it seems that essentially the two delay terms interact as if they are coupled oscillators. However, equation  is scalar with only one delay, and has no Hopf-Hopf bifurcations. Equation  is in the general class of problems $$\dot{u}(t)=-u(t) - \alpha h(u(t))+Ah(u(t-\tau),$$ where $h(u)$ is a unimodal function. Problems of this form, have been studied in the case $\alpha=0$, but we are not aware of systemic theoretical studies of the more general case with $\alpha\ne0$. It seems likely to us that the dynamics reminiscent of coupled oscillators are generated by an interaction between the two instances of the nonlinearity evaluated at the current time $t$ and the delayed time $t-\tau$. We found many examples of bistability in . These include bistability between pairs of periodic orbits (Figures \[fig:KappaCont\], \[fig2\_HSC\_Biftool1D\_KappaDA\]), a periodic orbit and a stable steady state (Figures \[fig:KappaCont\], \[fig:GammaCont\], \[fig:TauCont\], \[fig2\_HSC\_Biftool1D\_KappaDA\]), a periodic orbit and a torus (Figures \[fig2\_HSC\_Biftool1D\_KappaDA\], \[fig1\_DdeStemKappaQTori01\]), as well as bistability between chaotic and nonchaotic solutions (Figures \[fig.chaos0\]). Bistability of periodic orbits is caused by pairs of fold bifurcations of limit cycle which originate in cusp bifurcations (seen in Figures \[fig:KappaTau\_2D\_Cont\], \[fig:GammaKappa\_2D\_Cont\]). We found both subcritical and supercritical Hopf bifurcations and bistability between a stable limit cycle and a steady state is associated with the Bautin or generalised Hopf bifurcation (denoted in Figures \[fig:KappaTau\_2D\_Cont\], \[fig:GammaTau\_2D\_Cont\]) where the criticality of the Hopf bifurcation changes. A curve of fold bifurcations of limit cycles emerges from this point which results in the interval of bistability seen in the one-parameter continuations between the fold and the subcritical Hopf bifurcation. Few previous studies have been sufficiently detailed to detect the criticality of the Hopf bifurcations, but those that were only found supercritical Hopf bifurcations [@Bernard_2003; @Bernard_2004; @Milton_1989], though Bernard *et al* [@Bernard_2003] did find Hopf bifurcations which were close to a criticality change. Mathematical studies of differential equations that model hematopoiesis frequently focus on existence and stability of a nontrivial solution. Once a Hopf bifurcation is found, the steady state becomes unstable, and secondary bifurcations to more complex dynamical structures are often not pursued. On the other hand, peripheral blood samples are often only taken for a few days at a time during a hospital stay, and are otherwise not taken, or taken at widely and irregular spaced intervals. The data, even when well sampled, frequently appears noisy, and it is unheard of to see solutions that are exactly periodic. Often, a periodic signature is only revealed by a frequency test, such as the Lomb periodogram. In this context, the bistable, long period, quasi-periodic, and transient and persistent chaotic dynamics that we find are very interesting. The same individual can present very different looking dynamics during different sampling intervals. It might be that the dynamics are actually periodic, but the period is so long that different parts of the periodic solution are revealed by different sampling intervals. Another possibility is that the dynamics are actually chaotic (but not random), and different parts of the chaotic attractor are revealed at different times. As we saw in Figure \[fig.chaos1\] a time series of chaotic dynamics can appear to be surprisingly regular, while in Figure \[fig.chaos0\] where the time series of the dynamics was clearly not regular, the system actually spends significant time near to the attractor of the first example. In a period doubling cascade to chaos the initial seed orbit and its low order period-doublings continue to exist after they become unstable, and can be expected to have some organising influence on the structure of the dynamics. Thus it is natural to expect there to be some periodic signal contained in the time series, even of a chaotic solution, and it is very unlikely that sufficient blood measurements would be taken from a single subject to discern genuinely chaotic dynamics. A widely used strategy for determining perturbed parameters associated with dynamical diseases is to try to find parameters which generate a periodic solution which is closest to the data [@Langlois2017]. Given the difficulty in distinguishing between chaotic and periodic dynamics, and the ability of the mathematical models to generate both, this strategy may not be optimal, and we should consider that the disease dynamics may not generate a simple periodic orbit, but that there may be multiple bifurcations between the homeostasis and diseased states, leading to more complex dynamics. Acknowledgments {#acknowledgments .unnumbered} =============== We are grateful to Mike Mackey for useful discussions and feedback on a draft of this work. We also thank John Mitry for our discussions on canards. We are grateful to Dimitri Breda for sharing his code for the computation of Lyapunov exponents in DDEs. Finally, we wish to thank the referees and editor for their many constructive suggestions. DCS was supported by National Council for Scientific and Technological Development of Brazil (CNPq) postdoctoral fellowship 201105/2014-4. 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--- abstract: 'Practical structural engineering problems often exhibit a significant degree of uncertainty in the material properties being used, the dimensions of the modeled structures, the magnitude of loading forces, etc. In this paper, we consider two beam models: a cantilever beam clamped at one end and a beam clamped at both ends. We consider a static and a dynamic load. The material uncertainty resides in the Young’s modulus, which is either modeled by means of one random variable sampled from a univariate Gamma distribution or with multiple random variables sampled from a Gamma random field. The Gamma random field is generated starting from a truncated Karhunen-Loève expansion of a Gaussian random field, followed by a transformation. Three different responses are considered: the static elastic response, the dynamic elastic response and the static elastoplastic response. The first two respectively simulate the spatial displacement of a concrete beam and its frequency response function in the elastic domain. The third one simulates the spatial displacement of a steel beam in the elastic as well as in the plastic domain. The plastic region is governed by the von Mises yield criterion with linear isotropic hardening. In order to compute the statistical quantities of the static deflection and frequency response function, Multilevel Monte Carlo (MLMC) is combined with a Finite Element solver. This recent sampling method is based on the idea of variance reduction, and employs a hierarchy of finite element discretizations of the structural engineering model. The good performance of MLMC arises from its ability to take many computationally cheap samples on the coarser meshes of the hierarchy, and only few computationally expensive samples on the finer meshes. In this paper, the computational costs and run times of the MLMC method are compared with those of the classical Monte Carlo method, demonstrating a significant speedup of up to several orders of magnitude for the studied cases. For the static elastic response, the deflection of the beam including uncertainty bounds in the spatial domain is visualized. For the static elastoplastic response, the focus lies on the visualization of a force deflection curve including uncertainty bounds. For the dynamic elastic response, the visualizations consist of a frequency response function, including uncertainty bounds.' address: - 'KU Leuven, Department of Computer Science, NUMA Section, Celestijnenlaan 200A, 3001 Leuven, Belgium' - 'KU Leuven, Department of Civil Engineering, Kasteelpark Arenberg 40, 3001 Leuven, Belgium' author: - 'P. Blondeel' - 'P. Robbe' - 'C. Van hoorickx' - 'G. Lombaert' - 'S. Vandewalle' bibliography: - 'Bib\_ref.bib' title: Multilevel Monte Carlo for uncertainty quantification in structural engineering --- Multilevel Monte Carlo, Uncertainty Quantification, Structural Engineering 65C05 Introduction ============ There is an increasing need to accurately simulate and compute solutions to engineering problems whilst taking into account model uncertainties. Methods for such uncertainty quantification and propagation in structural engineering can roughly be categorized into two groups: non-sampling methods and sampling methods. Examples of non-sampling methods are the perturbation method and the Stochastic Galerkin Finite Element method. The perturbation method is based on a Taylor series expansion approximating the mean and variance of the solution [@Kleiber]. The method is quite effective, but its use is restricted to models with a limited number of relatively small uncertainties. The Stochastic Galerkin method, first proposed by Ghanem and Spanos [@GhanemAndSpanos], is based on a spectral representation in the stochastic space. It transforms the uncertain coefficient partial differential equation (PDE) problem by means of a Galerkin projection technique into a large coupled system of deterministic PDEs. This method allows for somewhat larger numbers of uncertainties and is quite accurate. However, it is highly intrusive and memory demanding, making its implementation cumbersome and restricting its use to rather low stochastic dimensions. Sampling methods, on the other hand, are typically non-intrusive. Each sample corresponds to a deterministic solve for a set of specified parameter values. Two particularly popular examples are the Stochastic Collocation method [@Bab] and the Monte Carlo (MC) method [@Fishman]. The former samples a stochastic PDE at a carefully selected multidimensional set of collocation points. After this sampling, a Lagrange interpolation is performed leading to a polynomial response surface. From this, the relevant stochastic characteristics can easily be computed in a post-processing step. However, as is also the case for Stochastic Galerkin, the Stochastic Collocation method suffers from the curse of dimensionality; the computational cost grows exponentially with the number of random variables considered in the problem. The MC method on the other hand, selects its samples randomly and does not suffer from the curse of dimensionality. A drawback is its slow convergence as a function of the number of samples taken. The convergence of Monte Carlo can be accelerated in a variety of ways. For example, alternative non-random selections of sampling points can be used, as in Quasi-Monte Carlo [@Caflish; @Niederreiter], and Latin Hypercube [@Loh] sampling methods. Also, variance reduction techniques such as Multilevel Monte Carlo (MLMC) [@Giles] and its generalizations, see, e.g., [@PJ; @PJ2], can speed up the method. Note that there also exist hybrid variants which exhibit both a sampling and non-sampling character. This type of methods combines, for example, the Stochastic Finite Element methodology with Monte Carlo sampling or a multi-dimensional cubature method, see, e.g., [@Ghanem_Hyrbid; @Acharjee]. Monte Carlo methods have since long been used in the field of structural engineering, for example in problems of structural dynamics [@Masanobu] or in elastoplastic problems where the structure’s reliability is assessed [@Pulido]. The focus of this paper is to apply the MLMC method to a structural engineering problem, discretized by means of the Finite Element method. A comparison with the standard MC method will show a significant reduction in computational cost. The structure of the paper is as follows. In section 2, we formulate the mathematical model, introduce the problem statement and describe how the uncertainty is modeled. Section 3 recalls the MLMC method, and provides some additional algorithmic implementation details. In Section 4, numerical results are presented. First, a performance comparison is made between standard MC and MLMC for a static elastic and elastoplastic load case. Next, results showing the uncertainty propagation for the static elastic, static elastoplastic and the dynamic elastic response are shown for when the Young’s modulus is modeled by means of a random variable coming from a univariate Gamma distribution and as multiple random variables sampled from a Gamma random field. The fifth and last section offers concluding remarks and details some paths for further research. The mathematical model ====================== Beam models and material parameters ----------------------------------- The considered engineering problem is the response of a cantilever beam clamped at one side and a beam clamped at both sides as seen in Fig.\[fig:bean\_configurations\], assuming plane stress. The following responses are considered: the static elastic response, the dynamic elastic response and the static elastoplastic response. The first two respectively simulate the spatial displacement and the frequency response function of a concrete beam model solely in the elastic region. The third one simulates the spatial displacement of a steel beam in the elastic as well as in the plastic region. The dimensions of the beam are $2.5\,\mathrm{m}$ (length) by $0.25\,\mathrm{m}$ (height) by $1\,\mathrm{m}$ (width) for the elastic responses and $10^{-3}\,\mathrm{m}$ (width) for the elastoplastic response. The material parameters of the concrete are as follows: a mass density of $2500\,\mathrm{kg/m^3}$, a Poisson ratio of $0.15$ and a Young’s modulus subjected to some uncertainty with a mean value of $30\,\mathrm{GPa}$. The material parameters of the steel are as follows: a yield strength of $240\,\mathrm{MPa}$, a Poisson ratio of $0.25$ and a Young’s modulus subjected to a certain degree of uncertainty with a mean value of $200\,\mathrm{GPa}$. In order to model the material uncertainty, two uncertainty models will be considered. The first model consists of a homogeneous Young’s modulus characterized by means of a single random variable. The second model is a heterogeneous Young’s modulus represented as a random field. Both uncertainty models will be used to compute the stochastic characteristics of the static elastic and elastoplastic response of a beam clamped at both sides as seen in Fig.\[fig:bean\_configurations\] (right) and the dynamic response of a cantilever beam clamped at one side as seen in Fig.\[fig:bean\_configurations\] (left). [0.54]{} ![Cantilever beam loaded on its right end (left) and beam clamped at both sides loaded in the middle (right).[]{data-label="fig:bean_configurations"}](Beam_nogrid.pdf "fig:"){height="1.9cm"} [0.45]{} ![Cantilever beam loaded on its right end (left) and beam clamped at both sides loaded in the middle (right).[]{data-label="fig:bean_configurations"}](DoubleClampedBeam.pdf "fig:"){height="1.9cm"} ### The homogeneous model Following [@YongLiu], we opt to describe the Young’s modulus in the homogeneous model by means of a univariate Gamma distribution. This distribution is characterized by a shape parameter $\alpha$ and a scale parameter $\beta$: $$f(x|\alpha,\beta) = \dfrac{1}{\beta^{\alpha} \Gamma(\alpha)} x^{\alpha-1} e^{\left(-\dfrac{x}{\beta}\right)}.$$ The corresponding mean value and variance can be computed respectively as $\mu=\alpha \beta$ and $\sigma^2=\alpha \beta^2$. In this paper, we will select $\alpha\!=\!7.1633$ and $\beta\!=\!4.1880 \times 10^9$ in order to model the material uncertainty for the concrete beam, which are based on values coming from [@Ellen]. This leads to a mean of $30\,\mathrm{GPa}$ and a standard deviation of $11.2\,\mathrm{GPa}$. For modeling the material uncertainty in the steel beam, we select $\alpha\!=\!934.2$ and $\beta\!=\!0.214\times 10^9$, see [@Hess]. This gives a mean of $200\,\mathrm{GPa}$ and a standard deviation of $6.543\,\mathrm{GPa}$. The Gamma distribution for both materials is plotted in Fig.\[fig:pdf\_gam\_prior\]. [0.54]{} [0.45]{} ### The heterogeneous model The Young’s modulus with spatially varying uncertainty will be represented by means of a (truncated) Gamma random field. The construction of this random field is done by means of a classical, two step process. First, a (truncated) Gaussian random field is generated, using a Karhunen-Loève (KL) expansion [@Loeve]. Next, this Gaussian random field is transformed into a Gamma random field with a memoryless transformation [@Grigoriu]. Consider a Gaussian random field $Z(\mathbf{x},\omega)$, where $\omega$ is a random variable, with exponential covariance kernel, $$\label{eq:covariance_kernel} C(\mathbf{x},\mathbf{y}):= \sigma^2 \exp\left(-\dfrac{{\left\lVert\mathbf{x}-\mathbf{y}\right\rVert}_p}{\lambda}\right)\,.$$ We selected the 1-norm ($p\!=\!1$), a correlation length $\lambda\!=\!0.3$ and a standard deviation $\sigma \!=\!1.0$. The corresponding KL expansion can then be formulated as follows: $$Z(\mathbf{x},\omega)=\overline{Z}(\mathbf{x},.)+\sum_{n=1}^{\infty} \sqrt{\theta_n} \xi_n(\omega) b_n(\mathbf{x})\,. \label{eq:KLExpansion}$$ $\overline{Z}(\mathbf{x},.)$ denotes the mean of the field, and is set to zero. The $\xi_n(\omega)$ denote i.i.d.standard normal random variables. The symbols $\theta_n$ and $b_n(\mathbf{x})$ denote, respectively, the eigenvalues and eigenfunctions of the covariance kernel . For the parameter values selected above, those eigenvalues and eigenfunctions can be computed analytically, see [@GhanemAndSpanos; @Cliffe; @Ghanem_prob; @Zhang]. For a one-dimensional domain $\left[0 , 1\right]$, they are given by $$\label{eq:1D_eigVec} \theta_n^{1\text{D}}= \dfrac{2 \lambda}{\lambda^2 w_n^2 +1} ~~~~\mbox{and}~~~ b^{1\text{D}}_n(x) = A_n \left(\sin(w_n x) +\lambda w_n \cos(w_n x)\right)\,.$$ The normalizing constants $A_n$ are chosen such that $ {\left\lVertb_n\right\rVert}_2 = 1$. The constants $w_n$ represent the real solutions, in increasing order, of the transcendental equation $$\label{eq:transcendental} \tan(w)=\dfrac{2 \lambda w}{\lambda^2 w^2 -1}\,.$$ For the two-dimensional case, the eigenvalues and functions are obtained in a tensorproduct way, $$\label{eq:2d_eigenVal}\theta_n^{2\text{D}} = \theta_{i_{n}}^{1\text{D}} \theta_{j_n}^{1\text{D}} ~~~\mbox{and}~~ b_n^{2\text{D}} (\mathbf{x}) = b_{i_n}^{1\text{D}} (x_1) b_{j_n}^{1\text{D}} (x_2)~~ \mbox{with}~~n = (i_n, j_n)\,.$$ In an actual implementation, the number of KL-terms in is truncated to a finite value. This value depends on the magnitude and, more precisely, on the decay of successive eigenvalues. Those eigenvalues are plotted in Fig.\[fig:Eigenvalf\] (left). Inclusion of the first 101 KL-terms is sufficient in order to represent 90% of the variance of the random field. The percentage of the variance that is accounted for as a function of the number of included eigenvalues, corresponds to the cumulative sum of the eigenvalues, and is also illustrated in the figure. A cumulative sum of 1.0 corresponds to 100% of the variance of the field being accounted for. [0.54]{} [0.45]{} Once the Gaussian field has been generated, a memoryless transformation is applied pointwise, $$\label{eq:MemTrans} g(y) = F^{-1}\left[\Phi(y)\right],$$ in order to obtain the Gamma random field [@Grigoriu]. Here, $F$ denotes the marginal cumulative density function (CDF) of the target distribution and $\Phi$ the marginal CDF of the standard normal distribution. This transformation is depicted in Fig.\[fig:Eigenvalf\] (right) with the full line representing a realization of $F$, and the dashed line representing a realization of $\Phi$. Contour plots of a realization of a Gaussian random field and the corresponding Gamma random field are presented for illustration purposes in Fig.\[fig:contour\]. [0.54]{} \[fig:Gaussian Random field\] [0.45]{} \[fig:Gamma random field\] The Finite Element method ------------------------- The Finite Element method will be used to compute the responses of the beam assuming plane stress. An equidistant, regular rectangular mesh is applied with bilinear quadrilateral elements. The implemented solver for the elastic and the elastoplastic response is different. The underlying equations and solution methods are reviewed hereunder.\ \ For the static elastic response, the system equation is of the form \[Displacement\_eq\] $$\mathbf{K} \mathbf{\underline{u}} = \mathbf{\underline{f}},$$ with $\mathbf{K}$ the global stiffness matrix, $\mathbf{\underline{f}}$ the global nodal force vector and $\mathbf{\underline{u}}$ the displacement. The global stiffness matrix and nodal force vector are obtained from the element stiffness matrices $\mathbf{K^e}$ and the element force vectors $\mathbf{\underline{f}^e}$. These are computed analytically by evaluation of the following integrals: $$\label{eq:K}\mathbf{K^e} = \int_{\Omega} \mathbf{B}^T \mathbf{D} \mathbf{B} d \Omega ~~~~\mbox{and}~~~ \mathbf{\underline{f}^e} = \int_{\Gamma_t} \mathbf{N}^T \mathbf{\overline{t}}_n d \Gamma_t.$$ The element nodal force vector $\mathbf{\underline{f}^e}$ is modeled as a Neumann boundary condition, where $\mathbf{\overline{t}}_n$ stands for the surface traction specified as a force per unit area and $\mathbf{N}$ the element shape function matrix, integrated over the free element boundary $\Gamma_t$. The element stiffness matrix $\mathbf{K^e}$ is obtained by integrating the matrix $\mathbf{B^TDB}$ over the element’s surface $\Omega$. Matrix $\mathbf{B}$ is defined as $\mathbf{LN}$ with $\mathbf{L}$ the derivative matrix specified below, and with $\mathbf{D}$ the elastic constitutive matrix for plane stress, containing the element wise material parameters, $$\mathbf{L} = \begin{bmatrix} \pdv{}{x} & 0 \\ 0 & \pdv{}{y} \\ \pdv{}{y} & \pdv{}{x} \\ \end{bmatrix}\, ~~~~\mbox{and}~~~ \mathbf{D} = \dfrac{E}{1-\nu^2} \begin{bmatrix} 1 & \nu & 0 \\ \nu & 1 & 0 \\ 0 & 0 & \dfrac{1-\nu}{2} \\ \end{bmatrix}\,. \label{eq:BDmatrix}$$ For the dynamic response case the following equation is obtained, $$\label{eq:dyn} \left(\mathbf{K}(1+\imath \,\eta) - (2 \pi f)^2 \mathbf{M}\right)\mathbf{\underline{u}} = \mathbf{\underline{f}} ~~~\mbox{with}~~\mathbf{M}^e= \int_{\Omega} \mathbf{N}^T \rho \mathbf{N} d \Omega\,.$$ Matrix $\mathbf{M}$ denotes the system mass matrix obtained from the assembly of the element mass matrices $\mathbf{M^e}$. $f$ denotes the frequency and $\rho$ the volumetric mass density of the material. The multiplication of the system stiffness matrix $\mathbf{K}$ with the imaginary unit $\imath$ and the constant $\eta$, denotes the damping matrix.\ \ The approach for solving the static elastoplastic case differs due to the nonlinear stress-strain relation in the plastic domain. The plastic region is governed by the von Mises yield criterion with isotropic linear hardening. An incremental load approach is used starting with a force of $0\,\mathrm{N}$. The methods used to solve the elastoplastic problem are based on Chapter 2 $\S$4 and Chapter 7 $\S$3 and $\S$4 of [@Borst]. For this case, the system equation takes the following form: $$\label{Displacement_eq_plast} \mathbf{K} \Delta\mathbf{\underline{u}} = \mathbf{\underline{r}},$$ where $\Delta\mathbf{\underline{u}}$ stands for the resulting displacement increment. The vector $\mathbf{\underline{r}}$ is the residual, $$\mathbf{\underline{r}}=\mathbf{\underline{f}}+\Delta\mathbf{\underline{f}}-\mathbf{\underline{q}},$$ where $\mathbf{\underline{f}}$ stands for the sum of the external force increments applied in the previous steps, $\Delta\mathbf{\underline{f}}$ for the applied load increment of the current step and $\mathbf{\underline{q}}$ for the internal force resulting from the stresses $$\mathbf{\underline{q}}=\int_{\Omega} \mathbf{B}^T \bm{\sigma} d\Omega.$$ First the displacement increment of all the nodes is computed according to , with an initial system stiffness matrix $\mathbf{K}$ resulting from the assembly of the element stiffness matrix $\mathbf{K^e}$, computed by means of a Gauss quadrature $$\label{K_el_plast} \mathbf{K^e} = \int_{\Omega} \mathbf{B}^T \mathbf{D}^{ep} \mathbf{B} d \Omega,$$ where $\mathbf{D}^{ep}$ denotes the elastoplastic constitutive matrix. The initial state of $\mathbf{D}^{ep}$ is the elastic constitutive matrix from . Secondly, the strain increment $\Delta \varepsilon$ is computed, $$\Delta \varepsilon = \mathbf{B}\Delta\mathbf{\underline{u}}.$$ Thirdly, the nonlinear stress-strain relationship, $$d\bm{\sigma} = \mathbf{D}^{ep} d\varepsilon,$$ is integrated by means of a backward Euler method. The backward Euler method essentially acts as an elastic predictor-plastic corrector; an initial stress state that is purely elastic is computed and then projected in the direction of the yield surface as to obtain the plastic stress state. Due to the implicit nature of the integrated stress-strain relation, this equation must be supplemented with the integrated form of the hardening rule and the yield condition. This system of nonlinear equations is then solved with an iterative Newton-Raphson method. Afterwards, the consistent tangent stiffness matrix is computed [@Agusti]. This matrix is then used to compute the updated element stiffness matrix , resulting in an updated system stiffness matrix $\mathbf{K}$. The inner iteration step of solving the stress-strain relation and the updated system stiffness matrix is repeated for each outer iteration step which solves equation . The outer step consists in balancing the internal forces with the external ones as to satisfy the residual, which in our case equals $10^{-4}$ times the load increment. The procedure used is incremental-iterative, relying on the iterative Newton-Raphson method. This process is repeated for each load increment. Each finite element is assigned a value of the Young’s modulus. This is accomplished by means of the midpoint approach, i.e., the value is taken constant within each individual element and equal to the value of the realization of the random field at the center point of the element [@jie_lie]. The Multilevel Monte Carlo method ================================= Method overview {#sec:MethodOverview} --------------- The Multilevel Monte Carlo method (MLMC) is an extension of the standard Monte Carlo (MC) method, see, e.g., [@Giles; @Giles2]. The method relies on a clever combination of many computationally cheap low resolution samples and a relatively small number of higher resolution, but computationally more expensive samples. The application of MLMC to our problem will be based on a hierarchy of nested finite element meshes. These meshes will be indexed from $0$ to $L$, with $0$ indicating the coarsest mesh and $L$ the finest mesh. An example of such a hierarchy is shown in Fig.\[fig:gridrefinement\]. It is common in the PDE setting to use a geometric relation for the number of degrees of freedom between the different levels. We set the number of finite elements for a mesh at level $\ell$ proportional to $2^{d\ell}$, where $d$ is the dimension of the problem. ![Illustrative example of a hierarchy used in the MLMC method.[]{data-label="fig:gridrefinement"}](Grid.pdf){height="1.9cm"} Let $\operatorname{\mathbb{E}}[P_L(\omega)]$, or $\operatorname{\mathbb{E}}[P_L]$ for short, be the expected value of a particular quantity of interest $P$ depending on a random variable $\omega$, discretized on mesh $L$. The standard MC estimator for $\operatorname{\mathbb{E}}[P_L]$ using $N_L$ samples on mesh $L$, denoted as $Q^{\textrm{MC}}_L$, can be written as $$Q^{\textrm{MC}}_L={\frac{1}{N_L}}\sum_{n=1}^{N_L} P_L(\omega^n)\,. \label{eq:SimpleMC}$$ Multilevel Monte Carlo, on the other hand, starts from a reformulation of $\operatorname{\mathbb{E}}[P_L]$ as a telescoping sum. The expected value of the quantity of interest on the finest mesh is expressed as the expected value of the quantity of interest on the coarsest mesh, plus a series of correction terms (or *differences*): $$\operatorname{\mathbb{E}}[P_L]=\operatorname{\mathbb{E}}[P_0]+\sum_{\ell=1}^L \operatorname{\mathbb{E}}[P_\ell -P_{\ell-1}]\,. \label{Eq:TelescopingSum}$$ Each term in the right-hand side is then estimated separately by a standard Monte Carlo estimator with $N_\ell$ samples, i.e., $$Q^{\textrm{MLMC}}_L=\frac{1}{N_0}\sum_{n=1}^{N_0} P_0(\omega^{n}) + \sum_{\ell=1}^L \left \{ \frac{1}{N_\ell} \sum_{n=1}^{N_\ell} \left( P_\ell(\omega^{n})-P_{\ell-1}(\omega^{n})\right) \right \}, \label{eq:MLMC}$$ where $Q^{{\textrm{MLMC}}}_L$ is the Multilevel Monte Carlo estimator for the expected value $\operatorname{\mathbb{E}}[P_L]$, which is a discrete approximation of the continuous solution $\operatorname{\mathbb{E}}[P]$. The mean square error (MSE) is defined as $$\label{eq:MSE} \begin{split} \textrm{MSE}(Q^{\textrm{MLMC}}_L) & := \operatorname{\mathbb{E}}\left[\left(Q^{\textrm{MLMC}}_L - \operatorname{\mathbb{E}}\left[P\right]\right)^2\right] \\ & := \operatorname{\mathbb{V}}\left[Q^{\textrm{MLMC}}_L\right] + \left(\operatorname{\mathbb{E}}\left[Q^{\textrm{MLMC}}_L\right] - \operatorname{\mathbb{E}}\left[P\right]\right)^2, \end{split}$$ with $\operatorname{\mathbb{V}}\left[\mathrm{x}\right]$ denoting the variance of a random variable $\mathrm{x}$. The MLMC estimator in can be written as a sum of $L+1$ estimators for the expected value of the difference on each level, i.e., $$Q^{\textrm{MLMC}}_L = \sum_{\ell = 0}^L Y_{\ell}, \quad \text{where} \quad Y_{\ell} = \frac{1}{N_\ell} \sum_{n=1}^{N_\ell} \left( P_\ell(\omega^{n})-P_{\ell-1}(\omega^{n})\right).$$ where we defined $P_{-1}\coloneqq0$. Because of the telescoping sum, the MLMC estimator is an unbiased estimator for the quantity of interest on the finest mesh, i.e., $$\operatorname{\mathbb{E}}[P_L] = \operatorname{\mathbb{E}}[Q^{\textrm{MLMC}}_L].$$ Denoting by $V_{\ell}$ the variance of the difference $P_{\ell} - P_{\ell-1}$, the variance of the estimator can be written as $$\operatorname{\mathbb{V}}[Q^{\textrm{MLMC}}_L] = \sum_{\ell=0}^L \frac{V_{\ell}}{N_{\ell}}.$$ In order to ensure that the MSE is below a given tolerance $\epsilon^2$, it is sufficient to enforce that the variance $\operatorname{\mathbb{V}}[Q^{\textrm{MLMC}}_L]$ and the squared bias $(\operatorname{\mathbb{E}}[P_L-P])^2$ are both less than $\epsilon^2/2$. The condition on the variance of the estimator can be used to determine the number of samples needed on each level $\ell$. Following the classic argument by Giles in [@Giles], we minimize the total cost of the MLMC estimator $$\text{cost}(Q^{\textrm{MLMC}}) = \sum_{\ell=0}^{L} N_\ell C_\ell,$$ where $C_\ell$ denotes the cost to compute a single realization of the difference $P_\ell-P_{\ell-1}$, subject to the constraint $$\sum_{\ell=0}^L \frac{V_{\ell}}{N_{\ell}} \leq \frac{\epsilon^2}{2}.$$ Treating the $N_\ell$ as continuous variables, we find $$\label{eq:nopt} N_\ell = \frac{2}{\epsilon^2} \sqrt{\frac{V_\ell}{C_\ell}} \sum_{\ell=0}^L \sqrt{V_\ell C_\ell} .$$ Note that if $\operatorname{\mathbb{E}}[P_\ell]\rightarrow\operatorname{\mathbb{E}}[P]$, then $V_\ell\rightarrow0$ as $\ell$ increases. Hence, the number of samples $N_\ell$ will be a decreasing function of $\ell$. This means that most samples will be taken on the coarse mesh, where samples are cheap, and increasingly fewer samples are required on the finer, but more expensive meshes. In practice, the number of samples must be truncated to $\lceil N_\ell \rceil$, the least integer larger than or equal to $N_\ell$. Using , the total cost of the MLMC estimator can be written as $$\label{eq:cost} \text{cost}(Q^{\textrm{MLMC}}) = \frac{2}{\epsilon^2}\left(\sum_{\ell=0}^L \sqrt{V_\ell C_\ell}\right)^{2}.$$ This can be interpreted as follows. When the variance $V_\ell$ decreases faster with increasing level $\ell$ than the cost increases, the dominant computational cost is located on the coarsest level. The computational cost is then proportional to $V_0 C_0$, which is small because $C_0$ is small. Conversely, if the variance decreases slower with increasing level $\ell$ than the cost increases, the dominant computational cost will be located on the finest level $L$, and proportional to $V_L C_L$. This quantity is small because $V_L$ is small. For comparison, the computational cost of a Monte Carlo simulation that reaches the same accuracy is proportional to $V_0 C_L$. In our numerical results presented next, we will compare the cost of the MLMC estimator to the cost of a standard MC simulation, both in actual runtime (seconds) and in some normalized cost measure. This cost is chosen such that the time needed to obtain a sample on the coarsest mesh (level zero) is equal to a cost of one unit. When using the standard geometric mesh hierarchy, the cost for a single sample on level $\ell$ is proportional to $2^{\gamma \ell}$. The factor $\gamma$ is determined by the efficiency of the solver. The advantage of using this normalized cost is that it decouples the cost of the simulation from the specific computer hardware. The second term in is used to determine the maximum number of levels $L$. A typical MLMC implementation is level-adaptive, i.e., starting from a coarse finite element mesh, finer meshes are only added if required to reach a certain accuracy. Assume that the convergence $\operatorname{\mathbb{E}}[P_\ell]\rightarrow\operatorname{\mathbb{E}}[P]$ is bounded as $|\operatorname{\mathbb{E}}[P_{\ell} - P]| = \mathcal{O}(2^{-\alpha \ell})$. Then, we can use the heuristic $$\label{eq:bias_constraint} \abs{\operatorname{\mathbb{E}}[P_L-P]} = \abs{\sum_{\ell=L+1}^{\infty} \operatorname{\mathbb{E}}[P_\ell-P_{\ell-1}]} \approx \frac{\abs{\operatorname{\mathbb{E}}[P_L-P_{L-1}]}}{2^\alpha-1}$$ and check for convergence using $|\operatorname{\mathbb{E}}[P_L-P_{L-1}]|/(2^\alpha-1)\leq\epsilon/\sqrt{2}$, see [@Giles] for details. For completion, we now mention the central MLMC complexity theorem. We refer to [@Giles2] for a proof. Given the positive constants $\alpha, \beta, \gamma, c_1, c_2, c_3$ such that $\alpha \geq \dfrac{1}{2} \mathrm{min}\left(\beta,\gamma\right)$ and assume that the following conditions hold: 1. $\lvert \operatorname{\mathbb{E}}[P_{\ell} - P] \rvert \leq c_1 2^{-\alpha \ell}$, 2. $V_{\ell} \leq c_2 2^{-\beta \ell}$ and 3. $C_{\ell} \leq c_3 2^{\gamma \ell}$. Then, there exists a positive constant $c_4$ such that for any $\epsilon < \exp(-1)$ there exists an $L$ and a sequence $\{N_{\ell}\}_{\ell=0}^L$ for which the multilevel estimator, $Q^{\mathrm{MLMC}}_L$ has an $\mathrm{MSE}\leq\epsilon^2$, and $$\mathrm{cost}(Q^{\mathrm{MLMC}}) \leq \left\{ \begin{aligned} & c_4 \epsilon^{-2} && \mathrm{if} \quad \beta > \gamma, \\ & c_4 \epsilon^{-2}\left(\log \;\epsilon \right)^2 && \mathrm{if} \quad \beta = \gamma, \\ & c_4 \epsilon^{-2-\left(\gamma-\beta\right)/\alpha} && \mathrm{if} \quad \beta < \gamma. \\ \end{aligned} \right. \label{eq:Algo_regime}$$ \[Theorem\_1\] Implementation aspects {#Implementation_aspects} ---------------------- The MLMC method is non-intrusive. Only an interface between the Finite Element solver routine and the multilevel routine is necessary. All modules used in the simulation are written in <span style="font-variant:small-caps;">Matlab</span>. The elastic computations are run in sequential mode, without making use of parallelization. The elastoplastic computations are run in parallel mode because of their significantly larger computational cost. In particular, the computation of the individual samples is parallelized in order to get a speedup. This is possible because of the *embarrassingly parallel* nature of the MLMC and the MC methods. For the aforementioned configuration, a number of $12$ samples can be computed concurrently. For more details on load balancing of MLMC samplers, we refer to [@Scheichl] Computation of the optimal number of samples per level according to is based on the variances of one finite element node on these levels. The node chosen for this task, is the node with the biggest variance of all the nodes which make up the finite element mesh. This way, the variance constraint is guaranteed to be satisfied for all nodes that constitute the finite element mesh. The first estimation of these variances is done by computing a trial sample set on levels 0, 1 and 2. The size of this sample set is $200$ for the elastic responses and $24$ for the elastoplastic responses. Variances on additional levels are estimated according to the second condition from Theorem \[Theorem\_1\], following [@Giles; @Giles2]. These trial samples are not included in the results presenting the number of samples per level. They are, however, included in the total runtime in seconds and the normalized cost of the algorithm. For MC, the number of trial samples taken for the elastic and elastoplastic response is identical as for level 0 in MLMC. As in the MLMC case, these samples will not be included in the tables. Numerical Results ================= In this section, we discuss our numerical experiments with the MLMC method. We consider the static and dynamic response, using both a homogeneous and a heterogeneous uncertain Young’s modulus. First a comparison will be made between the MC method and the MLMC method. Secondly, the rates from Theorem 1 will be estimated. For these first two parts, only the static responses (elastic and elastoplastic) are considered. This is because solutions for the static responses require only one MLMC simulation and one MC simulation for comparison. In contrast, solutions for the dynamic response require multiple individual MLMC simulations and multiple MC simulations: one for each individual frequency of the frequency response function. The third and last part visually presents solutions and their uncertainty. For both static responses, the solutions consist of displacements of the beam in the spatial domain. For the dynamic response, the solution is a frequency response function. All the results have been computed on a workstation equipped with 12 physical (24 logical) cores, Intel Xeon E5645 CPU’s, clocked at 2.40 GHz, and a total of 49 GB RAM. Comparison between MC and MLMC {#par:comparison} ------------------------------ First, we compare the efficiency of MLMC and MC for the static elastic and elastoplastic response of the beam clamped at both sides and loaded in the middle as shown in Fig.\[fig:bean\_configurations\] (right). For both the elastic and the elastoplastic case, the load is modeled as a distributed load acting on each of the vertical middle nodes of the beam. The sum of all these individual loads is independent of the refinement of the mesh. For the elastic case, the total load equals $10000\,\mathrm{kN}$. For the elastoplastic case, the beam is loaded in steps of $135\,\mathrm{N}$ from $0\,\mathrm{N}$ to $13.5\,\mathrm{kN}$. The quantity of interest is the node with the maximum transversal deflection, which coincidently also corresponds to the node with the largest variance. The coarsest finite element mesh (level 0) consists of 410 degrees of freedom with a square element size of $0.0625\,\mathrm{m}$, while the finest finite element mesh considered (level 4) consists of 83330 degrees of freedom with a square element size of $0.0039\,\mathrm{m}$. The coarsest mesh is chosen so as to discretize the beam in the height by at least four elements. ### Elastic Response Fig.\[fig:Times\] compares the actual simulation time needed to reach a certain tolerance $\epsilon$ on the root mean square error (RMSE) for both standard MC and MLMC, for a homogeneous (left) and heterogeneous uncertain Young’s modulus (right). Our MLMC method consistently outperforms the standard MC method, with speedups up to a factor ten or more. In Tab.\[table:times\], we summarize the results expressed in actual simulation time and in standard cost. For MC, some values of the RMSE have been omitted. This is due to the long computation time, which is of the order of days. Also note that the MC simulation is run at the finest level $L$ of the corresponding MLMC simulation, where $L$ is chosen according to . This explains the sudden jump in the MC simulation time in Fig.\[fig:Times\] (right). As stated in $\S$\[Implementation\_aspects\], $200$ samples are taken on the first three levels. The average time needed for this is $35$ seconds in case of a homogeneous Young’s modulus and $48$ seconds in case of a heterogeneous Young’s modulus. For the lowest listed tolerance of 2.5E-4, the percentage of time used to compute these trial samples amounts to $55$ percent of the total time. For finer tolerances, however, the relative cost of this preparatory work rapidly drops and becomes negligible compared to the cost of the remaining part of the algorithm. For the finest tolerance listed of 2.5E-5, this percentage is below $1$ percent. [0.54]{} [0.45]{} [0.54]{} [0.45]{} Fig.\[fig:sample\_sizes\] shows the number of samples over the different levels in function of the desired tolerance. Note that the number of samples is decreasing as the level $\ell$ increases, as required. Numerical values for $N_\ell$ are repeated in Tab.\[tab:Samples\]. In order to make a better comparison between the cost of MC and MLMC, we also list, under the column label “equivalent Max", the equivalent number of samples on the finest level $L$, that is, the number of samples on level $L$ that would yield a cost equal to the MLMC cost without including the cost of the trial samples. These results show a considerably lower sample size in favor of MLMC. Comparing the actual number of samples of MLMC with MC, we notice that when the Young’s modulus is homogeneous, the number of samples on the coarsest level in the MLMC simulation is of the same order of magnitude as the number of samples on the finest level $L$ in the MC simulation. This would allow the estimation of the number of samples needed for a MC simulation given the results of a MLMC simulation if required. However, when the Young’s modulus is heterogeneous, this number of samples differs. This stems from the fact that on this coarsest level a very rough approximation of the random field is used for the MLMC simulation, while the MC simulation uses a random field corresponding to the finest level of the MLMC simulation. It is thus probable that on the coarsest level, the variability of the random field is not fully and adequately captured. In order to better illustrate this point, the same Gaussian random field is shown in Fig.\[fig:level\_overview\] at different resolution levels. As can be seen, a sufficiently high number of points is required to appropriately capture the variability. In order to validate this hypothesis, MLMC simulations with a heterogeneous Young’s modulus have been rerun with a finer coarsest level. The results are presented in Tab.\[tab:Samples\_GRF\_refinement\]. As can be seen from this table, the number of samples of MC is of similar magnitude as the number of samples on the coarsest level of MLMC. [0.32]{} [0.32]{} [0.32]{} ### Elastoplastic Response For the elastic response, the MLMC simulations are level adaptive. The number of extra levels to be added depends on the condition of the bias, . However, for the elastoplastic response we chose to manually set the maximum level. This level is chosen based on a mesh convergence analysis. This is done because for the tolerances considered in Tab.\[table:times\_Plast\], samples would have to be taken on levels up to 5 or more. This would constitute a considerable time cost, which results from the iterative nature of the Finite Element implementation. The results of a mesh convergence study are shown in Fig.\[fig:mesh\_convergence\], respectively for the elastoplastic response (left) and the elastic response (right), with a beam configuration that is clamped at both sides and loaded in the middle. The figures show the transverse deflection of the middle node located on the beam’s top layer per level, represented as a full line, and the absolute value of its difference over the levels, represented as a dashed line. Comparing the results for both responses, it shows that for the elastic response the absolute value of the deflection’s difference on the finest mesh is much smaller than the one for the elastoplastic response. For the elastoplastic response, the deflection starts stagnating at around level three. Following these results, we thus state that the bias condition for the elastoplastic response is fulfilled at level three; no more than four MLMC levels are used. [0.54]{} [0.45]{} Fig.\[fig:Times\_plast\] lists the actual simulation time needed to reach a user specified tolerance $\epsilon$ on the root mean square error (RMSE) for MLMC for a homogeneous and a heterogeneous Young’s modulus for the elastoplatic response, i.e., the steel beam clamped at both sides. These values are presented in Tab.\[table:times\_Plast\]. As can be observed, the MLMC simulation outperforms the MC simulation in terms of computational speed and cost. A speedup of up to a factor ten in favor of MLMC is observed. As can be seen in Tab.\[table:times\_Plast\], the costs in case of lower tolerances for MLMC and MC are close together, while for finer tolerances, MLMC is significantly cheaper than MC. As elaborated in $\S$\[Implementation\_aspects\], $24$ samples are taken on the first three levels in order to obtain an initial estimate for the variances on the coarsest levels. The average time needed for this is $3728$ seconds in case of a homogeneous Young’s modulus and $3927$ seconds in case of a heterogeneous Young’s modulus. It can be observed that for the lowest tolerance of 2.5E-5 listed here, the percentage of time taken to compute these trial samples is about $33$ percent. For the finest tolerance of 2.5E-6 listed here, this percentage drops below $6$ percent. [0.54]{} [0.45]{} Fig.\[fig:sample\_sizes\_plast\] shows again the number of samples over the different levels in function of the desired tolerance. Numerical values are listed in Tab.\[tab:Samples\_plast\]. As is the case for the elastic response, the sample size decreases with increasing level. Note that on the finest level, only a few samples are required. For completeness, we need to mention a technical implementation detail. When suggests a value of $N_{\ell}$ equal to or less than 2 on an extra level, i.e., a level greater than 2, our algorithm still enforces a minimal number of 3 samples. This explains the lower limit of 3 samples in Tab.\[tab:Samples\_plast\]. As stated before, the parallelization enables the algorithm to compute $12$ samples concurrently. This is highly efficient for levels with a large number of samples, as is the case for coarser levels, and less efficient for finer levels. This is due to the load imbalance in the latter case. [0.54]{} [0.45]{} Rate verification ----------------- In this section, we estimate the rates $\alpha$, $\beta$ and $\gamma$ from Theorem 1, first for the static elastic response, then for the static elastoplastic response. ### Elastic Response We list the online estimated rates, i.e., the rates estimated during the run of the algorithm, for various tolerances $\epsilon$ on the RMSE, see Tab.\[tab:Parameters for the MLMC algorithm\], corresponding to the tolerances considered in Fig.\[fig:Times\]. Following the results in this table, it is possible to estimate the asymptotic cost of the MLMC estimator, using the different regimes from Eq.. Since $\beta>\gamma$ in all cases, we expect an optimal cost proportional to $\epsilon^{-2}$. This is indeed what we observed in the numerical experiments in Fig.\[fig:Times\]. Fig.\[fig:rates\] shows the behavior of the expected value and the variance of the quantity of interest $P_\ell$, and of the difference $P_\ell-P_{\ell-1}$, in case of a tolerance $\epsilon$ equal to 2.5E-5. Note that the mean and the variance of $P_\ell$ over the different levels remains constant while the variance and the mean of the differences between two successive levels continuously decreases. The rates $\alpha$ and $\beta$ can be read off as the slope of the lines labeled with $\triangledown$. In case where the Young’s modulus is homogeneous, Fig.\[fig:rates\] (left), the variance corresponding to the finest level is lower than the variance on the previous levels. This phenomenon can be explained by the number of samples on that level, which is used to estimate the variance. Only 7 samples were taken. Taking a higher number of samples on that level would bring the variance to equal the variance of the other levels. [0.54]{} [0.45]{} ### Elastoplastic Response As for the elastic response, we list the online rates for various tolerances $\epsilon$ on the $\text{RMSE}$ in Tab.\[tab:Parameters for the MLMC algorithm plast\]. For all tolerances, $\beta>\gamma$. Fig.\[fig:rates\_plast\] shows the behavior of the expected value and the variance of the quantity of interest $P_\ell$, and of the difference $P_\ell-P_{\ell-1}$, in case of a RMSE equal to 2.5E-6. Comparing Fig.\[fig:rates\] with Fig.\[fig:rates\_plast\], one notices that for the elastoplastic response the mean is not perfectly constant over the levels. This slight increase stems from the number of elements used for the discretization; the displacement is indeed dependent upon the discretization of the mesh, Fig.\[fig:mesh\_convergence\], for the mesh sizes considered here. [0.54]{} [0.45]{} Visualization of the solution ----------------------------- In this part, the static response, i.e., the displacement of the beam in the spatial domain, and the dynamic response, i.e., the frequency response functions (FRF), are shown. First, the static elastic response is presented. This consists of a visualization of the transverse displacement of the nodes along the top side of the beam. Secondly, results for the static elastoplastic response will be shown. The results consists of a visualization of a force deflection curve of the middle top side node of the beam. Thirdly and lastly, solutions for the dynamic response are presented. These consist of frequency responses functions for a single node of the finite element mesh. This node is chosen as the one that has the largest variance of all the nodes that make up the mesh. ### Static Elastic Response {#Sec:Viz_Sta_El} In this part we show the displacement of the concrete beam under a static load including uncertainty bounds in the elastic domain. The beam configuration is a beam clamped at both ends, loaded at mid span, as shown in Fig.\[fig:bean\_configurations\] (right). The load is modeled as a distributed load acting on each of the vertical middle nodes of the beam. The force on each node is such that the sum of these forces equals $10000\,\mathrm{kN}$, regardless of the refinement of the mesh. [0.54]{} [0.45]{} Fig.\[fig:deflection\_total\] shows the deflection of the beam with a homogeneous Young’s modulus (left) and with a heterogeneous Young’s modulus (right). The full orange line represents the average of the displacement, the dashed orange lines are the 1$\sigma$ bounds equidistant around the average, which are only relevant in case of a Normal distribution. The shades of blue represent the PDF, with the dark blue line corresponding to the most probable value. As can be observed from Fig.\[fig:deflection\_total\] (right), the average value and the most probable value of the PDF tend to coincide for the heterogeneous case. Here, the PDF of the displacement closely resembles that of a Normal distribution. This is however not the case when the Young’s modulus is homogeneous. Then the distribution of the solution has a non-negligible skewness. The homogeneous Young’s modulus case exhibits a larger uncertainty on its displacement due to the fact that the Young’s modulus is uncertain for each individual computed sample but uniform in each point for that sample. Averaging all these individual samples gives rise to wider uncertainty bounds. While for the heterogeneous Young’s modulus case, each individual computed sample is also uncertain but non-uniform; in each individual point the value for the Young’s modulus is different. This means that in different locations of the beam the Young’s modulus will be different. These locations will tend to compensate each other so that beam does not become overly stiff or weak. Fig.\[fig:indiv\_samples\_elast\] shows ten samples in case of a homogeneous Young’s modulus (left) and a heterogeneous Young’s modulus (right). [0.54]{} [0.45]{} [0.54]{} [0.45]{} [0.54]{} [0.45]{} Fig.\[fig:Construction of the PDF Gam Dist\] and Fig.\[fig:Construction of the PDF Gam Field\] show a cut-through in order to better illustrate how the shades of blue represent the PDF. ### Static Elastoplastic Response The solution is presented as a force deflection curve, Fig.\[fig:deflection\_total\_plast\]. The beam configuration is the same as for the static elastic response but with a different material; a steel beam clamped at both sides loaded at mid span, Fig.\[fig:bean\_configurations\] (right). The load is modeled in the same way as for the static elastic response. The beam is loaded from $0\,\mathrm{N}$ to $13.5\,\mathrm{kN}$ in steps of $135\,\mathrm{N}$. [0.54]{} [0.45]{} The line style and color convention is the same as for the static elastic response. As can be observed, the uncertainty bounds in case of a homogeneous Young’s modulus, Fig.\[fig:deflection\_total\_plast\] (left) are wider and more spread out than in case of a heterogeneous modulus (right). This behavior corroborates with the ones from the static elastic response. In Fig.\[fig:indiv\_samples\_plast\], ten individual samples are shown for a homogeneous Young’s modulus (left) and a heterogeneous Young’s modulus (right). Comparing these two sets of figures, it shows that, the deflection corresponding to the highest applied force in Fig.\[fig:indiv\_samples\_plast\] (left and right) is less than the deflection shown in Fig.\[fig:deflection\_total\_plast\] (left and right). This stems from the fact that these ten samples per Young’s modulus are generated on the coarsest mesh. These results illustrate even better the power of the MLMC method; even whilst taking most of the samples on coarser meshes, one still gets the result as if the simulation was run on the finest mesh. [0.54]{} [0.45]{} ### Dynamic Elastic Response For the dynamic elastic response, the solution is a frequency response function (FRF). The beam configuration we consider is a concrete cantilever beam clamped on the left side and loaded on the right side, as shown in Fig.\[fig:bean\_configurations\] (left), with a dynamic load. The load is modeled in the same way as for the static elastic response with the same magnitude. In order to ensure the correct representation of the dynamic response, and to determine the minimum number of finite elements required, the bending wavelength, $\lambda_{\mathrm{min}}$, must be evaluated as $$\lambda_{\mathrm{min}} = \sqrt{\dfrac{2 \pi}{f_{\mathrm{max}}}} \sqrt[4]{\dfrac{E I}{\rho A}}, \label{eq:MinElements}$$ with $E$ the mean Young’s modulus, $I$ the moment of inertia, $A$ the area, $\rho$ the density, $f_{\mathrm{max}}$ the highest simulated frequency, and $\lambda_{\mathrm{min}}$ the smallest obtained wavelength for the highest input frequency. For the considered beam configuration, it has been checked that at least six elements are used to represent the wavelength on the coarsest grid for the highest simulated frequency, which in this case is 400 Hz. [0.54]{} [0.45]{} The FRF results are presented in Fig.\[fig:dynamic\_total\]. As was the case for the static elastic and elastoplastic response, the shades of blue represent the PDF, with the blue line being the most probable value, and the orange line the average value. As can be observed, the uncertainty bounds for the FRF are wider and more spread out when the Young’s modulus is homogeneous, Fig.\[fig:dynamic\_total\] (left), as opposed to a heterogeneous Young’s modulus, Fig.\[fig:dynamic\_total\] (right). This discrepancy is due to the fact that in case of a homogeneous Young’s modulus, the resonance frequency will be shifted for each different sample. Averaging all these samples gives rise to a broad and wide uncertainty bound. In case of a heterogeneous Young’s modulus, the different samples compensate each other, in analogy with the explanation given in $\S$\[Sec:Viz\_Sta\_El\]. This gives rise to much smaller uncertainty bounds. Fig.\[fig:dynamic\_samples\] shows the resulting FRF for ten realizations. [0.54]{} [0.45]{} An important inequality that must hold for MLMC to work well is $$\operatorname{\mathbb{V}}[P_1-P_0] \ll \operatorname{\mathbb{V}}[P_1]. \label{eq:Variance_ineq}$$ It has been observed empirically that is not necessarily fulfilled for resonance frequencies and frequencies close by those resonance frequencies for the dynamic elastic response in case of a heterogeneous Young’s modulus. The reason for this phenomenon appears to be closely linked to the resolution of random fields on the different levels, as discussed in §\[par:comparison\]. In order to remedy to this, we use the following strategy. Using a small number of samples, the magnitude of $\operatorname{\mathbb{V}}[P_1] - \operatorname{\mathbb{V}}[P_1-P_0]$ is estimated. When the estimation is below a certain threshold $T$, the coarsest level is discarded and the algorithm is started on a finer mesh. We write this condition in a $\log_2$ base, in order not to lose consistency with Fig.\[fig:rates\], which depict the rates in a $\log_2$ base. The estimation after simplification is written as $$\log_2\left(\dfrac{\operatorname{\mathbb{V}}[P_1]}{\operatorname{\mathbb{V}}[P_1-P_0]}\right) > \log_2{T}. \label{eq:tresh}$$ For the experiments reported here, we selected the right-hand side of heuristically to be equal to $2.3$. Conclusion ========== In this work, it has been demonstrated that the Multilevel Monte Carlo method provides a significant computational cost reduction and speedup compared to the standard Monte Carlo method for computing the stochastics of the response in a structural engineering problem. This has been shown by means of actual computing times and normalized computational costs. Speedups are typically of the order of a factor ten for moderate accuracies and can be even much higher for higher accuracies. In addition, figures depicting the static elastic, the static elastoplastic and the dynamic elastic response have been presented with uncertainty bounds. To model the uncertainty in the Young’s modulus, we considered both a homogeneous model, represented by a single random variable, and a heterogeneous model, modeled by means of a random field. We observed that the nature of the uncertainty in the Young’s modulus has a big impact on the uncertainty characteristics of the simulation results. For the homogeneous Young’s modulus, the variance in the solution is larger than for the heterogeneous Young’s modulus. This is apparent from the wider and more spread out uncertainty bounds in $\S$\[Sec:Viz\_Sta\_El\]. Hence, for realistic applications, the appropriate choice of the uncertainty model is of great importance. Further paths of research will focus on ways to exploit the similarities between the responses for neighboring frequencies and to couple the Finite Element solvers with an existing Multilevel/Multi-index framework [@PJ; @PJ2]. Acknowledgements {#acknowledgements .unnumbered} ================ This research was funded by project IWT/SBO EUFORIA: “Efficient Uncertainty quantification For Optimization in Robust desgn of Industrial Applications" (IWT-140068) of the Agency for Innovation by Science and Technology, Flanders, Belgium. The authors gratefully acknowledge the support from the research council of KU Leuven though the funding of project C16/17/008 “Efficient methods for large-scale PDE-constrained optimization in the presence of uncertainty and complex technological constraints". The authors also would like to thank the Structural Mechanics Section of the KU Leuven and Jef Wambacq for supplying their StaBIL code and providing support. References {#references .unnumbered} ==========
--- abstract: 'We prove a sharp integral inequality for the dyadic maximal function of $\phi\in L^p$, when we are given the $L^1$ and $L^q$ norms of $\phi$, for some $q$, such that $1<q<p$. This way we are able to compute the Bellman function of three integral variables associated to the dyadic maximal operator, on a subset of its domain. Additionally, using the aforementioned inequality we manage to compute the Bellman function of two variables in a new way.' author: - 'Anastasios D. Delis, Eleftherios N. Nikolidakis' title: Sharp integral inequalities for the dyadic maximal operator and applications to its Bellman function of three variables --- Introduction {#sec:1} ============ It is well known that the dyadic maximal operator on ${\mathbb{R}}^n$ is a useful tool in analysis and is defined by $$\label{eq:1p1} \mathcal{M}_d \phi(x) = \sup \left\{ \frac{1}{|Q|}\int_Q|\phi(y)|\,{\mathrm{d}}y: x\in Q,\ Q\subseteq {\mathbb{R}}^n\ \text{is a dyadic cube}\right\},$$ for every $\phi\in L_\text{loc}^1({\mathbb{R}}^n)$, where the dyadic cubes are those formed by the grids $2^{-N}{\mathbb{Z}}^n$, for $N=0, 1, 2,\ldots$. It is also well known that it satisfies the following weak type (1,1) inequality $$\label{eq:1p2} \left|\left\{x\in{\mathbb{R}}^n : \mathcal{M}_d \phi(x) > \lambda\right\}\right| \leq \frac{1}{\lambda} \int_{\{\mathcal{M}_d\phi>\lambda\}}|\phi(y)|\,{\mathrm{d}}y,$$ for every $\phi\in L^1({\mathbb{R}}^n)$ and every $\lambda>0$, and which is easily proved to be best possible. Further refinements of can be seen in [@9] and [@10]. Then by using and the well known Doob’s method it is not difficult to prove that the following $L^p$ inequality is also true $$\label{eq:1p3} \|\mathcal{M}_d\phi\|_p \leq \frac{p}{p-1}\|\phi\|_p,$$ for every $p>1$ and $\phi\in L^p({\mathbb{R}}^n)$. Inequality turns out to be best possible and its sharpness is proved in [@16] (for general martingales see [@1] and [@2]). One way to study inequalities satisfied by maximal operators is by using the so called Bellman function technique. For example, in order to refine we can insert the $L^1$-norm of $\phi$ as an independent variable in , and try to find the best possible upper bound of $\|\mathcal{M}_d\phi\|_p$, when both the $L^1$ and $L^p$ norms of $\phi$ are given, by evaluating the (Bellman) function of two variables $$\label{eq:1p4} B^{(p)}(f,F) = \sup\left\{ \frac{1}{|Q|} \int_Q (\mathcal{M}_d \phi)^p : \phi \geq 0,\ \frac{1}{|Q|}\int_Q\phi=f,\ \frac{1}{|Q|}\int_Q\phi^p=F\right\},$$ where $Q$ is a fixed but otherwise arbitrary dyadic cube and $f, F$ are such that $0< f^p\leq F$. The approach of studying maximal operators by the introduction of the corresponding Bellman function was first seen in the work of Nazarov and Treil, [@5], where the authors defined the function $$\begin{gathered} \label{eq:1p5} B_p(f,F,L) =\\ \sup\left\{ \frac{1}{|Q|} \int_Q (\mathcal{M}_d \phi)^p : \frac{1}{|Q|}\int_Q\phi=f,\ \frac{1}{|Q|}\int_Q\phi^p=F, \sup_{R:Q\subseteq R}\frac{1}{|R|}\int_R\phi=L\right\},\end{gathered}$$ with $p>1$ (as an example they examine the case $p=2$), $Q$ is as above and $\phi$ is non negative in $L^p(Q),$ $R$ runs over all dyadic cubes containing $Q$ and the variables $F, f, L$ satisfy $0\leq f\leq L,$ $f^p\leq F.$ Exploiting a certain “pseudoconcavity” inequality it satisfies, they construct the function $4F-4fL+2L^2$ which has the same properties as and provides a good $L^p$ bound for the operator $\mathcal{M}_d$ (see [@5] for details). Both of the above Bellman functions were explicitly computed for the first time by Melas in [@3]. In fact this was done in the much more general setting of a non-atomic probability space $(X,\mu)$ equipped with a tree structure ${\mathcal{T}}$, which is similar to the structure of the dyadic subcubes of $[0,1]^n$ (see the definition in Section \[sec:2\]). Then we define the associated maximal operator by $$\label{eq:1p6} \operatorname{\mathcal{M}_\mathcal{T}}\phi(x) = \sup\left\{ \frac{1}{\mu(I)}\int_I |\phi|\,{\mathrm{d}}\mu: x\in I\in {\mathcal{T}}\right\},$$ for every $\phi\in L^1(X,\mu)$. Moreover and still hold in this setting and remain sharp. Now if we wish to refine for the general case of a tree $\mathcal{T},$ we should introduce the Bellman function of two variables related to the above maximal operator, which is given by $$\label{eq:1p7} B_{{\mathcal{T}}}^{(p)}(f,F) = \sup\left\{ \int_X(\operatorname{\mathcal{M}_\mathcal{T}}\phi)^p\,{\mathrm{d}}\mu: \phi\geq 0,\ \int_X\phi\,{\mathrm{d}}\mu = f,\ \int_X\phi^p\,{\mathrm{d}}\mu = F\right\},$$ where $0< f^p\leq F$. This function of course generalizes . In [@3] it is proved that $$\label{eq:1p8} B_{{\mathcal{T}}}^{(p)}(f,F) = F\,\omega_p\!\left(\frac{f^p}{F}\right)^p,$$ where $\omega_p : [0,1] \to \bigl[1,\frac{p}{p-1}\bigr]$, is defined by $\omega_p(z) = H_p^{-1}(z)$, and $H_p(z)$ is given by $H_p(z) = -(p-1)z^p + pz^{p-1}$. As a consequence $B_{{\mathcal{T}}}^{(p)}(f,F)$ does not depend on the tree ${\mathcal{T}}$. The technique for the evaluation of , that is used in [@3], is based on an effective linearization of the dyadic maximal operator that holds on an adequate class of functions called ${\mathcal{T}}$-good (see the definition in Section 2), which is enough to describe the problem as is settled on . Using this result on suitable subsets of $X$ and several calculus arguments, the author also managed to precisely evaluate the corresponding to Bellman function in this context, $$\begin{gathered} \label{eq:1p9} B_{{\mathcal{T}}}^{p}(f,F,L) = \sup\bigg\{\int_X (\max(\mathcal{M}_{{\mathcal{T}}} \phi, L)^p {\, \mathrm{d}}\mu :\phi\geq0, \phi \in L^p(X,\mu),\\ \int_X\phi {\, \mathrm{d}}\mu=f, \int_X\phi^p {\, \mathrm{d}}\mu=F, \bigg\}.\end{gathered}$$ Now and were computed in [@8] in a different way that avoids the calculus arguments involved in [@3]. A crucial intermediate result the authors obtain there, in this direction, is the following. [A]{} \[thm:a\] Let $\phi\in L^p(X,\mu)$ be non-negative, with $\int_X\phi\,{\mathrm{d}}\mu=f$. Then the following inequality is true $$\label{eq:1p10} \int_X(\operatorname{\mathcal{M}_\mathcal{T}}\phi)^p\,{\mathrm{d}}\mu \leq -\frac{1}{p-1}f^p + \frac{p}{p-1}\int_X \phi\,(\operatorname{\mathcal{M}_\mathcal{T}}\phi)^{p-1}\,{\mathrm{d}}\mu.$$ The motivation for our work here comes from our wish to refine even further by also considering the $q$-norm, $1<q<p,$ of the function $\phi$ as fixed and to compute the corresponding Bellman function. In particular, our goal was the evaluation of $$\begin{gathered} \label{eq:1p11} B_{{\mathcal{T}}}^{p,q}(f,A,F) = \sup\bigg\{\int_X (\mathcal{M}_{{\mathcal{T}}} \phi)^p {\, \mathrm{d}}\mu :\phi\geq0, \phi \in L^p(X,\mu),\\ \int_X\phi {\, \mathrm{d}}\mu=f, \int_X\phi^q {\, \mathrm{d}}\mu=A \int_X\phi^p {\, \mathrm{d}}\mu=F, \bigg\},\end{gathered}$$ where $1<q<p,$ and for $f, A, F$ we have $f^q < A < F^{\frac{q}{p}}.$ The new integral variable makes the problem considerably more difficult and thus makes it interesting to compute this Bellman function in a subdomain of its original domain. Our first attempt towards the evaluation of can be seen in Section 3, where exploiting techniques similar to those used in [@8] we find an upper bound for . In Sections 4 and 5 we prove our main result, stated in Theorem 1 below. To accomplish this, we use techniques presented in [@3] in a different, and better suited to our purposes, way (see Section 4), and we also exploit a property of the family ${\mathcal{S}}_\alpha$ (see Section 5 for the definition), that we prove in Lemma \[lem:5p3\]. Using techniques involved in this proof we compute on a subset of its domain. This will be carried out in Section 6. Moreover, also in Section 6, we shall present two more applications of Theorem \[thm:1\]. The first one, whose proof will require us to entangle a result from [@3], is Corollary \[cor:2\] below which strengthens and generalizes Theorem \[thm:a\] in the present context, while the second is to use this result in order to evaluate in a new way. So our main result is the following. [1]{} \[thm:1\] Let $q\in(1,p)$, $f>0$ and $\phi\in L^p(X,\mu)$ non-negative, with $\int_X\phi\,{\mathrm{d}}\mu=f.$ Then the inequality $$\begin{aligned} \label{eq:1p12} \int_X (\mathcal{M}_{\mathcal{T}}\phi)^p {\, \mathrm{d}}\mu & \leq \frac{p(\beta +1)^q}{G(p,q,\beta)} \int_X (\mathcal{M}_{\mathcal{T}}\phi)^{p-q}\phi^q {\, \mathrm{d}}\mu +\frac{(p-q)(\beta+1)}{G(p,q,\beta)}f^p \nonumber \\& + \frac{p(q-1)\beta}{G(p,q,\beta)} f^{p-q}\int_X (\mathcal{M}_{\mathcal{T}}\phi)^q {\, \mathrm{d}}\mu - \frac{p(\beta +1)^q}{G(p,q,\beta)}f^{p-q}\int_X\phi^q\,{\mathrm{d}}\mu,\end{aligned}$$ where $G(p,q,\beta)=p(q-1)\beta+(p-q)(\beta+1),$ is sharp for every $\beta>0.$ If we also assume that $\int_X\phi^q\,{\mathrm{d}}\mu=A,$ $f^q<A,$ then is best possible for $\beta=\omega_q(\frac{f^q}{A})-1,$ where $\omega_q$ is defined as above, with $q$ in place of $p.$ We now formulate our result regarding as follows. With $\omega_p$ as defined above (and $\omega_q$ in an analogous way), the following is true. \[cor:1\] For $f, A$ such that $0<f^q<A$ and $F=F(f,A)$ satisfying $\omega_p(\frac{f^p}{F})=\omega_q(\frac{f^q}{A}),$ we have that $$\label{eq:1p13} B_{{\mathcal{T}}}^{p,q}(f,A,F)=\omega_q\bigg(\frac{f^q}{A}\bigg)^pF.$$ Finally as mentioned above, the proof of Theorem \[thm:1\], together with results from [@3], will allow us to prove the following generalization of Theorem \[thm:a\], which in turn will lead to the evaluation of in a new way. \[cor:2\] Let $q\in(1,p)$, $f>0$, and $\phi\in L^p(X,\mu)$ non-negative, with $\int_X\phi\,{\mathrm{d}}\mu=f.$ Then the inequality $$\label{eq:1p14} \int_X(\operatorname{\mathcal{M}_\mathcal{T}}\phi)^p\,{\mathrm{d}}\mu \leq -\frac{q(\beta+1)}{G(p,q,\beta)}f^p + \frac{p(\beta+1)^q}{G(p,q,\beta)}\int_X (\mathcal{M}_{\mathcal{T}}\phi)^{p-q}\phi^q {\, \mathrm{d}}\mu,$$ is sharp for every $\beta>0,$ where $G(p,q,\beta)$ as above. If we also assume that $\int_X\phi^q\,{\mathrm{d}}\mu=A,$ $f^q<A,$ then is best possible for $\beta=\omega_q(\frac{f^q}{A})-1,$ We remark here that there are several problems in Harmonic Analysis were Bellman functions arise. Such problems (including the dyadic Carleson imbedding theorem and weighted inequalities) are described in [@7] (see also [@5], [@6]) and also connections to Stochastic Optimal Control are provided, from which it follows that the corresponding Bellman functions satisfy certain nonlinear second-order PDEs. The exact evaluation of a Bellman function is a difficult task which is connected with the deeper structure of the corresponding Harmonic Analysis problem. Until now several Bellman functions have been computed (see [@1], [@3], [@5], [@12], [@13], [@14]). The exact computation of has also been given in [@11] by L. Slavin, A. Stokolos and V. Vasyunin, which linked the computation of it to solving certain PDEs of the Monge-Ampère type, and in this way they obtained an alternative proof of the results in [@3] for the Bellman function related to the dyadic maximal operator. Also in [@15], using the Monge- Ampère equation approach, a more general Bellman function than the one related to the Carleson imbedding theorem has been precisely evaluated thus generalizing the corresponding result in [@3]. It would be an interesting problem to discover if the Bellman function of three variables defined in can be computed using such PDE-based methods. Preliminaries {#sec:2} ============= In this section we present the backgroung we need from \[3\], that will be used in all that follows, and we repeat only the proofs of those results that are essential in the sequel, in a way that will be useful to us. Let $(X,\mu)$ be a non-atomic probability space. Two measurable subsets A, B of $X$ will be called almost disjoint if $\mu(A\cap B)=0$. \[definition:2p1\] A set $\mathcal{T}$ of measurable subsets of $X$ will be called a tree if the following conditions are satisfied: 1. $X \in \mathcal{T}$ and for every $I \in \mathcal{T}$ we have $\mu(I)>0$. 2. For every $I\in \mathcal{T}$ there corresponds a finite or countable subset $\mathcal{C}(I) \subseteq \mathcal{T}$ containing at least two elements such that: - the elements of $\mathcal{C}(I)$ are pairwise almost disjoint subsets of $I$, - $I=\bigcup \mathcal{C}(I)$. 3. $\mathcal{T}= \bigcup_{m\geq 0}\mathcal{T}_{m}$ where $\mathcal{T}_{(0)}=\{X\}$ and $\mathcal{T}_{(m+1)}=\bigcup_{I\in \mathcal{T}_{(m)}} \mathcal{C}(I)$. 4. We have $\lim_{m \to \infty} \sup_{I \in \mathcal{T}_{(m)}}\mu(I)=0$. 5. $\mathcal{T}$ differentiates $L^1(X,\mu)$ This last condition means exactly that the Lesbesgue differentiation theorem holds in the space $L^1(X,\mu)$, with respect to the tree $\mathcal{T}$. Now we define for any tree $\mathcal{T}$ its exceptional set $E=E(\mathcal{T})$ as follows: $$\label{eq:2p1} E(\mathcal{T})= \bigcup_{I \in \mathcal{T}} \bigcup_{\substack{J_{1}, J_{2} \in \mathcal{C}(I) \\ J_{1}\neq J_{2}}} (J_{1}\cap J_{2}).$$ It is easy to see that $E(\mathcal{T})$ has measure $0$. By induction it can be seen that each family $\mathcal{T}_{(m)}$ consists of pairwise almost disjoint sets whose union is $X$. Moreover if $x \in X\setminus E(\mathcal{T})$ then for each $m$ there exists exactly one $I_{m}(x)$ in $\mathcal{T}_{(m)}$ containing x. For every $m>0$ there is a $J \in \mathcal{T}_{(m-1)}$ such that $I_{m}(x) \in \mathcal{C}(J)$. Then, since $x \in J,$ we must have $J=I_{m-1}(x)$, because $x$ does not belong to $E(\mathcal{T})$. Hence the set $\mathcal{A}=\{I \in \mathcal{T}:x \in I\}$ forms a chain $I_{0}(x)=X\varsupsetneq I_{1}(x)\varsupsetneq \dots$ with $I_{m} \in \mathcal{C}(I_{m-1}(x))$ for every $m>0$. From this remark it follows that if $I, J \in \mathcal{T}$ and $I \cap J \cap (X\setminus E(\mathcal{T}))$ is nonempty, then $I\subseteq J$ or $J \subseteq I$. In particular for any $I, J \in \mathcal{T}$, either $\mu(I\cap J)=0$ or one of them is contained in the other.\ We now need a Lemma from \[3\]. Some elements of its proof will be used later on the proofs of our main Theorems. \[lem:2p1\] For every $I \in \mathcal{T}$ and every $a$ such that $0<a<1$ there exists a subfamily $\mathcal{F}(I)\subseteq \mathcal{T}$ consisting of pairwise almost disjoint subsets of $I$ such that $$\label{eq:2p2} \mu \left(\bigcup_{J \in \mathcal{F}(I)}J \right) = \sum_{J \in \mathcal{F}(I)}\mu(J)= (1-a)\mu(I).$$ Let $I\in \mathcal{T}$. By Definition \[definition:2p1\](4) we choose $m_{1}^{I}$ such that $\sup_{J \in \mathcal{T}_{(m_{1}^{I})}}\mu (J)$\ $ < \frac{1-a}{2}\mu(I)$ and then choose a subfamily $\mathcal{F}_{1}(I) \subseteq \mathcal{G}_{1}(I) = \mathcal{T}_{(m_{1}^{I})}$ with\ $\sum_{J \in \mathcal{F}_{1}(I)}\mu(J)$ maximal and such that $$\label{eq:2p3} \sum_{J \in \mathcal{F}_{1}(I)}\mu(J) \leq (1-a)\mu(I).$$ If $\sum_{J \in \mathcal{F}_{1}(I)}\mu(J) = (1-a)\mu(I)$ we stop. Otherwise, since $\sum_{J \in \mathcal{G}_{1}(I)}\mu(J) = \mu(I),$ there exists $I_1 \in \mathcal{G}_{1}(I) \setminus \mathcal{F}_{1}(I)$ and so, by the maximality of $\mathcal{F}_{1}(I),$ we must have $\mu(I_1) + \sum_{J \in \mathcal{F}_{1}(I)}\mu(J) \geq (1-a)\mu(I)$ implying $$\label{eq:2p4} 0<\varepsilon_1 = (1-a)\mu(I) - \sum_{J \in \mathcal{F}_{1}(I)}\mu(J) < \frac{1-a}{2}\mu(I).$$ Next, in view of Definition \[definition:2p1\] (4) again, choose $m_2^I > m_1^I$ with $\sup_{J \in \mathcal{T}_{(m_{2}^{I})}}\mu (J) < \frac{\varepsilon_1}{2}$ and let $\mathcal{G}_{2}(I)$ be the family of all $J \subseteq I$ such that $J \in \mathcal{T}_{m_2^I}$ that are almost disjoint from $\cup \mathcal{F}_1(I).$ Choose a subfamily $\mathcal{F}_{2}(I) \subseteq \mathcal{G}_{2}(I)$ maximal under $$\label{eq:2p5} \sum_{J \in \mathcal{F}_{2}(I)}\mu(J) \leq \varepsilon_1.$$ Again we have $$\label{eq:2p6} 0 \leq \varepsilon_2 = \varepsilon_1 - \sum_{J \in \mathcal{F}_{2}(I)}\mu(J) < \frac{\varepsilon_1}{2}< \frac{1-a}{2^2}\mu(I)$$ and therefore $$\label{eq:2p7} 0\leq (1-a)\mu(I) - \sum_{J \in \mathcal{F}_{1}(I)\cup \mathcal{F}_{2}(I)}\mu(J) < \frac{1-a} {2^2}\mu(I).$$ If equality holds in the first inequality in , we stop. Otherwise, continuing this way we obtain the families $\mathcal{F}_{1}(I), \mathcal{F}_{2}(I), \cdots, \mathcal{F}_{s}(I), \cdots$ and it is easy to see that $\mathcal{F}(I)=\cup_{s=1}^{\infty}\mathcal{F}_{s}(I)$ has the desired properties. Given any tree $\mathcal{T}$ we remind that the maximal operator associated to it is defined as follows: $$\label{eq:2p8} \mathcal{M}_{\mathcal{T}}\phi(x)= \sup \lbrace \frac{1}{\mu(I)} \int_{I} \abs{\phi} {\, \mathrm{d}}\mu \: :\: x \in I \in \mathcal{T} \rbrace$$ for every $\phi \in L^1(X,\mu)$. Next we describe the linearization procedure for the operator $\mathcal{M}_{\mathcal{T}}$. Let $\phi \in L^1(X,\mu)$ be a nonnegative function and for any $I \in \mathcal{T}$ let $$\label{eq:2p9} \operatorname{Av}_I (\phi) = \frac{1}{\mu(I)}\int_{I} \phi {\, \mathrm{d}}\mu.$$ We will say that $\phi$ is $\mathcal{T}$-good if the set $$\label{eq:2p10} \Lambda_{\phi}=\lbrace x \in X\setminus E(\mathcal{T}) \: : \: \mathcal{M}_{\mathcal{T}}\phi(x) >\operatorname{Av}_I(\phi) \; for \; all \; I \; \in \mathcal{T} \; such \; that \; x \in I \rbrace$$ has $\mu$-measure zero. For any such function and every $x \in X\setminus (E(\mathcal{T})\cup \Lambda_{\phi})$ (i.e. for $\mu$-almost every $x$ in X) we define $I_{\phi}(x)$ to be the largest element in the nonempty set $\{ I \in \mathcal{T}: x \in I \; \text{and} \; \mathcal{M}_{\mathcal{T}}\phi(x)=\operatorname{Av}_I(\phi)\}$. Also given any $I \in \mathcal{T}$ let $$\label{eq:2p11} A(\phi, I) = \{x \in X\setminus (E(\mathcal{T})\cup \Lambda_{\phi}) : I_{\phi}(x)=I\} \subseteq I$$ and $$\label{eq:2p12} \mathcal{S}_{\phi}= \{I \in \mathcal{T}: \mu(A(\phi, I)>0\} \cup \{X\}.$$ It is clear that $$\label{eq:2p13} \mathcal{M}_{\mathcal{T}}\phi = \sum_{I \in \mathcal{S}_{\phi}}Av_I(\phi)\chi_{A(\phi,I)}, \; \text{almost everywhere},$$ where $\chi_B$ denotes the characteristic function of $B\subset X.$ Now we define the correspondence $I \to I^*$ with respect to $\mathcal{S}_{\phi}$ for $I \neq X$ in the following manner: $I^*$ is the smallest element of $\{J \in \mathcal{S}_{\phi}: \:I \subsetneq J \}$. It is clear that the sets $A_I=A(\phi, I), \; I \in \mathcal{S}_{\phi},$ are pairwise disjoint and since $\mu(\cup_{J \notin \mathcal{S}_{\phi}}A_J )=0$ their union has full measure. In the following Lemma we present several important properties of the sets defined above. At this point we define two measurable sets $A$ and $B$ to be almost equal if $\mu(A \setminus B)=\mu(B\setminus A)=0$ and in this case we write $A \thickapprox B$ \[lem:2p2\] 1. If $I,J \in \mathcal{S}_{\phi}$ then either $A_J\cap I = \varnothing$ or $J \subseteq I$. 2. If $I \in \mathcal{S}_{\phi}$ then there exists $J \in \mathcal{C}(I)$ such that $J \notin \mathcal{S}_{\phi}$ 3. For every $I \in \mathcal{S}_{\phi}$ we have $I \thickapprox \bigcup_ {\mathcal{S}_{\phi} \ni J \subseteq I} A_J$. 4. For every $I \in \mathcal{S}_{\phi}$ we have $A_I \thickapprox I \setminus \bigcup_{J \in \mathcal{S}_{\phi}: \; J^*=I}J$ and so $$\label{eq:2p14} \mu(A_I)= \mu(I) - \sum_{J \in \mathcal{S}_{\phi}: \; J^*=I}\mu(J).$$ From the above we get $$\label{eq:2p15} \operatorname{Av}_I(\phi) = \frac{1}{\mu(I)}\sum_{J \in \mathcal{S}_{\phi} : \; J \subseteq I} \int_{A_J}\phi {\, \mathrm{d}}\mu .$$ Now we fix $q>1$. Following \[3\] we set $$\label{eq:2p16} x_I = a_I^{-1+\frac{1}{q}} \int_{A_I}\phi {\, \mathrm{d}}\mu$$ for every $I \in \mathcal{S}_{\phi}$ where $a_I = \mu(A_I)$ (in case where $\mu(A_X)=0$ we set $x_X=0$) and from H$\ddot{o}$lder’s inequality and Lemma \[lem:2p2\] we get $$\label{eq:2p17} \mathcal{M}_{\mathcal{T}}\phi= \sum_{I \in \mathcal{S}_{\phi}}\left(\frac{1}{\mu(I)} \sum_{J \in \mathcal{S}_{\phi} : \; J \subseteq I} a_J^{1/\acute{q}}x_J \right) \chi_{A_I}$$ $\mu$-almost everywhere, where $\acute{q}=\frac{q}{q-1}$ is the dual exponent of q, and also $$\label{eq:2p18} \int_X \phi^q {\, \mathrm{d}}\mu= \sum_{I \in \mathcal{S}_{\phi}} \int_{A_I} \phi^q {\, \mathrm{d}}\mu \geq \sum_{I \in \mathcal{S}_{\phi}}x_I^q.$$ So we have $$\label{eq:2p19} \int_X (\mathcal{M}_{\mathcal{T}}\phi)^q {\, \mathrm{d}}\mu= \sum_{I \in \mathcal{S}_{\phi}} \left(\frac{1}{\mu(I)} \sum_{J \in \mathcal{S}_{\phi} : \; J \subseteq I} a_J^{1/\acute{q}}x_J \right)^q a_I = \sum_{I \in \mathcal{S}_{\phi}}a_I y_I^q$$ where $$\label{eq:2p20} y_I=\operatorname{Av}_I(\phi)=\sum_{J \in \mathcal{S}_{\phi} : \; J \subseteq I}a_J^{1/\acute{q}}x_J.$$ The last fact that we will use is the following Lemma (Lemma 3 in [@3]) which we present here without proof. Let $\omega_q$ be as defined in the Introduction (note that the subscript is q instead of p). \[lem:2p3\] Let $q>1$ and $\tau \in (0, 1]$ be fixed. Then for every $\alpha$ with $0<\alpha<1$ the equation $$\label{eq:2p21} -(z-\alpha)^q + (1-\alpha)^{q-1}z^q=\tau\alpha(1-\alpha)^{q-1}$$ has a unique solution $z=z(\alpha, \tau) \in [1, \infty)$ and moreover $$\label{eq:2p22} \lim_{\alpha \to 0^+}z(\alpha, \tau)=\omega_q(\tau).$$ Proof of in the case $\beta=0$ {#sec:3} =============================== In this Section we prove for $\beta=0$ (Lemma \[lem:3p1\]). For this proof we use a variation of the approach that arises in [@8], as can be seen below. This technique is not enough for us to provide a proof of in its general form. For this reason we will use (as we shall see in the next Section), the linearization procedure described in Section 2, in an effective way, which will give us the desired inequality. An application of the result of Lemma \[lem:3p1\], in case one also considers the $L^p$-norm of $\phi$ as given, provides us with an upper bound to . This is the content of Lemma \[lem:3p2\]. \[lem:3p1\] If $I=\int_X (\mathcal{M}_{\mathcal{T}}\phi)^p {\, \mathrm{d}}\mu$ and $\int_X\phi\,{\mathrm{d}}\mu=f,$ $\int_X\phi^q\,{\mathrm{d}}\mu=A$ then, $$I \leq f^p- \frac{p}{p-q}f^{p-q}A + \frac{p}{p-q}\int_X (\mathcal{M}_{\mathcal{T}}\phi)^{p-q}\phi^q {\, \mathrm{d}}\mu.$$ We have $$\begin{aligned} \label{eq:3p1} I&=\int_{\lambda=0}^{\infty}p\lambda^{p-1}\mu(\{\mathcal{M}_{\mathcal{T}}\phi\geq \lambda\}){\, \mathrm{d}}\lambda \nonumber \\&= \int_{\lambda=0}^{f}p\lambda^{p-1}\mu(\{\mathcal{M}_{\mathcal{T}}\phi\geq \lambda\}){\, \mathrm{d}}\lambda + \int_{\lambda=f}^{\infty}p\lambda^{p-1}\mu(\{\mathcal{M}_{\mathcal{T}}\phi\geq \lambda\}){\, \mathrm{d}}\lambda \nonumber \\ &=I_1+I_2.\end{aligned}$$ Then, since $\{\mathcal{M}_{\mathcal{T}}\phi\geq \lambda\}=X$, for every $\lambda<f$ (because $\mathcal{M}_{\mathcal{T}}\phi(x)\geq f$, for every $x \in X$), we have $$\label{eq:3p2} I_1=\int_{\lambda=0}^{f}p\lambda^{p-1}{\, \mathrm{d}}\lambda =f^p$$ and $$\label{eq:3p3} I_2=\int_{\lambda=f}^{\infty}p\lambda^{p-1}\mu(\{\mathcal{M}_{\mathcal{T}}\phi\geq \lambda\}){\, \mathrm{d}}\lambda.$$ For $\lambda>f$ set $E_\lambda=\{\mathcal{M}_{\mathcal{T}}\phi\geq \lambda\}=\cup I_j,$ where $I_j \in \mathcal{T}$ maximal with respect to the condition $\frac{1}{\mu(I_j)}\int_{I_j}\phi{\, \mathrm{d}}\mu \geq \lambda.$ Since by maximality the $I_j$’s are pairwise disjoint, $\mu(E_\lambda)=\sum\mu(I_j) \leq\sum_j\frac{1}{\lambda}\int_{I_j}\phi{\, \mathrm{d}}\mu= \frac{1}{\lambda} \int_{\cup I_j}\phi{\, \mathrm{d}}\mu=\frac{1}{\lambda} \int_{E_\lambda}\phi{\, \mathrm{d}}\mu.$ So $$\label{eq:3p4} \mu(E_\lambda)\leq \frac{1}{\lambda} \int_{E_\lambda}\phi{\, \mathrm{d}}\mu,$$ which is the general weak-type $(1,1)$ inequality for the dyadic maximal operator. From we conclude that $\frac{1}{\mu(E_\lambda)} \int_{E_\lambda}\phi{\, \mathrm{d}}\mu \geq \lambda$ and consequently, for $1<q<p,$ by using Hölder’s inequality, we obtain $\lambda^q \leq (\frac{1}{\mu(E_\lambda)} \int_{E_\lambda}\phi{\, \mathrm{d}}\mu)^q \leq \frac{1}{\mu(E_\lambda)} \int_{E_\lambda}\phi^q{\, \mathrm{d}}\mu$ and so $$\label{eq:3p5} \mu(E_\lambda)\leq\frac{1}{\lambda^q} \int_{E_\lambda}\phi^q{\, \mathrm{d}}\mu.$$ Plugging in , we get $$\begin{aligned} \label{eq:3p6} I_2 &\leq \int_{\lambda=f}^{\infty}p\lambda^{p-1}\frac{1}{\lambda^q} \int_{\{\mathcal{M}_{\mathcal{T}}\phi\geq \lambda\}}\phi^q{\, \mathrm{d}}\mu{\, \mathrm{d}}\lambda\nonumber\\&= \int_X p\phi(x)^q\big(\int_{\lambda=f}^{\mathcal{M}_{\mathcal{T}}\phi(x)}\lambda^{p-q-1}{\, \mathrm{d}}\lambda\big){\, \mathrm{d}}\mu(x)=\nonumber\\&= \frac{p}{p-q}\int_X \phi(x)^q [\lambda^{p-q}]_{\lambda=f}^ {\mathcal{M}_{\mathcal{T}}\phi(x)}{\, \mathrm{d}}\mu(x)\nonumber\\& =\frac{p}{p-q}\int_X (\mathcal{M}_{\mathcal{T}}\phi)^{p-q}\phi^q {\, \mathrm{d}}\mu -\frac{p}{p-q}f^{p-q}\int_X\phi^q{\, \mathrm{d}}\mu \nonumber\\&= \frac{p}{p-q}\int_X (\mathcal{M}_{\mathcal{T}}\phi)^{p-q}\phi^q {\, \mathrm{d}}\mu -\frac{p}{p-q}f^{p-q}A.\end{aligned}$$ From and , we finally get $$\label{eq:3p7} I=\int_X (\mathcal{M}_{\mathcal{T}}\phi)^p {\, \mathrm{d}}\mu \leq f^p- \frac{p}{p-q}f^{p-q}A + \frac{p}{p-q}\int_X (\mathcal{M}_{\mathcal{T}}\phi)^{p-q}\phi^q {\, \mathrm{d}}\mu$$ which is the desired inequality. \[lem:3p2\] If $\int_X\phi\,{\mathrm{d}}\mu=f,$ $\int_X\phi^q\,{\mathrm{d}}\mu=A$ and $\int_X\phi^p\,{\mathrm{d}}\mu=F,$ then $$\label{eq:3p8} I=\int_X (\mathcal{M}_{\mathcal{T}}\phi)^p {\, \mathrm{d}}\mu \leq F h^{-1}(\frac{p f^{p-q}A-(p-q)f^p}{F})^p,$$ where $h:[1,\infty) \to (-\infty,q]$ with $h(t)=pt^{p-q}-(p-q)t^p.$ From , after using Hölder inequality for the integral on the right hand side, we have $$I=\int_X (\mathcal{M}_{\mathcal{T}}\phi)^p {\, \mathrm{d}}\mu \leq f^p- \frac{p}{p-q}f^{p-q}A + \frac{p}{p-q}I^\frac{p-q}{p}F^\frac{q}{p},$$ which, if we divide by F becomes $$(p-q)\frac{I}{F} \leq \frac{(p-q)f^p-pf^{p-q}A}{F}+p\Big(\frac{I}{F}\Big)^\frac{p-q}{p}$$ and this gives $$p\Big(\frac{I}{F}\Big)^\frac{p-q}{p}-(p-q)\Big(\frac{I}{F}\Big) \geq k(f,A,F),$$ where $k(f,A,F)=(pf^{p-q}A-(p-q)f^p)/F.$ So $$p\Big[\Big(\frac{I}{F}\Big)^\frac{1}{p}\Big]^{p-q} - (p-q)\Big[\Big(\frac{I}{F}\Big)^\frac{1}{p}\Big]^p \geq k(f,A,F) \Rightarrow$$ $$\label{eq:3p9} h\Big[\Big(\frac{I}{F}\Big)^\frac{1}{p}\Big] \geq k(f,A,F).$$ We consider now the function $h(t)=pt^{p-q}-(p-q)t^p,$ $t>0.$ We observe that $$h'(t)=p(p-q)t^{p-q-1}-p(p-q)t^{p-1}=p(p-q)t^{p-q-1}(1-t^q).$$ So $h$ is strictly decreasing on $[1,\infty)$, and strictly increasing on $(0,1]$. Additionally, $h(1)=q$ while $\lim_{t\to \infty}h(t)=-\infty.$ Thus $h(t)\leq q$, for any $t>0.$ At the same time $$\begin{aligned} k(f,A,F)&=\frac{pf^{p-q}A-(p-q)f^p}{F}= \frac{-(p-q)\big(\frac{f}{A^{1/q}}\big)^p + p\big(\frac{f}{A^{1/q}}\big)^{p-q}}{F} A^{\frac{p}{q}}\nonumber\\&= \frac{-(p-q)\tau^p + p\tau^{p-q}}{F} A^{\frac{p}{q}}= \Big[\frac{h(\tau)}{F}\Big]A^{\frac{p}{q}} \leq q\frac{A^{\frac{p}{q}}}{F}\leq q,\end{aligned}$$ where $\tau=\frac{f}{A^{1/q}}$. So $$q\geq k(f,A,F)>0$$ and yields $$\Big(\frac{I}{F}\Big)^\frac{1}{p} \leq h^{-1}(k(f,A,F)),$$ which is what we wanted. Proof of {#sec:4} ========= We shall first prove for the class of $\mathcal{T}$-good functions. Let $\phi: (X,\mu)\to {\mathbb{R}}^+$ be ${\mathcal{T}}$-good and such that $\int_X\phi\,{\mathrm{d}}\mu=f$ and $\int_X\phi^q\,{\mathrm{d}}\mu=A$. We use the linearization technique mentioned in Section 2. From and , if we set $$F'=\int_X(\operatorname{\mathcal{M}_\mathcal{T}}\phi)^{p-q}\phi^q{\, \mathrm{d}}\mu,$$ we get $$\begin{aligned} \label{eq:4p1} F' & =\int_X \sum_{I\in \mathcal{S}}y_I^{p-q} \chi_{A_I}\phi^q {\, \mathrm{d}}\mu =\sum_{I\in \mathcal{S}}y_I^{p-q}\int_{A_I} \phi^q {\, \mathrm{d}}\mu \nonumber \\& =\sum_{\substack{I\in \mathcal{S}\\I \neq X}}y_I^{p-q}\int_{A_I} \phi^q {\, \mathrm{d}}\mu +y_X^{p-q}A - y_X^{p-q}\sum_{\substack{I\in \mathcal{S}\\I^*= X}}\int_I \phi^q {\, \mathrm{d}}\mu,\end{aligned}$$ where for the last equality in we used Lemma \[lem:2p2\](4). Lemma \[lem:2p2\](3) and the definition of the correspondence $I\to I^*$ imply $$\label{eq:4p2} \sum_{\substack{I\in \mathcal{S}\\I^*= X}}\int_I \phi^q {\, \mathrm{d}}\mu =\sum_{\substack{I\in \mathcal{S}\\I^*= X}}\sum_{\substack{J\in \mathcal{S}\\J \subseteq I}} \int_{A_J} \phi^q {\, \mathrm{d}}\mu =\sum_{\substack{I\in \mathcal{S}\\I\neq X}}\int_{A_I} \phi^q {\, \mathrm{d}}\mu.$$ Moreover, it is easy to see that $$x_I^q= a_I^{q-1}(y_I\mu(I) -\sum_{\substack{J\in \mathcal{S}\\J^*=I}}y_J\mu(J))^q$$ and $$\int_{A_I}\phi^q{\, \mathrm{d}}\mu \geq x_I^q.$$ So using Hölder’s inequality in the form $$\label{eq:4p3} \frac{(\lambda_1 + \lambda_2 + \ldots + \lambda_m)^q}{(\sigma_1 + \sigma_2 + \ldots + \sigma_m)^{q-1}} \leq \frac{\lambda_1^q}{\sigma_1^{q-1}} + \frac{\lambda_2^q}{\sigma_2^{q-1}} + \ldots + \frac{\lambda_m^q}{\sigma_m^{q-1}},$$ which holds for every $\lambda_i\geq 0,\ \sigma_i>0$, since $q>1,$ and the properties of the correspondence $I\to I^*,$ becomes $$\begin{aligned} \label{eq:4p4} F'&=\sum_{\substack{I\in \mathcal{S}\\I \neq X}}y_I^{p-q}\int_{A_I} \phi^q {\, \mathrm{d}}\mu +y_X^{p-q}A- y_X^{p-q}\sum_{\substack{I\in \mathcal{S}\\I\neq X}}\int_{A_I} \phi^q {\, \mathrm{d}}\mu \nonumber \\ &=\sum_{I\in \mathcal{S}}(y_I^{p-q} - y_X^{p-q})\int_{A_I} \phi^q {\, \mathrm{d}}\mu +y_X^{p-q}A \nonumber\\ &\geq \sum_{I\in \mathcal{S}}(y_I^{p-q} - y_X^{p-q})x_I^q +y_X^{p-q}A \nonumber \\ &= \sum_{I\in \mathcal{S}}(y_I^{p-q} - y_X^{p-q}) \frac{(y_I\mu(I) -\sum_{\substack{J\in \mathcal{S}\\J^*=I}}y_J\mu(J))^q} {(\mu(I)-\sum_{\substack{J\in \mathcal{S}\\J^*=I}}\mu(J))^{q-1}} +y_X^{p-q}A\nonumber \\ &\geq \sum_{I\in \mathcal{S}}(y_I^{p-q} - y_X^{p-q}) \Big(\frac{(y_I\mu(I))^q}{(\tau_I\mu(I))^{q-1}}-\sum_{\substack{J\in\mathcal{S}\\J^*=I}} \frac{(y_J\mu(J))^q}{((\beta+1)\mu(J))^{q-1}}\Big) +y_X^{p-q}A\nonumber \\ &= K -\sum_{\substack{I\in \mathcal{S}\\I \neq X}}(y_{I^*}^{p-q} - y_X^{p-q}) \frac{(y_I\mu(I))^q}{((\beta+1)\mu(I))^{q-1}}\notag \\&= K -\sum_{\substack{I\in \mathcal{S}\\I \neq X}}y_{I^*}^{p-q}y_I^q\frac{\mu(I)^q}{((\beta+1)\mu(I))^{q-1}} + y_X^{p-q}\sum_{\substack{I\in \mathcal{S}\\I \neq X}}\frac{(y_I\mu(I))^q}{((\beta+1)\mu(I))^{q-1}}\end{aligned}$$ provided that the $\tau_I>0$ satisfy $\tau_I\mu(I)-(\beta+1)\sum_{J^*=I}\mu(J))=\mu(I)-\sum_{J^*=I}\mu(J)$, which in turn gives $$\label{eq:4p5} \tau_I=\beta+1-\beta \rho_I,$$ with $\rho_I=\frac{a_I}{\mu(I)},$ and $$\label{eq:4p6} K=\sum_{I\in \mathcal{S}}(y_I^{p-q} - y_X^{p-q})\frac{(y_I\mu(I))^q}{(\tau_I\mu(I))^{q-1}} +y_X^{p-q}A,$$ We now use the following elementary inequality, $$p x^q\!\cdot\! y^{p-q} \leq q x^p + (p-q) y^p,$$ which holds since $1< q < p$, for any $x, y>0,$ to get $$\label{eq:4p7} F' \geq K-\frac{p-q}{p}\sum_{\substack{I\in \mathcal{S}\\I \neq X}}y_{I^*}^p\frac{\mu(I)}{(\beta+1)^{q-1}} -\frac{q}{p}\sum_{\substack{I\in \mathcal{S}\\I \neq X}}y_{I}^p\frac{\mu(I)}{(\beta+1)^{q-1}} +y_X^{p-q}\sum_{\substack{I\in \mathcal{S}\\I \neq X}}\frac{y_I^q\mu(I)}{(\beta+1)^{q-1}}.$$ From Lemma \[lem:2p2\] (4), $$\begin{aligned} \label{eq:4p8} \sum_{\substack{I\in S\\ I\neq X}} \mu(I)y_{I^*}^p &=\sum_{I\in \mathcal{S}}\sum_{\substack{J\in S \\J^*= I}}\mu(J)y_I^p= \sum_{I\in S}(\mu(I)-a_I)y_I^p \nonumber \\& =y_X^p +\sum_{\substack{I\in S\\ I\neq X}}\mu(I)y_I^p - \sum_{I\in S}a_I y_I^p.\end{aligned}$$ So $$\begin{aligned} \label{eq:4p9} F' &\geq K-\sum_{\substack{I\in \mathcal{S}\\I \neq X}}(y_I^{p-q} - y_X^{p-q}) \frac{y_{I}^q\mu(I)}{(\beta+1)^{q-1}}-\frac{(p-q)y_X^p}{p(\beta+1)^{q-1}} +\frac{p-q}{p}\sum_{I\in \mathcal{S}}\frac{a_Iy_{I}^p}{(\beta+1)^{q-1}} \nonumber \\ &= \sum_{I\in \mathcal{S}}(y_I^{p-q} - y_X^{p-q})\frac{1}{\rho_I} \left(\frac{1}{(\beta+1-\beta\rho_I)^{q-1}}-\frac{1}{(\beta+1)^{q-1}}\right)a_Iy_{I}^q\nonumber \\ &-\frac{p-q}{p}\frac{y_X^p}{(\beta+1)^{q-1}} +\frac{p-q}{p}\sum_{I\in \mathcal{S}}\frac{a_Iy_{I}^p}{(\beta+1)^{q-1}} +y_X^{p-q}A,\end{aligned}$$ after we have expanded $K.$ Note now that $$\frac{1}{(\beta+1-\beta x)^{q-1}} - \frac{1}{(\beta+1)^{q-1}} \geq \frac{(q-1)\beta x}{(\beta+1)^q},$$ by the mean value theorem on derivatives for all $x\in [0,1]$, so becomes $$\begin{aligned} \label{eq:4p10} F' &\geq \left(\frac{(q-1)\beta}{(\beta+1)^{q}}+ \frac{p-q}{p(\beta+1)^{q-1}}\right) \int_X (\mathcal{M}_{\mathcal{T}}\phi)^p {\, \mathrm{d}}\mu \nonumber \\& -\frac{p-q}{p}\frac{f^p}{(\beta+1)^{q-1}} -f^{p-q}\frac{(q-1)\beta}{(\beta+1)^{q}}\int_X(\mathcal{M}_{\mathcal{T}}\phi)^q {\, \mathrm{d}}\mu +f^{p-q}A\end{aligned}$$ for every $\beta > 0$. Rearranging the terms, we get for ${\mathcal{T}}$- good functions. For the general $\phi \in L^p(X,\mu)$ with $\int_X \phi=f$ and $\int_X \phi^q=A$, $f^q<A$, $1<q<p,$ is proved as follows. We consider the sequence $\{\phi_m\}$, where $\phi_m = \sum_{I\in \mathcal{T}_{(m)}}\operatorname{Av}_I(\phi)\chi_I$ and we set $$\Phi_m= \sum_{I\in \mathcal{T}_{(m)}}\max\{\operatorname{Av}_I(\phi):\; I\subseteq J \in \mathcal{T}\}\chi_I= \mathcal{M}_\mathcal{T}\phi_m,$$ since $\operatorname{Av}_J(\phi)=\operatorname{Av}_J(\phi_m)$ whenever $I\subseteq J \in \mathcal{T}$. It is easy to see that $$\label{eq:4p11} \int_X \phi_m {\, \mathrm{d}}\mu=\int_X \phi {\, \mathrm{d}}\mu=f, \qquad \int_X \phi_m^q {\, \mathrm{d}}\mu \leq \int_X \phi^q {\, \mathrm{d}}\mu$$ for all m and that $\Phi_m$ converges monotonically almost everywhere to $\mathcal{M}_\mathcal{T}\phi$. Since $\phi_m$ is easily seen to be ${\mathcal{T}}$-good, from what we have just shown, $$\begin{aligned} \label{eq:4p12} \int_X \Phi_m^p {\, \mathrm{d}}\mu & \leq \frac{p(\beta +1)^q}{G(p,q,\beta)} \int_X \Phi_m^{p-q}\phi_m^q {\, \mathrm{d}}\mu +\frac{(p-q)(\beta+1)}{G(p,q,\beta)}f^p \nonumber \\& + \frac{p(q-1)\beta}{G(p,q,\beta)} f^{p-q}\int_X \Phi_m^q {\, \mathrm{d}}\mu - \frac{p(\beta +1)^q}{G(p,q,\beta)}f^{p-q}\int_X \phi_m^q.\end{aligned}$$ Since $\mathcal{T}$ differentiates $L^1(X,\mu)$ and by the definition of $ \phi_m,$ if $\{I_m(x)\}$ is the chain of elements of $\mathcal{T}$ which contain $x\in X,$ then $$\label{eq:4p13} \lim_{m \to \infty}\phi_m(x) = \lim_{m \to \infty}\operatorname{Av}_{I_m(x)}(\phi) = \phi(x)$$ and $\phi_m \leq \Phi_m$ Taking limits using the monotone and dominated convergence theorems and Fatou’s lemma, we obtain for the general $\phi \in L^p(X,\mu).$ Proof of Theorem \[thm:1\] {#sec:5} ========================== Following [@3], we choose $\alpha$ with $0<\alpha <1$ and using Lemma \[lem:2p1\], for every $I \in \mathcal{T}$ we choose a family $\mathcal{F}(I)\subseteq \mathcal{T}$ of pairwise almost disjoint subsets of I such that $$\label{eq:5p1} \mu \left(\bigcup_{J \in \mathcal{F}(I)}J \right) = \sum_{J \in \mathcal{F}(I)}\mu(J)= (1-a)\mu(I).$$ Then we define $\mathcal{S}={\mathcal{S}}_\alpha$ to be the smallest subset of $ \mathcal{T}$ such that $X \in \mathcal{S}$ and for every $I \in \mathcal{S}$, $\mathcal{F}(I) \subseteq \mathcal{S}$ and the correspondence $I \to I^*$ (it should not be confused with the one described in Section 2, although it plays the same role) with respect to this $\mathcal{S},$ by setting $J^*=I \in \mathcal{S}$ if and only if $J \in \mathcal{F}(I)$ and so writing $$\label{eq:5p2} A_I =I \setminus \bigcup_{J \in \mathcal{S}:\; J^*=I}J,$$ we have $a_I=\mu(A_I)=\mu (I) - \sum_{J \in \mathcal{S}:\; J^*=I}\mu (J)=\alpha\mu(I)$ for every $I \in \mathcal{S}$. Also it is easy to see that $$\label{eq:5p3} \mathcal{S}=\bigcup_{m \geq 0}\mathcal{S}_{(m)}, \ \text{where} \ \mathcal{S}_{(0)}=\{X\} \ \text{and} \ \mathcal{S}_{m+1}=\bigcup_{I \in \mathcal{S}_{(m)}}\mathcal{F}(I).$$ The rank $r(I)$ of any $I \in \mathcal{S}$ is now defined to be the unique integer m such that $I \in \mathcal{S}_{(m)}$. For $\alpha \in (0,1)$ and the family $\mathcal{S}_\alpha$ defined as above, we set $$\label{eq:5p4} x_I =\lambda\gamma^{r(I)}\mu(I)^{1/q}$$ where $\lambda=f\alpha^{-1/\acute{q}}(1-\gamma(1-\alpha)), \; 1/q +1/\acute{q} = 1, \; \gamma=\frac{\beta+1}{\beta+1-\beta\alpha}, \; \beta >0,$ and consider the function $$\label{eq:5p5} \varphi_\alpha=\sum_{I \in \mathcal{S}}\frac{x_I}{a_I^{1/q}}\chi_{A_I},$$ $I \in \mathcal{S}$. So $$\label{eq:5p6} \int_X \phi_\alpha^q {\, \mathrm{d}}\mu= \sum_{I \in \mathcal{S}}x_I^q.$$ For every $I \in \mathcal{S}$ and every $m \geq 0$ we write $$\label{eq:5p7} b_m(I)= \sum_{\substack{\mathcal{S}\ni J \subseteq I \\ r(J)=r(I)+m}}\mu(J)$$ and observing that $$\label{eq:5p8} b_{m+1}(I)=\sum_{\substack{\mathcal{S}\ni J \subseteq I, \\ r(J)=r(I)+m}} \sum_{L \in \mathcal{F}(J)}\mu(L)=(1-\alpha)b_m(I)$$ we get $$\label{eq:5p9} b_m(I)=(1-\alpha)^m \mu(I).$$ Hence $$\begin{aligned} \label{eq:5p10} \sum_{I \in \mathcal{S}_\alpha}x_I^q &=\lambda^q\sum_{m\geq 0}\sum_{I \in {\mathcal{S}}_{(m)}}\gamma^{mq}\mu(I) =\lambda^q\sum_{m\geq 0}\gamma^{mq}b_m(X)\nonumber \\ &=\lambda^q\sum_{m\geq 0}[\gamma^q(1-\alpha)]^m =\frac{\lambda^q}{1-\gamma^q(1-\alpha)},\end{aligned}$$ (since $\gamma^q(1-\alpha)<1$). Additionally, $$\begin{aligned} \label{eq:5p11} a_I^{1/q} y_I & = \frac{1}{\mu(I)}\sum_{\substack{J \in \mathcal{S} \\ J \subseteq I}}a_I^{1/q}a_J^{1/\acute{q}}x_J \nonumber \\ &=\frac{\lambda}{\mu(I)}\sum_{\substack{J \in \mathcal{S} \\ J \subseteq I}}(\alpha \mu(I))^{1/q}(\alpha \mu(J))^{1/\acute{q}}\gamma^{r(J)}\mu(J)^{1/q}\nonumber \\ &=\frac{\alpha\lambda}{\mu(I)^{1-\frac{1}{q}}}\sum_{m\geq 0}\gamma^{m+r(I)} \sum_{\substack{\mathcal{S}\ni J \subseteq I \\ r(J)=r(I)+m}}\mu(J)\nonumber \\ &=\frac{\alpha\lambda}{\mu(I)^{1-\frac{1}{q}}}\gamma^{r(I)}\sum_{m\geq 0} \gamma^m\mu(I)(1-\alpha)^m =\frac{\alpha}{(1-\gamma(1-\alpha))}x_I,\end{aligned}$$ where the $y_I$’s are defined by the first equality above and so, using , after replacing $a_I$ with $\alpha\mu(I),$ $$\label{eq:5p12} y_I= \gamma^{r(I)}f.$$ and consequently $$\label{eq:5p13} \int_X \phi_\alpha {\, \mathrm{d}}\mu = \sum_{I \in \mathcal{S}}a_I^{1/\acute{q}}x_I=y_X=f.$$ Now we need the following. \[lem:5p1\] $$\begin{aligned} \label{eq:5p14} \int_X &(\mathcal{M}_{\mathcal{T}}\phi_\alpha)^p {\, \mathrm{d}}\mu \geq\frac{A(\alpha,\beta)} {B(\alpha,\beta)}\bigg(\gamma^{-(p-q)}\int_X (\mathcal{M}_{\mathcal{T}}\phi_\alpha)^{p-q}\phi^q {\, \mathrm{d}}\mu \nonumber \\& + \gamma^{-q}C(\alpha,\beta) f^{p-q}\int_X (\mathcal{M}_{\mathcal{T}}\phi_\alpha)^q {\, \mathrm{d}}\mu -f^{p-q}\int_X \phi_\alpha^q {\, \mathrm{d}}\mu +D(\alpha,\beta)f^p\bigg)\end{aligned}$$ where $$A(\alpha, \beta)= p\alpha(\beta+1-\beta\alpha)^q(\beta+1)^{p-1},$$ $$B(\alpha, \beta)=p(\beta+1)^{p-1}(\beta+1-\beta\alpha)-(p-q)(\beta+1)^p(1-\alpha) -q(\beta+1-\beta\alpha)^p,$$ $$C(\alpha, \beta)=\frac{(\beta+1)^{q-1}-(\beta+1-\beta\alpha)^{q-1}} {\alpha(\beta+1-\beta\alpha)^{q-1}(\beta+1)^{q-1}}$$ and $$D(\alpha, \beta)=\frac{p(\beta+1)^{p-q}-q(\beta+1-\beta\alpha)^{p-q}}{p(\beta+1)^{p-1}}$$ We provide two Lemmas (\[lem:5p2\] and \[lem:5p3\]), which will enable us to prove Lemma \[lem:5p1\]. We first observe that from , if $I \in \mathcal{S}_\alpha$, then $$\label{eq:5p15} I=A_I \cup \bigcup_{J\in \cup_{i=1}^{\infty}\mathcal{F}_i(I)}J$$ where $\bigcup_{J\in\cup_{i=1}^{\infty}\mathcal{F}_i(I)}J$ is possibly a finite union, disjoint from $A_I$ and the ${\mathcal{F}}_i(I)$’s are constructed as in the proof of Lemma \[lem:2p1\]. In any case we have the following: \[lem:5p2\] Let $I \in \mathcal{S}_\alpha$ and $K \subseteq I,$ $K\in {\mathcal{T}}$ with $(K\setminus A_I) \cap (X\setminus E(\mathcal{T})) \neq \varnothing $ and $K\cap A_I \cap (X\setminus E(\mathcal{T})) \neq \varnothing$. Then $$\label{eq:5p16} (K\setminus A_I) \cap (X\setminus E(\mathcal{T}))= \bigcup_{J \in G_{K,I}}(J\cap (X\setminus E(\mathcal{T}))$$ where $G_{K,I}=\{J \in \cup_{i=1}^\infty \mathcal{F}_i(I): \; J \subseteq K\}.$ Since $(K\setminus A_I) \cap (X\setminus E(\mathcal{T})) \neq \varnothing$, from we have $$(K\setminus A_I) \cap (X\setminus E(\mathcal{T}))\subseteq \bigcup_{J \in \cup_{i=1}^\infty \mathcal{F}_i(I)}(J \cap(X\setminus E(\mathcal{T})).$$ If $(K\setminus A_I) \cap (X\setminus E(\mathcal{T}))\cap J \neq \varnothing$ for some $J \in \cup_{i=1}^\infty \mathcal{F}_i(I),$ from the comments after Definition \[definition:2p1\], either $K \subseteq J$ or $J \subseteq K$. Now if $K \subseteq J$ then by $K\cap A_I \cap (X\setminus E(\mathcal{T})) = \varnothing$ which contradicts our hypothesis. So $J \subseteq K$ and $$(K\setminus A_I) \cap (X\setminus E(\mathcal{T}))\subseteq \bigcup_{J \in G_{K,I}}(J\cap (X\setminus E(\mathcal{T})).$$ The converse inclusion is immediate from . \[lem:5p3\] For $\mathcal{S}_\alpha,$ $\phi_\alpha$ and $\gamma$ as defined above we have that $$\label{eq:5p17} \sum_{I \in \mathcal{S}_\alpha}y_I\chi_{A_I} \leq \mathcal{M}_\mathcal{T}\phi_\alpha < \gamma \sum_{I \in \mathcal{S}_\alpha}y_I\chi_{A_I}$$ almost everywhere in $X.$ The first inequality follows immediately from the definition of $\mathcal{M}_\mathcal{T}\phi_\alpha$. For the second one, let $x \in A_I \cap (X\setminus E(\mathcal{T}))$ and consider the sequence $\mathcal{A}(x) = \{I_m(x)\}$ of elements of $\mathcal{T}$ that contain $x$. As we have already mentioned in Section 2, this sequence forms a chain $I_{0}(x)=X\varsupsetneq I_{1}(x)\varsupsetneq \dots$ with $I_{m} \in \mathcal{C}(I_{m-1}(x))$. So let $J \in \mathcal{A}(x)$ with $J \cap (X\setminus E(\mathcal{T})) \subseteq A_I \cap (X\setminus E(\mathcal{T}))$. Then, from and , $$\begin{aligned} \label{eq:5p18} \frac{1}{\mu(J)}\int_J \phi_\alpha {\, \mathrm{d}}\mu &= \frac{x_I}{a_I^{1/q}} =\frac{y_I}{\beta +1 - \beta\alpha} < \gamma y_I\end{aligned}$$ since $\gamma=\frac{\beta+1}{\beta+1-\beta\alpha}.$ Now if $J \in \mathcal{A}(x)$, $J \subseteq I$ with $(J\setminus A_I) \cap (X\setminus E(\mathcal{T})) \neq \varnothing,$ then from Lemma \[lem:5p1\] we have that $$(J\setminus A_I) \cap (X\setminus E(\mathcal{T}))= \bigcup_{K \in G_{J,I}}(K \cap (X\setminus E(\mathcal{T})).$$ So $$\begin{aligned} \frac{1}{\mu(J)}\int_J \varphi_\alpha {\, \mathrm{d}}\mu &= \frac{1}{\mu(J)}\left(\int_{(J\setminus A_I) \cap (X\setminus E(\mathcal{T}))} \varphi_\alpha {\, \mathrm{d}}\mu + \int_{J\cap A_I \cap (X\setminus E(\mathcal{T}))} \varphi_\alpha {\, \mathrm{d}}\mu\right) \nonumber \\ &= \frac{1}{\mu(J)}\left(\sum_{K \in G_{J,I}}\int_{K \cap (X\setminus E(\mathcal{T}))}\varphi_\alpha {\, \mathrm{d}}\mu + \frac{x_I}{a_I^{1/q}}\mu(J\cap A_I)\right) \nonumber \\ &= \frac{1}{\mu(J)}\left(\sum_{K \in G_{J,I}}\int_{K}\varphi_\alpha {\, \mathrm{d}}\mu + \frac{y_I}{\beta +1 - \beta\alpha}\mu(J\cap A_I)\right) \nonumber \\ &= \frac{1}{\mu(J)}\left(\sum_{K \in G_{J,I}}\mu(K)y_K + \frac{y_I}{\beta +1 - \beta\alpha}\mu(J\cap A_I)\right) \nonumber.\end{aligned}$$ Moreover by the construstion of $\mathcal{S}_\alpha$ and , for every $K \in \cup_{i=1}^\infty \mathcal{F}_i(I)$ we have $y_K = \gamma y_I.$ Thus $$\begin{aligned} \label{eq:5p19} \frac{1}{\mu(J)}\int_J \phi_\alpha {\, \mathrm{d}}\mu &= \frac{1}{\mu(J)}\left(\gamma y_I\sum_{K \in G_{J,I}}\mu(K) + \frac{y_I}{\beta +1 - \beta\alpha}\mu(J\cap A_I)\right) \nonumber \\ & < \frac{\gamma y_I}{\mu(J)}\left(\sum_{K \in G_{J,I}}\mu(K)+\mu(J\cap A_I)\right) = \gamma y_I.\end{aligned}$$ For the case $J \in \mathcal{A}(x)$ and $I \subsetneq J$, then either $J\cap A_K \cap (X\setminus E(\mathcal{T})) \neq \varnothing$ for some $K \in \mathcal{S}_\alpha$ with $K \supseteq J$ or $J\cap A_K \cap (X\setminus E(\mathcal{T})) = \varnothing$ for every such K. In the first case, and since $A_I \subseteq J$ implies $(J\setminus A_K) \cap (X\setminus E(\mathcal{T})) \neq \varnothing$ ($A_I \cap A_K = \varnothing$ if $I\neq K,$ by the construction of ${\mathcal{S}}_\alpha$), from what we have shown above, we get $y_J < \gamma y_K < \gamma y_I$ with the second inequality coming from the fact that for any $I, K \in \mathcal{S}_\alpha$ with $I \subsetneq K$, from and since $\gamma>1,$ we have that $y_K= \gamma^{r(K)}f < \gamma^{r(I)}f = y_I$. For the second case, let $\tilde{I}$ be the smallest element of the set $ \{ K \in \mathcal{S}_\alpha : \; J \subseteq K \} $. This is non-empty since $X \in \mathcal{S}_\alpha$ by construction. It is clear that in this case $J \notin \mathcal{S}_\alpha$ and so $J \subsetneq \tilde{I}$. Since $J\cap A_{\tilde{I}} \cap (X\setminus E(\mathcal{T}))= \varnothing$ we get that $J \cap (X\setminus E(\mathcal{T})) \subseteq \bigcup_{L \in \cup_{i=1}^\infty \mathcal{F}_i(\tilde{I})}(L \cap (X\setminus E(\mathcal{T})))$. Actually something more is true: $$\label{eq:5p20} J \cap (X\setminus E(\mathcal{T}))= \bigcup_{\substack{L \in \cup_{i=1}^\infty \mathcal{F}_i(\tilde{I}) \\ L \subsetneq J}}(L \cap (X\setminus E(\mathcal{T}))).$$ Indeed, if $J \cap (X\setminus E(\mathcal{T})) \cap L \neq \varnothing$ for some $ L \in \cup_{i=1}^\infty \mathcal{F}_i(\tilde{I})$, then either $J \subsetneq L$ or $L \subsetneq J$. But if $J \subsetneq L$, then we would have $J \subsetneq L \subsetneq \tilde{I}$, because $L \in \cup_{i=1}^\infty \mathcal{F}_i(\tilde{I})$ and $L, \tilde{I} \in \mathcal{S}_\alpha$, which is impossible from the choice of $\tilde{I}$. So $L \subsetneq J$. The converse is immediate. So in this case we also have: $$\begin{aligned} \label{eq:5p21} \frac{1}{\mu(J)}\int_J \varphi_\alpha {\, \mathrm{d}}\mu &=\frac{1}{\mu(J)}\int_{J\cap(X\setminus E(\mathcal{T}))} \varphi_\alpha \nonumber \\ &= \frac{1}{\mu(J)}\sum_{\substack{L \in \cup_{i=1}^\infty \mathcal{F}_i(\tilde{I}) \\L \subsetneq J}}\int_L \varphi_\alpha =\frac{1}{\mu(J)}\sum_{\substack{L \in \cup_{i=1}^\infty \mathcal{F}_i(\tilde{I}) \\L \subsetneq J}} \mu(L)y_L \nonumber \\ &= \frac{\gamma y_{\tilde{I}}}{\mu(J)} \sum_{\substack{L \in \cup_{i=1}^\infty \mathcal{F}_i(\tilde{I}) \\L \subsetneq J}}\mu(L) =\gamma y_{\tilde{I}} < \gamma y_I\end{aligned}$$ since $I \subsetneq \tilde{I}.$ We conclude that for every $J \in \mathcal{A}(x)$, $y_J < \gamma y_I$ and this immediately implies the second inequality of our Lemma.  \ Set $F_\alpha =\int_X (\mathcal{M}_\mathcal{T}\phi_\alpha)^{p-q}\varphi_\alpha^q {\, \mathrm{d}}\mu.$ Using the above Lemma we have $$\begin{aligned} \label{eq:5p22} F_\alpha &\leq \int_X (\sum_{I \in \mathcal{S}_\alpha}\gamma^{p-q}y_I^{p-q}\chi_{A_I}) (\sum_{I \in \mathcal{S}_\alpha}\frac{x_I^q}{a_I}\chi_{A_I}) {\, \mathrm{d}}\mu =\gamma^{p-q}\sum_{I \in \mathcal{S}_\alpha}y_I^{p-q}x_I^q\nonumber \\ &=\gamma^{p-q}\big(\sum_{I \in \mathcal{S}_\alpha}(y_I^{p-q}-y_X^{p-q})x_I^q + y_X^{p-q}\sum_{I \in \mathcal{S}_\alpha}x_I^q\big).\end{aligned}$$ From the definition of $\phi_\alpha$ and $y_I,$ $x_I^q= \frac{1}{a_I^{q-1}}(\int_{A_I}\phi_\alpha {\, \mathrm{d}}\mu)^q$ and $y_I=\operatorname{Av}_I(\phi_\alpha)$ and thus and imply $$\begin{aligned} \label{eq:5p23} x_I^q&=\frac{(y_I\mu(I) -\sum_{\substack{J\in \mathcal{S}_\alpha\\J^*=I}}y_J\mu(J))^q} {(\mu(I)-\sum_{\substack{J\in \mathcal{S}_\alpha \\J^*=I}}\mu(J))^{q-1}} \nonumber \\ &= \frac{(y_I\mu(I))^q}{(\tau_I\mu(I))^{q-1}} -\sum_{\substack{J\in\mathcal{S}_\alpha\\J^*=I}} \frac{(y_J\mu(J))^q}{((\beta+1)\mu(J))^{q-1}},\end{aligned}$$ where the $\tau_I$’s are as in with the difference now, that for every $I \in \mathcal{S}_\alpha$, $\rho_I=\frac{a_I}{\mu(I)}=\alpha.$ If we set $K$ as in , with $\sum_{I \in \mathcal{S}_\alpha}x_I^q$ in place of $A,$ we easily conclude that $$\label{eq:5p24} F_\alpha \leq \gamma^{p-q}\bigg(K - \sum_{\substack{I\in \mathcal{S}_\alpha\\I \neq X}}(y_{I^*}^{p-q} - y_X^{p-q})\frac{(y_I\mu(I))^q}{((\beta+1)\mu(I))^{q-1}}\bigg),$$ while from , and since $\alpha\mu(I)=a_I$ (arguing as in ), $$\begin{aligned} \label{eq:5p25} \sum_{\substack{I\in \mathcal{S}\\I \neq X}}y_{I^*}^{p-q}y_I^q\frac{\mu(I)} {(\beta+1)^{q-1}}& =\sum_{\substack{I\in \mathcal{S}\\I \neq X}} y_{I^*}^{p-q}\frac{(\gamma y_{I^*})^q\mu(I)}{(\beta+1)^{q-1}}\notag \\ &=\gamma^q\big(\frac{p-q}{p}\sum_{\substack{I\in \mathcal{S}\\I \neq X}} y_{I^*}^p\frac{\mu(I)}{(\beta+1)^{q-1}} + \frac{q}{p}\sum_{\substack{I\in \mathcal{S}\\I \neq X}}y_{I^*}^p\frac{\mu(I)}{(\beta+1)^{q-1}}\big) \nonumber \\ &=\gamma^q\big(\frac{p-q}{p}\sum_{I\in \mathcal{S}}y_{I}^p\frac{\mu(I)-a_I}{(\beta+1)^{q-1}} + \frac{q}{p}\gamma^{-p}\sum_{\substack{I\in \mathcal{S}\\I \neq X}} y_I^p\frac{\mu(I)}{(\beta+1)^{q-1}}\big)\nonumber \\ &=\frac{(p-q)(1-\alpha)(\beta+1)^p + q(\beta+1-\beta\alpha)^p}{\alpha p(\beta+1)^{p-1}(\beta+1-\beta\alpha)^q}\sum_{I\in\mathcal{S}}a_I y_{I}^p \nonumber \\ &- \frac{q(\beta+1-\beta\alpha)^{p-q}}{p(\beta+1)^{p-1}}y_X^p.\end{aligned}$$ At the same time $$K= \frac{1}{\alpha(\beta+1-\beta\alpha)^{q-1}}\big(\sum_{I\in\mathcal{S}}a_I y_{I}^p - y_X^{p-q}\sum_{I\in\mathcal{S}}a_I y_{I}^q\big) + y_X^{p-q}\sum_{I\in\mathcal{S}}x_I^q.$$ So, after straightforward calculations, becomes $$\begin{aligned} \label{eq:5p26} F_\alpha \leq \gamma^{p-q}\big(&\frac{B(\alpha, \beta)}{A(\alpha, \beta)}\sum_{I\in\mathcal{S}}a_I y_{I}^p - C(\alpha, \beta)f^{p-q}\sum_{I\in\mathcal{S}}a_I y_{I}^q - D(\alpha, \beta)f^p \nonumber \\ &+ f^{p-q}\sum_{I \in \mathcal{S}_\alpha}x_I^q\big), \end{aligned}$$ and after involving , Lemma \[lem:5p3\] and rearranging the terms, we get . Now, to finish the proof of Theorem \[thm:1\], we first need to observe that $\gamma\to 1$ as $ \alpha \to 0^+$ $$\label{eq:5p27} \lim_{\alpha\to 0^+}\frac{A(\alpha,\beta)}{B(\alpha,\beta)} =\frac{p(\beta +1)^q}{G(p,q,\beta)}$$ $$\label{eq:5p28} \lim_{\alpha\to 0^+}D(\alpha,\beta)=\frac{(p-q)}{p(\beta+1)^{q-1}}$$ and $$\label{eq:5p29} \lim_{\alpha\to 0^+}C(\alpha,\beta) =\frac{(q-1)\beta}{(\beta+1)^q}.$$ So taking limits as $\alpha \to 0^+$ in we conclude the sharpness of for every $\beta>.0$ What remains to be shown is that we can choose $\beta$ or equivalently $z=z(\alpha)=\beta+1-\beta\alpha$ so, as to obtain $\int_X \phi_\alpha^q =A$ for every $0<\alpha<1,$ with $\phi_\alpha$ as defined in . From , $\int_X\phi_\alpha^q {\, \mathrm{d}}\mu =A$ is equivalent to $\sum_{I \in \mathcal{S}_\alpha}x_I^q=A$ and from , since $z=\beta+1-\beta\alpha$ (which means $\gamma=\frac{z-\alpha}{z(1-\alpha)}$), equivalent to $$\label{eq:5p30} \frac{f^q\alpha(1-\alpha)^{q-1}}{-(z-\alpha)^q + (1-\alpha)^{q-1}z^q}=A.$$ Lemma \[lem:2p3\] now, gives the appropriate value of $z$ which in the limit as $\alpha\to 0$ gives $\beta =\omega_q(\frac{f^q}{A})-1$ and the proof of Theorem 1 is complete. Evaluation of the Bellman functions {#sec:6} =================================== We are now in a place where we can prove . All quantities mentioned are defined as in the previous Section.  \ Let $\phi_\alpha$ be as in and $z=z(\alpha)$ such that holds. Thus $\int_X\phi_\alpha{\, \mathrm{d}}\mu =f$ and $\int_X\phi_\alpha^q\,{\mathrm{d}}\mu=A,$ $f^q<A.$ Keeping in mind that $z=\beta+1-\beta\alpha$ and $\gamma=\frac{\beta+1}{\beta+1-\beta\alpha}=\frac{z-\alpha}{z(1-\alpha)},$ from we have that $$\label{eq:6p1} \sum_{I\in\mathcal{S}}a_I y_{I}^q=z^q \sum_{I \in \mathcal{S}_\alpha}x_I^q,$$ and, using , we continue from , to conclude that $$\begin{aligned} \label{eq:6p2} F_\alpha \leq \gamma^{p-q}\bigg(&\frac{B(\alpha, \beta)}{A(\alpha, \beta)}\int_X({\mathcal{M}}_{\mathcal{T}}\phi_\alpha)^p{\, \mathrm{d}}\mu \notag \\& - \big(\frac{(C(\alpha, \beta)z^q-1)\alpha(1-\alpha)^{q-1}}{-(z-\alpha)^q + (1-\alpha)^{q-1}z^q} + D(\alpha, \beta)\big)\bigg)f^p\end{aligned}$$ and thus $$\label{eq:6p3} \int_X(\mathcal{M}_\mathcal{T}\phi_\alpha)^p {\, \mathrm{d}}\mu \geq \frac{A(\alpha, \beta)}{B(\alpha, \beta)}\bigg(\gamma^{q-p} \int_X (\mathcal{M}_\mathcal{T}\phi_\alpha)^{p-q}\varphi_\alpha^q {\, \mathrm{d}}\mu - \frac{q(\beta+1-\beta\alpha)^{p-q}}{p(\beta+1)^{p-1}}f^p\bigg).$$ Now, if we set $F(\alpha)=\int_X \phi_\alpha^p {\, \mathrm{d}}\mu,$ arguing as in , we get $$\label{eq:6p4} F(\alpha) = \sum_{I \in \mathcal{S}_\alpha}a_I^{-\frac{p-q}{q}} x_I^p = f^p\frac{\alpha(1-\alpha)^{p-1}}{z^p(1-\alpha)^{p-1}-(z-\alpha)^{p}},$$ and, together with , $$\label{eq:6p5} \int_X (\mathcal{M}_\mathcal{T}\phi_\alpha)^{p-q}\varphi_\alpha^q {\, \mathrm{d}}\mu \geq\sum_{I \in \mathcal{S}_\alpha}y_I^{p-q}x_I^q =z^{p-q}F(\alpha),$$ where for the inequality we used Lemma \[lem:5p3\]. Thus , after several calculations, takes the form $$\label{eq:6p6} \int_X(\mathcal{M}_\mathcal{T}\phi_\alpha)^p {\, \mathrm{d}}\mu \geq \frac{p\gamma^{q-p}(z-\alpha)^{p-1}F(\alpha)- qf^p(1-\alpha)^{p-1}}{p(z-\alpha)^{p-1}F(\alpha)-qf^p(1-\alpha)^{p-1}} z^p F(\alpha).$$ The fraction above tends to 1 while from Lemma \[lem:2p3\], $z \to \omega_q(\frac{f^q}{A})$ and $F(\alpha)\to \frac{f^p}{H_p(\omega_q(\frac{f^q}{A}))}=F(f,A)$ as $\alpha \to 0^+.$ So taking limits in , we conclude that $$\label{eq:6p7} B_{{\mathcal{T}}}^{p,q}(f,A,F(f,A))\geq \omega_q(\frac{f^q}{A})^p F(f,A).$$ To see that the converse inequality also holds, we use , our hypothesis on the relation between $f, A, F,$ and the inequality $$B_{{\mathcal{T}}}^{p,q}(f,A,F) \leq B_{{\mathcal{T}}}^{(p)}(f,F),$$ which holds by the corresponding definitions. So finally, $$\label{eq:6p8} B_{{\mathcal{T}}}^{p,q}(f,A,F(f,A))=\omega_q(\frac{f^q}{A})^p F(f,A).$$ which is what we wanted.  \ Since $\int_X \phi {\, \mathrm{d}}\mu =f$, from (4.25) in \[3\], we know that $$\label{eq:6p9} \int_X(\mathcal{M}_{\mathcal{T}}\phi)^q {\, \mathrm{d}}\mu \leq \frac{\beta+1}{\beta} \frac{(\beta+1)^{q-1}\int_X \phi^q {\, \mathrm{d}}\mu-f^q}{q-1}$$ for every $\beta>0,$ for $\phi$ a ${\mathcal{T}}$-good function. Plugging this into we get for ${\mathcal{T}}$-good functions and defining $\phi_m$ and $\Phi_m$ as in Section 4, we get for the general $\phi \in L^p(X,\mu),$ using the monotone convergence theorem. Now, to show it is sharp for every $\beta>0$, let $\phi_\alpha$ be as in so that $\int_X\phi_\alpha{\, \mathrm{d}}\mu=f$ for every $\alpha>0.$ Taking limits as $\alpha\to 0^+$ in we are done. Now, is valid if we also assume $\int_x\phi^q {\, \mathrm{d}}\mu=A,$ so by the above proof still holds in this case. Finally, sharpness is proved in exactly the same way as the corresponding case of Theorem \[thm:1\]. This completes the proof. Our final application is to derive the least upper bound for $\int_X\left(\operatorname{\mathcal{M}_\mathcal{T}}\phi\right)^p{\mathrm{d}}\mu$, when on $\phi$ we impose the conditions $\int_X\phi{\mathrm{d}}\mu=f$ and $\int_X\phi^p{\mathrm{d}}\mu=F$ (where $f,F$ are fixed, satisfying $0\leq f^p\leq F$), by using the proof of inequality , for an arbitrary $q$ belonging to $(1,p)$, and a suitable value of $\beta$, depending on $q, p, f$ and $F$. That is we find the main Bellman function of two variables, , associated to the dyadic maximal operator. We proceed to this as follows. Fix $q\in (1,p)$. First of all it is easy to see that for the above $f, F$, there exists $\beta \in (0,\frac{1}{p-1})$, such that $$\label{eq:6p10} h_{\beta}(\beta+1)F=\frac{q}{p}\frac{1}{(\beta+1)^{q-1}}f^p,$$ where $h_{\beta}(y)$ is defined, for every $y>1$, by $h_{\beta}(y)=y^{p-q}-A_{\beta}y^p$ and $A_{\beta}$ is defined by $$\label{eq:6p11} A_{\beta}=\frac{(q-1)\beta}{(\beta+1)^q}+\frac{p-q}{p}\frac{1}{(\beta+1)^{q-1}}.$$ For this existence, we just need to define the function $$G(\beta)=\frac{1}{(\beta+1)^{p-1}[1-\beta(p-1)]},$$ of $\beta\in (0,\frac{1}{p-1})$, and note that $G(0+)=1$ and $G(\frac{1}{p-1}-)=+\infty$. Thus there exists $\beta\in (0,\frac{1}{p-1})$, such that $G(\beta)=\frac{F}{f^p}\geq1$. If this last condition is true we easily see, after some simple calculations, that $h_{\beta}(\beta+1)F=\frac{q}{p}\frac{1}{(\beta+1)^{q-1}}f^p$, which is the equality we wish to have for $\beta$. Now, because of , for any $\phi \in L^p(X,\mu),$ and for this value of $\beta$, the following inequality is holds $$\int_X\phi^q\left(\operatorname{\mathcal{M}_\mathcal{T}}\phi\right)^{p-q}{\mathrm{d}}\mu\geq A_{\beta}\int_X\left(\operatorname{\mathcal{M}_\mathcal{T}}\phi\right)^p{\mathrm{d}}\mu + \frac{q}{p}\frac{1}{(\beta+1)^{q-1}}f^p$$ Applying Hölder’s inequality on the left side of the above inequality we obtain $$F^{q/p}\big(\int_X\left(\operatorname{\mathcal{M}_\mathcal{T}}\phi\right)^p{\mathrm{d}}\mu\big)^{(p-q)/p}\geq A_{\beta}\int_X\left(\operatorname{\mathcal{M}_\mathcal{T}}\phi\right)^p{\mathrm{d}}\mu+\frac{q}{p}\frac{1}{(\beta+1)^{q-1}}f^p$$ or equivalently, by dividing both sides by $F$, $$I_{\phi}^{(p-q)/p}\geq A_{\beta}I_{\phi}+\frac{q}{p}\frac{1}{(\beta+1)^{q-1}}\frac{f^p}{F},$$ where in the last inequality we denote $I_{\phi}=\frac{\int_X\left(\operatorname{\mathcal{M}_\mathcal{T}}\phi\right)^p{\mathrm{d}}\mu}{F}$, which in turn means that $$\label{eq:6p12} h_{\beta}(I_{\phi}^{1/p})\geq \frac{q}{p}\frac{1}{(\beta+1)^{q-1}}\frac{f^p}{F}.$$ Now for any $\beta\in (0,\frac{1}{p-1})$, we prove that the function $h_{\beta}$, with domain $(1,+\infty)$, is strictly decreasing. For this proof we proceed in the following way. We have that $\frac{d}{dy}h_{\beta}(y)=y^{p-1}[(p-q)y^{-q}-pA_{\beta}]<y^{p-1}[(p-q)-pA_{\beta}]$, where the inequality in the last relation is true due to the fact that $y$ is greater than $1$. Now we claim that $A_{\beta}>\frac{p-q}{p}$, for any $q\in[1,p]$ and $\beta\in (0,\frac{1}{p-1})$. For this reason, we consider $A_{\beta}$ as a function of $\beta$, in the above mentioned domain and denote it as $K(\beta)$. Then $K(0)=\frac{p-q}{p}$, so we just need to prove that $K(\beta)$ is strictly increasing. For this purpose we evaluate $\frac{d}{d\beta}K(\beta)$, which as can be easily seen by using is equal to $\frac{(q-1)q[1-\beta(p-1)]}{p(\beta+1)^{q+1}}$, which is positive for any $\beta$ as above. By the above discussion we conclude that $\frac{d}{dy}h_{\beta}(y)<0$, for any $y>1$. Thus from we have as a consequence that $I_{\phi}^{1/p}\leq h_{\beta}^{-1}(L)$, where $L=\frac{q}{p}\frac{1}{(\beta+1)^{q-1}}\frac{f^p}{F}$. This conclusion holds, if we suppose that $I_{\phi}>1$, which may be assumed, since in the opposite case we have nothing to prove. We finally reach the inequality $$\label{eq:6p13} \int_X\left(\operatorname{\mathcal{M}_\mathcal{T}}\phi\right)^p{\mathrm{d}}\mu\leq F(h_{\beta}^{-1}(L))^p$$ Having now in mind that holds, we show that $h_{\beta}^{-1}(L)=\omega_p\!\left(\frac{f^p}{F}\right)$, where $\omega_p$ is defined in the Introduction. Indeed, by , we immediately conclude that $h_{\beta}^{-1}(L)=\beta+1$, so we just need to prove that $\beta+1=\omega_p\!\left(\frac{f^p}{F}\right)$. Equivalently this means that $H_p(\beta+1)=\frac{f^p}{F}$. But by , we easily see that $$\frac{p}{q}(\beta+1)^{q-1}[(\beta+1)^{p-q}-A_{\beta}(\beta+1)^p]=\frac{f^p}{F}.$$ After simple calculations in the left side of the above equality, the real number $q$ is cancelled giving us the quantity $$-(p-1)(\beta+1)^p+ p(\beta+1)^{p-1},$$ which is exactly $H_p(\beta+1)$. 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--- abstract: 'A pseudo-triangle is a simple polygon with exactly three convex vertices, and a pseudo-triangulation is a face-to-face tiling of a planar region into pseudo-triangles. Pseudo-triangulations appear as data structures in computational geometry, as planar bar-and-joint frameworks in rigidity theory and as projections of locally convex surfaces. This survey of current literature includes combinatorial properties and counting of special classes, rigidity theoretical results, representations as polytopes, straight-line drawings from abstract versions called combinatorial pseudo-triangulations, algorithms and applications of pseudo-triangulations.' address: - 'Institut für Informatik, Freie Universität Berlin, Takustra[s]{}e 9, D-14195 Berlin, Germany. ' - 'Departamento de Matemáticas, Estadística y Computación, Universidad de Cantabria, E-39005 Santander, Spain' - 'Department of Computer Science, Smith College, Northampton, MA 01063, USA.' author: - 'G[ü]{}nter  Rote' - Francisco  Santos - Ileana  Streinu bibliography: - 'pt-survey.bib' title: 'Pseudo-Triangulations — a Survey' --- [^1] [^2] [^3] Acknowledgments. {#acknowledgments. .unnumbered} ---------------- We thank the referees for their extensive comments. [^1]: First author partly supported by the Deutsche Forschungsgemeinschaft (DFG) under grant RO 2338/2-1. [^2]: Second author supported by grant MTM2005-08618-C02-02 of the Spanish Ministry of Education and Science. [^3]: Third author supported by NSF grants CCR-0430990 and NSF-DARPA CARGO-0310661.
--- abstract: 'We study the halo mass function in the presence of a kurtosis type of primordial non-Gaussianity. The kurtosis corresponds to the trispectrum as defined in Fourier space. The primordial trispectrum is commonly characterized by two parameters, $\tau_{\rm NL}$ and $g_{\rm NL}$. We focus on $\tau_{\rm NL}$ which is an important parameter to test the physics of multi-field inflation models. As applications of the derived non-Gaussian mass function, we consider the effects on the abundance of void structure, on early star formation, and on formation of the most massive objects at high redshift. We show that by comparing the effects of primordial non-Gaussianity on cluster abundance with that on void abundance, we can distinguish between the skewness and the kurtosis types of primordial non-Gaussianity. As for early star formation, we show that the kurtosis type of primordial non-Gaussianity seems on the average not to affect the reionization history of the Universe. However, at high redshifts (up to $z\simeq 20$) such non-Gaussianity does somewhat affect the early stages of reionization.' author: - | Shuichiro Yokoyama$^{1}$, Naoshi Sugiyama$^{1,2}$, Saleem Zaroubi$^{3,4}$ and Joseph Silk$^{5}$\ $^{1}$ Department of Physics and Astrophysics, Nagoya University, Aichi 464-8602, Japan\ $^{2}$ Institute for the Physics and Mathematics of the Universe, University of Tokyo, Kashiwa, Chiba, 277-8568, Japan\ $^{3}$ Kapteyn Astronomical Institute, University of Groningen, P.O. Box 800, 9700 AV Groningen, the Netherlands\ $^{4}$ Physics Department, Technion, Haifa 32000, Israel\ $^{5}$ Oxford University, Astrophysics, Denys Wilkinson Building, Keble Road, Oxford, OX1, 3RH, UK title: 'Modification of the halo mass function by kurtosis associated with primordial non-Gaussianity' --- :Inflation, large scale structure of the Universe Introduction ============ The inflation paradigm has been well-known as a successful scenario for resolving several shortcomings of the standard Big Bang Model, in particular, the generation of primordial fluctuations which seed cosmic microwave background (CMB) fluctuations and structure formation of the Universe. In the standard inflationary scenario, the primordial density fluctuations are generated from quantum fluctuations of a scalar field and they have almost Gaussian statistics. In recent years it has been realized that studying the non-Gaussianity of the primordial density fluctuations can reveal valuable information about the dynamics of inflation [@Komatsu:2001rj; @Bartolo:2004if; @Bartolo:2010qu; @Komatsu:2010hc] (and references therein). Thanks to significant progress in cosmological observations, most notably the CMB observations, we may expect that a meaningful measurement of this quantity will become observationally available in the near future and will thereby allow several inflation models to be tested. In Ref. [@Komatsu:2001rj], the authors have introduced a simple new parameter which describes the deviation from Gaussianity of the statistics of the primordial curvature fluctuations, the so-called non-linearity parameter $f_{\rm NL}$, defined as [@Salopek:1990jq; @Gangui:1993tt; @Verde:1999ij] $$\begin{aligned} \zeta({\bf x}) = \zeta_{\rm G}({\bf x}) + {3 \over 5}f_{\rm NL}\left( \zeta_{\rm G}^2({\bf x}) - \langle \zeta_{\rm G}({\bf x})^2 \rangle \right) + O(\zeta_{\rm G}^3({\bf x}))~, \label{eq:originalfNL}\end{aligned}$$ where $\zeta$ represents the primordial curvature fluctuations on a uniform energy density hypersurface and $\zeta_{\rm G}$ denotes the Gaussian part. In the Probability Density Function (PDF) of the primordial fluctuations, the non-zero value of the non-linearity parameter $f_{\rm NL}$ may generate a non-zero value of the skewness (3rd order moment), the kurtosis(4-th order moment) and so on. Obviously, the skewness can be parametrized by the leading term using $f_{\rm NL}$. However, the kurtosis can be affected not only by the $f_{\rm NL}$ term but also by higher order terms, such as the $\zeta_{\rm G}^3({\bf x})$ term in the above expression (\[eq:originalfNL\]). In general, one needs two parameters in order to characterize the kurtosis in the PDF. These parameters are normally called $\tau_{\rm NL}$ and $g_{\rm NL}$, where the first is usually (although not always) related to $f_{\rm NL}$ and the second is the parameter that characterizes the third moment of $\zeta$. Such kind of non-linearity is the so-called local type of non-Gaussianity. Recently, other types of non-Gaussianity have been discussed in the literature, *e.g.*, equilateral and orthogonal types. Theoretically, the local type of non-Gaussianity can be generated from the super-horizon non-linear dynamics of primordial curvature perturbations. On the other hand, the equilateral and orthogonal types of non-Gaussianity can be generated when one considers a scalar field which has a non-canonical kinetic term or the higher order derivative correction terms. In this paper, we focus on the local type of non-Gaussianity and consider the case where the equilateral and orthogonal types are negligible. In the case where the primordial curvature fluctuations were generated from single field stochastic fluctuations (single-sourced case), *i.e.*, the primordial curvature fluctuations can be expressed as Eq. (\[eq:originalfNL\]), and $\tau_{\rm NL}$ can be described only by $f_{\rm NL}$. But in general, *e.g.*, if the primordial curvature fluctuations were generated from multi-stochastic fluctuations then $\tau_{\rm NL}$ and $f_{\rm NL}$ have no universal relation any more [@Suyama:2007bg; @Suyama:2010uj; @Sugiyama:2011jt]. Hence, it seems to be important to investigate the observational consequences of $\tau_{\rm NL}$ independently of $f_{\rm NL}$. In this paper, we focus on the effects of the kurtosis type of primordial non-Gaussianity on the Large Scale Structure (LSS), in particular, on the halo mass function. There are many studies of the effects of primordial non-Gaussianity on the LSS and also on the formulation of the non-Gaussian halo mass function [@Matarrese:2000iz; @Slosar:2008hx; @Maggiore:2009rx; @Verde:2010wp; @D'Amico:2010ta; @DeSimone:2010mu; @Wagner:2010me] (and references therein), which focus not only on $f_{\rm NL}$-type but also $g_{\rm NL}$-type [@Desjacques:2009jb; @Maggiore:2009hp; @Chongchitnan:2010xz; @Chongchitnan:2010hb; @Enqvist:2010bg]. Here, we study the effects of kurtosis of the non-Gaussian primordial fluctuations whose non-linearity is parameterized by the two free parameters, $g_{\rm NL}$ and $\tau_{\rm NL}$. Recently, a number of authors have studied non-Gaussian initial perturbations in two-field inflationary models [@Tseliakhovich:2010kf; @Smith:2010gx]. In these papers, the authors have considered the effect of non-Gaussianity on the halo bias. Although this type of primordial non-Gaussianity is similar to the one considered here, we study the effects on the halo mass function. This paper is organized as follows. In the next section, we briefly review the kurtosis type of primordial non-Gaussianity considered here. In section \[sec:nGmass\], we formulate halo mass functions with primordial non-Gaussianity, based on the Press-Schechter theory and Edgeworth expansion. In section \[sec:nGappl\], we apply the non-Gaussian halo mass function to the formation of astrophysical objects. We consider three applications: early star formation, the most massive object at high redshift and the abundance of voids. Section \[sec:sum\] provides a discussion and summary of our results. We adopt throughout the best fit cosmological parameters taken from WMAP 7-year data. Trispectrum of primordial non-Gaussian curvature fluctuations ============================================================= Here, we focus on the local-type non-Gaussianity. Following the notation commonly used, in the single-sourced case, up to the third order, the primordial curvature fluctuations can be expressed as $$\begin{aligned} \zeta = \zeta_{\rm G} + {3 \over 5}f_{\rm NL} \left( \zeta_{\rm G}^2 - \langle \zeta_{\rm G}^2 \rangle \right) + {9 \over 25} g_{\rm NL} \zeta_{\rm G}^3~. \label{eq:zetafNL}\end{aligned}$$ Based on this expression, the trispectrum of $\zeta$ is given by $$\begin{aligned} &&\langle \zeta({\bf k}_1) \zeta({\bf k}_2) \zeta({\bf k}_3) \zeta({\bf k}_4) \rangle = (2 \pi)^3 T_\zeta(k_1,k_2,k_3,k_4) \delta^{(3)}\left( {\bf k}_1 + {\bf k}_2 + {\bf k}_3 + {\bf k}_4 \right)~,\cr\cr && T_\zeta(k_1,k_2,k_3,k_4) = \tau_{\rm NL} \left( P_\zeta(k_1)P_\zeta(k_2)P_\zeta(k_{13}) + 11 {\rm perms.}\right) + {54 \over 25}g_{\rm NL} \left( P_\zeta(k_1)P_\zeta(k_2)P_\zeta(k_{3}) + 3 {\rm perms.}\right)~, \label{eq:trispectrum}\end{aligned}$$ where $k_{13}=\left| {\bf k}_1 + {\bf k}_3\right|$ and $P_\zeta(k_1)$ is a power spectrum of $\zeta$ given by $\langle \zeta({\bf k}_1) \zeta({\bf k}_2) \rangle = (2\pi)^3 P(k_1)\delta^{(3)} \left( {\bf k}_1 + {\bf k}_2 \right)$. For the above definition of $\tau_{\rm NL}$ and the form of the non-linearity of the curvature perturbation (\[eq:zetafNL\]), $\tau_{\rm NL}$ can be written in terms of the non-linearity parameter $f_{\rm NL}$ as $$\begin{aligned} \tau_{\rm NL} = {36 \over 25}f_{\rm NL}^2~.\end{aligned}$$ This consistency relation is satisfied only in the case where the primordial curvature fluctuations can be described by Eq. (\[eq:zetafNL\]), namely, the primordial curvature fluctuations are sourced only from the quantum fluctuations of a single scalar field, *e.g.*, curvaton [@Enqvist:2001zp; @Lyth:2001nq; @Moroi:2001ct]. However, if there are multiple sources of the primordial curvature fluctuations, then the above consistency relation is not satisfied [@Langlois:2004nn; @Ichikawa:2008iq; @Huang:2009vk; @Byrnes:2010em]. In general, it has been known that there exists an inequality between the local type non-linearity parameters $\tau_{\rm NL}$ and $f_{\rm NL}$ given by [@Suyama:2007bg; @Suyama:2010uj; @Sugiyama:2011jt] $$\begin{aligned} \tau_{\rm NL} > {1 \over 2}\left({6 \over 5}f_{\rm NL}\right)^2~.\end{aligned}$$ For example, let us consider the local-type non-Gaussianity given by $$\begin{aligned} \zeta = \phi_{\rm G} + {3 \over 5}f_{\rm NL} \left( \phi_{\rm G}^2 - \langle \phi_{\rm G}^2 \rangle \right) + t_{\rm NL} \phi_{\rm G} \psi_{\rm G}~, \label{eq:multi}\end{aligned}$$ where $\phi_{\rm G}$ and $\psi_{\rm G}$ are Gaussian fluctuations with $\langle \phi_{\rm G} \psi_{\rm G}\rangle = 0$ and $t_{\rm NL}$ is a non-linearity parameter, which represents the non-linear coupling between $\phi_{\rm G}$ and $\psi_{\rm G}$ in $\zeta_{\rm G}$. At leading order, the power spectrum of $\zeta$ is given by that of the Gaussian part $\phi_{\rm G}$ as $$\begin{aligned} \langle \zeta({\bf k}) \zeta({\bf k}') \rangle = \langle \phi_{\rm G}({\bf k}) \phi_{\rm G}({\bf k}') \rangle = (2\pi)^3 P_\phi(k) \delta^{(3)}({\bf k}+ {\bf k}')~, \label{eq:powerspectrum}\end{aligned}$$ and the bispectrum is given only by $f_{\rm NL}$ as $$\begin{aligned} \langle \zeta({\bf k}_1) \zeta({\bf k}_2) \zeta({\bf k}_3) \rangle = (2\pi)^3 {6 \over 5} f_{\rm NL} (P_\phi(k_1)P_\phi(k_2) + 2 {\rm perms.}) \delta^{(3)}({\bf k}_1 + {\bf k}_2 + {\bf k}_3)~, \label{eq:bispectrum}\end{aligned}$$ because of $\langle \phi_G \psi_G \rangle = 0$. In the single-source case which corresponds to the case of $t_{\rm NL} = 0$, as mentioned above, the trispectrum can be also parameterized only by $f_{\rm NL}$. However, for the above type of curvature fluctuations the trispectrum is given by $$\begin{aligned} T_\zeta(k_1,k_2,k_3,k_4) &=& \left(\frac{6}{5} f_{\rm NL}\right)^2 (P_\phi(k_1)P_\phi(k_2)P_\phi(k_{13}) + 11 {\rm perms.}) \cr\cr && + t_{\rm NL}^2\biggl( P_\phi(k_1)P_\phi(k_2)P_\psi(k_{13}) + P_\phi(k_1)P_\phi(k_2)P_\psi(k_{14}) + P_\phi(k_1)P_\phi(k_3)P_\psi(k_{12}) \cr\cr && \qquad\quad + P_\phi(k_1)P_\phi(k_3)P_\psi(k_{14}) + P_\phi(k_1)P_\phi(k_4)P_\psi(k_{12}) + P_\phi(k_1)P_\phi(k_4)P_\psi(k_{13}) \cr\cr && \qquad\quad + P_\phi(k_2)P_\phi(k_3)P_\psi(k_{12}) + P_\phi(k_2)P_\phi(k_3)P_\psi(k_{24}) + P_\phi(k_2)P_\phi(k_4)P_\psi(k_{12}) \cr\cr && \qquad\quad + P_\phi(k_2)P_\phi(k_4)P_\psi(k_{23}) + P_\phi(k_3)P_\phi(k_4)P_\psi(k_{13}) + P_\phi(k_3)P_\phi(k_4)P_\psi(k_{23}) \biggr) ~,\end{aligned}$$ where $k_{13} = |{\bf k}_1 + {\bf k}_3|$. We assume that the power spectra of random Gaussian fields $\phi_G$ and $\psi_G$ have only weak scale-dependence, that is, the power spectra are respectively given by $$\begin{aligned} P_\phi(k) \equiv {2\pi^2 \over k^3} A_\phi \left( {k \over k_0} \right)^{n_\phi-1}~, ~P_\psi(k) \equiv {2 \pi^2 \over k^3} A_\psi \left( {k \over k_0} \right)^{n_\psi-1}~,\end{aligned}$$ where $k_0$ is a pivot scale and $\left| n_\phi - 1 \right| \ll 1$ and $\left| n_\psi - 1 \right| \ll 1$. In such a case, we can rewrite the power spectrum of $\psi_G$ as $$\begin{aligned} P_\psi(k) \simeq \alpha P_\phi(k)~,~ \alpha \equiv A_\psi / A_\phi~,\end{aligned}$$ and then using the ratio of the amplitudes $\alpha$, the expression for the trispectrum can be reduced to $$\begin{aligned} \langle \zeta({\bf k}_1) \zeta({\bf k}_2) \zeta({\bf k}_3) \zeta({\bf k}_4) \rangle & \simeq & (2\pi)^3 \delta^{(3)}({\bf k}_1 + {\bf k}_2 + {\bf k}_3+{\bf k}_4)\left( {36 \over 25} f_{\rm NL}^2 + \alpha t_{\rm NL}^2 \right)(P_\phi(k_1)P_\phi(k_2)P_\phi(k_{13}) + 11 {\rm perms.}) ~. \nonumber\\\end{aligned}$$ From the above equation and Eq. (\[eq:trispectrum\]), we easily find that the non-linearity parameter $\tau_{\rm NL}$ is $$\begin{aligned} \tau_{\rm NL} = {36 \over 25} f_{\rm NL}^2 + \alpha t_{\rm NL}^2 \ge {36 \over 25} f_{\rm NL}^2~.\end{aligned}$$ Hence, in the following discussion, we consider $\tau_{\rm NL}$ independently of $f_{\rm NL}$. Non-Gaussian mass function induced from primordial non-Gaussianity {#sec:nGmass} ================================================================== In the previous section, we have shown that there is a strong theoretical motivation for considering $\tau_{\rm NL}$ to be independent of $f_{\rm NL}$. The parameter $\tau_{\rm NL}$ characterizes the amplitude of the trispectrum of primordial curvature fluctuations as well as $g_{\rm NL}$. Here, we briefly review the formula for the halo mass function with not only the non-zero primordial bispectrum but also the non-zero primordial trispectrum, based on Press-Schechter theory. Probability Density Function of the smoothed density field with primordial non-Gaussianity ------------------------------------------------------------------------------------------ The matter density linear fluctuations in Fourier space at redshift $z$, $\delta ({\bf k}, z)$, are given by the primordial curvature perturbation on a uniform energy density hypersurface $\zeta({\bf k})$ as $$\begin{aligned} &&\delta ({\bf k},z)={\cal M}(k)D(z)\zeta({\bf k})~,\\ && {\cal M}(k) = {2 \over 5}{1 \over \Omega_{m0}}{k^2 \over H_0^2} T(k)~,\end{aligned}$$ where $\Omega_{m0}$ is the present density parameter for total non-relativistic matter, $H_0$ is the Hubble constant, $D(z)$ is a linear growth function and $T(k)$ is a transfer function. Using these expressions, we can obtain the linear matter power spectrum as $$\begin{aligned} && \langle \delta ({\bf k},z) \delta ({\bf k}', z)\rangle \equiv (2\pi)^3 P_\delta (k,z) \delta^{(3)}({\bf k} + {\bf k}')~,\\ && P_\delta (k,z) = {2 \pi^2 \over k^3} {\cal M}(k)^2 D(z)^2 {\cal P}_\phi (k)~,\end{aligned}$$ where ${\cal P}_\phi (k) = k^3 P_\phi(k)/ ( 2 \pi^2 )$. Following the standard procedure, let us define the smoothed density fluctuation on a given length scale, $R$, as $$\begin{aligned} \delta_R = \int {d^3 {\bf k} \over \left(2\pi\right)^3}W_R(k) \delta ({\bf k},z)~,\end{aligned}$$ where $W_R(k)$ is the Fourier transform of a spherical top-hat window function given by $$\begin{aligned} W_R(k) = 3\left({\sin(kR) \over k^3 R^3} - {\cos(kR) \over k^2 R^2}\right)~.\end{aligned}$$ In order to take into account primordial non-Gaussianity in the smoothed density fluctuations, let us consider the PDF of $\delta_R$, $F(\delta_R) d\delta_R$. The $n$-th central moment for $F(\delta_R)d \delta_R$ is defined as $$\begin{aligned} \langle \delta_R^n\rangle \equiv \int^{\infty}_{-\infty} \delta_R^n F(\delta_R)d\delta_R~,\end{aligned}$$ and each reduced $p$-th cumulant can be defined as $$\begin{aligned} S_p(R) \equiv \frac{\langle \delta_R^p \rangle_c}{\langle \delta_R^2 \rangle_c^{p-1}}~, \label{eq:cum}\end{aligned}$$ where a subscript $c$ denotes the connected part of $p$-point function given by $$\begin{aligned} &&\langle \delta_R \rangle_c = 0~,~ \langle \delta_R^2 \rangle_c = \langle \delta_R^2 \rangle \equiv \sigma_R^2~,\cr\cr &&\langle \delta_R^3 \rangle_c = \langle \delta_R^3 \rangle ~, ~ \langle \delta_R^4 \rangle_c = \langle \delta_R^4 \rangle - 3 \langle \delta_R^2 \rangle_c^2 ~, \: {\rm etc.}, $$ with zero mean density field. Here, $\sigma_R^2$, $S_3(R)$ and $S_4(R)$ are the variance, the skewness and the kurtosis, respectively. Let us consider a non-Gaussian PDF of matter density fluctuations, based on the concept of the Edgeworth expansion. Here, we consider the expansion of the PDF of the density field $F(\nu)d\nu$ with $\nu \equiv \delta_R / \sigma_R$ in terms of the derivatives of the Gaussian PDF, $F_G(\nu)$, as [@Juszkiewicz:1993hm; @LoVerde:2007ri] $$\begin{aligned} F(\nu)d\nu = d\nu \left[ c_0 F_G(\nu) + \sum\limits_{m=1} {c_m \over m!} F_G^{(m)}(\nu) \right]~, \label{eq:PDF}\end{aligned}$$ with $$\begin{aligned} F_G(\nu) &\equiv& (2\pi)^{-1/2} \exp (-\nu^2 / 2)~, \\ F_G^{(m)}(\nu) &\equiv& {d^m \over d\nu^m} F_G(\nu) = (-1)^m H_m(\nu) F_G(\nu)~,\end{aligned}$$ where $H_m(\nu)$ is the Hermite polynomials; $$\begin{aligned} &&H_1(\nu) = \nu~,~H_2(\nu) = \nu^2 -1~,~H_3(\nu) = \nu^3 - 3 \nu~,\nonumber\\ &&H_4(\nu) = \nu^4 - 6 \nu^2 + 3~,~H_5(\nu) = \nu^5 - 10 \nu^3 + 15 \nu~,\cdots~. \label{eq:Hermite}\end{aligned}$$ From the above relation between the derivatives of the Gaussian PDF and Hermite polynomials, we can regard the expression (\[eq:PDF\]) as a non-Gaussian PDF expanded in terms of the Hermite polynomials. Since the Hermite polynomials satisfy orthogonal relations; $$\begin{aligned} \int^{\infty}_{-\infty} H_m(\nu)H_n(\nu) F_G(\nu) d\nu = \left\{ \begin{array}{cc} 0 &,~{\rm if}~m \neq n~, \\ m! &,~{\rm if}~m = n~, \\ \end{array} \right.\end{aligned}$$ we can evaluate the coefficients as $$\begin{aligned} c_m = (-1)^m \int^{\infty}_{-\infty} H_m(\nu) F(\nu)d\nu~.\end{aligned}$$ Then, we can obtain the expressions for the coefficients, $c_m$, in terms of the reduced cumulants (variance, skewness, kurtosis and so on) as $$\begin{aligned} &&c_0=1~,~c_1 = c_2 = 0~,~c_3 = - S_3(R) \sigma_R~,~c_4 = S_4(R) \sigma_R^2~,\cr\cr &&c_5 = - S_5(R)\sigma_R^3~,c_6= 10 S_3(R)^2\sigma_R^2 + S_6(R)\sigma_R^4~,\cdots~,\end{aligned}$$ and, as a result, the non-Gaussian PDF of the density field, $F(\nu)d\nu$, can be obtained as $$\begin{aligned} F(\nu)d\nu & = &{d\nu \over \sqrt{2\pi}} \exp \left(-\nu^2 / 2 \right) \Biggl[ 1 + {S_3(R) \sigma_R \over 6}H_3(\nu) + {1 \over 2}\left({S_3(R) \sigma_R \over 6} \right)^2 H_6(\nu) +{1 \over 6}\left({S_3(R) \sigma_R \over 6} \right)^3 H_9(\nu) \cr\cr &&\qquad\quad +{S_4(R) \sigma_R^2 \over 24}H_4(\nu) + {1 \over 2}\left({S_4(R) \sigma_R^2 \over 24} \right)^2 H_8(\nu) + {1 \over 6}\left({S_4(R) \sigma_R^2 \over 24} \right)^3 H_{12}(\nu) + \cdots \Biggr]~,\end{aligned}$$ up to the third order terms in $S_3(R)$ and $S_4(R)$ and neglect the contributions of the higher order cumulants; $S_n(R)$ $(n \ge 5)$. This derivation of the non-Gaussian PDF is based on the so-called Edgeworth expansion. Of course, the non-zero non-linearity parameters $f_{\rm NL}$, $\tau_{\rm NL}$ and $g_{\rm NL}$ also generate non-zero higher order cumulants; $S_n(R)$ $(n \ge 5)$. However, as far as considering the non-Gaussian curvature fluctuations given by Eq. (\[eq:multi\]) and current observational constraints on the non-linearity parameters [@Komatsu:2010fb; @Fergusson:2010gn], terms $S_n(R)$ $(n \ge 5)$ are greatly suppressed [@Enqvist:2010bg]. Hence, the assumption of neglecting the higher order cumulants seems to be reasonable. Halo mass function with non-Gaussian corrections {#subsec:halomass} ------------------------------------------------ Let us consider the halo mass function with non-Gaussian PDF of the smoothed density field as given in the previous subsection. Based on the spirit of the Press-Schechter formula, the halo mass function which gives the number density of collapsed structures (halos) with the mass between $M$$(= 4\pi \bar{\rho}R^3/3$ with $\bar{\rho}$ is the background matter density) and $M + dM$ at a redshift $z$, $(dn(M,z)/dM) dM$ is given by [@D'Amico:2010ta] $$\begin{aligned} {dn \over dM} (M,z) dM &=& -dM{2\bar{\rho} \over M} {d \over dM} \int^{\infty}_{\delta_c / \sigma_R}d\nu F(\nu) \nonumber\\ & = & - dM \sqrt{2 \over \pi} {\bar{\rho} \over M} \exp \left[ - {\nu_c^2 \over 2}\right] \Biggl\{ {d \ln \sigma_R \over dM}\nu_c \Biggl[ 1 \cr\cr &&\qquad\qquad + {S_3(R)\sigma_R \over 6}H_3(\nu_c) + {1 \over 2}\left({S_3(R) \sigma_R \over 6} \right)^2 H_6(\nu_c) +{1 \over 6}\left({S_3(R) \sigma_R \over 6} \right)^3 H_9(\nu_c) \cr\cr &&\qquad\qquad +{S_4(R) \sigma_R^2 \over 24}H_4(\nu_c) + {1 \over 2}\left({S_4(R) \sigma_R^2 \over 24} \right)^2 H_8(\nu_c) + {1 \over 6}\left({S_4(R) \sigma_R^2 \over 24} \right)^3 H_{12}(\nu_c) \Biggr] \cr\cr && \qquad + {d\over dM}\left({S_3(R) \sigma_R \over 6} \right) H_2(\nu_c) +{1 \over 2}{d\over dM}\left({S_3(R) \sigma_R \over 6} \right)^2 H_5(\nu_c) +{1 \over 6}{d\over dM}\left({S_3(R) \sigma_R \over 6} \right)^3 H_8(\nu_c) \cr\cr &&\qquad + {d\over dM}\left({S_4(R) \sigma_R^2 \over 24} \right) H_3(\nu_c) +{1 \over 2}{d\over dM}\left({S_4(R) \sigma_R^2 \over 24} \right)^2 H_7(\nu_c) +{1 \over 6}{d\over dM}\left({S_4(R) \sigma_R^2 \over 24} \right)^3 H_{11}(\nu_c) \Biggr\} + \cdots~, \label{eq:nonGaussmass}\end{aligned}$$ where $\nu_c = \delta_c / \sigma_R$ and $\delta_c$ denotes the threshold for collapse which is originally given by $\delta_c \approx 1.69$. However, in Ref. [@Grossi:2009an], the authors have suggested that using the correction $\delta_c \to \delta_c \sqrt{q}$ with $q=0.75$ puts the analytic predictions in good agreement with the numerical simulations. This is due to the more realistic case of ellipsoidal collapse. Hence $\delta_c = 1.69 \times \sqrt{q}$ is often referred to as the critical density of ellipsoidal collapse. Here we adopt this corrected density threshold $\delta_c = 1.69 \times \sqrt{0.75}$. In the following calculations, we use the above formula of the non-Gaussian mass functions up to the third order in terms of $S_3$ and $S_4$. For a Gaussian probability distribution, the mass function is given by $$\begin{aligned} {dn_{\rm G} \over dM} (M,z)dM = - \sqrt{2 \over \pi} {\bar{\rho} \over M} \exp \left[ - {\nu_c^2 \over 2}\right] {d \ln \sigma_R \over dM} \nu_c dM~, \label{eq:gaussmass}\end{aligned}$$ and we define the ratio between the non-Gaussian mass function and the Gaussian one as[^1] $$\begin{aligned} {\cal R}_{\rm NG}(M,z) \equiv {dn(M,z)/dM \over dn_{\rm G}(M,z)/dM}~. \label{eq:ratio}\end{aligned}$$ Let us focus on the redshift dependence of the above expression. From the definition of the reduced cumulants (\[eq:cum\]) and the fact that the redshift dependence of the density field is given by $\delta_R \propto D(z)$, we can easily find that $\sigma^{p-2}S_p(R)$ has no redshift-dependence. Hence, any remaining redshift dependence comes only from the term ${\delta_c \over \sigma_R}$. Here, following the literature, the redshift-dependence can be carried by $\delta_c$ as $\delta_c \to \delta_c(z) \propto D(z)^{-1}$ and then the variance $\sigma_R$ has no redshift-dependence. In the following discussion, we change the subscript $R$ to $M$ because $R$ and $M$ have a one-to-one correspondence through the equation $M = 4\pi R^3 \bar{\rho}/3$. ### Variance, skewness and kurtosis {#subsubsec:vsk} Let us consider the concrete expressions of the variance, skewness and kurtosis of the primordial curvature perturbations whose power-, bi- and tri-spectra are given by Eqs. (\[eq:powerspectrum\]), (\[eq:bispectrum\]) and (\[eq:trispectrum\]), respectively. The variance is given by $$\begin{aligned} \sigma_R^2 = \int {dk \over k} W_R^2(k){\cal M}(k)^2{\cal P}_\phi(k) ~, \label{eq:fullvariance}\end{aligned}$$ the skewness is [@Chongchitnan:2010xz] $$\begin{aligned} S_3(R) &\equiv& {6 \over 5}{f_{\rm NL} \over \sigma_R^4} \tilde{S}_3(R)~, \nonumber\\ \tilde{S}_3(R)&=& \int { d k_1 \over k_1} W_R(k_1){\cal M}(k_1) {\cal P}_\phi(k_1) \int { d k_2 \over k_2} W_R(k_2){\cal M}(k_2){\cal P}_\phi(k_2)~\nonumber\\ && \qquad \times \int {d \mu_{12} \over 2} W_R(k_{12}){\cal M}(k_{12}) \biggl[ 1 + {P_\phi(k_{12}) \over P_\phi(k_1)} + {P_\phi(k_{12}) \over P_\phi(k_2)} \biggr]~, \label{eq:fullskewness}\end{aligned}$$ where $k_{12} = \sqrt{k_1^2 + k_2^2 + 2k_1k_2 \mu_{12}}$ and $\mu_{12} = \cos \theta_{12}$, and the kurtosis which is proportional to the non-linearity parameter $\tau_{\rm NL}$ is given by $$\begin{aligned} S_4^{\tau}(R) & \equiv & {\tau_{\rm NL} \over \sigma_R^6} \tilde{S}_4^{\tau}(R)~, \nonumber\\ \tilde{S}_4^{\tau}(R) &=& \int {dk_1 \over k_1}W_R(k_1){\cal M}(k_1){\cal P}_\phi (k_1) \int {dk_2 \over k_2}W_R(k_2){\cal M}(k_2){\cal P}_\phi (k_2) \int { dk_3 \over k_3}W_R(k_{3}){\cal M}(k_3) {\cal P}_\phi (k_3) \cr\cr && \times \int^{1}_{-1} {d\mu_{12} \over 2} \int^{1}_{-1} {d\mu_{13} \over 2} \int^{2\pi}_0 {d\varphi_{13} \over 2\pi} W_R(k_{123}){\cal M}(k_{123}) \cr\cr && \times \Biggl\{P_{\phi}(k_{12}) \left[ {1 \over P_\phi(k_1)} + {1 \over P_\phi(k_2)} \right] \left[ 1 + {P_\phi(k_{123}) \over P_\phi(k_3)} \right] \cr\cr && + P_{\phi}(k_{23}) \left[ {1 \over P_\phi(k_2)} + {1 \over P_\phi(k_3)} \right] \left[ 1 + {P_\phi(k_{123}) \over P_\phi(k_1)} \right] + P_{\phi}(k_{31}) \left[ {1 \over P_\phi(k_3)} + {1 \over P_\phi(k_1)} \right] \left[ 1 + {P_\phi(k_{123}) \over P_\phi(k_2)} \right] \Biggr\}~. \label{eq:fullkurtosis}\end{aligned}$$ Here, we have fixed the three vectors, ${\bf k}_1$, ${\bf k}_2$ and ${\bf k}_3$ that appear in the expression of the trispectrum, as shown in Fig. \[fig:4\_point.eps\]. ![the three vectors, ${\bf k}_1$, ${\bf k}_2$ and ${\bf k}_3$ in the trispectrum.[]{data-label="fig:4_point.eps"}](4_point.eps) Hence, using the angular variables, $\theta_{12}$, $\theta_{13}$ and $\varphi_{13}$, we have $$\begin{aligned} && k_{12} = \sqrt{k_1^2 + k_2^2 + 2 k_1k_2\mu_{12}}~,\cr\cr && k_{23} = \sqrt{k_2^2 + k_3^2 + 2k_2 k_3 \left( \sqrt{(1 - \mu_{12}^2)(1 - \mu_{13}^2)} \cos \varphi_{13} + \mu_{12}\mu_{13}\right)}~, \cr\cr && k_{13} = \sqrt{k_1^2 + k_3^2 + 2 k_1k_3\mu_{13}}~,\end{aligned}$$ and $$\begin{aligned} k_{123} = \sqrt{k_1^2 + k_2^2 + k_3^2 + 2k_1k_2\mu_{12}+2k_1k_3\mu_{13} +2k_2k_3\left( \sqrt{(1 - \mu_{12}^2)(1 - \mu_{13}^2)} \cos \varphi_{13} + \mu_{12}\mu_{13}\right)}~,\end{aligned}$$ where $\mu_{ij} \equiv \cos \theta_{ij}$. In order to calculate the skewness more easily, let us consider the squeezed limit in momentum space, e.g., $k_1 \ll k_2 \simeq k_3$. In this limit, the equation for the skewness (\[eq:fullskewness\]) can be reduced to $$\begin{aligned} \tilde{S}_3 \biggr|_{k_1 \ll k_2 \simeq k_3} \simeq 2 \sigma_R^2 \int {dk_1 \over k_1}W_R(k_1){\cal M}(k_1){\cal P}_\phi (k_1)~, $$ and by considering other limiting cases, i.e., $k_2 \to 0$ and $k_3 \to 0$, we obtain $$\begin{aligned} \tilde{S}_3 \simeq 6 \sigma_R^2 \int {dk \over k}W_R(k){\cal M}(k){\cal P}_\phi (k)~. \label{eq:appskew}\end{aligned}$$ Based on the above approximate expression, we find a simple formula; $$\begin{aligned} \sigma_R S_3(R) = 4.3 \times 10^{-4} f_{\rm NL} \times \sigma_R^{0.13}~~( 10^{12} h^{-1} M_{\odot} < M < 2 \times 10^{15} h^{-1} M_{\odot})~. \label{eq:anaskew}\end{aligned}$$ This result seems to be close to those given in Refs. [@DeSimone:2010mu; @Enqvist:2010bg][^2]. Hence, we adopt the above expression in the following discussion. In a similar way, from the expression of the kurtosis (\[eq:fullkurtosis\]), we can easily find that the kurtosis induced from the non-linearity parameter $\tau_{\rm NL}$ becomes largest in the limit of $k_i \to 0 (i=1,2,3,4)$ or $k_{ij} \to 0 (i \neq j=1,2,3,4)$ (local type). Then, we have an approximate expression $$\begin{aligned} \tilde{S}_4^{\tau}(R) \simeq 8 \int {dk \over k} W_R(k){\cal M}(k) {\cal P}_\phi (k) \times \tilde{S}_3(R) + 12 A_\phi \sigma_R^4~.\nonumber\\ \label{eq:approxkurt}\end{aligned}$$ On the other hand, in the squeezed limit $k_i \to 0 (i=1,2,3,4)$, the kurtosis which is proportional to the non-linearity parameter $g_{\rm NL}$ can be also reduced to [@Chongchitnan:2010xz; @Enqvist:2010bg] $$\begin{aligned} S_4^{g} &\equiv& {54 \over 25}{ g_{\rm NL} \over \sigma_R^6} \tilde{S}_4^{g}~,\cr\cr \tilde{S}_4^{g} &\simeq& 2 \int {dk \over k} W_R(k){\cal M}(k) {\cal P}_\phi (k) \times \tilde{S}_3(R)~.\end{aligned}$$ From these approximate expressions, we respectively obtain simple formulae for the kurtosis in the form $$\begin{aligned} \sigma_R^2 S_4^{\tau}(R) &=& 1.9 \times 10^{-7} \tau_{\rm NL} \times \sigma_R^{0.25}~~( 10^{12} h^{-1} M_{\odot} < M < 2 \times 10^{15} h^{-1} M_{\odot})~, \cr\cr \sigma_R^2 S_4^g(R) &=& 9.4 \times 10^{-8} g_{\rm NL} \times \sigma_R^{0.27}~~( 10^{12} h^{-1} M_{\odot} < M < 2 \times 10^{15} h^{-1} M_{\odot})~. \label{eq:anakurt}\end{aligned}$$ The result for $S_4^{g}(R)$ also is close to that obtained in Ref. [@Enqvist:2010bg][^3]. Hence in the following discussion , we also adopt the above expressions for the kurtosis as well as that for the skewness. ### Difference between the Gaussian and the non-Gaussian mass functions {#subsub:dif} Based on the above calculations for the variance, $\sigma_R^2$, the skewness, $S_3$, and also the kurtosis, $S_4$, the mass function can now be calculated. In the following discussion, we take values of the non-linearity parameters as $f_{\rm NL} = 100$, $\tau_{\rm NL} = 10^6$ and $g_{\rm NL} = 0$. This value of $\tau_{\rm NL}$ may be inconsistent with the observational constraint obtained by Ref. [@Smidt:2010ra] as $-0.6 < \tau_{\rm NL} / 10^4 < 3.3$ at $95\%$ confidence level. However, there might be a caveat since in Ref. [@Fergusson:2010gn], the authors have claimed that the approach in Smidt et al. does not directly subtract the effect of anisotropic noise and other systematic effects which are important in obtaining an accurate and optimized result. Nonetheless, in order to emphasize the differences between the Gaussian mass functions and the non-Gaussian mass functions with the non-zero $f_{\rm NL}$ and the non-zero $\tau_{\rm NL}$ cases, we take the above values. In Fig. \[fig:mass\_func\_0.eps\], we show that the mass function in the mass range between $5.0 \times 10^{14} h^{-1} M_{\odot}$ and $2.0 \times 10^{15} h^{-1} M_\odot$ at the redshift $z=0$. The red thin line shows the mass function with the Gaussian density fluctuations given by Eq. (\[eq:gaussmass\]). The blue dashed and green thick lines show the non-Gaussian mass function given by Eq. (\[eq:nonGaussmass\]) in the cases with $f_{\rm NL}=100$ and $\tau_{\rm NL} = g_{\rm NL} = 0$ and $\tau_{\rm NL} = 10^6$ and $f_{\rm NL} = g_{\rm NL} = 0$, respectively. ![The mass function with the mass range between $4.5 \times 10^{14} h^{-1} M_{\odot}$ and $2.0 \times 10^{15} h^{-1} M_\odot$ at the redshift $z=0$. The red thin line shows the mass function with the Gaussian density fluctuations given by Eq. (\[eq:gaussmass\]). The blue dashed and green thick lines show the non-Gaussian mass function given by Eq. (\[eq:nonGaussmass\]) in the case with $f_{\rm NL}=100$ and $\tau_{\rm NL} = g_{\rm NL} = 0$ and the case with $\tau_{\rm NL} = 10^6$ and $f_{\rm NL} = g_{\rm NL} = 0$, respectively.[]{data-label="fig:mass_func_0.eps"}](mass_func_0.eps) From this figure, it is rather difficult to see the differences between the Gaussian and the non-Gaussian mass functions. In Fig. \[fig:mass\_func\_ratio.eps\] we show the ratios between the Gaussian and the non-Gaussian mass functions defined by Eq. (\[eq:ratio\]). The red dashed line shows ${\cal R}_{\rm NG}(M,0)$ with $f_{\rm NL} = 100$ and $\tau_{\rm NL} = g_{\rm NL} = 0$ and the blue solid line shows that with $f_{\rm NL} = g_{\rm NL} = 0$ and $\tau_{\rm NL} = 10^6$. The magenta dotted line is for the case with $\tau_{\rm NL} = g_{\rm NL} = 0$ and $f_{\rm NL} = 30$ which is corresponding to the mean value of the current WMAP data [@Komatsu:2010fb]. The black dashed-dotted line is for the case with $f_{\rm NL} = g_{\rm NL} = 0$ and $\tau_{\rm NL} = 10^4$ which is consistent with the maximum allowed value obtained by Ref. [@Smidt:2010ra]. ![The ratio between the Gaussian and the non-Gaussian mass functions. The red dashed line shows ${\cal R}_{\rm NG}(M,0)$ with $f_{\rm NL} = 100$ and $\tau_{\rm NL} = g_{\rm NL} = 0$ and the blue solid line shows that with $f_{\rm NL} = g_{\rm NL} = 0$ and $\tau_{\rm NL} = 10^6$. The magenta dotted line is for the case with $\tau_{\rm NL} = g_{\rm NL} = 0$ and $f_{\rm NL} = 30$ which is corresponding to the mean value of the current WMAP data [@Komatsu:2010fb]. The black dashed-dotted line is for the case with $f_{\rm NL} = g_{\rm NL} = 0$ and $\tau_{\rm NL} = 10^4$ which is consistent with the maximum allowed value obtained by Ref. [@Smidt:2010ra].[]{data-label="fig:mass_func_ratio.eps"}](mass_func_ratio.eps) ![The ratio between the non-Gaussian mass function and the Gaussian mass function v.s. the redshift ($0 < z < 5$) with changing the value of $\tau_{\rm NL}$. Here we have fixed the mass of halo as $M = 10^{14} h^{-1} M_{\odot}$. The solid line is for the case with $\tau_{\rm NL} = 10^5$, dashed line for $2.0 \times 10^5$, dotted line for $5.0 \times 10^5$ and dot-dashed line for $10^6$.[]{data-label="fig:mass_func_z.eps"}](tau_change.eps) From this figure, we infer that for both types of primordial non-Gaussianity, i.e., positive skewness and kurtosis, the mass functions can be systematically enhanced for more massive objects, as compared with the Gaussian case. The enhancement of the mass functions depends on the values of $\tau_{\rm NL}$ and $g_{\rm NL}$. We find that for the cases with $f_{\rm NL}=30$ and $\tau_{\rm NL} = 10^4$ ${\cal R}_{\rm NG}$ are respectively $1.06$ and $1.01$ for $M = 2 \times 10^{15} h^{-1} M_{\odot} $. Hence, in both cases the effects of the primordial non-Gaussianity on the mass functions seem to be too small to detect. We also find that the enhancement of the non-Gaussian mass function with the non-zero kurtosis type of primordial non-Gaussianity, i.e., non-zero $\tau_{\rm NL}$, depends more strongly on the mass of the collapsed objects than the case with the non-zero skewness type of primordial non-Gaussianity. This is because in the expression for the non-Gaussian mass function (\[eq:nonGaussmass\]), the $\delta_c/\sigma_R$-dependence of the term related with the kurtosis $S_4$ is stronger than that of the term related with the skewness $S_3$, namely, $S_4$-term $\propto (\delta_c/\sigma_R)^{5}$ and $S_3$-term $\propto (\delta_c/ \sigma_R)^4$. As the collapsed objects become more massive, the variance $\sigma_R$ becomes smaller and hence $\delta_c / \sigma_R$ becomes larger. Thus, if we would detect the enhancement of the mass function for massive collapsed objects and find its scale-dependence, then we might distinguish the kurtosis type of primordial non-Gaussianity from the skewness type. In Fig. \[fig:mass\_func\_z.eps\], we show the redshift dependence of the ratio between the non-Gaussian mass function and the Gaussian mass function as we change the value of $\tau_{\rm NL}$. Here we have fixed the mass of the halo as $M = 10^{14} h^{-1} M_{\odot}$. The solid line is for the case with $\tau_{\rm NL} = 10^5$, the dashed line for $2.0 \times 10^5$, dotted line for $5.0 \times 10^5$ and the dashed-dotted line for $10^6$. From this figure, we find that at higher redshift the enhancement of the mass function for massive collapsed objects increases. This is because the critical density $\delta_c(z) = \delta_c / D(z)$ becomes much larger at larger redshifts due to the smaller linear growth function $D(z)$. Hence, in order to observationally test the kurtosis type of primordial non-Gaussianity it will be useful to observe high-redshift rare objects. Applications {#sec:nGappl} ============ In this section, we consider applications of the mass function with both skewness and kurtosis types of primordial non-Gaussianity. Here, we also take values of the non-linearity parameters to be $f_{\rm NL} = 100$, $\tau_{\rm NL} = 10^6$ and $g_{\rm NL} = 0$. Early Star Formation {#sub:early} -------------------- Let us first investigate the effect of primordial non-Gaussianity on the epoch of reionization. As is well-known, in order to understand the mechanism of reionization, it is important to estimate the number of photons from Population III stars. Following Refs. [@Somerville:2003sk; @Somerville:2003sh; @Sugiyama:2003tc], the global star-formation-rate density denoted by $\dot{\rho}_*$ can be written as $$\begin{aligned} \dot{\rho}_* = e_* \rho_b {d \over dt} F_h (M_{\rm vir} > M > M_{\rm crit},t)~. \end{aligned}$$ Here, $\rho_b$ is the background baryon number density and $e_*$ denotes the star-formation efficiency usually taken to be 0.002 for $200 M_{\odot }$ Pop III stars and 0.001 for $ 100 M_{\odot}$. $F_h (M_{\rm vir} > M > M_{\rm crit},t)$ represents the fraction of the total mass in collapsed objects (halos) with masses greater than the minimum collapse mass scale $M_{\rm crit} = 10^6 h^{-1} M_{\odot}$ [@Yoshida:2003rw; @Fuller:2000vk] and lower than the virial mass $M_{\rm vir} = M(T_{\rm vir} = 10^4 {\rm K})$. The relation between the mass and the virial temperature is given by [@Barkana:2000fd; @Yoshida:2003rw] $$\begin{aligned} T_{\rm vir} = 4.7 \times 10^3 \left( {\mu \over 0.6} \right) \left( {M \over 10^8 h^{-1} M_{\odot}} \right)^{2/3} \left( {\Omega_{m0} \over 0.24}{\Delta_c(z) \over 18 \pi^2}\right)^{1/3} \left( {1+z \over 10} \right)~{\rm K}\end{aligned}$$ where $\mu$ is the mean molecular weight, and $\Delta_c(z)$ is the final overdensity relative to the critical density, which is given by a fitting formula [@Bryan:1997dn] $$\begin{aligned} \Delta_c = 18 \pi^2 + 82 \left( \Omega_m(z) - 1\right) -39 \left( \Omega_m(z) - 1\right)^2~,\end{aligned}$$ where $\Omega_{m}(z)$ is the density parameter of matter at redshift $z$; $$\begin{aligned} \Omega_m(z) = {\Omega_{m0}(1+z)^3 \over \Omega_{m0}(1+z)^3 + \Omega_{\Lambda}}~.\end{aligned}$$ Assuming that the photon number production-rate per $ M_{\odot}$ from Pop. III stars is $N_{\gamma} = 1.6 \times 10^{48} s^{-1} M_{\odot}^{-1}$ and that the life time of Pop. III star is $\tau_{\rm III} = 3.0 \times 10^6 {\rm yr}$, we can obtain the total production rate of ionizing photons at time $t$ as $$\begin{aligned} {d n_\gamma \over dt}(t) = e_* \rho_b N_{\gamma} \left(F_h(t) - F_h(t - \tau_{\rm III})\right)~,\end{aligned}$$ hence the cumulative number of photons per H atom is $$\begin{aligned} {n_\gamma \over n_H}(z) \simeq \mu m_p e_* N_\gamma F_h(M_{\rm vir} > M > M_{\rm crit},z)\tau_{\rm III}~, \label{eq:cumnum}\end{aligned}$$ with the proton mass $m_p$ and the hydrogen number density $n_H$. In the above expression, $F_h (M_{\rm vir} > M > M_{\rm crit},z)$ is given by Press-Schechter theory as $$\begin{aligned} F_h (M_{\rm vir} > M > M_{\rm crit}, z) = {1 \over \bar{\rho}} \int^{M_{\rm vir}}_{M_{\rm crit}} M {d n \over dM}(M,z)dM~.\end{aligned}$$ Substituting our expression (\[eq:nonGaussmass\]) for the non-Gaussian mass function into the above equation, we can estimate the effect of primordial non-Gaussianity on the number of photons emitted from Population III stars, which is one of the most important quantities during the epoch of reionization. ![ Cumulative photon number per $H$ atom as a function of the redshift for $8 < z < 18$. The black solid line shows $n_\gamma / n_H (z)$ for the case with the Gaussian fluctuations. The blue dashed line is for the case with the non-Gaussian fluctuations; $f_{\rm NL} = 100$ and $\tau_{\rm NL} = g_{\rm NL} = 0$. The red dot-dashed line is for the case with $\tau_{\rm NL} = 1.0 \times 10^6$ and $f_{\rm NL} = g_{\rm NL} = 0$. The thin black dashed line corresponds to $n_\gamma / n_H$ as a guide of the complete reionization. []{data-label="fig:photon_H_ratio.eps"}](cum_photon.eps){width="85mm"} In Fig. \[fig:photon\_H\_ratio.eps\], we show the cumulative photon number per $H$ atom given by Eq. (\[eq:cumnum\]) as a function of the redshift for $ 8 < z < 18$. The black solid line shows $n_\gamma / n_H (z)$ for the case with the Gaussian fluctuations, the blue dashed line is for the case with the non-Gaussian fluctuations; $f_{\rm NL} = 100$ and $\tau_{\rm NL} = g_{\rm NL} = 0$ and the red dot-dashed line for the case with $\tau_{\rm NL} = 1.0 \times 10^6$ and $f_{\rm NL} = g_{\rm NL} = 0$. The thin black dashed line corresponds to $n_\gamma / n_H = 10$ as a guide of the complete reionization on average. From this figure, we find that primordial non-Gaussianity seems not to affect the reionization history of the Universe on average which is characterized by the value of $n_\gamma / n_H = 10$. However, at higher redshift the effect of primordial non-Gaussianity seems to be significant on the cumulative photon number density. We evaluate this effect ![We plot the ratio between $n_\gamma (z)/ n_H$ in the pure Gaussian primordial fluctuation case and that in the non-Gaussian case for $5 < z < 20$. The blue dashed line is for the case with $f_{\rm NL} = 100$ and $\tau_{\rm NL} = g_{\rm NL} = 0$ and the red solid line for the case with $ \tau_{\rm NL} = 1.0 \times 10^6$ and $f_{\rm NL} = g_{\rm NL} = 0$. []{data-label="fig:photon_number.eps"}](photon_number.eps) in Fig. \[fig:photon\_number.eps\], where we plot the ratio between $n_\gamma / n_H (z)$ in the pure Gaussian primordial fluctuation case and that in the non-Gaussian case. The blue dashed line is for the case with $f_{\rm NL} = 100$ and $\tau_{\rm NL} = g_{\rm NL} = 0$ and the red solid line for the case with $ \tau_{\rm NL} = 1.0 \times 10^6$ and $f_{\rm NL} = g_{\rm NL} = 0$. From this figure, we find that compared to the Gaussian case the cumulative number of photons in the non-Gaussian case is larger at higher redshifts both in the non-zero $S_3$ and the non-zero $S_4$ cases. Moreover, as we have mentioned in the previous section the kurtosis type of primordial non-Gaussianity affects the enhancement of the photon number density more significantly at high redshift. That is, there seems to be the possibility of dramatically changing the history of the early stage of reionization due to the kurtosis type of primordial non-Gaussianity even for values in the range of the current limits obtained from CMB observations. Of course, the above rough estimate is not precise enough to enable us to estimate the exact cumulative number of the ionizing photons. However, we consider here that, in view of the completely ad hoc nature of the amount of non-Gaussianity due to the absence of a compelling inflationary model, it suffices for us to focus on the deviation of the photon number based on the non-Gaussian mass function from that based on the Gaussian mass function. High-Redshift Massive Clusters {#sub:mass} ------------------------------ Recently, the authors in Ref. [@Jee:2009nr; @Rosati:2009cm] have presented a weak lensing analysis of the galaxy cluster XMMU J2235.3-2557 which has a high redshift $z \approx 1.4$ and whose mass is $M_{324} = (6.4 \pm 1.2) \times 10^{14} M_{\odot}$[^4]. In $\Lambda$CDM model the formation of such a massive cluster at this redshift would be a rare event (at least $3\sigma$). In Ref. [@Cayon:2010mq], the authors have considered the effects of primordial non-Gaussianity parametrized by the non-linearity parameter $f_{\rm NL}$ which they found to be $f_{\rm NL} = 449 \pm 286$ at wave number of about $0.4 {\rm Mpc}^{-1}$ in order to explain the existence of such a massive cluster at high redshift. Considering scale-invariant $f_{\rm NL}$, this result contradicts the current CMB observational constraint $f_{\rm NL} < 100$. Therefore, the authors remarked that one would need to invoke scale-dependent $f_{\rm NL}$. In Ref. [@Enqvist:2010bg], the authors have considered non-zero $g_{\rm NL}$ case and found that $g_{\rm NL} = O(10^6)$ could explain the existence of high redshift massive clusters. Here, instead of considering the scale-dependence of $f_{\rm NL}$ or $g_{\rm NL}$, let us consider the effect of the kurtosis induced from the $\tau_{\rm NL}$-type primordial non-Gaussianity on the formation of massive clusters. Of course, for a more detailed analysis we need to calculate the probability of the massive clusters under the procedure done in Ref. [@Cayon:2010mq]. However, in order to give a naive estimation of the value of $\tau_{\rm NL}$ which can explain the existence of the massive cluster XMMU J2235.3-2557, we investigate the value of $\tau_{\rm NL}$ which gives the same value as does the non-Gaussian mass function, namely, ${\cal R}_{\rm NG}$ defined as Eq. (\[eq:ratio\]), including the effect of kurtosis $S_4$ on the corresponding scale at the corresponding redshift by including the effect of $f_{\rm NL}$, i.e., skewness. Here, we adopt $M_{\rm XMMU} = 6.4 \times 10^{14} M_{\odot}$ and $z_{\rm XMMU} = 1.4$ as the mass and the redshift of the massive cluster XMMU J2235.3-255, respectively. For the value of $f_{\rm NL}$, the best fit value derived in Ref. [@Cayon:2010mq] is adopted. For these parameters, we also find that this value can be realized in the case with $f_{\rm NL} = g_{\rm NL} = 0$ and $\tau_{\rm NL} = 1.7 \times 10^6$. As we have mentioned in Sec. \[subsub:dif\], this value may be ruled out by the result obtained by Ref. [@Smidt:2010ra]. Hence, if we believe this constraint, we need to consider the possibility such as scale-dependent $\tau_{\rm NL}$. Abundance of voids {#sub:void} ------------------ As another example, we study the void abundance with primordial non-Gaussian corrections. In Ref. [@Kamionkowski:2008sr], the authors showed that the void distribution function can be derived in the same way as the halo mass function using Press-Schechter theory. This is done by replacing the critical “overdensity” parameter, $\delta_c$, with the negative “underdensity” parameter, $\delta_v$. The precise value of $\delta_v$ depends on the definition of a void. For example, if the voids are regions having a density half of $\bar{\rho}$ , then we can estimate the critical value of underdensity as $\delta_v \simeq -0.7$ [@Kamionkowski:2008sr]. There are also several numerical studies about the value of $\delta_v$ which suggest $\delta_v \approx -0.8$ [@Shandarin:2005ea; @Park:2006wu; @Colberg:2008qg]. In any case, based on Press-Schechter theory, the abundance of voids which have radius between $R$ and $R + d R$ is given by [@Kamionkowski:2008sr] $$\begin{aligned} {dn^{\rm void}(R) \over dR} dR = - dR \times {6 \over 4 \pi R^3} {d \over dR}\int_{-\infty}^{\delta_v / \sigma_R} F(\nu)d\nu ~.\end{aligned}$$ For pure Gaussian PDF, we have $$\begin{aligned} {dn_{\rm G}^{\rm void}(R) \over dR} = \sqrt{2 \over \pi} {3 \over 4 \pi R^4} \exp \left[ -{\delta_v^2 \over 2 \sigma_R^2}\right] {\delta_v \over \sigma_R} {d \ln \sigma_R \over d \ln R}~.\end{aligned}$$ Up to the third order in terms of $S_3$ and $S_4$, the void abundance with primordial non-Gaussian corrections is also given by $$\begin{aligned} {dn^{\rm void} (R) \over dR} &=& \sqrt{2 \over \pi} {3 \over 4 \pi R^4} \exp \left[ -{\delta_v^2 \over 2 \sigma_R^2}\right] \Biggl\{ {d \ln \sigma_R \over d \ln R} {\delta_v \over \sigma_R} \Biggl[ 1 \cr\cr &&\qquad + {S_3(R)\sigma_R \over 6}H_3(\delta_v / \sigma_R) + {1 \over 2}\left({S_3(R) \sigma_R \over 6} \right)^2 H_6(\delta_v / \sigma_R) +{1 \over 6}\left({S_3(R) \sigma_R \over 6} \right)^3 H_9(\delta_v / \sigma_R) \cr\cr &&\qquad +{S_4(R) \sigma_R^2 \over 24}H_4(\delta_v / \sigma_R) + {1 \over 2}\left({S_4(R) \sigma_R^2 \over 24} \right)^2 H_8(\delta_v / \sigma_R) + {1 \over 6}\left({S_4(R) \sigma_R^2 \over 24} \right)^3 H_{12}(\delta_v / \sigma_R) \Biggr] \cr\cr && + {d\over d \ln R}\left({S_3(R) \sigma_R \over 6} \right) H_2(\delta_v / \sigma_R) +{1 \over 2}{d\over d\ln R}\left({S_3(R) \sigma_R \over 6} \right)^2 H_5(\delta_v / \sigma_R) +{1 \over 6}{d\over d\ln R}\left({S_3(R) \sigma_R \over 6} \right)^3 H_8(\delta_v / \sigma_R) \cr\cr && + {d\over d\ln R}\left({S_4(R) \sigma_R^2 \over 24} \right) H_3(\delta_v / \sigma_R) +{1 \over 2}{d\over d\ln R}\left({S_4(R) \sigma_R^2 \over 24} \right)^2 H_7(\delta_v / \sigma_R) +{1 \over 6}{d\over d\ln R}\left({S_4(R) \sigma_R^2 \over 24} \right)^3 H_{11}(\delta_v / \sigma_R)~. \Biggr\}\end{aligned}$$ Following the previous section, we define the ratio between the void abundance with the pure Gaussian PDF and that with the primordial non-Gaussian corrections as $$\begin{aligned} {\cal R}_{\rm NG}^{\rm void} \equiv {dn^{\rm void}(R)/dR \over dn^{\rm void}_{\rm G}(R) / dR}~. \nonumber\\ \label{eq:voidratio}\end{aligned}$$ In Fig. \[fig:void\_ratio.eps\], ![The ratio between the void abundance distribution function with non-zero primordial non-Gaussian correction and the Gaussian distribution function at redshifts $z=0.0$, $0.75$, $1.5$ given by Eq. (\[eq:voidratio\]). The red lines are for the case with $f_{\rm NL} = 100$ and $\tau_{\rm NL} = g_{\rm NL} = 0$, the blue thick lines for the case with $\tau_{\rm NL} = 10^6$ and $f_{\rm NL} = g_{\rm NL} = 0$ and the black thin lines for the case with $\tau_{\rm NL} = 10^4$ and $f_{\rm NL} = g_{\rm NL} = 0$. The solid lines are for the case with $z=0.0$, the dot-dashed lines for the case with $z=0.75$ and the dashed lines for the case with $z=1.5$. Here we took $\delta_v = -0.7$.[]{data-label="fig:void_ratio.eps"}](void_ab.eps) we show ${\cal R}_{\rm NG}^{\rm void}$ for the cases with $f_{\rm NL}=100, \tau_{\rm NL} = g_{\rm NL} = 0$ (red lines), $\tau_{\rm NL} = 10^6, f_{\rm NL} = g_{\rm NL} = 0$ (blue thick lines) and $\tau_{\rm NL} = 10^4, f_{\rm NL} = g_{\rm NL} = 0$ (black thin lines). We adopt $\delta_v = -0.7$. We also show the ratio with changing the redshift; the solid lines for the case with $z=0.0$, the dot-dashed lines for the case with $z=0.75$ and the dashed lines for the case with $z=1.5$. From this figure, we conclude that the non-Gaussian void abundance with non-zero $\tau_{\rm NL}$ becomes larger than the Gaussian one on relatively larger scales whereas that with the non-zero $f_{\rm NL}$ becomes smaller. On the other hand, as seen in the previous section, the halo abundance becomes larger not only with non-zero $\tau_{\rm NL}$ but also with non-zero $f_{\rm NL}$ in relatively more massive objects. Hence, from this discussion, we confirm that the non-Gaussian effects on both the halo and the void abundances allow as to distinguish the large kurtosis, i.e., large $\tau_{\rm NL}$, case from the large skewness, i.e., large $f_{\rm NL}$ case [@Chongchitnan:2010xz]. Summary and Discussion {#sec:sum} ====================== It has recently become clear that cosmological large-scale structure and CMB observations could provide stringent constraints on the PDF of primordial adiabatic curvature fluctuations. In particular, the high order moments of the PDF, such as its skewness and kurtosis, can give unique insights into the dynamics and conditions of the inflationary phase in the early Universe. In this paper, we have investigated the effects of the $\tau_{\rm NL}$-type of primordial non-Gaussianity on the halo mass function. In particular, we have obtained a formula for the halo mass function with the non-Gaussian corrections coming from the kurtosis induced by the non-zero $\tau_{\rm NL}$. We find that the deviations of the non-Gaussian mass function from the Gaussian one become larger for larger mass objects ($M \gtrsim10^{14} h^{-1} M_\odot$ for $z \sim 0$) as well as at higher redshifts ($z \gtrsim 1$ for $M \approx 10^{14} h^{-1} M_\odot$) in the case with $\tau_{\rm NL} = O(10^6)$. Such features are quite similar to those obtained from skewness-driven non-Gaussian corrections that are induced by the $f_{\rm NL}$-type of primordial non-Gaussianity. As examples of applications of our formulae, we have considered the effects on early star formation, formation of the most massive objects at high redshift, and the abundance of voids. For early star formation, we applied our formula for the non-Gaussian halo mass function in order to estimate of the redshift-dependence of the cumulative number of photons emitted from population III stars, a crucial quantity in considerations of the reionization history of the Universe. We found that primordial non-Gaussianity does not affect the reionization history of the Universe on the average, but at high redshift ($z\simeq 20$), namely the earliest stages of reionization, it is effective. We have also obtained an estimate of the value of $\tau_{\rm NL}$ needed to naturally explain the existence of the galaxy cluster XMMU J2235.3-2557, namely $\tau_{\rm NL} = 1.7 \times 10^6 $. Hence, in light of the result of Smidt et al., we might need to consider a possibility such as scale-dependent $\tau_{\rm NL}$ in the case with non-zero $f_{\rm NL}$. In Ref. [@Hoyle:2010ce], the authors have investigated $15$ high-mass and high-redshift galaxy clusters and found that such objects are extremely rare in the standard $\Lambda$CDM model with Gaussian primordial fluctuations. They derived a constraint on $f_{\rm NL}$ in order to explain the mere existence of these objects as $f_{\rm NL} > 475$ at $95\%$ confidence level, with the other cosmological parameters fixed to best fit values of WMAP data. In Ref, [@Enqvist:2010bg], the authors have extended the analysis of Ref. [@Hoyle:2010ce] to the case with non-zero $g_{\rm NL}$. It should clearly be of interest to derive a constraint on $\tau_{\rm NL}$ for these observed high-mass and high-redshift galaxy clusters. We will address this in future work. As mentioned in Refs. [@Kamionkowski:2008sr; @Chongchitnan:2010xz], the non-Gaussian correction coming from skewness reduces the abundance of voids on large scales when the non-linearity parameter $f_{\rm NL}$ is positive in contrast to the fact that positive $f_{\rm NL}$ enhances the number of more massive halo objects. On the other hand, the non-Gaussian correction coming from kurtosis enhances not only the numbers of more massive halo objects but also the abundances of voids on large scales. Hence, if one could also measure the void abundance as well as the halo mass function more precisely, one could potentially distinguish between the $f_{\rm NL}$ and the $\tau_{\rm NL}$-types of primordial non-Gaussianity. *NOTE;*  During the time that we were preparing this manuscript, Ref. [@LoVerde:2011iz] appeared on the arXiv. In Ref. [@LoVerde:2011iz], they considered the same type of primordial non-Gaussianity as in our study and obtained a useful analytic formula for the halo mass function with the kurtosis type primordial non-Gaussianity using N-body simulations. We find that our formula (\[eq:nonGaussmass\]) is in reasonably good agreement with their formula as far as the behavior of the halo mass function with the kurtosis type of primordial non-Gaussianity. acknowledgments =============== S.Y. thanks Yoshitaka Takeuchi and Shogo Masaki, and J.S. thanks Laura Cayon, Sirichai Chongchitnan, and Christopher Gordon for valuable discussions. This work is supported by the Grant-in-Aid for Scientific research from the Ministry of Education, Science, Sports, and Culture, Japan, No. 22340056. 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Phys. 10, 22 Yoshida N., Abel T., Hernquist L., N. Sugiyama N., 2003, ApJ, 592, 645 MVJ expression {#app:mvj} ============== In Ref. [@Matarrese:2000iz], the authors have given a formula for the ratio between the non-Gaussian mass function and the Gaussian mass function as $$\begin{aligned} R_{\rm NG}^{\rm MVJ} (M,z) &=& \exp \left[\nu_c^3 {S_3(R) \sigma_R \over 6} + \nu_c^4 {S_4(R) \sigma_R^2 \over 24} \right] \times \left[ \delta_3 + {\nu_c \over \delta_3} \left( - {S_3(R) \sigma_R \over 6} \right) + \left({ d \ln \sigma_R \over dM} \right)^{-1} {d \over dM}\left( {S_3(R) \sigma_R \over 6} \right)\right] \cr\cr && \qquad\qquad \times \left[ \delta_4 + {\nu_c^2 \over \delta_4} \left( - {S_4(R) \sigma_R^2 \over 12} \right) + \left({ d \ln \sigma_R \over dM} \right)^{-1} {d \over dM}\left( {S_4(R) \sigma_R^2 \over 24} \right)\right]~, \cr\cr \delta_3 & \equiv & \left( 1 - \nu_c {S_3(R) \sigma_R \over 3} \right)^{1/2}~,~ \delta_4 \equiv \left( 1 - \nu_c^2 {S_4(R) \sigma_R^2 \over 12} \right)^{1/2}~,\end{aligned}$$ which was not derived based on the Edgeworth expansion as mentioned in section \[sec:nGmass\]. In section \[sec:nGappl\], we have discussed some applications of the non-Gaussian halo mass function. As for the discussion in subsection \[sub:early\] about early star formation, the redshift-dependence of the critical value of the cumulative photon number per H atom ($n_\gamma / n_H = 10$) is not so sensitive to primordial non-Gaussianity. In order to estimate more precisely how $n_\gamma / n_H$ at high redshift is enhanced due to primordial non-Gaussianity, we should check which formula better describes the effect of primordial non-Gaussianity on the halo mass function. This is a future issue. In subsection \[sub:void\], we have discussed the void abundance and noted that the kurtosis type of primordial non-Gaussianity can enhance the abundance of the large voids as opposed to the skewness type of primordial non-Gaussianity. This is just qualitative discussion. On the other hand, the discussion in subsection \[sub:mass\] is so quantitative and hence we have investigated the difference of the estimated value of $\tau_{\rm NL}$ for the observation of XMMU J2235.3-2557 between the case with MVJ expression and that with Eq. (\[eq:nonGaussmass\]) given in section \[sec:nGmass\]. Our naive estimated value of $\tau_{\rm NL}$ given in subsection \[sub:mass\] is $\tau_{\rm NL} = 1.7 \times 10^6$. For the case by making use of MVJ expression, we obtained $\tau_{\rm NL} = 1.1 \times 10^6$. These values seem to be same order and hence the result does not extremely change. [^1]: In Refs. [@Enqvist:2010bg; @Matarrese:2000iz; @Verde:2000vr], the authors introduced the MVJ convention for defining the ratio given by $$\begin{aligned} \cal R_{\rm NG}^{\rm MVJ} (M,z) &=& \exp \left[\nu_c^3 {S_3(R) \sigma_R \over 6} + \nu_c^4 {S_4(R) \sigma_R^2 \over 24} \right] \times \left[ \delta_3 + {\nu_c \over \delta_3} \left( - {S_3(R) \sigma_R \over 6} \right) + \left({ d \ln \sigma_R \over dM} \right)^{-1} {d \over dM}\left( {S_3(R) \sigma_R \over 6} \right)\right] \cr\cr && \qquad\qquad \times \left[ \delta_4 + {\nu_c^2 \over \delta_4} \left( - {S_4(R) \sigma_R^2 \over 12} \right) + \left({ d \ln \sigma_R \over dM} \right)^{-1} {d \over dM}\left( {S_4(R) \sigma_R^2 \over 24} \right)\right]~, \cr\cr \delta_3 & \equiv & \left( 1 - \nu_c {S_3(R) \sigma_R \over 3} \right)^{1/2}~,~ \delta_4 \equiv \left( 1 - \nu_c^2 {S_4(R) \sigma_R^2 \over 12} \right)^{1/2}~,\end{aligned}$$ which is not based on the Edgeworth expansion. In our calculation, we have also checked the consistency between the above MVJ expression and Eq. (\[eq:ratio\]). This issue is discussed in Appendix \[app:mvj\]. [^2]: As mentioned in Ref. [@Enqvist:2010bg], this result is different from that in the published version of Ref. [@Chongchitnan:2010xz]. However, the authors in Ref. [@Chongchitnan:2010xz] have corrected the result in the arXiv version and their new derivation is now close to our result (\[eq:anaskew\]). [^3]: In addition to the expression derived for the skewness, the result for kurtosis in Ref. [@Chongchitnan:2010xz] has been corrected in the arXiv version and this is also close to our result (\[eq:anakurt\]). [^4]: The halo is defined as a spherical overdense region whose density is 324 times the mean matter density of the Universe.
\#1[[cite\#\#1\#\#2[\#\#1]{}]{}]{} \#1[cite[\#1]{}]{} tempcntc citex\[\#1\]\#2[@fileswauxout tempcnta@tempcntb@neciteacite[forciteb:=\#2citeo]{}[\#1]{}]{} citeo[tempcnta&gt;tempcntbciteacitea[,]{} tempcnta=tempcntbtempcnta]{} CERN–TH/96–313\ hep-ph/9611249 [ ${\boldsymbol{\mathrm{e}}^{\boldsymbol{+}}}{\boldsymbol{\mathrm{e}}^{\boldsymbol{-}}}$ [**collisions**]{}]{}\ [Gerhard A. Schuler$^a$]{}\ [*Theory Division, CERN,*]{}\ [*CH-1211 Geneva 23, Switzerland*]{}\ [ E-mail: Gerhard.Schuler@cern.ch]{} [**Abstract**]{}\ A Monte Carlo program is presented for the computation of the most general cross section for two-photon production in ${{{\mathrm{e}}}}^+{{{\mathrm{e}}}}^-$ collisions at fixed two-photon invariant mass $W$. Functions implemented for the five $\gamma^\star\gamma^\star$ structure functions include three models of the total hadronic cross section and the lepton-pair production cross section. Prospects of a structure-function determination through a study of the azimuthal dependence between the two scattering planes are outlined. All dependences on the electron mass and the photon virtualities $Q_i^2$ are fully kept. Special emphasis is put on a numerically stable evaluation of all variables over the full $Q_i^2$ range from $Q_{i\min}^2 \sim {m_{{{{\mathrm{e}}}}}}^2 (W/\sqrt{s})^4 \ll {m_{{{{\mathrm{e}}}}}}^2$ up to $Q_{i\max}^2 \sim s$. A comparison is made with an existing Monte Carlo program for lepton-pair production and an equivalent-photon approximation for hadronic cross sections. ------------------------------------------------------------------------ ${}^a$ Heisenberg Fellow. CERN–TH/96–313\ October 1996 Program Summary -------------------------------------------------------------------------------------- --------------------------------------------------------------------------------------- [*Title of program:*]{} GALUGA [*Program obtainable from:*]{} G.A. Schuler, CERN–TH, CH-1211 Geneva 23, Switzerland; Gerhard.Schuler@cern.ch [*Licensing provisions:*]{} none [*Computer for which the program is designed and others on which it is operable:*]{} all computers [*Operating system under which the program has been tested:*]{} UNIX [*Programming language used:*]{} FORTRAN 77 [*Number of lines:*]{} $1445$ [*Keywords:*]{} Monte Carlo, two-photon, ${{{\mathrm{e}}}}^+{{{\mathrm{e}}}}^-$, azimuthal dependence [*Subprograms used:*]{} VEGAS [@VEGAS] (included, 229 lines) RANLUX [@RANLUX] (included, 305 lines) HBOOK [@HBOOK] and DATIME [@DATIME] for the test program (367 lines) -------------------------------------------------------------------------------------- --------------------------------------------------------------------------------------- [*Nature of physical problem:*]{}\ Hadronic two-photon reactions in a new energy domain are becoming accessible with LEP2. Unlike purely electroweak processes, hadronic processes contain dominant non-perturbative components parametrized by suitable structure functions, which are functions of the two-photon invariant mass $W$ and the photon virtualities $Q_1$ and $Q_2$. It is hence advantageous to have a Monte Carlo program that can generate events at fixed, user-defined values of $W$ and, optionally, at fixed values of $Q_i$. Moreover, at least one program with an exact treatment of both the kinematics and the dynamics over the whole range $m^2 \gg m^2 (W/\sqrt{s})^4 {\raisebox{-0.8mm}{\hspace{1mm}$\stackrel{<}{\sim}$\hspace{1mm}}}Q_i^2 {\raisebox{-0.8mm}{\hspace{1mm}$\stackrel{<}{\sim}$\hspace{1mm}}}s$ ($m$ is the electron mass and $\sqrt{s}$ the ${{{\mathrm{e}}}}^+{{{\mathrm{e}}}}^-$ c.m. energy) is needed, (i) to check the various approximations used in other programs, and (ii) to be able to explore additional information on the hadronic physics, e.g. coded in azimuthal dependences. [*Method of solution:*]{}\ The differential cross section for ${{{\mathrm{e}}}}^+{{{\mathrm{e}}}}^- \rightarrow {{{\mathrm{e}}}}^+{{{\mathrm{e}}}}^- X$ at fixed two-photon invariant mass $W$ is rewritten in terms of four invariants with the photon virtualities $Q_i$ as the two outermost integration variables in order to simultaneously cope with antitag and tagged electron modes. Due care is taken of numerically stable expressions while keeping all electron-mass and $Q_i$ dependences. Special care is devoted to the azimuthal dependences of the cross section. Cuts on the scattered electrons are to a large extent incorporated analytically and suitable mappings introduced to deal with the peaking structure of the differential cross section. The event generation yields either weighted events or unweighted ones (i.e. equally weighted events with weight $1$), the latter based on the hit-or-miss technique. Optionally, VEGAS can be invoked to (i) obtain an accurate estimate of the integrated cross section and (ii) improve the event generation efficiency through additional variable mappings provided by the grid information of VEGAS. The program is set up so that additional hadronic (or leptonic) reactions can easily be added. [*Typical running time:*]{}\ The integration time depends on the required cross-section accuracy and the applied cuts. For instance, $13$ seconds on an IBM RS/6000 yields an accuracy of the VEGAS integration of about $0.1$% for the antitag mode or of about $0.2\%$ for a typical single-tag mode; within the same time the error of the simple Monte Carlo integration is about $0.5$% for either mode. Event generation with or without VEGAS improvement and for either tag mode takes about $4\times 10^{-4}$ ($2\times 10^{-3}$) seconds per event for weighted (unweighted) events. Introduction ============ Two-photon physics is facing a revival with the advent of LEP2. Measurements of two-photon processes in a new domain of $\gamma\gamma$ c.m. energies $W$ are ahead of us [@LEP2]. Any two-photon process is, in general, described [@Budnev] by five non-trivial structure functions (two more for polarized initial electrons). Purely QED (or electroweak) processes are fully calculable within perturbation theory. Several sophisticated Monte Carlo event generators exist [@vermaseren; @diag36; @Diberder] to simulate $4$-fermion production in ${{{\mathrm{e}}}}^+{{{\mathrm{e}}}}^-$ collisions. Indeed, the differential cross section is not explicitly decomposed as an expansion in the five $\gamma^\star\gamma^\star$ structure functions. Rather, the full matrix element for the reaction ${{{\mathrm{e}}}}^+{{{\mathrm{e}}}}^- \rightarrow {{{\mathrm{e}}}}^+{{{\mathrm{e}}}}^- \ell^+\ell^-$ is calculated as a whole, partly even including QED radiative corrections. Such a procedure is, however, not possible for hadronic two-photon reactions since the hadronic behaviour of the photon is of non-perturbative origin. The decomposition into the above-mentioned five structure functions (and their specification, of course) is hence mandatory for a full description of hadronic reactions. Monte Carlo event generators for hadronic two-photon processes can be divided into two classes. Programs of the first kind [@herwig; @pythia; @phojet; @minijet; @ggps1; @gghv01] put the emphasis on the QCD part but are (so far) restricted to the scattering of two [*real*]{} photons. The two-photon sub-processes are then embedded in an approximate way in the overall reaction of ${{{\mathrm{e}}}}^+{{{\mathrm{e}}}}^-$ collisions. A recent discussion of the so-called equivalent-photon approximation can be found in [@GAS]. The other type of programs [@twogam; @twogen] treat the kinematics of the vertex ${{{\mathrm{e}}}}^+{{{\mathrm{e}}}}^- \rightarrow {{{\mathrm{e}}}}^+{{{\mathrm{e}}}}^-\gamma\gamma$ more exactly, but they contain only simple models of the hadronic physics. Moreover, the event generation is done in the variables that are tailored for ${{{\mathrm{e}}}}{{{\mathrm{e}}}}\rightarrow {{{\mathrm{e}}}}{{{\mathrm{e}}}}\gamma\gamma$, namely the energies and angles (or virtualities) of the photons and the azimuthal angle $\phi$ between the two lepton-scattering planes in the [*laboratory*]{} system. Hence, both the hadronic energy $W$ and the azimuthal angle $\tilde{\phi}$ in the [*photon*]{} c.m.s. (which enters the decomposition of the ${{{\mathrm{e}}}}^+{{{\mathrm{e}}}}^-$ cross section into the five hadronic structure functions) are highly non-trivial functions of these variables[^1]. In the study of hadronic physics one prefers to study events at fixed values of $W$. Not only is $W$ the crucial variable that determines the nature of the hadronic physics, but through studies of events at fixed $W$ can $\gamma\gamma$ collisions be compared with $\gamma{{{\mathrm{p}}}}$ and ${{{\mathrm{p}}}}{\overline{{{\mathrm{p}}}}}$ ones [@SaS]. Next to $W$, the virtualities $Q_1$ and $Q_2$ of the two photons determine the hadronic physics. At fixed values of $W$ and one of the $Q$’s, $Q_1$ say, one obtains the cross section of deep-inelastic electron–photon scattering. Varying $Q_2$ one can investigate the so-called target-mass effects, i.e. the influence of non-zero values of $Q_2$ on the extraction of the photon structure function $F_2$. Hence it is desirable to have an event generator that keeps $W$ fixed and in which $Q_1$ and $Q_2$ are the outermost integration variables so that these can be held constant. The remaining two non-trivial integration variables, which complete the phase space of ${{{\mathrm{e}}}}^+{{{\mathrm{e}}}}^- \rightarrow {{{\mathrm{e}}}}^+{{{\mathrm{e}}}}^- X$, should be chosen such that three conditions are fulfilled. First, cuts on the scattered electrons are usually imposed in experimental analyses. Hence, the efficiency and accuracy of the program is improved if these can be treated explicitly rather than incorporated by a simple rejection of those events that fall outside the allowed region. Second, the peaking structure of the differential cross section should be reproduced as well as possible in order to reduce the estimated Monte Carlo error and to improve the efficiency of the event generation. And third, it should be possible to achieve a numerically stable evaluation of all variables needed for a complete event description. These three conditions are met to a large extent by the choice of subsystem squared invariant masses $s_1$ and $s_2$ as integration variables besides $Q_1^2$ and $Q_2^2$. In the laboratory frame, $s_i$ are related to the photon energies $\omega_i$ by $s_{1/2} - m^2 = 2 \omega_{2/1} \sqrt{s}$, where $m$ denotes the electron mass. In the interest of those readers not interested in calculational details, the paper starts with a presentation of a few results in section \[sec:results\]. The differential cross section for the reaction ${{{\mathrm{e}}}}^+{{{\mathrm{e}}}}^- \rightarrow {{{\mathrm{e}}}}^+{{{\mathrm{e}}}}^- X$ is rewritten in terms of the four invariants $Q_i^2$ and $s_i$ ($i=1,2$) in section \[sec:notation\] where also models for the cross section $\sigma(\gamma^\star\gamma^\star \rightarrow X)$ are described. The integration boundaries with $Q_1^2$ and $Q_2^2$ as the two outermost integration variables are specified in section \[sec:phase\]. The derivation of the integration limits is standard [@Byckling] but tedious. Here the emphasis is put on numerically stable expressions[^2]. To our knowledge, numerical stable forms of $\phi$ and $\tilde{\phi}$ are presented here for the first time. All dependences on the electron mass and the virtualities of the two photons are kept. The formulas are stable over the whole range from $Q_{i\min}^2 \sim m^2 (W/\sqrt{s})^4 \ll m^2$ up to $Q_{i\max}^2 \sim s$, i.e. the program covers smoothly the antitag and tag regions. An equivalent-photon approximation is also implemented (section \[sec:EPA\]). The complete representation of the four-momenta of the produced particles in terms of the integration variables is given in section \[sec:momenta\]. Section \[sec:cuts\] describes the incorporation of cuts on the scattered electrons. Details of the Monte Carlo program GALUGA are given in section \[sec:program\]. A few results {#sec:results} ============= In order to check GALUGA, we include the production of lepton pairs, for which several well-established Monte Carlo generators [@vermaseren; @diag36; @Diberder] exist. The five structure functions for $\gamma^\star\gamma^\star \rightarrow \ell^+ \ell^-$ as quoted in [@Budnev] have been implemented. For the comparison we have modified the two-photon part of the four-fermion program DIAG36 [@diag36] (i.e. DIAG36 restricted to the multiperipheral diagrams) in such a way that it can produce events at [*fixed*]{} values of $W$. The agreement is excellent. Two examples are shown in Fig. \[fig:blnthx\], the first corresponding to a no-tag setup and the second to a single tagging mode. Next we study the (integrated) total hadronic cross section. Figures \[fig:nocuts\] and \[fig:cutone\]–\[fig:cutthree\] compare different ansätze for the $Q_i^2$ behaviour of the various cross sections for transverse and longitudinal photons. The results of the two models of generalized-vector-meson-dominance type (GVMD (\[GVMDform\]) and VMDc (\[VMDcform\]), dash-dotted and dotted histograms, respectively) are hardly distinguishable in the no-tag case, but may deviate by more than $20\%$ in a single-tag case. In the contrast, the different $Q_i^2$ behaviour of a simple $\rho$-pole (dashed histograms) shows up already in the no-tag mode. Note that this model includes scalar photon contributions, but does not possess an $1/Q^2$ “continuum” term for transverse photons. These differences imply that effects of non-zero $Q_i^2$ values must not be neglected for a precision measurement of $\sigma_{\gamma\gamma}(W^2)$. During the course of the LEP2 workshop, sophisticated programs to generate the full (differential) hadronic final state in two-photon collisions have been developed [@gagaMC]. The description of hadronic physics with one (or both) photons off-shell by virtualities $Q_i^2 \ll W^2$ is still premature. Indeed, existing programs are thus far for [*real*]{} photons and hence use, in one way or another, the equivalent-photon approximation (EPA) to embed the two-photon reactions in the ${{{\mathrm{e}}}}^+{{{\mathrm{e}}}}^-$ environment. It is hence indispensable to check the uncertainties associated with the EPA. Hadronic physics is under much better theoretical control for deep-inelastic scattering, i.e. the setup of one almost real photon probed by the other that is off-shell by an amount $Q^2$ of the order of $W^2$. Corresponding event generators exist [@gagaMC] but also in this case it is desirable to check the equivalent-photon treatment of the probed photon. An improved EPA has recently been suggested in [@GAS]. In essence, the prescription consists in neglecting $Q_i^2$ w.r.t. $W^2$ in the kinematics but to keep the full $Q_i^2$ dependence in the $\gamma^\star\gamma^\star$ structure functions. In addition, non-logarithmic terms proportional to $m^2/Q_i^2$ in the luminosity functions are kept as well. The study [@GAS] shows that this improved EPA works rather well for the [*integrated*]{} ${{{\mathrm{e}}}}^+{{{\mathrm{e}}}}^-$ cross section. In Fig. \[fig:nocuts\] we show that this EPA (solid compared to dash-dotted histograms) works well also for differential distributions, with the exception of the polar-angle distribution of the hadronic system at large angles, where it can, in fact, fail by more than an order of magnitude! (There, of course, the cross section is down by several orders.) The EPA describes also rather well the dynamics of the scattered electrons in the single-tag mode except in the tails of the distributions (Fig. \[fig:cutone\]). The same holds for the distributions in the photon virtualities, see Fig. \[fig:cuttwo\]. Sizeable differences do, however, show up (Fig. \[fig:cuttwo\]) in the distributions of the subsystem invariant masses $\sqrt{s_i}$. These then lead to the wrong shapes for the energy and momentum distributions of the hadronic system shown in Fig. \[fig:cutthree\]. The EPA should, therefore, not be used for single-tag studies. Finally we study the prospects of a determination of additional structure functions besides $F_2$. One such possibility was outlined in [@LEP2], namely the study of the azimuthal dependence in the $\gamma\gamma$ c.m.s.  between the plane of the scattered (tagged) electron and the plane spanned by the beam axis and the outgoing muon or jet. Here we propose to study the azimuthal angle $\tilde\phi$ between the two electron scattering planes, again in the $\gamma\gamma$ c.m.s. Although such a study requires a double-tag setup, the event rates need not be small, since one can fully integrate out the hadronic system but for its invariant mass $W$. In order to demonstrate the sensitivity of such a measurement we show, as a preparatory exercise, the $\tilde\phi$ distribution for muon-pair production in Fig. \[fig:phtnvspht\]. Fitting to the functional form $$\frac{{{{\mathrm{d}}}}\sigma}{{{{\mathrm{d}}}}\tilde\phi} \propto 1 + A_1\, \cos\tilde\phi + A_2\, \cos 2\, \tilde\phi \ , \label{phitildeform}$$ we find $$A_1 = 0.098 \qquad , \qquad A_2 = -0.028 \ . \label{fitresult}$$ Let us emphasize that the selected tagging ranges have in no way been optimized for such a study. Nonetheless, given the magnitudes of $A_i$, a measurement appears feasible. All but one [@vermaseren] event generators for two-photon physics use the azimuthal angle $\phi$ between the two scattering planes in the [*laboratory*]{} frame as one of the integration variables. In fact, $\phi$ appears as a trivial variable in these programs. None of these up to now provides the calculation of $\tilde\phi$. An expression for $\tilde\phi$ in terms of $t_i$, $\phi$, and two other invariants is given in [@Budnev] (see (\[phibudnev\]) below) and, in principle, is available in TWOGAM [@twogam]. However, the factor $\sqrt{t_1 t_2}$ appears explicitly in the denominator of $\cos\tilde\phi$ but not in its numerator. Hence, at small values of $-t_i$ this factor will be the result of the cancellation of several much larger terms, rendering this expression for $\cos\tilde\phi$ numerically very unstable. (Recall that $|t_i|_{\min} \sim m^2(W/\sqrt{s})^4 \ll m^2$, while the numerator contains terms of order $s$.) In contrast, we use the numerically stable expression given in (\[phistable\])[^3]. An approximation for $\tilde\phi$ in terms of $\phi$ is proposed in [@Arteaga]: $$\cos\tilde\phi_{{\mathrm{approx}}} = \cos\phi + \sin^2\phi\, \frac{ Q_1 \, Q_2\, (2\, s - s_1 - s_2)} {(W^2 - t_1 - t_2) \sqrt{(s-s_1) (s-s_2)}} \ . \label{phitildeapprox}$$ Indeed, the correlation between $\tilde\phi$ and its approximation is very high in the no-tag case, where, however, the dependence on $\tilde\phi$ is almost trivial (i.e. flat). Figure \[fig:phtnvspht\] exhibits that there is still a correlation for a double-tag mode, but formula (\[phitildeapprox\]) fails to reproduce the correct $\tilde\phi$ dependence: a fit to (\[phitildeform\]) yields $A_1 = 0.084$ and $A_2 = 0.017$, quite different from (\[fitresult\]). Notation and cross sections {#sec:notation} =========================== Consider the reaction $${{{\mathrm{e}}}}^+(p_a) + {{{\mathrm{e}}}}^-(p_b) \rightarrow {{{\mathrm{e}}}}^+(p_1) + X(p_X) + {{{\mathrm{e}}}}^-(p_2) \label{eereact}$$ proceeding through the two-photon process $$\gamma(q_1) + \gamma(q_2) \rightarrow X(p_X) \label{ggreact} \ .$$ The cross section for (\[eereact\]) depends on six invariants, which we choose to be the ${{{\mathrm{e}}}}^+{{{\mathrm{e}}}}^-$ c.m. energy $\sqrt{s}$, the $\gamma\gamma$ c.m. (or hadronic) energy $W$, the photon virtualities $Q_i$, and the subsystem invariant masses $\sqrt{s_i}$: $$\begin{aligned} \,s = (p_a + p_b)^2 & & \quad , \quad W^2 = p_X^2 \ , \nonumber\\ s_1 = (p_1 + p_X)^2 = (p_a + q_2)^2 & & \quad , \quad - Q_1^2 = t_1 = q_1^2 \equiv (p_a - p_1)^2 \ , \nonumber\\ s_2 = (p_2 + p_X)^2 = (p_b + q_1)^2 & & \quad , \quad - Q_2^2 = t_2 = q_2^2 \equiv (p_b - p_2)^2 \ . \label{invdef}\end{aligned}$$ We find it convenient to introduce also the dependent variables: $$\begin{aligned} u_2 = s_1 - m^2 - t_2 & & \quad , \quad \nu = \frac{1}{2}\, \left(W^2 - t_1 - t_2 \right) \ , \nonumber\\ u_1 = s_2 - m^2 - t_1 & & \quad , \quad K = \frac{1}{2\, W}\, \sqrt{ \lambda(W^2,t_1,t_2) } = \frac{1}{W}\, \sqrt{ \nu^2 - t_1\, t_2 } \ , \nonumber\\ \, \beta = \sqrt{1 - \frac{4 m^2}{s}} & & \quad , \quad y_i = \sqrt{1 - \frac{4 m^2}{t_i}} \ , \label{notation}\end{aligned}$$ where $\lambda(x,y,z) = (x-y-z)^2-4yz$ and $m$ denotes the electron mass. Note that $K$ is the photon three-momentum in the $\gamma\gamma$ c.m.s. In terms of these variables the ${{{\mathrm{e}}}}^+{{{\mathrm{e}}}}^-$ cross section at fixed values of $\sqrt{s}$ and $\tau = W^2/s$ is given by: $$\frac{{{{\mathrm{d}}}}\sigma[ {{{\mathrm{e}}}}^+ {{{\mathrm{e}}}}^- \rightarrow {{{\mathrm{e}}}}^+ + {{{\mathrm{e}}}}^- + X]}{{{{\mathrm{d}}}}\tau} = \frac{\alpha^2\, K\,W}{2\, \pi^4\, Q_1^2\, Q_2^2 \beta}\, {{{\mathrm{d}}}}R_3\, \Sigma(W^2,Q_1^2,Q_2^2,s_1,s_2,\tilde{\phi};s,m^2) \ , \label{crossee}$$ where $R_3$ is the phase space for (\[eereact\]). The hadronic physics is fully encoded in five structure functions. Three of these can be expressed through the cross sections $\sigma_{ab}$ for scalar ($a,b=S$) and transverse photons ($a,b=T$) ($\sigma_{ST} = \sigma_{TS}(q_1 \leftrightarrow q_2)$). The other two structure functions $\tau_{TT}$ and $\tau_{TS}$ correspond to transitions with spin-flip for each of the photons with total helicity conservation. Introducing $\tilde{\phi}$, the angle between the scattering planes of the colliding ${{{\mathrm{e}}}}^+$ and ${{{\mathrm{e}}}}^-$ in the [*photon*]{} c.m.s., these structure functions enter the cross sections as: $$\begin{aligned} \Sigma & = & 2\, \rho_1^{++}\, 2\, \rho_2^{++}\, \sigma_{TT} + 2\, \rho_1^{++}\, \rho_2^{00}\, \sigma_{TS} + \rho_1^{00}\, 2\, \rho_2^{++}\, \sigma_{ST} + \rho_1^{00}\, \rho_2^{00}\, \sigma_{SS} \nonumber\\ & & +~ 2\, |\rho_1^{+-}\, \rho_2^{+-}|\, \tau_{TT}\, \cos2\, \tilde{\phi} - 8\, |\rho_1^{+0}\, \rho_2^{+0}|\, \tau_{TS}\, \cos\tilde{\phi} \ . \label{Sigmadef}\end{aligned}$$ The density matrices of the virtual photons in the $\gamma\gamma$-helicity basis are given by $$\begin{aligned} 2\, \rho_1^{++} & = & \frac{\left( u_2 - \nu\right)^2}{K^2\, W^2} + 1 + \frac{4\, m^2}{t_1} \nonumber\\ \rho_1^{00} & = & \frac{\left( u_2 - \nu\right)^2}{K^2\, W^2} - 1 \nonumber\\ |\rho_1^{+-}| & = & \rho_1^{++} - 1 \nonumber\\ |\rho_1^{+0}| & = & \sqrt{\left(\rho_1^{00} + 1\right)|\rho_1^{+-}|} = \frac{u_2 - \nu}{K\, W}\, \sqrt{ \rho_{1}^{++} - 1} \label{rhoidef}\end{aligned}$$ with analogous formulas for photon $2$. A few remarks about the numerical stability of the $\tilde\phi$-dependent terms are in order. Thus far, these terms are implemented solely in the TWOGAM [@twogam] event generator, using the formulas quoted in [@Budnev]. Given in [@Budnev] and coded in [@twogam] are the products $X_2 = 2\, |\rho_1^{+-}\, \rho_2^{+-}|\, \cos2\, \tilde{\phi}$ and $X_1 = 8\, |\rho_1^{+0}\, \rho_2^{+0}|\, \cos\tilde{\phi}$ in terms of invariants. Now, the expressions for $X_i$ contain explicit factors of $t_1 t_2$ ($X_2$) and $\sqrt{t_1 t_2}$ ($X_1$) in the denominators but not in the numerators. Clearly, the evaluation of $X_i$ becomes unstable for small values of $|t_i|$. On the other hand, the factors multiplying $\cos\tilde\phi$ and $\cos 2\tilde\phi$ in $X_i$ approach perfectly stable expressions in the limit $m^2/W^2 \rightarrow 0$ and $t_i/W^2 \rightarrow 0$: $$\begin{aligned} |\rho_1^{+-}| & \rightarrow & \frac{2}{x_1^2}\, (1 - x_1) + \frac{2\, m^2}{t_1} \nonumber\\ |\rho_1^{+0}| & \rightarrow & \frac{2 - x_1}{x_1}\, \sqrt{ |\rho_{1}^{+-}| } \ ,\end{aligned}$$ where $x_i = W^2/s_i \approx s_k/s$ ($i \neq k$). Hence a numerically stable evaluation of $\tilde\phi$ guarantees a correct evaluation of the $\tilde\phi$-dependent terms. The structure functions $\sigma_{ab}$ and $\tau_{ab}$ for lepton-pair production are well quoted in the literature; the formulas of [@Budnev] are implemented in the program. Much less is known about the structure functions for hadronic processes. Since we are not aware of a model for $\tau_{ab}$, the current version of the program assumes $$\tau_{TT} = 0 = \tau_{TS} \ . \label{taudef}$$ The cross sections $\sigma_{ab}$ are uncertain at small values of $Q_i$. Three models for $\sigma_{ab}$ are provided, all based upon the assumption $$\sigma_{ab}(W^2,Q_i^2) = h_a(Q_1^2)\, h_b(Q_2^2)\, \sigma_{\gamma\gamma}(W^2) \ , \label{sigapprox}$$ which is valid for $Q_i^2 \ll W^2$, which is justified in most applications. Note the cross section for the scattering of two real photons $\sigma_{\gamma\gamma}(W^2)$ that enters as a multiplicative factor in (\[sigapprox\]). The three models are defined as follows. The first one is based upon a parametrization [@Bezrukov] of the $\gamma^* {{{\mathrm{p}}}}$ cross section calculated in a model of generalized vector-meson dominance (GVMD): $$\begin{aligned} h_T(Q^2) & = & r\, P_1^{-2}(Q^2) + (1-r)\, P_2^{-1}(Q^2) \nonumber\\ h_S(Q^2) & = & \xi\, \left\{ r\, \frac{Q^2}{m_1^2}\, P_1^{-2}(Q^2) + (1-r)\, \left[ \frac{m_2^2}{Q^2} \, \ln P_2(Q^2) - P_2^{-1}(Q^2) \right] \right\} \nonumber\\ P_i(Q^2) & = & 1 + \frac{Q^2}{m_i^2} \ , \label{GVMDform}\end{aligned}$$ where we take $\xi=1/4$, $r=3/4$, $m_1^2 = 0.54\,$GeV$^2$ and $m_2^2 = 1.8\,$GeV$^2$. The second model [@Sakurai] adds a continuum contribution to simple (diagonal, three-mesons only) vector–meson dominance (VMDc): $$\begin{aligned} h_T(Q^2) & = & \sum_{V=\rho,\omega,\rho}\, r_V\, \left(\frac{m_V^2}{m_V^2 + Q^2}\right)^2 + r_c\, \frac{m_0^2}{m_0^2 + Q^2} \nonumber\\ h_S(Q^2) & = & \sum_{V=\rho,\omega,\rho}\, \frac{\xi\, Q^2}{m_V^2}\, r_V\, \left(\frac{m_V^2}{m_V^2 + Q^2}\right)^2 \ , \label{VMDcform}\end{aligned}$$ where $r_\rho=0.65$, $r_\omega = 0.08$, $r_\phi = 0 .05$, and $r_c = 1 - \sum_V r_V$. Since photon-virtuality effects are often estimated by using a simple $\rho$-pole only, we include also the model defined by ($\rho$-pole): $$h_T(Q^2) = \left(\frac{m_\rho^2}{m_\rho^2 + Q^2}\right)^2 \qquad , \qquad h_S(Q^2) = \frac{\xi\, Q^2}{m_\rho^2}\, \left(\frac{m_\rho^2}{m_\rho^2 + Q^2}\right)^2 \ . \label{rhoform}$$ Since the program is meant to be used at fixed $W$, we take $$\sigma_{\gamma\gamma}(W) = 1 \ .$$ Finally we give the relation between the cross section at fixed values of $\tau$ and $Q_2^2$ and the usual form used in deep-inelastic scattering: $$\frac{{{{\mathrm{d}}}}\sigma}{{{{\mathrm{d}}}}\tau\, {{{\mathrm{d}}}}t_2} = \frac{x^2\, s}{Q_2^2}\, \frac{{{{\mathrm{d}}}}\sigma}{{{{\mathrm{d}}}}x\, {{{\mathrm{d}}}}Q_2^2} \ , \label{DIScross}$$ where $x$ is the Bjorken-$x$ variable defined by $$x = \frac{Q_2^2}{2\, q_1\cdot q_2} = \frac{Q_2^2}{W^2 + Q_2^2 + Q_1^2} \ . \label{xBdef}$$ Phase space {#sec:phase} =========== The phase space can be expressed in terms of four invariants: $$\begin{aligned} {{{\mathrm{d}}}}R_3 & \equiv & \prod_{i=1,2,X}\, \int\, \frac{{{{\mathrm{d}}}}^3 p_i}{2\, E_i}\, \delta^4 \left( p_a + p_b - \sum_i\, p_i \right) \nonumber\\ & = & \frac{1}{16\,\beta\,s}\, \int\,{{{\mathrm{d}}}}t_2\, {{{\mathrm{d}}}}t_1\, {{{\mathrm{d}}}}s_1 \, {{{\mathrm{d}}}}s_2\, \frac{\pi}{\sqrt{ - \Delta_4}} \ ,\end{aligned}$$ where $\Delta_4$ is the $4\times 4$ symmetric Gram determinant of any four independent vectors formed out of $p_a$, $p_b$, $p_1$, $p_X$, $p_2$. The physical region in $t_2$, $t_1$, $s_1$, $s_2$ for fixed $s$ satisfies $\Delta_4 \leq 0$. Since $\Delta_4$ is a quadratic polynomial in any of its arguments, the boundary of the physical region, $\Delta_4 = 0$, is a quadratic equation and has two solutions. Picking $s_2$ as the innermost integration variable, the explicit evaluation of $\Delta_4$ yields $$16\, \Delta_4 = a\,s_2^2 + b\,s_2 + c = a\, (s_2 - s_{2+})\, (s_2 - s_{2-}) \ ,$$ where $$\begin{aligned} a & = & \lambda(s_1, t_2, m^2) \nonumber\\ b & = & -2\,s\ {m}^{2}t_1-2\,{m}^{2}s_1^{2}+8\, t_2\,{m}^{4}-2\,{m}^{2}t_2^{2}-2\,s\ s_1\,{W}^{2} +2\,{m}^{2}s\ {W}^{2}+2\,t_1\,s\ s_1+2\,s\ t_2\,s_1 \nonumber\\ & &~ +4\,{m}^{2}s_1\,{W}^{2}+4\,{m}^{4}s_1+2\,t_1\, t_2\,s-2\,t_2\,{m}^{2}t_1-2\,t_2^{2}s -2\,{m}^{2}t_2\,s+2\,t_1\,t_2\,s_1 \nonumber\\ & &~ -4\,{m}^{4}{W}^{2} -2\,t_1\,s_1^{2}+2\,s\ t_2\,{W}^{2}-2\,{m}^{6} +2\,{m}^{4}t_1 \nonumber\\ c & = & -2\,s\ {m}^{4}{W}^{2}-2\,t_1^{2}{m}^{2}s_1 -2\,t_1\,t_2\,{s}^{2}+2\,s\ t_1\,t_2\,s_1-2\, s\ t_1^{2}s_1+t_1^{2}{s}^{2}+t_1^{2}s_1^{2} +t_2^{2}{s}^{2} \nonumber\\ & &~ +{m}^{4}s_1^{2}+{m}^{4}t_1^{2}-6 \,{m}^{6}t_1-2\,{m}^{6}s_1-4\,{m}^{4}s_1\,{W}^{2}+2\, {m}^{4}t_2\,s+2\,{m}^{4}t_2\,s_1+8\,{m}^{4}t_1\,s_1 \nonumber\\ & &~ -2\,{s}^{2}t_2\,{W}^{2}-2\,t_1\,{s}^{2}{W}^{2}-2\,{m} ^{2}t_1\,s_1^{2}+{m}^{8}-2\,{m}^{2}s\ t_2\,s_1+4\, {m}^{6}{W}^{2}+{m}^{4}t_2^{2} \nonumber\\ & &~ +4\,{m}^{2}t_1\,t_2\,s-2 \,{m}^{2}t_1\,t_2\,s_1-6\,{m}^{6}t_2+{s}^{2}{W}^{4 }+6\,{m}^{2}s\ t_2\,{W}^{2}-4\,s\ {m}^{2}{W}^{4} \nonumber\\ & &~ -2\,s\ {m}^{2}t_1^{2}+2\,s\ t_1\,{m}^{4}-2\,s\ {m}^{2}t_2^{2} +2\,t_1\,t_2\,{m}^{4}+2\,s\ {m}^{2}s_1\,{W}^{2}-4\,s\ t_1\,t_2\, {W}^{2} \nonumber\\ & &~ -2\,s\ t_1\,{m}^{2}s_1+6\,s\ t_1\,{m}^{2}{W}^{2}+2\, s\ t_1\,s_1\,{W}^{2} \ . \label{abcdef}\end{aligned}$$ A numerical stable form for the $s_2$ limits is $$\begin{aligned} s_{2+} & = & \frac{-b + \sqrt{\Delta}}{2\, a} \nonumber\\ s_{2-} & = & \frac{c}{a\, s_{2+}} \ , \label{s2maxmin}\end{aligned}$$ where $\Delta = b^2 - 4\, a\, c$ is given below in a numerically stable form, in (\[Deltadef\]). In order to remove the singularity due to $(-\Delta_4)^{-1/2}$ (in the limit $|t_i|$, $m^2 \ll s_i$, $W^2$, the $s_2$ integration degenerates to an integration over the $\delta$-function $\delta(s_2 - s\, W^2/s_1)$), it is advisable to change variable from $s_2$ to $x_4$, $0 \leq x_4 \leq 1$: $$\begin{aligned} s_2 & = & \frac{1}{2\, a}\, \left\{ - b - \sqrt{\Delta}\, \cos\left( x_4\, \pi\right) \right\} \nonumber\\ \int_{s_{2-}}^{s_{2+}}\, \frac{ {{{\mathrm{d}}}}s_2}{\sqrt{-\Delta_4}} & = & \frac{4\, \pi}{\sqrt{a}}\, \int_0^1\, {{{\mathrm{d}}}}x_4 \ . \label{s2vartrafo}\end{aligned}$$ For later use we also need a numerically stable form of the Gram determinant, which reads $$16\, \Delta_4 = - \frac{\Delta\, \sin^2\left( x_4 \, \pi\right)}{4\, a} \ .$$ The $s_1$-integration limits follow from the requirement $\Delta > 0$. They are most easily derived when realizing that the discriminant $\Delta$ is given as the product of two $3\times 3$ symmetric Gram determinants or, equivalently, the product of two kinematic $G$ functions $$\frac{1}{4}\, \Delta = 4\, G_3\, G_4 = 64\, D_3\, D_4 \ , \label{Deltadef}$$ where $$\begin{aligned} -4\, D_3 \equiv -4\, \Delta_3(p_a,p_b,q_2) & = & G(s, t_2, s_1, m^2, m^2, m^2) \equiv G_3 \nonumber\\ -4\, D_4 \equiv -4\, \Delta_3(p_a,q_1,q_2) & = & G(t_1, s_1, t_2, m^2, m^2, W^2) \equiv G_4 \ . \label{Didef}\end{aligned}$$ Since any $3\times 3$ Gram determinant $\Delta_3$ satisfies $\Delta_3 \geq 0$, the physical region is that where both $G_3$ and $G_4$ are simultaneously negative. Solving $G_i$ for $s_1$ $$\begin{aligned} G_3 & = & m^2\, (s_1 - s_{11+})\, (s_1 - s_{11-}) \nonumber\\ & = & {m}^{2}s_1^{2}-2\,{m}^{4}s_1-s\ t_2\,s_1-3\,{m}^{2}t_2\,s\ +{m}^{6}+t_2\,{s}^{2}+t_2^{2}s \nonumber\\ G_4 & = & t_1\, (s_1 - s_{12+})\, (s_1 - s_{12-}) \nonumber\\ & = & -2\,t_1\,{m}^{2}s_1-t_2\,{m}^{2}t_1+{m}^{4}t_1-{m}^{2}{W}^{2}t_1 +{m}^{2}t_2^{2}+t_2\,{W}^{2}t_1-t_1\,s_1\,{W}^{2} \nonumber\\ & &~ -2\,{m}^{2}t_2\,{W}^{2}+{m}^{2}{W}^{4}+t_1\,s_1^{2}+t_1^{2} s_1-t_1\,t_2\,s_1 \label{Gidef}\end{aligned}$$ we find $$\begin{aligned} s_{11\pm}& = & {\frac {t_2\,S+2\,{m}^{4}\pm \sqrt {\lambda(S,{m}^{2} ,{m}^{2})\lambda(t_2,{m}^{2},{m}^{2})}}{2\,{m}^{2}}} \nonumber\\ s_{12\pm} & = & {\frac {t_2}{2}}+{m}^{2}+{\frac {{W}^{2}}{2}}-{ \frac {t_1}{2}} \pm {\frac {\sqrt {\lambda(t_1,t_2,{W}^{2}) \lambda(t_1,{m}^{2},{m}^{2})}}{2\,t_1}} \nonumber\\ s_{11+} s_{11-} & = & {\frac {t_2\,S\left (-3\,{m}^{2}+S+t_2 \right )}{{m}^{2}}}+{m}^{4} \nonumber\\ s_{12+} s_{12-} & = & \left( W^2 - m^2 \right)\, \left (-{m}^{2} +t_2\right )+{\frac {{m}^{2}\left ({W}^{2}-t_2\right )^{2}} {t_1}} \ . \label{sijdef}\end{aligned}$$ Note that $s_{12+} \leq s_{12-}$. Since $G_3$ is always negative between its two roots, the range of integration over $s_1$ is $s_{12-} \leq s_1 \leq s_{11+}$. Numerically it is more advantageous to calculate the limits as $$\begin{aligned} s_{1\min} & = & s_{12-} = m^2 + \frac{1}{2}\, \left(W^2-t_1+t_2+y_1\ \sqrt{ \lambda(W^2,t_1,t_2)} \right) \nonumber\\ s_{1\max} & = & s_{11+} = m^2 + \frac{2\ (s+t_2-4\ m^2)}{1+\beta\ y_2} \ . \label{s1minmax}\end{aligned}$$ The dominant behaviour of the $s_1$ integration is given by the factor $\lambda^{-1/2}(s_1,t_2,m^2)$, see (\[s2vartrafo\]). (In the limit $t_2$, $m^2 \ll s_1$, this becomes ${{{\mathrm{d}}}}s_1/s_1$ integration.) This factor can be transformed away by the variable transformation from $s_1$ to $x_3$, $0 \leq x_3 \leq 1$, $$\begin{aligned} s_1 & = & X_{1}/2 + m^2 + t_2 + 2\, m^2\, t_2/X_{1} \nonumber\\ X_{1} & = & (\nu + K\, W)\, (1+y_1)\, \exp{(\delta_1\, x_3)} \nonumber\\ \delta_1 & = & \ln\frac{s\, (1+\beta)^2} {(\nu+K\, W)\, (1+y_1)\, (1+y_2)} \ , \label{s1vartrafo}\end{aligned}$$ such that $$\int_{s_{1\min}}^{s_{1\max}}\, {{{\mathrm{d}}}}s_1\, \frac{4\,\pi}{\sqrt{a}} = 4\, \pi \, \delta_1\, \int_0^1\, {{{\mathrm{d}}}}x_3 \ . \label{x3def}$$ The physical region in the $t_1$–$t_2$ plane is defined by the requirement $G_i < 0$ for all $s_1$ values between the limits $(m+W)^2 \leq s_1 \leq (\sqrt{s}-m)^2$. Since for the reaction considered here the masses of the particles involved are such that the values $t_1 = (m_a - m_1)^2$, $t_2 = (m_b - m_2)^2$ cannot be reached and $t_2$ is never larger than zero, the boundary curve in the $t_1$–$t_2$ plane is simply given by $s_{12-} = s_{11+}$. Equivalently, the $t_1$ limits can be found by solving $G_4=0$ with $s_1 = s_{11+}$ for $t_1$: $$t_{1\min} = - \frac{1}{2}\, \left( \frac{b_1}{a_1} + \Delta t_1 \right) \qquad , \qquad t_{1\max} = \frac{ c_1}{a_1\ t_{1min} } \ , \label{t1maxmin}$$ where $$\begin{aligned} \Delta t_1 & = & \frac{ \sqrt{ \Delta_1}}{a_1} \nonumber\\ a_1 & = & 2\ (Q+t_2+2\ m^2+W^2) \nonumber\\ b_1 &= & Q^2-W^4+2\ W^2\ t_2-t_2^2-8\ m^2\ t_2-8\ m^2\ W^2 \nonumber\\ c_1 & = & 4\ m^2\, (W^2-t_2)^2 \nonumber\\ \Delta_1 & \equiv & b_1^2 - 4\, a_1\, c_1 \nonumber\\ & = & (Q+t_2-W^2+4\ m\ W)\, (Q+t_2-W^2-4\ m\ W) \nonumber\\ & &~ (Q^2-2\ Q\ t_2+2\ Q\ W^2+t_2^2+W^4-16\ m^2\ t_2-2\ W^2\ t_2) \nonumber\\ Q & = & \frac{1}{m^2}\, \left\{ t_2\,s-{m}^{2}t_2-{m}^{2}{W}^{2}+ \sqrt {\lambda(s,{m}^{2},{m}^{2})\lambda(t_2,{m}^{2},{m}^{2})} \right\} \nonumber\\ & = & \frac{ 4\ (s+t_2-4\ m^2)}{1+\beta\ y_2} - t_2 - W^2 \ . \label{t1hilf}\end{aligned}$$ Finally, the $t_2$-integration limits follow from requiring $\Delta_1 \geq 0$: $$\Delta_1 = \frac{ (t_2 - t_{21+})\, (t_2 - t_{21-})}{F_1} \, \frac{ (t_2 - t_{22+})\, (t_2 - t_{22-})}{F_2} \, \frac{ t_2\, (t_2 - t_{23})^2}{F_3} \ ,$$ where $$\begin{aligned} F_1& = & {\frac {4\,s}{t_2\,s\beta\,y_2+t_2\,s-2\,{W} ^{2}{m}^{2}+4\,{m}^{3}W}} \nonumber\\ F_2& = & {\frac {4\,s}{t_2\,s\beta\,y_2+t_2\,s-2\,{W} ^{2}{m}^{2}-4\,{m}^{3}W}} \nonumber\\ F_3& = & - 16\, m^4 / \left\{ -2\,t_2\,{s}^{2}\beta\,y_2+ 4\,t_2\,s\beta\,y_2\,{m}^{2}-t_2\,{s}^{2}-t_2\,{s} ^{2}{\beta}^{2}+4\,t_2\,s{m}^{2} \right. \nonumber\\ & &~ \left. +4\,{s}^{2}{\beta}^{2}{m}^{2}-4\, t_2\,{m}^{4}+16\,{m}^{6} \right\} \nonumber\\ t_{21\pm}& = & -{\frac {2\,mW-{W}^{2}+{s}-4\,{m}^{2}\pm \beta\, \sqrt {\left (s-{W}^{2}\right ) \left( s - W_-^2 \right) }}{2}} \nonumber\\ t_{22\pm}& = & -{\frac {-2\,mW-{W}^{2}+{s}-4\,{m}^{2} \pm \beta\, \sqrt { \left (s-{W}^{2}\right ) \left( s - W_+^2 \right) }}{2}} \nonumber\\ t_{23}& = & {\frac {\left (s-2\,{m}^{2}\right )^{2}}{{m}^{2}}} \nonumber\\ W_{\pm} & = & W \pm 2\, m \ .\end{aligned}$$ Equivalently, they are arrived at by solving $G_3=0$ with $s_1 = (m+W)^2$ for $t_2$: $$\begin{aligned} t_{2\min} & = & t_{22+} = -\frac{1}{2} \left(s - W^2 - 2\ m\ W - 4\ m^2 + \Delta t_2 \right) \nonumber\\ t_{2\max} & = & t_{22-} = \frac{m^2\ W^2\ W_+^ 2}{s\ t_{2\min}} \ , \label{t2maxmin}\end{aligned}$$ where $$\Delta t_2 = \beta \sqrt{(s - W^2)(s - W_+^2)} \ . \label{t2hilf}$$ The phase space finally becomes $${{{\mathrm{d}}}}R_3 = \frac{\pi^2}{4\, \beta\, s}\, \int_{t_{2\min}}^{t_{2\max}}\, {{{\mathrm{d}}}}t_2\ \int_{t_{1\min}}^{t_{1\max}}\, {{{\mathrm{d}}}}t_1\ \delta_1(t_1,t_2)\, \int_0^1 {{{\mathrm{d}}}}x_3\, \int_0^1 {{{\mathrm{d}}}}x_4 \ . \label{newdR3}$$ The dominant $t_i$ behaviour is taken into account through a logarithmic mapping, so that we end up with a cross section of the form $$\begin{aligned} \frac{{{{\mathrm{d}}}}\sigma}{{{{\mathrm{d}}}}\tau} & = & \prod_{i=1}^{4}\, \int {{{\mathrm{d}}}}x_i\ F(x_i) \nonumber\\ & \equiv & \prod_{i=1}^{4}\, \int {{{\mathrm{d}}}}x_i\ \ln\frac{t_{2\max}}{t_{2\min}}\ \ln\frac{t_{1\max}}{t_{1\min}}\ \ln\frac{s\, (1+\beta)^2}{(\nu + K\, W)\, (1+y_1)\, (1+y_2)}\ \frac{\alpha^2\, K\, W}{8\, \pi^2\, \beta^2\, s}\ \Sigma \ . \label{Fdef}\end{aligned}$$ Equivalent-photon approximation {#sec:EPA} =============================== An approximation is arrived at by neglecting as much as possible the electron-mass and $t_i$ dependences in the kinematics, but keeping the full dependence on $W$ and $Q_i$ in the hadronic cross sections $\sigma_{ab}(W^2,Q_1^2,Q_2^2)$ [@GAS]: $$\begin{aligned} \frac{{{{\mathrm{d}}}}\sigma}{{{{\mathrm{d}}}}\tau} & = & \int_{W^2}^{s}\, \frac{{{{\mathrm{d}}}}s_1}{s_1}\ \int_{t_{2a}}^{t_{2b}}\, \frac{{{{\mathrm{d}}}}t_2}{t_2}\ \int_{t_{1a}}^{t_{1b}}\, \frac{{{{\mathrm{d}}}}t_1}{t_1}\ \int_{W^2}^{s}\, {{{\mathrm{d}}}}s_2\ \delta \left( s_2 - \frac{s\, W^2}{s_1} \right)\ \frac{\alpha^2\, W^2}{16\, \pi^2\, s}\, \nonumber\\ & &~ \left\{ 2\, \rho_{1{\mathrm{approx}}}^{++}\, 2\, \rho_{2{\mathrm{approx}}}^{++}\, \sigma_{TT} + 2\, \rho_{1{\mathrm{approx}}}^{++}\, \rho_{2{\mathrm{approx}}}^{00}\, \sigma_{TS} \right. \nonumber\\ & &~~ \left. + \rho_{1{\mathrm{approx}}}^{00}\, 2\, \rho_{2{\mathrm{approx}}}^{++}\, \sigma_{ST} + \rho_{1{\mathrm{approx}}}^{00}\, \rho_{2{\mathrm{approx}}}^{00}\, \sigma_{SS} \right\} \ . \label{approxcross}\end{aligned}$$ The integration limits are given by: $$\begin{aligned} t_{ia} & = & - \frac{m^2\, x_i^2}{1 - x_i} - (1 - x_i)\, \sin^2 \frac{\theta_{i\max}}{2} \nonumber\\ t_{ib} & = & - \frac{m^2\, x_i^2}{1 - x_i} - (1 - x_i)\, \sin^2 \frac{\theta_{i\min}}{2} \ , \label{tiapproxlimits}\end{aligned}$$ where $x_1 = s_2/s$ and $x_2 = s_1/s$. The approximate forms of the photon density matrices read: $$\begin{aligned} 2\, \rho_{1{\mathrm{approx}}}^{++} & = & \frac{2}{x_1^2}\, \left\{ 1 + (1-x_1)^2 - \frac{2\, m^2\, x_1^2}{Q_1^2} \right\} \nonumber\\ \rho_{1{\mathrm{approx}}}^{00} & = & \frac{4}{x_1^2}\, \left( 1 - x_1 \right) \ .\end{aligned}$$ Momenta {#sec:momenta} ======= Here we present the particle momenta in the laboratory frame. The particle energies follow simply from $E_i = (p_a + p_b)\cdot p_i /\sqrt{s}$: $$\begin{aligned} E_1 & = & \frac{s + m^2 - s_2}{2\, \sqrt{s}} \nonumber\\ E_2 & = & \frac{s + m^2 - s_1}{2\, \sqrt{s}} \nonumber\\ E_X & = & \frac{s_1 + s_2 - 2\, m^2}{2\, \sqrt{s}}\end{aligned}$$ and the moduli of the three-momenta from $P_i^2 = E_i^2 - m_i^2$. The polar angles $\theta_i$ with respect to the beam axis could be calculated from $p_b \cdot p_i = E_b\, E_i - P_b\, P_i\, \cos\theta_i$ $$\begin{aligned} \cos\theta_1 & = & \frac{s- s_2+ 2\, t_1 - 3\, m^2} {2\,\beta\, \sqrt{s}\, P_1} \nonumber\\ \cos\theta_2 & = & \frac{s- s_1+ 2\, t_2 - 3\, m^2} {2\, \beta\, \sqrt{s}\, P_2} \nonumber\\ \cos\theta_X & = & - \frac{s_2-s_1+2\, (t_2-t_1)} {2\, \beta\, \sqrt{s}\, P_X} \ . \label{polarcosdef}\end{aligned}$$ Typically, the polar angles are very small and it is better to calculate them in a numerically stable form from $$\begin{aligned} \sin\theta_1 & = & \frac{2\, \sqrt{D_1}}{s\, \beta\, P_1} \nonumber\\ \sin\theta_2 & = & \frac{2\, \sqrt{D_3}}{s\, \beta\, P_2} \nonumber\\ \sin\theta_X & = & \frac{2\, \sqrt{D_5}}{s\, \beta\, P_X} \ .\end{aligned}$$ Equations (\[polarcosdef\]) are then only used to resolve the ambiguity $\theta_i \leftrightarrow \pi - \theta_i$. The quantity $D_3$ is defined in (\[Didef\]–\[sijdef\]); $D_1$ is obtained from $D_3$ by the interchange $t_1 \leftrightarrow t_2$ and $s_1 \leftrightarrow s_2$. The same interchange relates $D_2$, needed below, with $D_4$, given in (\[Didef\]–\[sijdef\]). Furthermore, we have: $$\begin{aligned} D_5 & = & D_1 + D_3 + 2\, D_6 \nonumber\\ D_6 & = & \frac{s}{8}\, \left[ -(s-4\, m^2)\, (W^2-t_1-t_2) +(s_1-t_2-m^2)\, (s_2-t_1-m^2)+t_1\, t_2 \right] \nonumber\\ & &~ - \frac{m^2}{4}\, (s_1-m^2)\, (s_2-m^2) \ .\end{aligned}$$ The polar angles $\phi_1$ ($\phi_2$) between the ${{{\mathrm{e}}}}^+$ (${{{\mathrm{e}}}}^-$) plane and the hadronic plane and the polar angle $\phi$ between the two lepton planes in the ${{{\mathrm{e}}}}^+{{{\mathrm{e}}}}^-$ c.m.s. are again best calculated using the numerically more stable form for the sinus function $$\begin{aligned} \cos\phi & = & \frac{D_6}{\sqrt{D_1\, D_3}} \nonumber\\ \sin\phi & = & \frac{s\, \beta\, \sqrt{-\Delta_4}}{2\, \sqrt{D_1\, D_3}} \nonumber\\ \sin\phi_1 & = & \frac{2\, \sqrt{-\Delta_4}} {s\, \beta\, P_X\, \sin\theta_X\, P_1\, \sin\theta_1} \nonumber\\ \sin\phi_2 & = & \frac{2\, \sqrt{-\Delta_4}} {s\, \beta\, P_X\, \sin\theta_X\, P_2\, \sin\theta_2} \nonumber\\ \cos\phi_1 & = & \frac{D_1+D_6}{\sqrt{D_1\, D_5}} \nonumber\\ \cos\phi_2 & = & \frac{D_3+D_6}{\sqrt{D_3\, D_5}} = \frac{ \sqrt{D_3} + \sqrt{D_1}\, \cos\phi} {\sqrt{D_3 + D_1 + 2\, \sqrt{D_1\, D_3}\, \cos\phi} } \ .\end{aligned}$$ An expression for the azimuthal angle between the lepton planes in the $\gamma\gamma$ c.m.s. can be deduced from the formulas given in [@Budnev]: $$\cos\tilde{\phi} = \frac{-2\, s+u_1+u_2-\nu+4\, m^2 + \nu(u_2-\nu) \, (u_1-\nu)/(K^2\, W^2)}{\sqrt{ t_1 \, t_2\, \left( 2\, \rho_1^{++}-2\right)\, \left( 2\, \rho_2^{++}-2\right)} } \ . \label{phibudnev}$$ Numerically more stable is the following form $$\sin\tilde{\phi} = \frac{K\, W\, \sqrt{- \Delta_4}} {\sqrt{D_2\, D_4}} \qquad {\mathrm{and}} \qquad \cos\tilde{\phi} = \frac{D_7}{\sqrt{D_2\, D_4}} \ , \label{phistable}$$ where $$\begin{aligned} 16\, D_7 & = & 2\,W^{2}\left (s_1\,s_2-sW^{2}\right ) -\, 2\,t_1\,\left (-t_1\,s_1+st_1+s_1\,s_2+ W^{2}s_1-2\,sW^{2}\right ) \nonumber\\ & &~ -\, 2\,t_2\,\left (-t_2\,s_2+t_2\,s+s_1\,s_2 -2\,sW^{2}+s_2\,W^{2}\right ) +\, 2\,t_1\,t_2\,\left (-s_1+2\,s+2\,W^{2}-s_2 \right ) \nonumber\\ & &~ -\, 2\,{m}^{2}\left ({m}^{2}t_2-{t_2}^{2}-{m}^{2}W^{2} +{m}^{2}t_1-2\,W^{4}-{t_1}^{2}+W^{2}s_1+2\,t_1\,t_2 \right. \nonumber\\ & &~ \left. +\, 3\,t_1\,W^{2}-t_1\,s_1+3\,t_2\,W^{2}- t_2\,s_2-t_2\,s_1+s_2\,W^{2}-t_1\,s_2 \right ) \ .\end{aligned}$$ A numerically stable relation between $\phi$ and $\tilde{\phi}$ at $-t_i$, $m^2 \ll W^2$ is provided by $$\begin{aligned} 4\, s^2 \, D_2 & = & 4\, s_2^2\, D_3 +\, 2\,t_2\,{s}^{2}r_2\,\cos\phi\,s_2 -\, 2\,t_1\,t_2\,ss_2\,\left (s-s_1\right ) \nonumber\\ & &~ -\, st_1\,t_2\,\left (-2\,t_1\,t_2-st_1+t_1 \,s_1+3\,t_2\,s_2-t_2\,s+2\,r_2\,\cos\phi \,s\right ) \nonumber\\ & &~ -\, 4\,{m}^{2}sr_2\,\cos\phi\,s_1\,s_2 +\, O\left( m^4\, s_1^2\, s_2^2/s, m^2\, t_i\, s\, s_1\, s_2 \right) \ , \label{numeristableone}\end{aligned}$$ an analogous expression for $D_4$, and $$\begin{aligned} 16\, s\, \sqrt{D_2 D_4}\cos\tilde{\phi} & = & 16\, W^2 \sqrt{D_1 D_3}\cos\phi -\, 4\,t_2\,{s}^{2}r_2\,\cos\phi -\, 4\,t_1\,{s}^{2}r_2\,\cos\phi \nonumber\\ & &~ +\, 4\,t_1\,t_2\,s\left (-s_1+2\,s-s_2+t_1 +t_2\right ) +\, 8\,{m}^{2}\cos\phi\,r_2\,s_1\,s_2 \nonumber\\ & &~ + \frac {2\,{m}^{2}t_2}{s}\,\left (-t_2\,{s}^{2}+4\,st_2\, s_2-2\,t_2\,{s_2}^{2}-4\,ss_1\,s_2+2\,s_1 \,{s_2}^{2} \right. \nonumber\\ & &~~ \left. -\, 8\,r_2\,\cos\phi\,ss_2 + {s}^{2}s_1+ {s}^{2}s_2+8\,r_2\,\cos\phi\,{s}^{2}\right ) \nonumber\\ & &~ +\, \frac {2\,{m}^{2}t_1}{s}\,\left (-t_1\,{s}^{2}+4\,t_1\,s s_1-2\,t_1\,{s_1}^{2}-4\,ss_1\,s_2+2\, {s_1}^{2}s_2 \right. \nonumber\\ & &~~ \left. +\, 8\,r_2\,\cos\phi\,{s}^{2} - 8\,r_2\, \cos\phi\,ss_1+{s}^{2}s_1+{s}^{2}s_2\right ) \nonumber\\ & &~ +\, O\left( m^4\, s_i^2\, s_j/s, m^2\, t_1\, t_2\, s\right) \ , \label{numeristabletwo}\end{aligned}$$ where $$r_2^2 = \left[ m^2\, \left({s_1\over s}\right)^2 + t_2\, \left(1+{t_2\over s}-{s_1\over s}\right) \right] \, \left[ m^2\, \left({s_2\over s}\right)^2 + t_1\, \left(1+{t_1\over s}-{s_2\over s}\right) \right] \ .$$ For $m\rightarrow 0$ and $t_i/W^2 \rightarrow 0$, (\[numeristableone\]) and (\[numeristabletwo\]) lead to the approximate relation (\[phitildeapprox\]). The four-momenta are now given by $$\begin{aligned} p_a & = & \frac{1}{2}\, \sqrt{s}\, (1, 0, 0, -\beta) \nonumber\\ p_b & = & \frac{1}{2}\, \sqrt{s}\, (1, 0, 0, \beta) \nonumber\\ p_1 & = & (E_1, - P_1\, \sin\theta_1\, \cos\phi_1, - P_1\, \sin\theta_1\, \sin\phi_1, - P_1\, \cos\theta_1) \nonumber\\ p_2 & = & (E_2, - P_2\, \sin\theta_2\, \cos\phi_2, P_2\, \sin\theta_2\, \sin\phi_2, P_2\, \cos\theta_2) \nonumber\\ p_X & = & (E_X, P_X\, \sin\theta_X, 0, P_X\, \cos\theta_X) \ .\end{aligned}$$ Experimental cuts {#sec:cuts} ================= If cuts on the angle $\theta_2$ and the energy $E_2$ of the scattered electron are applied, the $(s_1,t_2)$-integration region shrinks as follows (see Fig. \[fig:phasespace\]): $$\begin{aligned} s_{1{{{\mathrm{low}}}}} & = & \min\, \left\{ (m+W)^2, m^2 + s\, \left( 1 - \frac{2\, E_{2\max}}{\sqrt{s}} \right) \right\} \nonumber\\ s_{1{{{\mathrm{upp}}}}} & = & \max\, \left\{ (\sqrt{s}-m)^2, m^2 + s\, \left( 1 - \frac{2\, E_{2\min}}{\sqrt{s}}\right) \right\}\end{aligned}$$ and $$T_2(s_1,\theta_{2\max}) < t_2 < T_2(s_1,\theta_{2\min}) \ ,$$ where $$\begin{aligned} T_2(s_1,\theta_2) & = & \frac{1}{2}\, \left( 3\, m^2 - s + s_1 + \beta\, \cos\theta_2\, \sqrt{\lambda(s,s_1,m^2)} \right) \nonumber\\ & = & - \frac{2\, m^2\, (s_1 - m^2)^2}{s\, \left[ \beta\, \lambda^{1/2} (s,s_1,m^2) + s - s_1 - 3\, m^2 \right]} - \beta\, \lambda^{1/2}(s,s_1,m^2)\, \sin^2\frac{\theta_2}{2} \nonumber\\ & \rightarrow & - \left\{ \frac{m^2\, x_2^2}{1-x_2} + s\, (1-x_2) \, \sin^2\frac{\theta_2}{2} \right\} \ .\end{aligned}$$ The approximate form holds for $m^2 \ll s_1$ and a small angle $\theta_2$ and is used in (\[tiapproxlimits\]). If, as in our case, $t_2$ is the outer integration, then its lower limit becomes $$t_{2\min} = \min\, \left\{ T_2(s_{1{{{\mathrm{upp}}}}},\theta_{2\max}), T_2(s_{1{{{\mathrm{low}}}}},\theta_{2\max}) \right\} \ , \label{t2mincut}$$ while the upper limit is more complicated $$\begin{aligned} t_{2\max} = & T_2(s_{1{{{\mathrm{upp}}}}},\theta_{2\min}) \qquad & \hat{s}_1 > s_{1{{{\mathrm{upp}}}}} \nonumber\\ = & T_2(s_{1{{{\mathrm{low}}}}},\theta_{2\min}) \qquad & \hat{s}_1 < s_{1{{{\mathrm{low}}}}} \nonumber\\ = & \hat{t}_2 \qquad & s_{1{{{\mathrm{low}}}}} < \hat{s}_1 < s_{1{{{\mathrm{upp}}}}} \ , \label{t2maxcut}\end{aligned}$$ where $$\begin{aligned} \hat{s}_1 & = & s + m^2 - \frac{2\,m\, \sqrt{s}}{\sqrt{X}} \nonumber\\ & = & m^2 + \frac{s\, \beta^2\, \sin^2\theta_2} {X\, \left( 1 + 2\, m\, /\sqrt{s\, X} \right) } \nonumber\\ \hat{t}_2 & = & 2\, m^2 - m\, \sqrt{s\, X} \qquad (\theta_2 < \pi/2) \nonumber\\ & = & 2\, m^2 - m\, \sqrt{s}\, \frac{1 + \beta^2\, \cos^2\theta_2} {\sqrt{X}} \qquad (\theta_2 > \pi/2) \nonumber\\ X & = & \frac{4\, m^2}{s} + \beta^2\, \sin^2\theta_2 \ . \label{cuth1} \end{aligned}$$ The $s_1$-integration range is a rather complicated function of $t_2$ and may even consist of two separated ranges (Fig. \[fig:phasespace\]). Moreover, the $s_1$-integration range is affected by $t_1$ and cuts on $E_1$ and $\theta_1$. Then it is better to use the Monte Carlo method. In any case, since the $t_i$ integration are the most singular ones, the most important constraints are taken into account through (\[t2mincut\]) and (\[t2maxcut\]) and the analogous formulas for $t_1$. Details of the program {#sec:program} ====================== Common blocks ------------- The user can decide whether to keep $t_2$ at a fixed, user-defined value or to integrate over $t_2$, i.e. to calculate (\[crossee\]) or (\[DIScross\]). In the case of integration over all variables, the user can choose between the exact or an approximate treatment (\[approxcross\]) of the kinematics. Common /ggLapp/ t2user,iapprx,ivegas,iwaght ------------ ----------------------------------------------------------------- ` t2user ` Fixed value of $t_2$ chosen by user for ` iapprx = 1. ` ` iapprx ` $=1$: $t_2$ is kept fixed at the user value; $=2$: approximate kinematics is used, $t_2$ is integrated over; $=0$: all variables are integrated using exact kinematics. ` ivegas ` $=1$: VEGAS integration; $=0$: Simple integration. ` ivegas ` $=1$: Unweighted events, i.e. ` Weight` $=1$; $=0$: Weighted events \[2ex\] ------------ ----------------------------------------------------------------- Cuts on the scattered leptons are set in Common /ggLcut/th1min,th1max,E1min,E1max,th2min,th2max,E2min,E2max ------------------- ------------------------------------------------------------------------- ` th1min,th1max ` Minimum and maximum scattering angles of scattered ${{{\mathrm{e}}}}^+$ w.r.t. direction of incident ${{{\mathrm{e}}}}^+$. ` th2min,th2max ` Minimum and maximum scattering angles of scattered ${{{\mathrm{e}}}}^-$ w.r.t. direction of incident ${{{\mathrm{e}}}}^-$. Tighter cuts should be applied to the ${{{\mathrm{e}}}}^-$. ` E1min,E1max ` Minimum and maximum energies of scattered ${{{\mathrm{e}}}}^+$. ` E2min,E2max ` Minimum and maximum energies of scattered ${{{\mathrm{e}}}}^-$. \[2ex\] ------------------- ------------------------------------------------------------------------- Models for the $\gamma^\star\gamma^\star$ cross sections and their parameters are chosen in Common /ggLmod/ imodel ----------------------- --------------------------------------------------- ` imodel` $=~1\qquad$ GVMD model (\[GVMDform\]) ` imodel` $=~2$ VMDc model (\[VMDcform\]) ` imodel` $=20$ $\rho$-pole model (\[rhoform\]) ` imodel` $=~0$ $\rho$-pole model (\[rhoform\]) with $h_S(Q^2)=0$ ` imodel` $=~3$ Exact cross section for lepton-pair production. ----------------------- --------------------------------------------------- Common /ggLhad/ r,xi,m1s,m2s,rrho,romeg,rphi,rc,mrhos, & momegs,mphis,mzeros Parameters for (\[GVMDform\]–\[rhoform\]): $r$, $\xi$, $m_1^2$, $m_2^2$, $r_\rho$, $r_\omega$, $r_\phi$, $r_c$, $m_\rho^2$, $m_\omega^2$, $m_\phi^2$, $m_0^2$.\ The integration variables and the particle momenta are stored in Common /ggLvar/ &yar(4),t2,t1,s1,s2,E1,E2,EX,P1,P2,PX,th1,th2,thX,phi1,phi2,phi,pht ----------------------- -------------------------------------------------------------- ` yar(i) ` Integration variables for VEGAS. ` t2,t1,s1,s2 ` Invariants $t_2$, $t_1$, $s_1$, $s_2$. ` E1,E2,EX ` Energies $E_1$, $E_2$, $E_X$. ` P1,P2,PX ` Three-momenta $P_1$, $P_2$, $P_X$. ` th1,th2,thX ` Polar angles $\theta_1$, $\theta_2$, $\theta_X$. ` phi1,phi2,phi,pht ` Azimuthal angles $\phi_1$, $\phi_2$, $\phi$, $\tilde{\phi}$. ----------------------- -------------------------------------------------------------- Common /ggLvec/ mntum(7,5) Particle four-momenta ` mntum(i,k): ` $k=1 \ldots 5$ for $p_x$, $p_y$, $p_z$, $E$, ${{\mathrm{sign}}}(p^2) \times \sqrt{|p^2|}$; $i=1 \ldots 7$ for incident ${{{\mathrm{e}}}}^+$, incident ${{{\mathrm{e}}}}^-$, photon from ${{{\mathrm{e}}}}^+$, photon from ${{{\mathrm{e}}}}^-$, scattered ${{{\mathrm{e}}}}^+$, scattered ${{{\mathrm{e}}}}^-$, hadronic system $X$.\ Parameter for the simple integration and results of the integration and event generation are stored in Common /ggLuno/ cross,error,Fmax,Fmin,Weight,npts,nzero,ntrial ------------ ---------------------------------------------------------------- ` cross ` Estimate of luminosity. ` error ` Estimate of error on luminosity. ` Fmax ` Maximum function value, calculated in `ggLcrs`; checked in `ggLgen`. ` Fmin ` Minimum function value, calculated in ` ggLuF. ` ` Weight ` Weight if weighted events requested. ` npts ` Number of function evaluations for simple integration. ` nzero ` Number of cases where function was put to zero in ` ggLuF` because it failed the cuts; initialized to zero in ` ggLcrs`, ` ggLgen`. ` ntrail ` Number of trials necessary in ` ggLgen ` to generate an event; incremented by each call. \[2ex\] ------------ ---------------------------------------------------------------- Parameters for the VEGAS integration are set in Common /ggLvg1/ xl(10),xu(10),acc,ndim,nfcall,itmx,nprn ------------ ----------------------------------------------------------------------- ` acc ` VEGAS accuracy (Default (D): $10^{-4}$). ` ndim ` Number of integration variables (D: $4$). ` nfcall ` Maximum number of function calls per iteration for VEGAS (D: $10^5$). ` itmx ` Number of iterations for VEGAS (D: $4$). ` nprn ` Print flag for VEGAS (D: $2$). \[2ex\] ------------ ----------------------------------------------------------------------- Additional common blocks Common /ggLprm/ s,roots,Whad,m,Pi,alem ----------- ------------------------------------------------------------ ` s ` Overall c.m. energy square $s$. ` roots ` Overall c.m. energy $\sqrt{s}$ (twice the beam energy), set by user through call to ` ggLcrs. ` ` Whad ` Hadronic mass $W$, set by user through call to ` ggLcrs. ` ` m ` Electron mass (D: $511\,$keV). ` Pi ` $\pi$ ` alem ` $\alpha_{{{\mathrm{em}}}}$ (D: $1/137$). ----------- ------------------------------------------------------------ Common/ggLvg2/XI(50,10),SI,SI2,SWGT,SCHI,NDO,IT Common /ggLerr/ & it1,iD1,iD3,iD5,itX,iph,ip1,ip2,ia1,ia2,ia3,ia4,ie1,ie2,ipt,is Block Data ggLblk Subroutines ----------- -------------------------- ---------------------------------------------------------------------------------------------------------- ` ggLcrs(rs,W) ` Integrates ${{{\mathrm{d}}}}\sigma/{{{\mathrm{d}}}}\tau$ and finds ` Fmax`; ` rs` $=\sqrt{s}$, ` W` $=W$ \[1ex\] ` ggLmom ` Builds up four-momenta. \[1ex\] ` ggLprt ` Prints four-momenta and checks momentum sum. \[1ex\] ` ggLgen(Flag) ` Generates one event; `Flag=F` if a new maximum is found; then it is advisable to restart event generation with adjusted maximum. -------------------------- ---------------------------------------------------------------------------------------------------------- Double-precision functions -------------------------- --------------------------------------- ---------------------------------------- ` ggLint(W2,m2,Q1s,Q2s,s1,s2,phi,s) ` $\Sigma$ as defined in (\[Sigmadef\]). \[1ex\] --------------------------------------- ---------------------------------------- -------------------------------- --------------------------------------------------------------- ` ggLuF(xar,wgt) ` $F(x_i)$ as defined in (\[Fdef\]). \[1ex\] ` ggLhTT(W2,Q1s,Q2s) ` $\sigma_{TT}(W^2,Q_1^2,Q_2^2)$ \[1ex\] ` ggLhTS(W2,Q1s,Q2s) ` $\sigma_{TS}(W^2,Q_1^2,Q_2^2)$ \[1ex\] ` ggLhSS(W2,Q1s,Q2s) ` $\sigma_{SS}(W^2,Q_1^2,Q_2^2)$ \[1ex\] ` ggLrTS(W2,Q1s,Q2s) ` $\tau_{TS}(W^2,Q_1^2,Q_2^2)$ \[1ex\] ` ggLrTT(W2,Q1s,Q2s) ` $\tau_{TT}(W^2,Q_1^2,Q_2^2)$ \[1ex\] ` ggLhT(Qs) ` $h_{T}(Q^2)$ \[1ex\] ` ggLhS(Qs) ` $h_{S}(Q^2)$ \[1ex\] ` ggLgg(W2) ` $\sigma_{\gamma\gamma}(W^2)$ \[1ex\] ` ggLuG(z) ` Makes the variable transformation from $x_i$ in (\[Fdef\]) to those used by the simple or VEGAS integration. -------------------------------- --------------------------------------------------------------- Excerpt from the demonstration program -------------------------------------- * Initialize the random number generator RanLux Call rLuxGo(3,314159265,0,0) * * Initialize GALUGA; get luminosity within cuts Call ggLcrs(rs,W) * * Initialize plotting Call User(0) * * Timing: Call Timex(time1) Call rLuxGo(3,314159265,0,0) * * Event loop Do 10 i=1,Nev Call ggLgen(Flag) If(.not.Flag) Write(6,*) 'Caution: new maximum' * * Calculate 4-momenta Call ggLmom * * Display first 3 events If(i.le.3) call ggLprt * * Fill histrograms Call User(1) 10 Continue * Call Timex(Time2) Write(6,300) Nev,Time2-Time1,(Time2-Time1)/real(Nev), & iwaght,ntrial,nzero,Fmax * * Finalize plotting Call User(-Nev) * 300 Format(/,3x,'time to generate ',I8,' events is ',E12.5,/, &3x,'resulting in an average time per event of ',E12.5,/, &3x,'unweighted events requested if 1: ',I8,/, &3x,'the number of trials was: ',I8,/, &3x,'the number of zero f was: ',I8,/, &3x,'the (new) maximum f value was: ',E12.5) * Stop [99]{} G.P. Lepage, [*J. Comp. Phys. *]{} [**27**]{} (1978) 192 F. James, “RANLUX: A Fortran implementation of the high-quality pseudorandom number generator of Lüscher”, CERN Program Library V115, [[*Comput. Phys. Commun. *]{}[**79**]{}]{} (1994) 111;\ M. Lüscher, [[*Comput. Phys. Commun. *]{}[**79**]{}]{} (1994), 100 R. Brun and D. Lienart, “HBOOK User Guide – Version 4”, CERN Program Library Y250, 1988 J. 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However, the expressions taken from [@Budnev] are numerically very unstable at small $Q_i$; see the discussion following (\[rhoidef\]). Moreover, $\tilde{\phi}$ itself is not calculated. [^2]: A similar phase-space decomposition with $s_1$ replaced by $\Delta = [(s-2m^2) (W^2+Q_1^2+Q_2^2) -(s_1+Q_2^2-m^2)(s_2+Q_1^2-m^2)]/4$ is presented in [@vermaseren]. [^3]: This form of $\tilde\phi$ could, with only minor modifications, be implemented in [@vermaseren].
--- author: - | **Dissertation**\ \ \ \ \ \ \ \ date: '\' title: | Several Approaches to Break the\ Curse of Dimensionality --- [ ]{} I would like to express my deepest gratitude to my supervisor Professor Dr. Erich Novak for numerous hints, suggestions and remarks during the preparation of this work. Furthermore, I would like to thank all the members of the research groups “Theoretical numerics” and “Function spaces” in Jena for supporting me during the times of my Diploma thesis and my Ph.D. studies. Finally, I like to thank our friends from the IBC community for many fruitful discussions at several conferences during the last years. [ ]{} [ ]{} Preface {#preface .unnumbered} ======= In modern science the efficient numerical treatment of high-dimensional problems becomes more and more important. A fundamental insight of the theory of *information-based complexity* (*IBC* for short) is that the computational hardness of a problem can not be described properly only by the rate of convergence. An impressive example that illustrates this fact was given recently by Novak and Woźniakowski [@NW09]. They studied a problem for which an exponential number of information operations is needed in order to reduce the initial error, although there exist algorithms which provide an arbitrary large rate of convergence. Problems that yield this exponential dependence are said to suffer from the *curse of dimensionality*. While analyzing numerical problems it turns out that we can often vanquish this curse by exploiting additional structural properties. The aim of this thesis is to present several approaches of this type. A numerical problem $S$ is given by a sequence of compact linear operators $S_d$ acting between normed spaces ${\mathcal{F}}_d$ and ${\mathcal{G}}_d$, where $d\in{\mathbb{N}}$. In general we seek for algorithms $A_{n,d}$ that approximate $S_d$ while using at most $n\in{\mathbb{N}}_0$ pieces of information on the input elements $f\in{\mathcal{F}}_d$. The quality of this approximation is measured by the so-called *worst case error* $$\begin{gathered} \Delta^{\mathrm{wor}}(A_{n,d}; S_d) = \sup_{{\left\| f{\; \vrule \;}{\mathcal{F}}_d \right\|}\leq 1} {\left\| S_d(f)-A_{n,d}(f) {\; \vrule \;}{\mathcal{G}}_d \right\|}\end{gathered}$$ which we try to minimize. Problems based on tensor product structures, as well as linear algorithms that are easy to implement, are of particular interest. The minimal number of information operations needed to solve a given problem $S$ to within a threshold ${{\varepsilon}}>0$ is called *information complexity*: $$\begin{gathered} n({{\varepsilon}},d;S_d) = { \mathop{\mathrm{min}}\left\{n\in{\mathbb{N}}_0 {\; \vrule \;}\exists A_{n,d}\colon \Delta^{\mathrm{wor}}(A_{n,d}; S_d) \leq {{\varepsilon}}\right\} }, \quad {{\varepsilon}}>0, d\in{\mathbb{N}}.\end{gathered}$$ If this quantity grows exponentially fast with the dimension $d$ then $S$ suffers from the curse of dimensionality. In the case where $n({{\varepsilon}},d;S_d)$ is neither exponential in $d$, nor in ${{\varepsilon}}^{-1}$, the problem $S$ is said to be *weakly tractable*. A special case is described by the notion of *polynomial tractability* for which the information complexity needs to be bounded from above by a polynomial in $d$ and ${{\varepsilon}}^{-1}$, [i.e. ]{}$$\begin{gathered} n({{\varepsilon}},d;S_d) \leq C\, {{\varepsilon}}^{-p} \, d^q \quad \text{for some} \quad C,p>0,\, q\geq 0 \quad \text{and all} \quad {{\varepsilon}}\in(0,1], d\in{\mathbb{N}}.\end{gathered}$$ If the latter inequality is valid even for $q=0$ then $S$ is called *strongly polynomially tractable*.\ Next we present the three approaches to exploit structural properties we study in this thesis and we briefly summarize our main complexity results. A rather simple class of problems $S$ is given by the set of all compact linear operators between tensor products of Hilbert spaces. Especially the complexity of tensor product problems $S_d=\bigotimes_{k=1}^d S_1 \colon H_d {\rightarrow}{\mathcal{G}}_d$, induced by some operator $S_1\colon H_1{\rightarrow}{\mathcal{G}}_1$, is well-understood. It depends on the non-increasingly ordered sequence $\lambda=(\lambda_m)_{m\in{\mathbb{N}}}$ of the squares of the singular values of the underlying operator $S_1$. In particular, it is well-known that $S=(S_d)_{d\in{\mathbb{N}}}$ is not polynomially tractable if we have $\lambda_1 \geq 1$ and $\lambda_2>0$. Actually, we are faced with the curse of dimensionality if $\lambda_1$ is strictly larger than $1$ and $\lambda_2>0$, or if $\lambda_1\geq\lambda_2=1$; cf. .\ A first approach to modify such a problem is to scale the inner products of the source spaces $H_d$, $d\in{\mathbb{N}}$. We set $$\begin{gathered} {\left\langle \cdot, \cdot \right\rangle}_{{\mathcal{F}}_d} = \frac{1}{s_d} \, {\left\langle \cdot, \cdot \right\rangle}_{H_d} \quad \text{for some} \quad s_d>0 \quad \text{and all} \quad d\in{\mathbb{N}}\end{gathered}$$ and investigate the complexity of the problem operators $S_d$ interpreted as mappings between the Hilbert spaces ${\mathcal{F}}_d$ and ${\mathcal{G}}_d$, $d\in{\mathbb{N}}$. The resulting problem, scaled by factors from the sequence $s=(s_d)_{d\in{\mathbb{N}}}$, then is denoted by $S_{(s)}=(S_{d,s_d}\colon {\mathcal{F}}_d {\rightarrow}{\mathcal{G}}_d)_{d\in{\mathbb{N}}}$. We study the worst case setting with respect to the absolute error criterion and prove \[Z\_en\_thm:scaled\_PT\] Using the introduced notation and assuming that $\lambda_2>0$ the following assertions are equivalent: 1. \[Cond\_SPT\_z\_en\] $S_{(s)}$ is strongly polynomially tractable. 2. \[Cond\_PT\_z\_en\] $S_{(s)}$ is polynomially tractable. 3. \[Cond\_sup\_z\_en\] There exists $\tau\in(0,\infty)$ such that $\lambda\in{\ell}_\tau$ and $\sup_{d\in{\mathbb{N}}} s_d {\left\| \lambda {\; \vrule \;}{\ell}_\tau \right\|}^d < \infty$. 4. \[Cond\_limsup\_z\_en\] There exists ${\varrho}\in(0,\infty)$ such that $\lambda\in{\ell}_{\varrho}$ and $\limsup_{d{\rightarrow}\infty} s_d^{1/d} < \frac{1}{\lambda_1}$. If one of these (and hence all) conditions applies then the *exponent of strong polynomial tractability* is given by $p^* = \inf\{2\tau {\; \vrule \;}\tau \text{ fulfills condition {(\ref{Cond_sup_z_en})}} \}$. We refer to in . It is remarkable that similar to unscaled problems polynomial tractability of the problem $S_{(s)}$ already implies strong polynomial tractability, despite the fact that we can choose the sequence of scaling factors $(s_d)_{d\in{\mathbb{N}}}$ completely arbitrary.\ The less restrictive property weak tractability and the curse of dimensionality can be characterized, provided that we additionally assume a certain asymptotic behavior of the initial error ${{\varepsilon}}_d^{{\mathrm{init}}}=\sqrt{s_d\cdot \lambda_1^d}$; see in . \[Z\_en\_thm:scaled\_WT\] We study the scaled tensor product problem $S_{(s)} = (S_{d,s_d})_{d\in{\mathbb{N}}}$ in the worst case setting [w.r.t. ]{}the absolute error criterion and assume $\lambda_2>0$. Moreover, - let $\ln \!\left({\varepsilon}_d^{\mathrm{init}}\right)\notin o(d)$, as $d{\rightarrow}\infty$. Then we have the curse of dimensionality. - let ${\varepsilon}_d^{\mathrm{init}} \in \Theta(d^\alpha)$, as $d{\rightarrow}\infty$, for some $\alpha\geq 0$. - If $\lambda_1=\lambda_2$ then $S_{(s)}$ suffers from the curse of dimensionality. - In the case $\lambda_1>\lambda_2$ the problem $S_{(s)}$ is weakly tractable if and only if $\lambda_n \in o\!\left(\ln^{-2(1+\alpha)} n \right)$, as $n{\rightarrow}\infty$. - let ${\varepsilon}_d^{\mathrm{init}} {\rightarrow}0$, as $d$ approaches infinity. Then we are never faced with the curse of dimensionality. Furthermore, $S_{(s)}$ is weakly tractable if and only if 1. $\lambda_1=\lambda_2$ and $\lambda_n \in o\!\left(\ln^{-2} n\right)$, as $n{\rightarrow}\infty$, and ${\varepsilon}_d^{\mathrm{init}}\in o(1/d)$, as $d{\rightarrow}\infty$, or 2. $\lambda_1>\lambda_2$ and $\lambda_n \in o\!\left(\ln^{-2} n\right)$, as $n{\rightarrow}\infty$. Here the parameter $\alpha$ that controls the polynomial growth of the initial error is of particular interest. In the case where $\lambda_1>\lambda_2$ it directly enters the condition for the characterization of weak tractability. Moreover, the condition ${\varepsilon}_d^{\mathrm{init}}\in o(1/d)$, as $d{\rightarrow}\infty$, in the third part of the theorem is quite surprising. Since for unscaled problems the initial error only can grow or decline exponentially, or it equals one in any dimension, these phenomena can not occur in the classical theory, [i.e. ]{}in the case where $s_d=1$ for all $d\in{\mathbb{N}}$. Another approach to overcome the curse of dimensionality is related to problems defined between function spaces. Here we can make use of some a priori given knowledge about the influence of certain (groups of) variables on the functions in the source space, in order to approximate them efficiently. To this end, we endow these spaces with weighted norms. During the last years especially problems on function spaces that yield a Hilbert space structure, equipped with so-called product weights, attracted a lot of attention. Problems where the source and/or target spaces are allowed to be more general Banach spaces were studied less frequently within the IBC community.\ Among other things, in this thesis we consider the uniform approximation problem $$\begin{gathered} \mathrm{App} = \left(\mathrm{App}_d \colon F_d^\gamma {\rightarrow}{\mathrm{L}}_\infty([0,1]^d)\right)_{d\in{\mathbb{N}}} \quad \text{with} \quad \mathrm{App}_d(f)=f \quad \text{for} \quad d\in{\mathbb{N}}\end{gathered}$$ defined on certain classes of smooth functions $$\begin{gathered} F_d^\gamma = \left\{f\colon[0,1]^d{\rightarrow}{\mathbb{R}}{\; \vrule \;}f\in C^\infty([0,1]^d) \text{ with } {\left\| f {\; \vrule \;}F_d^\gamma \right\|}<\infty\right\}\end{gathered}$$ which are endowed with the weighted norms $$\begin{gathered} {\left\| f {\; \vrule \;}F_d^\gamma \right\|} = \sup_{\bm{\alpha}\in{\mathbb{N}}_0^d} \frac{1}{\gamma_{\bm{\alpha}}} {\left\| D^{\bm{\alpha}}f{\; \vrule \;}{\mathrm{L}}_\infty([0,1]^d) \right\|}.\end{gathered}$$ Here for every $\bm{\alpha}\in{\mathbb{N}}_0^d$, $d\in{\mathbb{N}}$, the *product weights* $\gamma_{\bm{\alpha}}=\prod_{j=1}^d (\gamma_{d,j})^{\alpha_j}$ are constructed out of a uniformly bounded sequence $C_\gamma\geq \gamma_{d,1}\geq \ldots \geq \gamma_{d,d}>0$ of so-called generator weights. It turns out that the complexity of the approximation problem depends on certain summability properties of these generators which also play an important role when dealing with problems on product-weighted Hilbert spaces. We define the quantities $$\begin{aligned} &p(\gamma)=\inf\left\{\kappa>0{\; \vrule \;}\limsup_{d{\rightarrow}\infty} \sum_{j=1}^d (\gamma_{d,j})^\kappa<\infty\right\}, \quad \text{as well as} \\ &q(\gamma)=\inf\left\{\kappa>0{\; \vrule \;}\limsup_{d{\rightarrow}\infty} \sum_{j=1}^d (\gamma_{d,j})^\kappa / \ln(d+1)<\infty\right\},\end{aligned}$$ and prove the following \[Z\_en\_thm:weighted\_PT\] For the worst case setting [w.r.t. ]{}the absolute error criterion we have: - If the problem $\mathrm{App}$ is polynomially tractable then $q(\gamma) \leq 1$. Moreover strong polynomial tractability implies the condition $p(\gamma) \leq 1$. - If $q(\gamma)<1$ or even $p(\gamma)<1$ then $\mathrm{App}$ is polynomially tractable or even strongly polynomially tractable, respectively. In fact, we show these necessary and sufficient criteria for a whole scale of weighted Banach spaces that fulfill certain embedding conditions; see and for details. The source space $F_d^\gamma$ as defined above appears as a special case within this scale. On the other hand, it generalizes a space considered by Novak und Woźniakowski [@NW09]. In addition, we prove that the sufficient conditions $q(\gamma)<1$ and $p(\gamma)<1$ are also necessary for (strong) polynomial tractability of the ${\mathrm{L}}_\infty$-approximation problem defined on a certain unanchored Sobolev space ${\mathcal{H}}_d^\gamma$; cf. .\ Weak tractability and the curse of dimensionality can be characterized as follows. \[Z\_en\_thm:weighted\_WT\] For $\mathrm{App}=(\mathrm{App}_d)_{d\in{\mathbb{N}}}$ the following assertions are equivalent: 1. The problem is weakly tractable. \[Cond\_WT\_en\] 2. The curse of dimensionality is not present.\[Cond\_Curse\_en\] 3. For all $\kappa > 0$ we have $\lim_{d{\rightarrow}\infty}\limits \frac{1}{d} \sum_{j=1}^d \left( \gamma_{d,j} \right)^\kappa = 0$. 4. There exists $\kappa \in (0,1)$ such that $\lim_{d{\rightarrow}\infty}\limits \frac{1}{d} \sum_{j=1}^d \left( \gamma_{d,j} \right)^\kappa = 0$.\[Cond\_Limit\_en\] This immediately follows from our in which we discuss a more general situation. Note that the implication [(\[Cond\_Curse\_en\])]{} $\Rightarrow$ [(\[Cond\_WT\_en\])]{} is not trivial. Moreover, the condition [(\[Cond\_Limit\_en\])]{} is typical for problems defined on Hilbert spaces equipped with product weights. Finally, our third approach to vanquish the curse is based on exploiting certain symmetry properties of the elements in the source space. For this purpose we again consider tensor product problems $S=(S_d\colon H_d {\rightarrow}{\mathcal{G}}_d)_{d\in{\mathbb{N}}}$ between Hilbert spaces. But now we restrict them to suitable subspaces which solely consist of (anti)symmetric elements. We illustrate this concept by considering the special case of problems defined between function spaces.\ For $d\in{\mathbb{N}}$ and $I\subseteq\{1,\ldots,d\}$ let ${\mathcal{S}}_I$ denote the collection of all permutations $\pi$ of the coordinate set $\{1,\ldots,d\}$ that leave the complement $I^c=\{1,\ldots,d\}\setminus I$ of $I$ fixed. Then a real-valued function $f\in H_d=H_1\otimes\ldots\otimes H_1$ on $[0,1]^d$ is called *$I$-symmetric* if $$\begin{gathered} f(\bm{x}) = f(\bm{\pi(x)}) \quad \text{for every} \quad \bm{x}\in [0,1]^d \quad \text{and all} \quad \pi\in{\mathcal{S}}_I.\end{gathered}$$ In contrast, $f$ is called *$I$-antisymmetric* if the equality $f(\bm{x}) = (-1)^{{\left| \pi \right|}} f(\bm{\pi(x)})$ holds true for every $\bm{x}$ and $\pi$. In what follows we denote the corresponding linear subspaces of $H_d$ that exclusively contain symmetric or antisymmetric functions by ${\mathfrak{S}}_I(H_d)$ and ${\mathfrak{A}}_I(H_d)$, respectively. Particularly antisymmetric functions, [i.e. ]{}functions that change their sign when we exchange the variables $x_i$ and $x_j$, $i,j\in I$, turned out be of some practical interest; see, e.g., . For the restriction of a given tensor product problem $S=(S_d\colon H_d{\rightarrow}{\mathcal{G}}_d)_{d\in{\mathbb{N}}}$ to the subspaces $P_{I_d}(H_d)$, $d\in{\mathbb{N}}$, we write $S_{I}=(S_{d,I_d})_{d\in{\mathbb{N}}}$. Here the kind of symmetry $P\in\{{\mathfrak{S}},{\mathfrak{A}}\}$, as well as a sequence $(I_d)_{d\in{\mathbb{N}}}$ of subsets of the coordinates, is assumed to be fixed.\ Since for $d\in{\mathbb{N}}$ the operators $S_{d,I_d}$ can be interpreted as a composition of $S_d$ with suitable orthogonal projections, there exists a close relation of the singular values of $S_d$ with the corresponding singular values of the restricted operators $S_{d,I_d}$. These numbers essentially determine the minimal worst case error of the problem $S_I$. This knowledge furthermore allows the construction of an optimal (linear) algorithm that realizes this error; cf. .\ Consequently, we can conclude assertions that relate the information complexity of $S_I$ to the squares of the singular values of $S_1$ and to the number of (anti)symmetry conditions we impose. For the sake of simplicity we restrict ourselves again to the absolute error criterion and start by discussing the case of symmetric problems; see . \[Z\_en\_thm:sym\_PT\] Let $S_1 \colon H_1 {\rightarrow}{\mathcal{G}}_1$ denote a compact linear operator between Hilbert spaces and let $\lambda=(\lambda_m)_{m\in {\mathbb{N}}}$ be the sequence of eigenvalues of $W_1={S_1}^{\!\dagger} S_1$ [w.r.t. ]{}a non-increasing ordering. Assume $\lambda_2>0$ and for $d>1$ let ${\emptyset}\neq I_d \subseteq\{1,\ldots,d\}$ be fixed. We consider the restriction $S_I=(S_{d,I_d})_{d\in{\mathbb{N}}}$ of the tensor product problem $S=(S_d\colon H_d{\rightarrow}{\mathcal{G}}_d)_{d\in{\mathbb{N}}}$ to the $I_d$-symmetric subspaces ${\mathfrak{S}}_{I_d}(H_d)\subset H_d$, $d\in{\mathbb{N}}$. Then $S_I$ is strongly polynomially tractable if and only if $\lambda \in {\ell}_\tau$ for some $\tau\in(0,\infty)$ and - $\lambda_1<1$, or - $1=\lambda_1>\lambda_2$ and $(d-\#I_d) \in {\mathcal{O}}(1)$, as $d{\rightarrow}\infty$. Moreover, provided that $\lambda_1 \leq 1$ the problem is polynomially tractable if and only if $\lambda \in {\ell}_\tau$ for some $\tau\in(0,\infty)$ and - $\lambda_1<1$, or - $\lambda_1=1$ and $(d-\#I_d) \in {\mathcal{O}}(\ln d)$, as $d{\rightarrow}\infty$. It remains the open problem to find sufficient conditions for polynomial tractability in the case $\lambda_1>1$. However, our results show that the conditions $\lambda \in {\ell}_\tau$ and $(d-\#I_d) \in {\mathcal{O}}(\ln d)$ are necessary in this situation, too. In conclusion, we see that imposing sufficiently many additional symmetry assumptions, we can avoid the curse of dimensionality which we are faced with [e.g.]{} in the case $\lambda_1=\lambda_2=1$; see also .\ The complexity analysis of antisymmetric problems is more demanding. On the other hand, it turns out that here even weaker conditions are sufficient to conclude polynomial tractability and thus to vanquish the curse. One of the reasons is the structure of the initial error which is more complicated in this case. Similar to in we can summarize the main results on the complexity as follows: \[Z\_en\_thm:asy\_PT\] Let $S_1 \colon H_1 {\rightarrow}{\mathcal{G}}_1$ denote a compact linear operator between Hilbert spaces and let $\lambda=(\lambda_m)_{m\in {\mathbb{N}}}$ be the sequence of eigenvalues of $W_1={S_1}^{\!\dagger} S_1$ [w.r.t. ]{}a non-increasing ordering. Assume $\lambda_2>0$ and for $d>1$ let ${\emptyset}\neq I_d \subseteq\{1,\ldots,d\}$ be fixed. We consider the restriction $S_I=(S_{d,I_d})_{d\in{\mathbb{N}}}$ of the tensor product problem $S=(S_d\colon H_d{\rightarrow}{\mathcal{G}}_d)_{d\in{\mathbb{N}}}$ to the $I_d$-antisymmetric subspaces ${\mathfrak{A}}_{I_d}(H_d)\subset H_d$, $d\in{\mathbb{N}}$. Then for the case $\lambda_1 < 1$ the following statements are equivalent: - $S_I$ is strongly polynomially tractable. - $S_I$ is polynomially tractable. - There exists a constant $\tau \in (0,\infty)$ such that $\lambda \in {\ell}_\tau$. Moreover, the same equivalences hold true if $\lambda_1\geq 1$ and the number of antisymmetric coordinates $\#I_d$ grows linearly with the dimension $d$. Clearly, these assertions show that antisymmetric tensor product problems are significantly easier than their symmetric counterparts which on their part possess a lower information complexity than entire tensor product problems, as long as we impose enough (anti)symmetry conditions. On the other hand, there exist quite natural examples which show that even fully antisymmetric problems are not necessarily trivial or polynomially tractable, in general. For details we refer to . Let us briefly explain the structure of the present thesis. In the first chapter we settle some notational conventions and we define the abstract problem we are faced with in IBC. Furthermore, here we introduce the used cost model and recall the formal definitions of several complexity categories. In we discuss special classes of numerical problems, as well as elementary tools that we need to handle them. In particular, here we give a detailed introduction to the singular value decomposition (SVD) of compact operators between Hilbert spaces. In many cases it builds the basis for the construction of optimal algorithms. Hence it is of fundamental importance for the rest of our work. In addition, we discuss tensor product structures in Hilbert spaces and recall some well-known complexity assertions for problems related to this concept. Finally, we briefly introduce so-called reproducing kernel Hilbert spaces (RKHSs) and collect some of their properties. In the first two sections of the third chapter we derive the characterizations of the different types of tractability of scaled tensor product problems between Hilbert spaces we presented in and above. Moreover, from them we conclude a complete characterization for the normalized error criterion in . It turns out that here the scaling factors become irrelevant. Apart from formulas of the optimal algorithm and its worst case error, we additionally show that these new assertions generalize the known theory in a quite natural way. We conclude this chapter by the application of the obtained results to two simple examples. then deals with problems on function spaces endowed with weighted norms. Here we explain the concept of weighted spaces in full detail and illustrate it using the example of some unanchored Sobolev ${\mathcal{H}}_d^\gamma$ space equipped with product weights. For the uniform approximation problem on this space we present an algorithm $A_{n,d}^*$ that satisfies suitable upper error bounds. Together with corresponding lower bounds, which we prove for spaces of low-degree polynomials, the application of simple embedding arguments then leads us to complexity assertions for a whole scale of product-weighted Banach spaces. In particular, these assertions cover the results for the space $F_d^\gamma$ stated in and . Finally, the last section within this chapter, , presents some generalizations of the techniques developed before. Among other things, here we show how to handle ${\mathrm{L}}_p$-approximation problems, where $1\leq p<\infty$, defined on suitable spaces. Moreover, we show that the algorithm $A_{n,d}^*$ is essentially optimal for ${\mathrm{L}}_\infty$-approximation on ${\mathcal{H}}_d^\gamma$. For the proof we make use of arguments due to Kuo, Wasilkowski and Woźniakowski [@KWW08] that relate the uniform approximation problem in the worst case setting, defined on quite general reproducing kernel Hilbert spaces, to a certain average case ${\mathrm{L}}_2$-approximation problem.\ Some of the results presented in this chapter were already published in [@W12]. However, we were able to partially improve these assertions. We will explicitly emphasize generalizations and new results at the appropriate points. Finally, is devoted to problems with (anti)symmetry conditions. We start with the definition of (anti)symmetry in Hilbert function spaces. In particular, we focus our attention to tensor product structures and conclude fundamental properties of the respective projections and subspaces. At the end of we use these properties in order to generalize the notion of (anti)symmetry to tensor products of abstract Hilbert spaces. Afterwards we define (anti)symmetric numerical problems $S_{I}=(S_{d,I_d})_{d\in{\mathbb{N}}}$ by the restriction of a given tensor product problem $S=(S_d)_{d\in{\mathbb{N}}}$ to the subspaces of (anti)symmetric elements in the source spaces. We prove the commutativity of the operators $S_d$ with certain projections and conclude formulas for optimal algorithms and their worst case errors. This in hand, in we then discuss the complexity of (anti)symmetric numerical problems. We distinguish between symmetric and antisymmetric problems, as well as between the absolute and the normalized error criterion. Here we particularly derive the proofs of and . The chapter is concluded by a section which is devoted to several applications. On the one hand, we use simple examples to show that the additional knowledge about (anti)symmetry conditions can dramatically reduce the information complexity. On the other hand, we also discuss more advanced problems that play a role in computational practice. To this end, we illustrate the application of this new theory to the approximation problem of so-called wavefunctions that arise in certain models of quantum mechanics and theoretical chemistry.\ A major part of the results proven in this chapter was published in [@W12b]. However, at some points we use different proof techniques that allow slight generalizations. Within every chapter formulas are numbered consecutively. Moreover, we use a sequential numbering for lemmata, remarks, examples, propositions, and theorems; [e.g.]{} is followed by and . The symbols $\square$ and $\blacksquare$ are used to indicate the end of remarks and examples, as well as of proofs, respectively. [ ]{} Preliminaries {#chapt:1} ============= Apart from introducing some notational conventions, the aim of this first chapter is to define the general objects of interest in *information based complexity* (IBC). We give an abstract formulation of the general problem in . Afterwards we introduce some classes of algorithms and discuss the used cost model in . Finally, in , we recall the notions of tractability, as well as the definition of the curse of dimensionality. Basic notation -------------- General problem {#sect:General} --------------- Algorithms and cost model {#sect:Algos} ------------------------- Notions of tractability {#sect:TractDef} ----------------------- Properties and tools for special problem classes {#chapt:prop} ================================================ This chapter deals with basic properties of certain classes of problems and algorithms. We state simple consequences obtained from fundamental assumptions on the operators under consideration. Furthermore, we present more or less classical tools used in the framework of information-based complexity to acquire tractability results in a quite general context. In detail, we begin with a simple lower error bound in a very general setting which will be used on several occasions later on. In we then show that for our purposes it is reasonable to concentrate mainly on compact problems and linear, non-adaptive algorithms. Moreover, there we derive a formula for the initial error of the problems we are interested in. Afterwards, in , we turn to the important class of problems defined between Hilbert spaces. We recall well-known tools such as the singular value decomposition, conclude optimal algorithms and characterize several types of tractabilities of such problems. In we restrict ourselves further and assume an additional tensor product structure which will play an important role throughout the rest of this thesis. Finally we conclude this chapter with the discussion of so-called reproducing kernel Hilbert spaces. The main references for the functional analytic background needed in this part, as well as on the theory of $s$-numbers (or $n$-widths, respectively) are the monographs of Pinkus [@P85] and Pietsch [@P87; @P07]. For a detailed discussion of applications to tractability questions we refer again to Novak and Wo[ź]{}niakowski [@NW08; @NW10; @NW12] and to Math[é]{} [@M90]. Lower bounds on linear subspaces -------------------------------- Linearity and compactness {#sect:Linearity} ------------------------- General Hilbert space problems {#sect:General_HSP} ------------------------------ In this section we describe the *singular value decomposition* (SVD) which turns out to be the main tool when dealing with problems where both the source and the target spaces are Hilbert spaces. We prove well-known formulas for optimal linear algorithms using continuous linear functionals and calculate their worst case errors. Afterwards, we use the obtained assertions to give characterizations for (strong) polynomial tractability for these problems. Tensor product problems {#sect:TensorBasics} ----------------------- Reproducing kernel Hilbert spaces {#sect:RKHS} --------------------------------- Problems on Hilbert spaces with scaled norms {#chapt:ScaledNorms} ============================================ The present chapter deals with a generalization of tensor product problems $S=(S_d)_{d\in{\mathbb{N}}}$ between Hilbert spaces in the sense of . We introduce additional scaling factors $s_d$ to the norm of the source spaces ${\mathcal{F}}_d$ and analyze their influence on the squared singular values $\lambda_{d,s_d,i}$ of the new problem operators $S_{d,s_d}$. Using the techniques from we conclude optimal algorithms for these modified problems at the end of . Afterwards, in , we investigate tractability properties of this class of problems [w.r.t. ]{}the worst case setting. Finally we present some applications of the obtained results in . Definitions, eigenpairs and the optimal algorithm {#sect:Scaled_basics} ------------------------------------------------- Complexity {#sect:ScaledComplexity} ---------- Similar to we proceed with the analysis of the information complexity of scaled tensor product problems $S_{(s)}=(S_{d,s_d})_{d\in{\mathbb{N}}}$ in the worst case setting. We first take a look at necessary and sufficient conditions for (strong) polynomial tractability with respect to absolute errors. Afterwards, in , we complete these assertions and investigate respective conditions for weak tractability and the curse of dimensionality. Finally we will see in that the obtained improvements due to scaling are completely ruled out when we turn to the normalized error criterion. As usual $\lambda=(\lambda_m)_{m\in{\mathbb{N}}}$ denotes the (extended) sequence of squared singular values of the underlying operator $S_1 \colon H_1{\rightarrow}{\mathcal{G}}_1$. To avoid triviality we assume that $\lambda_2>0$ throughout the rest of this section. The reason for this assumption is explicitly stated in . ### Polynomial tractability {#sect:ScaledProbs_pt} ### Weak tractability and the curse {#sect:ScaledProbs_wt} ### Normalized errors {#sect:ScaledProbs_normed} Examples {#sect:ScaledProbs_Ex} -------- Problems on function spaces with weighted norms {#chapt:weighted} =============================================== In [@NW09] it is shown that the approximation problem defined on $C^\infty([0,1]^d)$ is intractable. In fact, Novak and Woźniakowski considered the linear space $F_d$ of all real-valued infinitely differentiable functions $f$ defined on the unit cube $[0,1]^d$ in $d$ dimensions for which the norm $$\begin{gathered} \label{normFd} {\left\| f {\; \vrule \;}F_d \right\|} = \sup_{\bm\alpha \in {\mathbb{N}}_0^d} {\left\| D^{\bm\alpha} f {\; \vrule \;}{\mathrm{L}}_\infty([0,1]^d) \right\|}\end{gathered}$$ of $f \in F_d$ is finite. In this case the (uniform) approximation problem is given by the sequence of solution operators $S=(S_d)_{d\in{\mathbb{N}}}$, $$\begin{gathered} \label{eq:uniform_app} S_d = {\mathrm{id}}_d \colon \widetilde{F}_d {\rightarrow}{\mathrm{L}}_{\infty}([0,1]^d), \quad f \mapsto {\mathrm{id}}_d(f)=f, \quad d\in{\mathbb{N}},\end{gathered}$$ defined on the unit ball $\widetilde{F}_d={\mathcal{B}}(F_d)$ of $F_d$. The authors studied this problem in the worst case setting using algorithms from the classes ${\mathcal{A}}_d^{n, \mathrm{cont}}$ and ${\mathcal{A}}_d^{n,\mathrm{adapt}}$ as defined in . The *initial error* of this problem is given by ${\varepsilon}_d^{{\mathrm{init}}} = e^{\mathrm{wor}}(0,d; {\mathrm{id}}_d)=1$, the norm of the embedding $F_d \hookrightarrow {\mathrm{L}}_\infty$, since $A_{0,d}\equiv 0$ is a valid choice of an algorithm which does not use any information of $f$; see . This means that the problem is well-scaled such that there is no difference in studying the absolute or the normalized error criterion. Now [@NW09 Theorem 1] yields that the $n$th minimal worst case error of ${\mathrm{L}}_\infty$-approximation defined on $F_d$ satisfies $$\begin{gathered} \label{eq:NW09_bound} e^{{\mathrm{wor}}}(n,d; {\mathrm{id}}_d)=1 \quad \text{for all} \quad n=0,1,\ldots, 2^{{\left\lfloor d/2 \right\rfloor}}-1.\end{gathered}$$ Therefore, for all $d\in {\mathbb{N}}$ and every ${\varepsilon}\in (0,1)$, the information complexity is bounded from below by $$\begin{gathered} n^{{\mathrm{wor}}}({\varepsilon},d; {\mathrm{id}}_d) \geq 2^{{\left\lfloor d/2 \right\rfloor}}.\end{gathered}$$ Hence the problem suffers from the curse of dimensionality; in particular it is intractable. One possibility to avoid this exponential dependence on $d$, [i.e. ]{}to break the curse, is to shrink the function space $F_d$ by introducing *weights*. In the present chapter we follow this idea. We show that turning to spaces equipped with *product weights* can dramatically improve the tractability behavior of certain problems such as uniform approximation. In we formally introduce the concept of weighted spaces by considering the examples of weighted Banach spaces of smooth functions and of weighted reproducing kernel Hilbert spaces. Uniform approximation in the latter class of spaces then is studied in . Afterwards, in , we show how to use the obtained upper error bounds for the ${\mathrm{L}}_\infty$-approximation problem defined on scales of smooth functions. Moreover we prove corresponding lower bounds on the information complexity which enable us to give necessary and sufficient conditions for several kinds of tractabilities in terms of the used weights. Most of the results stated in this chapter are published in the article [@W12]. The concept of weighted spaces {#sect:concept} ------------------------------ The idea to introduce weights directly into the norm of the function space appeared for the first time in a paper of Sloan and Woźniakowski in 1998; see [@SW98]. They studied the integration problem defined over some Sobolev Hilbert space, equipped with so-called *product weights*, to explain the overwhelming success of QMC integration rules. Thenceforth weighted problems attracted a lot of attention. For example it turned out that tractability of approximation of linear compact operators between Hilbert spaces can be fully characterized in terms of the weights and the singular values of the operators if we use information operations from the class $\Lambda^{{\mathrm{all}}}$. The proof of this kind of assertions is once again based on the singular value decomposition; see . One such result is given in below. But first let us illustrate the concept of weighted spaces by modifying the space $F_d$ we introduced before. ### Weighted Banach spaces of smooth functions {#sect:weightedSmooth} A closer look at the norm given in [(\[normFd\])]{} yields that for $f\in {\mathcal{B}}(F_d)$ we have $$\begin{gathered} \label{unitball} {\left\| D^{\bm\alpha} f {\; \vrule \;}{\mathrm{L}}_\infty([0,1]^d) \right\|} \leq 1 \quad \text{ for all } \quad {\bm\alpha} \in {\mathbb{N}}_0^d. \end{gathered}$$ Hence every derivative is equally important. In order to shrink the space, for each ${\bm\alpha} \in {\mathbb{N}}_0^d$ we replace the right-hand side of inequality [(\[unitball\])]{} by a non-negative weight $\gamma_{\bm\alpha}$. For $\bm\alpha$ with ${\left| \bm\alpha \right|}=1$ this means that we control the importance of every single variable. So, the norm in the weighted space $F_d^\gamma$ is now given by $$\begin{gathered} \label{eq:weightednorm} {\left\| f {\; \vrule \;}F_d^\gamma \right\|} = \sup_{\bm\alpha \in {\mathbb{N}}_0} \frac{1}{\gamma_{\bm\alpha}} {\left\| D^{\bm\alpha} f {\; \vrule \;}{\mathrm{L}}_\infty([0,1]^d) \right\|},\end{gathered}$$ where we demand $D^{\bm\alpha} f$ to be equal to zero if $\gamma_{\bm\alpha}=0$. It is clear from the construction that we indeed shrink the space if all $\gamma_{\bm\alpha}$ are chosen strictly less than one. Since this approach is quite general we restrict ourselves to so-called *product weights* (with uniformly bounded generators) in what follows. Thus we assume that for every $d\in{\mathbb{N}}$ there exists an ordered and uniformly bounded sequence $$\begin{gathered} C_\gamma \geq \gamma_{d,1} \geq \gamma_{d,2} \geq \ldots \geq \gamma_{d,d} \geq 0.\end{gathered}$$ Then for $d\in {\mathbb{N}}$ the product weight sequence $\gamma=\left( \gamma_{\bm\alpha} \right)_{\bm\alpha \in {\mathbb{N}}_0^d}$ is given by $$\begin{gathered} \label{ProdWeights} \gamma_{\bm\alpha} = \prod_{j=1}^d \left( \gamma_{d,j} \right)^{\alpha_j}, \quad \bm\alpha \in {\mathbb{N}}_0^d.\end{gathered}$$ Note that the dependence of $x_j$ on $f$ is now controlled by the so-called *generator weight* $\gamma_{d,j}$. Since $\gamma_{d,j}=0$ for some $j\in \{1,\ldots,d\}$ implies that $f$ does not depend on $x_j,\ldots,x_d$ we assume that $\gamma_{d,d}>0$ in the rest of this chapter. Moreover observe that the ordering of $\gamma_{d,j}$ is without loss of generality. Later on we will see that tractability of our problem will only depend on summability properties of the generator weights. Among other things, we show in that for the ${\mathrm{L}}_\infty$-approximation problem defined on the Banach spaces $F_d^\gamma$ with the norm given above and generator weights $\gamma_{d,j}\equiv \gamma^{(j)}\in\Theta \left(j^{-\beta} \right)$ we have - intractability for $\beta = 0$, - weak tractability but no polynomial tractability for $0 < \beta < 1$, - strong polynomial tractability if $1 < \beta$. Furthermore, we prove that for $\beta = 1$ the problem is not strongly polynomially tractable. ### Weighted Hilbert spaces and weighted RKHS {#sect:weightedSobolev} Let us briefly discuss the idea of weighted norms in the case of Hilbert (function) spaces, before we turn to weighted RKHSs. Our approach is based on a generalization of the so-called ANOVA[^1] decomposition of $d$-variate functions $f$, where $d$ is an arbitrary large integer. For the ease of presentation we follow the lines of [@NW08 Section 5.3.1]. Thus, we focus our attention on Hilbert function spaces constructed out of tensor products and equipped with some assumptions that can be significantly relaxed. For further information on more general settings the interested reader is referred to [@KSWW10b] and the references therein. Given a $d$-fold tensor product space $H_d=H_1\otimes\ldots\otimes H_1$, $d\in{\mathbb{N}}$, as well as an orthonormal basis $\{e_i {\; \vrule \;}i\in{\mathbb{N}}\}$ of the underlying univariate Hilbert space[^2] $H_1$ that contains the constant function $e_1 \equiv 1$, it is easy to see that every $f\in H_d$ can be represented as $$\begin{gathered} f = \sum_{{\mathfrak{u}}\subseteq \{1,\ldots,d\}} f_{{\mathfrak{u}}}.\end{gathered}$$ In this decomposition the (formally $d$-variate) functions $f_{\mathfrak{u}}$ solely depend on the variables $x_j$ with index $j\in{\mathfrak{u}}$. The main advantage of this kind of representation is that for fixed $f$ the collection of all $f_{\mathfrak{u}}$, ${\mathfrak{u}}\subseteq\{1,\ldots,d\}$, can be taken mutually orthogonal [w.r.t. ]{}the inner product ${\left\langle \cdot, \cdot \right\rangle}_{H_d}$ in $H_d$. Therefore the norm of $f\in H_d$ can be expressed by $$\begin{gathered} {\left\| f {\; \vrule \;}H_d \right\|}^2 = \sum_{{\mathfrak{u}}\subseteq\{1,\ldots,d\}} {\left\| f_{{\mathfrak{u}}} {\; \vrule \;}H_d \right\|}^2 = \sum_{{\mathfrak{u}}\subseteq\{1,\ldots,d\}} {\left\| f_{{\mathfrak{u}},1} {\; \vrule \;}H_{{\left| {\mathfrak{u}}\right|}} \right\|}^2,\end{gathered}$$ where $f_{{\mathfrak{u}},1}$ equals $f_{\mathfrak{u}}$ interpreted as an element of the ${\left| {\mathfrak{u}}\right|}$-fold tensor product space $H_{{\left| {\mathfrak{u}}\right|}}$ of the closed subspace $$\begin{gathered} H_1' = \left\{h \in H_1 {\; \vrule \;}{\left\langle h, e_1 \right\rangle}_{H_1}=0\right\} \subset H_1\end{gathered}$$ with itself. That is, in the unweighted situation the contribution of each $f_{\mathfrak{u}}$ to the norm of $f\in H_d$ is the same. Now suppose that we have some additional, a priori knowledge about the importance of some (groups of) variables in dimension $d$. This can be modeled by assigning positive[^3] weights $\gamma_{d,{\mathfrak{u}}}$ to each of the $2^d$ subsets ${\mathfrak{u}}$ of $\{1,\ldots,d\}$. We denote the collection of these weights by $\gamma_{(d)} = \{\gamma_{d,{\mathfrak{u}}} {\; \vrule \;}{\mathfrak{u}}\subseteq \{1,\ldots,d\} \}$. Then it can be verified that $$\begin{gathered} \label{eq:weighted_innerprod} {\left\langle f, g \right\rangle}_{\gamma_{(d)}} = \sum_{{\mathfrak{u}}\subseteq \{1,\ldots,d\}} \frac{1}{\gamma_{d,{\mathfrak{u}}}} {\left\langle f_{{\mathfrak{u}}}, g_{{\mathfrak{u}}} \right\rangle}_{H_d} = \sum_{{\mathfrak{u}}\subseteq \{1,\ldots,d\}} \frac{1}{\gamma_{d,{\mathfrak{u}}}} {\left\langle f_{{\mathfrak{u}},1}, g_{{\mathfrak{u}},1} \right\rangle}_{H_{{\left| {\mathfrak{u}}\right|}}}\end{gathered}$$ defines an inner product on the tensor product space $H_d$ which implies an equivalent norm depending on $\gamma_{(d)}$. The Hilbert space $H_d$ endowed with this new inner product will be denoted by $H_d^{\gamma_{(d)}}$. At this point we need to stress the fact that for general weights $\gamma_{(d)}$ these spaces are no longer tensor product spaces, although their construction is based on $H_d=H_1\otimes\ldots\otimes H_1$ and $H_{{\left| {\mathfrak{u}}\right|}}$, respectively. To overcome this problem we restrict ourselves to the case of *product weights* in the following. Thus we assume $$\begin{gathered} \label{eq:prod_weights_u} \gamma_{d,{\mathfrak{u}}} = \prod_{k\in{\mathfrak{u}}} \gamma_{d,k}\end{gathered}$$ for some positive $\gamma_{d,k}$, $k=1,\ldots,d$, and every ${\mathfrak{u}}\subseteq\{1,\ldots,d\}$. Then it can be checked that indeed $H_d^{\gamma_{(d)}}$ is again a tensor product space. For the study of other types of weights such as *finite-order*, *finite-diameter*, *order-dependent* or the recently developed *POD[^4] weights* we refer to Novak and Wo[ź]{}niakowski [@NW08 Section 5.3.2] and to Kuo, Schwab and Sloan [@KSS11]. In the last decade it turned out that weighted norms provide a powerful tool to vanquish the curse of dimensionality that we are often faced with. Since the $H_d^{\gamma_{(d)}}$’s are still Hilbert spaces the complexity analysis of weighted problems $S^{\gamma_{(d)}}=(S_d^{\gamma_{(d)}} \colon {\mathcal{B}}(H_d^{\gamma_{(d)}}) {\rightarrow}{\mathcal{G}}_d)_{d\in{\mathbb{N}}}$ again is based on the singular value decomposition presented in ; at least in the cases where the target spaces ${\mathcal{G}}_d$ are also Hilbert spaces. Fortunately, the introduced weights enter the spectrum of the operator $W_d^{\gamma_{(d)}}=\left(S_d^{\gamma_{(d)}}\right)^\dagger S_d^{\gamma_{(d)}}$ in a straightforward way. Therefore in many cases tractability properties of $S$ can be fully characterized in terms of the singular values and the introduced weights.\ For our purposes weighted Hilbert spaces that possess a reproducing kernel are of particular interest. Typical examples of such weighted RKHSs are the following unanchored Sobolev spaces endowed with product weights which will play an important role in our further argumentation; see also Sloan and Wo[ź]{}niakowski [@SW02]. Instead of applying the presented approach which is based on decompositions we use the common procedure and define them directly. \[ex:Unanchored\_Sobolev\] As usual we start with the definition for $d=1$ and $\gamma>0$. Then the space ${\mathcal{H}}_1^\gamma$ is nothing but the Sobolev space of all absolutely continuous real-valued functions $f$ defined on the unit interval $[0,1]$ whose first derivative[^5] $f'$ belongs to the space ${\mathrm{L}}_2([0,1])$. The difference to the classical Sobolev space is the inner product which here depends on the parameter $\gamma$: $$\begin{aligned} {\left\langle f, g \right\rangle}_{{\mathcal{H}}_1^\gamma} &= {\left\langle f, g \right\rangle}_{{\mathrm{L}}_2([0,1])} + \gamma^{-1} {\left\langle f', g' \right\rangle}_{{\mathrm{L}}_2([0,1])} \label{eq:uni_prod}\\ &= \int_0^1 f(x) \, g(x) \, {\,\mathrm{d}\uplambda}^1(x) + \gamma^{-1} \int_0^1 f'(x) \, g'(x) \, {\,\mathrm{d}\uplambda}^1(x), \qquad f,g\in{\mathcal{H}}_1^\gamma.\nonumber\end{aligned}$$ For the sake of completeness we define the space ${\mathcal{H}}_1^0$ as the limit of ${\mathcal{H}}_1^\gamma$ for $\gamma {\rightarrow}0$. Consequently the derivatives of $f\in{\mathcal{H}}_1^0$ need to vanish $\uplambda^1$-almost everywhere on $[0,1]$ which implies that the space ${\mathcal{H}}_1^0$ only consists of constant functions. This coincides with the common convention $0/0=0$. Note that the univariate space ${\mathcal{H}}_1^\gamma$ algebraically coincides with its anchored analogue $\widetilde{{\mathcal{H}}}_1^\gamma$ where the term ${\left\langle f, g \right\rangle}_{{\mathrm{L}}_2([0,1])}$ in [(\[eq:uni\_prod\])]{} is replaced by $f(a)\cdot g(a)$ for some *anchor point* $a\in[0,1]$. For details we refer to [@SW02] and [@W12]. Finally we mention that for positive parameters $\gamma$ all these definitions imply equivalent norms on the classical Sobolev space $W_2^1([0,1])$. Once more the $d$-variate spaces ${\mathcal{H}}_d^\gamma$ for $d>1$ are defined by a tensor product construction similar to . We set ${\mathcal{H}}_d^\gamma = \bigotimes_{k=1}^d {\mathcal{H}}_1^{\gamma_{d,k}}$, where now $\gamma$ denotes a (subset of a) product weight sequence $(\gamma_{\bm{\alpha}})_{\bm{\alpha}\in\{0,1\}^d}$ induced by some generator weights $\gamma_{d,k}$, $k=1,\ldots,d$; see [(\[ProdWeights\])]{}. Remember that at the beginning of this chapter we assumed $\gamma_{d,d}>0$ for all $d\in{\mathbb{N}}$. That is, we avoid to take the trivial spaces ${\mathcal{H}}_1^0$ as factors in the definition of ${\mathcal{H}}_d^\gamma$. How does the inner product of ${\mathcal{H}}_d^\gamma$ looks like? Following the lines of it is uniquely determined by the coordinate-wise inner products of the factors of simple tensors $f=\bigotimes_{k=1}^d f_k$ and $g=\bigotimes_{k=1}^d g_k$, where $f_k,g_k\in {\mathcal{H}}_1^{\gamma_{d,k}}$ for $k=1,\ldots,d$. Consequently, $$\begin{aligned} {\left\langle f, g \right\rangle}_{{\mathcal{H}}_d^{\gamma}} &= \prod_{k=1}^d {\left\langle f_k, g_k \right\rangle}_{{\mathcal{H}}_1^{\gamma_{d,k}}} = \prod_{k=1}^d \left( {\left\langle f_k, g_k \right\rangle}_{{\mathrm{L}}_2([0,1])} + \frac{1}{\gamma_{d,k}} \, {\left\langle f_k', g_k' \right\rangle}_{{\mathrm{L}}_2([0,1])} \right) \\ &= \sum_{{\mathfrak{u}}\subseteq \{1,\ldots,d\}} \prod_{k\in {\mathfrak{u}}} \frac{1}{\gamma_{d,k}} \cdot \prod_{k\in {\mathfrak{u}}} {\left\langle f_k', g_k' \right\rangle}_{{\mathrm{L}}_2([0,1])} \cdot \prod_{j\in \{1,\ldots,d \}\setminus{\mathfrak{u}}} {\left\langle f_j, g_j \right\rangle}_{{\mathrm{L}}_2([0,1])}\\ &= \sum_{{\mathfrak{u}}\subseteq \{1,\ldots,d\}} \prod_{k\in {\mathfrak{u}}} \frac{1}{\gamma_{d,k}} \cdot \int_{[0,1]^d} \prod_{k\in {\mathfrak{u}}} f_k'(x_k) \, g_k'(x_k) \prod_{j\in \{1,\ldots,d \}\setminus{\mathfrak{u}}} f_j(x_j) \, g_j(x_j) {\,\mathrm{d}\uplambda}^d(\bm{x}) \\ &= \sum_{{\mathfrak{u}}\subseteq \{1,\ldots,d\}} \frac{1}{\gamma_{d,{\mathfrak{u}}}} \cdot \int_{[0,1]^d} \frac{\partial^{{\left| {\mathfrak{u}}\right|}} f}{\partial x_{\mathfrak{u}}}(\bm{x}) \, \frac{\partial^{{\left| {\mathfrak{u}}\right|}} g}{\partial x_{\mathfrak{u}}}(\bm{x}) {\,\mathrm{d}\uplambda}^d(\bm{x}),\end{aligned}$$ where we used [(\[eq:prod\_weights\_u\])]{} and the shorthand notation $\partial^{{\left| {\mathfrak{u}}\right|}}/(\partial x_{\mathfrak{u}})$ for $\prod_{k\in{\mathfrak{u}}} \partial/(\partial x_k)$. Note that this representation resembles [(\[eq:weighted\_innerprod\])]{} from the general approach to weighted Hilbert spaces introduced at the beginning of this subsection. For our purposes it is more convenient to rewrite the subsets ${\mathfrak{u}}\subseteq \{1,\ldots,d\}$ in terms of multi-indices $\bm{\alpha}=(\alpha_1,\ldots,\alpha_d)\in\{0,1\}^d$. In detail, we set $\alpha_k=1$ if and only if $k\in{\mathfrak{u}}$ and $\alpha_k=0$ otherwise. Then we can express the norm of any $f\in {\mathcal{H}}_d^\gamma$ by $$\begin{gathered} \label{eq:norm_Sobol} {\left\| f {\; \vrule \;}{\mathcal{H}}_d^\gamma \right\|}^2 = \sum_{\bm{\alpha} \in \{0,1\}^d} \frac{1}{\gamma_{\bm{\alpha}}} \cdot \int_{[0,1]^d} {\left| D^{\bm{\alpha}}f(\bm{x}) \right|}^2 {\,\mathrm{d}\uplambda}^d(\bm{x})\end{gathered}$$ since then $\gamma_{\mathfrak{u}}=\gamma_{\bm{\alpha}}$. The inner products of the multivariate anchored spaces, $\widetilde{{\mathcal{H}}}_d^\gamma$, can be found by a similar reasoning; see [@W12 p. 67] for the final result. It is known (cf. Micchelli and Wahba [@MW81]) that the univariate spaces ${\mathcal{H}}_1^\gamma$ are reproducing kernel Hilbert spaces for any $\gamma>0$. Consequently, this property is transferred to the multivariate tensor product space. To stress this fact we write ${\mathcal{H}}(K_d^\gamma)$ for ${\mathcal{H}}_d^\gamma$ in what follows. Equation (5) in [@WW09] now states that the reproducing kernel $K_d^\gamma\colon [0,1]^d\times[0,1]^d{\rightarrow}{\mathbb{R}}$ in dimension $d\geq 1$ is given by[^6] $$\begin{aligned} &K_d^\gamma(\bm{x},\bm{y}) \\ &\quad = \prod_{k=1}^d \frac{\sqrt{\gamma_{d,k}}}{\sinh\!\left(\sqrt{\gamma_{d,k}}\right)} \, \cosh\!\left(\sqrt{\gamma_{d,k}} \, (1-{ \mathop{\mathrm{max}}\left\{x_k,y_k\right\} })\right) \, \cosh\!\left(\sqrt{\gamma_{d,k}}\, { \mathop{\mathrm{min}}\left\{x_k,y_k\right\} }\right),\end{aligned}$$ $\bm{x},\bm{y}\in[0,1]^d$. For $d=1$ this kernel formula follows from Thomas-Agnan [@TA96 Corollary 2] whereas the higher-dimensional generalization for product weights $\gamma$ results from the tensor product structure; see [(\[eq:tensor\_RKHS\])]{} in . In particular we note that $K_d^\gamma$ is continuous (and thus also bounded) along its diagonal $$\begin{gathered} \left\{(\bm{x},\bm{y})\in[0,1]^{2d} {\; \vrule \;}\bm{x}=\bm{y}\right\}.\end{gathered}$$ Moreover, from [@WW09 Lemma 4.1] we know that for $\gamma>0$ the set $$\begin{gathered} E_1(\gamma) = \left\{ e_{1,\gamma,i} \colon [0,1]{\rightarrow}{\mathbb{R}}{\; \vrule \;}i\in{\mathbb{N}}\right\}\end{gathered}$$ with $e_{1,\gamma,1} \equiv 1$ and $$\begin{gathered} e_{1,\gamma,i}(x) = \cos(\pi (i-1) x) \cdot \sqrt{\frac{2\gamma}{\gamma + \pi^2(i-1)^2}}, \qquad x\in[0,1], \quad i \geq 2,\end{gathered}$$ builds an orthonormal basis in the univariate space ${\mathcal{H}}(K_1^\gamma)$. Applying the arguments from this leads to an ONB $E_d(\gamma)$ of ${\mathcal{H}}(K_d^\gamma)=\bigotimes_{k=1}^d {\mathcal{H}}(K_1^{\gamma_{d,k}})$ that consists of tensor product functions $$\begin{gathered} \label{eq:basis_sobolev} \widetilde{e}_{d,\gamma,\bm{m}} = \bigotimes_{k=1}^d e_{1,\gamma_{d,k},m_k}, \qquad \bm{m}=(m_1,\ldots,m_d)\in{\mathbb{N}}^d.\end{gathered}$$ For a direct proof of this result we refer to [@NW08 Appendix A.2.1][^7] and to [@WW09 Lemma 4.2]. Actually, these proofs show a little bit more; namely that the functions $\widetilde{e}_{d,\gamma,\bm{m}}$ together with $$\begin{gathered} \label{eq:L2_eigenvalues} \widetilde{\lambda}_{d,\gamma,\bm{m}} = \prod_{k=1}^d \lambda_{1,\gamma_{d,k},m_k} = \prod_{k=1}^d \frac{\gamma_{d,k}}{\gamma_{d,k} + \pi^2 (m_k-1)^2}, \qquad \bm{m}\in{\mathbb{N}}^d,\end{gathered}$$ describe the full set of eigenpairs $\{(\widetilde{\lambda}_{d,\gamma,\bm{m}}, \widetilde{e}_{d,\gamma,\bm{m}}) \,|\, \bm{m}\in{\mathbb{N}}^d\}$ of the operator $W_d^\gamma = \left( S_d^\gamma \right)^\dagger S_d^\gamma$ where $S_d^\gamma \colon {\mathcal{H}}_d^\gamma \hookrightarrow {\mathrm{L}}_2([0,1]^d)$ denotes the solution operator of the ${\mathrm{L}}_2$-approximation problem on ${\mathcal{H}}_d^\gamma={\mathcal{H}}(K_d^\gamma)$. $\square$ Uniform approximation in reproducing kernel Hilbert spaces {#sect:uniformRKHS} ---------------------------------------------------------- The main result of this section is based on a paper of Kuo, Wasilkowski and Wo[ź]{}niakowski [@KWW08]. In contrast to the presentation given in [@W12] we decided to apply this result to the case of the unanchored Sobolev Space introduced in instead of the anchored analogue studied in [@KWW08]. This opens up the opportunity to explain the underlying ideas without literally repeating the proof given in [@KWW08] while obtaining a result which is (according to our knowledge) not published elsewhere so far. We start with an upper error bound which remains valid for any reproducing kernel Hilbert space ${\mathcal{H}}(K_d)$ of real-valued functions $f$ on $[0,1]^d$ with $$\begin{gathered} \label{eq:BoundedKernel} \operatorname*{ess-sup}_{\bm{x}\in[0,1]^d} K_d(\bm{x},\bm{x})<\infty.\end{gathered}$$ This condition guarantees that ${\mathcal{H}}(K_d)$ is continuously embedded into ${\mathrm{L}}_\infty([0,1]^d)$ since the reproducing property [(\[eq:reproducing\_prop\])]{}, together with the Hahn-Banach theorem (cf. [@Y80 IV.6 Cor.2]), yields that ${\left\| {\mathrm{id}}_d {\; \vrule \;}{\mathcal{L}}({\mathcal{H}}(K_d),{\mathrm{L}}_\infty([0,1]^d)) \right\|}$ is given by $$\begin{aligned} \sup_{f\in{\mathcal{B}}({\mathcal{H}}(K_d))} {\left\| f {\; \vrule \;}{\mathrm{L}}_\infty([0,1]^d) \right\|} &= \operatorname*{ess-sup}_{\bm{x}\in [0,1]^d} \sup_{f\in{\mathcal{B}}({\mathcal{H}}(K_d))} {\left| f(\bm{x}) \right|} \\ &= \operatorname*{ess-sup}_{\bm{x}\in [0,1]^d} \sup_{f\in{\mathcal{B}}({\mathcal{H}}(K_d))} {\left| {\left\langle f, K_d(\cdot,\bm{x}) \right\rangle}_{{\mathcal{H}}(K_d)} \right|} \\ &= \operatorname*{ess-sup}_{\bm{x}\in [0,1]^d} K_d(\bm{x},\bm{x})^{1/2}.\end{aligned}$$ Now the mentioned upper bound reads as follows: \[prop:L\_inftyAlgo\] For $d\in{\mathbb{N}}$ consider a RKHS ${\mathcal{H}}(K_d)$, where $K_d$ fulfills [(\[eq:BoundedKernel\])]{}, [i.e. ]{}${\mathcal{H}}(K_d)\hookrightarrow{\mathrm{L}}_\infty([0,1]^d)$. Furthermore, suppose $\Xi=\{ \xi_j \colon [0,1]^d{\rightarrow}{\mathbb{R}}{\; \vrule \;}j\in{\mathbb{N}}\}$ to be some orthonormal basis of ${\mathcal{H}}(K_d)$ and let $n\in{\mathbb{N}}_0$. Then the algorithm $A_{n,d}^{\Xi}\in{\mathcal{A}}_d^{n,{\mathrm{lin}}}(\Lambda^{{\mathrm{all}}})$, given by $$\begin{gathered} f\mapsto A_{n,d}^{\Xi} f = \sum_{j=1}^n {\left\langle f, \xi_j \right\rangle}_{{\mathcal{H}}(K_d)}\xi_j(\cdot), \end{gathered}$$ for uniform approximation on ${\mathcal{H}}(K_d)$ fulfills $$\begin{gathered} \label{eq:L_inftyErrorBound} \Delta^{{\mathrm{wor}}}(A^{\Xi}_{n,d}; {\mathrm{id}}_d \colon {\mathcal{B}}({\mathcal{H}}(K_d)) {\rightarrow}{\mathrm{L}}_\infty([0,1]^d)) \leq {\left\| \sum_{j=n+1}^\infty \xi_j(\cdot)^2 {\; \vrule \;}{\mathrm{L}}_{\infty}([0,1]^d) \right\|}^{1/2}. \end{gathered}$$ Since $\Xi$ builds an ONB we may represent any $f\in{\mathcal{H}}(K_d)$ by its basis expansion, $f=\sum_{j=1}^\infty {\left\langle f, \xi_j \right\rangle}_{{\mathcal{H}}(K_d)}\, \xi_j$. Therefore Parseval’s identity implies $$\begin{aligned} {\left| f(\bm{x}) - A^{\Xi}_{n,d}f(\bm{x}) \right|} &= {\left| (f - A^{\Xi}_{n,d}f)(\bm{x}) \right|} = {\left| \sum_{j=n+1}^\infty {\left\langle f, \xi_j \right\rangle}_{{\mathcal{H}}(K_d)}\, \xi_j(\bm{x}) \right|}\\ &={\left| {\left\langle f, \sum_{j=n+1}^\infty \xi_j(\bm{x})\,\xi_j \right\rangle}_{{\mathcal{H}}(K_d)} \right|}\end{aligned}$$ which can be estimated from above using the inequality of Cauchy and Schwarz. Thus we obtain $$\begin{aligned} {\left| f(\bm{x}) - A^{\Xi}_{n,d}f(\bm{x}) \right|} &\leq {\left\| f {\; \vrule \;}{\mathcal{H}}(K_d) \right\|} \cdot {\left\| \sum_{j=n+1}^\infty \xi_j(\bm{x})\,\xi_j {\; \vrule \;}{\mathcal{H}}(K_d) \right\|} \label{pw_est}\\ &= {\left\| f {\; \vrule \;}{\mathcal{H}}(K_d) \right\|} \cdot \left(\sum_{j=n+1}^\infty \xi_j(\bm{x})^2 \right)^{1/2} \nonumber\end{aligned}$$ for every $f\in{\mathcal{H}}(K_d)$ and all fixed $\bm{x}\in[0,1]^d$. Taking the (essential) supremum with respect to $\bm{x}$ in the $d$-dimensional unit cube and the supremum over all $f\in{\mathcal{B}}({\mathcal{H}}(K_d))$ gives the desired result. We note in passing that we can easily prove more than we stated in the latter assertion. In what follows we only need the given upper error bound such that we restrict ourselves to some brief comments on further results in the next remark. For fixed $\bm{x}\in[0,1]^d$ we see that the function $f^*=C\cdot \sum_{j=n+1}^\infty \xi_j(\bm{x})\,\xi_j$ with $C>0$ gives equality in [(\[pw\_est\])]{}. Of course, we can choose the constant $C$ such that ${\left\| f^* {\; \vrule \;}{\mathcal{H}}(K_d) \right\|}=1$ provided that $\bm{x}$ is not a common root of $\xi_j$ for all $j > n$. Hence, the upper bound in [(\[eq:L\_inftyErrorBound\])]{} is sharp. Moreover, [@KWW08 Theorem 2] shows that the $n$th minimal worst case error for ${\mathrm{L}}_\infty$-approximation on ${\mathcal{H}}(K_d)$ is given by $$\begin{gathered} e^{{\mathrm{wor}}}(n,d; {\mathrm{id}}_d \colon {\mathcal{B}}({\mathcal{H}}(K_d)) {\rightarrow}{\mathrm{L}}_\infty([0,1]^d)) = \inf_{\Xi=\{\xi_j {\; \vrule \;}j\in{\mathbb{N}}\}} {\left\| \sum_{j=n+1}^\infty \xi_j(\cdot)^2 {\; \vrule \;}{\mathrm{L}}_{\infty}([0,1]^d) \right\|}^{1/2},\end{gathered}$$ where the infimum is taken [w.r.t. ]{}all orthonormal bases $\Xi\subset{\mathcal{H}}(K_d)$. Thus, any clever choice of the basis $\Xi$ in leads to algorithms $A^{\Xi}_{n,d}$ with almost optimal worst case errors. $\square$ Next we apply to the weighted unanchored Sobolev spaces ${\mathcal{H}}_d^\gamma$ introduced in using the basis $\Xi=E_d(\gamma)$ given in [(\[eq:basis\_sobolev\])]{}. Since the ordering of the basis functions $\xi \in \Xi$ is essential for our application we rearrange them non-increasingly with respect to their ${\mathrm{L}}_\infty$-norm: $$\begin{gathered} \label{eq:OrderedBasis} {\left\| \xi_j {\; \vrule \;}{\mathrm{L}}_{\infty}([0,1]^d) \right\|} \geq {\left\| \xi_{j+1} {\; \vrule \;}{\mathrm{L}}_{\infty}([0,1]^d) \right\|} \quad \text{for all} \quad j\in{\mathbb{N}}.\end{gathered}$$ We obtain an estimate which resembles the corresponding result for the anchored case studied in [@W12 Proposition 2]. \[cor:optAlgoSobolev\] For $n\in{\mathbb{N}}_0$ and $d\in{\mathbb{N}}$ there exists an algorithm $A_{n,d}^*\in{\mathcal{A}}_d^{n,{\mathrm{lin}}}(\Lambda^{{\mathrm{all}}})$ for uniform approximation on ${\mathcal{H}}^\gamma_d$ such that for every $\tau \in (1/2,1)$ $$\begin{gathered} \Delta^{\mathrm{wor}}(A_{n,d}^*; {\mathrm{id}}_d\colon {\mathcal{B}}({\mathcal{H}}_d^\gamma){\rightarrow}{\mathrm{L}}_{\infty}([0,1]^d) ) < a_\tau { \mathop{\mathrm{exp}}\left( b_\tau \, \sum_{k=1}^d (\gamma_{d,k})^\tau \right) } \cdot n^{-(1-\tau)/(2\tau)}, \end{gathered}$$ where the constants $a_\tau, b_{\tau}>0$ are independent of $\gamma$, $n$, and $d$. To keep the notation as short as possible we abbreviate the ${\mathrm{L}}_\infty$-norm in $d$ dimensions, ${\left\| \cdot {\; \vrule \;}{\mathrm{L}}_{\infty}([0,1]^d) \right\|}$, by ${\left\| \cdot \right\|}_d$ within this proof. Following our plan we fix $n\in{\mathbb{N}}_0$, as well as $d\in{\mathbb{N}}$, and take $A_{n,d}^*=A_{n,d}^{\Xi}$ defined in with $\Xi=E_d(\gamma)$ as above. From [(\[eq:basis\_sobolev\])]{} we conclude for $d=1$ and any $\gamma>0$ that $$\begin{gathered} {\left\| e_{1,\gamma,1}^2 \right\|}_1 = 1 \quad \text{and} \quad {\left\| e_{1,\gamma,i}^2 \right\|}_1 = \frac{2\gamma}{\gamma + \pi^2 (i-1)^2} < \frac{2\,\gamma}{\pi^2} \cdot (i-1)^{-2}, \quad i\geq 2.\end{gathered}$$ Moreover, for every simple tensor $f = \bigotimes_{k=1}^d f_k \in {\mathcal{H}}(K_d^\gamma)$ we clearly have $$\begin{gathered} {\left\| f \right\|}_d = \prod_{k=1}^d {\left\| f_k \right\|}_1 \quad \text{and} \quad f(\bm{x})^2 = \prod_{k=1}^d f_k(x_k)^2, \quad \bm{x}\in[0,1]^d.\end{gathered}$$ Consequently, for any $j\in{\mathbb{N}}$ and all $\tau \in(1/2,\infty)$ the ordering of $\Xi$ given in [(\[eq:OrderedBasis\])]{} implies $$\begin{aligned} j \cdot {\left\| \xi_j^2 \right\|}_d^\tau &\leq \sum_{m=1}^\infty {\left\| \xi_m^2 \right\|}_d^\tau = \sum_{\bm{m}\in{\mathbb{N}}^d} {\left\| \widetilde{e}_{d,\gamma,\bm{m}}^2 \right\|}_d^\tau = \prod_{k=1}^d \sum_{i=1}^\infty {\left\| e_{1,\gamma_{d,k},i}^2 \right\|}_1^\tau \\ &=\prod_{k=1}^d \left( 1 + \sum_{i=2}^\infty {\left\| e_{1,\gamma_{d,k},i}^2 \right\|}_1^\tau \right) <\prod_{k=1}^d \left( 1 + \left(\frac{2\, \gamma_{d,k}}{\pi^2}\right)^\tau \sum_{i=2}^\infty (i-1)^{-2\tau} \right) \\ &= \prod_{k=1}^d \left( 1 + c_\tau \gamma_{d,k}^\tau \right),\end{aligned}$$ where we set $c_\tau = (2/\pi^2)^\tau \, \zeta(2\tau)$. Hence, if $\tau\in(1/2,1)$ then $$\begin{aligned} {\left\| \sum_{j=n+1}^\infty \xi_j^2 \right\|}_d \leq \sum_{j=n+1}^\infty {\left\| \xi_j^2 \right\|}_d < \sum_{j=n+1}^\infty j^{-1/\tau} \cdot \left(\prod_{k=1}^d \left( 1 + c_\tau \gamma_{d,k}^\tau \right) \right)^{1/\tau}<\infty.\end{aligned}$$ Since the first factor is no larger than $\int_n^\infty x^{-1/\tau} {\,\mathrm{d}\uplambda}^1(x)= \tau/(1-\tau)\cdot n^{-(1-\tau)/\tau}$ and the second factor can be bounded by ${ \mathop{\mathrm{exp}}\left( c_\tau/\tau \cdot \sum_{k=1}^d (\gamma_{d,k})^\tau \right) }$ we conclude $$\begin{gathered} {\left\| \sum_{j=n+1}^\infty \xi_j(\cdot)^2 {\; \vrule \;}{\mathrm{L}}_{\infty}([0,1]^d) \right\|}^{1/2} < a_\tau { \mathop{\mathrm{exp}}\left( b_\tau \, \sum_{k=1}^d (\gamma_{d,k})^\tau \right) } \cdot n^{-(1-\tau)/(2\tau)}\end{gathered}$$ with $a_\tau = \sqrt{\tau/(1-\tau)}$ and $b_\tau = c_\tau/(2\tau)=(2/\pi^2)^\tau \, \zeta(2\tau) / (2\tau)$. Now the claim follows from [(\[eq:L\_inftyErrorBound\])]{} in . Uniform approximation in Banach spaces of smooth functions {#sect:LinftyApprox} ---------------------------------------------------------- Our derivation of necessary and sufficient conditions for various kinds of tractability for the ${\mathrm{L}}_\infty$-approximation problem defined on the weighted spaces $F_d^\gamma$ introduced in is based on simple embedding arguments. To this end, we consider a whole scale of Banach spaces ${\mathcal{F}}_d^\gamma$ (where $F_d^\gamma$ is a special case of). Then we first study lower bounds on the $n$th minimal error on a space ${\mathcal{P}}_d^\gamma \hookrightarrow {\mathcal{F}}_d^\gamma$ which consists of $d$-variate polynomials of low degree. Afterwards, in , we use the results for ${\mathcal{H}}_d^\gamma \hookleftarrow {\mathcal{F}}_d^\gamma$ from to conclude corresponding upper bounds. Finally we discuss a couple of concrete examples in . ### Lower bounds for spaces of low-degree polynomials {#sect:pol_bound} Following the lines of [@W12 Section 4] we use to obtain a lower bound for the ${\mathrm{L}}_\infty$-approximation error for the space $$\begin{gathered} {\mathcal{P}}_d^\gamma = {\mathop{\mathrm{span}}\left\{p_{\bm{i}} \colon [0,1]^d {\rightarrow}{\mathbb{R}}, \, p_i(\bm{x}) = \bm{x}^{\bm{i}} = \prod_{j=1}^d \left( x_j\right)^{i_j} {\; \vrule \;}\bm{i}=(i_1,\dots,i_d)\in\{0,1\}^d\right\}}\end{gathered}$$ of all real-valued $d$-variate polynomials of degree at most one in each coordinate direction, defined on the unit cube $[0,1]^d$. We equip this linear space with the weighted norm $$\begin{gathered} \label{eq:norm_P} {\left\| f {\; \vrule \;}{\mathcal{P}}_d^\gamma \right\|} = {\mathop{\mathrm{max}}\displaylimits}_{\bm{\alpha} \in \{0,1\}^d} \frac{1}{\gamma_{\bm{\alpha}}} {\left\| D^{\bm{\alpha}} f {\; \vrule \;}{\mathrm{L}}_\infty([0,1]^d) \right\|}, \qquad f \in {\mathcal{P}}_d^\gamma,\end{gathered}$$ similar to [(\[eq:weightednorm\])]{}, where $\gamma$ is a product weight sequence as described in [(\[ProdWeights\])]{}, and study the worst case setting. \[Theorem\_Polynom\] For $d\in{\mathbb{N}}$ and $n\in{\mathbb{N}}_0$ assume $A_{n,d}\in{\mathcal{A}}_d^{n, \rm cont} \cup {\mathcal{A}}_d^{n, \rm adapt}$ to be an arbitrary algorithm for the uniform approximation problem defined on ${\mathcal{P}}_d^\gamma$. Then we have $$\begin{gathered} \Delta^{\mathrm{wor}}(A_{n,d}; {\mathrm{id}}_d \colon B_r({\mathcal{P}}_d^\gamma){\rightarrow}{\mathrm{L}}_{\infty}([0,1]^d)) \geq r \quad \text{for all} \quad r\geq 0 \end{gathered}$$ provided that $n<2^s$, where $s=s(\gamma,d)\in\{0,1,\ldots,d\}$ is some integer such that $$\begin{gathered} \label{estimate_s} s > \frac{1}{2+C_\gamma} \cdot \left(\sum_{j=1}^d \gamma_{d,j} - 2 \right). \end{gathered}$$ The proof of this lower error bound consists of several steps. First we fix $d\in{\mathbb{N}}$ and construct a partition of the set of coordinates $\{1,\ldots,d\}$ into $s+1$ parts which we will need later and with $s=s(\gamma,d)$ satisfying [(\[estimate\_s\])]{}. In a second step we define a special linear subspace $V \subseteq {\mathcal{P}}_d^{\gamma}$ with $\dim V = 2^s$. Step 3 then shows that $V$ satisfies the assumptions of . The proof is completed in Step 4. *Step 1*. For $k\in\{0,\ldots,d\}$ let us define inductively $m_0=0$ and $$\begin{gathered} m_k = \inf \left\{t \in {\mathbb{N}}{\; \vrule \;}m_{k-1} < t \leq d, \, \text{ with } \, 2 \leq \sum_{j=m_{k-1}+1}^t \gamma_{d,j} \right\}\end{gathered}$$ with the usual convention $\inf {\emptyset}= \infty$. Note that the infimum coincides with the minimum in the finite case, since then $m_k\in{\mathbb{N}}$. Moreover we set $$\begin{gathered} s = { \mathop{\mathrm{max}}\left\{ k\in\{0,\ldots,d\} {\; \vrule \;}m_k < \infty \right\} }.\end{gathered}$$ We denote $I_k = \{ m_{k-1}+1, m_{k-1}+2, \ldots, m_k \}$ for $k=1,\ldots,s$. Thus, this gives a uniquely defined disjoint partition of the set $$\begin{gathered} \{1,\ldots,d\} = \left( \bigcup_{k=1}^s I_k \right) \cup \{m_s+1,\ldots,d\}, \end{gathered}$$ and $m_k$ denotes the last element of the block $I_k$. For all $k=1,\ldots,s$ we conclude $$\begin{gathered} 2 \leq \sum_{j\in I_k} \gamma_{d,j} < 2 + \gamma_{d,m_k} \leq 2 + C_\gamma,\end{gathered}$$ where $C_\gamma$ is the uniform upper bound for $\gamma_{d,j}$; see . Finally, summation of these inequalities gives $$\begin{gathered} \sum_{j=1}^{d} \gamma_{d,j} < \sum_{k=1}^s \sum_{j\in I_k} \gamma_{d,j} + 2 < (2+C_\gamma) s + 2,\end{gathered}$$ and [(\[estimate\_s\])]{} follows immediately. If $s=0$ then we can stop at this point since the initial error is $1$ as the norm of the embedding ${\mathcal{P}}_d^\gamma\hookrightarrow {\mathrm{L}}_\infty$ (cf. ) and the remaining assertion is trivial. Hence, from now on we can assume that $s>0$ and thus $m_s \geq 1$. *Step 2*. To apply we have to construct a linear subspace $V$ of ${\mathcal{F}}= {\mathcal{P}}_d^\gamma$ such that the condition [(\[eq:NormCondition\])]{} holds for the target space ${\mathcal{G}}= {\mathrm{L}}_\infty([0,1]^d)$, the embedding operator $S={\mathrm{id}}_d$, and $a = 1$. Note that we restrict ourselves to the set $$\begin{gathered} \widehat{{\mathcal{F}}} = \left\{ f\in {\mathcal{F}}{\; \vrule \;}\ f \text{ depends only on } x_1,\ldots,x_{m_s} \right\},\end{gathered}$$ since we can interpret $\widehat{{\mathcal{F}}}$ as the space ${\mathcal{P}}_{m_s}^\gamma$ by a simple isometric isomorphism. We are ready to construct a suitable space $V$ using the partition from Step 1. We define $V$ as the span of all functions $g_{\bm{i}} \colon [0,1]^{m_s} {\rightarrow}{\mathbb{R}}$, $\bm{i}=(i_1,\dots,i_s) \in\{0,1\}^s$, of the form $$\begin{gathered} g_{\bm{i}}(\bm{x}) = \prod_{k=1}^s \left( \sum_{j\in I_k} \gamma_{d,j} \cdot x_j \right)^{i_k}, \quad \bm{x} \in X = [0,1]^{m_s}.\end{gathered}$$ Clearly, $V$ is a linear subspace of ${\mathcal{P}}_{m_s}^\gamma$ and with the interpretation above it is also a linear subspace of ${\mathcal{F}}$. Moreover it is easy to see that we have by construction $$\begin{gathered} {\left\| g {\; \vrule \;}{\mathcal{F}}\right\|} = {\left\| g {\; \vrule \;}{\mathcal{P}}_{m_s}^\gamma \right\|} \quad \text{and}\quad {\left\| g {\; \vrule \;}{\mathrm{L}}_\infty(X) \right\|} = {\left\| g {\; \vrule \;}{\mathrm{L}}_\infty([0,1]^d) \right\|} \quad \text{for} \quad g\in V.\end{gathered}$$ Finally we note that $\dim V = \# \{0,1\}^s = 2^s$. It remains to show that this subspace is the right choice to prove the claim using . *Step 3*. The proof of the needed condition [(\[eq:NormCondition\])]{}, $$\begin{gathered} {\left\| g {\; \vrule \;}{\mathcal{P}}_{m_s}^\gamma \right\|} \leq {\left\| g {\; \vrule \;}{\mathrm{L}}_\infty(X) \right\|} \quad \text{for all} \quad g\in V,\end{gathered}$$ is a little bit technical. Due to the special structure of the functions $g\in V$, the left-hand side reduces to ${ \mathop{\mathrm{max}}\left\{\gamma_{\bm{\alpha}}^{-1} {\left\| D^{\bm{\alpha}} g {\; \vrule \;}{\mathrm{L}}_\infty(X) \right\|} {\; \vrule \;}\bm{\alpha}\in \mathbb{M}\right\} }$, where the maximum is taken over all multi-indices $\bm{\alpha}$ in the set $$\begin{gathered} \mathbb{M}= \left\{ {\bm{\alpha}} \in \{0,1\}^{m_s} {\; \vrule \;}\sum_{j\in I_k} \alpha_j \leq 1 \text{ for all } k=1,\ldots,s \right\}.\end{gathered}$$ This is simply because for ${\bm{\alpha}} \notin \mathbb{M}$ we have $D^{\bm{\alpha}} g \equiv 0$ and then the inequality is trivial. To simplify the notation let us define $$\begin{gathered} T \colon \{0,1\}^{m_s} {\rightarrow}{\mathbb{N}}_0^s, \quad {\bm{\alpha}} \mapsto T({\bm{\alpha}}) = \bm{\sigma} = (\sigma_1,\dots,\sigma_s),\end{gathered}$$ where $$\begin{gathered} \sigma_k= \sum_{j\in I_k} \alpha_j \quad \text{for} \quad k=1,\ldots,s.\end{gathered}$$ Note that $T(\mathbb{M})=\{0,1\}^s$. Moreover, for every $g=\sum_{\bm{i}\in\{0,1\}^s} c_{\bm{i}} \, g_{\bm{i}}(\cdot) \in V$ we define a function $$\begin{gathered} h_g \colon Z= {{\mathop {\mathchoice {\raise-0.22em\hbox{\huge $\times$}} {\raise-0.05em\hbox{\Large $\times$}}{\hbox{\large $\times$}}{\times} }}}_{k=1}^s \left[ 0, \sum_{j\in I_k} \gamma_{d,j} \right] {\rightarrow}{\mathbb{R}}, \qquad \bm{z} \mapsto h_g(\bm{z}) = \sum_{\bm{i}\in \{0,1\}^s} c_{\bm{i}} \prod_{k=1}^s z_k^{i_k} = \sum_{\bm{i}\in \{0,1\}^s} c_{\bm{i}} \, \bm{z}^{\bm{i}}.\end{gathered}$$ Hence, $h_g(\bm{z})=g(\bm{x})$ under the transformation $\bm{x}\mapsto \bm{z}$ such that $$\begin{gathered} z_k = \sum_{j\in I_k} \gamma_{d,j} x_j \quad \text{for every}\quad k=1,\ldots,s \quad \text{and every}\quad \bm{x}\in X. \end{gathered}$$ The span, $W$, of all functions $h\colon Z {\rightarrow}{\mathbb{R}}$ with this structure is a linear space, too. Furthermore, easy calculus yields that $$\begin{gathered} \label{g_and_h} \left( D_{\bm{x}}^{\bm{\alpha}} g \right)(\bm{x}) = \left( \prod_{j=1}^{m_s} \left( \gamma_{d,j} \right)^{\alpha_j} \right) \left( D_{\bm{z}}^{T({\bm{\alpha}})} h_g \right)(\bm{z}) \end{gathered}$$ for all $g\in V$, $\bm{\alpha} \in \mathbb{M}$ and $\bm{x}\in X$. Here the $\bm{x}$ and $\bm{z}$ in $D_{\bm{x}}^{\bm{\alpha}}$ and $D_{\bm{z}}^{T({\bm{\alpha}})}$ indicate differentiation with respect to $\bm{x}$ and $\bm{z}$, respectively. Since the mapping $\bm{x} \mapsto \bm{z}$ is surjective we obtain ${\left\| D^{\bm{\alpha}} g {\; \vrule \;}{\mathrm{L}}_\infty(X) \right\|} = \gamma_{\bm{\alpha}} {\left\| D^{T({\bm{\alpha}})} h_g {\; \vrule \;}{\mathrm{L}}_\infty(Z) \right\|}$ by the form of $\gamma$ given by [(\[ProdWeights\])]{}. Thus, $$\begin{gathered} {\mathop{\mathrm{max}}\displaylimits}_{{\bm{\alpha}} \in \mathbb{M}} \frac{1}{\gamma_{\bm{\alpha}}} {\left\| D^{\bm{\alpha}} g {\; \vrule \;}{\mathrm{L}}_\infty(X) \right\|} = {\mathop{\mathrm{max}}\displaylimits}_{\bm{\sigma} \in \{0,1\}^s} {\left\| D^{\bm{\sigma}} h_g {\; \vrule \;}{\mathrm{L}}_\infty(Z) \right\|}.\end{gathered}$$ Observe that [(\[g\_and\_h\])]{} with $\bm{\alpha}=0$ particularly yields that ${\left\| g {\; \vrule \;}{\mathrm{L}}_\infty(X) \right\|} = {\left\| h_g {\; \vrule \;}{\mathrm{L}}_\infty(Z) \right\|}$. Therefore the claim reduces to $$\begin{gathered} {\mathop{\mathrm{max}}\displaylimits}_{\bm{\sigma} \in \{0,1\}^s} {\left\| D^{\bm{\sigma}} h_g {\; \vrule \;}{\mathrm{L}}_\infty(Z) \right\|} \leq {\left\| h_g {\; \vrule \;}{\mathrm{L}}_\infty(Z) \right\|} \quad \text{for every} \quad g\in V.\end{gathered}$$ We show this estimate for every $h \in W$, i.e., $$\begin{gathered} \label{estimate_norm_h_1} {\left\| D^{\bm{\sigma}} h {\; \vrule \;}{\mathrm{L}}_\infty(Z) \right\|} \leq {\left\| h {\; \vrule \;}{\mathrm{L}}_\infty(Z) \right\|} \quad \text{for all} \quad \bm{\sigma} \in \{0,1\}^s.\end{gathered}$$ We start with the special case of one derivative. That is, we first consider $\bm{\sigma} = \bm{e_k}$ for a certain $k\in \{1,\ldots,s\}$. Since $h$ is affine in each coordinate we can represent it as $$\begin{gathered} h(\bm{z}) = a({\bm{z_{(k)}}}) \cdot z_k + b({\bm{z_{(k)}}})\end{gathered}$$ with functions $a$ and $b$ which only depend on ${\bm{z_{(k)}}}=(z_1,\ldots,z_{k-1},z_{k+1},\ldots,z_s)$. Hence we have $(D^{\bm{e_k}} h)(\bm{z}) = a({\bm{z_{(k)}}})$ and we need to show that $$\begin{gathered} \label{estimate_norm_h_2} {\left| a({\bm{z_{(k)}}}) \right|} \leq { \mathop{\mathrm{max}}\left\{ {\left| b({\bm{z_{(k)}}}) \right|}, {\left| a({\bm{z_{(k)}}}) \cdot \sum_{j\in I_k} \gamma_{d,j} + b({\bm{z_{(k)}}}) \right|}\right\} }.\end{gathered}$$ This is obviously true for every $\bm{z} \in Z$ with $a({\bm{z_{(k)}}})=0$. For $a({\bm{z_{(k)}}}) \neq 0$ we can divide by ${\left| a({\bm{z_{(k)}}}) \right|}$ to get $$\begin{gathered} 1 \leq { \mathop{\mathrm{max}}\left\{ {\left| t \right|}, {\left| \sum_{j\in I_k} \gamma_{d,j} - t \right|}\right\} }\end{gathered}$$ if we set $t=-b({\bm{z_{(k)}}})/a({\bm{z_{(k)}}})$. The last maximum is minimal if both of its entries coincide. This is for $t=\frac{1}{2} \sum_{j\in I_k} \gamma_{d,j}$. Consequently, we need to ensure that $$\begin{gathered} 2 \leq \sum_{j\in I_k} \gamma_{d,j}\end{gathered}$$ to conclude [(\[estimate\_norm\_h\_2\])]{} for all admissible $\bm{z}\in Z$. But this is true for every $k\in \{1,\ldots,s\}$ by definition of the sets $I_k$ in Step 1. Thus we have shown [(\[estimate\_norm\_h\_1\])]{} for the special case $\bm{\sigma}=\bm{e_k}$ for all $k\in \{1,\ldots,s\}$. The inequality [(\[estimate\_norm\_h\_1\])]{} also holds true for every $\bm{\sigma} \in\{0,1\}^s$ by an easy inductive argument on the cardinality of ${\left| \bm{\sigma} \right|}$. Indeed, if ${\left| \bm{\sigma} \right|} \geq 2$ then $\bm{\sigma}=\bm{\sigma'}+\bm{e_k}$ with ${\left| \bm{\sigma'} \right|} = {\left| \bm{\sigma} \right|}-1$. We now need to estimate ${\left\| D^{\bm{\sigma'}+\bm{e_k}}h {\; \vrule \;}{\mathrm{L}}_\infty(Z) \right\|}$. Since $(D^{\bm{e_k}} h)(\bm{z}) = a({\bm{z_{(k)}}})$ has the same structure as the function $h$ itself, we see that ${\left\| D^{\bm{\sigma'}+\bm{e_k}} h {\; \vrule \;}{\mathrm{L}}_\infty(Z) \right\|}$ equals ${\left\| D^{\bm{\sigma'}} a({\bm{z_{(k)}}}) {\; \vrule \;}{\mathrm{L}}_\infty(Z) \right\|}$ and the proof of [(\[estimate\_norm\_h\_1\])]{} then is completed by the inductive step. *Step 4*. Collecting the previous equalities and estimates we obtain $$\begin{aligned} {\left\| g {\; \vrule \;}{\mathcal{P}}_d^\gamma \right\|} &= {\left\| g {\; \vrule \;}{\mathcal{P}}_{m_s}^\gamma \right\|} = {\mathop{\mathrm{max}}\displaylimits}_{\substack{{\bm{\alpha}}\in\{0,1\}^{m_s}\\ T({\bm{\alpha}}) \in \{0,1\}^s}} \frac{1}{\gamma_{{\bm{\alpha}}}} {\left\| D^{\bm{\alpha}} g {\; \vrule \;}{\mathrm{L}}_\infty(X) \right\|} = {\mathop{\mathrm{max}}\displaylimits}_{\bm{\sigma} \in \{0,1\}^s} {\left\| D^{\bm{\sigma}} h_g {\; \vrule \;}{\mathrm{L}}_\infty(Z) \right\|} \\ &\leq {\left\| h_g {\; \vrule \;}{\mathrm{L}}_\infty(Z) \right\|} = {\left\| g {\; \vrule \;}{\mathrm{L}}_\infty(X) \right\|} = {\left\| g {\; \vrule \;}{\mathrm{L}}_\infty([0,1]^d) \right\|}\end{aligned}$$ for every $g \in V$, where $V$ is a linear subspace of ${\mathcal{F}}= {\mathcal{P}}_d^\gamma$ with $\dim V = 2^s$. Therefore with $a = 1$ yields that for $n < \dim V$ the worst case error $$\begin{gathered} \Delta^{\mathrm{wor}}(A_{n,d}; {\mathrm{id}}_d \colon B_r({\mathcal{P}}_d^\gamma){\rightarrow}{\mathrm{L}}_\infty([0,1]^d))\end{gathered}$$ of any algorithm $A_{n,d}$ from the class ${\mathcal{A}}_d^{n,\rm cont} \cup {\mathcal{A}}_d^{n,\rm adapt}$ is lower bounded by $r$, the radius of the centered ball $B_r({\mathcal{P}}_d^\gamma)$. ### Complexity results via embeddings {#sect:embeddings} Keeping in mind the assertions shown in the previous sections, we are ready to give conditions for tractability of the uniform approximation problem $$\begin{gathered} \mathrm{App} = (\mathrm{App}_d)_{d\in{\mathbb{N}}}, \qquad \mathrm{App}_d \colon {\mathcal{B}}({\mathcal{F}}_d^\gamma) {\rightarrow}{\mathrm{L}}_\infty([0,1]^d), \quad \mathrm{App}_d(f) ={\mathrm{id}}_d(f)= f.\end{gathered}$$ We suppose $({\mathcal{F}}_d^\gamma)_{d\in{\mathbb{N}}}$ to be a sequence of Banach spaces of real-valued functions $f$ defined on the unit cube $[0,1]^d$. We further assume that this sequence depends on product weights $\gamma=(\gamma_{\bm{\alpha}})_{\bm{\alpha}\in{\mathbb{N}}_0^d}$ and fulfills one of the following simple assumptions: 1. \[Assumption1\] ${\mathcal{P}}_d^{\gamma} \hookrightarrow {\mathcal{F}}_d^\gamma$ with norm $$\begin{gathered} C_{1,d}\leq c \cdot d^{q_1} \quad \text{for all}\quad d\in{\mathbb{N}}\end{gathered}$$ and some absolute constants $c,q_1 \geq 0$, 2. \[Assumption2\] ${\mathcal{F}}_d^\gamma \hookrightarrow {\mathcal{H}}_d^\gamma$ with norm $$\begin{gathered} \label{eq:bound_c2} C_{2,d} \leq a \cdot { \mathop{\mathrm{exp}}\left( b\cdot \sum_{j=1}^d (\gamma_{d,j})^t \right) } \quad \text{for all} \quad d\in{\mathbb{N}}\end{gathered}$$ and some absolute constants $a>0$, $b\geq 0$, as well as a parameter $t\in(0,1]$ independent of $d$ and $\gamma$. Here the spaces ${\mathcal{P}}_d^{\gamma}$ and ${\mathcal{H}}_d^\gamma={\mathcal{H}}(K_d^\gamma)$ are defined as in and , respectively. To simplify the notation we use the commonly known definitions of the so-called *sum exponents*[^8] for the product weight sequence $\gamma=(\gamma_{\bm{\alpha}})_{\bm{\alpha}\in{\mathbb{N}}_0^d}$, $d\in{\mathbb{N}}$, induced by uniformly bounded generator weights $0<\gamma_{d,j}\leq C_\gamma$, $j=1,\ldots,d$; see [(\[ProdWeights\])]{}. We set $$\begin{gathered} p(\gamma) = \inf\left\{ \kappa \geq 0 {\; \vrule \;}P_\kappa(\gamma) = \limsup_{d{\rightarrow}\infty} \sum_{j=1}^d \left( \gamma_{d,j} \right)^\kappa < \infty \right\},\end{gathered}$$ as well as $$\begin{gathered} q(\gamma) = \inf\left\{ \kappa \geq 0 {\; \vrule \;}Q_\kappa(\gamma) = \limsup_{d{\rightarrow}\infty} \frac{\sum_{j=1}^d \left( \gamma_{d,j} \right)^\kappa}{\ln(d+1)} < \infty \right\},\end{gathered}$$ with the usual convention that $\inf {\emptyset}= \infty$. The following necessary conditions for (strong) polynomial tractability slightly generalize Theorem 2 of [@W12]. \[Thm\_Necessary\] Assume that [(\[Assumption1\])]{} holds true with some $q_1\geq 0$. Consider ${\mathrm{L}}_\infty$-approximation over $({\mathcal{F}}_d^\gamma)_{d\in{\mathbb{N}}}$ in the worst case setting with respect to the class of algorithms ${\mathcal{A}}_d^{n,\rm cont} \cup {\mathcal{A}}_d^{n,\rm adapt}$ and the absolute error criterion. Then $$\begin{gathered} \label{LowerBound_n} n^{\mathrm{wor}}({\varepsilon},d; \mathrm{App}_d) > \frac{1}{2} \cdot 2^\wedge\!\!\left( \frac{1}{2+C_\gamma} \sum_{j=1}^d \gamma_{d,j} \right) \end{gathered}$$ for all $d\in {\mathbb{N}}$ and every ${\varepsilon}\in (0,C_{1,d}^{-1})$. Hence, - if the problem $\mathrm{App}$ is polynomially tractable then $q(\gamma) \leq 1$, - if $q_1=0$ and the problem is strongly polynomially tractable then $p(\gamma) \leq 1$. Let $d\in{\mathbb{N}}$. Due to [(\[Assumption1\])]{}, every algorithm $A_{n,d}\in {\mathcal{A}}_d^{n,\rm cont} \cup {\mathcal{A}}_d^{n,\rm adapt}$ for ${\mathrm{L}}_\infty$-approximation defined on ${\mathcal{F}}_d^\gamma$ also applies to the embedded space ${\mathcal{P}}_d^\gamma$. Furthermore the embedding constant $C_{1,d}$ implies that the ball $B_r({\mathcal{P}}_d^\gamma)$ of radius $r=C_{1,d}^{-1}$ in ${\mathcal{P}}_d^\gamma$ is completely contained in the unit ball ${\mathcal{B}}({\mathcal{F}}_d^\gamma)$ of ${\mathcal{F}}_d^\gamma$. Therefore, $$\begin{aligned} &\Delta^{{\mathrm{wor}}}(A_{n,d}; \mathrm{App}_d\colon {\mathcal{B}}({\mathcal{F}}_d^\gamma) {\rightarrow}{\mathrm{L}}_\infty([0,1]^d)) \\ &\qquad\qquad\geq \Delta^{{\mathrm{wor}}}\left(A_{n,d} \big|_{{\mathcal{P}}_d^\gamma}; {\mathrm{id}}_d\colon B_r({\mathcal{P}}_d^\gamma){\rightarrow}{\mathrm{L}}_\infty([0,1]^d)\right).\end{aligned}$$ From we have that the latter quantity is lower bounded by $r=C_{1,d}^{-1}$ provided that $n<2^s$, where $s=s(\gamma,d)\in\{0,\ldots,d\}$ satisfies [(\[estimate\_s\])]{}. Since this lower bound holds for any such $A_{n,d}$ it remains valid for the $n$th minimal error, [i.e. ]{}$$\begin{gathered} e^{\mathrm{wor}}(n,d;\mathrm{App}_d) \geq C_{1,d}^{-1} \quad \text{for all} \quad n < 2^s.\end{gathered}$$ Hence we obtain $n^{\mathrm{wor}}({\varepsilon},d; \mathrm{App}_d) \geq 2^s$ for all $d\in{\mathbb{N}}$ and every ${\varepsilon}\in (0,C_{1,d}^{-1})$ which implies [(\[LowerBound\_n\])]{} using [(\[estimate\_s\])]{}. Now suppose the problem $\mathrm{App} = (\mathrm{App}_d)_{d\in{\mathbb{N}}}$ to be polynomially tractable. Then there are constants $C,p>0$ and $q_2\geq0$ such that $$\begin{gathered} n^{\mathrm{wor}}({\varepsilon},d; \mathrm{App}_d) \leq C \, {\varepsilon}^{-p} \, d^{q_2} \quad \text{for all} \quad d\in{\mathbb{N}}\quad \text{and} \quad {\varepsilon}\in (0,1].\end{gathered}$$ For any given $d\in{\mathbb{N}}$ we can take, say, ${\varepsilon}= {\varepsilon}(d) = \frac{1}{2} \cdot { \mathop{\mathrm{min}}\left\{1, C_{1,d}^{-1}\right\} }$ to conclude $$\begin{gathered} \label{eq:strong_res} 2^\wedge\!\!\left( \frac{1}{2+C_\gamma} \sum_{j=1}^d \gamma_{d,j} \right) < C' \, { \mathop{\mathrm{max}}\left\{1,C_{1,d}^p\right\} }\, d^{q_2}\end{gathered}$$ for some $C'>0$ independent of $d$. If we now assume that $C_{1,d}\in{\mathcal{O}}(d^{q_1})$ then the right-hand side of the last inequality belongs to ${\mathcal{O}}(d^{p q_1+q_2})$, as $d{\rightarrow}\infty$. Provided that ${ \mathop{\mathrm{max}}\left\{q_1,q_2\right\} }>0$ this is equivalent to the boundedness of $\sum_{j=1}^d \gamma_{d,j} / \ln(d+1)$ such that we arrive at $q(\gamma) \leq 1$, as claimed. Finally, the case of strong polynomial tractability can be treated similarly by setting $q_1=q_2=0$ in the latter bounds. Then we obtain that $\sum_{j=1}^d \gamma_{d,j}$ is uniformly bounded in $d$ which implies $p(\gamma) \leq 1$. Of course, the conditions $q(\gamma)\leq 1$ and $p(\gamma)\leq 1$ are also necessary for polynomial and strong polynomial tractability with respect to smaller classes of algorithms such as, e.g., ${\mathcal{A}}_d^{n,\mathrm{lin}}(\Lambda^{{\mathrm{all}}})$. Observe that one of the improvements compared to [@W12 Theorem 2] is the possibility to choose the uniform upper bound for the generator weights, $C_\gamma$, different than $1$. Moreover, now we have weaker conditions on the embedding constant $C_{1,d}$. For the application we have in mind we will see that there still $C_{1,d}=1$. But we note in passing that the stated conclusions for (strong) polynomial tractability are only special instances of the more general bound [(\[eq:strong\_res\])]{} obtained in the latter proof which we will not investigate further. We next assume [(\[Assumption2\])]{} and show that slightly stronger conditions on the product weights $\gamma$ than in are sufficient for polynomial and strong polynomial tractability, respectively. This is stated in the next assertion which can be found as Theorem 3 in [@W12]. \[Theorem\_Sufficient\] Suppose that [(\[Assumption2\])]{} holds true with some $t\in(0,1]$. Consider ${\mathrm{L}}_\infty$-approximation over $({\mathcal{F}}_d^\gamma)_{d\in{\mathbb{N}}}$ in the worst case setting with respect to the class of linear algorithms ${\mathcal{A}}_d^{n,\mathrm{lin}}(\Lambda^{{\mathrm{all}}})$ and the absolute error criterion. Then - $q(\gamma)<t$ implies polynomial tractability, - $p(\gamma)<t$ implies strong polynomial tractability. Due to [(\[Assumption2\])]{}, the restriction of the algorithm $A_{n,d}^*$ in from ${\mathcal{H}}_d^\gamma$ to ${\mathcal{F}}_d^\gamma$ is admissible for ${\mathrm{L}}_\infty$-approximation over ${\mathcal{F}}_d^\gamma$. Furthermore, due to the linearity of $A_{n,d}^*$, we have $$\begin{aligned} {\left\| f-A^*_{n,d}f {\; \vrule \;}{\mathrm{L}}_\infty([0,1]^d) \right\|} &\leq \Delta^{{\mathrm{wor}}}(A_{n,d}^*; {\mathrm{id}}_d\colon{\mathcal{B}}({\mathcal{H}}_d^\gamma){\rightarrow}{\mathrm{L}}_\infty([0,1]^d)) \cdot {\left\| f{\; \vrule \;}{\mathcal{H}}_d^\gamma \right\|} \\ &\leq \Delta^{{\mathrm{wor}}}(A_{n,d}^*;{\mathrm{id}}_d\colon{\mathcal{B}}({\mathcal{H}}_d^\gamma){\rightarrow}{\mathrm{L}}_\infty([0,1]^d)) \cdot C_{2,d} \cdot {\left\| f {\; \vrule \;}{\mathcal{F}}_d^\gamma \right\|}\end{aligned}$$ for all $f\in {\mathcal{F}}_d^\gamma$. Therefore we can estimate the $n$th minimal error by $$\begin{aligned} e^{\mathrm{wor}}(n,d;\mathrm{App}_d) &\leq \Delta^{{\mathrm{wor}}}\left(A_{n,d}^* \big|_{{\mathcal{F}}_d^\gamma}; \mathrm{App}_d\colon{\mathcal{B}}({\mathcal{F}}_d^\gamma){\rightarrow}{\mathrm{L}}_\infty([0,1]^d) \right) \\ &\leq C_{2,d} \cdot \Delta^{{\mathrm{wor}}}(A_{n,d}^*; {\mathrm{id}}_d\colon{\mathcal{B}}({\mathcal{H}}_d^\gamma){\rightarrow}{\mathrm{L}}_\infty([0,1]^d) \\ &\leq a \cdot a_\tau \cdot { \mathop{\mathrm{exp}}\left( b \sum_{j=1}^d \left( \gamma_{d,j}\right)^t + b_\tau \sum_{j=1}^d \left( \gamma_{d,j}\right)^\tau \right) } \cdot n^{-(1-\tau)/(2\tau)},\end{aligned}$$ where $\tau$ is an arbitrary number from $(1/2, 1)$. Choosing $n$ such that the right-hand side is not greater than a given ${\varepsilon}\in(0,1]$, we obtain an estimate for the information complexity with respect to the class of linear algorithms, $$\begin{gathered} \label{UpperBound_tau} n^{\mathrm{wor}}({\varepsilon},d;\mathrm{App}_d) \leq c_1 \cdot {\varepsilon}^{-2\tau/(1-\tau)} \cdot { \mathop{\mathrm{exp}}\left( c_2 \sum_{j=1}^d \left( \gamma_{d,j} \right)^t + c_3 \sum_{j=1}^d \left( \gamma_{d,j} \right)^{\tau} \right) },\end{gathered}$$ where the non-negative constants $c_1$, $c_2$ and $c_3$ only depend on $\tau$, $a$ and $b$. Suppose that $q(\gamma)<t$. Then $Q_\kappa(\gamma)$ is finite for every $\kappa >q(\gamma)$. Taking $\kappa=t$ we obtain $$\begin{gathered} \frac{\sum_{j=1}^d \left( \gamma_{d,j} \right)^t}{\ln(d+1)} \cdot \ln(d+1) \leq (Q_t(\gamma)+\delta) \cdot \ln(d+1) = \ln (d+1)^{Q_t(\gamma) + \delta}\end{gathered}$$ for every $\delta>0$ whenever $d$ is larger than a certain $d_\delta\in{\mathbb{N}}$. This means that the factor ${ \mathop{\mathrm{exp}}\left( c_2\sum_{j=1}^d(\gamma_{d,j})^t \right) }$ in [(\[UpperBound\_tau\])]{} is polynomially dependent on $d$. On the other hand, we can choose $\tau \in ({ \mathop{\mathrm{max}}\left\{q(\gamma),1/2\right\} },1)$ such that $Q_\tau(\gamma)$ is finite and thus the factor ${ \mathop{\mathrm{exp}}\left( c_3\sum_{j=1}^d(\gamma_{d,j})^\tau \right) }$ in [(\[UpperBound\_tau\])]{} is also polynomially dependent on $d$. So, for this value of $\tau$ we can rewrite [(\[UpperBound\_tau\])]{} as $$\begin{gathered} n^{\mathrm{wor}}({\varepsilon},d;\mathrm{App}_d) \in {\mathcal{O}}\left({\varepsilon}^{-2\tau/(1-\tau)} \cdot (d+1)^{c_4} \right),\end{gathered}$$ with $c_4$, as well as the implied factor in the ${\mathcal{O}}$-notation, independent of $d$ and ${\varepsilon}$ which means that the problem is polynomially tractable, as claimed. Suppose finally that $p(\gamma)<t$. Then the sums $\sum_{j=1}^d(\gamma_{d,j})^t$ and $\sum_{j=1}^d(\gamma_{d,j})^\tau$ for $\tau\in({ \mathop{\mathrm{max}}\left\{p(\gamma),1/2\right\} },1)$ are both uniformly bounded in $d$. Consequently [(\[UpperBound\_tau\])]{} yields strong polynomial tractability, and completes the proof. The conditions in are obviously also sufficient if we consider larger classes of algorithms such as, [e.g.]{}, ${\mathcal{A}}_d^{n,\mathrm{cont}} \cup {\mathcal{A}}_d^{n,\mathrm{adapt}}$. Moreover note that the given proof also provides explicit upper bounds for the exponents of tractability. Let us briefly discuss the different roles of the assumptions [(\[Assumption1\])]{} and [(\[Assumption2\])]{} in the following remark. Assumption [(\[Assumption1\])]{} is used to find a lower bound on the information complexity for the space ${\mathcal{F}}_d^\gamma$ as long the space ${\mathcal{P}}_d^\gamma$ is continuously embedded in ${\mathcal{F}}_d^\gamma$ with an embedding constant which grows at most polynomially with the dimension $d$. Such an embedding can be shown for several different classes of functions. On the other hand, assumption [(\[Assumption2\])]{} is used to find an upper bound on the information complexity for the space ${\mathcal{F}}_d^\gamma$ as long as it is continuously embedded in the unanchored weighted Sobolev space ${\mathcal{H}}_d^\gamma={\mathcal{H}}(K_d^\gamma)$ with an embedding constant depending exponentially on the sum of some power of the generators $\gamma_{d,j}$ of the product weights $\gamma$. This considerably restricts the choice of ${\mathcal{F}}_d^\gamma$. We need this assumption in order to use the linear algorithm $A_{n,d}^*$ defined on the space ${\mathcal{H}}_d^\gamma$ and the error bound given in . Obviously, we can replace the space ${\mathcal{H}}_d^\gamma$ in [(\[Assumption2\])]{} by any other space which contains at least ${\mathcal{P}}_d^\gamma$ and for which we know a linear algorithm using $n$ linear functionals whose worst case error is polynomial in $n^{-1}$ with an explicit dependence on the product weights $\gamma$. $\square$ We now show that the assumptions [(\[Assumption1\])]{} and [(\[Assumption2\])]{} allow us to characterize weak tractability and the curse of dimensionality. \[Theorem\_Equivalence\] Suppose that for a sequence of Banach spaces $({\mathcal{F}}_d^\gamma)_{d\in{\mathbb{N}}}$ equipped with product weights $\gamma$ the assumptions [(\[Assumption1\])]{} and [(\[Assumption2\])]{} hold true with some parameter $t\in(0,1]$. Consider the ${\mathrm{L}}_\infty$-approximation problem $\mathrm{App}$ in the worst case setting and with respect to the absolute error criterion. Then the following statements are equivalent: 1. The problem is weakly tractable with respect to the class ${\mathcal{A}}_d^{n, {\mathrm{lin}}}(\Lambda^{{\mathrm{all}}})$. \[Equi\_1\] 2. The problem is weakly tractable with respect to the class ${\mathcal{A}}_d^{n,\mathrm{cont}} \cup {\mathcal{A}}_d^{n,\mathrm{adapt}}$. \[Equi\_2\] 3. There is no curse of dimensionality for the class ${\mathcal{A}}_d^{n, {\mathrm{lin}}}(\Lambda^{{\mathrm{all}}})$. \[Equi\_3\] 4. There is no curse of dimensionality for the class ${\mathcal{A}}_d^{n,\mathrm{cont}} \cup {\mathcal{A}}_d^{n,\mathrm{adapt}}$. \[Equi\_4\] 5. For all $\kappa > 0$ we have $\lim_{d{\rightarrow}\infty}\limits \frac{1}{d} \sum_{j=1}^d \left( \gamma_{d,j} \right)^\kappa = 0$. \[Equi\_5\] 6. There exists $\kappa \in (0,t)$ such that $\lim_{d{\rightarrow}\infty}\limits \frac{1}{d} \sum_{j=1}^d \left( \gamma_{d,j} \right)^\kappa = 0$. \[Equi\_6\] We start by showing that [(\[Equi\_6\])]{} implies [(\[Equi\_1\])]{}, i.e., $$\begin{gathered} \lim_{{\varepsilon}^{-1}+d {\rightarrow}\infty} \frac{\ln \left(n^{\mathrm{wor}}({\varepsilon},d; \mathrm{App}_d)\right)}{{\varepsilon}^{-1}+d} = 0,\end{gathered}$$ where the information complexity is taken with respect to the class ${\mathcal{A}}_d^{n, {\mathrm{lin}}}(\Lambda^{{\mathrm{all}}})$ of linear algorithms that use continuous linear functionals. By the arguments used in the proof of we obtain estimate [(\[UpperBound\_tau\])]{} for all ${\varepsilon}$ in $(0,1]$, as well as for every $d\in {\mathbb{N}}$, and all $\tau \in (1/2,1)$, due to assumption [(\[Assumption2\])]{}. Clearly, for $\kappa \in (0,t)$ as in the hypothesis and $t\in(0,1]$ as in the embedding condition, we find $\tau \in (1/2, 1)$ such that $\kappa < { \mathop{\mathrm{min}}\left\{t,\tau\right\} }$. So, since $\gamma_{d,j} \leq C_\gamma$, we can estimate $$\begin{gathered} \sum_{j=1}^d \left( \gamma_{d,j} \right)^s = C_\gamma^s \cdot \sum_{j=1}^d \left(\frac{\gamma_{d,j}}{C_\gamma}\right)^s \leq C_\gamma^{s-\kappa} \cdot \sum_{j=1}^d \left( \gamma_{d,j} \right)^\kappa \leq C \cdot \sum_{j=1}^d \left( \gamma_{d,j} \right)^\kappa,\end{gathered}$$ where $s$ either equals $t$ or $\tau$ and $C={ \mathop{\mathrm{max}}\left\{1,C_\gamma\right\} }$. Therefore the right-hand side of [(\[UpperBound\_tau\])]{} can be estimated from above and thus $$\begin{gathered} \frac{\ln \left( n^{\mathrm{wor}}({\varepsilon},d;\mathrm{App}_d) \right)}{{\varepsilon}^{-1}+d} \leq \frac{\ln(c_1)}{{\varepsilon}^{-1}+d} + \frac{2\tau}{1-\tau} \cdot \frac{\ln \left( {\varepsilon}^{-1} \right)}{{\varepsilon}^{-1}+d} + C \cdot { \mathop{\mathrm{max}}\left\{c_2,c_3\right\} } \cdot \frac{\sum_{j=1}^d \left( \gamma_{d,j} \right)^\kappa}{{\varepsilon}^{-1}+d}\end{gathered}$$ tends to zero when ${\varepsilon}^{-1}+d$ approaches infinity, as claimed. Clearly, [(\[Equi\_1\])]{} $\Rightarrow$ [(\[Equi\_2\])]{} $\Rightarrow$ [(\[Equi\_4\])]{} and [(\[Equi\_1\])]{} $\Rightarrow$ [(\[Equi\_3\])]{} $\Rightarrow$ [(\[Equi\_4\])]{}. Moreover the implication from [(\[Equi\_5\])]{} to [(\[Equi\_6\])]{} is obvious. Hence, it only remains to show that [(\[Equi\_4\])]{} $\Rightarrow$ [(\[Equi\_5\])]{}. From [(\[Assumption1\])]{} we have estimate [(\[LowerBound\_n\])]{}. Then the absence of the curse of dimensionality implies $$\begin{gathered} \lim_{d\to\infty}\frac1d\,\sum_{j=1}^d\gamma_{d,j}=0.\end{gathered}$$ Now Jensen’s inequality yields that $$\begin{gathered} \frac{1}{d} \,\sum_{j=1}^d \gamma_{d,j} \geq \left( \frac{1}{d} \,\sum_{j=1}^d \left( \gamma_{d,j} \right)^\kappa \right)^{1/\kappa} \quad \text{for} \quad 0 < \kappa \leq 1,\end{gathered}$$ because $f(y) = y^{\kappa}$ is a concave function for $y > 0$. This shows $$\begin{gathered} \lim_{d\to\infty}\frac{1}{d}\,\sum_{j=1}^d \left(\gamma_{d,j} \right)^\kappa=0 \quad \text{for all}\quad 0 < \kappa \le 1.\end{gathered}$$ Finally, for every $\kappa \geq 1$ we can estimate $\gamma_{d,j} \geq C_\gamma^{1-\kappa} \left( \gamma_{d,j} \right)^\kappa$ since $\gamma_{d,j}\leq C_\gamma$ for $j=1,\ldots,d$. Therefore $\lim_{d{\rightarrow}\infty} d^{-1} \sum_{j=1}^d (\gamma_{d,j})^\kappa = 0$ also holds true for $\kappa>1$, and the proof is complete. ### Conclusions and applications {#sect:conclusions} In this last part of the current section we give some examples to illustrate the obtained complexity results. To this end, we only have to prove the corresponding embeddings, [i.e. ]{}we need to verify assumption [(\[Assumption1\])]{} and/or [(\[Assumption2\])]{} from the beginning of . To begin with, we check the case where ${\mathcal{F}}_d^\gamma = {\mathcal{P}}_d^\gamma$ for every $d\in{\mathbb{N}}$. Then [(\[Assumption1\])]{} obviously holds with $C_{1,d}=1$, [i.e. ]{}$c=1$ and $q_1=0$. To prove [(\[Assumption2\])]{}, note that the algebraical inclusion ${\mathcal{F}}_d^\gamma \subset {\mathcal{H}}_d^\gamma$ is trivial by the definition of ${\mathcal{H}}_d^\gamma = {\mathcal{H}}(K_d^\gamma)$ given in . For $f \in {\mathcal{F}}_d^\gamma = {\mathcal{P}}_d^\gamma$ we calculate $$\begin{gathered} {\left\| f {\; \vrule \;}{\mathcal{H}}_d^\gamma \right\|}^2 \leq \sum_{\bm{\alpha} \in \{0,1\}^d} \frac{1}{\gamma_{\bm{\alpha}}} \int_{[0,1]^d} {\left\| D^{\bm{\alpha}} f {\; \vrule \;}{\mathrm{L}}_\infty([0,1]^d) \right\|}^2 {\,\mathrm{d}\uplambda}^d(\bm{x}) \leq {\left\| f {\; \vrule \;}{\mathcal{F}}_d^\gamma \right\|}^2 \cdot \sum_{\bm{\alpha} \in \{0,1\}^d} \gamma_{\bm{\alpha}}\end{gathered}$$ using [(\[eq:norm\_Sobol\])]{}, as well as [(\[eq:norm\_P\])]{}. Hence the norm of the embedding ${\mathcal{F}}_d^\gamma \hookrightarrow {\mathcal{H}}_d^\gamma$ is bounded by $$\begin{gathered} \left( \sum_{\bm{\alpha} \in \{0,1\}^d} \gamma_{\bm{\alpha}} \right)^{1/2} = \left( \prod_{j=1}^d (1+\gamma_{d,j}) \right)^{1/2} \leq { \mathop{\mathrm{exp}}\left( \frac{1}{2} \sum_{j=1}^d \gamma_{d,j} \right) }.\end{gathered}$$ So, with $a=1$, $b=1/2$, and $t=1$ the assumption [(\[Assumption2\])]{} is also fulfilled and we can apply the stated assertions from for the spaces ${\mathcal{F}}_d^\gamma = {\mathcal{P}}_d^\gamma$, $d\in{\mathbb{N}}$. We now turn to the case ${\mathcal{F}}_d^\gamma = {\mathcal{H}}_d^\gamma$. Unfortunately, the estimate above indicates that [(\[Assumption1\])]{} may not hold for ${\mathcal{F}}_d^\gamma={\mathcal{H}}_d^\gamma$ with $C_{1,d} \in {\mathcal{O}}(d^{q_1})$ without imposing additional conditions on the product weights $\gamma$. Nevertheless, in this case assumption [(\[Assumption2\])]{} is trivially true with $C_{2,d}=1$, i.e., $a=1$, $b=0$, and $t=1$. Therefore we can apply for this space. Thus the problem is polynomially tractable if $q(\gamma)<1$ and we have strong polynomial tractability if $p(\gamma)<1$. It can be shown that these conditions are also necessary; see . $\square$ Next we discuss a more advanced sequence of Banach function spaces. For every $d\in{\mathbb{N}}$ consider the space $$\begin{gathered} {\mathcal{F}}_d^\gamma = \left\{ f \colon [0,1]^d {\rightarrow}{\mathbb{R}}{\; \vrule \;}f \in C^{(1,\ldots,1)}([0,1]^d), \, \text{ where } \, {\left\| f {\; \vrule \;}{\mathcal{F}}_d^\gamma \right\|} < \infty \right\}\end{gathered}$$ of functions which are once continuously differentiable in every coordinate direction, where $$\begin{gathered} {\left\| f {\; \vrule \;}{\mathcal{F}}_d^\gamma \right\|} = {\mathop{\mathrm{max}}\displaylimits}_{\bm{\alpha} \in \{0,1\}^d} \frac{1}{\gamma_{\bm{\alpha}}} {\left\| D^{\bm{\alpha}} f {\; \vrule \;}{\mathrm{L}}_\infty([0,1]^d) \right\|}.\end{gathered}$$ Since ${\mathcal{P}}_d^\gamma$ is a linear subset of ${\mathcal{F}}_d^\gamma$ and, due to [(\[eq:norm\_P\])]{}, the norm ${\left\| \cdot {\; \vrule \;}{\mathcal{P}}_d^\gamma \right\|}$ is simply the restriction of ${\left\| \cdot {\; \vrule \;}{\mathcal{F}}_d^\gamma \right\|}$ we have ${\mathcal{P}}_d^\gamma \hookrightarrow {\mathcal{F}}_d^\gamma$ with an embedding factor $C_{1,d}=1$ and hence [(\[Assumption1\])]{} holds true. For the norm $C_{2,d}$ of the embedding ${\mathcal{F}}_d^\gamma \hookrightarrow {\mathcal{H}}_d^\gamma$, the same estimates hold exactly as in the previous example and, moreover, the set inclusion is obvious. Therefore also assumption [(\[Assumption2\])]{} is fulfilled and we can apply the propositions and theorems of to the sequence $({\mathcal{F}}_d^\gamma)_{d\in{\mathbb{N}}}$. $\square$ Our last example ${\mathcal{F}}_d^\gamma = F_d^\gamma$, for all $d\in{\mathbb{N}}$, finally shows that even very high smoothness does not improve the conditions for tractability. For $d\in{\mathbb{N}}$ and product weights $\gamma$ let $$\begin{gathered} {\mathcal{F}}_d^\gamma = F_d^\gamma = \left\{ f \colon [0,1]^d {\rightarrow}{\mathbb{R}}{\; \vrule \;}f \in C^{\infty}([0,1]^d) \, \text{ with } \, {\left\| f {\; \vrule \;}F_d^\gamma \right\|} < \infty\right\},\end{gathered}$$ where the norm is given by [(\[eq:weightednorm\])]{}. Obviously, ${\mathcal{P}}_d^\gamma \subset C^\infty$, because functions from ${\mathcal{P}}_d^\gamma$ are at most linear in each coordinate. This moreover implies that $D^{\bm{\alpha}} f \equiv 0$ for all $\bm{\alpha} \in {\mathbb{N}}_0^d \setminus \{0,1\}^d$. Therefore, once again we have $$\begin{gathered} {\left\| f {\; \vrule \;}{\mathcal{P}}_d^\gamma \right\|} = {\mathop{\mathrm{max}}\displaylimits}_{\bm{\alpha} \in \{0,1\}^d} \frac{1}{\gamma_{\bm{\alpha}}} {\left\| D^{\bm{\alpha}} f {\; \vrule \;}{\mathrm{L}}_\infty([0,1]^d) \right\|} = {\left\| f {\; \vrule \;}{\mathcal{F}}_d^\gamma \right\|} \quad \text{for all} \quad f \in {\mathcal{P}}_d^\gamma.\end{gathered}$$ Together this yields ${\mathcal{P}}_d^\gamma \hookrightarrow {\mathcal{F}}_d^\gamma$ with an embedding constant $C_{1,d}=1$ for all $d\in{\mathbb{N}}$. In addition, also [(\[Assumption2\])]{} can be concluded as in the examples above. So, even infinite smoothness leads to the the same conditions for tractability and the curse of dimensionality as before. $\square$ Note that in the latter example we do not need to claim a product structure for the weights according to multi-indices $\bm{\alpha} \in {\mathbb{N}}_0^d \setminus \{0,1\}^d$. Furthermore, this example is a generalization of the space $F_d$ studied in [@NW09]. For $\gamma_{\bm{\alpha}} \equiv 1$ we reproduce the intractability result stated there because then $F_d^\gamma$ equals $F_d$ for each $d\in{\mathbb{N}}$.\ In conclusion we discuss the tractability behavior of uniform approximation defined on one of the spaces ${\mathcal{F}}_d^\gamma$ above using a special class of product weights $\gamma$ which are *independent of the dimension* $d$. That is, for the generator weights we claim that $$\begin{gathered} \label{eq:pol_generator} \gamma_{d,j} \equiv \gamma^{(j)} \in \Theta(j^{-\beta}) \quad \text{for some} \quad \beta \geq 0,\end{gathered}$$ and all $j$ and $d\in{\mathbb{N}}$. The imposed polynomial behavior of $\gamma^{(j)}$ is a typical example in the theory of product weights. Clearly, $p(\gamma)$ is finite if and only if $\beta>0$, and if so then $p(\gamma)=1/\beta$. For details see [@NW08 Section 5.3.4]. If $\beta = 0$ then the ${\mathrm{L}}_\infty$-approximation problem $\mathrm{App}=(\mathrm{App}_d)_{d\in{\mathbb{N}}}$ is intractable (more precisely it suffers from the curse of dimensionality) due to , assertion [(\[Equi\_5\])]{}, since then $d^{-1} \sum_{j=1}^d \gamma_{d,j}$ does not tend to zero. For $\beta \in (0,1)$, easy calculus yields $q(\gamma)>1$. So, using we conclude polynomial intractability in this case. On the other hand, for all $\delta$ and $\kappa$ with $0< \delta < \kappa \leq 1$, we have $$\begin{gathered} \frac{\sum_{j=1}^d j^{-\kappa}}{d} = \frac{\sum_{j=1}^d j^{-\kappa} d^{\kappa -(1+\delta)}}{ d^{\kappa - \delta}} \leq \frac{\sum_{j=1}^d j^{-(1+\delta)}}{d^{\kappa - \delta}} {\rightarrow}0 \quad \text{for} \quad d{\rightarrow}\infty\end{gathered}$$ and if $\kappa > 1$ then the most left fraction obviously tends to zero, too. Hence condition [(\[Equi\_6\])]{} of holds and the problem is weakly tractable for all $\beta>0$. For $\beta=1$ we use inequality [(\[LowerBound\_n\])]{} from and estimate $$\begin{gathered} \sum_{j=1}^d \gamma_{d,j} \geq c \cdot \ln (d+1) \end{gathered}$$ for some positive $c$. Therefore, for every ${{\varepsilon}}\in(0,1)$ the information complexity $n({\varepsilon}, d; \mathrm{App}_d)$ is lower bounded polynomially in $d\in{\mathbb{N}}$. This proves that strong polynomial tractability does not hold for $\beta=1$. Moreover, it is easy to show that in this case the sufficient condition $q(\gamma)<1$ for polynomial tractability is not fulfilled. So, we do not know whether polynomial tractability holds or not. Finally, consider $\beta>1$ in [(\[eq:pol\_generator\])]{}. Then we easily see that $p(\gamma) = \frac{1}{\beta}<1=t$. Thus provides strong polynomial tractability in this situation. In summary, we proved all the assertions we claimed at the end of . Possible extensions and further results {#sect:final_remarks} --------------------------------------- Note that the main result of this chapter, the lower bound given in , can be easily transferred from $[0,1]^d$ to more general domains $\Omega_d\subset{\mathbb{R}}^d$. Indeed, the case $\Omega_d = [c_1,c_2]^d$, where $c_1<c_2$, can be immediately obtained using the presented techniques. It turns out that in this case we have to modify estimate [(\[estimate\_s\])]{} by a constant which depends only on the length of the interval $[c_1,c_2]$. Consequently, the general tractability behavior does not change. Another extension of the obtained results is possible if we consider ${\mathrm{L}}_p$-norms ($1\leq p <\infty$) instead of the ${\mathrm{L}}_\infty$-norm. In we briefly discuss these norms for the unweighted case. Then the modifications for the weighted case are obvious and thus we leave it for the interested reader. In passing we correct a small mistake stated in [@NW09]. Finally, in , we show that the algorithm studied in is essentially optimal for the uniform approximation problem on the unanchored weighted Sobolev space ${\mathcal{H}}(K_d^\gamma)$ defined in . ### Lp-approximation {#sect:Lp_approx} As in [@W12 Section 7] we follow Novak and Woźniakowski [@NW09] and define the spaces $$\begin{gathered} F_{d,p} = \left\{ f \in C^{\infty}([c_1,c_2]^d) {\; \vrule \;}{\left\| f {\; \vrule \;}F_{d,p} \right\|} = \sup_{\bm{\alpha} \in {\mathbb{N}}_0^d} {\left\| D^{\bm{\alpha}} f {\; \vrule \;}{\mathrm{L}}_p([c_1,c_2]^d) \right\|} < \infty \right\}\end{gathered}$$ for $1\leq p < \infty$ and $d\in{\mathbb{N}}$, where we assume that $l=c_2-c_1>0$. In what follows we want to approximate $f\in F_{d,p}$ in the norm of ${\mathrm{L}}_p$. That is, we modify [(\[eq:uniform\_app\])]{} and consider the problem $S^p=(S^p_d)_{d\in{\mathbb{N}}}$ given by $$\begin{gathered} S_d^p = {\mathrm{id}}_{d}^p \colon {\mathcal{B}}(F_{d,p}) {\rightarrow}{\mathrm{L}}_p([c_1,c_2]^d), \quad f\mapsto {\mathrm{id}}^p_{d}(f) = f.\end{gathered}$$ Hence we try to minimize the $n$th minimal worst case error $$\begin{gathered} e_p^{\mathrm{wor}}(n,d; {\mathrm{id}}_d^p) = \inf_{A_{n,d}} \sup_{f \in {\mathcal{B}}(F_{d,p})} {\left\| f- A_{n,d}(f) {\; \vrule \;}{\mathrm{L}}_p([c_1,c_2]^d) \right\|} \end{gathered}$$ which now depends on the additional integrability parameter $p$. Observe that, without loss of generality, we can restrict ourselves to the case $[c_1,c_2]=[0,l]$. In order to conclude a lower bound analogue to [(\[eq:NW09\_bound\])]{} and , i.e., $e_p^{\mathrm{wor}}(n,d; {\mathrm{id}}_{d}^p) \geq 1$ for $n<2^{s}$, we once again use with ${\mathcal{F}}= F_{d,p}$ and ${\mathcal{G}}= {\mathrm{L}}_p([0,l]^d)$.[^9] The authors of [@NW09] suggest to use the subspace $V_d^{(k)} \subset F_{d,p}$ defined as $$\begin{gathered} V_d^{(k)}={\mathop{\mathrm{span}}\left\{ g_i \colon [0,l]^d {\rightarrow}{\mathbb{R}}, \, \bm{x} \mapsto g_{\bm{i}}(\bm{x})=\prod_{j=1}^s \left( \sum_{m=(j-1)k+1}^{jk} x_m \right)^{i_j} {\; \vrule \;}\bm{i}\in\{0,1\}^s \right\}},\end{gathered}$$ where $s={\left\lfloor d/k \right\rfloor}$ and $k\in{\mathbb{N}}$ such that $kl \geq 2(p+1)^{1/p}$. Hence, if $l<2(p+1)^{1/p}$ then we have to use blocks of variables with size $k>1$, in order to guarantee [(\[eq:NormCondition\])]{}. That is, to fulfill the condition $$\begin{gathered} \label{NormCondition_Lp} {\left\| g {\; \vrule \;}F_{d,p} \right\|} \leq {\left\| g {\; \vrule \;}{\mathrm{L}}_p([0,l]^d) \right\|} \quad \text{for all} \quad g\in V_d^{(k)}.\end{gathered}$$ Therefore Novak and Woźniakowski defined $k={\left\lceil 2(p+1)^{1/p}/l \right\rceil}$, but this is too small as the following example shows. For $d\geq 4$ take $l=1$, i.e. $[c_1,c_2]^d=[0,1]^d$, and $p=1$. Then $k=4$ should be a proper choice, but for $g^*(\bm{x})=(x_1+x_2+x_3+x_4)-2$ it can be checked (using a computer algebra system) that $$\begin{gathered} {\left\| g^* {\; \vrule \;}{\mathrm{L}}_1([0,1]^d) \right\|} = \frac{7}{15} < 1 = {\left\| \frac{\partial g^*}{\partial x_1} {\; \vrule \;}{\mathrm{L}}_1([0,1]^d) \right\|}.\end{gathered}$$ This obviously contradicts [(\[NormCondition\_Lp\])]{}. $\square$ For an exhaustive proof of an assertion which states that a slightly larger choice of $k\in{\mathbb{N}}$ suffices to conclude the desired intractability result we need to show the following technical lemma first. Its proof is based on some well-known arguments from Banach space geometry. \[lemma:int\_est\] Let $p\in[1,\infty)$ and $k\in{\mathbb{N}}$. Then $$\begin{gathered} \label{Estimate_Integrals} \mathfrak{I}_{k,p} = \int_{[-1/2,1/2]^k} {\left| \sum_{m=1}^k z_m \right|}^p {\,\mathrm{d}\uplambda}^k(\bm{z}) \geq C_p \cdot k^{p/2}\end{gathered}$$ with some $C_p \geq 1/[(2\sqrt{2})^p(1+p)]$ independent of $k$. For $k=1$ we easily calculate $\mathfrak{I}_{1,p} = 1 / [2^p(1+p)]$. Hence, without loss of generality we can assume $k\geq 2$ in what follows. To abbreviate the notation, let us define $$\begin{gathered} \label{eq:def_sum} f=f_k \colon {\mathbb{R}}^k {\rightarrow}{\mathbb{R}}, \qquad \bm{z}=(z_1,\ldots,z_k) \mapsto f(\bm{z})=\sum_{m=1}^k z_m\end{gathered}$$ for any fixed $k\geq 2$. Moreover, for given vectors $\bm{z},\bm{y} \in {\mathbb{R}}^k$, let ${\left\langle \bm{z}, \bm{y} \right\rangle}$ denote the inner product $\sum_{m=1}^k z_m y_m$ in ${\mathbb{R}}^k$. In the special case $\bm{y}=\bm{\xi} = 1/\sqrt{k} \cdot \bm{(1,\ldots,1)}\in \mathbb{S}^{k-1}$ it is ${\left\langle \bm{z}, \bm{\xi} \right\rangle} = t$ for a given $t\in {\mathbb{R}}$ if and only if $f(\bm{z}) = t\sqrt{k}$. Furthermore note that every $\bm{y}$ in the $k$-dimensional unit sphere $\mathbb{S}^{k-1}\subset {\mathbb{R}}^k$ uniquely defines a hyperplane $\bm{y}^{\bot} = \left\{\bm{z}\in{\mathbb{R}}^k {\; \vrule \;}{\left\langle \bm{z}, \bm{y} \right\rangle}=0 \right\}$ perpendicular to $\bm{y}$ which contains zero. Therefore, for $\bm{y}=\bm{\xi}$ and every $t\in[0,\infty)$, the set $$\begin{gathered} \mathfrak{H}_t=\bm{\xi}^\bot + t \cdot \bm{\xi} = \left\{\bm{z}\in{\mathbb{R}}^k {\; \vrule \;}{\left\langle \bm{z}, \bm{\xi} \right\rangle}=t \right\}\end{gathered}$$ describes a parallel shifted hyperplane in ${\mathbb{R}}^k$ with distance $t$ to the origin. Using Fubini’s theorem, this leads to the following representation: $$\begin{aligned} \mathfrak{I}_{k,p} &= \int_{[-1/2,1/2]^k} {\left| f(\bm{z}) \right|}^p {\,\mathrm{d}\uplambda}^k(\bm{z}) = 2 \cdot \int_{\substack{[-1/2, 1/2]^k\\{\left\langle \bm{z}, \bm{\xi} \right\rangle} \geq 0}} f(\bm{z})^p {\,\mathrm{d}\uplambda}^k(\bm{z}) \\ &= 2 \cdot k^{p/2} \cdot \int_0^\infty t^p \left( \int_{[-1/2, 1/2]^k\cap \mathfrak{H}_t} 1 \, {\,\mathrm{d}\uplambda}^k(\bm{z}) \right) {\,\mathrm{d}\uplambda}^1(t).\end{aligned}$$ Now we see that the inner integral describes the $(k-1)$-dimensional volume $$\begin{gathered} v(t) = \uplambda^{k-1}\!\left( [-1/2,1/2]^k \cap \mathfrak{H}_t \right)\end{gathered}$$ of the parallel section of the unit cube with the hyperplanes defined above. Because of Ball’s famous theorem we know that $v(0)\leq \sqrt{2}$ holds independently of $k$; see, e.g., Chapter 7 in the monograph of Koldobsky [@K05]. Moreover taking $\mathfrak{H}_0 = \bm{\xi}^\bot$ provides a central hyperplane section of the unit cube. From this observation we conclude that $$\begin{gathered} \int_0^\infty v(t) \, {\,\mathrm{d}\uplambda}^1(t) = \frac{1}{2} \cdot \uplambda^{k}([-1/2,1/2]^k) = \frac{1}{2}\end{gathered}$$ because of the symmetry of $[-1/2,1/2]^k$ [w.r.t. ]{}$\mathfrak{H}_0$. In addition, by Brunn’s theorem (cf. [@K05 Theorem 2.3]), the function $v$ is non-negative and non-increasing on the interval $[0,\infty)$. Thus $v$ is related to the distribution function of a certain non-negative real-valued random variable $X$, up to some normalizing factor, i.e. $v(t) = v(0) \cdot {\mathbb{P}}(\{X\geq t\})$. Using Hölder’s inequality[^10] we obtain ${\mathbb{E}}(X^{1+p}) \geq ( {\mathbb{E}}X )^{1+p}$ and, respectively, $$\begin{gathered} \mathfrak{I}_{k,p} = k^{p/2}\cdot 2\int_0^{\infty} t^p \, v(t) \, {\,\mathrm{d}\uplambda}^1(t) \geq k^{p/2}\cdot \frac{2}{v(0)^p \, (1+p)} \left( \int_0^{\infty} v(t) \, {\,\mathrm{d}\uplambda}^1(t) \right)^{1+p}\end{gathered}$$ by integration by parts. In summary we have shown [(\[Estimate\_Integrals\])]{} and hence the proof is complete. Now the mentioned intractability result reads as follows: \[prop:Lp\_intract\] Let $1\leq p < \infty$ and $l>0$. Moreover, choose $k\in{\mathbb{N}}$ such that $$\begin{gathered} \label{LowerBound_k} k \geq \kappa_{p,l}={\left\lceil 8(p+1)^{2/p}/l^2 \right\rceil}. \end{gathered}$$ Then condition [(\[NormCondition\_Lp\])]{} holds for $V_d^{(k)}\subset F_{d,p}$. Hence the ${\mathrm{L}}_p$-approximation problem $S^p=({\mathrm{id}}_d^p\colon F_{d,p}{\rightarrow}{\mathrm{L}}_p([0,l]^d))_{d\in{\mathbb{N}}}$ suffers from the curse of dimensionality since $$\begin{gathered} e_p^{\mathrm{wor}}(n,d; {\mathrm{id}}_d^p) \geq 1 \quad \text{for all} \quad n<2^{{\left\lfloor d/k \right\rfloor}} \end{gathered}$$ and every $d\in{\mathbb{N}}$. Due to the structure of the functions $g$ from $V_d^{(k)}$, it suffices to show that $$\begin{gathered} {\left\| D^{\bm{\alpha}} g {\; \vrule \;}{\mathrm{L}}_p([0,l]^{ks}) \right\|} \leq {\left\| g {\; \vrule \;}{\mathrm{L}}_p([0,l]^{ks}) \right\|} \quad \text{for all} \quad g \in V_d^{(k)} \quad \text{and every} \quad \bm{\alpha} \in \mathbb{M}_d^{(k)},\end{gathered}$$ where the set of multi-indices $\mathbb{M}_d^{(k)}$ is defined by $$\begin{gathered} \mathbb{M}_d^{(k)} = \left\{ \bm{\alpha}=(\alpha_1,\ldots,\alpha_{ks}) \in \{0,1\}^{ks} {\; \vrule \;}\sum_{m\in I_j} \alpha_m \leq 1 \, \text{ for all } \, j=1,\ldots,s \right\}\end{gathered}$$ and $I_j=\{(j-1)k+1, \ldots, jk\}$. Observe that $\mathbb{M}_d^{(k)}$ depends on $d$ via $s={\left\lfloor d/k \right\rfloor}$. Similar to the proof of , we only need to consider the case $\bm{\alpha} = \bm{e_t} \in \{0,1\}^{ks}$ with $t \in I_j$. The rest then follows by induction. Given $t\in I_j$ for some $j\in\{1,\ldots,s\}$ we can represent every fixed $g\in V_d^{(k)}$, as well as its partial derivative $D^{\bm{e_t}}g$, by some functions $a, b \colon [0,l]^{k(s-1)} {\rightarrow}{\mathbb{R}}$ (depending on $g$ and $j$) such that $$\begin{gathered} g(\bm{x}) = a(\bm{\widetilde{x}}) \sum_{m=1}^k y_m + b(\bm{\widetilde{x}}) \quad \text{and} \quad (D^{\bm{e_t}} g)(\bm{x}) = a(\bm{\widetilde{x}}), \quad \bm{x}\in[0,l]^{ks}.\end{gathered}$$ Here we split the $ks$-dimensional vector $\bm{x} = (\bm{x_{I_1}}, \ldots, \bm{x_{I_{j-1}}}, \bm{y}, \bm{x_{I_{j+1}}}, \ldots, \bm{x_{I_s}})$ into $\bm{\widetilde{x}} = (\bm{x_{I_1}}, \ldots, \bm{x_{I_{j-1}}}, \bm{x_{I_{j+1}}}, \ldots, \bm{x_{I_s}}) \in [0,l]^{k(s-1)}$ and $\bm{y}=(y_1,\ldots,y_k)\in [0,l]^k$, where $\bm{x_{I_j}}$ denotes the $k$-dimensional block of components $x_m$ in $\bm{x}$ with coordinates $m \in I_j$. Using this representation we can rewrite the inequality ${\left\| D^{\bm{e_t}} g {\; \vrule \;}{\mathrm{L}}_p([0,l]^{ks}) \right\|} \leq {\left\| g {\; \vrule \;}{\mathrm{L}}_p([0,l]^{ks}) \right\|}$ as $$\begin{aligned} &\int_{[0,l]^{k(s-1)}} \int_{[0,l]^k} {\left| a(\bm{\widetilde{x}}) \right|}^p \, {\,\mathrm{d}\uplambda}^k(\bm{y}) \, {\,\mathrm{d}\uplambda}^{k(s-1)}(\bm{\widetilde{x}}) \\ &\qquad \leq \int_{[0,l]^{k(s-1)}} \int_{[0,l]^k} {\left| a(\bm{\widetilde{x}}) \sum_{m=1}^k y_m + b(\bm{\widetilde{x}}) \right|}^p \, {\,\mathrm{d}\uplambda}^k(\bm{y}) \, {\,\mathrm{d}\uplambda}^{k(s-1)}(\bm{\widetilde{x}})\end{aligned}$$ such that it is enough to prove a pointwise estimate of the inner integrals for ($\uplambda^{k(s-1)}$-almost every) fixed $\bm{\widetilde{x}}\in [0,l]^{k(s-1)}$ with $a=a(\bm{\widetilde{x}}) \neq 0$. Easy calculus yields $$\begin{gathered} \int_{[0,l]^k} {\left| a \sum_{m=1}^k y_m + b \right|}^p {\,\mathrm{d}\uplambda}^k(\bm{y}) = l^{p+k} \cdot \int_{[-1/2,1/2]^k} {\left| a \sum_{m=1}^k z_m + b' \right|}^p {\,\mathrm{d}\uplambda}^k(\bm{z})\end{gathered}$$ for some constant $b'\in{\mathbb{R}}$ that depends on $b=b(\bm{\widetilde{x}})$. Note that the right-hand side of the latter equality is minimized for $b'=0$. Therefore we can estimate the left-hand side from below by $$\begin{aligned} \int_{[0,l]^k} {\left| a \sum_{m=1}^k y_m + b \right|}^p {\,\mathrm{d}\uplambda}^k(\bm{y}) &\geq {\left| a \right|}^p \cdot l^{p+k} \cdot \int_{[-1/2,1/2]^k} {\left| \sum_{m=1}^k z_m \right|}^p {\,\mathrm{d}\uplambda}^k(\bm{z}) \\ &= \int_{[0,l]^k} {\left| a \right|}^p {\,\mathrm{d}\uplambda}^k(\bm{y}) \cdot l^p \cdot \int_{[-1/2,1/2]^k} {\left| \sum_{m=1}^k z_m \right|}^p {\,\mathrm{d}\uplambda}^k(\bm{z}).\end{aligned}$$ To complete the proof it remains to show that our choice of $k\geq \kappa_{p,l}$, with $\kappa_{p,l}$ given in [(\[LowerBound\_k\])]{}, implies that $$\begin{gathered} \label{eq:int_bound} \int_{[-1/2,1/2]^k} {\left| \sum_{m=1}^k z_m \right|}^p {\,\mathrm{d}\uplambda}^k(\bm{z}) \geq l^{-p}\end{gathered}$$ but this easily follows from above. Actually, using other proof methods we can slightly improve the lower bound for $C_p$ in and thus also $\kappa_{p,l}$ in formula [(\[LowerBound\_k\])]{} of . This is the subject of our final remark within this subsection: Let $\bm{Y}=(Y_1,\ldots,Y_k)$ denote a random vector of $k\in{\mathbb{N}}$ independent copies of some uniformly $[-1/2,1/2]$-distributed random variable $Y_0$. Then $\mathfrak{I}_{k,p}$ can be interpreted as the $p$th absolute moment ${\mathbb{E}}({\left| f_k(\bm{Y}) \right|}^p)$ of $f_k(\bm{Y})$, where $f_k$ again is given by [(\[eq:def\_sum\])]{}. In the case of even $p=2N$, $N\in{\mathbb{N}}$, this can be calculated exactly using the multinomial theorem. For $k,N\in{\mathbb{N}}$ we obtain $$\begin{gathered} \mathfrak{I}_{k,2N} = {\mathbb{E}}\left( {\left| f_k(\bm{Y}) \right|}^{2N} \right) = 2^{-2N} \sum_{\substack{\bm{j}=(j_1,\ldots,j_k)\in{\mathbb{N}}_0^k\\j_1+\ldots+j_k=N}} \binom{2N}{2j_1,\ldots,2j_k} \prod_{m=1}^k \frac{1}{2j_m+1},\end{gathered}$$ where we used the independence of the $Y_m$’s and fact that $$\begin{gathered} {\mathbb{E}}(Y_0^n) = \int_{-1/2}^{1/2} y^n\, {\,\mathrm{d}\uplambda}^1(y) = \begin{cases} 0 , &\text{if } n = 2j+1,\\ (2j+1)^{-1} \cdot 2^{-2j} , &\text{if } n = 2j, \end{cases}\end{gathered}$$ and $j\in{\mathbb{N}}_0$. In particular, we conclude $$\begin{gathered} \mathfrak{I}_{k,2} = \frac{1}{2^2 \cdot 3} \cdot k \qquad \text{and} \qquad \mathfrak{I}_{k,4} = \frac{1}{48} \cdot k \left(k-\frac{2}{5} \right) \geq \frac{1}{2^4 \cdot 5} \cdot k^2.\end{gathered}$$ Since $\mathfrak{I}_{k,p}={\left\| f_k{\; \vrule \;}{\mathrm{L}}_p([-1/2,1/2]^k) \right\|}^p$ we can use the monotonicity of the Lebesgue spaces in order to estimate $C_p$ for the remaining powers $p$. For $k\in{\mathbb{N}}$ and $1 \leq q \leq p < \infty$ we obtain $\mathfrak{I}_{k,p} \geq (\mathfrak{I}_{k,q})^{p/q} \geq (C_q)^{p/q} \, k^{p/2}$, [i.e. ]{}$C_p \geq (C_q)^{p/q}$, provided that $\mathfrak{I}_{k,q} \geq C_q\, k^{q/2}$. Consequently, we can take $$\begin{gathered} k\geq \begin{cases} {\left\lceil 12/l^2 \right\rceil}, &\text{if } 2\leq p < 4,\\ {\left\lceil 4\sqrt{5}/l^2 \right\rceil}, &\text{if } 4\leq p \end{cases}\end{gathered}$$ to fulfill [(\[eq:int\_bound\])]{} in the proof of . This clearly improves the bound $k\geq \kappa_{p,l}$ in [(\[LowerBound\_k\])]{}. $\square$ Nevertheless, we want to stress the point that also with these improvements the lower bounds on $k$ are not sharp since we know from [@NW09] that in the limit case $p = \infty$ we can take $k = {\left\lceil 2/l \right\rceil}$. On the other hand, we note that Hoeffding’s inequality implies the existence of some universal constants $C_p'$ such that $\mathfrak{I}_{k,p} \leq C_p'\, k^{p/2}$ for all $p\in[1,\infty)$ and every $k\in{\mathbb{N}}$. Thus the estimates on the integrals $\mathfrak{I}_{k,p}$ are of the right order in $k$ such that we need other proof techniques to obtain a better dependence of $k$ on $l=c_2-c_1$. ### Uniform approximation in the weighted Sobolev space {#sect:opt_uniform_algo} To show that the linear algorithm $A_{n,d}^*$ studied in is essentially optimal for ${\mathrm{L}}_\infty$-approximation on the unanchored Sobolev space ${\mathcal{H}}_d^\gamma={\mathcal{H}}(K_d^\gamma)$ in the worst case setting we study (weighted) ${\mathrm{L}}_2$-approximation on a related Banach space ${\mathcal{F}}_d$ in the average case setting; see for details. The relation of these two problems is given by the assertion below which follows from [@KWW08 Theorem 1]. For $d\in{\mathbb{N}}$ let ${\mathcal{H}}(K_d)$ denote a RKHS induced by a kernel $K_d\colon[0,1]^d\times[0,1]^d{\rightarrow}{\mathbb{R}}$ that satisfies [(\[eq:BoundedKernel\])]{}.[^11] Moreover, define the set of non-vanishing probability density functions ${\varrho}$ on the unit cube by $$\begin{gathered} \mathcal{D}_d = \left\{ {\varrho}\colon [0,1]^d {\rightarrow}[0,\infty) {\; \vrule \;}\int_{[0,1]^d} {\varrho}(\bm{x}) {\,\mathrm{d}\uplambda}^d(\bm{x})=1 \quad \text{and} \quad {\varrho}>0 \text{ ($\uplambda^d$-a.e.)} \right\}.\end{gathered}$$ Then, for every $n\in{\mathbb{N}}_0$ and all $d\in{\mathbb{N}}$, $$\begin{aligned} &e^{\mathrm{wor}}\! \left(n,d; {\mathrm{id}}_d \colon {\mathcal{B}}({\mathcal{H}}(K_d)){\rightarrow}{\mathrm{L}}_\infty([0,1]^d) \right) \\ &\qquad\,\qquad \geq \sup_{{\varrho}\in \mathcal{D}_d} \, e^{\mathrm{avg}} \!\left(n,d; {\mathrm{id}}_d^{{\varrho}} \colon {\mathcal{F}}_d {\rightarrow}{\mathrm{L}}_2^{\varrho}([0,1]^d) \right). \end{aligned}$$ Here the $n$th minimal errors are taken with respect to all algorithms from the class ${\mathcal{A}}_d^{n,{\mathrm{lin}}}(\Lambda^{\mathrm{all}})$. In particular, it follows that the ($n$th minimal) worst case error for ${\mathrm{L}}_\infty$-approximation on the unit ball of the *Sobolev space* ${\mathcal{H}}_d^\gamma$ is lower bounded by the average case error of *unweighted* ${\mathrm{L}}_2$-approximation on the corresponding Banach space. That is, we set $K_d=K_d^\gamma$ and ${\varrho}= \chi_{[0,1]^d}\in\mathcal{D}_d$ in the following. In turn we have (strong) polynomial tractability for the uniform approximation problem [w.r.t. ]{}the worst case setting only if average case ${\mathrm{L}}_2$-approximation is polynomially tractable, as long as we consider the absolute error criterion. Due to [@NW08 Theorem 6.1] we know that the latter holds true if and only if there exist a positive constant $c_1$, non-negative $q_1$, $q_2$ and $\tau\in(0,1)$ such that $$\begin{gathered} c_2 = \sup_{d\in{\mathbb{N}}} \frac{1}{d^{q_2}} \left( \sum_{i={\left\lceil c_1\, d^{q_1} \right\rceil}}^\infty (\lambda_{d,i})^\tau \right)^{1/\tau} < \infty,\end{gathered}$$ where $(\lambda_{d,i})_{i=1}^\infty$ denotes the sequence of eigenvalues of the correlation operator $C_{\nu_d}$ with respect to a non-increasing ordering. Moreover we have strong polynomial tractability if and only if this holds with $q_1=q_2=0$. Because of the observation at the end of it suffices to consider the eigenvalues of $W_d^\gamma = \left( S_d^\gamma \right)^\dagger S_d^\gamma \colon {\mathcal{H}}_d^\gamma {\rightarrow}{\mathcal{H}}_d^\gamma$, where $S_d^\gamma$ describes the embedding ${\mathcal{H}}_d^\gamma \hookrightarrow {\mathrm{L}}_2([0,1]^d)$. Recall that these eigenvalues are given by $$\begin{gathered} \left\{ \widetilde{\lambda}_{d,\gamma,\bm{m}}=\prod_{k=1}^d \lambda_{1,\gamma_{d,k},m_k} =\prod_{k=1}^d \frac{\gamma_{d,k}}{\gamma_{d,k}+\pi^2\,(m_k-1)^2} {\; \vrule \;}\bm{m}=(m_1,\ldots,m_d)\in{\mathbb{N}}^d \right\};\end{gathered}$$ see [(\[eq:L2\_eigenvalues\])]{} at the end of . Thus we only need to reorder this set appropriately using a rearrangement $\psi_d \colon {\mathbb{N}}{\rightarrow}{\mathbb{N}}^d$ such that $$\begin{gathered} \lambda_{d,i} = \widetilde{\lambda}_{d,\gamma,\psi_d(i)} \geq \widetilde{\lambda}_{d,\gamma,\psi_d(i+1)} \quad \text{for all} \quad i\in{\mathbb{N}}.\end{gathered}$$ Given $d\in{\mathbb{N}}$, $\tau\in(0,1)$, as well as $c_1>0$, and $q_1\geq 0$ we estimate $$\begin{aligned} \sum_{i={\left\lceil c_1\, d^{q_1} \right\rceil}}^\infty (\lambda_{d,i})^\tau &=\sum_{\bm{m}\in{\mathbb{N}}^d} \left( \widetilde{\lambda}_{d,\gamma,\bm{m}} \right)^\tau - \sum_{i=1}^{{\left\lceil c_1 \, d^{q_1} \right\rceil}-1} (\lambda_{d,i})^\tau \\ &\geq \prod_{k=1}^d \sum_{m\in{\mathbb{N}}} \left( \lambda_{1,\gamma_{d,k},m} \right)^\tau - (\lambda_{d,1})^\tau \left( {\left\lceil c_1 \, d^{q_1} \right\rceil}-1 \right) \\ &\geq \prod_{k=1}^d \left( 1+ \sum_{m=2}^\infty \left( \frac{\gamma_{d,k}}{\gamma_{d,k}+\pi^2(m-1)^2} \right)^\tau \right) - c_1 \, d^{q_1},\end{aligned}$$ since $\lambda_{d,1}=\widetilde{\lambda}_{d,\gamma,\bm{(1,\ldots,1)}}=\prod_{k=1}^d \lambda_{d,\gamma_{d,k},1} = 1$. Due to the boundedness of the generator weights $\gamma_{d,k}\leq C_\gamma$ for every $k\in\{1,\ldots,d\}$, we can further estimate the sum by $$\begin{gathered} \sum_{m=2}^\infty \left( \frac{\gamma_{d,k}}{\gamma_{d,k}+\pi^2(m-1)^2} \right)^\tau \geq \gamma_{d,k}^\tau \sum_{i=1}^\infty \frac{(C_\gamma')^\tau}{i^{2\tau}} = \gamma_{d,k}^\tau \, (C_\gamma')^\tau\, \zeta(2\tau),\end{gathered}$$ where we set $C_\gamma' = (C_\gamma + \pi^2)^{-1}$. Because of $\ln(1+y)\geq y/(1+y)$ for all $y \geq 0$ we conclude that for $k=1,\ldots,d$ and some positive $C$ depending on $C_\gamma$ and $\tau$ $$\begin{gathered} \ln\left(1+\gamma_{d,k}^\tau \, (C_\gamma')^\tau\, \zeta(2\tau)\right) \geq \frac{(C_\gamma')^\tau\, \zeta(2\tau)}{1+\gamma_{d,k}^\tau \, (C_\gamma')^\tau\, \zeta(2\tau)} \cdot \gamma_{d,k}^\tau \geq C \cdot \gamma_{d,k}^\tau.\end{gathered}$$ Consequently, this yields $$\begin{gathered} \sum_{i={\left\lceil c_1\, d^{q_1} \right\rceil}}^\infty (\lambda_{d,i})^\tau \geq \prod_{k=1}^d { \mathop{\mathrm{exp}}\left( C\cdot \gamma_{d,k}^\tau \right) } - c_1 \, d^{q_1} = { \mathop{\mathrm{exp}}\left( C \sum_{k=1}^d \gamma_{d,k}^\tau \right) } - c_1 \, d^{q_1}.\end{gathered}$$ Therefore, polynomial tractability implies $q(\gamma)<1$ and strong polynomial tractability is possible only if $p(\gamma)<1$. Here $p$ and $q$ describe the sum exponents of the product weight sequence $\gamma=(\gamma_{\bm{\alpha}})_{\bm{\alpha}\in{\mathbb{N}}_0^d}$, $d\in{\mathbb{N}}$, defined at the beginning of . Together with this finally proves \[thm:tract\_Sobolev\] Consider the uniform approximation problem defined on the sequence of unanchored Sobolev spaces $({\mathcal{H}}_d^{\gamma})_{d\in{\mathbb{N}}}$, where the product weight sequence $\gamma$ is constructed out of a uniformly bounded generator sequence $C_\gamma \geq \gamma_{d,1}\geq\ldots\geq\gamma_{d,d}$, $d\in{\mathbb{N}}$. We study this problem in the worst case setting and with respect to the absolute error criterion. Then we have - polynomial tractability if and only if $q(\gamma)<1$ and - strong polynomial tractability if and only if $p(\gamma)<1$. Problems on Hilbert spaces with (anti)symmetry conditions {#chapt:antisym} ========================================================= In this last chapter we describe an essentially new kind of a priori knowledge which can help to overcome the curse of dimensionality. As in , we study compact linear problems $S=(S_d)_{d\in{\mathbb{N}}}$ defined between tensor products of Hilbert spaces but now we restrict our attention to problem elements which fulfill certain (anti)symmetry conditions. After investigating some basic properties of the related subspaces of (anti)symmetric problem elements in we construct a linear algorithm that uses finitely many continuous linear functionals and show an explicit formula for its worst case error in terms of the eigenvalues $\lambda=(\lambda_m)_{m\in{\mathbb{N}}}$ of the operator $W_1 = {S_1}^{\!\dagger} S_1$. Moreover, in we show that this algorithm is optimal [w.r.t. ]{}a wide class of algorithms. Next we clarify the influence of different (anti)symmetry conditions on the complexity, compared to the case for the classical unrestricted problem studied in . In particular, we give necessary and sufficient conditions for (strong) polynomial tractability of (anti)symmetric problems in . Apart from the absolute error criterion we also deal with normalized errors. Finally, in , we discuss several applications. particularly indicates how to apply our results to the approximation problem for wavefunctions. Most of the results stated in this chapter are already published in the articles [@W11] and [@W12b]. At some points we improve the known results and/or proof techniques slightly. In particular, the presented results also hold for problems defined on finite-dimensional or on non-separable source spaces. Basic definitions related to (anti)symmetry {#sect:basic_antisym_def} ------------------------------------------- Optimal algorithms for (anti)symmetric problems {#sect:opt_asym_algo} ----------------------------------------------- Complexity of (anti)symmetric problems {#sect:complexity_antisym} -------------------------------------- Applications {#sect:applications} ------------ This last section of the present chapter is devoted to applications of the theory developed previously. In we follow the lines of the introduction of [@W12b] and illustrate the power of imposing additional (anti)symmetry conditions to linear tensor product problems by using simple toy examples. Afterwards, in , we focus our attention to more advanced problems which we are faced with in practice. There we briefly introduce wavefunctions and show how our results allow it to handle the approximation problem for such classes of functions. [ ]{} [KSWW10b]{} <span style="font-variant:small-caps;">N. 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[^7]: Note the missing factor $1/2$ in [@NW08 p. 351, line 5]. [^8]: Note that some authors use the name *decay* for $1/p(\cdot)$. [^9]: Note that it is sufficient to restrict ourselves to the case $r=1$ since now we do not need to take care of embedding constants as in the proof of . [^10]: See also [@K05 Lemma 7.5]. [^11]: Note that [(\[eq:BoundedKernel\])]{} clearly implies that $\int_{[0,1]^d} K_d(\bm{x},\bm{x}) \, {\varrho}(\bm{x})\, {\,\mathrm{d}\uplambda}^d(\bm{x})$ is finite for every probability density function ${\varrho}$ on $[0,1]^d$.
--- abstract: 'We consider the origin of compact, short-period, Jupiter-mass planets. We propose that their diverse structure is caused by giant impacts of embryos and super-Earths or mergers with other gas giants during the formation and evolution of these hot Jupiters. Through a series of numerical simulations, we show that typical head-on collisions generally lead to total coalescence of impinging gas giants. Although extremely energetic collisions can disintegrate the envelope of gas giants, these events seldom occur. During oblique and moderately energetic collisions, the merger products retain higher fraction of the colliders’ cores than their envelopes. They can also deposit considerable amount of spin angular momentum to the gas giants and desynchronize their spins from their orbital mean motion. We find that the oblateness of gas giants can be used to infer the impact history. Subsequent dissipation of stellar tide inside the planets’ envelope can lead to runaway inflation and potentially a substantial loss of gas through Roche-lobe overflow. The impact of super-Earths on parabolic orbits can also enlarge gas giant planets’ envelope and elevates their tidal dissipation rate over $\sim $ 100 Myr time scale. Since giant impacts occur stochastically with a range of impactor sizes and energies, their diverse outcomes may account for the dispersion in the mass-radius relationship of hot Jupiters.' author: - | Shang-Fei Liu$^{1,2}$, Craig B. Agnor$^{3}$, D. N. C. Lin$^{1,4,5}$ and Shu-Lin Li$^{1,6}$\ \ $^1$ Kavli Institute for Astronomy and Astrophysics and Department of Astronomy, Peking University, Beijing 100871, China; E-mail: [liushangfei@pku.edu.cn](liushangfei@pku.edu.cn)\ $^2$ Department of Earth and Planetary Sciences, University of California, Santa Cruz, CA 95064, USA\ $^3$ Astronomy Unit, School of Physics and Astronomy, Queen Mary University of London, United Kingdom\ $^4$ Department of Astronomy and Astrophysics, University of California, Santa Cruz, CA 95064, USA\ $^5$ Institute for Advanced Studies, Tsinghua University, Beijing 100084, China\ $^6$ National Astronomical Observatory of China, Chinese Academy of Sciences Beijing 100012, China bibliography: - 'ref.bib' title: 'Embryo impacts and gas giant mergers II: Diversity of Hot Jupiters’ internal structure' --- Extrasolar planets; planetary formation; planetary dynamics; planetary structure Introduction ============ In the conventional sequential accretion hypothesis (SAH), gas giant planets acquire critical-mass ($\sim 10 M_\oplus$) cores prior to the onset of efficient gas accretion [@Pollack:1996zr; @Ida:2004ko]. This scenario can account for many observational properties of Jupiter, Saturn, and extra-solar planets [@Schlaufman:2009bh]. However, the observed large dispersion in the internal structures of gas giant planets was not anticipated, even taking into account the fact that the assumptions they made to get this critical value may not be universal. In the first paper of this series [@Li:2010fk hereafter Paper I], we attribute the dichotomy between the mass of Jupiter and Saturn’s cores to giant impacts by embryos and mergers of gas giants during their formation and early dynamical evolution. Using two numerical methods, we simulated a series of models with parabolic collisions and show that these collisions generally lead to the accretion of embryos and coalescence of gas giants. We also show that massive super-Earth impactors can survive passage deep into the envelope of a gas giant and release energy near the core. This may drive intense convective motion in the planet and core erosion. In contrast, sub-Earths embryos disintegrate higher in the gaseous envelope and their fragments will dissolve in the envelope and contribute to a heavy element enrichment. Transit surveys suggest that there is a remarkable diversity in hot Jupiters’ radii. Extrasolar planets with masses comparable to Jupiter range in size by a factor of more than two (Figure \[fig:f0\]). Some of these planets ([*e.g.*]{} HD209458 b) have Jupiter mass ($M_{\rm J}$) but more than 30 % larger radius. Several planets (such as Corot 8 b and WASP 29 b) have masses comparable to Saturn, but radii $R_{\rm p}$ that are much smaller than Saturn’s. These compact planets may require the presence of massive cores ([*e.g.*]{} up to 70-$M_\oplus$ core is needed for HD149026 b). The main focus of this paper is to apply the giant impact and merger scenario (GIMs) to account for large dispersion in hot Jupiters’ radii. In order to construct this scenario, we briefly cite observational evidence for the structural and atmospheric diversity of hot Jupiters. Through a brief recapitulation of their origin, we present arguments in §2, to suggest that GIMs occur frequently during and after orbital migration to the proximity of their host stars. In §3, we used two complementary approaches: a Lagrangian smoothed particle hydrodynamics (SPH) code, and the Eulerian adaptive mesh hydrodynamics code FLASH [@Fryxell:2000kx] to simulate both head-on and oblique collisions between a Saturn-mass giant planet and a 10-$M_\oplus$ or 25-$M_\oplus$ embryo. These models extends the previous models to the high energy limit. We show that most hyperbolic oblique encounters deposit spin angular momentum to gas giants. Highly energetic collisions can directly lead to the substantial loss of envelope gas beyond their Roche radii. SPH and FLASH simulations of merger between two Saturn-like gas giants are also presented. In order to expand the dynamical range for density and investigate the long term evolution in our simulated models, we adopted, in paper I, a one-dimensional Lagrangian hydrodynamic (LHD) scheme which can be used to study the effect of tidal interaction of inflated merger products with the host stars. With LHD, we show in §4, the energy released from modest giant impacts can also induce the expansion of the gas giants’ envelope. Subsequent dissipation of stellar tidal perturbation in the gas giants may lead to runaway inflation. Since gas in the envelope is preferentially lost, these encounters lead to the enhancement of the planet’s metallicity and asymptotic compact size. Finally, we summarize our results and discuss their observational implications in §5. Giant impact history of close-in gas giants. {#sec:justify} ============================================ In paper I, we discuss the possibility of GIMs for long-period gas giants in the proximity of the birth place as an avenue for the enrichment of their envelope and a cause for structural diversity between Jupiter and Saturn. It is difficult to directly detect the metallicity and accurately determine the internal structure of extrasolar planets. However, if the metallicity (\[Fe/H\]), envelope and core masses ($M_{\rm env}$ and $M_{\rm core}$), the orbit and spin state of any planet are known, it is possible to construct its contraction sequence, taking into account stellar irradiation and tidal dissipation [@Bodenheimer:2001qf; @Bodenheimer:2003gd]. Although the interior structure of extrasolar gas giant planets remains uncertain and is an area of active research, in this work we simply ignore complicating factors, assume a core-envelope structure and assume that the core of any planet can be inferred from its measured radius ($R_{\rm P}$) and the age of its host star ($\tau_\ast$). As of this writing there are more than 200 transiting gas giants with measured masses $M_{\rm p}$ and radii $R_{\rm p}$ [^1]. Due to an observational selection effect, most of the known transiting gas giant planets are close-in planets known as hot Jupiters with periods and masses in the range of a few days and Jupiter mass respectively. The diversity in the value of their measured radii $R_{\rm P}$ and densities motivated us to invoke the GIM scenario. Possible causes for inflated hot Jupiters. {#sec:hotjup} ------------------------------------------ Most of the transiting hot Jupiters have $R_{\rm p}$’s consistent with those expected from models constructed for their $M_{\rm p}$ including cores with $M_{\rm core} \sim 10-20~M_\oplus$. However, there are several transiting planets, such as HD209458b, Tres-4 [@Mandushev:2007qa], and WASP-12b [@Li:2010lr] which have observationally inferred radii much larger than that of Jupiter ($R_{\rm J}$) even though their mass is comparable to that of Jupiter. For the special class of inflated hot Jupiters, suggestions for their unusually low densities include: 1\) the thermal contraction of these planets is suppressed by the stellar irradiation [@Burrows:2000ye; @Burrows:2004tw], 2\) they are inflated by additional heating sources such as the dissipation due to the ongoing stellar tidal perturbation [@Bodenheimer:2001qf], 3\) irradiation-driven circulation [@Showman:2002ff], or associated Ohmic dissipation [@Batygin:2010ys; @Wu:2013kq] 4\) the sedimentation of heavy elements [@Baraffe:2008fu]. Most of these propositions have some difficulties in accounting for the difference between the exceptions and the other “normal” gas giant planets. Here we suggest another scenario. Following Paper I, we suggest that the energy dissipation during GIM’s and the tidal dissipation thereafter may be adequate to inflate some gas giant planets. The stochastic nature of GIM’s provides a possible explanation for the wide dispersion $M_{\rm p}-R_{\rm p}$ distribution of hot Jupiters. Hot Jupiters with massive cores or most metallic compositions. {#sub:hotjupearth} -------------------------------------------------------------- In paper I we showed that GIM’s may lead to transitory planetary inflation. Merger events of two or more gas giants and multiple impacts of earth-mass embryos may also lead to the formation of massive cores and the heavy metallicity contamination of planetary envelopes. In this paper, we explore the possibility that giant impacts and mergers may produce mature, compact, hot Jupiters. HD149026 b has an observed radius of $R_{\rm p} =0.73 R_{\rm J}$ and the mass of Saturn [@Sato:2005uq]. Structural analysis indicates that more than half ($\sim 70~M_\oplus$) of its total mass may be contained within a core of heavy elements [@Sato:2005uq]. Later models suggest that it is also possible for heavy elements to distribute throughout the interior of the planet [@Ikoma:2006kx; @Baraffe:2008fu]. The extraordinary compact radius of HD149026 b poses a challenge to both gravitational instability and sequential accretion hypothesis (SAH) for gas giant planet formation. The required metallicity of HD149026 b is more than an order of magnitude larger than that of its star (which has a mass $M_\ast =1.3~M_\odot$ and metallicity \[Fe/H\] = 0.36). If this planet formed through gravitational instability [@Boss:1997bh], this heavy elemental concentration would require either an enormous loss of its original gaseous envelope or an extensive acquisition of heavy elemental material through the post-formation accretion of solid planetesimals and proto-planetary embryos. If this planet acquired its present-day structure through core accretion [@Pollack:1996zr] such a massive core would be attainable in a gas-rich environment beyond the snow-line provided it can avoid initial dynamical isolation. In principle, at several AU’s from its host star, the isolation mass $M_{\rm iso}$ of proto-planetary solid embryos can reach this massive level provided the local surface density of heavy elements $\Sigma_{\rm g}$ is more than an order of magnitude larger than that inferred from the MMN model. In the disk around a host star with solar metallicity, the inferred $\Sigma_{\rm g}$ would render the augmented disk to be gravitationally unstable. However, if the core-building material is able to congregate in some confined locations such as the snow line, solid embryos with very large $M_{\rm iso}$ would be attainable in a gravitational stable disk, prior to the accretion of a relatively massive envelope. An additional requirement is that the planetesimal accretion rate must exceed the gas accretion rate until the core has already acquired most of its present-day mass. With its massive core, this requirement may be difficult to accomplish for HD149026 b because the heat loss is efficient and the gas accretion rate are likely to be large around any cores with $M_{\rm core} > 20 M_\oplus$. During the phase of gas accretion, the protoplanets’ feeding zones expand with their masses. Early models for core accretion are constructed under the assumption that all embryos in the expanding feeding zone are accreted by the growing protoplanet [@Pollack:1996zr]. However, these embryos have a tendency to migrate away from the protoplanets and form gaps around them during their initial phase of modest gas accretion. Only when the gas giants rapidly accrete massive envelopes do the orbits of their nearby residual embryos become destabilized and a modest fraction of the orbit-crossing embryos would merge with them [@Zhou:2007vn; @Shiraishi:2008ys]. However, the total mass of HD149026 b’s gaseous envelope is smaller than that contain in the heavy elements. The above discussion indicates that the formation of some isolated gas giants with massive cores cannot be ruled out [*a priori*]{}. But these compact planets are apparently exceptions rather than the general rule. Around a host star OGLE-TR-132, a transiting planet has been found with similar $M_{\rm P}$ and $a$ as those of HD149026 b. Despite the nearly identical \[Fe/H\] for both host stars, OGLE-TR-132 b has a radius nearly twice that of HD149026 b [@Gillon:2007fk]. Presumably, OGLE-TR-132 b has a much smaller core mass than HD149026 b. The large discrepancy between these two same-mass planets around similar host stars again points to stochastic origins of their internal structures. The large dispersion in the average density of known gas giants (Figure 1) is further evidence of this structure diversity. Here we adopt the proposition and examine the possibility that the massive core of compact planets such as HD149026 b are the byproduct of one or more giant impacts and mergers [@Sato:2005uq; @Ikoma:2006kx] in close proximity to its host star. Whereas those planets with unusually large radii may have been struck recently (within the last $\sim 100$ Myr) by modest-mass embryos (with masses up to $\sim M_\oplus$), the impactors for this compact planet are either another gas giant or very massive (up to $\sim 10-20 M_\oplus$) embryos. We also suggest that these collisions occur so long ($>$ 1 Gyr) ago that the internal structure of this planet has had adequate time to readjust to a new thermal equilibrium. Availabilities of residual embryos as potential impactors of emerging gas giants {#sec:origin} -------------------------------------------------------------------------------- We first consider the possibility of giant impacts during the formation of gas giants. In typical disks around classical T Tauri stars, super-Earth embryos are not sufficiently massive to open gaps [@Lin:1993hl]. Nonetheless, they tidally interact with the disk and undergo type I migration[@Goldreich:1980vn; @Ward:1984ys; @Ward:1997mw]. Their migration time scale and direction are determined by the sum of the planets’ corotation and Lindblad torques on the disk and by the disk structure [@Paardekooper:2011rr]. There is a tendency for the super-Earth embryos to converge toward some trapping radii where their enhenced surface density promotes the embryos’ oligarchic growth and the emergence of cores with sufficient mass to initiate efficient gas accretion. As they rapidly gain mass, emerging protoplanets destabilize the orbits of neighboring residual sub-critical embryos [@Zhou:2007vn]. Some of these embryos collide with the protoplanets (see paper I). With sufficient masses, proto gas giants open gaps. In extended protostellar disks, gas giants form interior to the half mass radius of the disk and undergo inward type II migration and become hot Jupiters[@Lin:1996ey]. Along the way, the migrating gas giants capture residual embryos in their sweeping mean motion resonances [@Zhou:2005pr]. Although the relatively low-mass embryos have not migrated extensively on their own, they are being shepherded to spiral toward their stars with the migrating gas giants (cf @Yu:2001sh). This snow-plough effect has led to the formation of the resonant gas giant planets systems around 55 Cnc and GJ 876 [@Lee:2002ef]. This process can excite embryos’ eccentricity, lead to orbit crossing, and enhance the possibility of giant impacts [@Zhou:2005pr; @Fogg:2007nx; @Mandell:2007kl]. Many gas giants are members of multiple planet systems. During the stage when two or more gas giants capture each other into their mutual mean motion resonances (such as those around GJ 876 and 55 Cnc) or enter into secular resonances (such as those around Ups And), their apsidal precession frequencies change greatly [@Murray:1999le]. The location of their secular resonances also sweeps over extensively wide regions during the depletion of the solar nebula [@Ward:1981wb], or during later gas-free planetesimal-driven or instability-driven giant planet migration [@Agnor:2012dn]. These resonances may excite eccentricities of super-Earths and result in giant impacts with gas giants. @Ketchum:2011aa studied an alternative scenario, in which super-Earths form and migrate inward after hot Jupiters complete inward migration and reside inside the inner edge of the disk. They found those super-Earths have a great chance to collide with the inner gas giant if the eccentricity damping is not efficient. Supply of potential impactors to hot Jupiters during the epoch of disk depletion. {#sec:superearths} --------------------------------------------------------------------------------- The trapping radius where super Earths converge depends the surface density and temperature distribution in the disk. These quantities are also functions of disks’ accretion rate, effective viscosity, and stellar luminosity [@Garaud:2007oq]. In models with a standard $\alpha$ prescription, super Earths’ trapping radius is located at the transition between the inner disk which is heated by viscous dissipation and the outer disk where stellar irradiation provides the dominant thermal energy [@Kretke:2012ai]. During the advanced stage of disk evolution when accretion diminishes with gas depletion, the trapping radius contracts and the magnetosphere truncation radius expands. Through this process, super Earths may accumulate in the proximity of their host stars. On the observational front, radial velocity and transit surveys indicate that solar-type stars bear super-Earths, with mass and period up to $\sim$ 20 $M_\oplus$ and a few months, are more common than those with Jupiter-mass gas giants. Radial velocity survey with HARPS reports that it may be in the range 39-58%, though only a few of these claims have actually been published (Mayor et al 2009, Udry private communication). Although the eta-of-Earth survey [@Howard:2010bs] suggests a lower fraction of stars with super-Earths, it nonetheless indicates that the frequency of stars with short-period planets is a rapidly decreasing function of planet mass. The common existence of super-Earths is also suggested by many detections in the Kepler transit survey. In many cases, these super-Earths are members of multiple-planet systems in which migration and dynamical interaction most likely have influenced their formation and evolution [@Papaloizou:2010qf]. After the disk depletion, mutual secular perturbation between co-existing hot Jupiters and close-in super Earths leads to eccentricity excitation and eventually dynamical instability. Numerical simulations [@Zhou:2007fk] indicate that multiple planets, with equal masses $M_{\rm p}$ and a normalized separation $k_0 = \Delta a / R_{\rm H}$ where $a$, $\Delta a$, and $R_{\rm H} = a(2 M_{\rm p} / 3 M_\ast)^{1/3}$ are the semi major axis, separation, and Hill radius, undergo orbit crossing on a time scale $${\rm log} (T_{\rm c}/P_{\rm k}) \simeq -5 + 2.2 k_0 \,, \label{eq:tcross}$$ where $P_{\rm k}$ is the mean orbital period. In §2 Paper I, we have indicated that self gravity of protostellar disks introduces precession in the eccentric orbits of gas giants. Secular resonances occur in regions of disks where precession rates due to the gravitational perturbation of these planets matches with their precession rate due to the disk potential. As $\Sigma_{\rm g}$ decreases, these secular resonances sweep across them. Passage of the secular resonances excites the eccentricity of residual planets and super-Earths [@Ward:1981wb; @Nagasawa:2005oa] including those with short periods (such as $\upsilon$ And b and $\mu$ Ari b [@Nagasawa:2005xz; @Zhou:2005pr]). This process can also lead to orbit crossings which destabilize multiple planet systems and induce potential giant impact events[@Thommes:2008jb]. Hot Jupiters’ potential impactors around mature stars ----------------------------------------------------- In compact systems of multiple hot Jupiters and close-in super-Earths the crossing time scale may be lengthened by the stabilizing contributions of relativistic precession and internal tidal dissipation [@Mardling:2004mq; @Mardling:2007wm; @Terquem:2007lp]. However, these planets undergo further decay[@Novak:2003ta; @Lee:2013ph] which may eventually lead to dynamical instabilities, orbit crossing, and eventually cohesive collisions [@Lin:1997ig]. Compact systems of long-period planets also undergo orbit crossing, on the time scale of $T_{\rm c}$ and excite each other eccentricities [@Weidenschilling:1996ij; @Rasio:1996kx; @Lin:1997ig; @Zhou:2007fk; @Juric:2008uq; @Chatterjee:2008gd]. Known gas giants with periods longer than 1-2 weeks have nearly uniform eccentricity distribution ranging from zero to nearly unity. Subsequent close encounters, secular chaos, Kozai resonances, and external perturbations [@Wu:2003lo; @Wu:2007bs], [@Laughlin:1998fv; @Adams:2001dz; @Spurzem:2009fu; @Nagasawa:2011dp; @Wu:2011df] may drive distant planets to highly eccentric orbits where they may begin nearly parabolic close encounters with their host stars. It has been suggested that a significant fraction of hot Jupiters, especially those with high obliquities, may be scattered to the proximity of their host stars with follow-up tidal circularization of their orbits [@Faber:2005sp; @Fabrycky:2007kl; @Narita:2007qa; @Winn:2007mi; @Wolf:2007pi]. There is also a suggestion that the mass-period distribution for the injected planets [@Nagasawa:2008ff] is consistent with that observed for hot Jupiters [@Marchi:2009lh]. However, a fractional loss of envelope during their periastron passage may lead to the ejection of gas giants on parabolic orbits [@Guillochon:2011rt; @Liu:2013uq]. Energy dissipation within the planets’ envelope during the orbital circularization may also lead to substantial envelope inflation, mass loss, or total disruption [@Gu:2003ez]. Around stars with pre-existing short-period gas giants (which may have migrated to the stellar proximity through disk-planet tidal interaction), a fresh injection of both gas giants and embryos raises the possibility of highly energetic merger and collision events. In the following sections, we present models based on the simulations of such events. SPH and grid-based simulations of Giant Impact and Merger of Gas Giants {#sec:sph} ======================================================================= Collisional Modelling Methods ----------------------------- In paper I, we considered only head-on parabolic collisions between embryos and gas giants. For long-period gas giants and for protoplanets near their ice-line natal site, their local Keplerian velocity ($v_{\rm kep} \sim 10$ km s$^{-1}$) is generally smaller than their surface escape speed $v_{\rm esc} = (2 {\rm G} M_{\rm p} /R_{\rm p})^{1/2}$, so that their collisions are unlikely to be much more energetic than parabolic impacts. For hot Jupiters, however, the relative speed between (some parabolic) projectiles and target planets may exceed the local $v_{\rm kep}$ which is $\geq$ 200 km s$^{-1}$. In anticipation on these possibilities, we expand the range of impact speed in the simulated models presented here. We model head-on and oblique collisions between planets using two complementary methods, smoothed particle hydrodynamics (SPH) and an Eulerian grid-based code FLASH. Our SPH code is a descendant of the one used in @Benz:1986mo and includes self-gravity but neglects internal strength. We use an ideal gas equation of state for the gaseous envelope and the Tillotson equation of state for iron and basalt for condensed materials. The Tillotson equation of state relates pressure, density and internal energy of condensed materials and models them through three regimes. At low pressure this EOS captures the linear-shock particle relations and at high pressure the EOS extrapolates to the Thomas-Fermi limit. The Tillotson EOS also treats condensed material at low density as either a high energy gas or low energy and pressure fluid. Tillotson EOS parameters for particular materials (e.g., basalt, iron) are tuned to match results of laboratory experiments and theoretical limits [see Appendix A1 of @Melosh:1989book for a brief discussion]. In addition, we use the Eulerian hydrodynamics code FLASH to construct several models with similar parameter used in SPH simulations. FLASH is a parallel, multidimensional hydrodynamics code based on block-structured adaptive mesh refinement (AMR). The center of the computational domain is fixed to the center of mass of the system and is $4\times 10^{13}$ cm on a side. We have made a careful choice of the refinement criteria, making our smallest cells only $3.818\times 10^{7}$ cm in width, which is less than 7 parts in 1000 of the planet’s radius. Combing the vast simulation domain with high resolution at the center, we are able to keep track of the envelope expansion after the impact and resolve the planet’s dense core simultaneously. Poisson equation is solved using a multipole expansion of the fluid with a maximum level at 40. Apart from from hydrodynamics and gravity, the two codes also differ in the equation of state (EOS). In FLASH simulations, we model the internal structure of the gas giant planet with composite polytropes with polytropic indices $n_1=0.5$ and $n_2=1$, which can characterize the distinct chemical composition of the core and envelope of a gas giant planet [@Liu:2013uq]. The embryo impactors are modeled with an $n=0.01$ polytrope with a central density $\rho_{\rm c} = 10$ g cm$^{-3}$. In this treatment, the polytropic constant $K$ is a free parameter, and the polytropic model can be determined by given values of $n$, $M$ and $R$ [@Kippenhahn:2013aa]. Here the choice of polytropic index $n=0.01$ over a more common value $n\simeq 1/3$ is made to give extra weight to the incompressible nature of the impactor. The polytropic index $n$ of the impactor has little effect on the post-impact thermal evolution of the target in the sense that the kinetic energy of the impactor dominates its internal energy and the target has a much larger heat capacity than that of the impactor. Our use of an ideal gas or polytrope EOS for the gaseous envelope simplifies the computation at the expense of omitting the processing of energy into internal degrees of freedom. Shock heating of the envelope resulting in dissociation and ionization of gas may act as a sink to mechanical impact energy and affect the final outcome of a collision [@Zeldovich:1967aa]. As a result, our use of simplified equations of state for the gaseous envelope may increase the fraction of energy retained in mechanical kinetic energy and the amount of material escaping a collision. Future work may address the significance of this issue with a more sophisticated equation of state and assess how of how this effect alters collisional processing of giant planets. However, our SPH and FLASH collisions models serve to illustrate the basic outcome morphologies and their implications for the collisional evolution of hot Jupiters. This section is organized as follows. We first lay out our model of giant impact and merger. We then show that both codes produce consistent results, although some discrepancies are found. Finally, we will discussion the causes for these discrepancies and use our results to predict observational signatures. Range of impact speeds {#sec:impactspeed} ---------------------- In the limit that the orbits of two planets with $\Delta a \ll a$ marginally cross within each other’s Roche radius, their relative speed is $$\Delta v \sim v_{\rm kep} (a + \Delta a) ( 1 + {\Delta a / a}) - v_{\rm kep} (a) \simeq v_{\rm kep} (a) (\Delta a / 2 a) \,. \label{eq:Deltav}$$ Since gas giants’ gravity also accelerates impactors, the impact speed associated with GIM’s is generally larger than this value. We first consider a system of planets which formed with nearly circular orbits. After the gas in the natal disks is depleted, their $\Delta v$ grow and their orbits eventually cross each other on a time scale $T_{\rm c}$ (see Eq \[eq:tcross\]) with $\Delta v \sim (G M_{\rm p} / r_{\rm H})^{1/2} k_0/{\sqrt 12} \sim k_0/ 12^{1/3} ( M_{\rm p}/M_\ast)^{1/3} v_{\rm kep}$. For systems of gas giants, formed at a few AU’s with $k_0 \sim 6$, $T_{\rm c}$ is a few Gyr (ie comparable to the main sequence life span of the solar type stars). For impactors with much lower masses, $\Delta a \sim 0.25 a$ and $\Delta v \sim 1-2$ km/s. Their values may increase by 25% for two gas giants with comparable masses. Orbit crossing introduces finite collision probability rather than certain GIM’s. The collision time scale depends on the differential orbital speed $\Delta v$ at which impactors enter into the gas giants’ $r_{\rm H}$. For initial $k_0 > 1$, $\Delta v \sim (G M_{\rm p} / r_{\rm H})^{1/2} \sim 3^{1/6} ( M_{\rm p}/M_\ast)^{1/3} v_{\rm kep} $ and most encounters do not necessarily lead to physical collisions. Nevertheless, since $\Delta v$ is much smaller than $v_{\rm esc}$, planet’s mutual gravitational attraction would accelerate them to attain impact speeds for parabolic encounters [*i.e.*]{} $v_{\rm imp} \simeq v_{\rm esc}$. Numerical experiments [@Zhou:2007fk] indicate that dynamical instability can be suppressed by eccentricity damping. In the proximity of their host stars, the eccentricity damping time scale due to planets’ internal tidal dissipation [@Goldreich:1966rp] is $$\tau_e=\frac{4Q'_{\rm p}}{63 n_{\rm p}}\left(\frac{M_{\rm p}}{M_*}\right) \left(\frac{a}{R_{\rm p}}\right)^5. \label{eq:taue}$$ The planets’ quality factor $Q'_{\rm p} \sim 10^6$ and $\tau_e \sim 0.1$ Gyr are infered for most transiting hot Jupiters from their observed eccentricity- mean motion ($e-n_p$) distribution [@Sasselov:2003zp; @Ogilvie:2004rw]. Most close-in super-Earths have smaller $Q'_{\rm p}$ and $n_p$, and comparable $\tau_e$. From equations \[eq:tcross\] and \[eq:taue\], we estimate $T_{\rm c} \sim \tau_e$ with $\Delta a \sim 0.3-0.4 a$ for $k_0 \sim 7$ for these planets. Their corresponding $\Delta v$ is comparable to their $v_{\rm esc}$ such that the speed ($v_{\rm imp}$) of their giant impacts may range from a fraction to several $v_{\rm esc}$. For our numerical models, we consider this entire range of possibilities. Impact Orientation and Dynamics ------------------------------- We describe collision dynamics using $v_{\rm imp}$ and impact angle ($\xi$). The magnitude of $v_{\rm imp}$ is a function of both the relative velocity at infinity ($v_{\infty}$) and the two-body escape velocity ($v_{\rm esc,2}$) with $v_{\rm imp}^2 = v_{\rm esc,2}^2 + v_{\infty}^2$. For an encounter involving a target and impactor with masses ($M_{\rm T}$, $M_{\rm I}$) and radii ($R_{\rm T}$, $R_{\rm I}$) respectively, the two-body escape velocity is $$v_{\rm esc,2}^2 = \frac{2G (M_{\rm I} + M_{\rm T}) } {(R_{\rm I} + R_{\rm T})} \,.$$ In the limit $M_{\rm I} < < M_{\rm T}$ and $R_{\rm I} < < R_{\rm T}$, $v_{\rm esc,2} \sim v_{\rm esc}$. We have performed simulations of a range of impact speeds with $v_{\rm imp}/v_{\rm esc,2} = 1, 1.4, 3,$ and $5$, which is consistent with the probable range of encounter velocities produced by multiple scattering events. We use an impact angle ($\xi$) defined as the angle between the relative position and velocity vectors when the surfaces of the two bodies are in contact (i.e. $\xi=0^{\circ}$ for a head-on collision and $\xi =90^{\circ}$ for a grazing, tangential encounter). Assuming an isotropic flux of impactors, the probability of a collision with $\xi$ in the range $\xi\rightarrow \xi+d\xi$ [@Shoemaker:1962wa] is $$dP = 2\sin\xi\cos\xi\; d\xi \,.$$ The angle $\xi=45^{\circ}$ is the median value and the impact angles $\xi = 30^{\circ}$, $45^{\circ}$, $60^{\circ}$, and $90^{\circ}$ divide the impact angle distribution into quartiles. We have performed simulations of both head-on $\xi=0^{\circ}$ collisions as well as more oblique collisions with impact angles of $\xi=21^{\circ}$, $30^{\circ}$, $45^{\circ}$, and $60^{\circ}$. SPH simulations were run for about $24$-hours of model time before analyzing the result. The simulations are first analyzed to determine the particles that are contiguous with the initial gas giant’s core. Following that we determine which particles are gravitationally bound in a two-body sense. For FLASH simulations, we stop the simulations at about $2 \times 10^{5}$ s. The determination of contiguous and bound material is more challenging in a grid code, and we describe our approach to measure these quantities in Section \[sec:discrepancy\]. Model parameters of SPH and FLASH simulations and their results are summarized in Tables \[tab:sph25\], \[tab:sph10\], and \[tab:saturn-100\]. Collisions between a 25-$M_\oplus$ embryo and a Saturn-mass gas giant --------------------------------------------------------------------- We simulate both head-on and oblique collisions between a Saturn-like 100-$M_{\oplus}$ gas giant planet and a 25-$M_\oplus$ embryo. SPH model parameters and results of this series of simulations are summarized in Table \[tab:sph25\]. These results are the generalization of the head-on parabolic collision in Paper I. In addition, four FLASH simulations with similar parameters are performed for comparison. Part of the analysis and visualization of the FLASH results in this work relies on the YT Python package [@Turk:2011aa]. [cccccccccc]{} Model & $\xi$ & $v_{\rm imp}/v_{\rm esc,2}$ & $M_{\rm bm}$ & $M_{\rm bm,c}$ & $R_{\rm bm}/R_{\rm T}$ & $M_{\rm cc}$ & $M_{\rm cc,c}$ & $R_{\rm cc}/R_{\rm T}$ & $J_{\rm cc}/J_{\rm *}$\ SPH\ SA1a & 0 & 1.0 & 125.0 & 35.0 & 1.08 & 107.9 & 35.0 & 0.95 & 0.001\ SA1b & 21 & 1.0 & 124.7 & 35.0 & 1.08 & 104.2 & 35.0 & 0.92 & 0.168\ SA1c & 30 & 1.0 & 124.7 & 35.0 & 1.08 & 101.9 & 35.0 & 0.90 & 0.219\ SA1d & 45 & 1.0 & 124.7 & 35.0 & 1.09 & 96.3 & 34.3 & 0.88 & 0.261\ SA1e & 60 & 1.0 & 125.1 & 35.0 & 1.03 & 90.5 & 10.4 & 0.94 & 0.069\ \ SA2a & 0 & 1.4 & 123.1 & 35.0 & 1.09 & 102.8 & 35.0 & 0.93 & 0.001\ SA2b & 21 & 1.4 & 122.9 & 35.0 & 1.09 & 97.2 & 35.0 & 0.89 & 0.210\ SA2c & 30 & 1.4 & 123.0 & 35.0 & 1.10 & 93.8 & 34.6 & 0.87 & 0.263\ SA2d & 45 & 1.4 & 99.1 & 10.8 & 1.02 & 88.5 & 10.4 & 0.93 & 0.065\ SA2e & 60 & 1.4 & 99.9 & 10.2 & 1.01 & 93.2 & 10.1 & 0.95 & 0.034\ \ SA3a & 0 & 3.0 & 87.9 & 35.0 & 0.84 & 80.3 & 34.7 & 0.78 & 0.001\ SA3b & 21 & 3.0 & 85.5 & 11.5 & 1.00 & 67.6 & 11.1 & 0.84 & 0.070\ SA3c & 30 & 3.0 & 92.6 & 10.5 & 1.03 & 74.6 & 10.3 & 0.88 & 0.064\ SA3d & 45 & 3.0 & 98.2 & 10.2 & 1.03 & 85.2 & 10.0 & 0.92 & 0.048\ SA3e & 60 & 3.0 & 99.8 & 10.0 & 1.01 & 92.8 & 10.0 & 0.95 & 0.021\ \ SA4a & 0 & 5.0 & 0.4 & 0.3 & 0.11 & 0.1 & 0.1 & 0.07 & 0.229\ SA4b & 21 & 5.0 & 69.5 & 10.3 & 0.90 & 58.0 & 10.1 & 0.79 & 0.059\ SA4c & 30 & 5.0 & 85.5 & 10.2 & 1.01 & 67.8 & 10.1 & 0.85 & 0.051\ SA4d & 45 & 5.0 & 96.4 & 10.1 & 1.04 & 80.7 & 10.0 & 0.91 & 0.048\ SA4e & 60 & 5.0 & 99.5 & 10.0 & 1.02 & 89.9 & 10.0 & 0.94 & 0.026\ FLASH\ FA1c & 30 & 1.0 & 121.1 & 34.4 & 12.67 & 117.1 & 34.4 & 3.34 & 0.066\ FA2a & 0 & 1.4 & 115.3 & 34.3 & 7.09 & 108.6 & 34.3 & 2.58 & $5\times10^{-5}$\ FA3a & 0 & 3.0 & 8.5 & 5.4 & 6.49 & 8.5 & 5.4 & 6.49 & $10^{-4}$\ FA3c & 30 & 3.0 & 83.2 & 10.0 & 8.44 & 76.9 & 10.0 & 2.34 & 0.003\ The SPH simulations show that head-on collisions (with $v_{\rm imp} = 1-3 v_{\rm esc,2}$) lead to the capture of the impactor by the gas giant and the coalescence of the impactor with the core. In addition, the envelope re-establishes a spherical symmetry within a dynamical time scale. These results justify the 1-D prescription and core-impactor coalescence assumption used in our LHD scheme (see Paper I). For sufficiently high velocity encounters, the energy dissipation near the core leads to a thermal expansion of the envelope. In model SA3a with $v_{\rm imp} = 3 v_{\rm esc,2}$, the impactor’s kinetic energy is comparable to the gas giant’s gravitational binding energy and a significant fraction of the envelope is lost. The requirement $$v_{\rm imp} > v_{\rm cri} = (2 (M_{\rm T}+M_{\rm I})/M_{\rm I})^{1/2} v_{\rm esc,2} \label{eq:breakup}$$ corresponds to the condition for total disruption of isolated gas giants. Note that in model SA4a with $v_{\rm imp} = 5 v_{\rm esc,2}$, the impact energy is so large that both the envelope and core totally disintegrated. @Anderson:2012aa predicted a penetration velocity for head-on collisions four orders of magnitude greater than the escape velocity. Such energetic collisions likely lead to destruction of the gas giant other than penetration according to our results. Despite the general agreement between Equation \[eq:breakup\] and the SPH results in Table \[tab:sph25\], some of the results obtained through these SPH simulations are possibly due to numerical artefacts. For example, the ratio of the final to initial planetary radius ($R_{\rm f}/R_{\rm i}$) remains close to unity for head-on collisions with $v_{\rm imp} < 3 v_{\rm esc,2}$. This negligible change in the gas giant’s envelope is partially due to the artificial lower density limit we have adopted for the SPH scheme ([*i.e.*]{} $\rho_{\rm l} = 0.5$ g cm$^{-3}$). This restriction prevents the dispersal of representative fluid elements into tenuous medium. This technical limitation is clearly shown by the sharp density fall off at the gas giant’s new radius. This prescription may also over suppress the loss of gas in the giant’s original envelope as a consequence of high-speed collisions (see §\[sec:comparison\]). The actual critical impact speed required for substantial collision-induced mass loss may be substantially reduced in the proximity of the host star. Without such restriction of lower density limit, the FLASH code is ideal to study the expansion of the gas giant’s envelope after the impact. Consequently, the determination of different boundaries in FLASH simulations is also challenging (see Section \[sec:discrepancy\]). In run FA2a we find much larger $R_{\rm cc}$ and $R_{\rm bm}$ than those in run SA2a, while the enclosed masses $M_{\rm cc}$ and $M_{\rm bm}$ are slightly less than that in SPH. Besides, in model FA3a with $v_{\rm imp}$ 3 times the $v_{\rm esc,2}$, the impactor is able to disintegrate the gas giant, while under the same circumstances, SPH model SA3c shows that most of the original mass of the gas giant is retained after the impact. We also illustrate the results of oblique collisions, first with the parabolic impact speed ($v_{\rm imp} = v_{\rm esc,2}$). In this low-energy limit, the impactor is always consumed by the gas giants. Over most range of $\xi$, say $\xi < 60^{\circ}$, the impactor is able to penetrate through the gas giant’s envelope, to merge with its core in a few dynamical timescales. This tendency is well illustrated in Figure \[fig:f1\], which shows the SPH particles distribution of model SA1c and density slice of model FA1c in the orbital plane (designated to be the [*X*]{}-[*Y*]{} plane with the initial motion along the [*X*]{}-axis) for four different epochs. We also plot the particles distribution and density slice in the the [*X*]{}-[*Z*]{} plane (where [*Z*]{}-axis is the axis normal to the orbital plane). We notice that the asymptotic core radius increases with $\xi$ (in comparison with the Figure 1 in Paper I). In the limit of collisions with relatively large $\xi$, the impactor is disintegrated in the envelope well outside the iron core. If the debris of the impactor remains separated from gaseous material in the envelope of the gas giant, double-diffusive convection may kick in, and the compositional transport rate could be either very small or relatively large [@Rosenblum:2011aa]. In other word, whether the debris will dissolve in the envelope or eventually sediment into the core is not clear yet. Either way, the time scale to re-establish a hydrostatic equilibrium is considerably longer than the dynamical timescale. This fractionation process provides a protracted source of energy to sustain a relatively large planet radius $R_{\rm p}$ after the impact. The FLASH run FA1c shows similar consequences of the impact except that the final structure is much puffier and mass loss is slightly higher, as we have noticed in the head-on cases. The bottom panel of Figure \[fig:f1\] is an edge-on view of the planet after the impact, which indicates a rotational flattening structure in the gas giant which has been struck by an embryo at a modest grazing angles $30^{\circ}$. The capture of an embryo modifies the spin angular momentum of the gas giant by an amount $\Delta J \simeq M_{\rm I}\; R_{\rm T}\; v_{\rm imp} \sin \xi$. If the gas giant does not have any initial spin, the ratio of its final angular moment $J_{\rm f} = \Delta J$ and that corresponds to a rotational break up ($J_{\rm c} \sim (M_{\rm T} + M_{\rm I}) R_{\rm p} v_{\rm esc,2}$) would be $$\frac{J_{\rm f}}{J_{\rm c}} \simeq \frac{M_{\rm I}}{(M_{\rm T}+M_{\rm I})} \left( \frac{v_{\rm imp}}{v_{\rm esc,2}}\right) \; \sin \xi \,, \label{eq:angularmomentum}$$ where we have assumed $ R_{\rm p} \simeq R_{\rm T}$. This estimate is in good agreement with those listed in Table \[tab:sph25\]. In general, the angular momentum deposited by the impactor is not uniformly distributed throughout the planet. And one may notice that the ellipticity of isodensity contours of the edge-on view of run FA1c slightly decreases from inside to outside. Furthermore, different types of giant impact produce distinguished signatures of oblateness (see Section \[sec:oblateness\]). Here we simply assume this spin angular momentum is smoothly distributed throughout the gas giant’s envelope as a first-order approximation, it would speed with a frequency $\Omega_{\rm f} \sim (J_{\rm f}/J_{\rm c}) (v_{\rm esc, 2}/\alpha_{\rm p} R_{\rm p})$, where $\alpha_{\rm p}$ is the coefficient for the moment of inertia. For isolated long-period gas giants, a parabolic GIM by a super-Earth impactor with $M_{\rm I} < 0.1 M_{\rm T}$ would not lead to rotational break up. But, in the limit that $J_{\rm f} = \Delta J$, $$\frac{\Omega_{\rm f}}{ n_{\rm p}} \simeq \frac{M_{\rm I}}{(M_{\rm T}+ M_{\rm I})} \left( \frac{v_{\rm imp}}{ v_{\rm esc,2 }}\right) \left( \frac{3 R_{\rm H}}{R_{\rm p}} \right)^{3/2}\; \sin \xi \,,$$ where $R_{\rm H}$ is the Hill radius of the giant planet. For hot Jupiters, $\Omega_{\rm f}$ can be significantly larger than the rotational frequency associated with the spin-orbit synchronization ($\sim n_{\rm p}$). Thus, stochastic parabolic collisions by super-Earths can offset synchronous spin generally associated with star-planet tidal interaction. For high-energy oblique collisions, impactor’s capture probability decreases with both $\xi$ and $v_{\rm imp}$. During impactor’s first passage, the total amount of kinetic energy lost due to hydrodynamic drag alone is $$\Delta E_{\rm k} = M_{\rm I}(v_{\rm imp}^2-v_{\rm final}^2)/2 \simeq M_{\rm I} v_{\rm imp} \Delta v /2 \,,$$ where $v_{\rm final}$ is the impactor’s velocity after it passes through the target and $\Delta v = v_{\rm imp} - v_{\rm final}$ is the change of the impactor’s velocity. Because momentum is conserved, we have $M_{\rm I} \Delta v \simeq M_{\rm enc} v_{\rm imp}$ and $$M_{\rm enc} (\xi) \simeq \int_L \rho A dl \simeq {2 A R_{\rm p} \cos \xi \over 1 - \sin \xi } \int_{\sin \xi} ^1 \rho (x) dx \label{eq:menc}$$ is the “air mass” encountered by the impactor in the gas giant’s envelope, $L$ is impactor’s trajectory inside the gas giant and the gas density $\rho$ is a function of distance from the gas giant’s center $R$ or equivalent a function of $x \equiv R/R_{\rm p}$. For modest impacts, impactor’s effective cross section $A$ is comparable to its physical radius $R_{\rm I}$. In the above expression $M_{\rm enc}$ is a rapidly decreasing function of the impact angle $\xi$. We now determine the capture cross section of embryos by the gas giant. An impactor is unlikely to be captured in the limit that $\Delta E_{\rm k} < M_{\rm I} (v_{\rm imp}^2 - v_{\rm esc, 2}^2)/2$ or equivalently $$M_{\rm enc} < M_{\rm I} \left( 1 - (v_{\rm esc, 2}/v_{\rm imp})^2\right)/2 \,. \label{eq:airmass}$$ The air mass required for capturing a intruding embryo with low impact speed is a small fraction of $M_{\rm I}$. Provided the impact parameter is smaller than the gas giant’s radius $R_{\rm p}$ ([*i.e.*]{} $\cos \xi$ is non-negligible), most parabolic encounters leads to coalescence. But, the required air mass $M_{\rm enc} \sim M_{\rm I}/2$ for high impact speed collisions. Capture of relatively massive embryos would be possible only for relatively small $\xi$, i.e. nearly head-on collisions. Similar arguments have been used to estimate the fraction of material lifted into the proto-lunar disk via giant impact [@Canup:2008sph] and the mechanical energy dissipated in collisions between stars [@Freitag:2005sc] and planets [@Leinhardt:2012a]. The capture condition in equation \[eq:airmass\] is in good agreement with the simulation results in Table \[tab:sph25\], [*i.e.*]{} collisions with a high impact speeds and angles generally do not lead to capture whereas embryos which impinges onto gas giants with relative low impact speeds and angle generally merge with them. These general outcome morphologies have also been observed in collisions between stars and planets [@Agnor:2004ec; @Freitag:2005sc; @Asphaug:2006hr; @Marcus:2009se; @Leinhardt:2012a] and appear a scale-invariant feature of gravity-dominated collisions. With a high relative speed, the impactor also disintegrates when it encounters an air mass comparable to its own. Because when $M_{\rm enc} \sim M_{\rm I}$, the change of the impactor’s total energy is comparable to its gravitational binding energy. Thus, in the limit of high velocity encounters, captured embryos rapidly disintegrate whereas the flyby embryos essentially retain their initial mass. Finally, we point out that among all the simulations in which the planet is not destroyed, the mass loss is preferentially comes from the envelope other than the core. In fact, there is only very tiny mass loss contributed by the core. This result indicates that giant impacts may lead to the enhanced metallicity in the envelopes of gas giants. Collisions between a 10-$M_\oplus$ embryo and a Saturn-mass gas giant --------------------------------------------------------------------- Here, we consider a series of models which simulate the collision between a Saturn-like giant planet and a 10-$M_\oplus$ embryo. Model parameters and results of this series of simulations are summarized in Table \[tab:sph10\]. We emphasize here that relatively low-mass embryos disintegrate in the envelope before reaching the core. [cccccccccc]{} Model & $\xi$ & $v_{\rm imp}/v_{\rm esc,2}$ & $M_{\rm bm}$ & $M_{\rm bm,c}$ & $R_{\rm bm}/R_{\rm T}$ & $M_{\rm cc}$ & $M_{\rm cc,c}$ & $R_{\rm cc}/R_{\rm T}$ & $J_{\rm cc}/J_{\rm *}$\ SPH\ SB1a & 0 & 1.0 & 110.1 & 20.0 & 1.06 & 96.9 & 20.0 & 0.95 & 0.001\ SB1b & 21 & 1.0 & 110.1 & 20.0 & 1.05 & 97.0 & 20.0 & 0.95 & 0.097\ SB1c & 30 & 1.0 & 110.0 & 20.0 & 1.05 & 96.2 & 20.0 & 0.94 & 0.132\ SB1d & 45 & 1.0 & 109.7 & 20.0 & 1.05 & 94.3 & 18.7 & 0.93 & 0.155\ SB1e & 60 & 1.0 & 109.8 & 20.0 & 1.03 & 100.2 & 19.6 & 0.96 & 0.246\ \ SB2a & 0 & 1.4 & 110.1 & 20.0 & 1.07 & 92.8 & 20.0 & 0.94 & 0.001\ SB2b & 21 & 1.4 & 109.5 & 20.0 & 1.07 & 92.2 & 20.0 & 0.93 & 0.122\ SB2c & 30 & 1.4 & 109.1 & 20.0 & 1.06 & 91.0 & 19.7 & 0.92 & 0.163\ SB2d & 45 & 1.4 & 100.3 & 11.4 & 1.02 & 91.2 & 11.0 & 0.94 & 0.056\ SB2e & 60 & 1.4 & 100.2 & 10.4 & 1.00 & 94.0 & 10.1 & 0.95 & 0.023\ \ SB3a & 0 & 3.0 & 100.5 & 20.0 & 1.06 & 79.4 & 20.0 & 0.89 & 0.002\ SB3b & 21 & 3.0 & 93.3 & 11.6 & 1.04 & 73.7 & 11.1 & 0.88 & 0.070\ SB3c & 30 & 3.0 & 95.6 & 10.6 & 1.04 & 79.8 & 10.3 & 0.91 & 0.061\ SB3d & 45 & 3.0 & 98.7 & 10.1 & 1.02 & 87.3 & 10.0 & 0.93 & 0.038\ SB3e & 60 & 3.0 & 99.9 & 10.1 & 1.01 & 93.2 & 10.0 & 0.95 & 0.017\ \ SB4a & 0 & 5.0 & 64.9 & 19.7 & 0.78 & 55.7 & 19.3 & 0.69 & 0.003\ SB4b & 21 & 5.0 & 82.0 & 10.5 & 0.99 & 64.5 & 10.2 & 0.83 & 0.049\ SB4c & 30 & 5.0 & 90.8 & 10.2 & 1.03 & 72.4 & 10.1 & 0.88 & 0.049\ SB4d & 45 & 5.0 & 97.5 & 10.0 & 1.03 & 83.6 & 10.0 & 0.92 & 0.042\ SB4e & 60 & 5.0 & 99.6 & 10.0 & 1.01 & 91.4 & 10.0 & 0.95 & 0.022\ FLASH\ FB3a & 0 & 3.0 & 65.9 & 14.9 & 4.52 & 65.9 & 14.9 & 4.52 & 10$^{-4}$\ FB3c & 30 & 3.0 & 88.9 & 10.1 & 8.42 & 80.5 & 10.0 & 1.55 & 0.004\ FB4a & 0 & 5.0 & - & - & - & - & - & - & -\ The results in Table \[tab:sph10\] are qualitatively similar to those in Table \[tab:sph25\], [*i.e.*]{} all head-on and all low-speed collisions lead to the capture of this relatively low-mass embryo and oblique high-velocity embryos pass through and escape from the gas giant’s envelope. In contrast to the 25-$M_\oplus$ models, this lower mass (10-$M_\oplus$) impactor’s kinetic energy is inadequate to disrupt gas giant’s envelope in all SPH simulations. Note that in FLASH simulations, only the high-speed head-on collision (FB4a) can disrupt the gas giant. In some high-speed impacts (such as model SB4b), sufficient energy is deposited to the envelope to induces fractional mass loss, even though the impactor is not captured during the encounter. Nevertheless, it can carry sufficient amount of energy, especially through high-velocity encounters, to perturb the structure of the gas giant (albeit SPH scheme may not be able to resolve these changes), Captured embryos can also deposit a significant amount of spin angular momentum to the gas giant. Model parameters for the low-mass embryo to be marginally captured are in good agreement with equation (\[eq:airmass\]). The captured models indicate that spin angular momentum deposited into the envelope is greater than that needed for some hot Jupiters to uniformly rotate with an angular frequency which synchronizes with their orbital frequency. In contrast, gas giant’s mass and angular momentum are not significantly affected by fly-by encounters. But, in some models (such as SB2d and SB3b) a fraction of the impactor’s mass can be stripped from it during its passage through (rather than capture by) the gas giant’s envelope. As a result, the final core mass slightly increases. Model SB2c indicates that during oblique collisions with modest energy, this modest size embryo is mostly disrupted, which is similar to model SA1c except that most of the mass is deposited in the gas giant’s envelope, slightly outside the iron core. The kinetic energy of the impactor is also deposited in the envelope rather than near the core. Eventual sedimentation of impactor’s fragments releases additional energy on a much longer time scale (see §\[sec:lhd\]). In contrast, model SB3c shows that highly energetic embryos may not be retained by the envelope of the gas giant that they have oblique collisions with. In FLASH simulation FB3c, the impactor penetrate through the gas giant’s envelope as well, and causing over 10% of gas giant’s mass is unbound and it outer envelope becomes very extensive but its inner region remains unchanged (Figure \[fig:f2\]). The amount of angular momentum deposition is also correspondingly less that the total-coalescence models with the same impact parameter. These figures also indicate that although spherical symmetry is temporarily destroyed during the impact, it is quickly restored both in the envelope and the core after 1-2 dynamical time scales. Despite the injection of spin angular momentum to the gas giant’s envelope, the asymptotic distribution of the debris is approximately spherically symmetric. Merger between two Saturn-mass gas giants ----------------------------------------- In this subsection, we present simulations of collisions between two identical gas giants. Similar to the previous section, we adopt an initial model with a 90-$M_\oplus$ envelope around a 10-$M_\oplus$ core. We also explore a similar series of initial conditions which are listed in Table \[tab:saturn-100\]. [cccccccccc]{} Model & $\xi$ & $v_{\rm imp}/v_{\rm esc,2}$ & $M_{\rm bm}$ & $M_{\rm bm,c}$ & $R_{\rm bm}/R_{\rm T}$ & $M_{\rm cc}$ & $M_{\rm cc,c}$ & $R_{\rm cc}/R_{\rm T}$ & $J_{\rm cc}/J_{\rm *}$\ SPH\ SC1a & 0 & 1.0 & 199.6 & 20.0 & 1.25 & 178.0 & 20.0 & 1.13 & 0.000\ SC1b & 21 & 1.0 & 200.1 & 20.0 & 1.27 & 159.8 & 20.0 & 1.05 & 0.258\ SC1c & 30 & 1.0 & 200.3 & 20.0 & 1.32 & 144.9 & 20.0 & 1.02 & 0.299\ SC1d & 45 & 1.0 & 200.3 & 20.0 & 1.27 & 151.5 & 20.0 & 1.02 & 0.744\ SC1e & 60 & 1.0 & 200.3 & 20.0 & 1.29 & 87.6 & 10.0 & 0.92 & 0.118\ \ SC2a & 0 & 1.4 & 197.8 & 20.0 & 1.27 & 168.0 & 20.0 & 1.11 & 0.001\ SC2b & 21 & 1.4 & 199.3 & 20.0 & 1.33 & 138.7 & 20.0 & 1.00 & 0.276\ SC2c & 30 & 1.4 & 199.9 & 20.0 & 1.34 & 77.3 & 10.0 & 0.88 & 0.150\ SC2d & 45 & 1.4 & 100.0 & 10.0 & 1.04 & 85.4 & 10.0 & 0.92 & 0.087\ SC2e & 60 & 1.4 & 100.0 & 10.0 & 1.02 & 91.0 & 10.0 & 0.94 & 0.063\ \ SC3a & 0 & 3.0 & 112.1 & 20.0 & 0.89 & 98.1 & 19.5 & 0.82 & 0.001\ SC3b & 21 & 3.0 & 69.2 & 10.0 & 0.84 & 62.1 & 10.0 & 0.77 & 0.155\ SC3c & 30 & 3.0 & 87.4 & 10.0 & 1.01 & 70.7 & 10.0 & 0.86 & 0.069\ SC3d & 45 & 3.0 & 97.5 & 10.0 & 1.04 & 81.0 & 10.0 & 0.91 & 0.061\ SC3e & 60 & 3.0 & 99.6 & 10.0 & 1.01 & 91.3 & 10.0 & 0.95 & 0.034\ \ SC4a & 0 & 5.0 & 41.7 & 13.1 & 0.55 & 36.3 & 12.4 & 0.51 & 0.005\ SC4b & 21 & 5.0 & 48.0 & 10.0 & 0.62 & 44.0 & 10.0 & 0.57 & 0.126\ SC4c & 30 & 5.0 & 70.0 & 10.0 & 0.87 & 62.9 & 10.0 & 0.80 & 0.084\ SC4d & 45 & 5.0 & 93.6 & 10.0 & 1.04 & 74.9 & 10.0 & 0.88 & 0.055\ SC4e & 60 & 5.0 & 99.2 & 10.0 & 1.03 & 87.3 & 10.0 & 0.93 & 0.037\ FLASH\ FC1b & 21 & 1.0 & 196.3 & 20.0 & 9.76 & 188.1 & 20.0 & 3.43 & 0.181\ FC2a & 0 & 1.4 & 169.2 & 20.0 & 12.47 & 148.0 & 20.0 & 3.49 & $2\times 10^{-4}$\ These models again indicate that parabolic encounters lead to the total coalescence of gas giants without any loss of heavy elements in the cores and gas in the envelopes. In order to compare with results of the head-on parabolic merger, we present the results of model SC1b and FC1b in Figure \[fig:f3\]. These results indicate that the cores preserve their integrity until they are in contact with each other. During the collision, the iron cores of both gas giants are thoroughly mixed each other. These simulations also show that the internal structure of the merger product rapidly becomes symmetric about the Z-axis. Even with this small impact angle, the merger product acquires a substantial amount of spin angular momentum, primarily due to the relatively large mass of the impactor. In the SPH simulation, the asymptotic major axis (in the orbital planet) of the merger product is larger than the gas giant’s initial radius $R_{\rm p}$. The polar radius $R_{\rm po}$ (i.e. $R_{\rm p}$ in the [*X*]{}-[*Z*]{} plane) remains unaltered from the initial $R_{\rm p}$. But this difference between major and minor axes indicate that the enlargement of equatorial radius $R_{\rm eq}$ (i.e. $R_{\rm p}$ in the [*X*]{}-[*Y*]{} plane) is mainly due to the rotational effects rather than thermal expansion. Indeed, the amount of energy dissipation during the collision is actually a modest fraction of the gas giant’s initial internal energy. In the FLASH simulation, both $R_{\rm po}$ and $R_{\rm eq}$ of the merger product expand, but $R_{\rm eq}$ is also much larger than $R_{\rm po}$. We will discuss the oblateness of FLASH simulations in Section \[sec:oblateness\]. For highly oblique collisions (models SC1d and SC1e), the spin angular momentum of the merger product is comparable to the break up speed. There is likely to be substantial tidal evolution toward spin synchronization. The continuous influx of internal energy can lead to substantial tidal heating and possibly loss of envelope mass [@Gu:2003ez; @Gu:2004km]. For relatively high impact velocities ($v_{\rm imp}/v_{\rm esc,2} =1.4$) and small-impact angle collisions ($\xi<30^{\circ}$), the condensed materials of the impactor and target effectively merge with little loss of the gaseous envelope. However, for more oblique collisions (models SC2d and SC2e), the energy dissipation during the impact is insufficient to bind the colliding gas giants. They continue their original course with a small amount of energy loss and no mass loss. With a higher impact speed ($v_{\rm imp}/v_{\rm esc, 2} =3$), a head-on collision leads to the loss of most (more than half), but not all of initial envelope gas while the two cores merged (model SC3a). For collisions with modest impact angles ($\xi = 21^{\circ}$, model SC3b), merger of the cores is avoided while each gas giant loses a substantial (up to 1/3) fraction of its initial envelope. Given that half of collisions have non negligible $\xi$, these non-accretionary, envelope-eroding collisions are as probable as core-merging collisions. These intermediate-mass (between Jupiter and the Earth) merger products may account for the unexpected presence of some planets in the domain of “planet desert” [@Ida:2004ko; @Howard:2010bs; @Ida:2013fv]. For highly oblique encounters (with $\xi \ge 30^{\circ}$, models SC3c, SC3d and SC3e), gas giants retain their original envelope, albeit considerable angular momentum is deposited to the gas giant’s spin during gas giants’ brief passage through each other’s envelope. Hyper-velocity head-on collisions (with $v_{\rm imp}/v_{\rm esc, 2} =5$) lead to the non-equal fragmentation of both core and envelope. Finally, after high-velocity oblique collisions, cores continue their original fly paths with a fraction of their initial envelopes. In contrast to the substantial (but not entire) loses of gas giants’ initial envelope, most of the core material is retained by the merger product. Thus, the metallicity of the merger product always increases from that of the original gas giant planets. Finally, these collisions deposit not only energy and heavy elements but also angular momentum to the spin of the colliding gas giants. Comparison between SPH and FLASH results {#sec:discrepancy} ---------------------------------------- Both SPH and grid codes have been widely used to perform hydrodynamic simulations. Because of their inherent differences, SPH and grid methods are complementary to study the same problem. For example @Canup:2013aa compared these two methods in the context of the lunar-forming impacts, and they found a general agreement between the SPH and the grid code. In this work, we confirm that the SPH and the FLASH codes produce overall agreement on the critical impact speed for the envelope and core mass losses, even though they differ in the size of resultant collisions. Apart from the hydrodynamics, gravity and EOS, the two codes also differ in the treatment of the ambient medium. The SPH simulations incorporate a density limit $\rho_{\rm l} = 0.5$ g cm$^{-3}$ to prevent spreading of very low density gas (that is not well-resolved in an SPH model with equal particle mass). The density limit acts as an external pressure, and the expansion of the particle stops at the density limit and the particle then evolves due to gravity and whatever neighbors it is interacting with hydrodynamically. It is known that AMR codes are very suitable to deal with surrounding structure around a dense object given that the minimum-grid and refinement criteria are made to capture the dense region [@Tasker:2008aa]. In our grid scheme, we adopt a density $\rho = 10^{-19}$ g cm$^{-3}$ for the ambient medium, so gas can diffuse into tenuous interplanetary space. As a result, all the FLASH simulations produce much puffier gas giants than those observed in the SPH runs. However, to determine the corresponding contiguous and bound radii $R_{\rm cc}$ and $R_{\rm bm}$ in FLASH is less straightforward because the gas density gradually decreases all the way down to the ambient medium level at a fairly large distance (usually two orders of magnitude larger than the original size of the planet). Furthermore, simply prolong the simulation until the planet is fully relaxed is inefficient and may introduce other accumulative errors. Here we use the mass flux *M* to quantitatively analyze the structure of the planet. The *M* at a given radius is the amount of mass that is flowing out through the sphere with that radius, e.g., a positive *M* indicates an outflow (and vice versa). A post-impact sequence of the mass flux *M* as a function of the radius taken from FC1b is plotted in the left panel of Figure \[fig:f5\]. Ten successive snapshots of mass flux *M* with a constant separation of 5000 s are presented in different colors. Even almost 2 days after the impact, the FLASH result shows that large-amplitude oscillations of mass flow are still prevailing throughout the whole planet, which suggests that the relaxation time scale is much longer than the dynamical timescale. On the outskirts of the the planet where the density drops below $10^{-2}$ g cm$^{-3}$ (see Figure \[fig:f3\]), the amplitude of such oscillations gradually decreases. As the distance goes further, the amount of mass flux overturns representing the gas outflow due to the merger. The position of the peak increases with time whereas the value of the peak decrease, which suggests the inner part does not replenish the outflow. We plotted the time-averaged mass flux of the ten snapshots in the thick black line in the right panel of Figure \[fig:f5\]. We define the contiguous radius $R_{\rm cc}$ as the radius within which large-amplitude oscillations of *M* occur but the the averaged *M* is much reduced, so the net mass flow is orders of magnitude smaller than the instantaneous value. The bound radius $R_{\rm bm}$ is defined as the separation between the bound mass and unbound mass. Material in the region surrounded by spheres with $R_{\rm cc}$ and $R_{\rm bm}$ is very tenuous and but still bound to the planet. It may form a protosatellite disk or be stripped off due to other perturbations such as tidal interactions. The definitions of $R_{\rm cc}$ and $R_{\rm bm}$ are somewhat less stringent in the sense that these boundaries may evolve with time. Considering the planet will cool off due to thermal radiation, the contiguous radius $R_{\rm cc}$ is is likely to become smaller. As the planet adjusts its internal structure to cooling, it becomes more similar to that produced by SPH simulations. On the other hand, the removal of the outflow may also lead to the expansion of $R_{\rm bm}$. In the context of the planetary system, the $R_{\rm bm}$ is comparable to its Hill radius. However, we notice that the net mass flow is negligible - in other words, the enclosed mass does not change despite the evolution the $R_{\rm cc}$ and $R_{\rm bm}$. In this work, we measure the two radii at the time about 2 days after the impacts or mergers in all FLASH simulations. Oblateness as an observable signature of GIM {#sec:oblateness} -------------------------------------------- Oblique impacts or mergers deposit angular momentum into the gas giant planet and alter its shape as well as its spin axis. To describe the shape of a planet, it is useful to define the oblateness as $$f \equiv \frac{R_{\rm eq}-R_{\rm po}}{R_{\rm eq}},$$ where $R_{\rm eq}$ and $R_{\rm po}$ are the equatorial radius and the polar radius, respectively. In Figure \[fig:f6\], we plot the oblateness $f$ of isodensity surfaces from the interior to the surface of the planet. Three FLASH simulations FA1c, FB3c and FC1b are investigated. The oblateness profile is obtained by measuring the $R_{\rm eq}$ and $R_{\rm po}$ of many isodensity surfaces. Each simulation represents a possible scenario in giant impacts or mergers. In the simulation FB3c, a 10-$M_\oplus$ embryo penetrates the gas giant’s envelope rapidly leaving the bulk structure of the gas giant mostly unchanged. The time for the impactor to exert a torque to spin up the gas giant is short. As a result, the embryo does not deposit much angular momentum into the gas giant, so the oblateness $f$ remains close to 0 throughout. In the other giant impact simulation FA1c, a 25-$M_\oplus$ embryo is disintegrated near the gas giant’s core and deposit a large amount of angular momentum there. Therefore, the oblateness $f$ keeps a relatively high level in the interior (say $\rho>1$ g cm$^3$) and drops quickly to a sub-Saturn level at larger radii. In another words, the gas giant gains much more angular momentum than that in the previous case, but its observable shape does not differ very much from a sphere as $f$ is small in the surface layer. Nevertheless, the simulation FC1b shows an opposite pattern of $f$, in which $f$ is small in the interior and becomes prominent near the surface. That is because during the oblique merger of two equal mass gas giants, torque exerted on different parts varies a lot, which introduces differential rotation to the planet. As a result, the envelope spins faster than the core and $f$ peaks near the surface and remains significant at the outer boundary. Inspired by these dramatic differences, we propose that the oblateness may serve as a proxy for probing the impact history of gas giant planets given that the exchange of angular momentum between the planet and host star can be ignored. Giant impacts generally do not lead to large oblateness, while gas giant mergers do. An oblate transiting extrasolar planet would have a discernible effect on transit light curves [@Hui:2002aa; @Seager:2002aa; @Barnes:2003aa]. Combing the oblateness information with other physical properties like mass, radius and metallicity, we may be able to constrain the formation history of a giant planet. LHD simulations of Giant Impacts and Mergers of the Gas Giants {#sec:lhd} ============================================================== Hydrodynamic simulations of GIM can only last for a few dynamic timescales. After the establishment of hydrostatic equilibrium, the giant planet undergoes thermal evolution on a timescale much longer than the dynamical timescale. Here we following the long-term thermal evolution of the remnant gas giant using a one-dimensional Lagrangian Hydrodynamic (LHD) model (see Paper I for model details). Combinations of SPH impact models and 1-D long-term thermal models have been used to examine the luminosity and core-accretion of gas-giants [@Broeg:2012ca] and long-term luminosity of gas giants [@Anic:2007gg]. The details of our one-dimensional LHD model are described in paper I. Here we use similar initial models ($M_{\rm tot}=100~M_{\oplus}$, $M_{\rm core}=10~M_{\oplus}$, $R_{\rm tot}=7\times 10^9$ cm, $R_{\rm core}=2.2~R_{\oplus}$), but consider impacts into short-period planets and calculate models with a much larger range of $v_{\rm imp}$. In order to take into account the host stars’ tidal potential, we adopt a prescription for the gravitation potential (see §3 in Paper I). In most LHD models presented here, we set the semi major axis of the planet to be at $a = 0.042$ AU and calculate $R_{\rm R}$ ($\sim 3 \times 10^{10}$ cm = 4.3 $R_{\rm J}$) with $M_\ast$ being that of HD149026 ([*i.e.*]{} 1.3 $M_\odot$). We take the stellar irradiation into account by assuming the surface temperature of the planet to be 1500 K. For comparison, we also simulate other models (LB3e, LA3a, LA3b, LC2a, LC2b) in which, we place the gas giant at $a=0.1$ and 5 AU respectively. The planet’s surface temperature is changed accordingly. LHD models of GIMs on to hot Jupiters ------------------------------------- In this series of simulations, we adopt the same model parameters which we used to simulate head-on collisions with the SPH method. The results of 9 sets of simulations are summarized in Table \[tab:LHDGIM\]. They are three sets of models in which the impactor is a Saturn-mass gas giant, a 25 $M_\oplus$ and a 10 $M_\oplus$ embryo. All of the initial models are assumed to be in hydrostatic equilibrium prior to the impact. At the onset of the simulation, in order to take into account the impact energy released in the vicinity of the core, we impose a burst of thermal energy, $E_{\rm a} = M_{\rm I} v_{\rm imp}^2/2$, so that the total energy becomes $W^\prime = E_{\rm a} + E_{\rm g}/2$. We also adopt the assumption that cores and embryos are completely merged during head-on impacts. This assumption is supported by the results of SPH models SA1a-SA3a, SB1a-SB4a, and SC1a-SC4a. For an impactor with $M_{\rm I}=25~M_{\oplus}$, a parabolic collision (model LA1) leads to a merger without any loss of gaseous envelope. The gas giant’s photospheric radius expanded slightly. For a higher-speed collision (model LA2), the envelope expands by a larger factor. Although it is now comparable to some of the inflated transiting planets, this gas giant will contract as it loses its internal energy on a Kelvin Helmholtz time scale. For a hyperbolic encounter (model LA3), the total energy $E_{\rm a}$ deposited into the gas giant is comparable to its total gravitational potential energy $E_{\rm g}$ prior to the collision. With a positive total energy after the impact ($W^\prime >0$), the gas giant’s envelope expands and disintegrates rapidly. Analogues dynamical parameters are imposed for a 10 $M_\oplus$ impactor. In these models, the final $R_{\rm f}$ is slightly smaller for the same impact velocity compared with those models impacted by a larger impactor (25 $M_{\oplus}$). Still, we find total loss of the gaseous envelope in the model (LB3) with high velocity impact. For the gas giant merger models, LHD simulations yield greater loss of the gaseous envelope. In the low-velocity parabolic model LC1, the merger produces a much larger photospheric radius than the gas giants’ initial size. Nevertheless, the entire envelope is retained due to a comparable increase in the merger’s thermal and gravitational binding energy. The new Roche lobe of the merger is also enlarged so that $R_{\rm f}$ remains well inside $R_{\rm R}$. However, with a slightly larger impact speed (model LC2), the thermal energy dissipated is adequate to enlarge $R_{\rm f}$ beyond the new $R_{\rm R}$. Outflow across $R_{\rm R}$ leads to runaway mass loss. The results in this subsection indicate a bimodal distribution of GIM outcomes. With high impact speeds, gas giants lose their envelope entirely. Shortly after modest-speeds, gas giants retain their enlarged envelopes. It is possible that subsequent tidal evolution can lead to additional heating and modest mass loss [@Gu:2003ez; @Gu:2004km]. This process can account for the smaller observed masses for hot Jupiters versus long-period gas giants. But it cannot produce intermediate-mass (with $20 M_\oplus < M_{\rm p} < 100 M_\oplus$) hot Jupiters in a prolific manner. Thus, the bimodal mass distribution of planets anticipated by planetary synthesis simulations [@Ida:2008dq; @Ida:2013fv] is likely to be preserved despite the high probability of GIMs in the stellar proximity (see §\[sec:justify\]). Model $M_{\rm I}$ $M_{\rm I,c}$ $v_{\rm imp}/v_{\rm esc,2}$ $M_{\rm f,c}$ $M_{\rm f,g}$ $M_{\rm f,g}^{\rm S}$ $R_{\rm f}/R_{\rm i}$ $R_{\rm f}/R_{\rm S}$ ------- ------------- --------------- ----------------------------- --------------- --------------- ----------------------- ----------------------- ----------------------- LA1 25 25 1.0 35 90 90 1.08 1.25 LA2 25 25 1.4 35 90 88 1.22 1.42 LA3 25 25 3.0 35 0 53 - - LB1 10 10 1.0 20 90 90 1.10 1.28 LB2 10 10 1.4 20 90 90 1.12 1.30 LB3 10 10 3.0 20 0 81 - - LC1 100 10 1.0 20 180 180 1.94 2.25 LC2 100 10 1.4 20 0 180 - - LC3 100 10 3.0 20 0 92 - - : LHD models of collisions of a 100 $M_\oplus$ hot Jupiter with a 25 $M_\oplus$ embryo, a 10 $M_\oplus$ embryo, and another identical gas giant\[tab:LHDGIM\]. Each gas giant has a 10 $M_\oplus$ core. In all of these models, the semi major axis of the hot Jupiter is set to be 0.04 AU. All the symbols are identical to previous tables. For each model, we list the asymptotic mass $M_{f,g}^S$ of remaining gas envelope of the merger product obtained with the SPH simulations (see Tables \[tab:sph25\], \[tab:sph10\], and \[tab:saturn-100\]). Comparisons between the SPH and LHD models {#sec:comparison} ------------------------------------------ Similar to the results obtained from LHD calculations, the SPH models in §\[sec:sph\] clearly demonstrate a sharp transition between GIMs without much mass loss and total disruption. Modest events hardly modify gas giant’s radius whereas highly energetic GIMs can lead to their total dispersal. However, for some models, LHD calculations generate more significant mass loss after the impact compare with SPH results. This may be caused by the artefact of a density lower limit ($\rho_l = 0.5$ g cm$^{-3}$) which we have adopted for the SPH scheme (see Paper I). While the LHD scheme uses a freely expanding outer boundary condition. In this subsection, we carry out a series of models to demonstrate how the mass loss rate may depend sensitively on the outer boundary condition (see Table \[tab:LHDE3\]). We choose the parameters for an intermediate model SB3a to highlight the difference between SPH and LHD simulations in marginal cases. In the SPH simulations, nearly 90% of the envelope’s original mass was retained after a Saturnian gas giant was impacted by a 10 $M_\oplus$ embryo during a head-on high-speed collision. ------- ------------------------------ ------- -------------------- --------------- --------------- ----------------------- ----------------------- Model $v_{\rm imp}/v_{\rm esc, 2}$ $a$ $\rho_\infty$ $M_{\rm f,c}$ $M_{\rm f,g}$ $R_{\rm f}/R_{\rm i}$ $R_{\rm f}/R_{\rm S}$ (AU) g cm $^{-3}$ ($M_\oplus$) ($M_\oplus$) SB3a 3 — 0.5 20 80.5 1.06 1.05 LB3 3 0.042 free 20 0 - - LB3c 3 0.042 $2 \times 10^{-5}$ 20 0 - - LB3d 3 0.042 $5 \times 10^{-3}$ 20 0 - - LB3e 3 0.042 $5 \times 10^{-2}$ 20 90 2.80 3.25 LB3a 3 0.1 free 20 90 2.38 2.61 ------- ------------------------------ ------- -------------------- --------------- --------------- ----------------------- ----------------------- : Comparison of LHD models with a SPH model of collisions between a 100 $M_\oplus$ gas giant planet and a 10 $M_\oplus$ embryo\[tab:LHDE3\]. All models in this table are computed for $v_{\rm imp} = 3 v_{\rm esc, 2}$. In the SPH model, we consider the gas giant in isolation. In LHD models, tidal effect of a 1.3 $M_\ast$ host star is taken into account. For the LHD models, various outer boundary conditions are imposed to demonstrate the numerical artefacts introduced by the lower limit on the external density assumed in the SPH scheme. For the LHD models, the final mass of envelope is computed to be that inside the Roche radius $R_{\rm R}$ and the quantity $R_{\rm f}$ refers to the photospheric radius when the post-impact expansion of the planetary envelope is stalled. LHD simulations suggest there may be substantial mass loss associated with this set of parameters. In model LB3 (and most models), the pressure at outer boundary of the planet has a smooth transition from the planetary surface to the disk. The evolution of the density distribution $\rho$ (at four epochs) after the impact is plotted on the left panel of Figure \[fig:LB3den\]. Due to the dissipation of impactor’s kinetic energy, thermal energy is released near the core. A jump in the pressure drives a rapid initial expansion of the envelope (at around $5 \times 10^5$ s). Due to the $P{\rm d}V$ work, the expansion speed gradually decreases below the sound speed. Nevertheless, gas in the envelope is able to reach $R_{\rm R}$ (where the gravitational potential has a local maximum) with a modest density ($\rho_{\rm R} \sim 10^{-4}$ g cm$^{-3}$) at $\sim 10^6$ s after the impact. Gas outside $R_R$ is accelerated outward by the tidal force of the central star with a flux ${\dot M}_{\rm out} \sim 4 \pi \rho_{\rm R} R_{\rm R}^2 v_{\rm g} > 10^{23}$ g s$^{-1}$. Gas depletion just beyond $R_{\rm R}$ reduces the local pressure and a negative pressure gradient drives an outward gas flow to replenish that region. Since $R_{\rm R}$ is proportional to $M_{\rm p} (R_{\rm R})$, mass loss also decreases the size of $R_{\rm R}$. A continuous flow across $R_{\rm R}$ leads to the eventual loss of the entire envelope. There are some differences in the results in models LB3 and SB3a. Although the same model parameters are used, these two models are computed with LHD and SPH methods respectively. In order to account for these differences, we modify the outer boundary condition such that the planet is assumed to be surrounded by an external medium with a density $\rho_\infty$ and a temperature $T =1,500$ K. We impose a passive external density $\rho_\infty$ which is set to be $2 \times 10^{-5}$, $5 \times 10^{-3}$, $5 \times 10^{-2}$ g cm$^{-3}$ for models LB3c, d, and e respectively (This external material does not accumulate beyond the planet as it expands). The results in Table \[tab:LHDE3\] indicate that, with sufficiently high $\rho_\infty$, expansion of the envelope can be stalled before it reaches $R_R$. On the right panel of Figure \[fig:LB3den\], we plot the initial and stalled density distribution for models LB3d and LB3e. In both models LB3d and LB3e, the external density (or pressure) has stalled the expansion of the envelope. The main difference is that in model LB3d, the envelope continues to expand beyond $R_{\rm R}$ whereas in model LB3e, it is stalled and falling back inside $R_{\rm R}$. Due to the outward-directed tidal force, gas outside $R_{\rm R}$ does not return to the gas giant. In contrast, the build-up of a dense shell just inside $R_{\rm R}$ (in model LB3e) will settle back to the core region. However, the time scale for the dense shell to resettle onto a point mass potential may be larger than the local dynamical time scale, because the magnitude of gravity near $R_{\rm R}$ is reduced by the host star’s tidal force. The approximate agreements suggest that our adopted value of $\rho _{\rm l}$ in the SPH simulations can suppress the effect of envelope’s mass loss, especially for marginally disruptive models. In all SPH models, $a$ and $R_{\rm R}$ are set to be infinite. It is therefore not unfavorable for any gas giant to lose a substantial fraction of its envelope unless GIM’s occur with a kinetic energy which at least exceeds its initial gravitational binding energy. Some of the hyper-velocity models do have adequate kinetic energy to disintegrate the gas giant planets. Despite the detachment and high velocity of disposal of a part (but not all) of gas in the original envelopes, a residual amount of gas falls back to re-establish a much reduced envelope (see models SC4a-e). The LHD calculations take into account the stellar tide. When the gaseous envelope of the planet expands beyond its $R_{\rm R}$ after the GIM, the gas outside $R_{\rm R}$ would not return back to the planetary surface due to the stellar gravity. Therefore, the change of planets’ $a$ (and accordingly $R_{\rm R}$) should have influences on the retention efficiency of the planetary atmosphere after the impact. We calculate marginal models with different $a$’s and $R_{\rm R}$’s and present comparisons in the subsection below. GIM’s on to intermediate and long-period gas giants {#sec:weakentide} --------------------------------------------------- We have shown in our LHD models that in the stellar proximity where hot Jupiters’ $R_{\rm R}$ is only a few times Jovian radii, energetic GIM’s lead to catastrophic loss of their gaseous envelopes. During some GIM’s (such as those in models LA3 and LB3), although the initial expansion of a gas giant’s gaseous envelope’s is stalled with its $R_{\rm f}$ interior to $R_{\rm R}$, there is sufficient density at $R_{\rm R}$ to enable a high outflow flux. In these marginal cases, it is natural to explore whether a fraction of the initial envelope may be retained in the limit of larger $R_{\rm R}$’s. Take model LB3 for example, the total kinetic energy carried by the impactor is comparable to half of the total gravitational binding energy of the pre-impact gas giant. In the limit of negligible stellar tide, we anticipate that the planet’s envelope may double its half mass radius and increases the photospheric radius to even larger extent. The magnitude of $R_{\rm R}$ increases with $a$. Since gas giants probably formed outside the snow line at a few AU’s from their host stars [@Ida:2004ko], their $R_{\rm R}$ is two orders of magnitude larger than that of the hot Jupiters. In order to explore dependence of the envelope retention on the tidal perturbation of the host star, we consider variations of models LA3 and LB3 in which $a$ is chosen to be 0.1 AU and 5 AU. The surface temperature of these models are set to be 1000 K and 150 K accordingly. We list the models parameters and results in Table \[tab:LHDCOMP\]. In the choice of these model parameters, the target Saturn-mass gas giant had $v_{\rm esc} = 25$ km s$^{-1}$ while the Keplerian speeds at 0.1 and 5 AU around HD149026 are $>100$ km s$^{-1}$ and $\sim 15$ km s$^{-1}$ respectively. Although it is possible for GIMs to occur with $v_{\rm imp} = 3 v_{\rm esc, 2}$ at 0.1 AU (models LA3a and LB3a), it is highly unlikely for $v_{\rm imp}$ to be much larger than $v_{\rm esc, 2}$ at 5 AU (in models LA3b and LB3b). Therefore, the simulation of highly energetic GIM’s onto long-period exoplanets should be considered theoretical toy models albeit they can also be used to represent GIM’s by more massive super-Earths with somewhat smaller $v_{\rm imp}'$s. The total energy deposited during a hyper-velocity impact in the hot-Jupiter model LB3 is around half of the gas giant’s initial gravitational binding energy or comparable to its total energy. In principle, gas giant’s envelope should be more than double after it is fully virialized, which is consistent with our FLASH simulations. However, $R_{\rm R}$ for this hot Jupiter is only five times $R_{\rm p}$ prior to the impact. The initial expansion overshot causes sufficiently dense ($\rho > 10^{-5}$g cm$^{-3}$) outer region of the envelope to extend outside the gas giant’s Roche lobe such that it leads to rapid mass loss. In model LB3a, however, both $a$ and $R_{\rm R}$ are set to be more than twice as large as their values in model LA3. Consequently, the envelope expansion is stalled well inside $R_{\rm R}$ with very little mass loss. At later stages, the envelope contracts slightly as a quasi hydrostatic equilibrium is re-established while the radius of the photosphere is doubled from its pre-impact value. For comparison, we overplotted the stalled density profile of this model on the right panel of Figure \[fig:LB3den\]. In order to demonstrate the locations of planetary photosphere and Roche radius after the GIM, we marked them with filled and open circles respectively. We also consider a model located at larger value of $a$ (= 5 AU). In this case, the gaseous envelope is still well preserved. For the models LAa and LCa, the Roche lobe is still sufficiently compact, that there is considerable outflow and depletion of the gas envelope. In long-period models (LAb and LCb), the $R_{\rm R}$’s are so large that the expanding envelopes can hardly reach them. Even though, the gaseous envelopes in these models are dispersed as their expanding velocities have already exceeded the escape velocity. The total thermal energy input $E_{\rm a}$ (or kinetic energy dissipation) at the onset of these two models is comparable to the initial gravitational binding energy of the gas giant $E_{\rm g}$. Consequently, the net total energy ($W^\prime$) is slightly positive. During the initial expansion, this newly added internal energy is converted into a self similar outflow everywhere analogous to the Sedov-Taylor solution for supernova explosions. In the absence of energy losses, the magnitude of the gravitational binding energy would reduce with that of the internal energy while there would be adequate kinetic energy to disintegrate the entire envelope with an outflow velocity $\sim (2 W^\prime/ M_{\rm p}) ^{1/2}$. In the limit that sufficient thermal energy is deposited into these long-period gas giants, it is possible for a fraction of the envelope be dispersed while the rest is retained around the merged core. During the expansion of the gaseous envelope after an energetic impact, a fraction of the envelope remains around the core, albeit the asymptotic photosphere $R_{\rm f}$ is more than doubled. The radius of the photosphere $R_{\rm f}$ cannot expand indefinitely because the column density along the radial direction decreases with the expansion of the gas giant. For self similar expansions, the total opacity of the gaseous envelope is $\tau_{\rm tot} \propto R_{\rm exp}^{-2}$ where $R_{\rm exp}$ is an expansion factor. For sufficiently large $R_{\rm exp}$, the envelope becomes optically thin and the total energy can no longer be conserved. Rapid radiative loss can reduce $W^\prime$ below zero. This transition is equivalent to that from Sedov to snowplough phase of supernova shells. The envelope would be totally disrupted if $W^\prime$ is sufficiently large that at the transition phase, the outflow has already acquired a velocity in excess of the escape speed. Otherwise, a fraction of the envelope may be retained in the form of “fall back” gas. Our models indicate that partial retention of a gas giant’s envelope after a giant embryo impact requires a narrow range of energy deposition. This energy is generally unattainable from embryos and super-Earths near the sites of gas giant formation. Thus the mass of long-period gas giants is generally retained while their heavy elemental content is being enriched by GIM’s. ------- -------------- --------------- ------ ----------------------------- --------------- --------------- ----------------------- ----------------------- Model $M_{\rm I}$ $M_{\rm I,c}$ $a$ $v_{\rm imp}/v_{\rm esc,2}$ $M_{\rm f,c}$ $M_{\rm f,g}$ $R_{\rm f}/R_{\rm i}$ $R_{\rm f}/R_{\rm S}$ ($M_\oplus$) ($M_\oplus$) (AU) ($M_\oplus$) ($M_\oplus$) LA3 25 25 0.04 3.0 35 0 - - LA3a 25 25 0.1 3.0 35 0 - - LA3b 25 25 5.0 3.0 35 0 - - LB3 10 10 0.04 3.0 20 0 - - LB3a 10 10 0.1 3.0 20 90 2.38 2.61 LB3b 10 10 5.0 3.0 20 90 1.68 1.73 LC2 100 10 0.04 1.4 20 0 - - LC2a 100 10 0.1 1.4 20 0 - - LC2b 100 10 5.0 1.4 20 0 - - ------- -------------- --------------- ------ ----------------------------- --------------- --------------- ----------------------- ----------------------- De-synchronization of the envelope and further inflation -------------------------------------------------------- If a state of spin-orbit synchronization has already been accomplished by tidal evolution prior to the giant impact, both off-center collisions and the expansion of the envelope would introduce asynchronous spin and perhaps modest orbital eccentricity in general. As an example, we simulate in model SC1b , the merger event of two identical planets during which their cores also coalesce. Although the envelope experienced a more significant expansion, most of the envelope remains intact. In this case, the final spin rate would differ substantially from its initial values. This departure from synchronous rotation may have observable consequences for atmospheric flows [@Rauscher:2014pf] and thermal evolution. Subsequent tidal dissipation would lead to an additional source of heating for the planet [@Dobbs-Dixon:2004fu] such that $${\dot E}_{\rm t} = { G M_\ast M_{\rm p} e_{\rm p} ^2 \over a ( 1 - e_{\rm p} ^2) \tau_e} + {\alpha_{\rm p} M_{\rm p} R_{\rm p}^2 (n_{\rm p} - \Omega_{\rm p}) ^2 \over \tau_\Omega} \,,$$ where $\tau_\Omega= (7 \alpha_{\rm p} / 2) (R_{\rm p}/a)^2 \tau_e$ is the synchronization time scale and $\tau_e$ is the eccentricity damping time scale. The rate of energy dissipation depends not only on the differential angular frequency but also on the radius and internal core-envelope structure of the planet through its $Q_{\rm p} ^\prime$ value [@Ogilvie:2004rw]. Although the amount of extractable energy in the planet’s spin is limited, for a brief duration $\tau_\Omega$, the rate of tidal dissipation due to synchronization is comparable to that due to circularization. Under some circumstances, the planet’s envelope would further expand and perhaps overflow its Roche lobe, if the tidal dissipation rate exceeds the energy loss rate at the planet’s surface [@Gu:2003ez; @Gu:2004km]. We shall present an analysis of this possibility elsewhere. Consequence of giant impacts on HD149026 ---------------------------------------- Based on the results of the previous subsections, we propose the progenitor of HD149026b formed with a Saturn-like internal structure. When it migrated to the proximity of its host star, it may have merged with another short-period gas giant or a population of close-in super earths. If these collisions occurred with relatively low impact speeds, phase transition of the core material would be avoided. Nevertheless, heavy elements in the core would mix with an inflated envelope. Subsequent tidal evolution provides additional heating which may have led to the partial loss of its gaseous envelope through Roche-lobe overflow. We note that HD149026b is less massive than most known hot Jupiters and could have lost some mass. The metal-rich debris material would stream into the host star and significantly increase the metallicity of its shallow outer envelope [@Li:2008vn]. Angular momentum transfer between the planet and the debris stream would also enlarge the planet’s orbital semi major axis and Roche lobe so that the envelope loss would be limited. On a time scale comparable to the age of its host star, this planet would contract to its present-day radius. We anticipate common occurrence of similar type of merger events. Since it is difficult to assess the optimum initial conditions for this model, we now present a series of simulations with a HD149026b-like planet as our initial model. The total mass of the initial model is 110-$M_{\oplus}$, and the core mass is around 73-$M_{\oplus}$. According to the observations of HD149026b [@Sato:2005uq], the initial model has $R_{\rm p}=0.73 R_{\rm J}$, $a=0.042$ AU, and $T_{\rm e}=1500$ K accordingly. We calculate models struck by embryos with different masses and speeds. The model parameters and results are listed in Table \[tab:LHDBIGCORE\]. For the models struck by a 10-$M_\oplus$ or a 25-$M_\oplus$ embryo with low speed ($v_{\rm imp}/v_{\rm esc, 2}=1.0$), there is still inflation of its gaseous envelope. For larger impact velocity $v_{\rm imp}/v_{\rm esc, 2}=1.4$, the gaseous envelope of the model impacted by 25-$M_\oplus$ embryo has already expanded beyond its $R_{\rm R}$. When comparing the two series of models (Saturn-like and HD149026b-like models), we find that Saturn-like models are more resilient in retaining the gas envelope during the impacts. Even though some of the Saturn-like models (LA2) are hit by impactor with masses several times greater than that in the catastrophic HD149026b-like model LB2S, there is no significant mass loss in these models. This dichotomy may be explained by the following reasons. In the calculations, we assume that all the impactors can reach the core, with ablated mass and gravitational energy deposited in a region around the core. Thus, the deposited energy by the impactor is not only function of the mass, but also related its path and the internal structure of the target. For the same amount of material deposited at the same location, the deposited energy would be larger in the HD149026b-like models than that in the Saturn-like models, because the former have much larger solid core than the latter. And a side effect of the massive impactors is that the Hill radius will increase with the increasing mass of impactor, which will make it more difficult for the envelope to expand out of its Hill radius. Furthermore, the Saturn-like models have very massive gaseous envelope compared with the HD149026b-like models. They obviously require much larger energy deposition to overcome the gravitational binding for the envelope to escape out of the Hill radius. ------- -------------- --------------- ------------------------------ --------------- --------------- ----------------------- ----------------------- ----------------------- Model $M_{\rm I}$ $M_{\rm I,c}$ $v_{\rm imp}/v_{\rm esc, 2}$ $M_{\rm f,c}$ $M_{\rm f,g}$ $M_{\rm f,g}^{\rm N}$ $R_{\rm f}/R_{\rm i}$ $R_{\rm f}/R_{\rm S}$ ($M_\oplus$) ($M_\oplus$) ($M_\oplus$) ($M_\oplus$) ($M_\oplus$) LA1S 25 25 1.0 98 37 90 2.00 1.76 LA2S 25 25 1.4 98 0 90 - - LA3S 25 25 3.0 98 0 0 - - LB1S 10 10 1.0 83 37 90 1.13 1.00 LB2S 10 10 1.4 83 37 90 1.47 1.29 LB3S 10 10 3.0 83 0 0 - - ------- -------------- --------------- ------------------------------ --------------- --------------- ----------------------- ----------------------- ----------------------- Summary and Discussions {#sec:summary} ======================= A brief summary --------------- During the formation and dynamic evolution of the planetary system, there are many avenues of GIMs by embryos onto hot Jupiters. The supply of these large solid building blocks includes: 1) planetesimals along the paths of migrating gas giants, 2) super-Earths stalled at barriers outside the asymptotic destiny of hot Jupiters, 3) terrestrial bodies cleared by sweeping secular resonances and dynamical instabilities, and 4) short-period super Earths subjected to tidal orbital evolution. There are also several processes which can lead to the merger of gas giants: 1) orbit crossing triggered by run-away migration, 2) congregation of hot Jupiters near their host stars, and 3) long-term dynamical instability in the stellar proximity. Based on these consideration and the observation that a large fraction of stars bear super-Earths and a large fraction of known gas giants are members of multiple-planet systems, we infer common occurrence of close encounters among these relatively massive gaseous and solid planets. These events can lead to the observed Rayleigh distribution of gas giants’ orbital eccentricity [@Zhou:2007fk; @Juric:2008uq; @Chatterjee:2008gd]. Potential outcomes of GIM events -------------------------------- Here, we suggest that some of these close encounters results in GIMs. The energy released during such collisions may lead to expansion of the planets’ envelope. If this GIM scenario is applicable for the origin of the exceptional inflated close-in planets, their fraction (relative to the normal-size planets) would correspond to the duty cycle of a planet’s expanded state following a major impacting event. On the Kelvin-Holmheltz time scale ($\sim 10^8$ yr), the inflated planets return to their initial state of thermal equilibrium. Under the assumption that the residual planetesimals and embryos are sufficiently depleted that their characteristic collision time scale with close-in gas giants is comparable to the age of their host stars, the population of inflated planets is expected to be an order of magnitude less than that of the normal size gas giants. This inference is consistent with the relatively small fraction of inflated to normal close-in gas giants. In the proximity of their host stars, a substantial amount of the envelope gas may be lost, after each giant impact, through Roche-lobe overflow [@Gu:2004km], leading to a large metallicity enhancement of the planets. The intermediate-mass merger products may account for a population of planets found in the “planetary desert” which was predicted by the population synthesis models [@Ida:2004ko; @Howard:2010bs; @Ida:2013fv]. The metal-rich planetary debris may also be accreted by their host stars, leading to modest enrichment of the stellar outermost envelope [@Li:2008vn]. These events of planetary coalescence and the pollution of their host stars are unlikely to occur if gas giants are formed through gravitational instability. The verification of this GIM scenario may provide a distinguishing test for these competing scenarios of planet formation. For planet with longer orbital periods, tidal evolution toward spin synchronization is weak, angular momentum deposited by impacts can be reserved. In the future, oblateness measurements may be used to constrain impact history of gas giant planets. HD149026b --------- In this paper, we considered the possibility that the large core of hot Jupiter, HD149026b formed through giant impacts onto a gas giant planet with either residual proto-planetary embryos or merger with other gas giant planets in the proximity of its host star. In order to examine the outcome of these collisions, we carried out a series of numerical simulations with a SPH algorithm. These calculations demonstrate gentle collisions always lead to coalescence whereas nearly parabolic direct collisions can result in a preferential loss of gaseous envelope material while the heavy elements in the core are mostly retained. But, most of the core material may not merge or be retained as a consequence of oblique high-velocity collisions. We assume that a substantial fraction of the original residual planetesimals in the neighborhood of HD149026b may have been scattered into its host star which is an F star with a very shallow convective layer. We carried out evolution calculations to take into account of the effect of stellar pollution. We suggest that in this and other F star with planets, the extent of stellar pollution by residual planetesimals may be tested by an accurate determination of their mass, luminosity, and effective temperature. Such a determination will provide a firm support on the once existence of terrestrial-planet-building material. Acknowledgments {#acknowledgments .unnumbered} =============== We thank Drs F. Adams, E. Asphaug, P. Bodenheimer, J. Guillochon, T. Guillot, Y. Hori, S. Ida, M. Kouwenhoven and G. Laughlin for useful conversations. We also thank the referee David Stevenson for constructive comments. The software used in the hydrodynamic simulations was in part developed by the DOE-supported ASCI/Alliance Center for Astrophysical Thermonuclear Flashes at the University of Chicago. Computations were performed on the Laohu computer cluster at NAOC and the Hyades clusters at UCSC. This work is supported by UC/Lab Fee grants. S.-F. L. was sponsored by NASA NNX13AR66G “Collisional Accretion of Similar Sized Bodies”. This work is also supported by the Astronomy Unit’s STFC Consolidated Grant. \[sec:ref\] [^1]: see, e.g., <http://exoplanets.org/>.
--- abstract: 'Extreme Scattering Events are sometimes manifest in the light-curves of compact radio-quasars at frequencies of a few GHz. These events are not understood. The model which appears to offer the best explanation requires a new population of AU-sized, neutral gas clouds; these clouds would then make up a large fraction of the Galaxy’s dark matter. Independent of the question of which theoretical model is correct, if we extrapolate the observed behaviour to low radio-frequencies, we expect that the sky should be criss-crossed by a network of narrow caustics, at frequencies below about 700 MHz. Consequently at these frequencies sources should typically manifest additional, faint images which are substantially delayed with respect to the primary image. Although some examples of this type of behaviour are already known, it is expected that these are just the tip of the iceberg, with strong selection biases having been imposed by the instrumentation employed to date.' author: - 'Mark A. Walker' date: 'July 17th, 2000' title: Interpretation of Extreme Scattering Events --- Introduction ============ Extreme Scattering Events (ESEs), Fiedler et al (1987: F87), were discovered more than a decade ago but are still not understood. It is generally agreed that these events are due to refraction by intervening, ionised, Galactic gas (e.g. F87; Romani, Blandford & Cordes 1987: RBC87), but there is no agreement on the astrophysical context in which this gas arises. Most models attempt to explain ESEs with the minimum possible extrapolation from conventional astrophysical pictures (see, especially, Deshpande & Radhakrishnan 2000); this is the most conservative approach, and it seems very likely that at least one of these conventional models will prove relevant. There are, however, real difficulties in trying to explain some of the observed events – in particular the ESE in Q0954+658 – with conventional astrophysics, and this has motivated one rather exotic model in which a new population of dense, neutral gas clouds is invoked (Walker & Wardle 1998). The current lack of consensus on the correct physical picture for ESEs persists principally because the existing data have a fairly low information content. At this point the field is badly in need of some new observational initiatives; some ideas are presented in §6 (see also Walker 2000). Basic constraints on lenses =========================== The refracting structures which give rise to ESEs are conveniently referred to as “lenses”, although we should bear in mind that they might not be well-defined physical entities (model (iii) of §5). There are three basic properties of the individual lenses which are dictated fairly directly by the data on ESEs: (i) their transverse dimensions should be a few AU, (ii) the peak electron column-density should be of order $10^{17}\;{\rm cm^{-2}}$, and (iii) they should be symmetric. Point (i) follows immediately from the observed event durations (months) together with an assumed transverse speed of order $10^2\;{\rm km\,s^{-1}}$. Point (ii) is deduced by requiring a strong lens which can magnify a large fraction of a source which is of order a milli-arcsecond in size. These points were recognised at the time of discovery of the ESEs (F87). One detail deserves clarification however: an upper limit on the distance of the lenses follows from their transverse dimensions in combination with the requirement that they be larger in angular size than the source. This reasoning is correct, but the angular size of the source has previously been taken as the scatter-broadened size, leading to a distance upper limit of order one kpc, and this is overly restrictive. For distances of a kpc or more, at high Galactic latitude, the lens is beyond the majority of the scattering material in the Galactic disk, and the relevant angular size is then the intrinsic source size; this can be substantially smaller than the scatter-broadened size, thereby relaxing the distance limit. The third point has not previously been emphasised; it arises simply because the ESE light-curves are, crudely speaking, time-symmetric. At first sight this statement appears to have little value, because of the qualifying phrase “crudely speaking”, but this is not the case — most of the models which have been proposed for ESEs incorporate no lens symmetry whatsoever, and are therefore not good starting points for explaining even an approximate time-symmetry. That’s not to say that such models are excluded, because it might be possible to construct versions in which the lenses do yield such behaviour, but the point remains that this property must be explained somehow. Some symmetry might be effected by the process of averaging over the source structure, but this is true only for the angular/temporal scales corresponding to the source size, below which flux variations are suppressed. One might argue that models which involve symmetric lenses should, in turn, explain why the observed time-symmetry is only approximate. This, of course, is trivial, because real astrophysical entities never conform exactly to the symmetries which are employed in modelling them. Finally, if the approximate time-symmetry of the ESE light-curves is not accidental, it requires that any straight line drawn across the lens plane (representing the apparent path of the background source) should manifest a reflection symmetry about one point. In turn this indicates that the lens itself should have either mirror-symmetry and translational-invariance, or else it should be axisymmetric. Further constraints: the case of Q0954+658 ========================================== The ESE observed in Q0954+658 (F87) is by far the most spectacular event observed to date and deserves particular attention. This event exhibits a number of sharp peaks in the high-frequency (8.1 GHz) light-curve; these peaks are generally interpreted as being due to caustics. While there are four large peaks evident, there are roughly seven smaller, sharp peaks in this same light-curve, making eleven in total. Now caustic curves are closed curves, so that during a lensing event a source which crosses from the exterior to the interior of this boundary must later cross to the exterior again, giving rise to two peaks in the light curve. Furthermore, for a diverging lens the caustic curves come in pairs — one pair for every peak in electron column-density (provided the peak is sufficiently sharp). Thus, even if the source structure is a single-component only, the 8.1 GHz light-curve could be reproduced with as few as three column-density peaks, implying that the column density profile of the lens is likely to be very simple. Nature of the lens symmetry =========================== It is straightforward to decide which of the two possible lens symmetries (§2) is preferred; it is the axisymmetric lens. This can be seen immediately from figure 1, which shows examples of the low-frequency light-curves arising from axisymmetric/mirror-symmetric lenses for which, in both cases, the source passes behind two peaks in electron column-density. (The mirror-symmetric lens consists of two parallel filaments, with Gaussian cross-sections, while the axisymmetric lens is a simple ring, again with a Gaussian cross-section.) A single Gaussian component is used for the source structure in these calculations. Caustic crossings are seen as the peaks in the light-curves; only seven are visible because the central peak contains an unresolved pair in each case. Both light-curves exhibit the same deep flux depression, when the source is nearly on-axis. Conservation of energy demands that this power appears somewhere else in the observer’s plane, and it is in this respect that the two lenses differ greatly: for the translationally-invariant lens the light-curve actually manifests this flux conservation, in the sense that the flux averaged over the whole event is equal to the unlensed flux, whereas this is not true for the axisymmetric lens. The data for Q0954+658 clearly favour the axisymmetric model. [**Figure 1.**]{} Low frequency light-curves for ESEs produced by mirror-symmetric (dashed) and axisymmetric (solid) lenses. Overview of models ================== A number of models have been proposed to explain ESEs, in which the lenses are identified with a variety of physical phenomena. It should be borne in mind that the defining criteria for ESEs (F87, Fiedler et al 1994: F94) have been very loosely framed (“periods of unusual variability”), and this may have created heterogeneity in the class — more than one phenomenon could be represented amongst the events which have been dubbed ESEs. More than one lens model may therefore be relevant. A brief summary of lens models follows:\ (i) Random refracting elements (F87, F94)\ (ii) Magnetically confined filaments (RBC87)\ (iii) Steep spectrum turbulence (Deshpande & Radhakrishnan 2000)\ (iv) Shock waves (RBC87; Clegg, Chernoff & Cordes 1988)\ (v) Photo-ionised surfaces of giant clouds (Rickett, Lyne & Gupta 1997)\ (vi) Photo-ionised winds from AU-sized clouds (Walker & Wardle 1998).\ Of these models, (i) is at present a purely phenomenological model whose physical viability cannot be readily assessed. Models (ii) and (iii) possess no particular symmetry, and are therefore disfavoured, while models (iv) and (v) involve strongly asymmetric lenses and are strongly disfavoured in this respect; only model (vi) generates the observed quasi-symmetry in a natural way. The necessary electron column densities and scale-sizes may in principle be realised by any of models (ii–vi), but models (ii), (iv) and (v) need to be developed further before meaningful assessments can be made. Model (vi) appeared, initially, to yield the necessary column/scale-size combination in a very natural way, but McKee (2000) has since pointed out that a photo-evaporated wind would, in this context, have a modest ionisation fraction, so the calculation of the ionised column-density needs to be revisited for this model. Notwithstanding a huge reduction in the predicted column of ionised gas, model (vi) currently appears to offer the best explanation for some of the ESEs – notably Q0954+658 – and the main issue is whether or not a population of dense, neutral clouds actually exists. Indeed this is a question with ramifications throughout astrophysics, because the neutral clouds would have to constitute a major component of the Galactic dark matter. The putative clouds cannot be excluded on the basis of any existing data (Walker & Wardle 1999), and this model provides a strong motivation for intensive study of the ESE phenomenon. Future work =========== How can we make progress in this field? A key aspect of the problem is the fact that ESEs are rare. This difficulty can most easily be addressed by working at low frequencies, where the refraction angles are larger and the cross-section for multiple imaging is increased. Indeed, for a lens which is localised in both transverse dimensions, the optical depth for multiple-imaging should scale as $\lambda^4$, independent of the actual lens model. Taking the optical depth for Extreme Scattering (extragalactic sources) to be of order $5\times10^{-3}$ at 2.7 GHz (F94), it is straightforward to predict that at frequencies below about 700 MHz there will be multiple images present most of the time. At these frequencies, then, the sky should exhibit a network of caustics. This does [*not*]{} mean that large flux changes will be happening continuously below this frequency, because the caustics are very narrow and in total cover only a tiny fraction of the sky; rather it means that there should typically be some extra [*faint*]{} images present. This phenomenon is, in fact, well known from pulsar studies (Cordes & Wolszczan 1986; Rickett 1990), where it manifests itself as interference fringes in the dynamic spectra; it is also very common, occurring for more than 10% of the time for some pulsars (J.M. Cordes, 2000, personal communication). However, while the connection to ESEs has long been recognised (e.g. RBC87), the exact relationship between the two effects remains to be understood. Regrettably, the multiple imaging phenomenon has not yet been exploited in any systematic way to learn about the lenses. There are a number of possible avenues to improving the current situation by working with these multiple images. For example: one could gain some information on the structure of the lens simply by counting the number of images present; lens symmetry could be studied via VLBI observations through the course of a multiple imaging event; for long-duration events the evolution of the image delays could yield a “parallax” measurement; and magnetic fields in the lenses could be studied by comparing the fringe patterns in different polarisations. It is important to note that the faint, “extra” images can be substantially delayed with respect to the main image, and the magnitude of the delay is roughly proportional to the geometric area covered by the images, hence proportional to the optical depth. Now refraction through an angle of order a milli-arcsecond should, over a distance of order one kpc, introduce a geometric delay of order $10^{-6}$ sec. Thus for lenses of order a milli-arcsecond in size, a strong lensing event (ESE) should introduce image delays of this magnitude, while multiple imaging at frequencies below 700 MHz will introduce delays hundreds of times larger. Such images would be extremely difficult to detect with conventional techniques, because the interference fringes would be so fine that they could not be resolved with existing spectrometers. It is therefore to be expected that observations of mutiple imaging phenomena have, to date, been subject to strong instrumental biases which allow us to see only images with relatively small delays, with the typical secondary images being censored. This bias increases in severity very rapidly as the observing frequency is decreased, with the maximum delay scaling roughly as $\lambda^4$ in this regime. To be confident that we are not introducing a bias, the only way forward appears to be the use of base-band recording, from which the temporal auto-correlation of the electric field, for example, can be computed out to large lags. Clegg, A., Chernoff, D. and Cordes, J. 1988, [*AIP Conf. Proc.*]{} 174, 174 Cordes, J. and Wolszczan, A. 1986, [*ApJL*]{} 307, L27 Deshpande, A. and Radhakrishnan, V. 2000, [*In preparation*]{} Fiedler, R. et al 1987, [*Nature*]{} 326, 675 \[F87\] Fiedler, R. et al 1994, [*ApJ*]{} 430, 581 \[F94\] McKee, C. 2000 [*astro-ph/0008046*]{} Rickett, B. 1990, [*ARAA*]{} 28, 561 Rickett, B., Lyne, A. and Gupta, Y. 1997, [*MNRAS*]{} 287, 739 Romani, R., Blandford, R. and Cordes, J. 1987, [*Nature*]{} 328, 324 \[RBC87\] Walker, M. 2000, [*ASP Conf. Ser.*]{} 202, 561 Walker, M. and Wardle, M. 1998, [*ApJL*]{} 498, L125 Walker, M. and Wardle, M. 1999, [*Pub. Ast. Soc. Aus.*]{} 16(3), 262
--- abstract: 'We image local structural rearrangements in soft colloidal glasses under small periodic perturbations induced by thermal cycling. Local structural entropy $S_{2}$ positively correlates with observed rearrangements in colloidal glasses. The high $S_{2}$ values of the rearranging clusters in glasses indicate that fragile regions in glasses are structurally less correlated, similar to structural defects in crystalline solids. Slow-evolving high $S_{2}$ spots are capable of predicting local rearrangements long before the relaxations occur, while fluctuation-created high $S_{2}$ spots best correlate with local deformations right before the rearrangement events. Local free volumes are also found to correlate with particle rearrangements at extreme values, although the ability to identify relaxation sites is substantially lower than $S_{2}$. Our experiments provide an efficient structural identifier for the fragile regions in glasses, and highlight the important role of structural correlations in the physics of glasses.' author: - Xiunan Yang - Rui Liu - Mingcheng Yang - 'Wei-Hua Wang$^*$' - 'Ke Chen$^*$' title: Structures of local rearrangements in soft colloidal glasses --- Understanding the structure-property connections in glasses is one of the most challenging problems in condensed matter physics. The lack of a clear characterization of the structures in glasses has hindered the formulation of general theories for glass transition and deformation of glasses [@stru1; @stru2; @stru3; @stru4; @stru5; @stru6; @stru7]. The most elementary relaxation events in glasses are the local atomic rearrangements that play similar roles in the mechanical properties of glasses as the structural defects in crystals. Local regions prone to rearrangements in glasses are often known as flow units [@Wang1; @Wang2], shear transformation zones (STZs) [@Langer1; @Argon1], soft spots [@Harrowell1; @Chen1; @Manning1; @Xu1], or geometrically unfavoured motifs (GUMs) [@Ma1]. Unlike crystalline solids whose defects are easily identified from a periodic lattice, no distinguishing structures have been found for the fragile regions of glasses in an apparently disordered background. Without a clear structural indicator for fragile regions in glasses, many studies rely on phenomenological models such as soft glassy rheology (SGR) to understand macroscopic properties of amorphous materials [@Sollich1]. Early studies attempt to connect a particle’s propensity to rearrange to its immediate environment. Spaepen proposed a free-volume theory that favors sites with large free volumes for local rearrangements in hard-sphere glasses [@Spaepen1]. For soft sphere systems, which include metallic glasses, Egami and co-workers suggest that regions with extreme local stresses, which correspond to particles with either extremely large or extremely small free volumes, are potential “defects” sites [@Egami1; @Egami2]. Direct experimental examinations of these scenarios, however, are rare due to the difficulty of measuring local free volumes or atomic stresses in glassy materials [@Huang1]. More recently, machine learning methods are employed to search through multi-dimensional parameter space and use a combination of structural features to identify fragile regions in disordered solids  [@AndreaLiu1; @AndreaLiu2]. Phonon modes, which reflect the collective excitations in glasses, are shown to be a powerful identifier of fragile regions in glasses. Soft spots defined by the low-frequency quasi-localized soft modes are shown to overlap significantly with rearranging regions in glasses [@Chen1; @Manning1; @Harrowell1]. However, measurements of the local geometric structures of soft spots did not find any distinctive structural characteristic for these regions [@Manning1]. Recent theories and experiments suggest that in glasses, there may exist amorphously correlated structures formed during the glass transition [@Wolynes1; @Zamponi1; @Tanaka1; @Han1; @Biroli1; @Durian1], and the dynamics-structure correlation goes beyond single-particle measures [@Berthier1]. Therefore, the search for “defects” in glasses needs to probe correlations (or lack thereof) on length scales greater than the first-neighbor shell. Recent simulations by Tong *et al.* [@Xu1], find strong correlations between the distributions of soft modes in glasses and structural entropy $S_{2}$, raising the possibility that this correlation-based structural parameter may also be able to predict elementary deformations in glasses. In this letter, we employ video microscopy to directly image local deformations in quasi-2D colloidal glasses consisting of thermo-sensitive microgel particles, and measure the structures of local rearranging regions under cyclic thermal perturbations. Thermal cycling tunes the sizes of soft colloidal particles, and generates uniform compression or dilation when the particles swell or shrink. Similar to mechanical shearing [@Weitz1; @Keim1; @Candelier1; @Regev1; @Fiocco1], rearrangements during thermal cyclings are the results of local stress imbalances. But perturbations from thermal cycling impose no directional bias or boundary effects, and allow simultaneous activation of defects with different orientational preferences. We find that local free volumes are weakly correlated with particle rearrangements; particles with either extremely high or low free volumes are equally likely to rearrange. Local structural entropy $S_{2}$ derived from local pair correlation functions strongly correlates with observed local rearrangements. Rearranging clusters emerge and evolve with regions of high $S_{2}$ values, which are structurally less correlated or more disordered compared to the more stable background, similar to the topological defects in polycrystalline solids. The results presented here are for deeply jammed samples under small periodic perturbations. We have also observed qualitatively the same correlations in samples of different packing fractions and under different perturbation patterns [@suppl.]. ![ Spatial distribution of rearranging clusters during thermal cycling. Rearranging clusters from different thermal cycle numbers, shown by large spheres in different colors. Small dots are background particles that did not experience neighbor changes during the experiment. Large blue dotted circles indicate regions with repeated rearranging events. Small red circles show the only 4 overlapping rearranging particles during the experiment. []{data-label="fig1"}](fig1){width="9cm"} The samples are prepared by loading a binary mixture of poly-N-isopropylacrylamide (PNIPAM) particles between two coverslips. The colloidal suspension spreads under capillary forces, and forms a dense monolayer between the glass plates. The samples are then hermetically sealed using optical glue (Norland 63). A binary mixture is used to frustrate crystallization. The diameters of the particles are measured to be $1$ and $1.3~\mu m$ at $22^{\circ}$C by dynamical light scattering; and the number ratio between large and small particles is close to 1. PNIPAM colloidal particles are temperature sensitive whose diameters decrease when temperature increases [@Yunker1; @Still1; @Chen2; @Chen3]. The interactions between PNIPAM particles are generally characterized to be repulsive soft-sphere with a hard core [@Han2]. Rearrangements are induced by locally changing the sample temperature. The colloidal suspension is mixed with a small amount of non-fluorescent dye (Chromatch-Chromatint black 2232 liquid, $0.2\%$ by volume). When illuminated by a mercury lamp, the dye absorbs the incident light and heats up a small area much larger than the field of view, while the rest of the sample remains at ambient temperature [@Han3; @Yunker2]. When the mercury lamp is turned off, the heated region rapidly recovers to its original temperature. The temperature increase from optical heating is calibrated to be about $0.2 K$; and the sample reaches new thermal equilibrium in less than 1 second [@suppl.]. For this small temperature change, the samples remain in jammed glassy states at both temperatures, with a brief transient period in between. The samples are aged on the microscope stage for 200 min. before being periodically heated and cooled for a total of 10 cycles. Each cycle lasts for 30 min., with 15 min. each at the elevated and ambient temperature [@suppl.]. The samples are continuously imaged using standard bright-field microscopy at 20 fps for the duration of the experiment. There are $\sim3700$ particles within the image frame, with a packing fraction of 0.88 at the ambient temperature. The trajectory of each particle is extracted by particle tracking techniques [@Grier1]. We limit our analyses on particles that are at least 3 diameters away from the image boundaries, which leaves $\sim2990$ particles in a reduced field of view. Information of particles outside of this reduced field of view is utilized only for the calculation of parameters for particles within. Most particle rearrangements are observed within 60 seconds of a temperature switch. A video of one local rearranging event can be found in the supplementary materials [@suppl.] We extract plastically rearranged regions by comparing particle positions at the last minute of each cycle with the positions of the particles 1 minute before the same cycle; at both points the sample is at thermal equilibrium with the ambient temperature. Separation cut-offs defined by the first minima on the particle pair correlation functions between different species are used to determine neighbor changes during thermal cycling [@Harrowell1; @Yunker2]. Distributions of particle separations over time are also considered to avoid accidental misidentification [@suppl.]. Particles that changed neighbors during one cycle are grouped into clusters based on nearest-neighbor pairings, and clusters containing fewer than 5 particles are ignored [@Chen1; @Manning1]. Most neighbor changes are permanent; only about 2% of rearranging particles regain their lost neighbors in subsequent thermal cycles. Fig. \[fig1\] shows all rearranging clusters observed during 10 thermal cycles. Clusters from different cycles are shown in different colors. For a single cycle, the spatial distribution of rearranging clusters appears uncorrelated, with cluster separations much larger than typical cluster sizes. However, when the distributions of rearranging clusters from different thermal cycles are compared, it is clear that the clusters are concentrated in certain regions of the sample, as indicated by the large blue dotted circles in Fig. \[fig1\], instead of evenly scattered throughout the sample. Overlapping between neighboring clusters are rare, with only 4 overlapping particles for the duration of the experiment (small red circles in Fig. \[fig1\]). The rest of the rearranging particles did not rearrange again during our experiment. We examine the rearranging clusters in search for distinctive structural characteristics from the rest of the colloidal glasses. We first measure the particle free volumes ($V_{f}$). We employ radical Voronoi tessellation to define a polygon for each particle for our binary mixture [@volume1; @volume2; @volume3]. The radical Voronoi tessellation avoids cutting through large spheres by taking into account the relative particle sizes. The $V_{f}$ is determined by subtracting the cross-section area of a particle from the area of the polygon. The soft-sphere interaction between the colloidal particles allows negative $V_{f}$ for compressed particles. The average $V_{f}$ for particles in a rearranging cluster before and after local rearrangements is effectively the same as that of all the particles in the sample, with slightly larger widths, as plotted in Fig. \[fig2\]a. Fig. \[fig2\]b plots the measured rearranging probability as a function of initial $V_{f}$ for individual particles. Particles with extremely high or extremely low $V_{f}$ are those most likely to rearrange compared to particles with average $V_{f}$. The curve in Fig. \[fig2\]b is roughly symmetric to the system average, suggesting that loose and compact regions are equally prone to rearrangement. This symmetry may explain why previous studies that search for monotonic correlations between local free volumes and rearrangements did not find a significant overlap [@Manning1]. ![ Free volumes and local structural entropy in colloidal glasses. (a) Distribution of $V_{f}$ for all particles (black squares), initial $V_{f}$ for rearranging particles (red circles), and $V_{f}$ of rearranging particles after deformations (blue triangles). (b) Rearranging probability as a function of initial $V_{f}$ (subtracted by the average $V_{f}$ ). (c) $S_{2}$ distribution for all particles (black squares), for rearranging particles before deformations (red circles), and for rearranging particles after deformations (blue triangles). (d) Rearranging probability as a function of initial $S_{2}$. Dashed lines and arrows show the average value and standard deviation of $V_{f}$ and $S_{2}$ for the distributions for all particles. All distributions are obtained from the combined data of 10 cycles, and are individually normalized.[]{data-label="fig2"}](fig2){width="9cm"} The rearranging probability for particles with the most extreme free volumes is about 3 times higher than that of the least active particles, which gives free volume relatively weak ability to predict rearranging regions, compared to other known parameters such as soft spots [@Chen1; @Manning1; @Harrowell1]. As a highly local structural parameter, particle free volume focuses only on the contacting neighbors of a particle. Rearrangements in glasses, however, require the cooperation between particles over a longer range [@Harrowell1; @Harrowell2]. Thus the structural correlations beyond the first-neighbor shell need to be considered to better identify rearranging regions. Pair correlation function $g(r)$ is a simple measure of structural correlations in glasses. Using local $g(r)$, the structural entropy of a particle *i* can be defined as $S_{2,i}=-1/2\sum_{\nu}\rho_{\nu}\int d{\vec{r}}\left\{g^{\mu\nu}_i({\vec{r}}){\rm ln}g^{\mu\nu}_i({\vec{r}})-\left[g^{\mu\nu}_i({\vec{r}})-1\right]\right\}$ [@Xu1; @Tanaka1; @Ghosh1], where $\mu$ and $\nu$ denote the type of particles (large or small), $\rho_{\nu}$ is the number density of $\nu$ particles, and $g^{\mu\nu}_i(\vec{r})$ is the pair correlation function between particle $i$ of type $\mu$ and particles of $\nu$ type. Structural entropy measures the loss of entropy due to positional correlations [@Wallace1]; high $S_{2}$ values indicate less correlated local structures and vice versa. In our experiments, the integration is truncated at the $3^{rd}$ neighbor shell to avoid the loss of a large fraction of particles due to boundary effect. We verify, for particles near the center of the field of view, that the $S_{2}$ obtained from 3 shells of neighbor are highly correlated to $S_{2}$ obtained from 5 neighbor shell integrations, with correlation coefficients about 0.9. Beyond the $5^{th}$ neighbor shell, local $g(r)$ becomes very close to 1, thus contributing only negligibly to measured $S_{2}$. For each particle, time-average is performed for local $g(r)$ before integration to remove short-time fluctuations, although in principle, this averaging is not critical to the measurements of $S_{2}$ [@Tanaka2]. We limit the averaging time to the last 60 seconds (1200 frames) of each thermal cycle, within the $\beta$ relaxation time of the sample, to avoid direct coupling to diffusive dynamics [@Han1; @Tanaka2]. [![ Spatial distribution of particle $S_{2}$ and rearranging clusters. (a) Colored contour plot: $S_{2}$ distribution *before* thermal cycling, and rearranging clusters observed in the following 10 cycles (black circles). (b1-b3) Contour plots: local $S_{2}$ distributions before the $1^{st}$,$2^{nd}$, and $3^{rd}$ thermal cycle in a sub-region in the sample; black circles: rearranging clusters from $1^{st}$,$2^{nd}$ and $3^{rd}$ in the same region. (c1-c6) Contour plots: local $S_{2}$ distributions from the $3^{rd}$ to $8^{th}$ thermal cycle in another sub-region in the sample; black circles: rearranging clusters from the $4^{th}$ cycle (c2), and the $7^{th}$ cycle (c5).[]{data-label="fig3"}](fig3 "fig:"){width="9cm"}]{} The structural entropy of rearranging particles are qualitatively different from those of the general population in the colloidal glass. Fig. \[fig2\]c plots the distribution of $S_{2}$ for all the particles in the colloidal glass (black squares) and those for rearranging particles (red circles). Compared to $V_{f}$ distributions, the distribution of $S_{2}$ for rearranging particles peaks at a significantly higher value than the distribution from all particles, indicating less correlated structures in rearranging clusters. After rearrangements, the $S_{2}$ values of particles in rearranging clusters are decreased, but remain elevated compared to the system average, as shown by blue triangles in Fig. \[fig2\]c. Fig. \[fig2\]d plots the rearranging probability as a function of initial $S_{2}$ for individual particles. The probability to rearrange increases almost monotonically with $S_{2}$, with two orders of magnitude difference between the most stable particles and most unstable particles, revealing intrinsically different responses from local structures in colloidal glasses. Qualitatively the same correlations as shown in Fig. 2 have been observed in samples of different packing fractions and under different perturbation patterns [@suppl.]. Similar to soft modes, the structural entropy for individual particles can be employed to predict regions prone to rearrangements. The colored contour plot in Fig. \[fig3\]a shows the initial distribution of $S_{2}$ in the colloidal glass *before* thermal cycling, black circles overplotted on the contour plot are the rearranging clusters observed in the following 10 cycles. Remarkable overlap is observed between regions with high $S_{2}$ values and rearranging clusters, which demonstrates that local $S_{2}$ is able to predict rearranging regions long before they actually occur in colloidal glasses under thermal cycling. This correlation is enhanced when the system is closer to the relaxation events, as shown in the video of the evolution of the $S_{2}$ distributions and the rearranging clusters from each thermal cycle in the supplementary materials [@suppl.]. The overlap between the $S_{2}$ distribution before thermal cycling and rearranging clusters in later cycles suggests that the distribution of high $S_{2}$ regions evolves slowly under thermal cycling, which can be clearly seen in the video in the supplementary materials [@suppl.]. An example of the stability of local distribution of $S_{2}$ is shown in Fig. \[fig3\](b1-b3) (down-left corner of the video in the supplementary materials [@suppl.]). During the 3 consecutive local rearrangements, the area with high $S_{2}$ values only changes slightly. On the other hand, high $S_{2}$ regions can be created through local fluctuations. Some rearranging clusters that have small overlaps with high $S_{2}$ regions in Fig. \[fig3\]a are found to significantly overlap with high $S_{2}$ spots that emerge during the thermal cycling. An example is shown in Fig. \[fig3\](c1-c6) (mid-left region of the video in the supplementary materials [@suppl.]). Before the 4th and 7th cycles, two high $S_{2}$ spots are developed with no preceding rearranging clusters in its immediate neighborhood. The emergences of these two spots are followed by two rearranging events in the same region, after which local $S_{2}$ distribution falls back to the background. We quantitatively evaluate the correlations between high $S_{2}$ regions and local rearrangements during thermal cycling. High $S_{2}$ clusters are identified to the particle level by applying a cut-off to the distribution of local structural entropy. Averaged correlation between high $S_{2}$ clusters and rearranging clusters is measured to be 0.34 for a cutoff value of -1.5 [@suppl.], comparable to the correlations between soft spots and local rearrangements in colloidal experiments [@Chen1]. The highest single-cycle correlation is measured to be 0.69 [@suppl.]. We also measure the direct correlations between non-affine displacement coefficient $D_{min}^2$ and $S_{2}$ for all particles, regardless of neighbor changes. $D_{min}^2$ measures the particle level non-affine strain, and is defined as the minimum of $D^2(t_{1},t_{2})=\sum\limits_{n}\sum\limits_{i}[r^i_{n,t_{2}}-r^i_{0,t_{2}}-\sum\limits_{j}(\delta_{ij}+\varepsilon_{ij})\times(r^j_{n,t_{1}}-r^j_{0,t_{1}})]^2$ , where $r^i_{n,t}$ is the $i$-th ($x$ or $y$) component of the position of the $n$-th particle at time t (before or after a single cycle) and index $n$ runs over the particles within the interaction range of the reference particle $n=0$. The $\delta_{ij}+\varepsilon_{ij}$ that minimizes $D^2$ are calculated based on $r^i_{n,t}$ [@Langer1]. Direct correlation between $D_{min}^2$ and particle $S_{2}$ for all particles at each cycle yields an averaged correlation coefficient of 0.30 [@suppl.], suggesting that particles with higher $S_{2}$ values tend to experience larger strains under external loading. The noise levels for the above two correlation coefficients are both on the order of $\sim1/\sqrt{N}$, where $N$ is the number of particles in the field of view. On the other hand, structurally stable regions associated with low $S_{2}$ values in the colloidal glasses show strong resistance to the migration of rearranging clusters. Fig. \[fig4\] plots the particles whose $S_{2}$ remain below -1.6 (the $90^{th}$ percentile in the $S_{2}$) for the duration of the experiment (large gray circles). These grey particles form a large percolating network that only overlaps with rearranging clusters on the boundaries with high $S_{2}$ areas (small blue dots). The structural stability of these correlated domains may be related to the amorphously ordered structures predicted by random first-order transitions theories [@Wolynes1; @Zamponi1], and the dynamical slowdown during glass transitions [@Tanaka1; @Han1; @T1], which are fundamental challenges in the physics of glasses. And our results suggest $S_{2}$ may be employed to search for such structures. [![ Stable regions in colloidal glasses. Spatial distributions of particles with $S_{2}$ values below -1.6 during the experiment (large filled gray circles), particles with $S_{2}$ values exceeding -1.6 for at least one cycle (small blue dots), and rearranging clusters (red empty circles).[]{data-label="fig4"}](fig4 "fig:"){width="8cm"}]{} We can now make the analogy between structural defects in crystals to high $S_{2}$ regions in colloidal glasses. In polycrystalline solids, structural correlation is strong between atoms belonging to the same domain, and weak for atoms on defect sites. Similarly, the high $S_{2}$ spots in colloidal glasses are structurally less correlated compare to the percolating low $S_{2}$ domains. Thus high $S_{2}$ regions can be viewed as equivalent “structural defects” in glasses. This analogy is supported by the mechanical susceptibility of these regions and their creation and migration dynamics observed in the experiments. To summarize, we measure the structures of local rearranging clusters in colloidal glasses under thermal cycling. The local structural entropy of the rearranging regions is distinctively higher than that of the more stable background in glasses. Equivalent defects, that share the essential characteristics of structural defects in crystals, can be defined by the high $S_{2}$ values. As a structural parameter, $S_{2}$ provides a sensible measure of the apparently disordered structures of glasses. 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--- abstract: | Many events are followed by an absolute refractory state, when for some time after the event a repetition of a similar event is impossible. If uniform events, each of which is followed by the same period of absolute refractoriness, occur randomly, as in the Bernoulli scheme, then the event probability as a function of time can exhibit damped transient oscillations caused by a specific initial condition. Here we give an exact analytical description of the oscillations, with a focus on application within neuroscience. The resulting formulas stand out for their relative simplicity, enabling analytical calculation of the damping coefficients for the second and third peaks of the event probability. Keywords: renewal point process, absolute refractory period, damped oscillations, neuron, stochastic spiking author: - 'A.V. Paraskevov$^{1,2}$, A.S. Minkin$^{1}$' title: Damped oscillations of the probability of random events followed by absolute refractory period --- **1. Introduction** Many natural and technical events are followed by a refractory period, when for some time after the event a repetition of a similar event is unlikely (relative refractory period) or even impossible (absolute refractory period). A characteristic natural example is the neuron and the refractory nature of its ability to generate electrical impulses - spikes. In turn, a typical example from the technique is the existence of so-called dead time for some types of detectors (so-called the Type 1 counters [@Cox62]), especially for photodetectors. The dead time is a fixed period of time after the detector triggering during which it becomes inoperative. Both the neuronal refractoriness and the detector dead time influence essentially counting distributions. If instant events of the same type, each of which is followed by the same absolute refractory period $\tau_{ref}$, occur randomly, the average interval $T$ between the events can be represented as the sum $T=\tau_{ref}+T_{0}$, where $T_{0}$ is the value of the average interval if refractoriness is absent. A more nontrivial consequence of the refractory period is damped oscillations of the event occurrence probability as a function of time [@Perk67; @WCBJ72; @Kin06], which are strongly pronounced at $T_{0}\lesssim \tau_{ref}$. Such transient oscillations arise due to (i) a specific initial condition and (ii) the fact that random instant events ordered in time in the presence of refractoriness become dependent on each other. In particular, the probability of a subsequent event depends on the time elapsed from the preceding event. Random point processes of this kind are called renewal processes and are the subject of study of the renewal theory [@Cox62]. This theory has a method of getting an explicit analytical description for the damped oscillations of the event probability by finding of so-called renewal density [@Cox62; @Perk67]. For instance, this has been repeatedly done for the classic example of the Poisson process modulated by the dead time or absolute refractory period [@Malm47; @Ric66; @Muller73; @Muller74; @Cant75; @John83; @John86; @Pom99; @Picib08; @PRE2010; @JCN2012; @NC2018]. However, the resulting analytical formula for the time-dependent probability of the event is quite cumbersome, and that complicates further analysis. In this paper, for the Bernoulli process modulated by absolute refractoriness we give a compact analytical description of the damped oscillations without invoking the renewal theory. The description is presented in four equivalent forms (three kinds of a recurrence formula and one explicit formula) and is quantitatively consistent with both the results of numerical simulations and those of the renewal theory. One kind of the recurrence formula is especially simple, enabling accurate analytical calculation of the damping coefficients. Surprisingly, these are quite robust against changing the values of the model parameters. Finally, ready-to-use MATLAB/Octave codes for performing simulations and plotting the graphs of the obtained formulas are included as supplementary material. **2. Formulation of the model problem** For certainty, consider a model neuron that stochastically emits spikes, each of which is followed by an absolute refractory period. In particular, the neuron can spontaneously emit a spike with probability $p_{s}$ per unit time (i.e., in a given elementary time interval $\triangle t$) so that after the spike emission the neuron becomes temporarily inactive, i.e. it cannot emit spikes during the refractory period $\tau_{ref}=n_{ref}\triangle t$, where $n_{ref}$ is a positive integer. Then the mean rate of occurrence of events in the absence of refractoriness $\nu_{0}=1/T_{0}=p_{s}/\triangle t$, and at $\tau_{ref}\neq 0$ from the equality $T=\tau_{ref}+T_{0}$ for the mean rate $\nu=1/T$ one gets $\nu=\nu_{0}/(1+\tau_{ref}\nu_{0})$. In fact, this mean rate is equal to the ratio of the total number of spikes to observation time $T_{obs}$, given that $T_{obs} >> \tau_{ref}$, $T_{0}$. It is also useful to introduce the asymptotic value of the average probability of spike generation at each step, $\bar{p}_{s}=\nu \triangle t=p_{s}/(1+n_{ref}p_{s})$, such that, by the analogy with the formula for $\nu_{0}$, $$\nu=\bar{p}_{s}/\triangle t=p_{s}/(\triangle t+p_{s}\tau_{ref}).\label{nu}$$ An algorithm for simulating the neuron’s dynamics is extremely simple and as follows. Dividing the observation interval $T_{obs}$ by $N$ equal steps $\triangle t$, $T_{obs}=N\triangle t$, these time steps are numbered by a sequence of natural numbers starting with 1. At each step, a random number $\xi$, uniformly distributed from zero to one, is generated and compared with the given probability $p_{s}$ of generating a spike. If $\xi\leq p_{s}$, it is assumed that spike has been generated at this time step. After the spike generation, the neuron cannot emit a next spike during refractory period $\tau_{ref}=n_{ref}\triangle t$. The event of spike generation at an arbitrary $k$-th step is further denoted by $A_{k}$. For definiteness, the initial state of the neuron is chosen as a moment when the neuron has just left the refractory state after emitting a spike. Performing either a large number of repeated passes of the observation interval $T_{obs}$ for a single neuron or a single pass for the large ensemble of independent neurons, one gets a statistical distribution of the occurrence of events in the entire sequence of $N$ time intervals. Normalizing this distribution by the number of either the passes or the neurons in the ensemble, one obtains the probability distribution $P_{k}$ of spike generation at $k$-th elementary time step of the observation interval. Due to the refractory period, the event probability, as a function of time, exhibits damped oscillations with the average period equal to $\tau_{ref}$ (Fig. 1). In turn, the analytical problem consists in finding the probability $P_{k}\equiv P(A_{k})$ of spike generation at each $k$-th elementary time step so that in the asymptotic limit $k\rightarrow\infty$ one would obtain $P_{k}\rightarrow\bar{p}_{s}$. ![Numerical simulation of time dependence of the event probability, where an event is spike generation, for $10^{4}$ disconnected stochastically-spiking neurons at time step $\triangle t$ = 0.01 ms. Top: Raster of events (gray dots, scale on the right) and the corresponding time dependence of the statistical probability of an event (blue line, scale on the left) at $p_{s}$ = 0.1 and $\tau_{ref}$ = 2 ms. It is seen that the period of damped oscillations of probability is $\tau_{ref}$. The asymptotic probability value to which the damped oscillations converge corresponds to the calculated value $\bar{p}_{s}=4.76\cdot 10^{-3}$ (or the average event frequency $\nu = 476$ Hz, see (\[nu\])). Bottom: Similar graphs for $p_{s}$ = 0.01 and $\tau_{ref}$ = 5 ms. For this case, $\bar{p}_{s}=1.67\cdot 10^{-3}$ and $\nu = 167$ Hz. Note that the parameters of the neuron model (in particular, nonphysiologically large value of $p_{s}$) are chosen solely for the illustrative purpose.[]{data-label="Fig1"}](Fig1.eps){width="75.00000%"} **3. Derivation of the exact analytical formula for $P_{k}$** The probability of spike generation at $k$-th time step ($k=\overline{1,N}$), $P_{k}=P(A_{k})$, is equal to the product of $p_{s}$ and the probability that in the interval from $k-n_{ref}$ to $k-1$ inclusively no spike was emitted, $$P_{k}=p_{s}\left[ 1-P(A_{k-n_{ref}}+A_{k-n_{ref}+1}+...+A_{k-1})\right].\label{1}$$ Spike generation events in $n_{ref}$ consecutive time intervals are pairwise incompatible events. Therefore, according to the summation theorem for the probabilities of pairwise incompatible events, the probability of the sum in Eq. (\[1\]) equals the sum of probabilities $$P(A_{k-n_{ref}}+A_{k-n_{ref}+1}+...+A_{k-1})={\displaystyle\sum\limits_{j=k-n_{ref}}^{k-1}}P(A_{j}),\label{2}$$ and the sought-for probability at the $k$-th step is determined in a recurrent manner, with the recursion period equal to the refractory period, $$P_{k}=p_{s}(1-% {\displaystyle\sum\limits_{j=k-n_{ref}}^{k-1}}P_{j})=p_{s}\left[ 1-(S_{k-1}-S_{k-n_{ref}-1})\right],\label{3}$$ where, by definition, $$S_{m}= \begin{dcases} 0, \text{ } m \le 0, \\ {\displaystyle\sum\limits_{j=1}^{m}}P_{j}, \text{ } m>0. \end{dcases}\label{4}$$ Such a definition of $S_{m}$ allows us to directly generalize the formula for $P_{k}$ to the range $1\leq k\leq n_{ref}$. The closed formula for $S_{k}$ has the form of a linear recurrent sequence $$S_{k}=P_{k}+S_{k-1}=p_{s}(1+S_{k-n_{ref}-1})+(1-p_{s})S_{k-1}.\label{5}%$$ The resulting formula (\[3\]) accurately describes the numerical statistics (Fig. 2). ![Comparison of the results of numerical simulations (see Fig. 1) with the calculation by mutually equivalent analytical formulas (\[3\]), (\[Pk\]), (\[pol\]), (\[8\]) and (\[9\]). Top: Time dependence of the statistical probability of an event (blue line, data taken from the top panel of Fig. 1) and the corresponding analytic curve $P_{k}$ (red line) at $p_{s}$ = 0.1 and $\tau_{ref}$ = 2 ms. The asymptotic value $P_{\infty}=\bar{p}_{s}=4.76\cdot 10^{-3}$ is shown by the green horizontal line. Inset: Partial analytical curves for the first three refractory intervals calculated by formulas (\[Pk1\]), (\[Pk2\]) and (\[Pk3\]). Bottom: Similar graphs (main graph and inset) for the parameter values $p_{s}$ = 0.01 and $\tau_{ref}$ = 5 ms, giving the asymptotic probability $P_{\infty}=1.67\cdot 10^{-3}$.[]{data-label="Fig2"}](Fig2.eps){width="75.00000%"} One should note three important consequences. First, for $1\leq k\leq n_{ref}+1$, where $$S_{k}=p_{s}+(1-p_{s})S_{k-1},$$ the sum $S_{k}$ can be easily found explicitly, as it is an arithmetic-geometric progression of the form $$S_{k}=rS_{k-1}+d,$$ where $S_{1}=p_{s}$, $r=1-p_{s}$ è $d=p_{s}$. According to the formula for the explicit form of the $k$-th term of this progression, $$S_{k}=r^{k-1}\left[ S_{1}+\frac{d}{r-1}\right]-\frac{d}{r-1} = 1-(1-p_{s})^{k}.$$ Hence, the probability of spike generation in the interval $1\leq k\leq n_{ref}+1$ is equal to $$P_{k}\equiv P_{k}^{(1)}=p_{s}\left[ 1-S_{k-1}\right] =p_{s}(1-p_{s})^{k-1}.\label{Pk1}$$ Here and below, the upper index in parentheses indicates the number of the refractoriness interval, counted from the initial moment $t=0$. Given the initial condition, this probability naturally coincides with the probability of the first spike generation at an arbitrary $k$-th step, such that ${\displaystyle\sum\limits_{k=1}^{\infty}}P_{k}^{(1)}=1$, and refers to the geometric distribution. Second, the formula (\[3\]) makes it easy to obtain the asymptotic probability value $P_{k}$ at $k\rightarrow\infty$. The difference $S_{k-1}-S_{k-n_{ref}-1}$ contains $n_{ref}$ terms. At $k\rightarrow\infty$ the probability at the $k$-th step remains practically unchanged. Denoting it as $P_{\infty}$, from the general formula (\[3\]) one gets $$P_{\infty}=p_{s}\left[ 1-n_{ref}P_{\infty}\right],$$ whence $$P_{\infty}=p_{s}/(1+n_{ref}p_{s}).\label{7}$$ Third, calculating the adjacent terms $P_{k+1}$ or $P_{k-1}$ similarly to the formula (\[5\]), one can exclude sums (\[4\]) from the formula (\[3\]) and obtain a linear recurrent sequence for $P_{k}$: $$P_{k}= \begin{dcases} p_{s}(1-p_{s})^{k-1}, \text{ } k \leq n_{ref}+1, \\ p_{s}P_{k-n_{ref}-1}+(1-p_{s})P_{k-1}, \text{ } k>n_{ref}+1. \end{dcases}\label{Pk}$$ This formula is completely equivalent with Eq. (\[3\]) and using it one can easily find the expression for $P_{k}$ in an explicit form within the intervals of $k$ multiples of $n_{ref}+1$. For example, at $n_{ref}+1 \leq k \leq 2(n_{ref}+1)$ one gets $$P_{k}^{(2)}=(k-n_{ref}-1)p_{s}^{2}(1-p_{s})^{k-n_{ref}-2} + P_{k}^{(1)}=P_{k}^{(1)}[1+(k-n_{ref}-1)q],\label{Pk2}$$ where $q = p_{s}(1-p_{s})^{-(n_{ref}+1)}$. Next, at $2(n_{ref}+1) \leq k \leq 3(n_{ref}+1)$ one gets $$\begin{aligned} P_{k}^{(3)} & =\frac{1}{2}(k-2n_{ref}-2)(k-2n_{ref}-1)p_{s}^{3}(1-p_{s})^{k-2n_{ref}-3} + P_{k}^{(2)}=\\ & =P_{k}^{(1)}[1+(k-n_{ref}-1)q+\frac{1}{2}(k-2n_{ref}-2)(k-2n_{ref}-1)q^{2}].\label{Pk3}\end{aligned}$$ It is worth noting that the numerical coefficient in the highest-order term with respect to $q$ is the so-called triangular number $j(j+1)/2$ at $j=k-2n_{ref}-2$. The inset in Fig. 2 shows the plots for $P_{k}^{(1)}$, $P_{k}^{(2)}$ and $P_{k}^{(3)}$. Using the induction method, one can obtain a formula for $P_{k}^{(m+1)}$, valid within the range $m(n_{ref}+1) \leq k \leq (m+1)(n_{ref}+1)$: $$P_{k}^{(m+1)}=[\frac{1}{m!}\prod\limits_{j=1}^{m}(k-m \cdot n_{ref}-j)]q^{m}P_{k}^{(1)} + P_{k}^{(m)}.\label{Pkmp1}$$ The general explicit expression for $P_{k}$ in the polynomial form is as follows: $$\begin{aligned} P_{k} & =P_{k}^{(1)}[1+a_{k}^{(1)}p_{s}(1-p_{s})^{-(n_{ref}+1)}+a_{k}^{(2)}p_{s}^{2}(1-p_{s})^{-2(n_{ref}+1)}+\ldots]=\\ & =P_{k}^{(1)}[1+{\displaystyle\sum\limits_{i=1}^{m}}a_{k}^{(i)}q^{i}],\label{pol}\end{aligned}$$ where $m$ is the integer part of the rational number $k/n_{ref}$ rounded off to a smaller value, $q = p_{s}(1-p_{s})^{-(n_{ref}+1)}$, and the coefficients $$a_{k}^{(m)}\equiv\frac{\theta(k-m(n_{ref}+1))}{m!}{\displaystyle\prod\limits_{j=1}^{m}}(k-m\cdot n_{ref}-j),$$ where the unit step function $\theta(x) = 1$ at $x \geq 0$ and $\theta(x) = 0$ otherwise. Taking into account the equality $$\frac{1}{m!}{\displaystyle\prod\limits_{j=1}^{m}}(n-j)=\frac{(m+1)}{n}C_{n}^{m+1}=C_{n-1}^{m},\text{ } n>m \geq 1,$$ where $C_{n}^{k}$ is the standard binomial coefficient, $$C_{n}^{k}=\frac{n!}{k!(n-k)!}=\dbinom{n}{k},$$ the coefficients $a_{k}^{(m)}$ are directly expressed through the binomial coefficients: $$a_{k}^{(m)}=C_{k-m\cdot n_{ref}-1}^{m}\theta(k-m(n_{ref}+1)).$$ Another equivalent formula for $P_{k}$ can be found in a different way, as follows. Denote $P(A_{k}|A_{j})$ the conditional probability of spike generation at the $k$-th step, provided that the previous spike was generated at the $j$-th step. At times greater than the refractory period, i.e. at $k > n_{ref}$, the probability of generating a subsequent spike depends only on the moment of a previous spike. Therefore, taking into account the initial condition, $P(A_{k}|A_{j})=P_{k-j-n_{ref}}^{(1)}$, if $k-j>n_{ref}$, and $P(A_{k}|A_{j})=0$, if $k-j\leq n_{ref}$. At $k>n_{ref}+1$, the probability $P_{k}$ can be written as the sum of two terms: the probability $P_{k}^{(1)}$ that a spike will be emitted for the first time at the $k$-th step and the probability that at least one spike has been emitted previously. The latter has the form of a convolution and follows from the total probability formula. $$P_{k}=P_{k}^{(1)}+\sum\limits_{j=1}^{k-1}P_{j}\cdot P(A_{k}|A_{j})=P_{k}^{(1)}+\sum\limits_{j=1}^{k-n_{ref}-1}P_{j}\cdot P_{k-j-n_{ref}}^{(1)}. \label{8}$$ Notably, using substitution $i=k-j-n_{ref}$, one can virtually swap the indices of the multipliers under the sign of the sum in (\[8\]), while the formula does not change its numerical value: $$P_{k}=P_{k}^{(1)}+\sum\limits_{i=1}^{k-n_{ref}-1}P_{k-i-n_{ref}}\cdot P_{i}^{(1)}. \label{9}$$ Despite the different appearance in relation to the formulas (\[3\]) and (\[Pk\]), the formulas (\[8\]), (\[9\]) lead to the same numerical results and can be derived from the recurrent formula (\[Pk\]). In particular, the formula (\[9\]) can be straightforwardly obtained from (\[Pk\]) by successively substituting in the latter the values $P_{k-1}$, $P_{k-2}$, $\ldots$ , $P_{k-n_{ref}-1}$ and taking into account the definition (\[Pk1\]) for $P_{k}^{(1)}$. **4. Damping of the oscillations** The formulas (\[Pk2\]) and (\[Pk3\]) allow one to analytically calculate the relative damping of the second and third peaks of $P_{k}$. Finding the location of these peaks corresponds to solving a linear and quadratic algebraic equation, respectively. In particular, for $(n_{ref}+1)\leq k\leq 2(n_{ref}+1)$, calculating $dP_{k}^{(2)}/dk = 0$ by the explicit formula (\[Pk2\]) we get the location of the second peak, $$k_{\max2}=n_{ref}+1+R,$$ where $$\begin{aligned} R &=& 1/u-1/q, \label{R} \\ q &=& p_{s}(1-p_{s})^{-(n_{ref}+1)}, \label{q} \\ u &=& \ln\frac{1}{(1-p_{s})}. \label{u}\end{aligned}$$ Substituting $k_{\max2}$ into Eq. (\[Pk2\]), we get the amplitude of the second peak $$P_{\max}^{(2)}\equiv P_{k_{\max2}}^{(2)}=P_{k_{\max2}}^{(1)}\frac{q}{u}=\frac{p_{s}^{2}}{u}(1-p_{s})^{R-1}.\label{Pmax2}$$ The damping can be traced simply by the ratio of amplitudes for the adjacent peaks, $$D_{i+1}=P_{\max}^{(i+1)}/P_{\max}^{(i)}.$$ Given that $k_{\max1} = 1$ and $P_{k_{\max1}}^{(1)}=p_{s}$, for the second peak we get $$D_{2}=\frac{p_{s}}{u}(1-p_{s})^{R-1}.\label{D2}$$ For the third refractory interval, $2(n_{ref}+1)\leq k\leq3(n_{ref}+1)$, calculation of $dP_{k}^{(3)}/dk =0$ by the formula (\[Pk3\]) results in a quadratic equation, $$xk^{2}-yk-z=0,$$ where $$\begin{aligned} x & =\frac{1}{2}q^{2}u,\\ y & =q^{2}+(2n_{ref}+\frac{3}{2})q^{2}u-qu,\\ z & =-(2n_{ref}+\frac{3}{2})q^{2}-(n_{ref}+1)(2n_{ref}+1)q^{2}u+q+(n_{ref}+1)qu-u.\end{aligned}$$ A suitable solution of this equation is the root $$k_{\max3}=\frac{y+\sqrt{y^{2}+4xz}}{2x}=2(n_{ref}+1)+R+X,\label{k_max3}$$ where $$X=-\frac{1}{2}+\sqrt{\frac{1}{4}+\frac{1}{u^{2}}-\frac{(2n_{ref}+1)}{q}-\frac{1}{q^{2}}}.\label{X}$$ Substituting $k_{\max3}$ into the formula (\[Pk3\]) for $P_{k}^{(3)}$ yields $$P_{k_{\max3}}^{(3)}=P_{k_{\max3}}^{(1)}[1+(n_{ref}+1+R+X)q+\frac{1}{2}(R+X)(1+R+X)q^{2}].$$ After elementary but cumbersome calculations one can get a compact analytical expression for the damping coefficient of the third peak, $$D_{3}=P_{k_{\max3}}^{(3)}/P_{k_{\max2}}^{(2)}=p_{s}(1-p_{s})^{X}\cdot(\frac{1}{u}+X+\frac{1}{2}).\label{D3}$$ Numerical calculations have confirmed the validity of the obtained formulas. Below we have also listed the numerical values of the relevant quantities, calculated by the above formulas, for the two examples shown in Figs. 1 and 2. For the first example, at $\tau_{ref}=2$ ms ($n_{ref}=\tau_{ref}/\triangle t=200$ at $\triangle t=0.01$ ms) and $p_{s}=0.1$, we get $q \approx 1.6\times10^{8}$, $u \approx 0.1$, $R \approx 9.5$, $k_{\max2} = 210$, $P_{k_{\max2}}^{(2)} \approx 0.04$, $k_{\max3} = 420$, $P_{k_{\max3}}^{(3)} \approx 0.03$, $D_{2} \approx 0.39$, and $D_{3} \approx 0.74$. For the second example, at $\tau_{ref}=5$ ms ($n_{ref}=500$) and $p_{s}=0.01$, we get $q \approx 1.5$, $u \approx 0.01$, $R \approx 99$, $k_{\max2} = 599$, $P_{k_{\max2}}^{(2)} \approx 0.004$, $k_{\max3} = 1196$, $P_{k_{\max3}}^{(3)} \approx 0.003$, $D_{2} \approx 0.37$, and $D_{3} \approx 0.75$. It is seen that the numerical values $D_{2} \approx 0.4$ and $D_{3} \approx 0.75$ are fairly robust against changing the parameters. Thus, the magnitude of the second peak is approximately equal to 40% of the magnitude of the first. In turn, the value of the third peak is approximately 75% of the value of the second or 30% of the value of the first peak. **5. Comparison with the renewal theory** In the framework of the renewal theory [@Cox62; @Perk67], when the intervals between events are independent random variables, knowing the distribution density of such intervals, one can find the time dependence of the event probability, provided that an event occurred at the initial moment in time. In particular, the sought-for probability is expressed through the so-called renewal density $h(t)$, $$P^{(rd)}_{k}=h(k\triangle t)\triangle t.\label{h1}$$ In turn, the renewal density $h(t)$ is determined by the density $f(\tau)$ of the distribution of intervals $\tau$ between successive events [@Cox62; @Perk67], $$h(t)={\displaystyle\sum\limits_{n=1}^{\infty}}f_{n}(t),\label{h2}$$ where $f_{1}(t)\equiv f(t)$ and for $n\geq2$ functions $f_{n}(t)$ are given by the recursive convolution $$f_{n}(t)={\displaystyle\int\limits_{0}^{+\infty}}f_{n-1}(x)f(t-x)dx.\label{h4}$$ Qualitatively, functions $f_{n}(t)$ are the distribution densities of so-called $n$-th order intervals between events [@Perk67]: denoting as a first-order interval the elapsed time from some given event to the next following event, the second-order interval is defined as the elapsed time between the given event and the second following event, etc. An $n$-th order interval is therefore the sum of $n$ consecutive first-order intervals and is spanned by $(n + 1)$ consecutive events. At asymptotically large time $t\rightarrow+\infty$, $h(t)$ is saturated [@Cox62], $$\lim_{t\rightarrow+\infty}h(t)=\rho.\label{hlim}$$ Here $\rho$ is the mean rate of events, defined as the inverse mean interval between the successive events, $$\rho^{-1}={\displaystyle\int\limits_{0}^{+\infty}}\tau f(\tau)d\tau.\label{h5}$$ There are many, likely independent, examples of applying the renewal theory results to the case, where the events occurrence is the Poisson process modulated by the constant time of inoperativeness (dead time or, in our notations, absolute refractory period $\tau_{ref}$) or, equivalently, the distribution density of intervals between events has the form of a displaced exponential distribution, $$f(t)=\nu_{0}\exp(-\nu_{0}(t-\tau_{ref}))\theta(t-\tau_{ref}),\label{h6}$$ where $\nu_{0}=p_{s}/\triangle t$ and $\theta(\ldots)$ is the Heaviside step function. In particular, to the best of our knowledge, the first relevant paper dates back to 1947 [@Malm47] and has been followed by both in-depth studies [@Cox62; @Ric66; @Muller73; @Muller74; @Cant75; @Pom99; @Picib08; @PRE2010] and applied studies for neuroscience [@Perk67; @TMC78; @John83; @JASA85; @John86; @Koch93; @Berry98; @JCN2012; @NC2018] (see also [@John96] and [@Gerst02]). Below, we briefly outline and compare the previous results with our findings. ![Comparison of the time dependencies for the event probability $P_{k}$ calculated by mutually equivalent formulas (\[3\]), (\[Pk\]), (\[pol\]), (\[8\]), (\[9\]) in Section 3 (red curves) and the event probability $P^{(rd)}_{k}$ calculated within the renewal theory approach by formula (\[h1\]) in Section 5 (blue curves). The red and blue curves, accurate to an offset equal to the refractory period, completely coincide. The offset arises due to the different initial condition (exit from the refractory period for the red curves and spike generation for the blue ones) and is left intentionally in order to make the curves distinguishable. Top plot is for $p_{s}$ = 0.1 and $\tau_{ref}$ = 2 ms. Bottom plot is for $p_{s}$ = 0.01 and $\tau_{ref}$ = 5 ms.[]{data-label="Fig3"}](Fig3.eps){width="75.00000%"} For $f(t)$ given by (\[h6\]), using the Laplace transform, one can reduce the formula (\[h4\]) to the following expression $$f_{n}(t)=\nu_{0}\frac{(\nu_{0}(t-n\tau_{ref}))^{n-1}}{(n-1)!}\exp(-\nu_{0}(t-n\tau_{ref}))\theta(t-n\tau_{ref}),\label{h7}$$ which is the probability density function for the Erlang/gamma distribution, enabling computation of the renewal density $h(t)$ and the sought-for probability (\[h1\]). The plot of the function $P^{(rd)}_{k}$ with $$h(t)={\displaystyle\sum\limits_{n=1}^{n_{\max}}}f_{n}(t),\label{h8}$$ where $n_{\max}=10^{3}$ and $f_{n}(t)$ is determined by (\[h7\]), is shown in Fig. 3. The time dependence, accurate to an offset equal to the refractory period, completely coincides with that of $P_{k}$ calculated by mutually equivalent formulas (\[3\]), (\[Pk\]), (\[pol\]), (\[8\]), (\[9\]) in Section 3. The offset arises due to the different initial condition: an exit from the refractory period at $t=0$ in our model and an event occurrence at $t=0$ in the standard renewal-density approach. Finally, using (\[hlim\]), (\[h5\]) and (\[h6\]), one gets the asymptotic value of $h(t)$, $$\lim_{t\rightarrow+\infty}h(t)=\nu_{0}/(1+\nu_{0}\tau_{ref}),\label{h9}$$ which is consistent with formulas (\[nu\]) and (\[7\]) for $\nu$ and $P_{\infty}$, respectively. **6. Collation with spiking activity data for real neurons** Though stochastic spiking activity of the neuron serves only as an illustration of the obtained mathematical results, it seems worthwhile to provide the reader with references to experimental data and to discuss the conformity. Qualitatively, the model studied in the paper can be applied to the case where the neuron is externally stimulated by a relatively strong direct current implemented via a standard patch-clamp interface. The spike train variability-inducing noise could be either added to the stimulating current or be naturally ascribed to fluctuations of the neuronal potential due to incoming synaptic signals, imperfect ion channels etc. Alternatively to the stimulation by current, the neuron can have receptive field subject to an intense and nearly constant stimulus (a similar case of the fly photoreceptor is considered in [@Song17], see Figs. 5 and 6 there). Finally, the neuron can be a pacemaker with spontaneous high-frequency tonic spiking activity. For the reference, some examples of relevant experimental data are as follows: Fig. 4 in [@Rod67], Fig. 1 in [@Gray67], Figs. 1, 3, 4, 5, and 6 in [@Moor70], Fig. 3 in [@Rob87], Fig. 2B in [@Nin95], Figs. 2H and 8E in [@Kita04], Figs. 4A and 4B in [@Kita06], Fig. 2A in [@Beat12], Fig. 4A in [@Lee14], and Fig. 4A in [@Amit17]. In turn, a quantitative comparison of the obtained results with the experimental findings is hindered by such additional factors as (i) the relative refractory period, which may be time-dependent, and (ii) uncontrollable interaction of the neuron with the surrounding cells [@EBR11; @Coh11]. These factors often have a dominant influence on the heights, widths and locations of the experimental autocorrelogram peaks, especially if the stimulus magnitude is relatively small. In general, the damped oscillations studied in this paper are well-pronounced only if the absolute refractory period is a dominantly large timescale in the system’s dynamics responsible for the event occurrence. It should also be noted that in most cases of multi-unit systems, such as neuronal networks with neurons as the units, the absolute refractory period is not exactly the same for each unit. For instance, it may be randomly distributed so that each neuron has its own value of the absolute refractory period (e.g., see Fig. 2a in [@Avi13]). Alternatively, the absolute refractory period could be updated randomly after each event. Then increasing the variance of the refractory period distribution would apparently blur the damped oscillations. Solving this problem was beyond the scope of the present paper. **7. Conclusion** The model considered in this paper is a Bernoulli scheme supplemented by the condition of absolute refractoriness. This formally refers to the renewal theory. However, being quite simple, the model allows obtaining useful results without invoking this formalism. In particular, four equivalent analytical descriptions of the damped oscillations of the event probability have been given: (i) recurrent formula (\[3\]) through the difference of two sums, (ii) closed recurrent formula (\[Pk\]), (iii) explicit formula (\[pol\]) in the form of a polynomial, and (iv) recurrent convolution-type formulas (\[8\]), (\[9\]). It has also been shown that for the Poisson approximation these results accurately coincide with that of the renewal theory. Finally, using the closed recurrent formula, the relative damping coefficients for the second and third peaks of the event probability have been found in the exact analytical form. 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--- abstract: 'Bayesian optimization is a class of global optimization techniques. It regards the underlying objective function as a realization of a Gaussian process. Although the outputs of Bayesian optimization are random according to the Gaussian process assumption, quantification of this uncertainty is rarely studied in the literature. In this work, we propose a novel approach to assess the output uncertainty of Bayesian optimization algorithms, in terms of constructing confidence regions of the maximum point or value of the objective function. These regions can be computed efficiently, and their confidence levels are guaranteed by newly developed uniform error bounds for sequential Gaussian process regression. Our theory provides a unified uncertainty quantification framework for all existing sequential sampling policies and stopping criteria.' author: - | Rui Tuo$^*$\ Department of Industrial and\   Systems Engineering\ Texas A&M University\ College Station, TX 77843\ USA\ Wenjia Wang[^1]\ The Statistical and Applied Mathematical Sciences Institute\ Durham, NC 27709\ USA\ bibliography: - 'bib.bib' title: Uncertainty Quantification for Bayesian Optimization --- Introduction ============ The empirical and data-driven nature of data science field makes uncertainty quantification one of the central questions that need to be addressed in order to guide and safeguard decision makings. In this work, we focus on Bayesian optimization, which is effective in solving global optimization problems for complex blackbox functions. Our objective is to quantify the uncertainty of Bayesian optimization outputs. Such uncertainty comes from the Gaussian process prior, random input and stopping time. Closed-form solution of the output uncertainty is usually intractable because of the complicated sampling scheme and stopping criterion. Problem of interest ------------------- Let $f$ be an underlying *deterministic* continuous function over $\Omega$, a compact subset of ${\mathbb{R}}^p$. The goal of global optimization is to find the maximum of $f$, denoted by $\max_{x\in \Omega} f(x)$, or the point $x_{max}$ which satisfies $f(x_{max})=\max_{x\in \Omega} f(x)$. In many scenarios, objective functions can be expensive to evaluate. For example, $f$ defined by a complex computer model may take a long time to run. Bayesian optimization is a powerful technique to deal with this type of problems, and has been widely used in areas including designing engineering systems [@forrester2008engineering; @jones1998efficient; @mockus1978application], materials and drug design [@frazier2016bayesian; @negoescu2011knowledge; @solomou2018multi], chemistry [@hase2018phoenics], deep neural networks [@diaz2017effective; @klein2016fast], and reinforcement learning [@marco2017virtual; @wilson2014using]. In Bayesian optimization, $f$ is treated as a realization of a stochastic process, denoted by $Z$. Usually, people assume that $Z$ is a Gaussian process. Every Bayesian optimization algorithm defines a sequential sampling procedure, which successively generates new input points, based on the acquired function evaluations over all previous input points. Usually, the next input point is determined by maximizing an acquisition function. Examples of acquisition functions include probability of improvement [@kushner1964new], expected improvement [@huang2006global; @jones1998efficient; @mockus1978application; @picheny2013quantile], Gaussian process upper confidence bound [@azimi2010batch; @contal2013parallel; @desautels2014parallelizing; @srinivas2009gaussian], predictive entropy search [@hernandez2014predictive], entropy search portfolio [@shahriari2014entropy], knowledge gradient [@scott2011correlated; @wu2016parallel; @wu2017bayesian], etc. We refer to [@frazier2018tutorial] for an introduction to popular Bayesian optimization methods. Although Bayesian optimization has received considerable attention and numerous techniques have emerged in recent years, how to quantify the uncertainty of the outputs from a Bayesian optimization algorithm is rarely discussed in the literature. Since we assume that $f$ is a random realization of $Z$, $x_{max}$ and $f(x_{max})$ should also be random. However, the highly nontrivial distributions of $x_{max}$ and $f(x_{max})$ make uncertainty quantification rather challenging. Monte Carlo approaches can be employed to compute the posterior distributions of $x_{max}$ and $f(x_{max})$, but they are usually computationally expensive, especially when a large number of observations are available. It is worth noting that uncertainty quantification typically differs from convergence analysis of algorithms. In Bayesian optimization, the latter topic has been studied more often. See, for instance, [@bect2016supermartingale; @bull2011convergence; @calvin2005one; @calvin1997average; @ryzhov2016convergence; @vazquez2010convergence; @yarotsky2013univariate]. These analyses do not naturally produce uncertainty quantification techniques. Our results ----------- Inspired by a recent theoretical advance of Gaussian process regression [@wang2019prediction], we develop efficient methods to construct confidence regions of $x_{max}$ and $f(x_{max})$ for Bayesian optimization algorithms. **Uniform confidence intervals for Gaussian process regression.** We find sharp uniform error bounds for Gaussian process regression predictor, given by Theorem \[thm:undersmooth\]. To the best of our knowledge, this is the *first* theoretical result of this kind. Compared with the traditional point-wise predictive standard deviation of Gaussian process regression, denoted by $\sigma(x)$, our uniform bound is only inflated by a factor proportional to $\sqrt{\log (e \sigma/\sigma(x))}$, where $\sigma$ is the prior standard deviation. This bound leads to a uniform confidence upper limit of the reconstructed function, given by Algorithm \[alg:upper\]. Subsequently, we construct confidence regions of $x_{max}$ and $f(x_{max})$ based on the confidence upper limit and show the theoretical guarantees of their confidence level in Corollary \[coro:1\]. The results in this part work for fixed designs and will serve as a fundamental tool in the development of the general results under sequential sampling schemes. **Uncertainty quantification under sequential samplings.** There exist various Bayesian optimization approaches, based on different sequential sampling schemes. Our uncertainty quantification method *does not* rely on the specific formulae or strategies. We introduce a rather general framework called *abstract Bayesian optimization* (ABO), which covers all existing methods in an abstract sense. We show that by using the collected data of any instance algorithm of ABO, Algorithm \[alg:upper\] also gives a confidence upper limit with the same theoretical properties. Although sequential sampling is more complicated and has many different approaches, our theory shows that uncertainty quantification for sequential sampling done in the same way as that under fixed designs achieves the same confidence level. Optimization with Gaussian process regression under fixed designs ================================================================= In Bayesian optimization, we evaluate $f$ over a set of input points, denoted by $x_1,\ldots,x_n$. We call them the *design points*, because these points can be chosen according to our will. There are two categories of strategies to choose design points. We can choose all the design points before we evaluate $f$ at any of them. Such a design set is call a *fixed design*. An alternative strategy is called *sequential sampling*, in which the design points are not fully determined at the beginning. Instead, points are added sequentially, guided by the information from the previous input points and the corresponding acquired function values. In Bayesian optimization, sequential samplings are more popular, because such approaches can utilize the information from the previous responses and choose new design points in the area which is more likely to contain the maximum points. Before investigating the more important sequential sampling schemes, we shall first consider fixed designs in this section, because the latter situation is simpler and will serve as an important intermediate step to the general problem in Section \[Sec:UQ\]. Preliminaries ------------- In this work, we suppose that the Gaussian process $Z$ has mean zero, variance $\sigma^2$ and correlation function $\Psi$, i.e., $\text{Cov}(Z(x),Z(x'))=\sigma^2\Psi(x-x')$ with $\Psi(0)=1$. Under certain regularity conditions, Bochner’s theorem [@wendland2004scattered] suggests that the Fourier transform (with a specific choice of the constant factor) of $\Psi$, denoted by $\tilde{\Psi}$, is a probability density function and satisfies the inversion formula $\Psi(x)=\int_{\mathbb{R}^p}\cos(\omega^T x)\tilde{\Psi}(\omega) d \omega$. We call $\tilde{\Psi}$ the *spectral density* of $\Psi$. Some popular choices of correlation functions and their spectral densities are discussed in Section \[Sec:A0\]. Suppose the set of design points $X=(x_1,\ldots,x_n)$ is given. Then $f$ can be reconstructed via Gaussian process regression. Let $Y=(Z(x_1),\ldots,Z(x_n))^T$ be the vector of evaluations of the Gaussian process at the design points. The following results are well-known and can be found in [@rasmussen2006gaussian]. For any untried point $x$, conditional on $Y$, $Z(x)$ follows a normal distribution. The conditional mean and variance of $Z(x)$ are $$\begin{aligned} \mu(x) &:=& \mathbb{E}[Z(x)|Y]=r^T(x) K^{-1} Y,\label{mean}\\ \sigma^2(x) &:=& \text{Var}[Z(x)|Y]=\sigma^2(1-r^T(x) K^{-1} r(x)),\label{variance}\end{aligned}$$ where $r(x)=(\Psi(x-x_1),\ldots,\Psi(x-x_n))^T, K=(\Psi(x_j-x_k))_{j k}$. Since we assume that $f$ is a realization of $Z$, $\mu(x)$ can serve as a reconstruction of $f$. Uniform error bound ------------------- Although the conditional distribution of $Z(x)$ is simple as shown in (\[mean\])-(\[variance\]), those for $x_{max}$ and $Z(x_{max})$ are highly non-trivial because they are nonlinear functionals of $Z$. In this work, we construct confidence regions for the maximum points and values using a uniform error bound for Gaussian process regression, given in Theorem \[thm:undersmooth\]. We will use the notion $a\vee b:=\max(a,b)$. Also, we shall use the convention $0/0=0$ in all statements in this article related to error bounds. We need the following condition. For a vector $\omega=(\omega_1,\ldots,\omega_p)^T$, define its $l_1$-norm as $\|\omega\|_1=|\omega_1|+\ldots+|\omega_p|$. \[C1\] The correlation $\Psi$ has a spectral density, denoted by $\tilde{\Psi}$, and $$\begin{aligned} \label{A0} A_0=\int_{\mathbb{R}^p} \|\omega\|_1\tilde{\Psi}(\omega) d \omega<+\infty.\end{aligned}$$ \[thm:undersmooth\] Suppose Condition \[C1\] holds. Let $M=\sup_{x\in \Omega}\frac{Z(x) - \mu(x)}{\sigma(x)\log^{1/2} (e \sigma/\sigma(x))}$, where $\mu(x)$ and $\sigma(x)$ are given in (\[mean\]) and (\[variance\]), respectively. Then the followings are true. 1. $\mathbb{E}M\leq C_0 \sqrt{p(1 \vee \log(A_0 D_\Omega))}$, where $C_0$ is a universal constant, $A_0$ is as in Condition \[C1\], and $D_\Omega = \text{diam}(\Omega)$ is the Euclidean diameter of $\Omega$. 2. For any $t > 0$, $\mathbb{P}(M-\mathbb{E}M>t)\leq e^{-t^2/2}.$ The $l_1$-norm in (\[A0\]) can be replaced by the usual Euclidean norm. However, we use the former here because they usually have explicit expressions. See Section \[Sec:A0\] for details. In practice, Part 2 of Theorem \[thm:undersmooth\] is hard to use directly because $\mathbb{E}M$ is difficult to calculate accurately. Instead, we can replace $\mathbb{E}M$ by its upper bound in Part 1 of Theorem \[thm:undersmooth\]. We state such a result in Corollary \[coro:practice\]. Its proof is trivial. \[coro:practice\] Under the conditions and notation of Theorem \[thm:undersmooth\], for any constant $C$ such that $\mathbb{E}M\leq C \sqrt{p(1 \vee \log(A_0 D_\Omega))}$, we have $$\mathbb{P}(M-C \sqrt{p(1 \vee \log(A_0 D_\Omega))}>t)\leq e^{-t^2/2},$$ for any $t>0$, where the constants $A_0$ and $D_\Omega$ are the same as those in Theorem \[thm:undersmooth\]. To use Theorem \[thm:undersmooth\], we need to determine the universal constant $C$ and the moment of the spectral density $A_0$. We shall discuss the calculation of $A_0$ in Section \[Sec:A0\]. According to our numerical simulations in Section \[Sec:K\] and Section \[App:simu\] of the Supplementary material, we recommend using $C=1$ in practice. Uncertainty quantification -------------------------- In light of Theorem \[thm:undersmooth\], we can construct a confidence upper limit of $f$. Algorithm \[alg:upper\] describes how to compute the confidence upper limit of $f$ at a given untried $x$. For notational simplicity, we regard the dimension $p$, the variance $\sigma^2$, the moment $A_0$ and the universal constant $C$ as global variables so that Algorithm \[alg:upper\] has access to all of them. **Input:** Untried point $x$, significance parameter $t$, data $X=(x_1,\ldots,x_n)^T, Y$. Set $r=(\Psi(x-x_1),\ldots,\Psi(x-x_n))^T, K=(\Psi(x_j-x_k))_{j k}$. Calculate $$\begin{aligned} \mu&=&r^T K^{-1} Y,\\ s &=&\sqrt{\sigma^2(1-r^T K^{-1} r)}.\end{aligned}$$ **Output:** $\mu+s\sqrt{\log (e\sigma/s)}\left(C \sqrt{p(1 \vee \log(A_0 D_\Omega))}+t\right)$ \[alg:upper\] Based on the [[<span style="font-variant:small-caps;">UpperCL</span>]{}]{} function in Algorithm \[alg:upper\], we can construct a confidence region for $x_{max}$ and a confidence interval for $f(x_{max})$. These regions do not have explicit expressions. However, they can be approximated by calling [[<span style="font-variant:small-caps;">UpperCL</span>]{}]{} with many different $x$’s. Let $Y=(f(x_1),\ldots,f(x_n))^T$. The confidence region for $x_{max}$ is defined as $$\begin{aligned} \label{CR} CR_t:=\left\{x\in\Omega:{{\textsc{UpperCL}}}(x,t,X,Y)\geq \max_{1\leq i\leq n}f(x_i)\right\}.\end{aligned}$$ The confidence interval for $f(x_{max})$ is defined as $$\begin{aligned} \label{CI} CI_t:=\left[\max_{1\leq i\leq n}f(x_i), \max_{x\in\Omega}{{\textsc{UpperCL}}}(x,t,X,Y)\right].\end{aligned}$$ It is worth noting that the probability in Corollary \[coro:practice\] is *not* a posterior probability. Therefore, the regions given by (\[CR\]) and (\[CI\]) should be regarded as frequentist confidence regions under the Gaussian process model, rather than Bayesian credible regions. Such a frequentist nature has an alternative interpretation, shown in Corollary \[coro:1\]. Corollary \[coro:1\] simply translates Corollary \[coro:practice\] from the language of stochastic processes to a deterministic function approximation setting, which fits the Bayesian optimization framework better. It shows that $CR_t$ in (\[CR\]) and $CI_t$ in (\[CI\]) are confidence region of $x_{max}$ and $f(x_{max})$ with confidence level $1-e^{-t^2/2}$, respectively. In particular, to obtain a $95\%$ confidence region, we use $t=2.448$. \[coro:1\] Let $C(\Omega)$ be the space of continuous functions on $\Omega$ and $\mathbb{P}_Z$ be the law of $Z$. Then there exists a set $B\subset C(\Omega)$ so that $\mathbb{P}_Z(B)\geq 1-e^{-t^2/2}$ and for any $f\in B$, its maximum point $x_{max}$ is contained in $CR_t$ defined in (\[CR\]), and $f(x_{max})$ is contained in $CI_t$ defined in (\[CI\]). In practice, the shape of $CR_t$ can be highly irregular and representing the region of $CR_t$ can be challenging. If $\Omega$ is of one or two dimensions, we can choose a fine mesh over $\Omega$ and call [[<span style="font-variant:small-caps;">UpperCL</span>]{}]{}$(x,t,X,Y)$ for each mesh grid point $x$. In a general situation, we suggest calling [[<span style="font-variant:small-caps;">UpperCL</span>]{}]{}$(x,t,X,Y)$ with randomly chosen $x$’s and using the $k$-nearest neighbors algorithm to represent $CR_t$. Calculating $A_0$ {#Sec:A0} ----------------- For an arbitrary $\Psi$, calculation of $A_0$ in (\[A0\]) can be challenging. Fortunately, for two most popular correlation functions in one dimension, namely the Gaussian and the Matérn correlation functions [@rasmussen2006gaussian; @santner2013design], $A_0$ can be calculated in closed form. The results are summarized in Table \[tab:kernels\]. Correlation family Gaussian Matérn ---------------------- --------------------------------------------------------- ----------------------------------------------------------------------------------------------------------------------------------------------- Correlation function $\exp\{-(x/\theta)^2\}$ $\frac{1}{\Gamma(\nu)2^{\nu-1}}\left(\frac{2\sqrt{\nu}|x|}{\theta}\right)^\nu K_\nu\left(\frac{2\sqrt{\nu}|x|}{\theta}\right)$ Spectral density $\frac{\theta}{2\sqrt{\pi}}\exp\{-\omega^2\theta^2/4\}$ $\frac{\Gamma(\nu+1/2)}{\Gamma(\nu)\sqrt{\pi}}\left(\frac{4\nu}{\theta^2}\right)^\nu\left(\omega^2+\frac{4\nu}{\theta^2}\right)^{-(\nu+1/2)}$ $A_0$ $\frac{2}{\sqrt{\pi}\theta}$ $\frac{4\sqrt{\nu}\Gamma(\nu+1/2)}{\sqrt{\pi}(2\nu-1)\theta\Gamma(\nu)}$ for $\nu>1/2$ : Gaussian and Matérn correlation families, where $\Gamma(\cdot)$ is the Gamma function and $K_\nu(\cdot)$ is the modified Bessel function of the second kind.[]{data-label="tab:kernels"} For multi-dimensional problems, a common practice is to use *product* correlation functions. Specifically, suppose $\Psi_1,\ldots,\Psi_p$ are one-dimensional correlation functions. Then their product $\Psi(x)=\prod_{i=1}^{p}\Psi(x_i)$ forms a $p$-dimensional correlation function, where $x=(x_1,\ldots,x_p)^T$. If a product correlation function is used, the calculation of $A_0$ is easy. It follows from the elementary properties of Fourier transform that $\tilde{\Psi}(x)=\prod_{i=1}^p\tilde{\Psi}_i(x_i)$. Let $X_i$ be a random variable with probability density function $\Psi_i$. Then $A_0=\sum_{i=1}^p \mathbb{E}|X_i|$, i.e., the value of $A_0$ corresponding to a product correlation function is the sum of those given by the marginal correlation functions. If each $\Psi_i$ is either a Gaussian or Matérn correlation function, then $\mathbb{E}|X_i|$’s can be read from Table \[tab:kernels\]. Sequential samplings ==================== We now turn to sequential samplings. In Section \[Sec:existing\], we will review some existing methods in Bayesian optimization. In Section \[Sec:ABO\], we will propose a general Bayesian optimization framework called *abstract Bayesian optimization (ABO)*, which, in an abstract sense, covers all existing methods. Our uncertainty quantification methodology proposed in Section \[Sec:UQ\] works for all instance algorithms under ABO. Existing methods {#Sec:existing} ---------------- In general, a Bayesian optimization algorithm defines a sequential sampling scheme which determines the next input point $x_{n+1}$ by maximizing an *acquisition function* $a(x;X_n,Y_n)$, where $X_n=(x_1,...,x_n)^T$ consists of previous input points, and $Y_n =(f(x_1),...,f(x_n))^T$ consists of corresponding outputs. The acquisition function can be either deterministic or random given $X_n,Y_n$. A general Bayesian optimization procedure is shown in Algorithm \[alg:bayesopt\]. **Input:** A Gaussian process prior of $f$, initial observation data $X_{n_0},Y_{n_0}$. **for** $n=1,2...,$ **do**\ Find $x_{n+1} = {\operatorname*{argmax}}_{x\in \Omega}a(x;X_n,Y_n)$, evaluate $f(x_{n+1})$, update data and the posterior probability distribution on $f$. **Output:** The point evaluated with the largest $f(x)$. \[alg:bayesopt\] A number of acquisition functions are proposed in the literature, for example: 1. Expected improvement (EI) [@jones1998efficient; @mockus1978application], with $a_{\text{EI}}(x;X_n,Y_n) := {\mathbb{E}}((Z(x) - y_n^*) \mathbf{1}(Z(x) - y_n^*)|X_n, Y_n),$ where $\mathbf{1}(\cdot)$ is the indicator function, and $y_n^* = \max_{1\leq i\leq n} f(x_i)$. 2. Gaussian process upper confidence bound [@srinivas2009gaussian], with $a_{\text{UCB}}(x;X_n,Y_n) := \mu_n(x) + \beta_n \sigma_n(x)$, where $\beta_n$ is a parameter, and $\mu_n(x)$ and $\sigma_n(x)$ are as in and , respectively. 3. Predictive entropy search [@hernandez2014predictive], with $a_{\text{PES}}(x;X_n,Y_n) := f^{(n)}(x)$, where $f^{(n)}$ is an approximate simulation via spectral sampling [@quia2010sparse; @rahimi2008random] from $\text{GP}(0,\Psi|X_n,Y_n)$. Among the above acquisition functions, $a_{\text{EI}}$ and $a_{\text{UCB}}$ are deterministic functions of $(x,X_n,Y_n)$, whereas $a_{\text{PES}}$ is random because it depends on a random sample from the posterior Gaussian process. We refer to [@shahriari2016taking] for general discussions and popular methods in Bayesian optimization. In practice, one also needs to determine when to stop Algorithm \[alg:bayesopt\]. Usually, decisions are made in consideration of the budget and the accuracy requirement. For instance, practitioners can stop Algorithm \[alg:bayesopt\] after finishing a fixed number of iterations [@frazier2018tutorial] or no further significant improvement of function values can be made [@acerbi2017practical]. Although stopping criteria plays no role in the analysis of the algorithms’ asymptotic behaviors, they can greatly affect the output uncertainty. Abstract Bayesian optimization {#Sec:ABO} ------------------------------ We now propose a general framework of Bayesian optimization on which our uncertainty quantification methodology relies. This framework does not involve any specific formulae or criteria. In contrast, we extract from the existing methods the following three basic concepts: *information*, *policy* and *stopping rule*. These concepts and their minimum requirements are described as follows. 1. **Information**. Information consists of previous input and output points. At the beginning, the information is empty, denoted by $\varnothing$. As we evaluate the objective function at increasing number of input points, the set of information increases. 2. **Policy**. A policy is a sequential sampling strategy which determines the next set of input data based on the current information. A policy may be random. However, its random-sampling mechanism should not depend on unobserved data. The mathematical formulae of a policy may vary at different stages of the process. For example, one can choose initial designs and subsequent sampling points with different strategies. A policy can sample one point or a batch of points at a time. 3. **Stopping rule**. A Bayesian optimization algorithm should stop at some point. After each sampling-evaluation iteration, a stopping criterion is checked and to determine whether to terminate the algorithm. A stopping decision should depend only on the current information and/or prespecified values such as computational budget, and should not depend on unobserved data. Algorithm \[alg:ABO\] describes the procedure of the abstract Bayesian optimization (ABO). Algorithm \[alg:ABO\] is said abstract because it accepts a policy ${{\textsc{p}}}$ and a stopping rule ${{\textsc{t}}}$ as its inputs. Once ${{\textsc{p}}}$ and ${{\textsc{t}}}$ are specified, we obtain a concrete algorithm, called an *instance algorithm* of ABO. As before, we denote the objective function as $f$. For notational simplicity, we denote $f(X)=(f(x_1),\ldots,f(x_m))^T$ for a set of input points $X=(x_1,\ldots,x_m)$. The policy ${{\textsc{p}}}$ returns the new input point(s), and the stopping rule ${{\textsc{t}}}$ returns either $\mathbf{true}$ or $\mathbf{false}$. The algorithm stops when ${{\textsc{t}}}$ returns $\mathbf{true}$. **Input:** Policy ${{\textsc{p}}}$, stopping rule ${{\textsc{t}}}$, objective function $f$ Set $I=\varnothing$ Set $X={{\textsc{p}}}(I)$ Set $Y=f(X)$ Set $I=I\cup \{(X,Y)\}$ **return** $I$ \[alg:ABO\] Clearly, all existing methods discussed in Section \[Sec:existing\] are instance algorithms of ABO. Although Algorithm \[alg:ABO\] is call abstract Bayesian optimization, it does not involve any specific optimization strategies. Actually this framework also covers algorithms with other purposes such as root finding. Uncertainty quantification for sequential samplings {#Sec:UQ} =================================================== In this section, we extend our uncertainty quantification methodology for Bayesian optimization under fixed designs to that under sequential samplings. Theory ------ To facilitate our mathematical analysis, we now state the ABO framework in a rigorous manner. Recall that we assume that $f$ is a realization of a Gaussian process $Z$. From this Bayesian point of view, we shall not differentiate $f$ and $Z$ in this section. Denote the vectors of input and output points in the $i$th iteration as $X_i$ and $Y_i$, respectively. Denote $X_{1:n}=(X_1,\ldots,X_n)$ and $Y_{1:n}=(Y_1^T,\ldots,Y_n^T)^T$. Then, we can write the information we obtain after the $i$th iteration as $I_n=(X_{1:n},Y^T_{1:n})$. Because $X_{1:n}$ and $Y_{1:n}$ are random, the information $I_n$ is associated with the $\sigma$-algebra $\mathcal{F}_n$, defined as the $\sigma$-algebra generated by $(X_{1:n},Y^T_{1:n})$. When the algorithm just starts, no information is gain and we set $I_0=\varnothing$. The empty $I_0$ is associated with the trivial $\sigma$-algebra $\mathcal{F}_0$, which consists of only the empty set and the entire probability space. Let $T$ be the number of iterations when the algorithm stops, i.e., $T=\min_{n\geq 1}\{{{\textsc{t}}}(I_n)=\mathbf{true}\}$. Then an ABO algorithm must satisfy the following conditions. 1. Conditional on $\mathcal{F}_{n-1}$, $X_n$ and $Z$ are mutually independent for $n=1,2,\ldots$. 2. $T$ is a stopping time with respect to the filtration $\{\mathcal{F}_n\}_{n=0}^\infty$. We further require $1\leq T<+\infty, a.s.,$ to ensure a meaningful Bayesian optimization procedure. For every policy ${{\textsc{p}}}$ and stopping time $T$, the pair $({{\textsc{p}}},T)$ defines an instance algorithm of ABO. We shall establish a generic theory that bounds the uniform prediction error of any instance algorithms of ABO. Given $n$, we denote $$\begin{aligned} \label{mn} X_{1:n}=(x_1,\ldots,x_{m_n}),\end{aligned}$$ where each $x_i$ is corresponding to one data point and $m_n$ is the number of sampled points after $n$ iterations of the algorithm. Inspired by the Gaussian process regression method (\[mean\])-(\[variance\]), we define $$\begin{aligned} \mu_n(x) &:=& r^T_n(x) K^{-1}_n Y_{1:n},\label{seqmean}\\ \sigma^2_n(x) &:=& \sigma^2(1-r^T_n(x) K_n^{-1} r_n(x)),\label{seqvariance}\end{aligned}$$ where $r_n(x)=\Psi(x-x_1),\ldots,\Psi(x-x_{m_n}))^T, K_n=(\Psi(x_j-x_k))_{j k}$. The main objective of Theorem \[thm:sequential\] is to quantify the uncertainty of $Z(\cdot)-\mu_T(\cdot)$. Note that $Z(\cdot)-\mu_T(\cdot)$ is generally *not* a Gaussian process. Nonetheless, an error bound similar to that in Corollary \[coro:practice\] is still valid. \[thm:sequential\] Suppose Condition \[C1\] holds. Given $({{\textsc{p}}},T)$, let $$M_n=\sup_{x\in \Omega}\frac{Z(x) - \mu_n(x)}{\sigma_n(x)\log^{1/2} (e \sigma/\sigma_n(x))},$$ where $\mu_n(x)$ and $\sigma_n(x)$ are given in (\[seqmean\]) and (\[seqvariance\]), respectively. Then for any $t>0$, $$\begin{aligned} \label{boundsequential} \mathbb{P}(M_T-C \sqrt{p(1 \vee \log(A_0 D_\Omega))}>t)\leq e^{-t^2/2},\end{aligned}$$ where $C,A_0,D_\Omega$ are the same as in Corollary \[coro:practice\]. The probability bound (\[boundsequential\]) has a major advantage: the constant $C$ is independent of the specific $({{\textsc{p}}},T)$, and it can be chosen the same as that for fixed designs. As we shall calibrate $C$ with numerical simulations (see Section \[Sec:K\] and Section \[App:simu\] of the Supplementary material), Theorem \[thm:sequential\] suggests that we only need to simulate for fixed-design problems, and the resulting constant $C$ can be used for the uncertainty quantification of all past and possible future Bayesian optimization algorithms. Methodology ----------- The conclusion of Theorem \[thm:sequential\] is similar to that of Corollary \[coro:practice\]. Consequently, we can again apply Algorithm \[alg:upper\] to get a uniform confidence upper limit of the underlying function. Given $({{\textsc{p}}},T)$, similar to (\[CR\]) and (\[CI\]), we define $$\begin{aligned} CR_t^{\mathbf{seq}}&:=&\left\{x\in\Omega:{{\textsc{UpperCL}}}(x,t,X_{1:T},Y_{1:T})\geq \max_{1\leq i\leq m_T}f(x_i)\right\},\label{seqR}\\ CI_t^{\mathbf{seq}}&:=&\left[\max_{1\leq i\leq m_T}f(x_i), \max_{x\in\Omega}{{\textsc{UpperCL}}}(x,t,X_{1:T},Y_{1:T})\right],\label{seqI}\end{aligned}$$ where $m_T$ is defined in (\[mn\]) with $n$ replaced by the stopping time $T$. Then Theorem \[thm:sequential\] implies that, under the condition that $f$ is a realization of $Z(\cdot)$, $CR_t^{\mathbf{seq}}$ and $CI_t^{\mathbf{seq}}$ are confidence regions of $x_{max}$ and $f(x_{max})$, respectively, with a simultaneous confidence level $e^{-t^2/2}$. Analogous to Corollary \[coro:1\], we can restate Theorem \[thm:sequential\] under a deterministic setting in terms of Corollary \[coro:sequential\]. In this situation, we have to restrict ourselves to *deterministic ABO algorithms*, in the sense that the policy ${{\textsc{p}}}$ is a deterministic map, such as the first two examples in Section \[Sec:existing\]. \[coro:sequential\] Let $C(\Omega)$ be the space of continuous functions on $\Omega$ and $\mathbb{P}_Z$ be the law of $Z$. Given a deterministic ABO algorithm $({{\textsc{p}}},T)$, there exists a set $B\subset C(\Omega)$ so that $\mathbb{P}_Z(B)\geq 1-e^{-t^2/2}$ and for any $f\in B$, its maximum point $x_{max}$ is contained in $CR_t^{\mathbf{seq}}$ defined in (\[seqR\]), and $f(x_{max})$ is contained in $CI_t^{\mathbf{seq}}$ defined in (\[seqI\]). Calibrating $C_0$ via simulation studies {#Sec:K} ======================================== To use (\[seqR\]) and (\[seqI\]), we need to specify the constant $C$ in Theorem \[thm:sequential\], which relies on $C_0$ in Theorem \[thm:undersmooth\]. In this work, we identify $C_0$ by numerical simulations. The details are presented in Section \[App:simu\] of the Supplementary material. Here we outline the main conclusions of our simulation studies. Our main conclusions are: 1) $C_0 = 1$ is a robust choice for most of the cases; 2) for the cases with Gaussian correlation functions or small $A_0D_\Omega$, choosing $C_0=1$ may lead to very conservative confidence regions. We suggest practitioners first consider $C_0 = 1$ to obtain robust confidence regions. When users believe that this robust confidence region is too conservative, they can use the value in Table \[p1tab\] or \[p2tab\] corresponding to their specific setting, or run similar numerical studies as in Section \[App:simu\] of the Supplementary material to calibrate their own $C_0$. Numerical experiments {#Sec:numexp} ===================== We compare the performance between the proposed confidence interval $CI_t^{\mathbf{seq}}$ as in and the naive bound of Gaussian process, denoted by $CI_G$. The nominal confidence levels are 95% for both methods. Other technical details are given in Section \[App:numeg\] of the Supplementary material. The comparison results are shown in Figure \[figcr\]. Figure \[figcr\] shows the coverage rates and the width of the confidence intervals under different smoothness with $\nu=1.5,2.5,3.5$. From the left plot in Figure \[figcr\], we find that the coverage rate of $CI_t^{\mathbf{seq}}$ is almost 100% for all the experiments, while $CI_G$ has a lower coverage rate no more than $82\%$. Thus the proposed method is conservative while the naive one is permissive. Such a result shows that using the naive method may be risky in practice. The coverage results support our theory and conclusions made in Sections \[Sec:UQ\]-\[Sec:K\]. As shown by the right plot in Figure \[figcr\], the widths of $CI_t^{\mathbf{seq}}$ are about five times of $CI_G$. The ratio decrease as the number of iterations increases. The inflation in the width of confidence intervals is the cost of gaining confidence. ![Coverage rates and widths of $CI_t^{\mathbf{seq}}$ and $CI_G$[]{data-label="figcr"}](crall95){height="2.7in"} ![Coverage rates and widths of $CI_t^{\mathbf{seq}}$ and $CI_G$[]{data-label="figcr"}](crall95wid){height="2.7in"} Discussion ========== In this work we propose a novel methodology to construct confidence regions for the outputs given by any Bayesian optimization algorithm with theoretical guarantees. To the best of our knowledge, this is the *first* result of this kind. The confidence regions may be conservative, because they are constructed based on probability inequalities that may not be tight enough. Given the fact that naive methods may be highly permissive, the proposed method can be useful when a conservative approach is preferred, such as in reliability assessments. To improve the power of the proposed method, one needs to seek for more accurate inequalities in a future work. Inequalities for Gaussian processes =================================== In this section, we review some inequalities on the maximum of a Gaussian process. Let $G(x)$ be a separable zero-mean Gaussian process with $x\in \Gamma$. Define the metric on $\Gamma$ by $$\begin{aligned} \mathfrak{d}_g(G(x_1), G(x_2))=\sqrt{{\mathbb{E}}(G(x_1) - G(x_2))^2}.\end{aligned}$$ The $\epsilon$-covering number of the metric space $(\Gamma,\mathfrak{d}_g)$, denoted as $N(\epsilon,\Gamma,\mathfrak{d}_g)$, is the minimum integer $N$ so that there exist $N$ distinct balls in $(\Gamma,\mathfrak{d}_g)$ with radius $\epsilon$, and the union of these balls covers $\Gamma$. Let $D$ be the diameter of $\Gamma$ with respect to the metric $\mathfrak{d}_g$. The supremum of a Gaussian process is closely tied to a quantity called the *entropy integral*, defined as $$\begin{aligned} \label{entropy} \int_0^{D/2} \sqrt{\log N(\epsilon,\Gamma,\mathfrak{d}_g)}d\epsilon.\end{aligned}$$ For detailed discussion of entropy integral, we refer to [@adler2009random]. Lemma \[thm133\] provides an upper bound on the expectation of the maximum value of a Gaussian process, which is Theorem 1.3.3 of [@adler2009random]. \[thm133\] Let $G(x)$ be a separable zero-mean Gaussian process with $x$ lying in a $\mathfrak{d}_g$-compact set $\Gamma$, where $\mathfrak{d}_g$ is the metric. Let $N$ be the $\epsilon$-covering number. Then there exists a universal constant $\eta$ such that $$\begin{aligned} \label{thm133neq} \mathbb{E}\left(\sup_{x\in \Gamma} G(x)\right) \leq \eta\int_0^{D/2} \sqrt{\log N(\epsilon,\Gamma,\mathfrak{d}_g)}d\epsilon.\end{aligned}$$ Lemma \[thm211\], which is Theorem 2.1.1 of [@adler2009random], presents a concentration inequality. \[thm211\] Let $G$ be a separable Gaussian process on a $\mathfrak{d}_g$-compact $\Gamma$ with mean zero, then for all $u>0$, $$\begin{aligned} \label{thm211neq} \mathbb{P}\left(\sup_{x\in \Gamma} G(x) - \mathbb{E} (\sup_{x\in \Gamma} G(x)) > u\right) \leq e^{-u^2/2\sigma^2_\Gamma},\end{aligned}$$ where $\sigma^2_\Gamma = \sup_{x\in \Gamma} \mathbb{E}G(x)^2$. Theorem \[ourthmjasa\] is a slightly strengthened version of Theorem 1 of [@wang2019prediction]. Its proof, in Section \[App:jasaprof\], is based on Lemmas \[thm133\]-\[thm211\] and some machinery from scattered data approximation [@wendland2004scattered]. \[ourthmjasa\] Suppose Condition \[C1\] holds. Let $\mu(x)$ and $\sigma(x)$ be as in (\[mean\]) and (\[variance\]), respectively, and $D_\Omega = \text{diam}(\Omega)$ be the Euclidean diameter of $\Omega$. Then for any $u>0$, and any closed deterministic subset $A\subset \Omega$, with probability at least $1-\exp\{-u^2/(2\sigma_A^2)\}$, the kriging prediction error has the upper bound $$\begin{aligned} \label{eq:thmBound} \sup_{x\in A}Z(x)-\mu(x)\leq \eta_1 \sigma_A \sqrt{p(1 \vee \log(A_0 D_\Omega))} \sqrt{\log(e\sigma/\sigma_A))} + u,\end{aligned}$$ where $A_0$ is defined in Condition \[C1\], $\eta_1$ is a universal constant, and $\sigma_A = \sup_{x\in A}\sigma(x)$. Proof of Theorem \[thm:undersmooth\] ==================================== We proof Theorem \[thm:undersmooth\] by partitioning $\Omega$ into subregions, and applying Theorem \[ourthmjasa\] on each of them. Let $\Omega_i = \{x\in \Omega | \sigma e^{-i} \leqslant \sigma(x) \leqslant \sigma e^{-i+1}\},$ for $i=1,...$. Let $\sigma_i = \sup_{x\in \Omega_i} \sigma(x)$. Take $\eta_2 = \eta_1 \sqrt{2}e$. By Theorem \[ourthmjasa\], we have $$\begin{aligned} & P\left(\sup_{x\in \Omega}\frac{Z(x) - \mu(x)}{\sigma(x) \log^{1/2}(e\sigma/\sigma(x) )} > \eta_2 \sqrt{p(1 \vee \log(A_0 D_\Omega))} + u \right) \nonumber\\ \leqslant & \sum_{i=1}^\infty P\left(\sup_{x\in \Omega_i}\frac{Z(x) - \mu(x)}{\sigma(x)\log^{1/2}(e\sigma/\sigma(x) )} > \eta_2 \sqrt{p(1 \vee \log(A_0 D_\Omega))} + u \right)\nonumber\\ \leqslant & \sum_{i=1}^\infty P\left(\sup_{x\in \Omega_i}Z(x) - \mu(x) > ( \eta_2 \sqrt{p(1 \vee \log(A_0 D_\Omega))} + u)\sigma e^{-i}\sqrt{i} \right)\nonumber\\ \leqslant & \sum_{i=1}^\infty P\left(\sup_{x\in \Omega_i}Z(x) - \mu(x) > ( \eta_2 \sqrt{p(1 \vee \log(A_0 D_\Omega))} + u) \sigma_{i}\log^{1/2} (e\sigma/\sigma_{i})/(\sqrt{2}e) \right)\nonumber\\ \leqslant & \sum_{i=1}^\infty \exp\left\{ - u^2\log (e\sigma/\sigma_{i})/(4 e^2) \right\}\nonumber\\ \leqslant & \sum_{i=1}^\infty \exp\left\{ - i u^2/(4 e^2) \right\} = \frac{\exp\left\{ - u^2/(4 e^2) \right\}}{1 - \exp\left\{ - u^2/(4 e^2) \right\}},\nonumber\end{aligned}$$ which, together with the fact that $M\geq 0$, implies the following upper bound of $\mathbb{E}M$ $$\begin{aligned} \mathbb{E} M & = \int_0^\infty \mathbb{P}(M > x) dx\nonumber\\ & \leq \bigg(\int_0^{ \eta_2 \sqrt{p(1 \vee \log(A_0 D_\Omega))} + 1} + \int_{ \eta_2 \sqrt{p(1 \vee \log(A_0 D_\Omega))} + 1}^\infty\bigg)\mathbb{P}(M > x) dx\nonumber\\ & \leq \eta_2 \sqrt{p(1 \vee \log(A_0 D_\Omega))} + 1 + \int_1^\infty \frac{2\exp\left\{ - x^2/(4 e^2) \right\}}{1 - \exp\left\{ - x^2/(4 e^2) \right\}} dx\nonumber\\ & \leq C_0 \sqrt{p(1 \vee \log(A_0 D_\Omega))}.\end{aligned}$$ To access the tail probability, we note that $M-\mathbb{E}M$ is also a Gaussian process with mean zero. Thus by Lemma \[thm211\], we have $$\begin{aligned} \mathbb{P}(M-\mathbb{E}M>t)\leq e^{-t^2/2\sigma^2_M},\end{aligned}$$ where $$\sigma^2_M = \sup_{x\in \Omega}\mathbb{E}\frac{(Z(x) - \mu(x))^2}{\sigma(x)^2\log (e\sigma/\sigma(x))} \leq 1.$$ Hence, we complete the proof. Independence in sequential Gaussian process modeling ==================================================== The proof of Theorem \[thm:sequential\] relies on certain independence properties of sequential Gaussian process modeling shown in Lemmas \[lemma:twosets\]-\[thm:indenpendence\]. First we introduce some notation. For an arbitrary function $f$, and $X=(x_1,...,x_n)$, define $f(X)=(f(x_1),...,f(x_n))^T$, and $$\begin{aligned} \label{interp} \mathcal{I}_{\Psi,X}f( x)=r^T(x) K^{-1} f(X),\end{aligned}$$ where $r=(\Psi(x-x_1),\ldots,\Psi(x-x_n))^T, K=(\Psi(x_j-x_k))_{j k}$. For notational convenience, we define $\mathcal{I}_{\Psi,\varnothing}f=0$. \[lemma:twosets\] Let $Z$ be a stationary Gaussian process with mean zero and correlation function $\Psi$. For two sets of scattered points $X'\subset X=(x_1,\ldots,x_n)$, we have $$\begin{aligned} \label{decomposition} Z-\mathcal{I}_{\Psi,X'}Z=(Z-\mathcal{I}_{\Psi,X}Z)+\mathcal{I}_{\Psi,X}(Z-\mathcal{I}_{\Psi,X'}Z).\end{aligned}$$ In addition, if $X$ and $X'$ are deterministic sets, then the residual $Z-\mathcal{I}_{\Psi,X}Z$ and the vector of observed data $(Z(x_1),\ldots,Z(x_n))^T$ are mutually independent Gaussian process and vector, respectively. It is easily seen that $\mathcal{I}_{\Psi,X}$ and $\mathcal{I}_{\Psi,X'}$ are linear operators and $\mathcal{I}_{\Psi,X'}\mathcal{I}_{\Psi,X}=\mathcal{I}_{\Psi,X}$, which implies (\[decomposition\]). The residual $Z-\mathcal{I}_{\Psi,X}Z$ is a Gaussian process because $\mathcal{I}_{\Psi,X}$ is linear. The independence between the Gaussian process and the vector can be proven by calculation the covariance $$\begin{aligned} &&\text{Cov}(Z(x')-\mathcal{I}_{\Psi,X'}Z(x'),Z(X))\\ &=&\text{Cov}(Z(x')-r^T(x') K^{-1} Z(X),Z(X))\\ &=& r(x')-r(x')=0,\end{aligned}$$ which completes the proof. \[thm:indenpendence\] For any instance algorithm of ABO, the following statements are true. 1. Conditional on $\mathcal{F}_{n-1}$ and $X_{n}$, the residual process $Z(\cdot)-\mu_n(\cdot)$ is independent of $\mathcal{F}_n$. 2. Conditional on $\mathcal{F}_{n}$, the residual process $Z(\cdot)-\mu_n(\cdot)$ is a Gaussian process with same law as $Z'(\cdot)-\mathcal{I}_{\Psi,X_{1:n}}Z'(\cdot)$, where $Z'$ is an independent copy of $Z$. We use induction on $n$. For $n=1$, the desired results are direct consequences of Lemma \[lemma:twosets\], because the design set is suppressed conditional on $\mathcal{F}_0$. Now suppose that we have proven already the assertion for $n$ and want to conclude it for $n+1$. First, we invoke the decomposition given by Lemma \[lemma:twosets\] to have $$\begin{aligned} \label{Zprime} Z'-\mathcal{I}_{\Psi,X_{1:n}}Z'=(Z'-\mathcal{I}_{\Psi,X_{1:(n+1)}}Z')+\mathcal{I}_{\Psi,X_{1:(n+1)}}(Z'-\mathcal{I}_{\Psi,X_{1:n}}Z').\end{aligned}$$ Because $\mu_n=\mathcal{I}_{\Psi,X_{1:n}}Z$, we also have $$\begin{aligned} \label{Zmu} Z-\mu_{n}=(Z-\mu_{n+1})+\mathcal{I}_{\Psi,X_{1:(n+1)}}(Z-\mu_n).\end{aligned}$$ By the inductive hypothesis, $Z-\mu_n$ has the same law as $Z'-\mathcal{I}_{\Psi,X_{1:n}}Z'$ conditional on $\mathcal{F}_{n}$, denoted by $Z-\mu_n{\stackrel{\mathclap{\mbox{d}}}{=}}Z'-\mathcal{I}_{\Psi,X_{1:n}}Z'|\mathcal{F}_n$. Our assumption that $X_{n+1}$ is independent of $(Z,Z')$ conditional on $\mathcal{F}_n$ implies that $X_{n+1}$ is independent of $(Z-\mu_n,Z'-\mathcal{I}_{\Psi,X_{1:n}}Z')$ as well. Thus, $$Z-\mu_n{\stackrel{\mathclap{\mbox{d}}}{=}}Z'-\mathcal{I}_{\Psi,X_{1:n}}Z'|\mathcal{F}_n,X_{n+1}.$$ Clearly, this equality in distribution is preserved by acting $\mathcal{I}_{\Psi,X_{1:(n+1)}}$ on both sides, which implies $$\left(Z-\mu_n,\mathcal{I}_{\Psi,X_{1:(n+1)}}(Z-\mu_n)\right){\stackrel{\mathclap{\mbox{d}}}{=}}\left(Z'-\mathcal{I}_{\Psi,X_{1:n}}Z',\mathcal{I}_{\Psi,X_{1:(n+1)}}(Z'-\mathcal{I}_{\Psi,X_{1:n}}Z')\right)|\mathcal{F}_n,X_{n+1}.$$ Incorporating the above equation with (\[Zprime\]) and (\[Zmu\]) yields $$\begin{aligned} \label{eqindist} \left(Z-\mu_{n+1},Z-\mu_n)\right){\stackrel{\mathclap{\mbox{d}}}{=}}\left(Z'-\mathcal{I}_{\Psi,X_{1:(n+1)}}Z',Z'-\mathcal{I}_{\Psi,X_{1:n}}Z'\right)|\mathcal{F}_n,X_{n+1}.\end{aligned}$$ Now we consider the vectors $V:=Z(X_{n+1})-\mu_n(X_{n+1})$ and $V'=Z'(X_{n+1})-\mathcal{I}_{\Psi,X_{1:n}}Z'(X_{n+1})$. Then (\[eqindist\]) implies $$\begin{aligned} \label{eqindist2} \left(Z-\mu_{n+1},V)\right){\stackrel{\mathclap{\mbox{d}}}{=}}\left(Z'-\mathcal{I}_{\Psi,X_{1:(n+1)}}Z',V'\right)|\mathcal{F}_n,X_{n+1}.\end{aligned}$$ Because $V'$ consists of observed data, we can apply Lemma \[lemma:twosets\] to obtain that, conditional on $\mathcal{F}_n$ and $X_{n+1}$, $Z'-\mathcal{I}_{\Psi,X_{1:(n+1)}}Z'$ is independent of $V'$, which, together with (\[eqindist2\]), implies that $Z-\mu_{n+1}$ and $V$ are independent conditional on $\mathcal{F}_n$ and $X_{n+1}$. Because $\mu_n(X_{n+1})$ is measurable with respect to the $\sigma$-algebra generated by $\mathcal{F}_n$ and $X_{n+1}$, we obtain that $Z-\mu_{n+1}$ is independent of $Z(X_{n+1})$ conditional on $\mathcal{F}_n$ and $X_{n+1}$, which proves Statement 1. Combining Statement 1 and (\[eqindist\]) yields Statement 2. Proof of Theorem \[thm:sequential\] =================================== The law of total probability implies $$\begin{aligned} &&\mathbb{P}(M_T-C \sqrt{p(1 \vee \log(A_0 D_\Omega))}>t)\\ &=&\sum_{i=n}^{\infty} \mathbb{P}(M_T-C \sqrt{p(1 \vee \log(A_0 D_\Omega))}>t|T=n)\mathbb{P}(T=n)\\&=&\sum_{n=1}^{\infty} \mathbb{P}(M_n-C \sqrt{p(1 \vee \log(A_0 D_\Omega))}>t|T=n)\mathbb{P}(T=n)\\ &=&\sum_{n=1}^{\infty} \mathbb{E}\left\{\mathbb{P}(M_n-C \sqrt{p(1 \vee \log(A_0 D_\Omega))}>t|\mathcal{F}_n)\Big|T=n\right\}\mathbb{P}(T=n),\end{aligned}$$ where the last equality follows from the fact that $\{T=n\}\in\mathcal{F}_n$, namely, $T$ is a stopping time. Clearly, the desired results are proven if we can show $\mathbb{P}(M_n-C \sqrt{p(1 \vee \log(A_0 D_\Omega))}>t|\mathcal{F}_n)<e^{-t^2/2}$. Now we resort to part 2 of Lemma \[thm:indenpendence\], which states that conditional on $\mathcal{F}_n$, $Z(\cdot)-\mu_n(\cdot)$ is identical in law to its independent copy $Z'(\cdot)-\mathcal{I}_{\Psi,X_{1:n}}Z'(\cdot)$. Although the event $\{M_n-C \sqrt{p(1 \vee \log(A_0 D_\Omega))}>t\}$ looks complicated, it is measurable with respect to $Z(\cdot)-\mu_n(\cdot)$. Thus, we arrive at $$\begin{aligned} &&\mathbb{P}(M_n-C \sqrt{p(1 \vee \log(A_0 D_\Omega))}>t|\mathcal{F}_n)\nonumber\\ &=&\mathbb{P}\left(\sup_{x\in \Omega}\frac{Z'(x) - \mathcal{I}_{\Phi,X_{1:n}}Z'(x)}{\sigma_n(x)\log^{1/2} (e \sigma/\sigma_n(x))}-C \sqrt{p(1 \vee \log(A_0 D_\Omega))}>t|\mathcal{F}_n\right).\label{condition}\end{aligned}$$ Because $Z'$ is independent of $Z$, the part of conditioning with respect to $Z(X_{1:n})$ in (\[condition\]) has no effect on $Z'$. The only thing that matters is the effect of the conditioning on the design points $X_{1:n}$. Hence, (\[condition\]) is reduced to $$\begin{aligned} \mathbb{P}\left(\sup_{x\in \Omega}\frac{Z'(x) - \mathcal{I}_{\Phi,X_{1:n}}Z'(x)}{\sigma_n(x)\log^{1/2} (e \sigma/\sigma_n(x))}-C \sqrt{p(1 \vee \log(A_0 D_\Omega))}>t|X_{1:n}\right).\label{probfixed}\end{aligned}$$ Clearly, we can regard the points $X_{1:n}$ in the formula above as a fixed design. Then the probability (\[probfixed\]) is bounded above by $e^{-t^2/2}$ as asserted by Corollary \[coro:practice\]. Proof of Theorem \[ourthmjasa\] {#App:jasaprof} =============================== This proof is similar to Theorem 1 of [@wang2019prediction] but with a few technical improvements. Because $\mu(x)$ is a linear combination of $Z(x_i)$’s, $\mu(x)$ is also a Gaussian process. The main idea of the proof is to invoke a maximum inequality for Gaussian processes, which states that the supremum of a Gaussian process is no more than a multiple of the integral of the covering number with respect to its natural distance $\mathfrak{d}$. See [@adler2009random; @van1996weak] for related discussions. Let $g(x) = Z(x)-\mu(x)$. For any $x,x'\in A$, because $A$ is deterministic, we have $$\begin{aligned} \mathfrak{d}(x, x')^2 = &\mathbb{E}(g(x)-g(x'))^2\\ = & \mathbb{E}(Z(x)-\mu(x)-(Z(x')-\mu(x')))^2 \\ = &\sigma^2(\Psi(x-x) - r^T(x)K^{-1}r(x) + \Psi(x'-x') - r^T(x')K^{-1}r(x')\\ & -2[\Psi(x-x')- r^T(x')K^{-1}r(x) ]),\end{aligned}$$ where $r(\cdot) = (\Psi(\cdot-x_1),...,\Psi(\cdot-x_n))^T$, $K = (\Psi(x_j-x_k))_{jk}$. The rest of our proof consists of the following steps. In step 1, we bound the covering number $N(\epsilon,A,\mathfrak{d})$. Next we bound the diameter $D$. In step 3, we obtain a bound for the entropy integral. In the last step, we invoke Lemmas \[thm133\] and \[thm211\] to obtain the desired results. **Step 1: Bounding the covering number** Let $h(\cdot) = \Psi(x-\cdot) - \Psi(x'-\cdot)$. It can verified that $$\begin{aligned} \begin{split} \mathfrak{d}(x, x')^2 = & - \sigma^2[h( x') - \mathcal{I}_{\Psi,X}h(x')]+\sigma^2[h(x) - \mathcal{I}_{\Psi,X}h( x)]. \end{split}\end{aligned}$$ By Theorem 11.4 of [@wendland2004scattered], $$\begin{aligned} \label{eq:dist2} \mathfrak{d}(x, x')^2 \leq 2 \sigma^2 (\sigma_A/\sigma\|h\|_{\mathcal{N}_\Psi(\mathbf{R}^d)}) = 2 \sigma \sigma_A\|h\|_{\mathcal{N}_\Psi(\mathbf{R}^d)},\end{aligned}$$ where $$\begin{aligned} \sigma_A^2 = \sup_{x\in A}\sigma(x)^2 = \sigma^2\sup _{x\in A} (\Psi(x-x) - r^T(x)K^{-1}r(x)).\end{aligned}$$ Denote the Euclidean norm by $\|\cdot\|$. Then, by the definition of the spectral density and the mean value theorem, we have $$\begin{aligned} \|h\|^2_{\mathcal{N}_\Psi(\mathbf{R}^d)} & = \Psi(x-x) - 2\Psi(x'-x) + \Psi(x'-x')\nonumber\\ & = 2\int_{\mathbf{R}^d}(1-\cos({(x-x')^T\omega}))\tilde \Psi(\bm\omega)d\bm\omega\nonumber\\ & \leq \bigg( 2\int_{\mathbf{R}^d}\|\omega\|\tilde \Psi(\bm\omega)d\bm\omega\bigg)\|x-x'\|\nonumber\\ &\leq 2A_0\|x-x'\|,\label{eq:entropyC1}\end{aligned}$$ where the last inequality follows from the fact that $\|\omega\|\leq \|\omega\|_1$. Combining (\[eq:dist2\]) and (\[eq:entropyC1\]) yields $$\begin{aligned} \label{eq:dist} \mathfrak{d}(x, x')^2 \leq 2A_0^{1/2} \sigma\sigma_A\|x-x'\|^{1/2}.\end{aligned}$$ Therefore, the covering number is bounded above by $$\begin{aligned} \label{e1} \log N(\epsilon,A,\mathfrak{d}) \leq \log N\bigg(\frac{\epsilon^{4}}{4A_0 \sigma^2\sigma_A^2},A,\|\cdot\|\bigg).\end{aligned}$$ The right side of (\[e1\]) involves the covering number of a Euclidean ball, which is well understood in the literature. See Lemma 4.1 of [@pollard1990empirical]. This result leads to the bound $$\begin{aligned} \label{eq:entropyBoundC1} \log N(\epsilon,A,\mathfrak{d})\leq p\log\bigg(\frac{48 A_0 D_A \sigma^2\sigma_A^2}{\epsilon^4} + 1\bigg) \leq p\log\bigg(\frac{48A_0 D_\Omega \sigma^2\sigma_A^2}{\epsilon^4} + 1\bigg),\end{aligned}$$ where $D_A = \text{diam}(A)$ and $D_\Omega = \text{diam}(\Omega)$ are the Euclidean diameter of $A$ and $\Omega$, respectively. **Step 2: Bounding the diameter $D$** Define the diameter under metric $\mathfrak{d}$ by $D = \sup_{x, x'\in A}\mathfrak{d}(x, x')$. For any $x,x'\in A$, $$\begin{aligned} \label{eq:distD1} \mathfrak{d}(x, x')^2 = &\mathbb{E}(g(x)-g(x'))^2 \leq 4\sup _{x\in A}\mathbb{E}(g(x))^2\nonumber\\ = & 4\sup _{x\in A}\mathbb{E}(Z(x)-\mathcal{I}_{\Psi,\mathbf{X}}Z(x))^2\nonumber\\ = & 4\sigma^2\sup _{x\in A} (\Psi(x-x) - r^T(x)K^{-1}r(x)) = 4 \sigma_A^2.\end{aligned}$$ Thus we conclude that $$\begin{aligned} \label{eq:diamC1} D\leq 2 \sigma_A.\end{aligned}$$ **Step 3: Bounding the entropy integral** By (\[eq:entropyBoundC1\]) and (\[eq:diamC1\]), $$\begin{aligned} \label{eq:boundEC1} \int_0^{D/2} \sqrt{\log N(\epsilon,A,\mathfrak{d})}d\epsilon & \leq \int_0^{\sigma_A} \sqrt{ p\log\bigg(\frac{48A_0 D_\Omega \sigma^2\sigma_A^{2}}{\epsilon^4} + 1\bigg)}d\epsilon\nonumber\\ & \leq \left(\int_0^{\sigma_A}d\epsilon \right)^{1/2}\left(\int_0^{\sigma_A} p\log\bigg(\frac{48A_0 D_\Omega \sigma^2\sigma_A^{2}}{\epsilon^4} + 1\bigg)d\epsilon\right)^{1/2}\nonumber\\ & = \left(\int_0^{\sigma_A}d\epsilon \right)^{1/2}\left(\sigma\int_0^{\sigma_A/\sigma} p\log\bigg(\frac{48A_0 D_\Omega \sigma_A^{2}}{u^4\sigma^2} + 1\bigg)du\right)^{1/2}\nonumber\\ & \leq \sigma_A^{1/2}\left(\sigma\int_0^{\sigma_A/\sigma} p\log\bigg(\frac{48A_0 D_\Omega \sigma_A^{2}}{u^4\sigma^2} + \frac{\sigma_A^{2}}{u^4\sigma^2}\bigg)du\right)^{1/2}\nonumber\\ & \leq \sqrt{2p} \sigma_A \sqrt{\log(e^2\sqrt{1 + 48A_0 D_\Omega}\sigma/\sigma_A))}\nonumber\\ & \leq \sqrt{4p} \sigma_A \sqrt{\log(e\sqrt{1 + 48A_0 D_\Omega})} \sqrt{\log(e\sigma/\sigma_A))}\nonumber\\ & \leq c\sqrt{p(1 \vee \log(A_0 D_\Omega))} \sigma_A \sqrt{\log(e\sigma/\sigma_A))},\end{aligned}$$ where $c = \sqrt{6\log(7e)}$. **Step 4: Bounding $\mathbb{P}(\sup_{x\in A} Z(x)-\mu(x) > \eta\int_0^{D/2} \sqrt{\log N(\epsilon,A,\mathfrak{d})}d\epsilon + u)$** By Lemmas \[thm133\] and \[thm211\], we have $$\begin{aligned} \label{BorellTIS} P\bigg(\sup_{x\in A} Z(x)-\mu(x) > \eta\int_0^{D/2} \sqrt{\log N(\epsilon,A,\mathfrak{d})}d\epsilon + t\bigg)\leq e^{-t^2/(2\sigma^2_A)}.\end{aligned}$$ By plugging (\[eq:boundEC1\]) into , we obtain the desired inequality with $\eta_1 = c\eta$, which completes the proof. Calibrating $C_0$ via simulation {#App:simu} ================================ An upper bound of the constant $C_0$ in Theorem \[thm:undersmooth\] can be obtained by examine the proof of Lemma \[thm133\] and Theorem \[ourthmjasa\]. However, this theoretical upper bound can be too large for practical use. In this section, we consider estimating $C_0$ via numerical simulation. According to Part 1 of Theorem \[thm:undersmooth\], $$C_0 \geq \mathbb{E}M/\sqrt{p(1 \vee \log(A_0 D_\Omega))},$$ where $M=\sup_{x\in \Omega}\frac{Z(x) - \mu(x)}{\sigma(x)\log^{1/2} (e \sigma/\sigma(x))}$, $A_0$ is as in , and $D_\Omega$ is the Euclidean diameter of $\Omega$. For a specific Gaussian process, $\mathbb{E}M/\sqrt{p(1 \vee \log(A_0 D_\Omega))}$ is a constant and can be obtained by Monte Carlo. Let $\mathcal{M}$ be the collection of Gaussian processes satisfying the conditions of Theorem \[thm:undersmooth\]. Then $$C_0 = \sup_{M \in\mathcal{M}}\mathbb{E}M/\sqrt{p(1 \vee \log(A_0 D_\Omega))} =: \sup_{M \in\mathcal{M}} H(M) .$$ The idea is to consider various Gaussian processes and find the maximum value of $\mathbb{E}M/\sqrt{p(1 \vee \log(A_0 D_\Omega))}$. This value can be close to $C_0$ when we cover a broad range of Gaussian processes. In the numerical studies, we consider $\Omega=[0,1]^p$ for $p=1,2,3$. We consider different $A_0$ values to get different $A_0D_\Omega$’s. In each Monte Carlo sampling, we approximate $M$ using $$M_1 = \sup_{x\in \Omega_1}\frac{Z(x) - \mu(x)}{\sigma(x)\log^{1/2} (e \sigma/\sigma(x))},$$ where $\Omega_1$ is the first $100, 1000, 2000$ points of the Halton sequence [@niederreiter1992random] for $p=1,2,3$, respectively. We calculate the average of $M_1/\sqrt{p(1 \vee \log(A_0 D_\Omega))}$ over all the simulated realizations of each Gaussian process. Specifically, We simulate $1000$ realizations of the Gaussian processes for $p=1$, $100$ realizations for $p=2,3$ and consider the following four cases. In Cases 1-3, we use maximin Latin hypercube designs [@santner2013design], and use independent samples from the uniform distribution in Case 4. **Case 1:** We consider $p=1$ with $20$ and $50$ design points. We consider the Gaussian correlation functions and Matérn correlations functions with $\nu = 1.5,2.5,3.5$. The results are presented in Table \[p1tab\]. **Case 2:** We consider $p=2$ with $20$, $50$, and $100$ design points. We consider the Gaussian correlation functions and product Matérn correlations functions with $\nu = 1.5,2.5,3.5$. The results are presented in Table \[p2tab\]. **Case 3:** We consider $p=3$ with $20$, $50$, $100$ and $500$ design points. We consider the product Matérn correlations functions with $\nu = 1.5,2.5,3.5$. The results are shown in Table \[p3table\]. **Case 4:** We consider $p=2$ with $20$, $50$, and $100$ design points. We consider the product Matérn correlations functions with $\nu = 1.5,2.5,3.5$. The results are shown in Table \[uniftab\]. design points $A_0 D_\Omega=1$ $A_0 D_\Omega=3$ $A_0 D_\Omega=5$ $A_0 D_\Omega=10$ $A_0 D_\Omega=25$ ------------- --------------- ------------------ ------------------ ------------------ ------------------- ------------------- -- -- Gaussian 20 0.11640290 0.1978563 0.2450737 0.4542654 0.859318 50 0.08102775 0.0916648 0.1206034 0.1683377 0.422786 $\nu = 1.5$ 20 0.9640650 1.065597 0.9537634 0.9429957 1.0197966 50 0.9442937 1.009187 0.8981430 0.8331926 0.8372607 $\nu = 2.5$ 20 0.7432965 0.8554707 0.7804686 0.8371662 1.0074204 50 0.7304104 0.8218710 0.7346077 0.6987832 0.7563067 $\nu = 3.5$ 20 0.6054239 0.7248086 0.6833789 0.7711124 0.9608837 50 0.3367513 0.6941391 0.6244660 0.6278185 0.6928741 : Simulation results of Case 1[]{data-label="p1tab"} design points $A_0 D_\Omega=1$ $A_0 D_\Omega=3$ $A_0 D_\Omega=5$ $A_0 D_\Omega=10$ $A_0 D_\Omega=25$ ------------- --------------- ------------------ ------------------ ------------------ ------------------- ------------------- -- -- Gaussian 20 0.2801128 0.4767259 0.5644628 0.7408401 1.0554507 50 0.1465512 0.2927036 0.3789438 0.5683807 0.9309326 100 0.1156139 0.1961319 0.2436626 0.4189444 0.7641615 $\nu = 1.5$ 20 0.8106718 0.9528429 0.8748865 0.9365989 1.0894451 50 0.8114071 0.9299506 0.8568070 0.8576984 0.9964256 100 0.8137517 0.9108342 0.8224467 0.7951887 0.9168643 $\nu = 2.5$ 20 0.6072854 0.7709362 0.7411921 0.8540687 1.0933120 50 0.6316136 0.7218077 0.7218077 0.7690956 0.9703693 100 0.5651732 0.6677120 0.6677120 0.7090934 0.8791792 $\nu = 3.5$ 20 0.5243251 0.6881401 0.6915576 0.8290974 1.0876019 50 0.3947094 0.6420423 0.6434791 0.7030224 0.9494486 100 0.2898865 0.6279639 0.6036111 0.6420049 0.8373886 : Simulation results of Case 2[]{data-label="p2tab"} Cases $H(M)$ -------------------------------------------------- -- ----------- 20 design points, $\nu=1.5$, $A_0 D_\Omega = 1$ 0.6977030 500 design points, $\nu=3.5$, $A_0 D_\Omega = 5$ 0.4961581 100 design points, $\nu=2.5$, $A_0 D_\Omega = 3$ 0.6628567 50 design points, $\nu=1.5$, $A_0 D_\Omega = 10$ 0.7632713 : Simulation results of Case 3[]{data-label="p3table"} Cases $H(M)$ --------------------------------------------------- -- ----------- 100 design points, $\nu=3$, $A_0 D_\Omega = 3$ 0.6778535 50 design points, $\nu=1.5$, $A_0 D_\Omega = 1$ 0.8144700 20 design points, $\nu=2.5$, $A_0 D_\Omega = 5$ 0.7735112 100 design points, $\nu=1.5$, $A_0 D_\Omega = 10$ 0.8164859 : Simulation results of Case 4[]{data-label="uniftab"} From Tables \[p1tab\]-\[uniftab\], we find the following patterns: - All numerical values ($H(M)$) in Tables \[p1tab\]-\[uniftab\] are less than 1.10. Only eight entries are greater than one. - In most scenarios, the obtained values are decreasing in $\nu$. This implies that $H(M)$ is smaller when $M$ is smoother. - $H(M)$ is not monotonic in $A_0D_\Omega$, which implies a more complicated function relationship between $H(M)$ and $A_0D_\Omega$. - In most scenarios, $H(M)$ decreases as the dimension $p$ increases. - The obtained values are decreasing in the number of design points. In summary, the largest $H(M)$ values are observed when the sample size is small, the smoothness is low and the dimension is low. Therefore, we believe that our simulation study covers the largest possible $H(M)$ values and our suggestion of choosing $C_0=1$ can be used in most practical situations. Numerical experiments {#App:numeg} ===================== We now compliment the technical details in the numerical experiments discussed in Section \[Sec:numexp\]. The naive 95% confidence upper bound is defined as the usual pointwise upper bound of Gaussian process, i.e., $$\begin{aligned} CI_G:=&\left[\max_{1\leq i\leq m_T}f(x_i), \max_{x\in \Omega}\mu_T(x) + q_{0.05}\sigma_T(x) \right],\end{aligned}$$ where $q_{0.05}$ is the 0.95 quantile of the standard normal distribution, $\mu_n(x)$ and $\sigma_n(x)$ are given in (\[seqmean\]) and (\[seqvariance\]), respectively. As suggested in Section \[Sec:K\], we use $C_0 = 1$ in $CI_t^{\mathbf{seq}}$. We consider the Matérn correlations functions with $\nu = 1.5,2.5,3.5$, and $A_0D_\Omega = 25$. We simulate Gaussian processes on $\Omega =[0,1]^2$ for each $\nu$. We use optimal Latin hypercube designs [@stocki2005method] to generate 5 initial points. We employs $a_{\text{UCB}}$ (defined in Section \[Sec:existing\]) as the acquisition function, in which the parameter $\beta_n$ is chosen as suggested by [@srinivas2009gaussian]. We repeat the above procedure 100 times to estimate the coverage rate by calculating the relative frequency of the event $f(x_{\max})\in CI_t^{\mathbf{seq}}$ or $f(x_{\max})\in CI_G$. In Figure \[figcr\], we plot the coverage rate and the width of $CI_t^{\mathbf{seq}}$ and $CI_G$ under 5, 10, 15, 20, 25, 30 iterations, respectively. [^1]: Two authors contributed equally to this work.
--- abstract: 'Retransmission based on packet acknowledgement (ACK/NAK) is a fundamental error control technique employed in IEEE 802.11-2007 unicast network. However the 802.11-2007 standard falls short of proposing a reliable MAC-level recovery protocol for multicast frames. In this paper we propose a latency and bandwidth efficient coding algorithm based on the principles of network coding for retransmitting lost packets in a single-hop wireless multicast network and demonstrate its effectiveness over previously proposed network coding based retransmission algorithms.' author: - | Jalaluddin Qureshi, Chuan Heng Foh and Jianfei Cai\ School of Computer Engineering\ Nanyang Technological University, Singapore\ {jala0001, aschfoh, asjfcai}@ntu.edu.sg title: An Efficient Network Coding based Retransmission Algorithm for Wireless Multicast --- Introduction {#sect:Introduction} ============ One-to-many (broadcast/multicast) transmission scheme is popular for many applications, and is widely implemented in Wireless Local Area Networks (WLANs) for its effectiveness in bandwidth consumption in a spectrum-limited wireless space. WLANs transmission is currently dictated by standards set out by IEEE 802.11-2007 [@ieee]. For one-to-one (unicast) wireless transmission, transmission reliability is achieved through Automatic Repeat Request (ARQ) variants or/and Forward Error Correction (FEC) schemes. Since broadcast is a special case of multicast, without loss of generality we will use the term multicast henceforth. However for a multicast, no consideration is made for ACK/NAK and RTS/CTS packet exchange in 802.11-2007 except for those frames sent with the To DS field set. Additionally, for multicast network where consideration for control packet is made, such packets are collected individually one-by-one, and so is the retransmission of the lost packets done, that is, one-by-one. As such for multicast network, the reliability problem is two-folded: 1) Efficient mechanism for the transmission of control packets (ACK/NAK, RTS/CTS), and 2) efficient retransmission of packets lost. As multicasting is gaining popularity for applications such as file distribution and multimedia conferencing, a more reliable scheme is needed for the fulfillment of future growth in multicast network. Motivated by promising applications of Network Coding (NC), recent works[@conext-start] -[@iccsc-end] have demonstrated the suitability of NC for retransmission of lost packets to improve bandwidth performance in a multicast network. Our algorithm is based on the concept of network coding [@nc], [@cope]. Network coding in its simplest form exploits the fact that rather than transmitting wireless packets individually to some receivers which may be ‘overheard’ by some other receivers already having those packets, and vice versa, it is often possible to combine those packets using bit-by-bit XOR (denoted by $\oplus$) and transmit it as a single coded packet, which can then be decoded by all (/most) of the receivers based on the packets they already have. For illustration consider that receiver $R_1$ has packet $c_1$ but not $c_2$, while $R_2$ has $c_2$ but not $c_1$. Rather than transmitting these two packets individually, the transmitter can encode $c_1$ and $c_2$ to generate $c_1\oplus c_2$, which is then multicast to both the receivers and decoded. The remaining paper is organized as follow: In Section \[sect:relatedwork\] we give an overview of related work, followed by the problem statement in Section \[sect:problem\]. Following that, we discuss previously proposed coding algorithm in Section \[sect:coding\] and our BENEFIT algorithm in Section \[sect:benefit\]. We then confirm the performance of BENEFIT with simulation results in Section \[sect:simulation\], and finally present conclusion in Section \[sect:conclusion\]. Related Work {#sect:relatedwork} ============ Packet retransmission based on network coding for a one-to-many, single-hop multicast network is a recent field of study, first proposed by D. Nguyen et al. [@netcod], which was later further elaborated into [@vt] by D. Nguyen et al. In [@vt] the authors demonstrate bandwidth effectiveness achieved by employing greedy network coding for retransmission over traditional ARQ schemes through simulation work. In [@conext-start] the authors follow up the work in [@netcod] by comparing various packet coding algorithms for packet retransmissions. While in [@majid], the authors presents an analytical work on the reliability performance of network coding compared with ARQ and FEC in a lossy network. Network Coded Piggy Back (NCPB) [@ncpb] demonstrates an efficient and practical testbed implemented random linear network coding based many-to-many reliable network model for real-time multi-player game network. Since our work primarily focuses on proposing an efficient network coding based retransmission algorithm for a one-to-many single-hop network, we will be comparing our results with the algorithm given in [@conext-start] which is the most closely related work. Our Contribution ---------------- The novelty of our work is the development of a computationally feasible network coding based retransmission algorithm whose gains are two-folded: 1) Our algorithm BENEFIT delivers better throughput with respect to the current best single-hop, NC based retransmission algorithm, and 2) we also demonstrate that our algorithm achieves minimum time to decode packets. None of the previous works [@conext-start] -[@iccsc-end] on NC based retransmission incorporates consideration of packet latency in their work. Here we will also show that it is no longer necessary to follow the packet coding rule [@cope], [@ncrouting] strictly. This relaxation in the coding rule has the potential for modification and development of other network coding based applications. Problem Statement {#sect:problem} ================= Consider a single-hop multicast network with $M$ fixed receiver stations $R_i$, ($i$ is the receiver station ID, $1\leq i\leq M$) with static membership and $M\geq 2$, and a single transmitting station $T_x$. Packet batch size is denoted by $N$. Packet reception at $R_i$ follows Bernoulli model, whereby a successful reception of packet $c_k$ ($k$ is the datagram packet ID, $1\leq k\leq N$) at $R_i$ is indicated by ‘0’ and packet loss by ‘1’ in the transmission matrix (see Table \[table:matrix\]). A transmission matrix is a 2-dimensional array table, where the rows represents $R_i$ and columns represents $c_k$. For a given packet, its packet utility $cu_k$ ($0\leq cu_k\leq M$) is defined as the number of receiver(s) which have not received the packet (i.e. the numbers of ‘1’s in a given column). While the receiver utility $ru_i$ indicates the number of packet(s) not received by the receiver $R_i$ from a given set of specified packet(s). For Bernoulli model, packet loss at all receivers is homogeneous, and is determined by a fixed loss probability $p_i$, which gives a packet successful reception probability of $1-p_i$. For a fixed batch size, $L_i$ denotes the number of lost packets for station $i$. $Q_j$ is the probability that after $N$ transmissions, the total number of packets lost is no more than $j$ ($1\leq j\leq N$). The time taken for one transmission is represented by one time slot. The *time to decode* a lost packet for a given $R_i$ is the total number of transmissions (original transmission, retransmission and transmission of coded packet) after which the lost packet is recovered by the given $R_i$. For simplicity we assume that there is a reliable control packet exchange mechanism in the network[^1] and that all coded/retransmitted packets are successfully received by the receivers. In the context of BENEFIT algorithm, a *benefit* value is generally defined as the number of ‘1’s in the transmission matrix which are converted to ‘0’s after the transmission of a coded packet or retransmission of the packet, hence the name of the algorithm: ‘BENEFIT’. Theoretical Numbers of Retransmissions -------------------------------------- The probability that $L_i\leq j$, for a single $R_i$ is given by $$P[L_i\leq j] = \sum_{c=0}^j (_{c}^N) p_{i}^c (1-p_i)^{N-c}.$$ The probability that all $M$ stations experience a packet loss rate no more than $j$ is given by $\prod_{i=1}^M P[L_i\leq j]$. Given this result, the probability that the total number retransmission is $j$, is given by $$Q_j = \prod_{i=1}^M P[L_i\leq j] - \prod_{i=1}^M P[L_i\leq j-1].$$ A more elaborative discussion of retransmission bandwidth for different transmission schemes compared with network coding is given in [@vt], [@majid]. Coding Algorithm {#sect:coding} ================ Previous coding algorithms (except random linear network coding, RLNC) were build on the foundation of a simple *packet coding rule* [@cope], [@ncrouting]: *For $T_x$ to transmit (/retransmit) $M$ packets $c_1$, ..., $c_M$ to $M$ receivers, $R_1$, ..., $R_M$ respectively, the coded packet obtained by coding $M$ packets $c_1$, ..., $c_M$ can only be decoded at $R_i$ if $R_i$ has ($M-1$) of $c_j$ packets, except $c_i$ ($j\neq i$).* We now discuss the major coding algorithm used in network coding literature. Greedy Network Coding --------------------- A greedy algorithm for coding packets has been traditionally used in several network coding based literature and still continues to be a dominant approach in many NC networks like IP-level routers in the Internet [@nc], wireless mesh network [@cope] and multi-hop wireless routing [@ncrouting]. A greedy network coding algorithm, like traditional greedy algorithm makes coding decisions which gives optimal local results. A greedy coding algorithm encodes current locally available packets iteratively as long as it can be decoded by all the intended receivers, without consideration whether its a ‘globally’ optimal solution or not. Random Linear Network Coding ---------------------------- RLNC [@ncpb] is a decentralized network coding approach, whereby the coded packet is given by $c_{coded}$ = $\sum_{k=1}^{k=N}$ $g(e) c_k$, where $g(e)$ is the global encoding vector, and is included in $c_{coded}$ as an overhead information in the packet header. Each of the receiver $R_i$ must successfully receive $N$ innovative packets (i.e. coded packets which are linearly independent of the previously received coded packet). Once the receivers have $N$ innovative packet, it can then decode $N$ packets using simple matrix inversion. Sort-by-Utility --------------- The coding algorithm which E. Rozner et. al. [@conext-start] proclaims to be the delivering the best performance for a one-hop multicast network is Sort-by-Utility. Therefore we will be comparing our BENEFIT algorithm with Sort-by-Utility for evaluation purposes. In a Sort-by-Utility coding algorithm, the $T_x$ first transmits $N$ packets, and then sorts the packets in descending order of their $cu_k$ values, using arrival time as tie-breaker for those packets have equal packet utility. Once the packets are sorted, the remaining operation of Sort-by-Utility is essentially a greedy coding algorithm, i.e. the $T_x$ then iteratively starts coding successive packets starting from packets having highest packet utilities and codes them with successive sorted packets as long as the coded packet can be decoded by *all* receivers. $R_i$/$c_k$ $ru_i$ $c_1$ $c_2$ $c_3$ $c_4$ $c_5$ ------------- -------- ------- ------- ------- ------- ------- $cu_k$ 10 2 3 1 2 2 $R_1$ 3 1 1 0 0 1 $R_2$ 2 0 1 0 1 0 $R_3$ 2 0 1 1 0 0 $R_4$ 3 1 0 0 1 1 : Transmission Matrix Example[]{data-label="table:matrix"} Consider as an example the matrix given in Table \[table:matrix\]. Sort-by-Utility algorithm after initial packet transmission ($c_1$ to $c_5$) would sort the transmitted packets based on its packet utilities $cu_k$, i.e. $c_2$, $c_1$, $c_4$, $c_5$, $c_3$. Then $T_x$ transmits the packets as follows: $c_2$, $c_1\oplus c_3$, $c_4$, $c_5$. Thus requiring a total of 4 retransmissions with an average *time to decode* a packet to be 4.4 time slots. BENEFIT Algorithm {#sect:benefit} ================= BENEFIT (Fig \[flowchart\]), unlike Sort-by-Utility does not need to wait until the end of the batch size ($N$ packet transmissions) before starting the retransmission process. It start transmitting coded packet once the prospective coding packets satisfy the following three conditions: $CodingBenefit()$, $ColumnsBenefit()$ and $CombinationBenefit()$ (see Table \[table:terms\]). Retransmitting as soon as the right conditions are met rather than wait till the end of the batch size in effect reduces the time to decode the packet. BENEFIT works on the basis that *it is not necessary for the coded packet to be decodable by all the receivers immediately*, assuming that the non-decodable coded packet can be decoded based on future transmission of coded packet(s). This principle is in essence the key strength of BENEFIT and thus, this way it contrasts the traditional *packet coding rule*. For the first scan cycle, to decide whether to transmit a packet $c_k$ or to scan the next packet (see the first step of Fig. \[flowchart\]) is decided based on the fact that if any previously transmitted packet has never been the first prospective coding packet (i.e. pros\_pks\[0\]) in that cycle, and the *current* value of $cu_k$ of that previously transmitted packet is $1\leq cu_k <M$, then the algorithm scan the next packet and stores it as the first prospective coding packet, else it transmits the next packet. For consecutive cycles, the algorithm only scans the packets. $CodingBenefit()$ ensures that packets are only coded, if the immediate benefit derived from such coding outweighs or equals the benefit derived from transmitting a single packet (uncoded) with minimum packet utility ($cu_k$) from the set of prospective coding packets and considered packet ($c_k$). $ColumnsBenefit()$ selects the most suitable (/fittest) packets for coding, by eliminating those packets which can not be decoded by at least one receiver STA immediately. If the packets satisfy $CodingBenefit()$ and $ColumnsBenefit()$ conditions but not $CombinationBenefit()$, then they are considered eligible prospective coding packets for combination with other packet(s), and thus the algorithm then searches (scans) for other packet(s), which in combination with the previous prospective coding packets will satisfy the three conditions for packet coding. If the algorithm reaches the end of the batch size and there are still ‘1’s in the transmission matrix, then the $CombinationBenefit()$ condition is relaxed by decrementing $DesiredBenefit$ and the algorithm then starts a new scan cycle (maximum of $M$-1 scan cycles) until the transmission matrix is composed of $M$\*$N$ ‘0’s. -------------------------------------------------------------------------------------- $CodingBenefit()$ Checks the following equality: $DecodeBenefit()\geq$ $MinimumBenefit()$ $DecodeBenefit()$ Finds out how many receivers STA will benefit *immediately* from the transmission of the coded packet, $0\leq DecodeBenefit()\leq M$. $MinimumBenefit()$ From the list of prospective coding packets, find $c_k$ with minimum current packet utility $cu_k$, $0\leq MinimumBenefit()\leq M$. $ColumnsBenefit()$ From the list of prospective coding packets, checks if every packet can be decoded by at least one receiver STA immediately. $CombinationBenefit()$ From the list of considered packets for coding, calculates the number of receiver STA which will benefit either *immediately* ($ru_i$=1) or in *future* ($ru_i\geq$ 2) from the coding of the prospective coding packets. And then checks if its equal to $DesiredBenefit$ (see Fig \[flowchart\]). $ru_i$ value in the context of $CombinationBenefit()$ is computed only for the packets in pros\_pks\[ \], $CombinationBenefit()\geq DecodeBenefit()$. $DecodeSearch()$ Decodes the arrived coded packet if it can. If the packet gets decoded, then search the memory for any previously non-decodable packet which can now be decoded based on the current decoded packet, and decode it. $Benefit$ $immediately$/ $Immediate$ $decoding$ The decoding of coded packet ‘on the spot,’ without the need for information from future transmission(s). -------------------------------------------------------------------------------------- : Definition of BENEFIT terms[]{data-label="table:terms"} Computational Complexity of BENEFIT ----------------------------------- The computational complexity of $CodingBenefit()$, $ColumnsBenefit()$ and $CombinationBenefit()$ all grow linearly with respect to the number of prospective coding packets and receivers. While $DecodeSearch()$ can be implemented using a binary search algorithm whose average complexity is logarithmic. Given that the number of prospective coding packets increases with the number of receivers, the computational complexity of BENEFIT can be considered to be linearly increasing with the number of receiver STAs. Illustrative example - BENEFIT ------------------------------ Consider the algorithm given in Fig. \[flowchart\], illustrated with the transmission matrix given in Table \[table:matrix\]. After the $T_x$ transmit $c_1$, $c_1$ is stored as the first prospective coding packet. $T_x$ then transmit $c_2$, $c_1$ and $c_2$ are then checked for $CodingBenefit()$ and $ColumnsBenefit()$ conditions, which they satisfy. Hence they are then checked for $CombinationBenefit()$ condition. Since both $c_1$ and $c_2$ satisfy the $CombinationBenefit()$ condition as well, the packets are coded and transmitted. Only $R_1$ is not able to decode $c_1\oplus c_2$ immediately (current value of $cu_1$=$cu_2$=1). The algorithm then scan the next packet $c_2$ and stores it as the first prospective coding packet, and the $T_x$ then transmit $c_3$. Since $c_2$ and $c_3$ satisfy $CodingBenefit()$ and $ColumnsBenefit()$ conditions but not $CombinationBenefit()$ condition, $c_3$ is therefore saved as a prospective coding packet. The $T_x$ then transmits $c_4$, which in addition to $c_2$ and $c_3$ satisfy $CombinationBenefit()$ condition. Hence the packets are coded and transmitted. All receivers are able to immediately benefit from the transmission of $c_2\oplus c_3\oplus c_4$. $DecodeSearch()$ function at $R_1$ after decoding $c_2\oplus c_3\oplus c_4$, decodes $c_1\oplus c_2$ using $c_2$ and obtains $c_1$. The last packet in the batch $c_5$ is then transmitted and stored as prospective coding packet, however as it is obvious, there is not any possibility of finding coding packets for $c_5$, the algorithm decrements the value of $DesiredBenefit$ twice, following which $c_5$ is retransmitted without any encoding in the third scan cycle. This example illustrates that BENEFIT requires a total of only 3 retransmissions ($c_1\oplus c_2$, $c_2\oplus c_3\oplus c_4$ and $c_5$) in contrast to 4 retransmissions used by Sort-by-Utility (see section \[sect:coding\]-C), with an average *time to decode* a packet of 1.9 time slots ($c_1\oplus c_2$ transmitted after the transmission of $c_2$, and $c_2\oplus c_3\oplus c_4$ transmitted after the transmission of $c_4$) in contrast to 4.4 time slots used by Sort-by-Utility. Hence for this example, it has been shown, that BENEFIT outperforms Sort-by-Utility both in terms of retransmission bandwidth and packet delay. Simulation Results {#sect:simulation} ================== We construct a C++ based discrete time simulator, using Random Number Generator to generate transmission table like the one given in Table \[table:matrix\]. The characteristics of the network shall be the same as mentioned in Section \[sect:problem\]. For each set of values, the simulation is repeated 1000 times. For performance evaluation, we use *retransmission ratio* (also used in [@conext-start]) which is defined as the total number of retransmissions using coding algorithm divided by the total number of retransmissions using traditional 802.11 retransmission scheme. Theory in Fig. \[fig:1\] and \[fig:2\] refers to retransmission ratio obtained by dividing $Q_j$ (derived in Section \[sect:problem\]-A) with the total number of retransmissions using traditional 802.11 retransmission scheme. Figure \[fig:1\] shows that BENEFIT consistently performs better than Sort-by-Utility. The initial trough in the graph is because of more coding opportunity available with an increase in number of STA. A simple heuristic explanation for this is that for 2 STA, there is scope for only 2 packets to be coded, however for 4 STA, there is scope for 2, 3 and 4 packets to be coded together based on opportunities. However as the number of STA increases further, retransmission ratio starts increasing as then, increase in conflict opportunities (as shown in Fig. \[fig:3\], as the number of STA increases, the time to search for prospective coding packet also increases) between coding packets outweighs increase in coding opportunities. Figure \[fig:2\] shows the performance of BENEFIT over a range of loss probability values. Figure \[fig:1\] and \[fig:2\] proves that for a small-medium network BENEFIT performs close to the theoretical bound for all ranges of $p_i$. While the bandwidth performance of BENEFIT and Sort-by-Utility is almost similar for low loss probability and/or small network, for such networks BENEFIT can still be useful for real-time applications which are highly delay sensitive. As Fig. \[fig:3\] shows that even for a small batch size, the average time BENEFIT takes to decode/retransmit a packet is far less than that of Sort-by-Utility. BENEFIT latency efficiency can be improved further by decrementing the initial value of $DesiredBeneft$, which will relax the $CombinationBenefit()$ condition and thus require the $T_x$ to spend less time searching for suitable coding packet. Decreasing the batch size also reduces packet retransmission delay as show in [@conext-start], [@vt]. However both these techniques will come at a tradeoff cost of an increase in retransmission ratio. Flexibility to balance throughput-delay tradeoff in BENEFIT allows the network designer to modify the algorithm based on the network requirements. Figure \[fig:4\] shows that the average time to decode packet gradually increases with $p_i$ for BENEFIT, however as $p_i$ crosses 0.8 the average time to decode packet starts decreasing as most of the packets satisfy $cu_k==DesiredBenefit$ condition and hence are retransmitted without any coding. The time saved searching for prospective coding packets reduces the average time to decode. This also explain an increase in standard deviation for BENEFIT in Fig. \[fig:4\], as some packets are retransmitted without any encoding (shorter waiting time), while other packets need to wait for longer to find suitable coding partners. Conclusion {#sect:conclusion} ========== In this paper we have demonstrated a computationally feasible, bandwidth and latency efficient retransmission coding algorithm, which doesn’t strictly follows the traditional coding rule. Selectively modifying the BENEFIT conditions would also allow the network designer to adjust the algorithm as per the throughput-delay requirements of the network. We believe that there is potential for research work to exploit relaxation in the coding rule, and study modifications to mechanisms like COPE [@cope] based on rules derived from BENEFIT. [1]{} Wireless LAN Medium Access Control (MAC) and Physical Layer (PHY) Specifications, June 2007, IEEE 802.11-2007. E. Rozner, A. Padmanabha, Y. Mehta, L. Qiu, M. Jafry, “ER: Efficient Retransmission Scheme For Wireless LANs” In ACM CoNEXT 2007, New York City, USA, Dec 2007. D. Nguyen, T. Nguyen, B. Bose, “Wireless Broadcast Using Network Coding,” In NetCod 2007 Workshop, San Diego, USA, Jan 2007. D. Nguyen, T. Tran, T. Nguyen, B. Bose,“Wireless Broadcast Using Network Coding,” In IEEE Transactions on Vehicular Technology, Vol 58, Issue 2, p 914 - 925, Feb. 2009. X. Xiao, Y. Lu-Ming, W. Wei-Ping, Z. 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Rayanchu, S. Banerjee, “An Analysis of Wireless Coding for Unicast Sessions: The Case for Coding-Aware Routing,” In IEEE INFOCOM 2007, Anchorage, USA, p 1028 - 1036, May 2007. [^1]: Control packets in multicast network can be implemented by designing ACK packets from multiple STA such that, upon reception of these simultaneously transmitted ACK packets, the original sender is able to efficiently decode the packet which is the superimposition of all ACK packets and infer which receiver STA have received the datagram packet [@epfl].
--- author: - | Hao Peng, Jianxin Li*, Member, IEEE*, Qiran Gong, Senzhang Wang, Lifang He,\ Bo Li, Lihong Wang, and Philip S. Yu*, Fellow, IEEE*, [^1] bibliography: - 'newref.bib' title: 'Hierarchical Taxonomy-Aware and Attentional Graph Capsule RCNNs for Large-Scale Multi-Label Text Classification' --- Introduction ============ As a fundamental text mining task, text classification aims to assign a text with one or several category labels such as topic labels and sentiment labels. Traditional approaches represent the text as sparse lexical features due to the simplicity and effectiveness  [@Aggarwal2012]. For example, bag-of-words and n-gram are widely used to extract textual features, and then a general machine learning model such as Bayesian, logistic regression or SVM is utilized for text classification. With the development of deep learning techniques  [@Lecun2015Deep; @goodfellow2016deep], variants of neural network based models have been exploited from a large body of innovations, such as recurrent neural networks [@Tai2015Improved; @Wang2016Attention; @Yang2017Hierarchical; @shen2018biblosan], diversified convolutional neural networks [@Kim2014Convolutional; @YaoGCN2018; @Liu:2017:DLE:3077136.3080834; @Peng:2018; @Conneau2016Very], capsule neural networks [@Zhao2018Investigating] and adversarial structures  [@Liu2017Adversarial; @miyato2016adversarial]. These deep models have achieved inspiring performance gains on text classification due to their powerful capacity in representing the text as a fix-size feature map with rich semantics information. Recently, three popular deep learning architectures have attracted increasing research attention for text data, i.e., recurrent neural networks (RNNs) [@Yang2017Hierarchical; @TangQL15; @shen2018biblosan; @shen2018disan], convolutional neural networks (CNNs) [@Kim2014Convolutional; @Conneau2016Very; @Liu:2017:DLE:3077136.3080834] and graph convolutional networks (GCNs) [@Peng:2018; @YaoGCN2018]. RNNs are more powerful on capturing the semantics of short text [@bengio2003neural], but are less effective to learn semantic features of long documents. Although the bi-directional block self-attention networks are proposed [@shen2018biblosan] to better model text or sentence, they consider documents as natural sequences of words, and ignore the long-distance semantic between paragraphs or sentences. CNNs simply evaluate the semantic composition of the consecutive words extracted with n-gram, while n-gram may lose the long-distance semantic dependency among the words [@Aggarwal2012]. Compared with RNNs and CNNs, GCNs can better capture the non-consecutive phrases and long-distance word dependency semantics [@Peng:2018; @YaoGCN2018], but ignore the sequential information. To sum up, there still lacks of a model that can simultaneously capture the non-consecutive, long-distance and sequential semantics of text. Meanwhile, as the text labels of some real-world text classification tasks are characterized by large hierarchies, there may exist strong dependency among the class labels [@sun2001hierarchical; @xue2008deep; @Gopal2012Bayesian]. Existing deep learning models cannot effectively and efficiently leverage the hierarchical dependencies among labels for improving the classification performance, either. It is non-trivial to obtain a desirable classification performance for large-scale multi-label text due to the following major challenges. First, although there are many methods for document modeling, how to represent a document by fully preserving its rich and complex semantic information still remains an open problem [@Berry:2003:STM:945832]. It is challenging to come up with a document modeling method that can fully capture the semantics of a document, including the non-consecutive, long-distance and sequential semantics of the words. Second, existing CNNs, RNNs and GCNs models usually can only capture partial textual features. It is challenging to design a deep learning model that can simultaneously capture multiple types of textual features mentioned above. Third, although some recursive regularization based hierarchical text classification models  [@Gopal:2015:HBI:2737800.2629585; @Gopal2013Recursive; @Peng:2018; @xie2013multilabel] consider the pair-wise relation between labels, they fail to consider their hierarchical relations. In addition, the computation of the above regularized models is expensive due to the use of Euclidean constraints. How to make full use of the hierarchical label-dependencies among labels to improve the classification accuracy and reduce the computational complexity is also challenging. To address the above challenges, we propose a novel ierarchical taxonomy-awar and ttentional raph apsule ecurrent s framework called HE-AGCRCNN for large-scale multi-label text classification. Specifically, our framework contains three major parts: word order preserved graph-of-words for document modeling, attentional capsule recurrent CNNs for features learning, and hierarchical taxonomy-aware weighted margin loss for multi-label text classification. Next we will elaborate the three parts as follows. **Word Order Preserved Graph-of-Words for Document Modeling.** We regard each word as a vertex, the word co-occurrence relationships within a sliding window as edges, and the positional index of a word appearing in the document as its attribute. In this way, we build a word order preserved graph-of-words to represent a document. Then we select top $N$ central words from the graph-of-words based on the *closeness centrality*, and construct a subgraph for each central word from neighbors by breadth first search (BFS) and depth first search (DFS). To preserve local sequential, non-consecutive and long-distance semantics, we next normalize each subgraph to blocks of word sequences that retains local word order information by utilizing the attribute of the vertex, and construct an arranged words-matrix for the $N$ sub-graphs. To incorporate more semantic information, we use a pre-trained word embedding vectors based on word2vec  [@Mikolov:2013:DRW:2999792.2999959; @Mikolov2013Efficient] as word representation in the arranged words-matrix. Finally, each document is represented as a corresponding 3-D tensor whose three dimensions are the selected central words, the ordered neighbor words sequence, and the embedding vector of each word, respectively. **Attentional Capsule Recurrent Convolutional Neural Networks.** An attentional capsule recurrent CNN (RCNN) model is designed to make use of the document tensor as input for document features learning. The proposal model first uses two attentional RCNN layers to learn different levels of text features with both non-consecutive, long-distance and local sequential semantics. Here, we not only guarantee the independence of the feature representation between sub-graphs, but also model different impacts among different blocks of word sequences. When the convolution kernel slides horizontally along the combining long-distance and local sequential ordering of words, the attentional RNN unit is used to encode the output of the previous step of CNN, and the output of current step of attentional RNN to produce the final output feature map in the RCNNs layer. Then a capsule network layer is used to implement an iterative routing process to learn the intrinsic spatial relationship between text features from lower to higher levels for each sub-graph. In the final DigitCaps layer, the activity vector of each capsule indicates the presence of an instance of each class and is used to calculate the classification loss. **Hierarchical Taxonomy-Aware Weighted Margin Loss.** Considering the hierarchical taxonomy of the labels, we design two types of meta-paths, and use them to conduct random walk on the hierarchical label taxonomy network to generate label sequences. Therefore, the hierarchical taxonomy relation among the labels can be encoded in a continuous vector space with the skip-gram [@Mikolov2013Efficient] on the sequences. In this way, the distance between two labels can be measured by calculating the cosine similarity of their label vectors. By taking the distance between labels into consideration, we design a new weighted margin loss to guide the training of proposed models in multi-label text classification. We conduct extensive evaluations on our proposed framework by comparing it with state-of-the-art methods on three benchmark datasets, comparing with traditional shallow models and recent deep learning models. The results show that our approach outperforms them by a large margin in both efficiency and effectiveness on large-scale multi-label text classification. The contributions of this paper are summarized below. - A novel hierarchical taxonomy-aware and attentional graph capsule recurrent CNNs framework is proposed for large-scale multi-label text classification. - A new word order preserved graph-of-words method is proposed to better model document and more effectively extract textual features. The new document modeling method preserves both non-consecutive, long-distance and local sequential semantics. - A new word sequence block level attention recurrent neural network is proposed to better learn local sequential semantics of text. - A novel hierarchical taxonomy-aware weighted margin loss is proposed to better measure the distance of classes in hierarchy and guide the proposed models training. - Extensive evaluations on three benchmark datasets demonstrate the efficiency and effectiveness of the proposal. ![image](Text-graph){width="90.00000%"} The rest of the paper is organized as follows. We first introduce the word order preserved graph-of-words based document modeling in Section \[sec:graphfordoc\]. Then we present the model architecture in Sections \[sec:dgcrcnn\] and  \[sec:labelembedding\]. The evaluation is conducted in Section \[sec:experi\]. Finally, we review related work in Section \[sec:relatedwork\] followed by the conclusion and future work in Section \[sec:conclu\]. Word order Preserved graph-of-words for Document Modeling {#sec:graphfordoc} ========================================================= In this section, we introduce how we model a document as a word order preserved graph-of-words, and how to extract central words and sub-graphs from it to preserve both non-consecutive, long-distance and local sequential semantics of the document. Formally, we denote the training document set as $\mathcal{D} = \{d_{s} ,T_{s}\}_{s=1}^{M}$, where $M$ is the total number of documents in $\mathcal{D}$, $d_{s}$ is a document, $T_{s}$ is the label set of $d_{s}$ and $T_{s} \subset \mathcal{S}$. We also denote the set of labels as $\mathcal{S} = \{v_{i}|i=1, 2, \cdots, L\}$, where $L$ is the total number of labels. Word Order Preserved Graph-of-Words Construction ------------------------------------------------ In order to preserve more semantic information of text, we model a document as a word order preserved graph-of-words. We regard each word as a vertex, the word co-occurrence relationships within a sliding window as edges, and the positional index appearing in the document as its attribute, as shown in the step 1 of Figure \[fig:text2norma\]. We first split a document into a set of sentences and extract tokens using Stanford CoreNLP tool[^2]. We also employ a lemmatization of each token using Stanford CoreNLP, and remove the stop words. Then we construct an edge between two word nodes if they co-occur in a pre-defined fixed-size sliding window, and the weight of the edge is the times of their co-occurrence. Meanwhile, we record all the positional indexes where a word appears in the document as its attribute. For example, for the first sentence “*Musk told the electric car company that...*” shown in the document of Figure  \[fig:text2norma\], we perform lemmatization on the second word “*told*” to get “*tell*” with attribute “2”, and build a directed edge from “*Musk*” to each of the words in the sliding window. As shown in the word order preserved graph-of-words of Figure  \[fig:text2norma\], the word “*Company*” appears at the $5$-th, $19$-th, $35$-th, $55$-th, $99$-th, etc. positions, respectively. Note that the word order preserved graph-of-words is a weighted directed graph with the positional indexes as the node attributes. For example, in the word order preserved graph-of-words of Figure  \[fig:text2norma\], the weight of the edge between nodes “*Company*” and “*Car*” is $6$ meaning that “*Company*” and “*Car*” has a total of 6 co-occurrences in the sliding window. Arranged Words-Matrix Generation -------------------------------- We denote the word order preserved graph-of-words as $\mathcal{G} = (V, E, W, A)$, where $V$ denotes the node set and $|V| = n$, $E$ denotes the edge set and $|E| = m$, $W$ denotes the weights of the edges and $A$ denotes the attributes of the nodes. We extract top $N$ central words from $\mathcal{G}$ based on node’s *closeness centrality* feature. Here, in order to calculate *closeness centrality* for each node, we use $d(v, u)$ to denote the shortest-path distance between nodes $v$ and $u$ by using the Dijkstra algorithm. For each node $v$, its *closeness centrality* can be calculated by $C_{v} ={(n-1)}/{\sum_{u\in{V}, u\neq v}d(v,u)}$. So we can arrange the nodes in order of largest to smallest according to their *closeness centrality* features. The larger the *closeness centrality*, more important the node is in the graph. As shown in the word order preserved graph-of-words of Figure  \[fig:text2norma\], the *closeness centrality* of word “*Company*” is the highest $0.3714$ among the words’ in the graph. Then we select the top $N$ central nodes from the node “*Company*” to the node “*Open*”, as shown in the step 2 of Figure \[fig:text2norma\]. Next, we introduce how to extract sub-graph $\mathcal{G}(v)$ for each selected central node $v$. ![image](Deep_Graph_Capsule_RCNNs){width="90.00000%"} First, we extract the nodes and edges from the neighborhood of each central node in the order of breadth first search (BFS), depth first search (DFS) and the node’s *closeness centrality* feature to build a subgraph. Meanwhile, we limit the number of nodes in the subgraph to be no more than $K$, as shown in the step 3 of Figure \[fig:text2norma\]. In this way, the sub-graph $\mathcal{G}(v)$ contains both the non-consecutive, long-distance and local sequential information of the central word $v$ in the document. To further save the above information of a subgraph, we order the words in the sub-graph $\mathcal{G}(v)$ by their nodes (words) attributes. For the most case where a subgraph contains multiple sentences, we guarantee that the long sentences are in the front and the short sentences are in the back. As shown in the first line of the arranged words-matrix in Figure \[fig:text2norma\], we convert the first sub-graph $\mathcal{G}(v)$ as sequences liking“*electric car company plan purchase million common* and *company expect* ”. As a result, we normalize each subgraph as sequences of nodes (words) that keep the same length $T$. If the number of words in the sequence is less than $T$, it is padded with zeros. Finally, we concatenate all the normalized sequences of the $N$ central words into an arranged words-matrix, as shown in step 4 of Figure \[fig:text2norma\]. The red nodes represent central words. Unified Representation of the Documents --------------------------------------- For better representing the original words in the words-matrix, we use word2vec [@Mikolov2013Efficient; @Mikolov:2013:DRW:2999792.2999959] to incorporate as much word semantic information as possible. Specifically, word2vec is trained on a larger corpus, i.e., Wikipedia. All parameters for word2vec are set to be default values. In this way, we have a 3-D tensor representation for each document, where the padded vectors are zero vectors with the same dimension. Then the convolution, recurrent and capsule networks introduced in the next section will be operated over the unified representations of the documents. Attentional Capsule Recurrent CNN {#sec:dgcrcnn} ================================= In this section, we introduce the proposed attentional capsule recurrent CNN model. After converting each document into a 3-D tensor representation, we design a three layers of attentional capsule recurrent CNN model to learn both the non-consecutive, long-distance and local sequential feature. From the input document to the output labels, the architecture is shown in Figure \[fig:deepcrcnn\]. Specifically, the three layers of neural networks contain two major parts: two layers of attentional recurrent convolution neural networks and one layer of capsule networks for rich feature learning. Note that this is a general framework and the number of attentional recurrent convolution layers can be adjusted based on specific dataset for classification, and the parameter configuration of self-attentional recurrent operators and capsule networks can be customized in different text classification tasks. Attentional Recurrent CNN ------------------------- Different from the architecture of existing recurrent convolutional neural networks [@Lai:2015:RCN:2886521.2886636], which encode sentences or document as a dense vector for classification, our proposed attentional recurrent convolution neural networks encode whole document as 3-D feature map. The attentional recurrent CNN model takes the $N\times T \times D$ size of 3-D tensor extracted from the document and word embedding as the input, where $N$ is the number of central words, $T$ is the length of normalized sequence of words and $D$ is the dimension of word embedding, as shown in Figure \[fig:deepcrcnn\]. The output of the two layers of attentional recurrent convolution networks is the other 3-D feature map as input of the proposed capsule network. In the first layer, the convolution operator filters the input tensor with $k1$ kernels of size $1\times 3 \times D$ with a horizontal stride of 2 elements and a vertical stride of 1 element, which is illustrated with the black convolution slide direction arrow in Figure \[fig:deepcrcnn\]. We use *ReLU* as the activation function to speed up the training process and avoid over-fitting. Here, convolution kernel serves as a composition of the semantics in the receptive field to extract the higher level semantic features. Meanwhile, we employ a masked attentional recurrent neural operator to capture the local sequential semantic for each sub-graph $\mathcal{G}(v)$. The attentional recurrent neural operator acts on each horizontal words sequence, which is illustrated with the red recurrent slide direction arrow in Figure \[fig:deepcrcnn\]. However, as we know, we convert the subgraph $\mathcal{G}(v)$ into blocks of word sequences according to the properties of the nodes. We give three different blocks, as shown in Figure \[fig:blocks\], to illustrate the blocks of word sequences, which consist of each line of the arranged words-matrix. In order to measure the different impacts of different number of blocks on the local sequential semantic learning, we customize the masked attentional parameter shared long short-term memory, namely Attention-LSTM, to learn the rich local sequential semantic for each sub-graph $\mathcal{G}(v)$. Since our proposed framework needs to learn feature for multiple documents, the attention-LSTM module guarantees that any subgraphs with the same order and number of blocks share the same attention parameters. For example, for the $T$-th subgraph $\mathcal{G}(u)$ from the document $d_i$, assuming that it contains the $q$ blocks of word sequences, the parameter of the masked attention module is the $\alpha_{B_{T, q, 1}},\alpha_{B_{T, q, 2}}, \dots, \alpha_{B_{T, q, q}}$. However, for the $T$-th subgraph $\mathcal{G}(v)$ from the document $d_j$, assuming that it contains the $p$ blocks of word sequences, the parameter of the masked attention module is the $\alpha_{B_{T, p, 1}},\alpha_{B_{T, p, 2}}, \dots, \alpha_{B_{T, p, p}}$. But, among any one block, each word shares the same attention parameter. For example, in Figure \[fig:blocks\], we assume they are converted from the top 3 subgraphs. In the $1$-th block of the $3$-th subgraph, the words *electric*, *car*, *company*, *plan*, and *purchase* share the same masked attention parameter $\alpha_{B_{3,2,1}}$. Then, after the first attentional recurrent convolution layer, there is an $N\times (T-2) \times k1$ size of feature map. Compared with traditional convolution and recurrent networks on text data  [@Kim2014Convolutional; @Conneau2016Very; @Liu:2017:DLE:3077136.3080834; @Peng:2018; @Lai:2015:RCN:2886521.2886636; @shen2018biblosan], the significant difference of our designed attentional recurrent CNN units is that it can integrate the long-distance, non-consecutive and local sequential semantics of the corresponding sub-graph $\mathcal{G}(v)$. ![Illustration of blocks of word sequences. The red word refers to the central word for each subgraph. The words in any block are the contexts of the central word at different locations in a document.[]{data-label="fig:blocks"}](Block-sentences){width="50.00000%"} The second attentional recurrent convolution layer takes the output of the first attentional recurrent convolution as its input, and filters it with $k2$ kernels of size $1\times 3 \times k1$ with a horizontal stride of 2 elements and a vertical stride of 1 element, which are illustrated with the black convolution slide direction arrow and red recurrent slide direction arrow in Figure \[fig:deepcrcnn\]. We still guarantee that each horizontal feature map characterizes the semantics of corresponding sub-graph $\mathcal{G}(v)$, and the attentional recurrent and convolution operators between different sub-graphs are independent. In the second layer, for each sub-graph $\mathcal{G}(v)$, the number of attentional parameter is same with the first layer, but they are separated in training. After the second attentional recurrent convolution layer’s operation, a $N\times (T-4)\times k2$ size of feature map is generated. More formally, we we give the definitions of convolution operator and Attentional-LSTM unit, respectively. The convolution operator can be defined as $$\label{eq:cnn_operators} \begin{aligned} \centering x_{j}^{l} = f(\sum_{i\in M_{j}}x_{i}^{l-1}\cdot k_{ij}^{l} + b_{j}^{l}), \end{aligned}$$ where $x_{j}^{l}$ represents the $j$-th feature map of the $l$-th layer of the convolution network, and $l\in\{1,2\}$. This formula shows the convolution operation and the summation for all the associated feature maps $x_{i}^{l-l}$ and the $j$-th convolution kernel $k_{ij}^{l}$ of layer $l$, and then add an offset parameter $b_{j}^{l}$. Finally, a *ReLU* activation function $f$ is applied. Meanwhile, the Attentional-LSTM unit can be defined as: $$\label{eq:rnn_operators} \begin{aligned} \centering f_{t} =& \sigma_{g}(W_{f}\alpha_{B}x_{t} + U_{f}c_{t-1} + b_{f}),\\ i_{t} =& \sigma_{g}(W_{i}\alpha_{B}x_{t} + U_{i}c_{t-1} + b_{i}),\\ o_{t} =& \sigma_{g}(W_{o}\alpha_{B}x_{t} + U_{o}c_{t-1} + b_{o}),\\ c_{t} =& f_{t}c_{t-1} + i_{t}\sigma_{c}(W_{c}\alpha_{B}x_{t} + b_{c}),\\ h_{t} =& o_{t}\sigma_{h}(c_{t}), \end{aligned}$$ where $t$ refers to the index of the horizontal convolution sequence, $f_{t}$ refers to the forgotten gate, $i_{t}$ refers to the input gate, $o_{t}$ refers to the output gate, $c_{t}$ is the cell state, and $\alpha_{B}$ refers to the attentional parameter. Since the output of the convolution network is input to the LSTM, and the output of the LSTM is the feature map, $x_{t} = x_{j}^{l}$ and $x_{t+1} = x_{j+1}^{l}$. Capsule Networks with Dynamic Routing ------------------------------------- Since the capsule network can effectively learn some aspect features of textual representation [@xiao2018mcapsnet], the output of the two layers of attentional recurrent convolution networks is $N\times (T-4)\times k2$ size of feature map and is input to the next capsule networks with dynamic routing layer. In order to independently learn the features of each subgraph into the corresponding capsule vectors, different from existing textual capsule networks [@Zhao2018Investigating], our proposed capsule networks guarantee the independence of feature between sub-graphs, as shown in Figure \[fig:deepcrcnn\]. The capsules contain groups of locally invariant neurons that learn to recognize the presence of features and encode their properties into vector outputs, with the vector length representing the presence of the features. The primary capsule layer is a convolution capsule layer with $M$ channels of capsules, as shown in Figure \[fig:deepcrcnn\]. Each primary capsule contains $m$ convolution units with a $\frac{T+12}{9}\times k2$ size of kernel and a vertical stride of 1, and can be seen as the output of all $N\times\frac{T+12}{9}\times k2$ convolution units. Here, we guarantee the independence of the representation of sub-graph $\mathcal{G}(v)$. In total, the primary capsules have $N\times M$ capsule outputs, and each output is a $m$-dimensional vector, as shown in Figure \[fig:deepcrcnn\]. We can see all the primary capsules as a convolution layer with Eq. \[eq:capoutput\] as its block non-linearity. $$\label{eq:capoutput} \begin{aligned} \centering v_{j} = \frac{\Arrowvert s_{j} \Arrowvert^{2}}{1 + \Arrowvert s_{j} \Arrowvert^{2}} \cdot \frac{s_{j}}{\Arrowvert s_{j} \Arrowvert}, \end{aligned}$$ where $v_{j}$ is the output of capsule $j$, and $s_{j}$ is its total input. For all but the first layer of capsules, the total input to a capsule $j$ is a weighted sum over all the prediction vectors $\hat{u}_{j|i}$ from the capsules in the layer below, and is calculated by multiplying the output $u_{i}$ of a capsule in the layer below by a weight matrix $W_{ij}$ as following $$\label{eq:weightedsum} \begin{aligned} \centering s_{j} = \sum_{i}c_{ij}\hat{u}_{j|i},\qquad \hat{u}_{j|i} = W_{ij}u_{i}, \end{aligned}$$ where $c_{ij}$ is the coupling coefficient that is determined by the iterative dynamic routing process. The coupling coefficients between capsule $i$ and all the other capsules in the layer above sum to 1. They are determined by a *routing softmax* whose initial logits $b_{ij}$ are the log prior probabilities that capsule $i$ should be coupled to capsule $j$. The $c_{ij}$ can be calculated as following $$\label{eq:capsoft} \centering c_{ij} = \frac{\exp(b_{ik})}{\sum_{k}\exp(b_{ik})}.$$ The final DigitCaps layer has $n$ capsules per digit class and each of these capsules receives input from all the other capsules in the layer below. $W_{ij}$ is a weight matrix between each $u_{i},i\in (1, M\times N)$ in primary capsules and $v_{j}$, $j\in (1,L)$, where $L$ refers to the number of classes. As the length of the capsule’s output vector represents the presence of a class, the length $\Arrowvert v_{k} \Arrowvert$ of each capsule in the final layer can then be viewed as the probability of the text belonging to a particular class $k$. The length of the activity vector of each capsule in DigitCaps layer is used to calculate the classification loss. This encourages the network to learn a more general representation of text with classification task. Different from the capsule networks [@Sabour2017Dynamic; @Hinton2018Matrix; @jimenez2018capsule] applied in the field of computational vision, we consider the distance between the raw text and the output of the reconstructed representation in the word embedding space to be relatively large in practice. We do not perform text reconstruction during training. Next, we introduce how to design a weighted margin loss to measure distance of classes in hierarchy and guide the training of Attentional Capsule Recurrent CNN model. Hierarchical Taxonomy-aware Weighted Margin Loss {#sec:labelembedding} ================================================ Intuitively, the distances between any two classes on the hierarchy are different, but popular margin loss in capsule network [@Sabour2017Dynamic] and other distance measures in multi-label learning [@zhang2014review] between classes didn’t consider the hierarchical relations among labels. So, we explore a hierarchical taxonomy-aware weighted margin loss to guide the training of the proposed model in Section \[sec:dgcrcnn\]. For more formally, we denote the hierarchical taxonomy structure of the labels as $\mathcal{HG} = \{\mathcal{V}, \mathcal{E}\}$, where vertices $\mathcal{V}$ are classes $\mathcal{S}$ and the directed edges $\mathcal{E}$ represent the hierarchical parent-child relationship among the labels. In large-scale multi-label text classification, for a document $d_s$ and the corresponding positive labels set $T_s$, the number of labels in $T_s$ is usually much smaller than the remaining negative ones in $S$. Therefore, it will lead to a large loss of the objective function. In fact, in hierarchical label network, the closer the edge relationship between nodes, the closer the semantic distance between labels. In order to conveniently capture the relationship between labels on the hierarchical label structure, we design two meta-paths to guide random walk on the label structure, and generate label sequences to learn label representation. ![Illustrations of the hierarchical taxonomy of the labels and the two meta-paths. Vertices represent classes, and edges represent hierarchical parent-child relations.[]{data-label="fig:label2vec"}](hierarchical_and_graphic_label_dependency){width="50.00000%"} Figure \[fig:label2vec\] illustrates the hierarchical taxonomy structure of the labels, where each node refers to a label/class, and each directed edge represents a parent-child relationship. Note that the taxonomy network is not a strict hierarchical structure, and may contain cycles. For example, the two hierarchical relations of “*Economic*”-“*International trade*”-“*Arms sales*" and “*Economic*”-“*Defense economy*”-“*Arms sales*” can form a cycle. Both hierarchical structure of taxonomy and graph structure of taxonomy are common in practice, and can be modeled as a hierarchical graph-of-labels. For conveniently calculating the distance between any two labels, we measure the discrete cosine distance through their representation vectors. As shown in Figure \[fig:label2vec\], we extract the following two types of meta-paths from the hierarchical graph-of-labels, “*Child1 - Father0 - Child2*” and “*Father1 - Child0 - Father2*”. Actually, the two types of meta-paths control the directions of the random walk to the upper and lower layers, respectively. So, we perform meta-paths guided random walk to generate sequences of labels. Here, we set that the probability of selecting two meta-paths is equal during random walking. Similar to metapath2vec [@dong2017metapath2vec] and Deepwalk [@perozzi2014deepwalk], we also use the skip-gram with negative sampling  [@Mikolov:2013:DRW:2999792.2999959; @Mikolov2013Efficient] to encode the relations among the labels/classes into a continuous vector space. We optimize the following objective function, which maximizes the log-probability of observing a network neighborhood $N_{S}(l)$ for a node $l$ conditioned on its feature representation, given by $g:$ $$\label{eq:node2vec} \begin{aligned} \centering \max_{g} \sum_{l\in V}[-\log Z_{l} + \sum_{n_{i}\in N_{S}(l)}f(n_{i})\cdot f(l)], \end{aligned}$$ where $g(x)=\frac{1}{1+\exp(-x)}$ is the sigmoid function. Since it will be time consuming to compute $Z_{l} = \sum_{nl\in V}\exp(g(l)\cdot g(nl))$ for large network, we approximate it using the negative sampling technology. We optimize the Eq. \[eq:node2vec\] by using stochastic gradient method. Thus, given a tag label/class $l\in \mathcal{V}$, we can approximate a semantic distance between any other label $l_i, i\in [1,L]$ by calculating their discrete cosine distance between their embedding vectors as following: $$\label{eq:disvec} \begin{aligned} \centering d(l,l_i) = 1 - \cos(vec(l),vec(l_i)). \end{aligned}$$ Next, in order to take advantage of dependencies among labels to guide the training of the proposed attentional capsule recurrent CNN model, we design a hierarchical taxonomy-aware weighted margin loss objective function: $$\label{eq:weightedloss} \begin{aligned} \centering \mathcal{L} = \sum_{k=1}^{L} [& T_{k} \max(0, m^{+}-\Arrowvert v_{k}\Arrowvert)^2 \\ & + \lambda \cdot p \cdot \alpha_{k} \cdot (1-T_{k})\cdot \max(0, \Arrowvert v_{k}\Arrowvert - m^{-})^2], \end{aligned}$$ where $T_{k} = 1$ if and only if a digit of class $k$ is present, and $m^{+}$ and $m^{-}$ are the given thresholds for the upper and lower bounds. The $\lambda$ down-weighting of the loss for absent digit classes stops the initial learning from shrinking the lengths of the activity vectors of all the digit capsules, such as 0.5 in the original capsule networks  [@Sabour2017Dynamic; @Hinton2018Matrix]. $\alpha_{k} \in [0,1]$ is the minimum semantic distance from negative label $k$ to the positive labels set in the hierarchical label network. The total loss is the sum of the losses of all the digit capsules. Formally, for a document $d_{s}$, the positive label set is $T_{s} \subset \mathcal{S}$. And for any negative label $k$, the $\alpha_{k}$ is: $$\label{eq:params} \begin{aligned} \centering \alpha_{k} = 1 - \underset{t\in T_{s}}{\max}(\cos(vec(t),vec(k))). \end{aligned}$$ Meanwhile, to make an unbiased and smooth overall objective function after integrating the weight, we add an adjustment factor $p$ that satisfies $\sum_{k=1}^{L} p \cdot \alpha_{k} = 1$. We can approximate the distribution of the semantic distance $\alpha_{k}$ by $1-e^{-x}, x\in [1, L]$ to obtain an approximation of the adjustment factor $p$ for different datasets. EXPERIMENTS {#sec:experi} =========== In this section, we conduct experiments to evaluate the performance of the proposed framework. We will first introduce the used datasets, the evaluation metrics, methods for comparison, and experimental settings. Then, we will compare our methods with baselines, and provide the analysis and discussions on the results. Datasets Training Development Testing Class-Labels Depth Words/Sample Labels/Sample Samples/Label --------------- ---------- ------------- --------- -------------- ------- -------------- --------------- --------------- RCV1 23,149 - 784,446 103 6 268.95 3.24 729.67 EUR-Lex 15,449 - 3,865 3,956 4 1229.77 5.32 15.59 Reuters-21578 5800 600 300 10 - 257.32 - - Datasets -------- We use two datasets RCV1 and EUR-Lex for large-scale multi-label text classification, and use the Reuters-21578 to evaluate the effectiveness of our proposed capsule network in transferring from single-label to multi-label classification. The statistics of the datasets is shown in Table  \[tab:data\_desc\]. $\bullet$ **Reuters Corpus Volume I (RCV1)** [@Lewis2004RCV1] is a manually labeled newswire collection of Reuters News from 1996-1997. It consists of over 800,000 manually categorized newswire stories by Reuters Ltd for research purposes. Multiple topics can be assigned to each newswire story and there are 103 topics in total. The news documents are categorized with respect to three controlled vocabularies: industries, topics and regions. The relations among the labels are typically graphic structure with self-loops. We use the topic-based hierarchical classification because it has been widely adopted in evaluation. $\bullet$ **EUR-Lex**  [@EUR-LEX] is a collection of documents about European Union law. It contains many different types of documents, including treaties, legislation, case-law and legislative proposals, which are indexed according to several orthogonal categorization schemes to allow for multiple search facilities. The most important categorization is provided by the EUROVOC descriptors, which forms a topic hierarchy with almost 4000 categories regarding different aspects of European law. Directory code classes are organized in a hierarchy of 4 levels with a typical tree structure. Since the dataset contains several European languages, we choose English version of documents. $\bullet$ **Reuters-21578** [@Lewis:1992:EPC:133160.133172; @Zhao2018Investigating] is a collection appeared on the Reuters newswire in 1987. We follow [@Zhao2018Investigating] [^3] to choose 6,700 documents from the Reuters financial newswire service, where each document contains either multiple labels or a single label. And we also focus 10 popular topics, and reprocess the corpus to evaluate the capability of capsule networks of transferring from single-label to multi-label text classification. For development and training, we only use the single-label documents in the development and training sets. For testing, we only uses the multi-label documents in testing dataset. Note that this dataset is only for testing the advantages of the transferring from single-label to multi-label classification task of our capsule network that incorporates multiple semantics. Evaluation Metrics and Baselines -------------------------------- We use the standard evaluation metrics [@yang1999evaluation] to measure the performance of all the methods. $\bullet$ **Micro-$F_{1}$** is a metric considering the overall precision and recall of all the labels. Let $TP_{t}$, $FP_{t}$, $FN_{t}$ denote the true-positives, false-positives and false-negatives for the $t$-$th$ label in label set $\mathcal{S}$ respectively. The $Micro$-$F_1$ is defined as: $$\begin{aligned} \centering Micro\text{-}F_{1} &= \frac{2PR}{P+R}, \label{eq:mi-f1}\end{aligned}$$ where: $$\begin{aligned} & Precision(P) = \frac{\sum_{t\in \mathcal{S}}TP_{t}}{\sum_{t\in \mathcal{S}}TP_{t}+FP_{t}}, \nonumber \\ & Recall(R) = \frac{\sum_{t\in \mathcal{S}}TP_{t}}{\sum_{t\in \mathcal{S}}TP_{t}+FN_{t}}. \nonumber\end{aligned}$$ $\bullet$ **Macro-$F_{1}$** is a metric which evaluates the averaged $F_1$ of all the different class-labels. Different from Micro-F1 that gives equal weight to all the instances, $Macro$-$F_{1}$ gives equal weight to each label in the averaging process. Formally, $Macro$-$F_1$ is defined as: $$\label{eq:ma-f1} \begin{aligned} \centering Macro-F_{1} =& \frac{1}{|\mathcal{S}|}\sum_{t\in \mathcal{S}}\frac{2P_{t}R_{t}}{P_{t}+R_{t}}, ~~ \text{where} \\ P_{t} =\frac{TP_{t}}{TP_{t}+FP_{t}},& \qquad R_{t} = \frac{TP_{t}}{TP_{t}+FN_{t}},\\ \end{aligned}$$ Models CNNs Sorting LSTM Attentional LSTM Capsule Weighted Margin Loss ------------- ------ --------- ------ ------------------ --------- ---------------------- -- TGCNN(No-R) TGCNN TGRCNN GCCNN TAGRCNN GCRCNN AGCRCNN HE-TGCNN HE-TGRCNN HE-GCCNN HE-TAGRCNN HE-GCRCNN HE-AGCRCNN Meanwhile, we compare our model with both traditional text classification methods and recent state-of-the-art deep learning based methods. $\bullet$ **Flat baselines.** This type of methods generally first extract the TF-IDF features from the document, and then input them into the classification model such as Logistic Regression ([**LR**]{}) and Support Vector Machines ([**SVM**]{}). We call them flat baselines since they ignore both the relations among the words and the relations among the labels, and simply train a multi-class classifier. $\bullet$ **N-gram, sequence-of-words or graph-of-words based models.** These methods extract N-gram features, sequence-of-words or graph-of-words from the document as the input of classification models. These features are suitable for deep learning models, such as **CNN-non-static**  [@Kim2014Convolutional], **RCNN**  [@Lai:2015:RCN:2886521.2886636], **Deep CNN**  [@Conneau2016Very], **XML-CNN**  [@Liu:2017:DLE:3077136.3080834], **DGCNN-3** [@Peng:2018], Hierarchical LSTM (**HLSTM**)  [@chen2016neural], Hierarchical Attention Network [@Yang2017Hierarchical] (**HAN**) and Bi-directional Block Self-Attention Network [@shen2018biblosan] (**Bi-BloSAN**) etc. For example, **HLSTM** model learns sentence representations based on words sequences, and then use RNN models to encode document representations based on the learned sentence representations. **HAN** uses a global attention mechanism to attend useful words and sentences. **Bi-BloSAN** further splits the sequence into several blocks and employs intra-block and inter-block self-attentions to capture both local and long-range context dependencies, respectively. $\bullet$ **Hierarchical models.** These methods make use of the hierarchical or graphical label network to design hierarchical classification classifiers, such as Top-down Support Vector Machines (**TD-SVM**)  [@Liu:2005:SVM:1089815.1089821], **Hierarchical SVMs**  [@Tsochantaridis2006Large], Hierarchically Regularized Logistic Regression (**HR-LR**), Hierarchically Regularized Support Vector Machines (**HR-SVM**)  [@Gopal2013Recursive; @Gopal:2015:HBI:2737800.2629585], and Hierarchically Regularized Deep Graph CNN (**HR-DGCNN-3**) [@Peng:2018], etc. $\bullet$ **Sequence generation model.** These methods view the multi-label classification task as a sequence generation problem, and apply a sequence generation model, such as **SGM+GE** [@yang2018sgm], with decoder structure to solve it. $\bullet$ **Capsule Neural Networks.** These methods are shown to be effective in capturing the spatial features of text, including Capsule Networks with Dynamic Routing (**Capsule-A** and **Capsule-B**)  [@Zhao2018Investigating]. These capsule networks rely on N-gram convolution networks to extract shallow features and then use dynamic or static routing to learn the relationships between features. **Capsule-B** model employs parallel networks with different sizes of filters. It has been proven to have a better effect than the Capsule-A. $\bullet$ **Variations of HE-AGCRCNN.** We implement the following several variants of our proposed method. Three layers of Graph CNN (**TGCNN(No-R)**): no-sorting normalization process and without the hierarchical weighted margin loss, attentional LSTM units and capsule network; Three layers of Graph CNN (**TGCNN**): without the hierarchical weighted margin loss, attentional LSTM units and capsule network; Three layers of Graph Recurrent CNN (**TGRCNN**): without the hierarchical weighted margin loss, attention units and capsule network; Graph Capsule CNN (**GCCNN**): without the hierarchical weighted margin loss and attention LSTM units; Three layers of Attentional Graph Recurrent CNN (**TAGRCNN**): without the hierarchical weighted margin loss and capsule network; Graph Capsule Recurrent CNN (**GCRCNN**): without the hierarchical weighted margin loss and attention units; Three layers of Attentional Graph Capsule Recurrent CNN (**AGCRCNN**: without the hierarchical weighted margin loss), and the hierarchical weighted margin loss based models ([**HE-TGCNN, HE-TGRCNN, HE-GCCNN, HE-TAGRCNN and HE-GCRCNN**]{}). All these models have 3 layers of convolutional layers. In order to clearly present the advantages of the variations of HE-AGCRCNN, we give a table \[tab:function\] of models that enhance functionality. For all the baselines, we use the implementations or open source codes of these models released by authors and other researchers, and report the best performance of the results in our experiments. \[fig:embedding\] Experimental Settings {#sec:exp-settings} --------------------- All our experiments were performed on 64 core Intel Xeon CPU E5-2680 v4@2.40GHz with 512GB RAM and $8\times$NVIDIA Tesla P100-PICE GPUs. The operating system and software platforms are Ubuntu 5.4.0, Python 3.5.2, and Pytorch 0.4.0. The training and testing datasets are shown in Table  \[tab:data\_desc\]. In document modeling, the top $N$ numbers of central words are set to 100 (RCV1 and Reuters-21578) and 200 (EUR-Lex). For the sub-graph, the upper bound value $K$ (the maximum number of vertices) is set to 25. The length $T$ of the normalized word sequence is set to 20. The dimension $D$ of word embedding is set to 50. Here, we use word2vec technology to train 50 dimensional word embedding over the 100 billion words from Wikipedia corpus based on Skip-gram with Negative Sample model with window size of 5. For the hierarchical taxonomy embedding, we employ 50 threads to execute the random walk in parallel, and for each walk we use 500 steps. The dimension of label embedding vector is set to 200. For all the deep learning based models, the common parameters of training the models are empirically set, such as batch size $ = 32$ and learning rate $ = 0.001$ with Adam optimization algorithm. For the non-capsule neural network models, such as TGCNN(No-R), TGCNN, TGRCNN, TAGRCNN, HE-TGCNN, HE-TGRCNN and HE-TAGRCNN, we use a 2-layer fully connected networks and a sigmoid layer, and the popular cross entropy or the hierarchical taxonomy embedding based weighted margin loss as objective function. For the capsule network models, we use the the original margin loss or hierarchical taxonomy embedding based weighted margin loss as the objective function. For capsule based models, the dimension of capsule vector $m$ is 16, the channel of convolution capsule $M$ is 64, the dimensions of DigitCaps are $32\times103$ for the RCV1 dataset, $64\times3956$ for the EUR-Lex, and $32\times10$ for the reprocessed Reuters-21578, and $m^{+} = 0.9$, $m^{-} = 0.1$, $\lambda =0.5$. Considering the number of the class labels and the average number of labels per sample, we set the adjustment factor $p$ to $0.01$ for RCV1, $0.001$ for EUR-Lex and $0.1$ for the reprocessed Reuters-21578 in the Eq. \[eq:weightedloss\]. All convolution kernels are $1\times3$ in size. The numbers of convolution kernels per layer are 64 and 128. The LSTM operator contains 128 hidden layer units. The numbers of neurons in the fully connected layers are 1024 and 512 in RCV1, and 2048 and 4096 in EUR-Lex. Our models can achieve the best performance results among 20 to 70 epoches. For the experiment of transferring single-label model to multi-label classification, on the one hand, in order to be consistent with baseline  [@Zhao2018Investigating], we select the same number of training, testing, and validation data as shown in Table  \[tab:data\_desc\]. On the other hand, as the labels of the reprocessed Reuters-21578 is part of RCV1’s, we reuse the vector representation of the RCV1 label of our the hierarchical taxonomy-aware weighted margin loss in the transferring experiment. Evaluation on Label Embedding ----------------------------- In order to study whether the proposed meta-paths based random walk can learn desirable label embedding that reflects the hierarchical taxonomy relations among them, we use the meta-paths guided random walk and traditional random walk to generate the two label sequences respectively, and then generate two label vectors by the same skip-gram method. After obtaining the two vectors, we calculate the cosine distance between them, and use it to reconstruct the relations among the labels. When the distance between two label vectors is larger than the threshold, we add a edge between the two labels. We employ the Macro-$F_{1}$ and Micro-$F_{1}$ to evaluate the performance of reconstructing the relations in hierarchy. Figure \[fig:rcv1\_embedding\] and figure \[fig:EUR\_embedding\] show the results of the two label embedding vectors on the relation reconstruction task in the two datasets. One can see that overall the meta-paths guided random walk approach performs better for capturing the hierarchical taxonomy semantics than the traditional random walk approach. For RCV1, the most suitable thresholds of meta-paths based taxonomy embedding are $0.660$ and $0.940$, and the highest Macro-$F_{1}$ and Micro-$F_{1}$ are $0.337$ and $0.310$, respectively. For the traditional random walk based taxonomy embedding, the highest Macro-$F_{1}$ and Micro-$F_{1}$ are $0.275$ and $0.272$, respectively. For EUR-Lex, the most suitable thresholds of meta-paths based taxonomy embedding are $0.440$ and $0.880$, and the highest Macro-$F_{1}$ and Micro-$F_{1}$ are $0.168$ and $0.406$, respectively. For the traditional random walk based taxonomy embedding, the highest Macro-$F_{1}$ and Micro-$F_{1}$ are $0.164$ and $0.405$, respectively. One can observe that the performance difference between the two random walk methods is relatively small on EUR-Lex. This is probably because the taxonomy label structure of EUR-Lex is a hierarchical tree. Although the method of measuring the discrete cosine distance between any two labels based on unsupervised heterogeneous network representation learning vector is approximate, it’s a convenient method to estimate label distance for any two labels. Performance Evaluation on RCV1 ------------------------------ Next, we evaluate the performance on the RCV1 dataset through the multi-label text classification task. RCV1 is a dataset that training samples are much fewer than testing samples, as shown in Table \[tab:data\_desc\]. The experiment results are shown in Table \[tab:mltc\_results\]. Among the traditional text classification algorithms, one can see that the HR-SVM performs better than TD-SVM, HSVM, SVM, HR-LR and LR. For deep learning approaches, one can see that the performance of RNN based algorithms HLSTM and HAN are comparable to SVM and LR. RCNN performs worse on both settings. For fine-grained topical classification, the above recurrent models may not have advantages because it compresses the whole document as a dense vector for classification. The RNN models are more suitable to sentiment classification for short text, but is not suitable to learn features for long document [@Peng:2018]. For CNN models, it is shown that XML-CNN does not perform very well on RCV1. However, the deeper model DCNN improves the performance by 9% in terms of Macro-F1 and 4% in terms of Micro-F1. For capsule network, one can see that the Capsule-B achieves comparable performance with DCNN model. For sequence generation model, the SGM+GE improves the performance by 2% in terms of Macro-F1 and Micro-F1 compared with the XML-CNN model. For GCNN models, both DGCNN-3 and HR-DGCNN-3 improve the performance by 4% in terms of Macro-F1 and 3% in terms of Micro-F1 compared with Capsule-B. It demonstrates that graph-of-words representation is effective in modeling documents in multi-label text classification. For the popular bi-directional block self-attention network, the Bi-BloSAN improves the performance by 8% in terms of Macro-F1 and 3% in terms of Micro-F1 compared with the HAN model. For the proposed models, we try different model configurations listed in Table  \[tab:function\]. The results are shown in Table \[tab:mltc\_results\]. One can see that the LSTM units, attentional LSTM units, capsule networks and hierarchical taxonomy-aware weighted margin loss are all helpful to improve the classification performance. The proposed HE-AGCRCNN model outperforms the HR-DGCNN-3 by 8% in terms of Macro-F1. The simplified model TGCNN(No-R) also achieves comparable Macro-F1 and Micro-F1 with DGCNN-3. Meanwhile, without attentional LSTM units, capsule network and hierarchical label dependencies, the TGCNN also outperforms most of the baselines. Based on the arranged words-matrix representation, LSTM units and attentional LSTM units, the TGRCNN and TAGRCNN models achieve 5%-6% improvements in terms of Macro-$F_{1}$ over HR-DGCNN-3. This improvements show the importance of local sequential semantics for text features. Among the proposed models, one can see that capsule networks averagely achieve 1% gain in both Macro-$F_{1}$ and Micro-$F_{1}$. Overall, the hierarchical taxonomy-aware weighted margin loss can also improve the performances by 2% in terms of Macro-$F_{1}$ and 1% in terms of Micro-$F_{1}$. Finally, the proposed HE-AGCRCNN model achieves the highest $0.513$ Macro-$F_{1}$ and $0.778$ Micro-$F_{1}$ performance. The results of the different document modeling methods show that by representing document as arranged words-matrix, the proposed model can gain performance improvement for multi-label text classification in RCV1. One can also see that HR-SVM, HR-DGCNN-3 and HE-AGCRCNN represent two different ways of using the hierarchical label dependencies, and both improve the classification performance over RCV1 dataset. We will present the timeliness analysis of the experiment in section \[sec:time-analysis\]. ------------- -------------------------------------------------------- --------------- --------------- --------------- Marco-$F_{1}$ Micro-$F_{1}$ Marco-$F_{1}$ Micro-$F_{1}$ LR 0.328 0.692 0.181 0.522 SVM 0.330 0.691 0.185 0.551 HSVM 0.333 0.693 0.189 0.567 TD-SVM 0.337 0.696 0.198 0.571 HR-LR 0.322 0.716 0.180 0.583 HR-SVM 0.386 0.728 0.223 0.609 HLSTM 0.310 0.673 0.183 0.562 HAN 0.327 0.696 0.184 0.566 RCNN 0.293 0.686 0.168 0.554 XML-CNN 0.301 0.695 0.179 0.583 DCNN 0.399 0.732 0.231 0.611 DGCNN-3 0.432 0.761 0.237 0.632 HR-DGCNN-3 0.433 0.762 0.241 0.649 SGM+GE 0.348 0.719 0.216 0.628 Bi-BloSAN 0.401 0.720 0.219 0.619 Capsule-B 0.399 0.739 0.226 0.600 TGCNN(No-R) 0.443 0.745 0.244 0.648 TGCNN 0.472 0.747 0.257 0.655 GCCNN 0.480 0.749 0.261 0.658 TGRCNN 0.484 0.754 0.265 0.667 TAGRCNN 0.490 0.759 0.270 0.673 GCRCNN 0.488 0.765 0.275 0.668 AGCRCNN 0.494 0.769 0.283 0.675 HE-TGCNN 0.482 0.751 0.283 0.683 HE-GCCNN 0.491 0.754 0.290 0.688 HE-TGRCNN 0.495 0.762 0.292 0.680 HE-TAGRCNN 0.504 0.773 0.298 0.685 HE-GCRCNN 0.505 0.772 0.297 0.684 HE-AGCRCNN **[0.513]{} & **[0.778]{} & **[0.330]{} & **[0.688]{}\ ******** ------------- -------------------------------------------------------- --------------- --------------- --------------- : \[tab:mltc\_results\]Comparison of results on RCV1 and EUR-Lex. Performance Evaluation on EUR-Lex --------------------------------- As the number of labels in EUR-Lex is large, we use more neurons in the fully connected layers and set a larger dimension of capsule vector in the DigitCaps layer, as presented in Section \[sec:exp-settings\]. For the proposed models, we also try different configurations, and the results are shown in Table \[tab:mltc\_results\]. From the results one can see that LSTM units, attentional LSTM units, capsule networks and hierarchical taxonomy-aware weighted margin loss are all helpful to improve classification performance on the EUR-Lex dataset. HE-AGCRCNN model achieves about 6% improvements in terms of Macro-$F_{1}$ and 4% gains in terms of Micro-$F_{1}$ over the HR-DGCNN-3 model. Without using hierarchical label dependencies, LSTM units, attentional LSTM units and capsule networks, the TGCNN also performs better than HR-DGCNN-3, and the results are $0.257$ and $0.655$ for Macro-$F_{1}$ and Micro-$F_{1}$, respectively. When we do not order the words in the sub-graph, the results of TGCNN(No-R) are $0.244$ and $0.648$ for Macro-$F_{1}$ and Micro-$F_{1}$, respectively. The performance gap between TGCNN(No-R) and TGCNN shows the importance of local sequential semantics for text classification with the same three layers of Graph CNN models. Based on the arranged words-matrix representation, the LSTM units can help to improve 1% performance comparing GCCNN and GCRCNN. The performance gap between TGRCNN and TAGRCNN shows that the masked attentional units can help to improve about 0.5% performance. The hierarchical taxonomy-aware weighted margin loss also helps to improve about 1%-3% performances in terms of Macro-$F_{1}$ or Micro-$F_{1}$. Compared with the improvements of the hierarchical taxonomy-aware weighted margin loss in the RCV1 dataset, the improvements in the EUR-Lex dataset are greater by using the same weighted margin loss. Finally, HE-AGCRCNN model achieves $0.330$ Macro-$F_{1}$ and $0.688$ Micro-$F_{1}$, which are both the highest performance. The experimental results again demonstrate that by representing document as an arranged words-matrix and incorporating the proposed deep models, one can gain benefits from non-consecutive, long-distance and sequential semantics for topical multi-label text classification. Performance Evaluation on Reuters-21578 --------------------------------------- A significant advantage of capsule network is that it performs much better in the transferring single-label to multi-label classification task [@Zhao2018Investigating]. Different from traditional deep learning classification models that are based on fully connected network, capsule networks use activity vectors of each capsule in DigitCaps layer to indicate the presence of an instance of each class. We also perform the model transfer capacity experiment of the proposed capsule network on the reprocessed Reuters-21578 dataset [@Zhao2018Investigating]. The comparison results are shown in Table  \[tab:RCV-transferring\]. The baseline results are also reported from the work [@Zhao2018Investigating], and HE-AGCRCNN outperforms all the baselines. Compared with capsule-based models, the performances of LSTM, BiLSTM and CNN-non-static are the worst. From the results one can see that GCCNN, GCRCNN, AGCRCNN, HE-GCCNN, HE-GCRCNN and HE-AGCRCNN models all have achieved about 5%-7% improvements in terms of Micro-$F_{1}$ over the existing best baseline Capsule-B. Even without the attentional LSTM units, the simplified GCCNN can achieve $0.905$ performance in terms of Micro-$F_{1}$. Compared with the popular Capsule-A and Capsule-B models, our proposed GCCNN, GCRCNN, AGCRCNN, HE-GCCNN, HE-GCRCNN and HE-AGCRCNN models integrate more non-consecutive, long-distance and local sequential semantics, and make use of the hierarchical label dependencies. Finally, the proposed HE-AGCRCNN model achieves the highest $0.927$ performance in terms of Micro-$F_{1}$. The experimental improvements again prove the effectiveness of our proposed capsule models in learning rich textual features. ----------------------------------------------------------------------------------- Models P R Micro-$F_{1}$ ---------------- ------------------------------------------ ------- --------------- LSTM 0.867 0.547 0.635 BiLSTM 0.823 0.559 0.643 CNN-non-static 0.920 0.597 0.704 Capsule-A 0.882 0.801 0.820 Capsule-B 0.954 0.820 0.858 GCCNN 0.962 0.856 0.905 GCRCNN 0.970 0.871 0.917 AGCRCNN 0.973 0.875 0.921 HE-GCCNN 0.965 0.862 0.910 HE-GCRCNN 0.974 0.879 0.924 HE-AGCRCNN **[0.978]{} & **[0.882]{} & **[0.927]{}\ ****** ----------------------------------------------------------------------------------- : \[tab:RCV-transferring\]Comparison of the transferring capacity from single-label to multi-label text classification on the reprocessed Reuters-21578 dataset. Training Efficiency Evaluation {#sec:time-analysis} ------------------------------ To evaluate the training efficiency of the model, we next show the training time of the proposed model and its variants on both RCV1 and EUR-LEX datasets in Table \[tab:GPU\]. Here, for the RCV1, since the number of samples in the test set is about 34 times larger than the number of samples in the training set, as shown in Table \[tab:data\_desc\], we perform the testing by using multi-core CPUs. One can observe that most of these models can quickly achieve a promising classification result with less than 3 hours except for the models with the LSTM or capsule unites. For example, the TGCNN and HE-TGCNN models converge quickly with less than 0.2 hour on RCV1 dataset and less than 1.2 hours on EUR-Lex dataset. Meanwhile, the training time on RCV1 dataset is much less than EUR-Lex. This is mainly because the EUR-Lex dataset has a larger document representation and more parameters, according to Table  \[tab:data\_desc\]. We also verify that the models integrating more feature extraction operators, such as LSTM units, attentional LSTM units and capsule networks, will take longer time to train for achieving a desirable classification performance. Although HE-AGCRCNN model takes $1.119$ hours and $6.335$ hours to train for RCV1 and EUR-Lex datasets, respectively, it achieves the highest classification performance. One can also see that the hierarchical taxonomy embedding based weighted margin loss does not add much computational time compared with the recursive regularized optimization models [@Peng:2018; @Gopal2013Recursive; @Gopal:2015:HBI:2737800.2629585]. Usually, the time consumptions of the above recursive regularization based models are expensive for the large number of parameters and constraints on the Euclidean distance of the parameters. In particular, the time consumptions of recursive regularization optimized deep learning model, such as HR-DGCNN-3, is generally measured in days [@Peng:2018]. Models RCV1(hr.) EUR-Lex(hr.) ------------ ----------- -------------- TGCNN 0.166 1.100 GCCNN 0.537 3.415 TGRCNN 0.381 2.579 TAGRCNN 0.382 2.580 GCRCNN 1.116 6.327 AGCRCNN 1.117 6.328 HE-TGCNN 0.167 1.167 HE-GCCNN 0.542 3.421 HE-TGRCNN 0.385 2.583 HE-TAGRCNN 0.386 2.584 HE-GCRCNN 1.118 6.334 HE-AGCRCNN 1.119 6.335 : \[tab:GPU\]Comparison of training time on GPUs. ![image](Case_Study){width="100.00000%"} Case study ---------- To gain a closer view of what’s the two attention layers and output capsules in a document captured by our models, we visualize parts of the attention probability or alignment score by heatmaps in Figure \[fig:casestudy\]. The red words are central words in the document, each block of word sequence is context of central words. For each central word, there are two layers of masked attention. We choose the blocks of word sequences from the $1$-th, $2$-th and $3$-th central words. One can see that the weights of the upper left parts are higher than other places. This is probably because the contextual semantics of the front central words are more representative of the subject of the article. For the output capsule vectors, there are 4 vectors whose modulus length is greater than 0.9, corresponding to the category of output. Note that Sports category comes out due to the words game, match, defeat and win, although chess is not really a physical activity sport. Related Work {#sec:relatedwork} ============ As our work is closely related to text classification, textual deep learning models and graph convolution networks, in this section we will review related works from the three aspects. Tradition text classification models use feature engineering and feature selection to obtain features for text classification [@Aggarwal2012]. For example, Latent Dirichlet Allocation [@BleiNJ03] has been widely used to extract “topics” from corpus, and then represent documents in the topic space. It performs better than Bag-Of-Word (BOW) when the feature numbers are small. However, when the size of words in vocabulary increases, it does not show advantage over BOW on text classification  [@BleiNJ03]. There are also some existing work that tried to convert texts to graphs  [@Rousseau2015Text]. Similar to our proposed methods, they used word co-occurrence to construct graphs from texts, and then they applied similarity measure on graph to define new document similarity and features for text  [@Rousseau2015Text]. For hierarchical large-scale multi-label text classification, many efforts have been put on how to leverage the hierarchy of labels to improve the classification results. Recently, a recursive regularization of weight euclidean constraint with classifiers has been developed, and shown to be the out-performance in large-scale hierarchical text classification problems [@Gopal2013Recursive; @Gopal:2015:HBI:2737800.2629585]. For deep learning models, there have been RNNs, CNNs, and capsule models applied to text classification. For example, hierarchical RNN has been proposed for long document classification [@TangQL15] and later attention model is also introduced to emphasize important sentences and words [@Yang2017Hierarchical]. Similar to RNNs, the recently proposed self-attention based sentence embedding technologies [@lin2017structured; @shen2018disan; @shen2018biblosan] have shown effectively capturing both long-range and local dependencies in sentiment-level tasks. For example, Bi-BloSAN [@shen2018biblosan] is a bi-directional block self-attention network to learn text representation and models text as sequences. For CNNs models, Kalchbrenner et al.  [@kalchbrenner2014convolutional] and Kim et al. [@Kim2014Convolutional] used simpler CNN for text classification, and showed significant improvements over traditional texts classification methods. Zhang et al.  [@Zhang2015Character] and Conneau et al.  [@Conneau2016Very] used a character level CNN with very deep architecture to compete with traditional BOW or n-gram models. The combination of CNNs and RNNs are also developed which shows improvements over topical and sentiment classification problems  [@Lai:2015:RCN:2886521.2886636]. Capsule networks were proposed by Hinton et. [@Hinton2011Transforming; @Sabour2017Dynamic; @Hinton2018Matrix] as a kind of supervised representation learning methods, in which groups of neurons are called capsules. Capsule network has been proved effective in learning the intrinsic spatial relationship between features [@Zhao2018Investigating; @xiao2018mcapsnet; @zhang2018attention].  [@Zhao2018Investigating] showed that Capsule networks can help to improve low-data and label transfer learning. However, as mentioned in the introduction, existing textual deep learning models are not compatible with diverse text semantic coherently learning. Compared with our work, these previous studies only considered N-gram or sequential text modeling, but ignored high level of non-consecutive and long-distance semantics of text. They did not study and utilize the dependency among the the labels, either. Although there are prior works [@Gopal2013Recursive; @Peng:2018; @xie2013multilabel; @garg2015exploring] on modeling pair-wise relation between labels for multi-label classification, they fail to consider their hierarchical relations, and the computation of the above models is expensive due to the use of euclidean constraints in their regularization. GCN derived from graph signal processing  [@Shuman2013The; @Bruna2013Spectral], and the graph convolution operation has been recognized as the problem of learning filter parameters that were replaced by a self-loop graph adjacency matrix, updating network weights, and extended by utilizing fast localized spectral filers and efficient pooling operations in  [@Defferrard2016Convolutional; @Kipf2016Semi; @Duvenaud2015Convolutional]. With the development of GCN technologies, graphs embedding approaches, such as PSCN [@Niepert:2016:LCN:3045390.3045603] and GCAPS-CNN [@Verma2018Graph], have been developed in graph classification tasks. Recently, the recursively regularized deep graph-cnn [@Peng:2018] has been proposed to combine graph-of-words representation, graph CNN, and hierarchical label dependency for large-scale text classification. Then the Text GCN model [@YaoGCN2018] has been proposed to capture global word co-occurrence information and perform text classification without word embeddings or other external knowledge. Although long-distance and non-continuous text features are fully considered in the two models, the existing graph convolutional neural network models ignore the continuous and sequential semantics of words in the text. In addition, the recursive regularization is usually time consuming due to the euclidean constraint. Conclusion and Future Work {#sec:conclu} ========================== In this paper, we present a novel end-to-end hierarchical taxonomy-aware and attentional graph capsule recurrent CNN framework for large-scale multi-label text classification. We first propose to convert each document as an arranged words-matrix that preserves both the non-consecutive, long-distance and local sequential semantics for fully representing the document. Based on our document modeling, we next propose a HE-AGCRCNN model to coherently learn multiple types of textual features. In order to better learn local sequential semantics, we design a masked attentional LSTM to model the different impacts among different blocks of word sequences, and enhance the sequential features learning. To incorporate the hierarchical relations among the labels, we further propose a novel hierarchical taxonomy-aware weighted margin loss to improve the performance of multi-label text classification. The advantageous performance of our proposed models over other competing methods is evident as it obtained the best results on all the RCV1 and EUR-Lex datasets in our comparative evaluation. Compared to the N-gram based textual capsule networks, we verify the effectiveness of our proposed capsule models in learning rich textual features in transferring single-label to multi-label classification task. The experimental results show the effectiveness and efficiency of our model in multi-label text classification. In the future, we plan to invest subgraph-level attention capsule network, and upgrade our hierarchical taxonomy-aware and attentional graph capsule recurrent CNN to self attention rnn/cnn models [@shen2018biblosan; @shen2018disan], variable-size convolution kernels [@yin2015multichannel] and BERT [@devlin2018bert] pre-trains based models, and popularize to more sophisticated text classification datasets and applications. \[[![image](penghao.jpg){width="1in" height="1.2in"}]{}\] [Hao Peng]{} is currently a Ph.D. candidate at the State Key Laboratory of Software Development Environment, and Beijing Advanced Innovation Center for Big Data and Brain Computing in Beihang University. His research interests include representation learning, text mining and urban computing. \[[![image](ljx.jpg){width="1in" height="1.2in"}]{}\] [Jianxin Li]{} is currently a Professor with the State Key Laboratory of Software Development Environment, and Beijing Advanced Innovation Center for Big Data and Brain Computing in Beihang University. His current research interests include social network, machine learning, distributed system, virtualization, big data, trust management and network security. \[[![image](gqr.jpg){width="1in" height="1.2in"}]{}\] [Qiran Gong]{} is currently a B.E. candidate at the State Key Laboratory of Software Development Environment in Beihang University, Beijing, China. His research interests include social network mining and text mining. \[[![image](wsz.jpg){width="1in" height="1.2in"}]{}\] [Senzhang Wang]{} is currently an Associate Professor with the Collage of Computer Science and Technology, Nanjing University of Aeronautics and Astronautics, Nanjing. His current research interests include data mining, urban computing and social network analysis. \[[![image](he.jpg){width="1in" height="1.2in"}]{}\] [Lifang He]{} is currently a Postdoctoral Research Associate at the Department of Biostatistics and Epidemiology at the University of Pennsylvania. Her current research interests include machine learning, data mining, tensor analysis, biomedical informatics. \[[![image](libo.png){width="1in" height="1.2in"}]{}\] [Bo Li]{}, is currently an Associate Professor with the State Key Laboratory of Software Development Environment, and Beijing Advanced Innovation Center for Big Data and Brain Computing in Beihang University. His current research interests include big data computing theory, machine learning and computer security. \[[![image](wlh.jpg){width="1in" height="1.2in"}]{}\] [Lihong Wang]{} is a professor in National Computer Network Emergency Response Technical Team/Coordination Center of China. Her current research interests include information security, cloud computing, big data mining and analytics, information retrieval and data mining. \[[![image](psy.jpg){width="1in" height="1.2in"}]{}\] [Philip S. Yu]{} is a Distinguished Professor and the Wexler Chair in Information Technology at the Department of Computer Science, University of Illinois at Chicago. Before joining UIC, he was at the IBM Watson Research Center, where he built a world-renowned data mining and database department. He is a Fellow of the ACM and IEEE. Dr. Yu is the recipient of ACM SIGKDD 2016 Innovation Award for his influential research and scientific contributions on mining, fusion and anonymization of big data, the IEEE Computer Society’s 2013 Technical Achievement Award for “pioneering and fundamentally innovative contributions to the scalable indexing, querying, searching, mining and anonymization of big data” and the Research Contributions Award from IEEE Intl. Conference on Data Mining (ICDM) in 2003 for his pioneering contributions to the field of data mining. Dr. Yu has published more than 1,100 referred conference and journal papers cited more than 103,000 times with an H-index of 152. He has applied for more than 300 patents. Dr. Yu was the Editor-in-Chiefs of ACM Transactions on Knowledge Discovery from Data (2011-2017) and IEEE Transactions on Knowledge and Data Engineering (2001-2004). [^1]: Manuscript received May 6, 2019. (Corresponding author: Jianxin Li.) [^2]: http://stanfordnlp.github.io/CoreNLP/ [^3]: https://github.com/andyweizhao/capsule\_text\_classification
--- abstract: 'We observed five giant molecular clouds (GMCs) in the Large Magellanic Cloud (LMC) in the $^{12}{\rm CO}$ $J=1\mbox{--}0$ line using the Atacama Large Millimeter/submillimeter Array (ALMA). The sample includes four GMCs with some signs of star formation – either YSOs, regions, and/or young clusters – and one quiescent GMC without any sign of massive star formation. The data from the ALMA 12 m, 7 m, and Total-Power arrays are jointly deconvolved to obtain high-fidelity images at high spatial resolution ($3\arcsec = 0.7\;{\rm pc}$). The four star-forming GMCs show very complex structures with clumps and filaments. The quiescent GMC shows a relatively diffuse, extended emission distribution without prominent clumps or filaments. This difference is similar to that between structured molecular gas in Milky Way spiral arms and unstructured gas in the inter-arm regions. We characterize the difference with the brightness distribution function and brightness distribution index. In conjunction with other ALMA studies of GMCs in the LMC, the five GMCs tentatively form an evolutionary trend: from less structured, quiescent GMCs to more structured, actively star-forming GMCs. A future ALMA study will be able to map molecular clouds over the LMC and reveal the evolutionary sequence of molecular clouds.' author: - Tsuyoshi Sawada - Jin Koda - Tetsuo Hasegawa title: Internal Structures of Molecular Clouds in the LMC Revealed by ALMA --- Introduction {#sec:intro} ============ Molecular clouds are the site of star formation. Their internal structures and evolution are the key to understanding the process of star formation. Our recent studies of molecular gas in the Milky Way (MW) revealed the structural evolution of the molecular gas when a 1 pc scale is resolved. @sawada2012 [@sawada2012let] found that the gas in the MW spiral arms is [*structured*]{} (i.e., bright and spatially confined, parsec-sized, emission is prominent), while [*unstructured*]{} gas (i.e., faint and diffuse) dominates in the inter-arm regions. A similar trend has been found among local star-forming and quiescent molecular clouds [e.g., @kainulainen2009]. In order to characterize these internal structures, @sawada2012 introduced two simple tools: the brightness distribution function (BDF) and brightness distribution index (BDI; see Section \[sec:BDF\]). An extension of such study to external galaxies was, however, limited by the low spatial resolution, typically comparable to the typical size of giant molecular clouds (GMCs), several tens of parsecs, or even worse [e.g., @koda2009; @schinnerer2014]. Recently, the Atacama Large Millimeter/submillimeter Array [ALMA; @hills2010] has revolutionized such studies in the Large and Small Magellanic Clouds [LMC and SMC; e.g., @indebetouw2013]. It can resolve the spatial scale relevant to star formation [$\sim 1$ pc; e.g., @lada2003] at the distances of the LMC and SMC. Most studies so far have focused on the areas around star-forming regions, e.g., 30 Doradus, the most active starburst region in the Local Group [@indebetouw2013; @fukui2015; @saigo2017; @nayak2018], and the region N55, which is located within the largest supergiant shell (SGS) in the LMC [@naslim2018]. Molecular clouds in these star-forming environments show complex internal structures, such as clumps and filaments. @muraoka2017 also showed the presence of small clumps in the star-forming cloud N83C in the SMC. On the contrary, @wong2017 found a less structured emission distribution in a quiescent molecular cloud in the LMC. This paper aims to bridge the gap between these two extremes, clouds with very active star formation and those in the quiescent phase, and investigate the evolutionary trend. We observed five clouds in the LMC, which show a range of star formation activity. The properties of the interstellar medium and star formation in the LMC have been thoroughly studied at various wavelengths, thanks to its proximity [$\approx 50$ kpc; @pietrzynski2013] and nearly face-on geometry [$i\approx 35\arcdeg$; @vandermarel2001]. For this study, it was essential to recover both compact and extended structures. Thus, we combine and jointly deconvolve the data from the ALMA 12 m, 7 m, and Total-Power (TP) arrays. The presence of CO-dark ${\rm H_2}$ is being actively discussed [@greiner2005; @planck2011]. It should be predominantly in the outskirts of GMCs, where the CO is photodissociated by ultraviolet radiation from the outside, while ${\rm H_2}$ is optically thick against Lyman-Werner photons and protected. The ALMA observations and discussions in this paper are based on CO observations, and hence are about CO-bright parts of GMCs, not the outskirts. Target Selections {#sec:targets} ================= The goal of this paper is to study the evolutionary sequence in cloud structure from relatively quiescent to actively star-forming molecular clouds. Obviously, we can only get a peek at the trend with the limited amount of observing time. Our sample of clouds and field coverages for each cloud are therefore not complete by any means. Nevertheless, we selected five molecular clouds based on the cloud evolutionary stages defined by @kawamura2009. Their classification is based on the master catalog by @fukui2008, which consists of 272 clouds in the LMC from the $^{12}{\rm CO}$ $J=1\mbox{--}0$ survey with the NANTEN 4 m telescope at a $2\farcm 6$ ($\approx 40$ pc) resolution. They classified the GMCs into three phases of evolution: Types I (no sign of massive star formation), II (associated with regions, i.e., at an early stage of star formation), and III (associated with regions and stellar clusters, i.e., at a late stage of star formation). We selected five GMCs at three evolutionary stages; i.e., GMCs 2, 55, and 225 from Type I, GMC 216 from Type II, and GMC 197 from Type III (see Table \[tbl:params\]). Figure \[fig:overall\] illustrates the distribution of our GMCs, as well as the other clouds in the literature. The green circles show the locations of regions [@henize1956] whose surrounding molecular gas was observed with ALMA: 30 Doradus, N159, and N55. These clouds show clumps and filaments [@indebetouw2013; @fukui2015; @saigo2017; @naslim2018; @nayak2018]. The quiescent cloud, the “Planck cold cloud” (PCC) located near the southern edge of the LMC (green square) was also observed with ALMA [@wong2017]. Our five GMCs, 2, 55, 225, 216, and 197, are distributed from the edge of the disk to near 30 Doradus. We note that GMCs are sometimes referred to by the names of nearby regions (e.g., N55, N159), while our GMC IDs (GMC 2, 55, 225, 216, and 197) are from @fukui2008. Considerations -------------- Our target selection and observation strategies were built solely on @kawamura2009 and @fukui2008. Their results were based on the best information available at the time of their work. Of course, our knowledge has improved, and one may be tempted to modify their GMC classification with respect to star formation. Table \[tbl:params\] lists some other parameters that characterize the star formation activity of our GMCs, in addition to their Kawamura’s types based on their association with regions and young stellar clusters. For example, the number of young stellar objects (YSOs) within the 30 and 45 pc radii from the center of our ALMA mosaics are zero for GMC 2, but one to four for the other GMCs. With this criterion, GMC 2 (Type I) is the only GMC in our sample that has no sign of star formation, although GMCs 55 and 225 are also classified as Type I. For this, we counted only the YSOs from @seale2014 with a high likelihood (“probable” ones) of being in the LMC. The average dust temperature within a field of view (FoV) traces the radiation field. It is also higher ($T_{\rm dust}>20$ K) for GMCs 55, 197, and 216 than for GMCs 2 and 225 [@gordon2014]. The 8 $\micron$ flux is mainly from the interstellar polycyclic aromatic hydrocarbons (PAHs) excited by the background radiation. Hence, it also indicates the strength of the background radiation field. The background 8 $\micron$ flux in the table is calculated from the [*Spitzer Space Telescope*]{} 8 $\micron$ images [@meixner2006] after removing point sources (i.e., stars). Again, it is higher for GMCs 55, 197, and 216 than for GMCs 2 and 225 (Figure \[fig:intandmax1\]). Depending on the adopted indicator of star formation activity, the classification of the GMCs may be altered. This study has this type of uncertainty. From the above parameters, GMC 2 is the only target that does not show any sign of star formation. The other four GMCs show some sign, though some may be at an earlier stage of star formation than the others. In addition, our ALMA FoVs were set around the peak positions in the CO $J=1\mbox{--}0$ maps at the $2\farcm 6$ resolution [@kawamura2009]. The follow-up CO $J=1\mbox{--}0$ observations at a higher resolution [$45\arcsec$; @wong2011] resolved structures further. It turned out that in some cases, the emission peaks at the $2\farcm 6$ resolution are not the peaks at the $45\arcsec$ resolution (Figure \[fig:intandmax1\]). Therefore, our ALMA FoVs are not optimal in terms of the peak positions in the $45\arcsec$ resolution maps. Despite these retrospective considerations, the new ALMA data presented here show a tentative trend of cloud evolution. [ccccccccc]{} 2 & 97 & 10 & I & 0/0 & 16.5 & 0.59 & 0.79 & No\ 55 & 39 & 5 & I & 1/3 & 20.4 & 1.07 & 0.83 & Rim\ 197 & 220 & 100 & III & 1/2 & 22.4 & 1.80 & 0.86 & Rim\ 216 & 80 & 20 & II & 2/4 & 24.4 & 1.51 & 0.89 & Inside\ 225 & 73 & 10 & I & 1/1 & 17.8 & 0.74 & 0.70 & No\ Observations {#sec:obs} ============ The target five GMCs were observed with ALMA using the Band 3 receivers [@claude2008] as the Early Science Cycle 1 project 2012.1.00641.S. The observations using the 12 m array, which consisted of twenty-seven 12 m antennas and the 64-input correlator [@escoffier2007], were made in 2013 December. A 469 MHz wide, 3840-channel, dual-polarization spectral window ($244\;{\rm kHz}= 0.64\;{\rm km\,s^{-1}}$ resolution) was placed at the frequency of the CO $J=1\mbox{--}0$ line (rest frequency 115.271 GHz) in one of the four basebands. Three 2 GHz wide low spectral resolution spectral windows were set up in the remaining basebands as a serendipitous search for continuum emission, but resulted in no detections. The FoV of $\approx 2\farcm 5 \times 2\farcm 5$ for each GMC was covered by a 27-pointing mosaic. The total on-source integration time per object was typically 540 s (20 s per mosaic pointing), and the $u\mbox{--}v$ distance typically ranged from 14 to 450 m. The gain calibration was made by observing the QSO J0635$-$7516 at a typical interval of 8 minutes. Either Uranus or the radio galaxy J0519$-$4546, whose flux was monitored by the observatory, was observed in each execution as the flux calibrator. The typical system noise temperature $T_{\rm sys}$ at the frequency of the CO line was 170 K. The shorter $u$–$v$ distance range was fulfilled by observations using the 7 m array, a part of the Atacama Compact Array [ACA; @iguchi2009]. These observations were done in 2013 November–December and 2014 April using eight to eleven 7 m antennas and the ACA correlator [@kamazaki2012]. The spectral setup was equivalent to that for the 12 m array. The number of mosaic pointings was 10 for each GMC. The typical total on-source integration time and $u$–$v$ distance were 1000 s (100 s per mosaic pointing) and 7–35 m, respectively. A gain calibrator, QSO J0635$-$7516 in most cases, was observed at a typical interval of 10 minutes. Either a solar system object (Mars, Uranus, Callisto, Ganymede, or Pallas) or J0519$-$4546 was used as the flux calibrator. The typical $T_{\rm sys}$ was 120 K. The even shorter $u\mbox{--}v$ spacing was complemented by filled-aperture (single-dish) observations with the TP array, which was also a part of the ACA. The TP array observations were carried out in 2015 May and June using two or three 12 m antennas and the ACA correlator. A $225\arcsec \times 225\arcsec$ FoV for each GMC was mapped using the on-the-fly observing technique. A line-free reference position was visited before every raster row. The QSO 3C 279 was also observed to determine the antenna gains (${\rm Jy\;K^{-1}}$). The typical total on-source time (per antenna) and $T_{\rm sys}$ were 60 minutes and 140 K, respectively. Data Reduction {#sec:reduction} ============== Interferometer visibilities and single-dish maps from the ALMA 12 m, 7 m, and TP arrays were delivered by the observatory and were calibrated using the Common Astronomy Software Applications [CASA; @mcmullin2007]. We subtracted spectral baselines from the delivered TP map by fitting straight lines to emission-free channels. We also applied the correction for the weights of 12 and 7 m visibilities with the [statwt]{} task in CASA as described in the CASA guide[^1]. A better-filled $u$–$v$ coverage is always advantageous in deconvolution. We combined the 12 m, 7 m, and TP data in $u$–$v$ space and deconvolved them jointly. We converted the TP map into visibilities by using the Total Power to Visibilities (<span style="font-variant:small-caps;">TP2VIS</span>) package that runs on the CASA platform[^2]. This procedure is discussed and tested in depth in our previous paper [@koda2011]. Here, we briefly describe the essence of this procedure. <span style="font-variant:small-caps;">TP2VIS</span> generates a Gaussian visibility distribution, so that when it is Fourier transformed its dirty image and beam represent the TP map and primary beam, respectively. The optimal weight of the TP visibilities with respect to 12 and 7 m ones can be debated. We adopted the TP weight that corresponds to the rms noise in the TP map. This way the weights represent the quality of data properly. For deconvolution, we used the Multichannel Image Reconstruction, Image Analysis, and Display (Miriad) software package [@sault1995], instead of CASA. The [clean]{} and [tclean]{} tasks in CASA (as of version 5.3) use only one dirty beam for a whole mosaic in their minor cycles, while it varies across the mosaic. This often causes a divergence in flux and/or artificial stripping patterns[^3]. Therefore, we could not use CASA for imaging. All of the visibilities were converted to the MIRIAD data format and were inverted to dirty map and beam with the natural weighting. We ran the [mossdi2]{} task for image-based clean. We set the number of iterations to 4 million and the [gain]{} parameter to 0.05. It is often a problem that the flux of the cleaned map is not consistent with that of the TP map. This is because the two components in the cleaned map use two different beams – the model component uses a convolution beam, while the residual component uses the dirty beam. The areas of the two beams are often not the same, which leads to errors in brightness and flux. To circumvent this problem, we used the residual scaling scheme of @Jorsater1995. The convolution beams for the five GMCs are typically an ellipse with a major axis size of $2\farcs 2\mbox{--}2\farcs 6$. For analyses on an equal basis, we smoothed the maps to a common $3\farcs 0$ resolution (0.73 pc) along both the major and minor axes of the beams. We use a velocity channel spacing of $1\;{\rm km\,s^{-1}}$, at which the sensitivity ranges between 0.47 and 0.67 K (45–66 ${\rm mJy\,beam^{-1}}$). Results {#sec:results} ======= The CO $J=1\mbox{--}0$ integrated and peak intensity maps are shown in Figure \[fig:intandmax1\]. Each CO $J=1\mbox{--}0$ map covers a $2\farcm 6 \times 2\farcm 8$ ($38 \times 41\rm\, pc^2$) region. A scale bar is in one of the bottom panels. We also show a pseudo-color image from [*Spitzer*]{} (24, 8.0, 3.6$\mu$m in R, G, B, respectively) and contours from the Mopra CO $J=1\mbox{--}0$ integrated intensity [@wong2011]. YSO candidates are also marked. The thick white lines indicate the areas of the ALMA CO $J=1\mbox{--}0$ observations. The centers of our ALMA mosaics were set at the peak positions in @kawamura2009, which later turned out to not coincide with the peaks at the higher-resolution Mopra CO $J=1\mbox{--}0$ map. The velocity channel maps are presented in Figures \[fig:chmap002\]–\[fig:chmap225\]. The parameters of the GMCs derived in this paper are listed in Table \[tbl:results\]. We adopt a CO-to-${\rm H_2}$ conversion factor of $7\times 10^{20}\;{\rm cm^{-2}\,(K\,km\,s^{-1})^{-1}}$ [@fukui2008]. The masses within the ALMA fields range between 1.1 and $4.9 \times 10^5 M_\sun$; 5–30% of the total GMC masses in the NANTEN catalog. Below, we note characteristics of individual GMCs. GMC 2 : : This cloud is classified as Type I, without sign of massive star formation, by @kawamura2009. Indeed, it does not have YSOs [@seale2014]. From the channel maps, the total line width (full-width zero intensity; FWZI) is $\approx 15\;{\rm km\,s^{-1}}$. We find no bright emission ($>$ 10 K). At the lower velocity (248–254 ${\rm km\,s^{-1}}$) fluffy emission is distributed over the FoV. The typical brightness temperature is 3–5 K. At 255–259 ${\rm km\,s^{-1}}$ a $\approx$ 15 pc blob dominates. It is rather featureless, i.e., its brightness temperature is more or less uniform (3–5 K). GMC 55 : : Type I, but having YSOs, and hence star forming. The total line width (FWZI) is $\approx 25\;{\rm km\,s^{-1}}$. Possibly, there are two velocity components. The most prominent structure in the lower velocity range (245–257 ${\rm km\,s^{-1}}$) is a filament, or a chain of compact clumps separated by $\approx$ 5 pc, stretching from the NE to W of the FoV. The width of the filament, or the typical size of the clumps, is $\approx$ 2 pc, and the length (in the FoV) is $\approx$ 30 pc. There is a velocity gradient along the filament (several ${\rm km\,s^{-1}}$ over 30 pc). The line width is a few ${\rm km\,s^{-1}}$ at each portion along the filament. There might be another filament, or a chain of clumps, near the SW corner running from the middle of the western edge toward the SE. At a higher velocity (255–269 ${\rm km\,s^{-1}}$) there is an elongated ($\approx 10 \times 2$ pc) bright ($\approx 14$ K) clump with a sharp boundary in the SE of the FoV. Its line width (FWZI) is $\approx 15\;{\rm km\,s^{-1}}$. GMC 197 : : Type III with both regions and young star clusters. The total line width (FWZI) is $\approx 25\;{\rm km\,s^{-1}}$. There may be two velocity components. Emission at the bottom of the FoV shows up at 218 ${\rm km\,s^{-1}}$ and this structure persists until 232 ${\rm km\,s^{-1}}$ or so (15 ${\rm km\,s^{-1}}$ width). At 224 ${\rm km\,s^{-1}}$, filamentary structures start to appear, generally elongated in the N–S direction. There is a bright spot in one of them (${\rm R.A.} \approx 5^{\rm h} 39^{\rm m} 52^{\rm s}$, ${\rm decl.} \approx -70\arcdeg 06\arcmin 45\arcsec$) that has $\approx 15\;{\rm km\,s^{-1}}$ line width and a NW-to-SE velocity gradient. It looks like filaments are spreading radially from around that spot at 228–229 ${\rm km\,s^{-1}}$. The filaments merge into a $\approx$ 20 pc blob seen at 224–234 ${\rm km\,s^{-1}}$. After that blob is gone, another blob appear in the south, with a several ${\rm km\,s^{-1}}$ line width. GMC 216 : : Type II with regions and YSOs. The total line width (FWZI) is $\approx 15\;{\rm km\,s^{-1}}$. At lower velocities (221–225 ${\rm km\,s^{-1}}$), at least four filamentary structures are apparent and form a web. Along the eastern boundary of the map, there is one filament stretching straight from N to S (${\rm R.A.}\approx 5^{\rm h} 44^{\rm m} 47^{\rm s}$). Two filamentary structures run from NEE to W and SE to W, and apparently merge near the western boundary to form a $\approx$ 15 pc blob with a line width of several ${\rm km\,s^{-1}}$. These structures evolve into more spatially extended blobs at higher velocities (226–230 ${\rm km\,s^{-1}}$). Hence, it is not clear if they are filaments, or parts of blobs with large velocity widths. They may be shearing surfaces of, e.g., cloud–cloud collisions. There is another shorter, 10 pc scale, filament that runs in parallel to and north of the NEE–W filamentary structure. At 225–231 ${\rm km\,s^{-1}}$ the emission mostly fills the FoV (typically 5–10 K, with some bright spots). GMC 225 : : Type I, but having YSOs and star formation. The total line width (FWZI) is $\approx 10\;{\rm km\,s^{-1}}$. There are two components, one in the N and the other in the S. One is a $\approx$ 15 pc blob in the NW (which may extend beyond the FoV). The velocity range is 213–223 ${\rm km\,s^{-1}}$. The brightness is moderately high ($\approx 10$ K) in its northern half and lower ($\approx 5$ K) in the south. The southern component is at 216–221 ${\rm km\,s^{-1}}$. It may consist of two filaments running parallel to each other. They have a roughly 15$\times$2 pc geometry within the FoV. [cccccccc]{} 2 & $4^{\rm h} 47^{\rm m} 35^{\rm s}$ & $-67\arcdeg 13\arcmin 00\arcsec$ & 0.46 & 1.3 & 0.1 & $-\infty$\ 55 & $5^{\rm h} 07^{\rm m} 45^{\rm s}$ & $-68\arcdeg 59\arcmin 30\arcsec$ & 0.54 & 1.4 & 0.3 & $-0.92$\ 197 & $5^{\rm h} 39^{\rm m} 50^{\rm s}$ & $-70\arcdeg 07\arcmin 00\arcsec$ & 0.57 & 4.7 & 0.05 & $-1.50$\ 216 & $5^{\rm h} 44^{\rm m} 35^{\rm s}$ & $-69\arcdeg 26\arcmin 00\arcsec$ & 0.47 & 4.9 & 0.2 & $-0.83$\ 225 & $5^{\rm h} 47^{\rm m} 20^{\rm s}$ & $-70\arcdeg 41\arcmin 30\arcsec$ & 0.67 & 1.1 & 0.1 & $-1.55$\ Discussion {#sec:discussion} ========== We observed five GMCs in the LMC along the GMC evolutionary stages classified by @kawamura2009. As discussed in Section \[sec:targets\], additional information became available after this work started, showing YSOs in GMC 55 and 225, which were originally classified as Type I (no sign of massive star formation). Among the five, only GMC 2 does not show any sign of star formation. In addition, it turned out that our FoVs do not optimally cover the target GMCs. Even though our ALMA mosaics are large ($2\farcm 6 \times 2\farcm 8$; or $38\times 41\;{\rm pc^2}$), the emission peaks at the $45\arcsec$ resolution [@wong2011] were considerably offset from the peaks at the $2\farcm 6$ resolution [@fukui2008]. With these limitations in mind, there still seems to be a trend of GMC evolution, when our observations are viewed in the context of work in the literature. The Brightness Distribution Function and Index {#sec:BDF} ---------------------------------------------- The maps of the five GMCs (Figure \[fig:intandmax1\]) clearly show less developed internal structures in GMC 2 than in the others. GMC 2 is the one without any sign of star formation, while the others have some associated YSOs. In order to characterize the visual difference in internal structures, @sawada2012 introduced simple tools: the BDF and BDI. The BDF is a histogram of the brightness of a line emission, which quantitatively represents the map appearance. It is defined in $\alpha$–$\delta$–$v$ space as the fraction of the “pixels” with brightness between $T$ and $T+dT$. The BDI is a single number that represents the characteristics of the BDF: the fraction of the bright, structured gas over the bulk unstructured, faint gas. It is written as the flux ratio of the bright emission to faint emission: $$\begin{aligned} {\rm BDI} &=& \log_{10} \left( \frac{\int_{T_2}^{T_3} T\cdot B(T) dT} {\int_{T_0}^{T_1} T\cdot B(T) dT} \right) \nonumber\\ &=& \log_{10} \left( \frac{\sum_{T_2<T[i]<T_3}T[i]}{\sum_{T_0<T[i]<T_1}T[i]} \right), \label{eq:bdi}\end{aligned}$$ where $B(T)$ denotes the BDF; $T_0$, $T_1$, $T_2$, and $T_3$ are the brightness thresholds. $T[i]$ is the brightness of the $i$th pixel in the $\alpha$–$\delta$–$v$ space. In this paper, we adopt $(T_0, T_1, T_2, T_3) = (3, 5, 10, \infty)$ \[K\], the same brightness thresholds used for the MW analysis in @sawada2012. A high BDI indicates that the gas is structured and has more compact, bright structures. The BDF/BDI analysis revealed that in the MW disk, structured gas, represented by the high BDI ($\sim -1$ in CO $J=1\mbox{--}0$ seen at $\simeq 0.7$ pc resolution), is distributed along the spiral arms, while unstructured (low BDI; $\sim -3$) gas exists in the inter-arm regions. Thus, there is an evolution in gas structure across the spiral arms. Figure \[fig:BDF\] (right) shows the BDFs of the molecular gas in the MW around the Galactic longitude of $38\arcdeg$ from @sawada2012. Assuming the locations of the Sagittarius arm (near and far sides) and inter-arm regions as discussed in @sawada2012, the spatial scales are smoothed to match the 0.7 pc resolution of this LMC study. The BDFs of the spiral arms show a tail toward high brightness temperature, while the inter-arm molecular gas does not show the tail. The red and blue shadings are displayed for comparisons with the GMCs in the LMC below. The BDF/BDI analysis has been also applied to external galaxies [@hughes2013]. We applied the BDF/BDI analysis to the five GMCs. The data are smoothed to a $2\;{\rm km\,s^{-1}}$ velocity resolution to achieve better sensitivity. The resultant spatial (0.7 pc) and velocity resolutions and sensitivity (0.37–0.48 K) are comparable to the MW study [@sawada2012]. Figure \[fig:BDF\] (left) shows the BDF of the GMCs. The red and blue shadings are the same as those in the right panel; the red indicates the locus of the BDFs of the spiral arms in the MW, while the blue is for the inter-arm region. GMC 2 is the only GMC without any sign of star formation, and its structure is similar to the quiescent molecular gas in the MW inter-arm regions. Its BDI is very low ($-\infty$; meaning no gas with high brightness $>10$ K). The other four GMCs show some signs of star formation, and their BDFs, $-1.5$ to $-0.8$, are similar to those of the spiral arms in the MW. The BDIs of the GMCs are shown in the figure legend and Table \[tbl:results\]. Although the correlation exists between the high BDI and star formation activity, the cause of the complex structure in molecular clouds is debatable as to whether they are a precursor of star formation or developed by stellar feedback. We, however, note that in the MW high BDI regions do not necessarily coincide with regions, and hence indicating the former [@sawada2012let see the next subsection]. Synthesis with Work in the Literature ------------------------------------- Using the BDF/BDI, @sawada2012let demonstrated that in the MW, structured gas, represented by a high BDI, is distributed along the spiral arms, while unstructured (low BDI) gas exists in the inter-arm regions (see Figure \[fig:BDF\] right). The overall distribution of the high BDI gas is similar to that of the regions. They also found some moderately high BDI regions, which are massive molecular concentrations and located in the spiral arms, but without regions. These regions may be in the phase between quiescent and active star-forming GMCs. Compact, and presumably dense, structures have already developed in molecular gas, but in which star formation has not yet started. They may be pre-star-forming complexes. Similar structural differences are present among local star-forming and quiescent molecular clouds. @kainulainen2009 analyzed the probability distribution function (PDF) of column density derived from the near-infrared dust extinction. They found that the quiescent clouds show a log-normal shape of the PDF as predicted for a turbulent medium, while the star-forming clouds show excess “wings” at higher column densities, presumably due to the self-gravity of dense regions [@klessen2000; @federrath2008]. The PDF and BDF appear very similar if the CO brightness is converted to column density using the CO-to-${\rm H_2}$ conversion factor. In fact, the PDFs of star-forming and quiescent clouds are [*quantitatively*]{} consistent with the BDFs of arm and inter-arm molecular gas, respectively [@sawada2012; @sawada2012let]. A caveat is that it is not guaranteed that the conversion factor is applicable on a pixel-by-pixel basis. Other optically thin lines, e.g., $^{13}{\rm CO}$ or ${\rm C^{18}O}$, may trace dense gas better than $^{12}{\rm CO}$, but may miss extended, diffuse gas components (i.e., the denominator of the BDI). It is also possible that the emission lines may not necessarily trace the column/volume density of the gas, but rather the excitation of the spectral lines. We, however, note that @sawada2012 found a consistent result with $^{12}{\rm CO}$ and $^{13}{\rm CO}$ in the BDF analysis in the MW. Most ALMA studies have been focused on GMCs with active star formation (Figure \[fig:overall\]). A quantitative analysis of their data is beyond the scope of this paper, but qualitatively we find a similar trend: structured gas in star-forming clouds and less structured in quiescent clouds. @indebetouw2013 observed the northern part of 30 Doradus, the most active star-forming region in the Local Group. In this region, a large fraction of the molecular gas – about 50% – has a brightness temperature above 10 K (see their Figure 4). This cloud has already developed dense structures. In our five GMCs, only the densest clumps have such high brightness temperatures. N159 is another active star-forming region near 30 Doradus that has also been observed with ALMA [@fukui2015; @saigo2017; @nayak2018]. The presence of dense clumps and filaments led the authors to speculate on collisions of clouds or filaments as a possible cause of the structures. N55 is an region in the largest SGS in the LMC. A cloud near N55 is also very clumpy [@naslim2018]. The same is true in the star-forming cloud N83C in the SMC [@muraoka2017]. @wong2017 presented the first ALMA study to contrast star-forming and quiescent molecular clouds in the LMC. They observed a quiescent cloud, designated as the Planck Galactic Cold Clump (PGCC) G282.98$-$32.40 [@planck2016 Wong et al. called this the “Planck cold cloud” or PCC]. It does not show any signs of active massive star formation, and the estimated dust temperature is low ($T_{\rm d}\sim$15 K). The CO brightness distribution shows a much lower contrast than that in the 30 Doradus cloud and is relatively uniform with only one hotspot. The quiescent cloud is less structured. By synthesizing our five GMCs, one quiescent and four star-forming, with those in the literature, a simple view of a GMC evolutionary sequence is tempting: clouds start from a relatively structureless state, develop complex internal structures, and, as a result, form stars. Caveats ------- Of course, this evolutionary sequence in star formation activity may be too simplistic, as other factors, such as environmental triggers, might be at a play. For example, interactions with SGSs may be another important factor [@meaburn1980; @kim1999; @yamaguchi2001; @book2009; @dawson2013; @fujii2014]. In fact, GMCs 55, 197, and 216 show relatively high BDIs and are located at the edges of SGS complexes identified by @dawson2013. The question here is what is the direct cause of the gas with high brightness temperature (perhaps, dense gas). If compression by an SGS directly changes the cloud structure (hence, BDF or PDF) without invoking gas self-gravity, we may need to consider its effects separately from the internal physics. The other possible factor is surrounding GMCs. Figure \[fig:intandmax1\] (left) shows that some of the GMCs in @fukui2008 are GMC complexes rather than single GMCs. These are some of the reasons why even the relatively large ALMA mosaics ($\approx40$ pc) are not enough to cover the entire emission. If the surrounding GMCs trigger compression, e.g., by tides or collisions, the environment would be another factor to consider. Studies into multiple factors clearly require a much larger sample of GMCs in the LMC. Such statistics should be possible with ALMA as it is approaching its operational maturity. This paper makes use of the following ALMA data: ADS/JAO.ALMA\#2012.1.00641.S. ALMA is a partnership of ESO (representing its member states), NSF (USA) and NINS (Japan), together with NRC (Canada), MOST and ASIAA (Taiwan), and KASI (Republic of Korea), in cooperation with the Republic of Chile. The Joint ALMA Observatory is operated by ESO, AUI/NRAO and NAOJ. This research made use of Astropy, a community-developed core Python package for Astronomy [@astropy2013]. This research made use of Montage. It is funded by the National Science Foundation under grant number ACI-1440620, and was previously funded by the National Aeronautics and Space Administration’s Earth Science Technology Office, Computation Technologies Project, under Cooperative Agreement Number NCC5-626 between NASA and the California Institute of Technology. J.K. acknowledge the support from the NSF under grant AST-1211680. We thank J. Barrett for improving the manuscript. Astropy Collaborations, Robitaille, T. P., Tollerud, E. J., et al. 2013, , 558, A33 Book, L. G., Chu, Y.-H., Gruendl, R. A., & Fukui, Y. 2009, , 137, 3599 Claude, S., Jiang, F., Niranjanan, P., et al. 2008, , 7020, 70201B Dawson, J. R., McClure-Griffiths, N. M., Wong, T., et al. 2013, , 763, 56 Escoffier, R. P., Comoretto, G., Webber, J. 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--- abstract: 'We present novel and simple estimation of a minimal dimension required for an effective reservoir in open quantum systems. Using a tensor network formalism we introduce a new object called a reservoir network (RN). The reservoir network is the tensor network in the form of a Matrix Product State, which contains all effects of open dynamics. This object is especially useful for understanding memory effects. We discuss possible applications of the reservoir network and the estimation of dimension to develop new numerical and machine learning based methods for open quantum systems.' author: - 'I. A. Luchnikov' - 'S. V. Vintskevich' - 'S. N. Filippov' title: Dimension truncation for open quantum systems in terms of tensor networks --- Introduction ============ One of the most challenging and interesting problems of modern theoretical physics is simulation of many-body quantum systems. The dimension of Hilbert space grows exponentially with number of particles. This makes direct simulations impossible. There are many analytical and numerical approaches, but all of them have limits of applicability. Very few models can be solved analytically, for example, using Bethe ansatz [@1]. Perturbation theory can be used only for systems with weak interactions. In case of systems with strong interaction, numerical approaches are very successful, but also have problems. For example, tensor networks and DMRG- based methods [@2; @3; @4; @5; @6; @7; @8] work well only for a $1d$ case, and cannot predict long-term time dynamics (because of Leib-Robinson boundary [@9]). Monte Carlo- based methods [@10] in fermionic case have sign problem. The many-body problem is still relevant and requires new approaches.\ We can divide this problem into two parts: stationary problem and dynamical problem. In this paper we focus on dynamic properties. In the case of real experiment we are very often interested only in the dynamics of small subsystem of the whole many-body system (open quantum systems dynamics). Problem of open quantum system dynamics [@11; @12; @13] without acceptance of Markov approximation is impossible to solve directly because of exponentially large dimension of a reservoir. In this work, we consider low-dimension reservoir approximation for an open quantum system. We provide a simple estimation of reservoir sufficient dimension in terms of **memory length, minimum time scale of reservoir, characteristic interaction constant** and **number of terms in the part of Hamiltonian describing interaction between system and reservoir**. The most remarkable thing here that this estimation point to existence of low-dimension effective reservoirs. This allow us to perform simulations on classical computer using effective reservoirs. It is possible to develop new numerical and machine learning-based methods using low-dimension structure of reservoir in principle. The examples of successful application of the machine learning methods to many-body quantum physics can be found in [@14]. Also we introduce a new object - **Reservoir Network**, which can be represented in Matrix Product State form [@2] or in Artificial Neural Network form [@14]. In the limit $\tau\rightarrow0$ this object is very similar to Continuous Matrix Product State [@21]. Finally, we estimate sufficient dimension of the reservoir applying tensor network formalism and providing estimation of an entangled entropy upper bound. At last, we can say that our work is another view of the Time-evolving block decimation algorithm [@15] at an angle of $90$ degrees. Tensor networks representation for quantum reservoir ==================================================== In this section we consider the most general model of open quantum system. The Hilbert space of the whole system plus reservoir ($S+R$) is ${\cal H}={\cal V}_S\otimes{\cal V}_R$. Usually, dimension of ${\cal V}_S$ is small, for instance it can be qbit space. On the other hand, a ${\cal V}_R$ is many-body quantum system with huge dimension. Naturally, it is assumed that dynamics of the whole system is completely dissipative, the memory length is finite and the Poincare recurrence time is infinite. All these assumptions lead to the Area law’s analog in the condensed matter physics [@16; @17] as shown further. The Hamiltonian of our system is $H=H_S\otimes I+I\otimes H_R + \gamma \sum_{i=1}^{n}A_i\otimes B_i$, where $\gamma$ is a characteristic interaction rate. Let us divide Hamiltonian into two parts, $H_0=H_S\otimes I+I\otimes H_R$ and $H_{int}=\gamma \sum_{i=1}^{n}A_i\otimes B_i$ , with initial density matrix of whole system $\rho(0)=\rho_S\otimes\rho_R$. Now let us consider the dynamical properties of whole system. Its evolution can be represented in terms of the Trotter decomposition: $$\begin{aligned} \rho(T)=\lim_{\tau\rightarrow 0}\ \underbrace{\Phi_0[\tau]\circ \Phi_{\rm int}[\tau]\circ \dots \circ\Phi_0[\tau]\circ \Phi_{\rm int}[\tau]}_{T/\tau}\rho(0).\end{aligned}$$ Where $\Phi_0[\tau]=\exp[\tau {\cal L}_0]$, ${\cal L}_0\rho=-i[H_0,\rho]$ and $\Phi_{\rm int}[\tau]=\exp[\tau {\cal L}_{\rm int}]$, ${\cal L}_{\rm int}\rho=-i[H_{\rm int},\rho]$. Superoperators $\Phi_0$ and $\Phi_{\rm int}$ describe free evolution and interaction dynamics respectively in the the time scale $\tau$. We can illustrate all objects in the Trotter decomposition in tensor networks terms (Figure \[fig1\]). ![Key elements of tensor network representing system-reservoir dynamics: initial density matrices, free-evolution and interaction superoperators respectively.[]{data-label="fig1"}](fig1-01.eps) Trotter decomposition in this terms takes the following form (Figure \[fig2\]). ![Tensor network. Edges represent convolutions.\ a) Tensor network representation of Trotter decomposition.\ b) The representation of $\Phi_{\rm int}(\tau)$ as a convolution of two tensors, details are shown in the Appendix 1.[]{data-label="fig2"}](fig2a3.eps) Finally, the partial density matrix of the system takes the following form (Figure \[fig3\]). ![Tensor networks representation of partial density matrix of system. A red loop on the right side represents trace taken over reservoir.[]{data-label="fig3"}](fig33-01.eps) At this stage we introduce new interesting and important object - **Reservoir Network (RN)**. The RN is schematically represented in (Figure \[fig4\]). It contains all information about quantum reservoir and about all open dynamics effects (memory effects, dissipation, etc.), and is obtained by cutting tensor network horizontally where the lower part is another tensor network. ![Reservoir network. It is exactly the same as a Matrix Product State network. (an analog of $1{\rm d}$ quantum chains).[]{data-label="fig4"}](fig44-01.eps) This object is exactly the same as the Matrix Product State and we can treat this object like a quantum state [@2]. Furthermore, the analog of a density matrix and partial density matrix for this object can be constructed in terms of tensor networks. Also, let us define the connection between a memory length and a mutual information for the RN. It is known that mutual information between two subsystems with density matrices $\rho_{A}$ and $\rho_{B}$ of a quantum system in non-critical $1d$ case satisfies an inequality $I(\rho_A,\rho_B)\leq C\exp\Big[-\frac{l}{L}\Big]$. Here $l$ is number of vertexes between $A$ and $B$ subsystems and $L$ is a correlation length. $I(\rho_A,\rho_B)=S(\rho_A)+S(\rho_B)-S(\rho_{AB})$. In our case correlation length $L$ is replaced by characteristic memory length. In this case, we can expect behavior similar to the case of $1{\rm d}$ low energy quantum chains with a local Hamiltonian. We can expect that there is an MPS approximation with low dimension auxiliary space (in our case auxiliary space is reservoir space). For a better understanding connection between the RN and the MPS is presented in the Table 1. [p[4.2cm]{}|p[4.2cm]{}]{} Reservoir Network & Matrix Product State\ \ Time & Spatial coordinate\ \ Memory length & Correlation length\ \ Dimension fo reservoir & Dimension of auxiliary space\ \ $n$ from $H_{\rm int}=\gamma \sum_{i=1}^{n}$\ $A_i\otimes B_i$ (see Appendix 1) & Dimension of chain subsystem\ \ Area law (because memory length is finite) & Area law\ \ Mutual Information & Entangled entropy \ Now let us do a non-formal estimation of sufficient dimension for auxiliary space. It is known that the MPS with small entangled entropy can be truncated. Let us give a remind how it can be done, for example, in the condensed matter physics and quantum information [@18; @19]. Suppose we have two component quantum system. Wave function of this system takes the following form $\ket{\psi}=\sum_{i,j}\Psi_{i,j}\ket{i}\otimes\ket{j}$. We can do Singular Value Decomposition (SVD) for $\Psi_{i,j}$ matrix. $\ket{\psi}=\sum_{i,j,\alpha}U_{i \alpha}\lambda_\alpha V^\dagger_{\alpha j}\ket{i}\otimes\ket{j}=\sum_\alpha \lambda_\alpha \ket{\alpha}\otimes\ket{\alpha}$, which is also known as the Schmidt decomposition. Number of non-zero singular values is called the Schmidt rank. We can truncate Schmidt rank neglecting the minimum singular values (optimal low-rank approximation). Let us define the new small Schmidt rank by considering partial density matrices for subsystems. $\rho_1=\sum_{j}\lambda_\alpha^2\ket{\alpha}\bra{\alpha}$ and $\rho_2=\sum_{j}\lambda_\alpha^2\ket{\alpha}\bra{\alpha}$ density matrices for the first and second subsystems respectively. Entangled entropy is $S(\rho_1)=-\sum_j\lambda_j\log\lambda_j$. For whole system with the Schmidt rank $d$, the maximal entangled entropy is defined by $S_{\rm max}=\log d$. Here we emphasize that a naturally sufficient dimension takes following form $\log(d_{\rm suff})=S$. The trick is that the Schmidt rank describes only the entangled entropy less or equal then $S$. As a result, for an auxiliary space, sufficient dimension is given by $d_{\rm suff}=\exp S$. We use this trick in Appendix $2$ for more precise sufficient dimension estimation.\ On the next step we consider **Reservoir Partial Density Matrix (RPDM)** for our RN (Fig.\[fig5\]). Notice that ${\rm Number\ of\ connections} \propto T$ in the considered PRDM and $d_{\rm suff}\propto\exp[{\rm Number\ of\ connections}]$. Finally, $d_{\rm suff}\propto\exp[\gamma T]$, where $\gamma$ appears from simple physical dimensional analysis. More precise estimation is: $$\begin{aligned} d_{\rm suff}\approx\exp\left(2n\gamma T\left[1-\log\left(\gamma t\right)\right]\right).\end{aligned}$$ The derivation is presented in the Appendix 2. ![image](fig5-01.eps) Discussion ========== Based on analogy between Matrix Product State and time decomposition of reservoir we developed new language for open quantum systems (OQS) which can be effectively applied to its numerical analysis. For instance, we showed existence of valid low dimension approximation of quantum reservoir for any open quantum systems with finite memory length. Let us consider a simple example to understand characteristic dimension of effective low-dimension reservoir. Suppose, that $\frac{1}{\gamma}\approx T$, $n=2$ and $t\approx\frac{1}{5\gamma}$. From eq. (2) one can find that $d_{\rm suff}\approx1000$. As one can see, it is a reservoir containing just $10$ qbits and can be easily simulated on a computer. Using this fact, we can develop innovative numerical and machine learning based methods of research for the OQS. For example, choosing ensemble of preliminary measurements, one can reconstruct a structure of quantum reservoir. Such reconstruction procedure is possible because the low dimension approximation is described by a small number of parameters. This technique is very similar to the Hidden Markov Models in the voice recognition problem [@20]. On the other hand, we can try to use variation principle and reconstruct a reservoir using optimization algorithms. The considered approximation becomes similar to optimization based reconstruction procedure of the MPS in the condensed matter physics. We suggest a new family of objects (RN and etc.) which can provide a convenient and useful formalism for open quantum systems. Furthermore, existence of such approximation is an interesting fundamental result. Appendix 1 ========== Here we present some detailed explanations of RN definition. The Hamiltonian of our system is given by $$\begin{aligned} H=H_{S}\otimes I+I\otimes H_{R}+\gamma\sum^n_{i=1} A_i \otimes B_i. \label{0}\end{aligned}$$ For better understanding let us introduce the new notation. First, we represent the density matrix in the column form: $$\begin{aligned} &&\varrho=\sum_{i,j}\rho_{i,j}\ket{i}\bra{j}\ \rightarrow \ \ket{\varrho}=\sum_{i,j}\rho_{i,j}\ket{i}\otimes \ket{j}, \nonumber \\ &&A\varrho B\ \rightarrow \ A\otimes B^T \ket{\varrho}.\end{aligned}$$ Second, the time dynamics of density matrix at time $\tau$ takes the following form: $$\begin{aligned} \ket{\varrho(\tau)}=\exp[-i\tau H]\otimes \exp[i\tau H^*]\ket{\varrho}.\end{aligned}$$ Using the Baker Hausdorff equality and assuming that $\tau$ is small, one can obtain the following expression: $$\begin{aligned} &&\exp[-i\tau H]\otimes \exp[i\tau H^*]=\nonumber\\&&=\exp[-i\tau H_{0}]\otimes\nonumber\\&&\otimes \exp[i\tau H_{0}^*]\exp[-i\tau H_{\rm int}]\otimes \exp[i\tau H_{\rm int}^*]+O(\tau^2).\end{aligned}$$ Where $H_0=H_{S}\otimes I+I\otimes H_{R}$ and $H_{\rm int}=\gamma\sum^n_{i=1} A_i \otimes B_i$. Here we divided dynamics of whole system into two parts $$\begin{aligned} \Phi_0=\exp[-i\tau H_{0}]\otimes \exp[i\tau H_{0}^*].\end{aligned}$$ $$\begin{aligned} &&\Phi_{\rm int}(\tau)=\exp[-i\tau H_{\rm int}]\otimes \exp[i\tau H_{\rm int}^*]=\nonumber\\&&=I\otimes I \otimes I \otimes I -i\tau\gamma\sum_{i=1}^n A_i\otimes B_i \otimes I \otimes I+\nonumber\\&&+i\tau\gamma\sum_{i=1}^n I \otimes I \otimes A_i^*\otimes B_i^*+O(\tau^2).\end{aligned}$$ For convenience, we rearrange the Hilbert spaces: $$\begin{aligned} &&\Phi_{\rm int}(\tau)=I\otimes I \otimes I \otimes I -i\tau\gamma\sum_{i=1}^n A_i\otimes I \otimes B_i \otimes I+\nonumber\\&&+i\tau\gamma\sum_{i=1}^n I \otimes A_i^* \otimes I \otimes B_i^*+O(\tau^2).\end{aligned}$$ Using this notation we can rewrite it in the short form: $$\begin{aligned} \Phi_{\rm int}(\tau)=\sum_{i=0}^{2n} {\cal A}_i \otimes {\cal B}_i.\end{aligned}$$ Where, $$\begin{aligned} &&{\cal A}_i=\begin{cases} I\otimes I,\ i=0 \\\sqrt{\gamma\tau}A_i \otimes I,\ 0<i<n+1 \\ \sqrt{\gamma\tau}I\otimes A_{i-n}^*,\ n<i, \end{cases}\nonumber\\ &&{\cal B}_i=\begin{cases} I\otimes I,\ i=0 \\-i\sqrt{\gamma\tau} B_i \otimes I,\ 0<i<n+1 \\i\sqrt{\gamma\tau} I\otimes B_{i-n}^*,\ n<i. \end{cases} \label{1}\end{aligned}$$ Such separation of $\gamma\tau$ is non-trivial. In order to explain this separation we consider a two point correlation function for the RN. This function is well-defined only for presented separation for $\gamma\tau \rightarrow \sqrt{\gamma\tau}$ and $\sqrt{\gamma\tau}$. Our goal is to represent the whole system evolution in terms of tensor networks (Fig\[fig6\]). ![Reservoir network. Notice that superscripts $S$ and $R$ correspond to upper and lower parts of the network respectively. For simplicity of diagrams, these superscripts are omitted further.[]{data-label="fig6"}](fig6.eps) We can give diagrammatic formulation for the density matrix of the system in terms of tensor network diagrams as well as for the RN. (Fig\[fig7\]). ![Density matrix of system $+$ reservoir in terms of the Tensor Network and the Reservoir Network.\ a) Tensor network representation for $\rho_{\rm S}(T)$.\ b) Tensor network representation for quantum reservoir. This object is actually multidimensional tensor in the MPS form. Any time marks $\tau, 2\tau, 3\tau$ etc. are equivalent to the network nodes of the MPS, whereas full number of nodes equals $T/\tau$.[]{data-label="fig7"}](fig7-01.eps) Owing to Reservoir Network (RN) can be interpreted as a quantum state (formal analogy: RN $\rightarrow$ $\ket{\psi}$) one can discern a generalization of this analogy further. Based on the RN new tensor network can be easily constructed, which we may treat as a reservoir density tensor network (RDN) in accordance with complete analogy of a density matrix. This idea is illustrated in the (Fig.\[fig8\]). ![Density matrix of system in terms of the Tensor Network and the Reservoir Network (RN). Diagrammatic construction of Reservoir Density Network (RDN) and Partial Reservoir Density Network.\ a) Reservoir network (RN) in terms of the MPS and Reservoir density network (RDN)\ b) The replacement and simplification of nodes of in the RDN\ c) The RDN after simplification\ d) The Partial Reservoir Density Network (PRDN). Connected indeces between upper and lower parts of network mean partial trace in selected subsystem, which starts from some node with number N.[]{data-label="fig8"}](fig8-01.eps) Appendix 2 ========== The Appendix shows the estimation of entangled entropy. Let us start with simple example: consider an arbitrary density matrix decomposition on the **not orthogonal** and **not normalize** set $\{\ket{j}\}$. In accordance with previous discussion more precise estimation of entangled entropy needs to be done. In our case for $\rho=\sum_j \ket{j}\bra{j}$ the entropy of this density matrix is $S\leq-\sum_j \braket{j|j}\log \braket{j|j}$. Let us prove this inequality. We can rewrite object $\ket{j}\bra{j}$ in the following form: $$\begin{aligned} \ket{j}\bra{j}=q_jP_j,\end{aligned}$$ where $P_j^2=P_j$ and $q_j=\braket{j|j}$. For the Renyi entropy with $\alpha\geq1$ we have: $$\begin{aligned} &&S_\alpha=\frac{1}{1-\alpha}\log\left[{\rm tr}\left(\sum_j q_jP_j\right)^\alpha\right]=\nonumber\\&&=\frac{1}{1-\alpha}\log\left[{\rm tr}\left(\sum_j q_jP_j\right)^\frac{\alpha}{2}\left(\sum_j q_jP_j\right)^\frac{\alpha}{2}\right]=\nonumber\\&&= \frac{1}{1-\alpha}\log\left[\Big\|\sum_j q_jP_j\Big\|_\alpha^\alpha\right]\leq\nonumber\\&&\leq\frac{1}{1-\alpha}\log\left[\Big(\sum_j q_j\|P_j\|_\alpha\Big)^\alpha\right]\leq\nonumber\\&&\leq\frac{1}{1-\alpha}\log\left[\sum_j q_j^\alpha\right]\end{aligned}$$ For $\alpha=1$ (Von Neumann entropy) the inequality is proved. Similarly we estimate upper bound of the entangled entropy for our PRDM, where the PRDM is assumed to be normalized. Following chain of transformations leads to cut of PRDM’s tail. Thus procedure is presented on (Fig\[fig9\]). The whole chain of transformation based on inequality $I(\rho_A,\rho_B)\leq C\exp\Big[-\frac{l}{T}\Big]$.Here $T$ is memory length, l is number of blocks between $A$ and $B$ subsystems, $C$ is just some constant. ![Chain of transformations for RDN.\ a) A selection subsystem of reservoir with some memory time $T$.\ b) Using assumption $I(\rho_A,\rho_B) \approx 0 $ for finite $T$, inner part can be replaced by ancillary tensor network without this inner part. The ancillary network doesn’t have any connection between remaining parts from both sides of the replaced one.\ c) The next step we stress that the right part of the network obtained from previous step b) essentially is a constant, which can be represented as an arbitrary convolution.\ d) Previous steps show that inner part can be replaced to the tensor without internal convolutions and vice versa. Such opportunity occurs because of absence of any correlations between parts on sides from inner one.[]{data-label="fig9"}](fig9-01.eps) Finally, using $S\leq-\sum_j \braket{j|j}\log \braket{j|j}$ we can represent entropy estimation in the diagrammatic form Fig.(\[fig10\]). ![Entropy estimation within diagrammatic representation.[]{data-label="fig10"}](fig10-01-01.eps) Estimation of logarithmic term : $$\begin{aligned} &&{\cal A}_i=\begin{cases} I\otimes I,\ i=0 \\\sqrt{\gamma\tau}A_i \otimes I,\ 0<i<n+1 \\ \sqrt{\gamma\tau}I\otimes A_{i-n}^*,\ n<i, \end{cases}\nonumber\\ &&{\cal B}_i=\begin{cases} I\otimes I,\ i=0 \\-i\sqrt{\gamma\tau} B_i \otimes I,\ 0<i<n+1 \\i\sqrt{\gamma\tau} I\otimes B_{i-n}^*,\ n<i. \end{cases} \label{1}\end{aligned}$$ We can state (see Appendix 1) $$\begin{aligned} {\rm Norm}_{i_1,i_2,\dots,i_{T/\tau}}\approx N (\gamma\tau)^{I(i_1)+I(i_2)+\dots+I(i_{T/\tau})},\end{aligned}$$ where ${\rm Norm_{i_1,i_2,\dots,i_{T/\tau}}}$ is the term under logarithm, $N$ is some constant and\ $I(i)=\begin{cases}1,\ \ i\neq0\\0,\ \ i=0\end{cases}$. Now we should derive constant $N$. We can use normalization condition: $$\begin{aligned} 1=\sum_{i_1,i_2,\dots,i_{T/\tau}}{\rm Norm}_{i_1,i_2,\dots,i_{T/\tau}}=N(1+2n\gamma\tau)^{T/\tau}.\end{aligned}$$ Now we can find entropy estimation $$\begin{aligned} &&S\approx -\frac{2n\gamma T}{1+2n\gamma\tau}\log\frac{\gamma\tau}{1+2n\gamma\tau}-\nonumber\\&&-\frac{1}{1+2n\gamma\tau}\log\frac{1}{1+2n\gamma\tau}\end{aligned}$$ For the small $\gamma\tau$ we have $$\begin{aligned} S\approx2n\gamma T\left[1-\log\left(\gamma\tau\right)\right]\end{aligned}$$ Unfortunately, this expression diverges in the limit $\tau\rightarrow0$. However, divergence has simple explanation and can be removed. This problem is similar to the problem of the divergence of the classical ideal gas entropy. The solution based on fact that the minimal phase volume for the particle state is equal $(2\pi\hbar)^3$. Hence the entropy of the classical ideal gas becomes finite. But what is the analog of this phase volume in this particular case? Let us consider some $\epsilon$ neighborhood in the time axis. If $\epsilon$ is small enough, we can replace any physical indices in this $\epsilon$ neighborhood Fig(\[fig11\]). ![Replacement indices in small time-neighbourhood of RN doesn’t change it much[]{data-label="fig11"}](fig11-01.eps) Thus, $\epsilon$ neighborhood is a direct analog of minimal phase volume for the gas. What is the physical meaning of this? We can say that this is **minimal time scale for the reservoir** - $\tau_{\rm min}$. Using this fact we can replace $\tau$ to $\tau_{\rm min}$ in our expressions and divergence consequently disappears . Finally, taken previous assumptions into account for the expression of the entropy we have: $$\begin{aligned} S\approx2n\gamma T\left[1-\log\left(\gamma t\right)\right].\end{aligned}$$ As a main result the sufficient dimension of reservoir is given by: $$\begin{aligned} d_{\rm suff}\approx\exp\left(2n\gamma T\left[1-\log\left(\gamma t\right)\right]\right).\end{aligned}$$ [9]{} H. Bethe, Z. Phys. A **71**, 205 (1931). R. Orus, “A practical introduction to tensor networks: Matrix product states and projected entangled pair states,” Annals of Physics **349**, 117–158 (2014), arXiv:1306.2164. R. Or´us, “Advances on tensor network theory: symmetries, fermions, entanglement, and holography,” European Physical Journal B **87**, 280 (2014), arXiv:1407.6552. F. Verstraete, V. Murg, and J. I. Cirac, “Matrix product states, projected entangled pair states, and variational renormalization group methods for quantum spin systems,” Advances in Physics **57**, 143–224 (2008), arXiv:0907.2796. I. 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--- abstract: 'The high planetary multiplicity revealed by *Kepler* implies that Transit Time Variations (TTVs) are intrinsically common. The usual procedure for detecting these TTVs is biased to long-period, deep transit planets whereas most transiting planets have short periods and shallow transits. Here we introduce the Spectral Approach to TTVs technique that allows expanding the TTVs catalog towards lower TTV amplitude, shorter orbital period, and shallower transit depth. In the Spectral Approach we assume that a sinusoidal TTV exists in the data and then calculate the improvement to $\chi^2$ this model allows over that of linear ephemeris model. This enables detection of TTVs even in cases where the transits are too shallow so individual transits cannot be timed. The Spectral Approach is more sensitive due to the reduced number of free parameters in its model. Using the Spectral Approach, we: (a) detect 131 new periodic TTVs in *Kepler* data (an increase of $\sim2/3$ over a previous TTV catalog); (b) Constrain the TTV periods of 34 long-period TTVs and reduce amplitude errors of known TTVs; (c) Identify cases of multi-periodic TTVs, for which absolute planetary mass determination may be possible. We further extend our analysis by using perturbation theory assuming small TTV amplitude at the detection stage, which greatly speeds up our detection (to a level of few seconds per star). Our extended TTVs sample shows no deficit of short period or low amplitude transits, in contrast to previous surveys in which the detection schemes were significantly biased against such systems.' author: - 'Aviv Ofir , Ji-Wei Xie, Chao-Feng Jiang, Re’em Sari' - Oded Aharonson title: A spectral approach to transit timing variations --- Introduction ============ A transiting exoplanet’s Keplerian orbits impart precisely evenly spaced transit events. Any deviation from such regularity indicates that other forces are at play. Importantly, if there are other massive bodies in the system they would interact with the transiting planet and perturb its orbit, imparting deviations from strict periodicity known as transit timing variations (TTVs). This important effect was predicted (Holman [*et al.* ]{}2005, Agol [*et al.* ]{}2005) and later observed (Kepler-9 system, Holman [*et al.* ]{}2010, Dreizler & Ofir 2014). TTVs are stronger and easier to detect in systems near mean motion resonances (MMRs) and generally appear as sine-like deviations. Inverting the observed TTVs back for the parameters of the physical system that produced them is difficult (Lithwick [*et al.* ]{}2012) primarily since the important parameters (e.g., planetary mass and eccentricity) are degenerate when higher-order effects are not significant ([*i.e.*]{} departures from exact sine-shaped TTVs). In fortuitous cases where higher-order effects are observed in multi-transiting systems, the TTVs may yield the absolute masses of the planets ([*e.g.*]{}: Kepler-9, Kepler-11, Kepler-87, Holman [*et al.* ]{}2010, Lissauer [*et al.* ]{}2011, Ofir [*et al.* ]{}2014, and others) without the use of high precision radial velocity (RV) measurements, allowing the determination of planetary masses even beyond current RV capabilities in some cases ([*e.g.*]{} Kipping [*et al.* ]{}2014). TTVs can be useful without higher-order effects, especially in multi-transiting systems, since anti-correlated TTVs constitute dynamical confirmation of the candidates as true planets in the same system, undergoing angular momentum exchange ([*e.g.*]{} Steffen [*et al.* ]{}2013, Xie 2013, Xie 2014) - albeit with weak mass constraints. Even TTVs that are observed in only one transiting planet in a system are still useful since they may reveal the presence of additional planets in the system that may not be transiting at all (Nesvorn[ý]{} [*et al.* ]{}2012, 2013). Finally, the fact that TTVs are observed mostly for planets near MMRs allows using TTVs to address questions related planet formation and migration - and the means by which planets are captured in MMRs ([*e.g.*]{} Xie [*et al.* ]{}2014, Mills [*et al.* ]{}2016). For these reasons, a significant amount of work has gone into identifying TTVs, particularly in the *Kepler* dataset ([*e.g.*]{} Ford [*et al.* ]{}2011, Xie 2013, Mazeh [*et al.* ]{}2013, Holczer [*et al.* ]{}2016). In this work, we develop and apply a new technique – the Spectral Approach – to detect TTVs. We focus on detecting low-amplitude TTVs (defined in § \[Domain\]), with the assumption that high-amplitude TTVs were already identified (at least in the *Kepler* data). We further increase the sensitivity of TTVs detection - both to the fundamental signal and to its higher-order components. To demonstrate this improvement we compare our work to the recent Holczer [*et al.* ]{}(2016) results, hereafter H16, using both their “long-term” and “short-term” TTVs combined (TTV periods of above and below 100 d, respectively), and treating multi-periodic TTV as separate signals. Below we will present the spectral approach in § \[GridSearch\] and its more computationally-efficient perturbative approximation in § \[Perturbative\]. We then describe the details of applying the spectral approach to *Kepler* data in §\[ApplicationToKepler\], the resultant TTVs catalog in § \[Results\], and conclude in § \[Conclusions\]. A spectral approach to TTVs {#GridSearch} =========================== Motivation and description {#Motivation} -------------------------- This paper is concerned with low-amplitude TTVs, so we assume that a candidate transiting planet signal was already identified using simple linear ephemeris. When described using the Mandel-Agol 2002 formalism (hereafter MA02), the transit model requires explicitly just two parameters: the normalized planetary radius $r$, and the normalized distance between the star and planet $d_i$ to calculate the observed normalized flux at the $i$th point along the orbit. At a given time $d_i$ is a function of four orbital parameters in the case of circular orbits: $P, T_{\mathrm{mid}}, a, b$ - for the planetary orbital period, time of mid-transit, normalized semi-major axis and normalized impact parameter – and two additional parameters for eccentric orbits (where all normalized parameters are relative to the stellar radius). Limb darkening parameters are also needed but are usually not fitted (except in the highest signal to noise ratio (SNR) cases) and depend on the selected limb darkening law. In the usual procedure, the TTV search begins by allowing the individual times of mid-transit to deviate from their linear ephemeris. The individual timings are then searched for excess scatter or periodic signals. We note that in this case there are a total of $N_{\mathrm{tr}}+4$ fitted parameters, where $N_{\mathrm{tr}}$ is the number of individual transit events in the data and four are the other circular-orbit parameters. Since $N_{\mathrm{tr}}$ can be several hundreds or more in the case of short-period planets observed throughout the four years of *Kepler*’s normal operation, the large number of fitted parameters reduces the sensitivity of this technique. To first order, planetary TTVs close to first-order resonances are theoretically expected to have sine-like shapes with a super-period significantly longer than the orbital period of the transiting planets: $P_{\mathrm{Sup}}={P_{\rm out}}/{(j\Delta)}$ where $\Delta=\frac{P_{out}}{P_{in}}\cdot\frac{j-1}{j}-1$ is the normalized distance to resonance (Lithwick [*et al.* ]{}2012). Indeed many of the detected TTVs ([*e.g.*]{} H16) are actually observed to be mainly sine-like, sometimes exhibiting other frequencies due to, for example, resonances of higher order or terms of higher order of a given resonance ([*e.g.*]{} Deck [*et al.* ]{}2014, Agol & Deck 2016), but all are still well-described by sine functions. We therefore do not attempt to measure the individual times of mid-transit and then fit these timings, but rather assume that a sinusoidal TTVs exists in the data and then calculate the improvement of the model under this assumption relative to the linear model. We refer to this procedure as a Spectral Approach to TTVs. We wish to scan all possible sine-like TTVs. This requires a three-dimensional search grid of TTV frequency $f$, TTV amplitude $A$ and TTV phase $\phi$ (to avoid ambiguity the last will later be translated to the time $T_0$ at which $\phi=0$). On one hand, the frequency search axis can be well defined: the minimum frequency $f_{\rm min}={{}^{1}\!/_{2s}}$ where $s$ is the span of the data. The maximum frequency is set by the Nyquist frequency $f_{\rm max}={{}^{1}\!/_{2P}}$. The critical frequency spacing scale is as usual for sine fitting $\Delta f_{\mathrm{crit}}=1/s$ and one may want to oversample this critical frequency spacing by a factor of a few to avoid missing local peaks. On the other hand, there is no natural upper limit to $A$ (other than $A<P$ which is not very informative) and no natural resolution to $A$ or $\phi$ - leaving the search grid undefined and the processing slow (the MA02 model function need to be invoked for each transit event separately, on each test position on the search grid). The procedure therefore appears plausible but numerically inefficient. Signal detection can be significantly accelerated, however, if one limits the search to low- or medium- amplitude TTVs (see § \[Domain\] for definition). This improvement is made by using perturbation theory, and is described in § \[Perturbative\], and will be henceforward called the Perturbative Approximation (PA) to the Spectral Approach to TTVs, as opposed to the full fit. We stress that PA is used for detection only. Indeed, and as done in § \[FinalFit\], a full non-perturbative model should be used for the determination of the final TTV signal parameters and their associated errors. Benefits of Spectral Approach ----------------------------- The spectral approach to TTVs offers several advantages. These include: (i) It encapsulates all the TTV information in 3 parameters – the sinusoid’s period, amplitude and phase – regardless of $N_{\mathrm{tr}}$. (ii) Since the entire light curve is fitted with a single TTV model, the TTV detection sensitivity scales closely with the precision of the linear $T_{\mathrm{mid}}$. The reason is that of the four linear ephemeris parameters, $T_{\mathrm{mid}}$ is the only one affected by adding small TTVs. This results in elimination of the bias to long periods since the absolute precision on $T_{\mathrm{mid}}$ of the linear ephemeris (and thus the TTV detection sensitivity) actually improves with decreasing orbital period. (iii) Short-period signals (of few days period or shorter) have few points in-transit in each individual event, making transit time fitting nearly impossible. The extreme case is KOI 1843.03 (Ofir & Dreizler 2013) which has an orbital period of $\sim4.25$ hr and total transit duration not much longer than a single *Kepler* long cadence. Ultra-short period planet candidates are also typically too shallow to be detected at all as single transit events. Searching for TTVs using the spectral approach presents no such limitations. (iv) The resultant TTV spectra are compatible with later stages of system interpretation: theoretical considerations can predict the TTV frequencies and their amplitude ratios ([*e.g.*]{} Lithwick [*et al.* ]{}2012, Deck [*et al.* ]{}2014, Agol & Deck 2016). Therefore this information is useful even when no single transit event is above the noise. The classical approach to TTV detection is biased to long-period planets and deep transits (Steffen 2016) whereas most transiting planets have short periods and shallow transits. Indeed, in the the H16 catalog of *Kepler* TTVs the median orbital period for planets that exhibit TTVs is 29.8 d, while the median period of all *Kepler* candidates is less than a third of that number – about 9.5 d. This bias is therefore relevant to a large number of objects, and the proposed Spectral Approach can all but remove it. We note that some TTVs are not sine-shaped: TTVs which originate from non-gravitational processes ([*e.g.*]{} instrumental effects, starspots) may have different morphologies, and their description as sinusoids or sum of a few sinusoids may be inadequate. Theoretical comparison with the classical technique --------------------------------------------------- The probability density function (PDF) of the timing of an individual transit event is a delta function at the true time, which is then broadened by observational noise. In high SNR data, the PDF of transit timing fits is close to a gaussian centered on the true transit time, and with a width that depends on the SNR. As SNR decreases, the PDF of the transit time fits reduces in amplitude, broadens, and no longer resembles the true PDF. Instead, there is an almost uniform probability of finding the transit at any time in the search interval, with only a slight increase at the true transit time. Thus fitting these individual transit PDFs with individual gaussians, as in past work, results in poor fits that cannot approximate the true PDF when the SNR is low. In other words, when the SNR is low the PDF is so wide and shallow that many of the individual transits are detected far from the true signal. Thus they do not contribute to the coherent addition that allows searching for a periodic variation. Using a single global fit avoids this problem. In order to illustrate this effect, we generated $N_s$ segments of duration unity on either side of a gaussian-shaped transit-like signal of height $\epsilon$, each segment with $N_p$ points including normally distributed noise of unit amplitude. The simulated mid-transit time was chosen as a random (uniform) location within the central half of the domain. We then determined the best-fitting mid-transit time in two ways. We first fit a periodic gaussian shape to all the segments simultaneously. We also took the average of individual times measured by fitting each segment separately. The difference in time between the best fitting and the actual mid-transit location is $\delta t_c$. We repeated this experiment $N_r$ times, and averaged the results to produce smooth curves. Figure \[gaussgame1\] shows clearly that the a global fit outperforms the average of individual fits, [*i.e.*]{}, the RMS of the timing error, $\left<\delta t_c^2\right>^{1/2}$, remains small up to lower values of the signal amplitude, $\epsilon$, relative to the noise. The mean timing error $\left<\delta t_c^2\right>^{1/2}$ rises sharply and exceeds the transit duration at signal amplitudes much lower (an order of magnitude for the representative parameters chosen) for the global fit than the individual fits. This again demonstrates the general effect that should be unsurprising: when the signal to noise is low enough to mask the individual transit such that its typical timing error exceeds the event duration, averaging many such fits does not improve the detection. However a global fit remains sensitive to lower SNR. ![Illustration of improvement in timing fits that use global fitting of a periodic gaussian signal of amplitude $\epsilon$ embedded in unit noise. Simulation using $N_p$=200 points, in $N_s$ segments shown in color, run over $N_r=200$ trials. The gray line is the width assumed for the simulated gaussian transit; the unit of time is chosen such that on average the segments have unit duration on either side of the simulated transit.[]{data-label="gaussgame1"}](gaussgame1a.pdf){width="\columnwidth"} Perturbative approximation to the spectral approach {#Perturbative} =================================================== Basic perturbation theory ------------------------- To some general data set of $N$ points $\{t_i,F_i\}$ and $F$-errors $\sigma_i$ we fit a model $m_i=m(x_i)$. We wish to examine the sensitivity of the model to perturbation $g_i$ in the *independent* parameter $t_i$ assuming the perturbation’s functional shape is known ([*e.g.*]{}: linear trend, periodic variation, etc.). This perturbing function is known and normalized, only its scaling parameter $S$ is sought after. We therefore define the effective independent parameter: $$t'_i \equiv t_i + Sg_i$$ The perturbations are assumed to be small in the sense defined in § \[Domain\], and in this limit a linear approximation is possible: $$\begin{array}{l} m_i(t'_i) \approx m_i + Sg_im'_i, \\ m'_i\equiv\frac{dm}{dt}(t_i), \quad \quad m_i\equiv m(t_i).\\ \end{array} \label{Linearization}$$ Since we are interested in TTVs, the independent parameter $t_i$ is the time $F_i$ is the normalized flux, and $m_i$ is the Mandel-Agol (2002) transit model. The derivative of the transit model $m'_i$ is computed with respect to time, but application to sparse data (such as *Kepler*’s long-cadence light curves) should be done with care (see §\[ApplicatioToTTVs\]). Thus the amplitude $S$ has the desired meaning of TTV amplitude, and $m_i(t'_i)$ is the unknown TTV-inclusive model for which we solve perturbatively. We use least-squares to fit the new model: $$\chi^2=\sum_i^N{\frac{\big(F_i-m_i(t'_i)\big)^2}{\sigma_i^2}}.$$ Differentiating with respect to $S$, we obtain the best-fitting amplitude for the perturbation $g_i$: $$\label{PA_amp_eq} S=\sum^N_i{\frac{(F_i-m_i)g_im'_i}{\sigma^2_i}} \Bigg/ \sum_i^N{\bigg(\frac{g_im'_i}{\sigma_i}\bigg)^2 }.$$ It is useful to generalize this result to two-parameter optimization: now two perturbations $g_i$ and $h_i$ perturb the model function $m(x)$ so that the effective independent parameter is now $$t'_i \equiv t_i + S_g g_i + S_h h_i, \label{TwoParameterZ}$$ giving the set of linear equations $\bf{MX}=\bf{V}$ where: $$\bf{M}=\left( \begin{array}{cc} \sum_i^N{\left(\frac{g_im'_i}{\sigma_i}\right)^2} & \sum_i^N{\frac{g_ih_im'^2_i}{\sigma_i^2}} \\ \sum_i^N{\frac{g_ih_im'^2_i}{\sigma_i^2}} & \sum_i^N{\left(\frac{h_im'_i}{\sigma_i}\right)^2} \\ \end{array} \right),\\$$ $$\bf{X}=\left( \begin{array}{c} S_g \\ S_h \\ \end{array} \right),$$ $$\bf{V}=\left( \begin{array}{c} \sum_i^N{\frac{\left(F_i-m_i\right)m'_ig_i}{\sigma_i^2}}\\ \sum_i^N{\frac{\left(F_i-m_i\right)m'_ih_i}{\sigma_i^2}}\\ \end{array} \right).$$ These linear equations may be solved for $\bf{X}$, the amplitudes of the two perturbations. Application to TTVs {#ApplicatioToTTVs} ------------------- We now proceed to apply the analytical method above to TTVs. In this context, the model function $m$ is the linear ephemeris MA02 model and we assume that a linear model was already fitted to the light curve. At this point each data point has an assigned time from mid-transit $t_i$ which corresponds to a normalized distance $d_i$ for the MA02 model. We note that finding the true global minimum $\chi_{\mathrm{linear}}^2$ is of special importance since the TTVs analysis depend on the exact shape of the linear signal. We took great care in performing this step, as described in § \[ApplicationToKepler\]. Furthermore, *Kepler*’s long cadence means that $\frac{dm}{dt}$ cannot be computed simply from the time series as $m_i(x)=({m(t_i)-m(t_{i-1})})/({t_i-t_{i-1}})$ since the details of the ingress/egress are all but erased from the individual transits. To correct this effect $m'$ can be estimated either from a densely re-sampled flux model, or from the folded model light curve (in time modulo the period and not in phase, in order to be consistent with the TTV units of time and Eq. \[Linearization\]). In the folded light curve the ingress/egress region is much better sampled than in individual events, enabling a more precise evaluation of ${dm}/{dt}$. We then add a perturbation - a sine in time which has a frequency $f$, amplitude $A$ and phase zero at time $T_0$. The effective independent parameter is therefor: $$\label{TwoComponentModel} \begin{array}{l c l} t'_i & = & t_i + A \cdot \sin\left[2\pi f (t_i-T_0) \right] \\ & = & t_i + A\left[ \sin(2\pi f t_i) \cos(2 \pi T_0 f) \right. \\ & & \left. - \cos(2\pi f t_i ) \sin(2 \pi f T_0)\right] \\ & = & t_i + S_1 \sin(2\pi f t_i ) + S_2 \cos(2\pi f t_i) \\ \end{array}$$ which encapsulate $\{A,T_0\}$ in $\{S_1,S_2\}$ in the form described by Eq. \[TwoParameterZ\]. This linearization allows us to analytically solve for the best fitting amplitude and phase of a sine-shaped TTV, removing the problems identified above and leaving only the TTV frequency as a single searched parameter (which remains well defined with clear boundaries and resolution as explained in § \[Motivation\]). The resultant perturbed model (Eq. \[TwoComponentModel\]) appears similar to a true TTV-shifted signal but it is only a mathematical construct and not a physical model ([*e.g.*]{} it may have flux values greater than unity, unless these are clipped, see § \[ModelClipping\]). ![image](PA_visualization.pdf){width="100.00000%"} Figure \[PA\_visualization\] visualizes the process: the folded linear ephemeris model is cropped to just a region surrounding the linear-ephemeris transit, and the model derivative $dm/dt$ is calculated. The perturbed model is generated by adding $dm/dt$, scaled by some amplitude, to the linear model above – to mimic a TTV-affected light curve. While the perturbed model usually appears just as a time-shifted model, some points can obtain non-physical values in the perturbed model: above unity and below $\textrm{min}(m)$. Model clipping (see § \[ModelClipping\]) can be applied to counter this effect. Domain of validity {#Domain} ------------------ PA approximates the light curve of TTV-affected planets assuming these TTVs are small. Indeed, the ability of the PA to find a better model than the linear model arises from the non-zero derivative of the model $m'_i$ in Eq. \[PA\_amp\_eq\], which allows adjustment of the linear model to better fit the data. Transit light curves are roughly trapeze-shaped, so non-zero derivatives are significant mostly during ingress and egress. We can therefore divide the possible TTV amplitudes to three broad regimes of validity within the PA: \(i) Low-amplitude TTVs are those with amplitude lower than the ingress/egress duration. In this regime the TTVs are small enough for $m'_i$ to be able to approximate well the true TTV signal. We therefore expect here that the PA will both find the true TTV frequency and reproduce the TTV amplitude with optimal sensitivity. \(ii) Medium-amplitude TTVs have amplitudes larger than the ingress/egress duration, but smaller than the full transit duration. In this regime the PA-detected amplitude saturates as $(F_i-m_i)m'_i$ factor is nonzero but nearly independent of the TTV amplitude – so the PA can only compensate for the ingress/egress duration part of the actual TTV amplitude. Here the correct TTV frequency would likely be identified but with a lower amplitude than the correct one. \(iii) High-amplitude TTVs have amplitude comparable to the transit duration or larger. In this case either $m'_i$ or $y_i-m_i$ in Eq. \[PA\_amp\_eq\] average to zero at all times, and so the perturbative approximation fails and it is unable to detect the TTVs. Note the full search without the linear approximation would still apply well. These domains are simulated and shown in § \[SimulatedDetection\]. The PA approximation is valid for the majority of TTVs – especially those yet undetected – since the median TTV amplitude in the H16 catalog is $<23$ min while the median transit duration of the same objects $>300$ min. In other TTV catalogs, such as those of Xie (2013, 2014), the case is even clearer with median amplitude of $<14$ min and median duration of $270$ min. While rare, high-amplitude TTVs do exist with amplitudes that may even exceed the duration of the transit itself. These are better detected by other techniques ([*e.g.*]{} QATS (Carter & Agol 2013), human eyes (Schmitt [*et al.* ]{}2014), or single event detection schemes (e.g. Osborn [*et al.* ]{}2016). Model clipping {#ModelClipping} -------------- The PA produces a perturbed model which is a mathematical construct (not a physical model). This construct can be made nearly indistinguishable from a physical model by clipping the perturbed model on both the upper and lower ends, to zero flux decrement and to $\textrm{min}(m(x))$, respectively, before $\chi^2$ is calculated (see Figure \[PA\_visualization\]). Model clipping is not used here to refine the PA-detected amplitude nor frequency of the signal, but improves the reliability of the $\chi^2$ values in discriminating true from spurious TTV signals. Correction to the Mandel-Agol (2002) model ------------------------------------------ We use the standard MA02 model to calculate the size of the flux decrement during transit. This model is a set of a few mathematical formulae, each valid in its own section of the parameter space and with mathematically accurate transitions between these regions. While the computational implementation of these formulae has finite precision and thus some discontinuities in the transition between sections exist, the effect is small enough ($10^{-4}$ times the transit depth itself) to be largely ignored. However, in the PA the MA02 *derivative* with respect to the planet-star separation $z$ is needed - and here the finite precision can no longer be neglected for points close to two problematic transitions at $z\approx r$ and $z\approx 1-r$. These numerical errors are small in amplitude but they also occur over a small range of $z$ values, causing their $dm/dz$ amplitude to rival that of the main signal (and actually diverge at the limit). This is demonstrated in Figure \[MA02\_correction\]: we sample normalized distances of a $r=0.1$ planet in the MA02 model at a resolution of $dz\approx10^{-4}$. The small miscalculation of the model around the $z\approx r$ transition is observed with an amplitude $<10^{-5}$ (see insert on the top panel). At this resolution the anomaly near $z\approx 1-r$ is $\approx 100$ times smaller still (and thus not visible). Yet, the model’s derivative has a large anomaly at $z\approx r$ (the one near $z\approx 1-r$ is still small). As the discontinuity is approached this becomes more severe: at higher resolution ([*e.g.*]{} $dz=10^{-5}$) the anomaly near $z\approx 1-r$ becomes dominant - reaching amplitudes several orders of magnitude higher than the main $dm/dz$ signal. These numerical errors are inherent to the MA02 model and will persist even if numerical convergence criteria used in the MA02 are made stricter. We therefore propose an ad-hoc correction to the model. The $z\approx r$ anomaly has continuous slopes on either side (there is no physical change at $z=r$), while the $z\approx 1-r$ anomaly does reflect a physical change at $z=1-r$ (the II & III contacts). We therefore correct the former by polynomial interpolation from both sides, and the latter by polynomial extrapolation from either side, separately. In both cases the corrected region is $\Delta z=2\cdot10^{-4}$ from the anomaly, and the polynomials are fitted using a region up to $\Delta z=10^{-3}$ from it. The polynomials are of $2^{nd}$ order, unless $r>0.5$ in which case they are of $3^{rd}$ order. While satisfactory, this procedure is imperfect: each transition from MA02 model to polynomial interpolation/extrapolation involves a “stitching” point that produces an outlier point in the model’s derivative - but these are points (vs. finite regions in the uncorrected model) and with far reduced amplitude, regardless of the proximity to the transition points. ![A scan though the MA02 model for a planet (with $r=0.1$) at a resolution of $dz=10^{-4}$. **Top panel:** The original and corrected model functions (indistinguishable on the main panel). The insert shows the region near the $z\approx r$ anomaly. **Bottom panel:** the derivative to the model amplifies the anomaly. At this resolution the $z\approx r$ anomaly has amplitude larger than the main $dm/dz$ signal, and the $z\approx 1-r$ is still not visible. At finer resolutions still (e.g. $10^{-5}$) both anomalies have much higher amplitude than the main signal.[]{data-label="MA02_correction"}](MA02_correction3.pdf){width="50.00000%"} Model sampling -------------- A further consequence of using the model’s derivative concerns the model oversampling needed due to *Kepler*’s finite integration time. The commonly used criterion for computing the needed oversampling is given by Eq. 40 of Kipping [*et al.* ]{}(2010), which calculates the inaccuracy caused by sampling the instantaneous model at cadence $\mathcal{I}$ relative to integrating it over intervals with cadence $\mathcal{I}$ (the latter is of course closer to a real measurement). If the signal has a depth of $\delta$ and ingress/egress duration of $\tau$ that inaccuracy is found to be $\sigma(F_{\textrm{Resampling}})\propto \delta \mathcal{I}/\tau$ (The $\sigma$ sign is used since the inaccuracy can be viewed as a type of modeling noise). However, this computation is not valid in the case of PA since now the sampling noise must be compared not with the model flux but with the model’s flux derivative. The slope which Kipping [*et al.* ]{}2010 sampled was $\delta/\tau$. Analogously, the slope that needs to be well-sampled in PA is approximately $\delta/\tau^2$. We therefore update Eq. 40 of Kipping 2010 for use in PA by dividing it with yet another ingress/egress duration, or: $$\sigma(F_{\textrm{Resampling}})=\frac{\delta}{\tau^2}\frac{\mathcal{I}}{8N^2},$$ where $N$ is the number of subsamples computed. In practice, we impose a lower limit of $N=10$ to ensure adequate subsampling. Speed ----- Linearly fitting two of the three search dimensions allows a significant speedup in calculating TTVs. This part of the calculation for a typical system now takes 10 seconds or less on a single-threaded CPU (a far larger amount of time is required to fit the linear model of MA02) as zero calls to the MA02 model function are needed. Detection of simulated signals {#SimulatedDetection} ------------------------------ We simulated *Kepler*-like data that included a transiting planet with sinusoidal TTVs with a TTV period shorter than the time baseline of the simulated data, and tried to detect those TTVs using PA. We considered a detection successful when the frequency of the most significant peak on the TTV spectrum was close to the simulated one (*i.e.* $|\Delta f|<(\mathrm{time\,\,baseline})^{-1}$), and calculated the detection efficiency of the algorithm while scanning along various parameters. Notably, we also recorded the amplitude of the best-detected peak, [*i.e.*]{} the $\chi^2$ improvement (as a positive number) over the best-fit linear model: $\Delta\chi^2\equiv\chi^2_{linear}-\chi^2_{PA}$. In Figure \[DetectionEfficiencyFig\] we show the results of three such parameter scans – each along the transit SNR axis but at different TTV amplitude. Here, the SNR is defined per transit event, [*i.e.*]{}: the depth of the transit divided by the per-point uncertainty and the square root of the average number of points in a transit event. As expected, at a given transit SNR (left column of panels) detection becomes easier when the TTV amplitude increases. Scans of other parameters show similar and expected trends. Detection efficiency is higher when the planet is larger or when the noise or planetary orbital period are smaller. These different dependencies can make it challenging to establish an exact TTV detection limit of a given system. On the other hand, the panels on right column of Figure \[DetectionEfficiencyFig\] show the same data sorted by the $\Delta\chi^2$ score - and the red line is an empirical function (using the error function: $f(x)=\frac{1}{2}[1+\textrm{erf}((x-x_0)/\sigma)]$ with $x_0=11,\,\,\sigma=8$). The three panels show similar behavior across significantly different parameter values. Importantly, nearly identical $f(x)$ is obtained when varying all other parameters of the problem. We conclude that the spectral approach and its PA approximation allow TTVs to become uniformly detectable – always becoming detectable with high efficiency when $\Delta\chi^2 \gtrapprox 20$ (which is a plausible threshold: $4-5 \sigma$ detections are usually regarded as reliable). This uniformity allows the spectral approach to detect TTVs using a full dataset even if no single transit is detectable, as is often the case for small planets. As expected, higher TTV amplitude does not always increase the detection efficiency: as explained in § \[Domain\], Figure \[AmplitudeTestGrid\] shows that high amplitude TTV have low detection efficiencies, if at all detected, using this technique. We note that the parameters chosen for the above test, especially the single-event SNR$<2.5$, is such that this planet would not even qualify for analysis by some catalogs (Mazeh [*et al.* ]{}2013) - but here PA obtains high detection efficiency over a wide range of values. ![Transit detection efficiency of simulated light curves. The three rows show different TTV amplitudes of otherwise identical systems. **Left column:** detection efficiency at various single-transit SNR ratios . **Right column:** the same data as the left column now sorted according the best-fit $\Delta\chi^2$. The red solid line is an empirical function (the error function), identical among the three panels, emphasizing the consistent behavior.[]{data-label="DetectionEfficiencyFig"}](DetectionEfficiency_vs_SNR_Chi2_v2.pdf){width="50.00000%"} ![Transit detection efficiency of simulated light curves as a function of TTV amplitude. The simulated light curve has its noise sized to produce a low SNR of 2.5 on individual transit events if no TTVs are present - and TTVs would make the effective SNR even lower. Still, TTVs are detectable with high efficiency over a wide range / amplitudes - all the way to the expected limit of high-amplitude TTVs at TTV amplitude equal the transit duration (vertical dashed line).[]{data-label="AmplitudeTestGrid"}](AmplitudeTestGrid.pdf){width="50.00000%"} The full spectral approach fit {#FinalFit} ------------------------------ PA is a fast approximation to the full, but slower, spectral approach. Therefore, a full spectral approach model is computed based on the PA results for cases where significant TTVs are detected by PA. The full spectral approach model fits the non-periodic planet similarly to classical techniques of fitting the transit parameters simultaneously with the times of mid-transit ([*e.g.*]{} H16), except that the series of times of mid-transit is given by a sine function and not by a numerical vector. Grid-searching, as presented in §\[GridSearch\], is sub-optimal. We therefore use a well-tested MCMC code ([*e.g.*]{} Ofir [*et al.* ]{}2014) to find the best-fit solution and the error ranges of eight parameters: four linear-ephemeris orbital parameters, normalized planet radius and three sine-TTV parameters. The results of the full fit usually agree well with the PA results and have somewhat better $\Delta \chi^2$ than PA (see also Figure \[KOI\_209\_system\]). Mismatches between the PA and the full fit are common when the TTV period is longer than the baseline of the data - causing the solutions range to be highly correlated (see Fig \[KOI\_75.01\]) - something that is not captured by the PA model. The adopted values are therefore the result of the full spectral approach fit. Application to *Kepler* candidates {#ApplicationToKepler} ================================== Data and pre-processing ----------------------- We used *Kepler* Data Release 24 as the source data, and *Kepler* Objects of Interest (KOIs) table downloaded from the NExSci archive[^1] on 25 December 2015 as the source of list of candidate signals, and processed 4706 object not dispositioned as “false positive”. Many of the false alarm KOIs are eclipsing binaries (EBs), and eclipse timing variations in EBs may allow the detection of additional bodies in the systems and even circumbinary planets (e.g. Borkovits [*et al.* ]{}2016), and so are ostensibly closely related to the project at hand. However, high-precision modeling of EBs, and especially non-detached EBs, is significantly more complicated than modeling planetary transits. We therefore identify the likely EBs as KOIs that appear on the *Kepler* Eclipsing Binaries Catalog (hereafter KEBC, Kirk [*et al.* ]{}2016) or have eclipse depth &gt;10% and are not otherwise cataloged as confirmed planets. We remove these EBs from the candidates list, and defer work on eclipse timing using the spectral approach to the future. Having good linear ephemeris in advance is useful for optimizing the extraction of the transits. Transits of each KOI were extracted by fitting several polynomials (orders zero through five) to a small region bracketing each transit. Each such region extended three times the transit duration before and after the linear ephemeris expected time of mid-transit, and excluded the points in transit and two extra data points before and after it (since the transits were suspected to have some TTVs). Regions that did not include at least two data points before and two data points after the transit were not accepted, and these transits were ignored. Each fit was iteratively clipped to four sigma, and the common set of all remaining points in all six polynomial fits were compared such that the fit with the lowest Bayesian Information Statistic (BIS) was selected (where BIS$\equiv\chi^2+k\mathrm{ln}(n)$, $k$ is the number of model parameters and $n$ is the number of data points). The selected polynomial was then interpolated to the in-transit points to generate the background flux on each transit. Notably, variable stars with variability time scales similar to the duration of the transit itself (or shorter) will not be well-modeled using this procedure and the resultant background will be severely affected by this variability. PA results on such variable stars may not be reliable as is, and relevant information is given in our results below when such a case is identified. Each normalized light curve was fitted with the MA02 model after accounting for *Kepler*’s finite exposure time (Kipping 2010), using the MCMC code. We adjusted the five usual parameters $P, T_{\mathrm{mid}}, r, a, b$ for most KOIs, adding also the stellar limb darkening coefficients $u_1$, $u_2$ for planets with NExSci-reported model SNR &gt;100. After this fit we rejected in-transit outliers and then calculated the $\chi^2$ of every transit event. Events that had $\chi^2$ larger by more than $4\sigma$ from the median (where $\sigma$ was assessed ignoring the event with the highest $\chi^2$) were rejected. If indeed either in-transit points or transit events were rejected then we re-fitted the remaining data. We note that multi-planet systems where modeled such that before analyzing a given KOI all the other transit signals on the same host star were modeled-out using the NExSci linear ephemeris parameters. This has the effect that systems may be inaccurately modeled if more than one TTV-bearing planet exist and some of the transits are nearly overlapping in time. In such cases the TTVs of the *other* planet (not the KOI of interest) are imperfectly modeled out by our pipeline, potentially hampering the analysis of the KOI of interest. Selection of significant TTVs ----------------------------- In order to be selected as a significant TTV a candidate signal should pass a number of statistical tests: the boostrap analysis confidence test, and a set of tests based on cumulative $\Delta\chi^2$ (below). ### Boostrap analysis confidence test {#Bootstrap} For each KOI we performed a bootstrap analysis confidence test (Press [*et al.* ]{}1992): we generated $10^3$ artificial light curves, each by sampling with replacement the residuals to the linear model and then adding these re-sampled residuals back to the linear model. Each artificial light curve was then analyzed using PA and the peak of its TTV $\Delta\chi^2$ spectrum was recorded. The Confidence metric, equivalent to 1 minus the false alarm probability, is the fraction of those random artificial light curves that had a lower peak $\Delta\chi^2$ than the real data. Indeed, previously detected TTV-bearing KOIs clearly cluster around high Confidence metric ($\geq 0.999$) with most of the exceptions easily understood as either high-amplitude TTVs (for which PA is not suitable in the first place) or variable stars with special characteristics ([*e.g.*]{} at time scales similar to the transit duration, making the polynomial background estimation inadequate). We therefore use this Confidence metric as the main threshold for the selection of significant TTVs. Note that this test checks for the reality of the most significant peak. Since the degeneracy in TTV inversion can be broken in cases where additional super-frequencies are detected, we use this test also as a rough guide to the significance of other peaks in the PA spectrum. ### $\Delta\chi^2$ test As shown in §\[SimulatedDetection\], high efficiency detection is expected for $\Delta\chi^2\simeq20$ or larger. We therefore checked if any of the high-confidence signals had lower $\Delta\chi^2$; we found only one such case, with only a slightly lower $\Delta\chi^2$, confirming $\Delta\chi^2\simeq20$ is a good discriminant value. ### Cumulative $\Delta\chi^2$ definition and tests {#CumChi2} It can be difficult to judge if a $\Delta\chi^2$ for a given system is due to a real signal or just some outliers as $\Delta\chi^2$ is only summary total of the difference between the linear and the perturbed models. However, the two scenarios above differ in their rate of accumulating $\Delta\chi^2$ in time - a real signal would gradually accumulate $\Delta\chi^2$ whereas if due to an outlier event much of the $\Delta\chi^2$ would accumulate over a small number of data points. Below we detail a few statistical tests one can perform to identify false positive TTV signals when using the PA technique by distinguishing the cases with and without TTVs. The observed cumulative $\Delta\chi^2$ curve is: $$\mathrm{Obs}_i =\sum_{j\le i}{\left(\frac{y_j-m_{T,j}}{\sigma_j}\right)^2 - \left(\frac{y_j-m_{L,j}}{\sigma_j}\right)^2}$$ where $\{m_L\}$ is the linear ephemeris model and $\{m_T\}$ is the best-fit TTV model for the data and errors $\{y, \sigma\}$, and where the best-fit model $m_T$ always has better (lower) $\chi^2$ than $m_L$. Moreover, if the best-fit model is correct ([*i.e.*]{} in the limit of $y\rightarrow{m_T}$) then there is specific shape to this $\Delta\chi^2$-gain curve which we call the “expected” curve: $$\mathrm{Exp}_i =-\sum_{j\le i}{\left(\frac{m_{L,j}-m_{T,j}}{\sigma_j}\right)^2}$$ If the TTV model is correct, then $y\rightarrow m_T$ and $\mathrm{Obs}_i$ reduces to $\mathrm{Exp}_i$. On the other hand, if there are no TTVs in the data then $y\rightarrow{m_L}$ and $\mathrm{Obs}_i$ will differ from $\mathrm{Exp}_i$. Generally, a wrong TTV model would manifest itself by exhibiting significantly different observed/expected cumulative $\Delta\chi^2$ curves. In Figure \[Cumulative\_DeltaChi2\] we demonstrate this metric on the well-known TTV bearing star KOI 103.01, and on another signal, apparently with high-confidence and high-$\Delta\chi^2$, that fails some of the cumulative $\chi^2$ tests. The cumulative $\Delta\chi^2$ curves are information-rich and we use them to design the following tests for the reality of a PA-detected TTV signals: **(i) Normalized area between curves:** the normalized area between the $Obs_i$ and $Exp_i$ curves can be defined as: $\sum_i{|\mathrm{Obs}_i-\mathrm{Exp}_i|}/\sum_i{\mathrm{Exp}_i}$. False positive detections are expected to have large normalized area between the two curves. **(ii) maximum single-point $\Delta\chi^2$:** a real TTV signal would accumulate $\Delta\chi^2$ slowly over time, while an attempt of PA to minimize $\chi^2$ due to some outliers would accumulate most of the $\Delta\chi^2$ in a small region. By recording the highest single-point $\Delta\chi^2$ gain (absolute value) for each KOI one may detect irregular behaviour. We note that this test bears similarity to the KS test. **(iii) RMS of difference:** the RMS of the difference between the $\mathrm{Obs}_i$ and $\mathrm{Exp}_i$ curves is computed. True TTV signals should have a low value for this RMS, so we required that all new TTV signals will have a smaller RMS value than the ones found on the H16 objects (that are not high amplitude). **(iv) Correlation coefficient:** The correlation coefficient between the $\mathrm{Obs}_i$ and $\mathrm{Exp}_i$ curves is computed. True TTV signals should have high correlation between $\mathrm{Obs}_i$ and $\mathrm{Exp}_i$, so we required that the correlation coefficient for all new TTV signals will be larger than the ones found on the H16 objects (that are not high amplitude). ------------------------------------------------------------- -------------------------------------------------------------- (a) (b) ![image](KOI_103_01_Cumulative_Chi2.pdf){width="50.00000%"} ![image](KOI_4341_01_Cumulative_Chi2.pdf){width="50.00000%"} (c) (d) ![image](KOI_103_01_Visualization.pdf){width="50.00000%"} ![image](KOI_4341_01_Visualization.pdf){width="50.00000%"} ------------------------------------------------------------- -------------------------------------------------------------- We computed these statistical measures above for all KOIs, and required that all new PA detections will be in the range spanned by the H16 high significance candidates. We found 8 KOIs that passed all other criteria but failed the normalized area test, and indeed all of them were found to be false positives by visual inspection (seven strongly pulsating stars and one EB). The remaining signals passed all the additional tests. ### Visualization of the TTV signal: By rearranging Eq. \[Linearization\] to express the model derivative, and applying it to the measurement set $F_i$, one obtains an observational estimation of the flux derivative. This can be used to visualize the PA-detected TTV signal, by comparing the observed shape to the expected one, [*i.e.*]{} a transit derivative. $$F'_i=\left(F_i-m_i\right)/(S g_i).$$ The lower panels of Figure \[Cumulative\_DeltaChi2\] depicts this residual. Note that this is not the quantity that is fitted, but is only used for visualization. Since the oscillating function $g_i$ is in the denominator, some points diverge on the plot. Results of application to *Kepler* candidates {#Results} ============================================= General statistics and comparison with H16 {#GenStat} ------------------------------------------ We analyzed 4605 KOIs that met the rather minimal requirements of not being marked as false positives by the NExSci archive, and including at least three usable transits after preprocessing. We found 527 objects with Confidence $\geq0.999$, all of them but one also with $\Delta\chi^2>20.7$ (see Figure \[Chi2\_vs\_Confidence\]), similar to the expected threshold of $\sim20$ from the tests on simulated data. False positive comprise 198 of these object, arising due to the following reasons: (i) they appear in the current KEBC; (ii) they have depth $>10\%$; (c) they have high-amplitude TTVs (making the PA-detected value likely wrong); (d) The TTV frequency was the maximal one (see discussion on § \[Recurring\]) (e) they failed any of the cumulative $\Delta\chi^2$ test (§ \[CumChi2\]) (f) they had scatter-to-error ratio $>50$. Collectively these objects will be termed the EBs/FPs (eclipsing binaries and false positives) sample, while the sample of objects with Confidence $\geq0.999$ that are not EBs/FPs will be termed the High Confidence TTVs. The list of 329 remaining KOIs with high-confidence TTVs was cross-referenced to the TTV catalogs of: H16, Xie (2013), Xie (2014), Hadden & Lithwick (2014, 2016) (hereafter HL14, HL16), Jontof-Hutter [*et al.* ]{}(2016) (hereafter JH16) and Van Eylen & Albrecht (2015). In all cases we required that the reported TTV amplitude in these catalogs will be significant to at least $3\sigma$ as determined by each catalog, in order to be considered as a significant TTV detection by these catalogs, *i.e.,* a “known object”. After removing the known objects, 165 remaining KOIs with new periodic TTVs were identified, of which 34 are present in the H16 catalog with long-term and unconstrained periods (dubbed “polynomial TTVs”), Thus 131 TTV signals found here are considered new. Our results are given in Table \[ResultsTable\]. The power of the spectral approach is seen by comparing the distribution of TTV-bearing KOIs detected using PA and the classical techniques ([*e.g.*]{} H16 in Figure \[Compare\_P\_and\_A\_TTV\_and\_r\]). As predicted, we see that PA can detect TTVs for planets with shorter orbital periods, lower amplitude and smaller size than was previously possible, and the higher object count indicates that PA is also more sensitive. Numerically, as mentioned above the median orbital period for planets that exhibit TTVs in the H16 catalog is more than three times the $\sim9.5$ d median period of all *Kepler* candidates, while the median period of the Spectral Approach new TTV detections is $\sim10.8$ d - eliminating the period bias. Similarly, the median transit depth of the H16 objects is $\sim1075$ ppm while the same number for of all *Kepler* candidates and the Spectral Approach detections is $\sim428$ ppm and $\sim458$ ppm, respectively - eliminating the depth bias. Additionally, in Figure \[Tmid\_TTVamp\_connection\] we see that the precision in the amplitude of the detected TTVs (and thus, the sensitivity to them) scales exactly as the precision on the linear $T_{\mathrm{mid}}$ over more than five decades of amplitude. ![High Confidence TTVs detected by PA (light blue), the TTVs detected by H16 that have amplitude significant to more than $3\sigma$ and are not High Amplitude (cyan) and the overlap between the two sets (darker shade), focusing on the lower range of all parameters. **Upper panel:** Orbital periods distribution: PA overdetects TTVs on short-period planets. **Middle panel:** TTV amplitude distribution: PA overdetects TTVs of lower amplitude. Note that identical objects may have somewhat differently determined TTV amplitudes in each catalog, hence the slightly different overlap regions in similar bins. **Lower panel:** relative planets size distribution: PA overdetects TTVs on planets with smaller relative ratii. Note that since H16 does not quote this parameter, all $r_p/R_*$ were taken from the MAST database for both sets of KOIs.[]{data-label="Compare_P_and_A_TTV_and_r"}](Compare_With_Holczer16-P_orb_and_TTV_A.pdf){width="50.00000%"} ![The $1\sigma$ error range for the TTV amplitude vs. the $1\sigma$ error range for the linear $T_{\mathrm{mid}}$ – both taken from the non-linear fit. The correlation is clear over multiple decades. The locus of the 1:1 ratio is shown as a thin line.[]{data-label="Tmid_TTVamp_connection"}](Tmid-to-TTV_amp-connection.pdf){width="50.00000%"} During this work, we noticed a few issues with the H16 catalog to which we compare: (i) the TTV detection and final fits are separate steps, and their results are occasionally incompatible. In particular, 22 of the reportedly significant TTVs (p-values $<10^{-4}$, see H16) have low amplitude significance $\mathrm{A}/\sigma_{\mathrm{A}}<3$. These 22 TTV signals are questionable: we only detected high-confidence TTVs on two of these objects (KOIs 282.01, 1783.01), and only the first one is consistent with H16. We therefore term the remaining 20 signals as low-significance H16 KOIs, and do not include them in the following comparison with H16. (ii) The uncertainty of the linear ephemeris orbital period was found to be systematically too low in H16: while we obtained similar uncertainties for short-period KOIs, the H16 uncertainties were progressively smaller than ours towards longer period, typically about an order of magnitude smaller and approaching three orders of magnitude smaller at the long-period end. Examination of the individual signals confirms that these uncertainties are underestimated in H16. (iii) the H16 uncertainty on the linear-ephemeris time of mid-transit is almost always larger than our own. The difference is far smaller than in point (ii) above (median factor of 2.35) and thus is probably unrelated to it. ![The distribution of FAP (false alarm probability) vs. $\Delta\chi^2$ for all analyzed KOIs (blue dots). Higher confidence object are appear on the bottom of the plot. The axes were chosen to highlight the distribution near the transition from low- to high- confidence objects – most high-confidence TTVs have $\Delta\chi^2>100$, [*i.e.*]{} beyond the scale of this figure.[]{data-label="Chi2_vs_Confidence"}](DeltaChi2_vs_Confidence.pdf){width="50.00000%"} Many of the TTV-bearing KOIs we found were previously identified in the literature. Indeed, of the 144 objects in common with the H16 catalog, 132 have TTV period consistent within $3\sigma$ between the catalogs, demonstrating the compatibility of the spectral approach with the classical TTV identification techniques used in H16. It is noteworthy that due to the lower number of free parameters the spectral approach allows improving the precision of the determination of the TTV parameters. The uncertainty on the TTV period was reduced by a median factor of $\sim10\%$ and the uncertainty on the TTV amplitude was reduced by a median factor of $\sim40\%$. There are 30 stars with periodic and significant TTVs identified by H16 that were not detected by PA. We evaluated each of these stars manually, and found that about half of them reside in active stars with high frequency variability (scale of transit duration or shorter) making TTV detection a more subtle issue. Nearly all the rest were either low-significance to begin with (amplitude significant to less than $4\sigma-5\sigma$ according to H16) or that the same TTV frequency was detected by PA and H16 but different confidence levels were attributed to the signals. Indeed only in a two cases (KOIs 1581.02, 4519.01) no good explanation was identified for the PA missing robust H16 signals. Discussion {#Recurring} ---------- Below we list a number of recurring phenomena that can be useful in understanding the results given in Table \[ResultsTable\]. **The chosen cutoff levels are somewhat arbitrary:** We provide in Table \[ResultsTable\] information on more than just the high-confidence targets (&gt;0.999), to help identify the ones that may, with relatively little additional input data or detailed analysis, become high confidence. **Best fit TTV frequency is the maximal one:** If an EB with two similar eclipses is misidentified as a transiting planet, any difference between the odd and even eclipses (that may actually stem from non-zero eccentricity, different surface brightness of the stars, etc.) may cause the PA fit to incorrectly add high-frequency TTV at twice the “planetary” orbital period. We found 75 High-Confidence TTV signal that are close to the maximal one, and these KOIs are suspected as EBs or otherwise false positives. Indeed, 60 of these were already in the EBs/FPs sample, and the rest (but one) appear in H16’s suspected false positives list (their Table 1). In practice, the maximal frequency at which we searched for TTVs was limited by the orbital period $P$ of the transiting planet: $f_{\mathrm{max}}={1}/{(2P)}$. Since the frequency resolution scales with the time span of the data, we labeled systems with best-fit TTV frequency within $s^{-1}$ of $f_\mathrm{max}$ as suspect. We note that sometimes stellar pulsations that survived filtering also caused such high-frequency apparent TTVs. To summarize, in cases where the best fit TTV frequency is consistent with the maximal one, it is a strong sign of a misidentified EB and those objects are labeled likely false positives. However they are given in Table \[ResultsTable\] for completeness of high-confidence signals. **Very long TTV periods:** Usually the error ranges are symmetrical in frequency space. However, on very long periods $P_{\mathrm{TTV}}>s$ where $s$ is the span of the data, the period error range is highly correlated with TTV amplitude. This is expected at such long TTV periods as the data does not allow seeing even a single complete TTV period, making the observed amplitude either close to the total real amplitude (if $f_{\mathrm{TTV,true}}\leq s^{-1}$) or just a fraction of it (if $f_{\mathrm{TTV,true}}\ll s^{-1}$). Importantly, in such cases the best constraint is on some function of $f_{TTV}$ *and* $A_{TTV}$ and not on each of them individually (see Figure \[KOI\_75.01\]). For this reason some objects in Table \[ResultsTable\] appear to have low significance to either $f_{TTV}$ or $A_{TTV}$ however these objects all have very long TTV period and so the detection of some long-period TTVs on these objects is correct, but the exact amplitude and frequency of these TTVs are more poorly constrained. ![Posterior distributions of the spectral approach optimization for KOI 19.01 (Left) and KOI 75.01 (right) in the TTV period and TTV amplitude plane. Error ranges are the location of constant $\Delta\chi^2$ relative to the best-fit position, which is marked in a black ’+’. Stars that have TTV periods smaller than the data time span show little or no correlation between the TTV period and its amplitude, while significant correlation exists for the longest TTV periods. For e.g., KOI 75.01 was determined by H16 to have polynomial TTVs but here we show that the TTV frequency can be well constrained (better than $3\sigma$) using the spectral approach - albeit with significant amplitude correlation. We note that the actual MCMC jump parameter is the TTV frequency, which is more uniformly distributed, but we presented here the TTV period which is more intuitive.[]{data-label="KOI_75.01"}](SuperPeriod_Amplitude_correlations.pdf){width="50.00000%"} **Stroboscopic frequency:** *Kepler*’s finite exposure time may also produce a stroboscopic effect, exhibiting apparent TTVs on strictly periodic transiting planets, when the orbital planet’s period of happen to be close to an integer multiple of the exposure time (Szab[ó]{} [*et al.* ]{}2013, Mazeh [*et al.* ]{}2013). We therefore also provide in Table \[ResultsTable\] the expected stroboscopic frequency and note that it was detected in practice multiple times. **Unexplained significant TTV signals:** TTV signal with no apparent connection to any other known object in the system were detected multiple times. Such TTVs are very likely due to interaction with additional planets in the system that are not transiting. Such systems are fertile ground to RV surveys to connect the inner and outer parts of multi-planet systems. **Multi-periodic TTVs:** More than half (277 objedts) of all stars with high-confidence TTVs were found to have more than one significant TTV frequency. Additional TTV frequencies allow breaking the degeneracy in the TTV inversion back to absolute masses and are therefore very useful - provided they are real. Indeed, Pulsating/variable stars can create spurious TTV signals (see also the following paragraph). On the other hand, a high number of apparently significant TTV frequencies (we adopted the &gt;5 threshold) is likely a sign of imperfect filtering of a variable star and not of multiple dynamical phenomena - and 91 stars exhibit it. **Pulsating/variable stars:** These are more difficult to filter, and sometimes residuals of the variability signal remain, especially when the variability time is close to- or shorter than- the transit duration itself. Such stars frequently exhibit TTVs but those are difficult to judge for reliability without individually-tailored filtering, spot- or pulsations- modeling, etc. To indicate this as well as cases that may have not been filtered and/or modeled well, we provide in table \[ResultsTable\] the ratio of scatter around the linear model to the median error. As a general rule, ratios lower than two usually mean good filtering and modeling (unity being the white noise limit), and ratios larger than three should be reviewed on a case-by-case basis unmodeled phenomena likely exist. None of the objects with scatter-to-error ration $>50$ seemed to be actually reasonably modeled with linear ephemeris and thus these objects were removed from further analysis as strongly pulsating. We also visually inspected all high-confidence signals and commented on systems that appeared to be affected by such effects. When computing the bootstrap analysis, we saved in addition to the best $\chi^2$ of each mock data, also its entire PA spectrum. This in turn allows us to build a smaller bootstrap test for each frequency individually: how often did test frequency $f$ had higher $\Delta\chi^2$ than our final $\Delta\chi^2$ cutoff? by counting these frequencies one can easily identify systems that likely include multi-frequency information (and thus possibly enable inversion for masses) even if this is not visible by eye. We note that poorly modeled systems that exhibit large ratio of scatter to the median error (also reported on Table \[ResultsTable\]) are prone to exhibiting unrealistically large number of apparently-significant frequencies. Specific systems ---------------- Here we discuss some of systems for which new information was gained by applying the above analysis. Our goal was to flag interesting systems and not to fully characterize them in depth (given the scope of this paper), and our main tool for analyzing the systems are plots as shown in Figure \[KOI\_209\_system\]: in each such figure we superimpose the PA spectra of all transit signals in given system as well as all expected and TTV frequencies, which are of four types: (1) all possible super-frequencies expected from all MMRs with $j:j-N$ period-ratio up to $j=9$ (arbitrarily) and $N=j-1$ using equation from §5 of Deck & Agol 2016. (2) The orbital frequencies. (3) The so-called “chopping frequencies” which are the frequency of conjunctions between any planet pair: $f_\textrm{C}=f_\textrm{in}-f_\textrm{out}$. (4) The expected stroboscopic frequencies of individual planets. This rather dense figure allowed to quickly assign an observed PA peak with a possible physical meaning, even in high-multiplicity systems. Plots like Figure \[KOI\_209\_system\] show only those expected frequencies which are found to be relevant to a given system and discussed in the text. ![PA spectra of all KOIs in the KOI 209 system, each planet/signal is plotted in a different color. Relevant theoretically-expected super-frequencies between any two interacting planets that are discussed in the text are marked at the top in a colored dashed vertical line in the colors of the relevant pair and the letter “M”, as well as an indication of the relevant MMR. Similarly, the chopping frequencies associated with each pair may also be indicated with a “C”. The peak of the PA spectrum was the starting point for non-linear optimization, the results of which are given as a horizontal error bar in the appropriate color for each TTV signal that was subjected to non-linear optimization - usually directly above it and with somewhat better significance (higher $\Delta\chi^2$). The different orbital frequencies are indicated at the bottom with vertical black dotted lines marked with an “f” and the relevant signal’s number. If a the stroboscopic frequency of a particular planet is in the scanned frequency range, it is similarly marked with a dashed line and the “S” label (not relevant for KOI 209). In the textbook-like case of KOI 209.02 it is a clearly simultaneously affected by two different nearby resonances and and the orbital period of 209.01.[]{data-label="KOI_209_system"}](KOI_209_system.pdf){width="50.00000%"} - **KOI 89 / Kepler-462:** (Figure \[KOI\_89\_system\]) KOI 89.02 was detected by the PA to have significant TTVs with $f_{\mathrm{TTV}}\approx9\cdot 10^{-4} d^{-1}$. The non-linear fit revised this value to $f_{\mathrm{TTV}}=(5.39_{-0.74}^{+0.67}) \cdot10^{-4}$ - consistent with the predicted 5:2 MMR with KOI 89.01 at $f_{\mathrm{Sup}}=4.704\cdot 10^{-4} d^{-1}$, while previous analyses (e.g. H16) had a $>4\sigma$ discrepancy. This, together with the high-confidence of the TTV detection, confirms KOI 89.02, hitherto just a candidate, as a bona-fide planet. ![Similar to Figure \[KOI\_209\_system\], for the KOI 89 system.[]{data-label="KOI_89_system"}](KOI_89_system.pdf){width="50.00000%"} - **KOI 108 / Kepler-103:** (Figure \[KOI\_108\_system\]) The PA approach detects two significant frequencies in KOI 108.02, and no TTVs in KOI 108.01. The frequencies detected do not correspond to any known interaction frequency between the known planets, and hence may suggest the presence of additional objects in the system. These TTVs are consistent with those reported by H16, but are inconsistent with those first identified by Van Eylen & Albrecht (2015), possibly owing to their use of only part of the data used here. ![Similar to Figure \[KOI\_209\_system\], for the KOI 108 system.[]{data-label="KOI_108_system"}](KOI_108_system.pdf){width="50.00000%"} - **KOI 185:** (Figure \[KOI\_185.01\]) This system presents the longest TTV period we were able to constrain (at $>3\sigma$). A TTV frequency of $f_{\mathrm{TTV}}=(1.75_{-0.45}^{+0.79}) \cdot10^{-4}$ (TTV period of $15.7 \pm 4.1$ years). Note however that KOI-185.01 may not be due to a planet: with a $\sim3\%$ deep grazing transit on a $\sim0.8R_\odot$ star the occulting object may be too large for a planet. ![Similar to Figure \[KOI\_75.01\], but for KOI 185.01 - the longest TTV period (but still constrained to better than $3\sigma$) TTV signal of about 18yrs.[]{data-label="KOI_185.01"}](KOI_185_01_TV_Optimized_Figure_2.pdf){width="50.00000%"} - **KOI 209 / Kepler-117:** (Figure \[KOI\_209\_system\]) This system is included for illustrative purposes. There are three distinct frequencies in the inner planet’s TTVs (KOI 209.02): one matches the 2:1 MMR, another matches the 3:1 MMR, and the last matches the orbital period of the outer KOI 209.01 (Bruno [*et al.* ]{}2015) [^2]. - **KOI 262 / Kepler-50:** (Figure \[KOI\_262\_system\]) Steffen [*et al.* ]{}2013 confirmed this two-planet system based on anti-correlated TTVs spanning $\approx700$ d. The data available today clearly shows that the two planets have TTVs of different frequencies: $f_{\mathrm{TTV,01}}=(9.74_{-0.11}^{+0.12}) \cdot10^{-4}$ and $f_{\mathrm{TTV,02}}=(15.42_{-0.14}^{+0.15}) \cdot10^{-4}$ Interpreting these TTV signals is not trivial: their period ratio is very close to the 6:5 MMR ($\Delta\simeq0.000131$), however, the expected super period is too long and we cannot resolve it using current data. The two different observed TTV periods could be a sign of separate interactions of each of the observed planets with yet another non-transiting planet in the system. For example, a planet on a $\sim11.7$ days orbital period could explain both observed TTV periods. We conclude that the confirmation above of the two planets by connecting the observed TTVs to mutual interaction between them is not correct, and speculate on the cause of the observed TTVs. ![Similar to Figure \[KOI\_209\_system\], for the KOI 262 system.[]{data-label="KOI_262_system"}](KOI_262_system.pdf){width="50.00000%"} - **KOI 271 / Kepler-127:** (Figure \[KOI\_271\_system\]) This system reveals dynamical interactions of KOI 271.02 with both KOI 271.01 and KOI 271.03. The main peak of KOI 271.02 is consistent with the expect 5:3 resonant frequency among .01 and .02. Additionally, the 2:1 between .02 and .03 appears is the most significant peak of .03, and possibly contributing a shoulder to the spectral peak of .02. All three planets were hitherto only statistically validated. ![Similar to Figure \[KOI\_209\_system\], for the KOI 271 system.[]{data-label="KOI_271_system"}](KOI_271_system.pdf){width="50.00000%"} - **KOI 282 / Kepler-130:** (Figure \[KOI\_282\_system\]) KOIs 282.01 and 282.03 are close to 3:1 MMR. The known TTV signal on 282.01 at $f_{TTV}=(20.55\pm0.44)\cdot 10^{-4} d^{-1}$ is consistent with the predicted $f_{\mathrm{Sup}}=20.73\cdot 10^{-4} d^{-1}$ for that $2^{nd}$ order resonance, confirming both planets (which were only statistically validated thus far but had no dynamical confirmation). ![Similar to Figure \[KOI\_209\_system\], for the KOI 282 system.[]{data-label="KOI_282_system"}](KOI_282_system.pdf){width="50.00000%"} - **KOI 312 / Kepler-136:** (Figure \[KOI\_312\_system\]) KOIs 312.01 and 312.02 are close to a 3:2 period commensurablity $\Delta=-0.056$, and both have significant TTVs, yet the TTVs are not close either the expected super-frequency or the expected chopping frequency. The system is not yet understood, and it is suspected there exists an additional undetected perturbing planet. ![Similar to Figure \[KOI\_209\_system\], for the KOI 312 system.[]{data-label="KOI_312_system"}](KOI_312_system.pdf){width="50.00000%"} - **KOI 464 / Kepler-561:** (Figure \[KOI\_464\_system\]: The two planets in the system are widely separated (periods of 5 and 58 days for KOIs 464.02 and 464.01, respectively) and thus likely not interacting. The outer transiting planet KOI 464.01 exhibits TTVs with two distinct and significant spectral peaks, that may be explained by one or more additional undetected planets. ![Similar to Figure \[KOI\_209\_system\], for the KOI 464 system.[]{data-label="KOI_464_system"}](KOI_464_system.pdf){width="50.00000%"} - **KOI 523 / Kepler-177:** (Figure \[KOI\_523\_system\]): This system shows that in addition to the known (Xie 2014) TTV signal of KOIs 523.01 (at $f_{TTV,01}=(4.33_{0.70}^{1.08})\cdot 10^{-4} d^{-1}$) which is consistent with the super-frequency of the 4:3 MMR with 523.02, there exists an additional significant spectral peak at $f_{TTV,2}\simeq62.8\cdot 10^{-4} d^{-1}$ in KOI 523.01 which may be related to the expected “chopping” frequency of $f_{Chop}=68.94 \cdot 10^{-4} d^{-1}$. Furthermore, hints of both of these peaks are seen in the PA spectrum of KOI 523.02. These new identifications may be used to improve on the current mass determination (Xie 2014). ![Similar to Figure \[KOI\_209\_system\], for the KOI 523 system.[]{data-label="KOI_523_system"}](KOI_523_system.pdf){width="50.00000%"}   - **KOI 775 / Kepler-52:** (Figure \[KOI\_775\_system\]): In addition to the previously known ([*e.g.*]{} H16, HL14) TTV peak of KOI 775.02 at a frequency close to the super-period associated with the 2:1 MMR, we detect a new peak in the spectrum of KOI 775.01 at nearly the same frequency ($f_{\mathrm{TTV}}=48.81 \cdot 10^{-4} d^{-1}$), though at a confidence of 0.998. This may allow improved mass determination for both planets. ![Similar to Figure \[KOI\_209\_system\], for the KOI 775 system.[]{data-label="KOI_775_system"}](KOI_775_system.pdf){width="50.00000%"} - **KOI 841 / Kepler-27:** (Figure \[KOI\_841\_system\]): We detect two previously unidentified frequencies in the PA spectrum of KOI 841.02, in addition to the known primary frequency (Steffen [*et al.* ]{}2012, HL14). These peaks may allow improved constraints on the masses. ![Similar to Figure \[KOI\_209\_system\], for the KOI 841 system.[]{data-label="KOI_841_system"}](KOI_841_system.pdf){width="50.00000%"} - **KOI 870 / Kepler-28:** (Figure \[KOI\_870\_system\]): KOI 870.01 and 870.02 are close to the 3:2 MMR and both exhibit significant TTVs at frequency consistent with the expected $f_{\mathrm{Sup}}=44.21 \cdot 10^{-4} d^{-1}$. Upper limits to the masses were given by Steffen [*et al.* ]{}(2013), and low-significance ($m/\Delta m\leq3$) detection of masses by HL14, but only using the most significant TTV frequency. Here we detect secondary frequencies that are just below the high-significance threshold at bootstrap Confidences of 0.983 and 0.998 for KOIs 870.01 and 870.02 respectively, which may allow a better mass determination for both planets. ![Similar to Figure \[KOI\_209\_system\], for the KOI 870 system.[]{data-label="KOI_870_system"}](KOI_870_system.pdf){width="50.00000%"} - **KOI 877 / Kepler-81:** (Figure \[KOI\_877\_system\]): The TTVs on KOI 877.02 and KOI 877.01 are both consistent with the expected 2:1 MMR between them at $f_{\mathrm{Sup}}=18.1465 \cdot 10^{-4} d^{-1}$, although only the former is high-confidence while the latter has near-threshold confidence of 0.996. HL14 analysed the system and constrained the component’s masses of KOI 877.01 and KOI 877.02 to $3\sigma$ or less, but used only the most significant TTV frequency, while KOI 841.02 exhibits a few more near-threshold frequencies. ![Similar to Figure \[KOI\_209\_system\], for the KOI 877 system.[]{data-label="KOI_877_system"}](KOI_877_system.pdf){width="50.00000%"} - **KOI 880 / Kepler-82:** (Figure \[KOI\_880\_system\]): There are prominent and well-known TTVs (e.g. HL14, H16) on both KOI 880.01 and KOI 880.02, where the primary peak of KOI 880.01 at the super-frequency corresponding to 2:1 MMR between them. Here we detect one additional significant PA-spectral peak for KOI 880.01 and three additional peaks in KOI 880.02, and note that the most prominent TTV frequency of the KOI 880.02 at $f_{\mathrm{TTV}}=(8.15\pm0.12) \cdot 10^{-4} d^{-1}$ is offset from both the predicted super-frequency and from the observed TTV frequency of KOI 880.01 by a significant margin. Finally, we find that the most significant TTV frequency of KOI 880.04 is just below the threshold in PA (confidence=0.996) but its $\Delta \chi^2$ increases to above-threshold in the full fit. This observed TTV frequency does not correspond to any expected of the ones. ![Similar to Figure \[KOI\_209\_system\], for the KOI 880 system.[]{data-label="KOI_880_system"}](KOI_880_system.pdf){width="50.00000%"} - **KOI 886 / Kepler-54:** (Figure \[KOI\_886\_system\]): TTV most significant TTV frequency on both KOI 886.01 and KOI 886.02 is well known (e.g. H16). Here we find additional peaks in the PA spectrum of KOI 886.01 at frequencies which are harmonics of the main peak. In addition, the primary peaks of both KOI 886.01 and KOI 886.02 are shifted by $4-6\sigma$ from the expected super-frequency of the 3:2 MMR between them. ![Similar to Figure \[KOI\_209\_system\], for the KOI 886 system.[]{data-label="KOI_886_system"}](KOI_886_system.pdf){width="50.00000%"} - **KOI 935 / Kepler-31:** (Figure \[KOI\_935\_system\]): HL14 identified the main TTV frequency of KOIs 935.01 and 935.02 as a 2:1 MMR at the expected $f_{\mathrm{Sup}}=10.27 \cdot 10^{-4} d^{-1}$ and provided a determination of the mass of KOI 935.02. Here we find the 935.01 also has a second high-significance peak close to the expected $f_{\mathrm{Sup}}=23.01 \cdot 10^{-4} d^{-1}$ of the 4:1 MMR with 935.03. Also the PA peak of KOI 935.02 is wider than expected (and wider than KOI 935.01’s peak) - close to the expected super-frequency of the 2:1 MMR between the 935.02 and 935.03. We therefore suspect that the PA spectrum of KOI 935.02 includes the sum of two blended and similar peaks tying the three outer planets in a 1:2:4 resonance chain. This may allow determination of other masses in the system. ![Similar to Figure \[KOI\_209\_system\], for the KOI 935 system.[]{data-label="KOI_935_system"}](KOI_935_system.pdf){width="50.00000%"} - **KOI 952 / Kepler-32:** (Figure \[KOI\_952\_system\]): We find a significant peak in the PA spectrum of KOI 952.02 at a low frequency, in addition to the known peak near the 3:2 MMR super-frequency ($f_{\mathrm{Sup}}=38.649 \cdot 10^{-4} d^{-1}$) between it and 952.01, previously identified by HL14. ![Similar to Figure \[KOI\_209\_system\], for the KOI 952 system.[]{data-label="KOI_952_system"}](KOI_952_system.pdf){width="50.00000%"} - **KOI 1102 / Kepler-24:** (Figure \[KOI\_1102\_system\]): KOI 1102.01 and KOI 1102.02 have both well known TTVs (e.g. HL14, H16) with TTV frequency consistent with the 3:2 MMR between them. Here we identify another significant TTV frequency for KOI 1102.01 - so a more precise determination of the masses seems possible. ![Similar to Figure \[KOI\_209\_system\], for the KOI 1102 system.[]{data-label="KOI_1102_system"}](KOI_1102_system.pdf){width="50.00000%"} - **KOI 1236 / Kepler-279:** (Figure \[KOI\_1236\_system\]): Xie 2014 identified the main TTV frequency of KOIs 1236.01 and 1236.03 as a 3:2 MMR at the expected $f_{\mathrm{Sup}}=8.4773\cdot 10^{-4} d^{-1}$ and provided a determination of their mass ($49.4^{+7.2}_{-5.9} m_\oplus$ and $37.5^{+5.5}_{-4.5} m_\oplus$ for KOI 1102.01 and KOI 1102.03, respectively). Xie 2014 did not use, however, the fact that there are more significant frequencies for both planets - including one peak for KOI 1236.03 near the expected 1:4 MMR super-frequency with KOI 1236.02, which has thus far no mass constraints, and a peaks for KOI 1236.01 near the expected “chopping” frequency with KOI 1236.03. A more precise determination of the masses seems possible. ![Similar to Figure \[KOI\_209\_system\], for the KOI 1236 system.[]{data-label="KOI_1236_system"}](KOI_1236_system.pdf){width="50.00000%"} - **KOI 1258 / Kepler-281:** (Figure \[KOI\_1258\_system\]): Here we detect a PA spectral peak for 1258.01 that is not consistent with any expected super-frequency of the previously known members of the system. The transit signals of KOIs 1258.01 and 1258.02 were statistically validated (Morton [*et al.* ]{}2016). One of the secondary (low confidence) peaks of PA spectrum is close to the orbital frequency of KOI 1258.03, possibly confirming the latter (currently classified as a candidate). ![Similar to Figure \[KOI\_209\_system\], for the KOI 1258 system.[]{data-label="KOI_1258_system"}](KOI_1258_system.pdf){width="50.00000%"} - **KOI 1366 / Kepler 293:** (Figure \[KOI\_1366\_system\]) We find a significant TTV signal for 1366.01 at $f_{TTV}=(33.43_{-0.63}^{+0.65})\cdot 10^{-4} d^{-1}$, consistent with the expected frequency for second-order 3:1 MMR with KOI 1366.02 at $f_{\mathrm{Sup}}=34.58 \cdot 10^{-4} d^{-1}$. This fequency also appears as a smaller peak in KOI’s 1366.02 PA spectrum. This confirms both planets (hitherto only having only statistical validation). ![Similar to Figure \[KOI\_209\_system\], for the KOI 1366 system.[]{data-label="KOI_1366_system"}](KOI_1366_system.pdf){width="50.00000%"} - **KOI 1426 / Kepler 297:** (Figure \[KOI\_1426\_system\]) The system is close to 1:2:4 MMR chain with normalized distance from resonances of $\Delta_{01,02}=-0.03748$ and $\Delta_{02,03}=0.001097$. As previously noted (HL14), the PA spectrum of 1426.02 exhibits evidence for the interaction between it and 1426.01 at the expected super-frequency and its first harmonic. The additional peaks of 1426.02 may be attributed to the orbital period of 1426.03 or the “chopping” frequency between 1426.02 and 1426.03. Moreover, the observed TTV frequency of 1426.03 and 1426.01 at $f_{TTV}=(8.91_{-0.28}^{+0.33})\cdot 10^{-4} d^{-1}$ is consistent with the 4:1 MMR between these objects. We note 1426.03 is currently a still a candidate, that no RV variation was detected in the system (Santerne [*et al.* ]{}2016). The system has a high SNR, so it is attractive for further analysis (previous analysis by Diamond-Lowe [*et al.* ]{}2015 is not publicly available). ![Similar to Figure \[KOI\_209\_system\], for the KOI 1426 system.[]{data-label="KOI_1426_system"}](KOI_1426_system.pdf){width="50.00000%"} - **KOI 1529 / Kepler-59:** (Figure \[KOI\_1599\_system\]) We detect a significant peak in 1529.02 at the same frequency as the known peak in 1529.01, both consistent with the 3:2 MMR super-frequency. Moreover, there are peaks in the PA spectrum of KOI 1529.01 that appear to be just below the adopted significance threshold - possibly enabling absolute mass determinations for these small planets (both have radii$<2R_\oplus$) ![Similar to Figure \[KOI\_209\_system\], for the KOI 1529 system.[]{data-label="KOI_1529_system"}](KOI_1529_system.pdf){width="50.00000%"} - **KOI 1599:** (Figure \[KOI\_1599\_system\]) KOI-1599.02 is found to have TTVs with $f_{TTV}=(6.00\pm0.67)\cdot 10^{-4} d^{-1}$ which is approximately consistent with that of KOI-1599.01, but both are not consistent with the expected 3:2 MMR super-frequency. This, together with the strong ($\Delta=-0.00081$) resonance suggests the system is in resonance (and not just near resonance), rendering the usual expression for the super-frequency irrelevant. In such a case the TTV frequency, in addition to the amplitude, can constrain the planetary masses. It is noteworthy that there are a few more significant frequencies in the PA spectrum of KOI-1599.01, but the signal is high-amplitude. Both KOIs are currently still neither confirmed or validates. ![Similar to Figure \[KOI\_209\_system\], for the KOI 1599 system. Note KOI-1599.01 has high-amplitude TTVs.[]{data-label="KOI_1599_system"}](KOI_1599_system.pdf){width="50.00000%"} - **KOI 1783:** (Figure \[KOI\_1783\_system\]) TTVs of both candidates were detected by H16: a low TTV frequency of $f_{H16}=7.19 \pm 0.67 10^{-4} d^{-1}$ and long-term (“polynomial”) TTVs for KOIs 1783.01 and 1783.02 respectively. We find (and indeed see clearly in H16’s figures) that the most significant TTV frequency for KOI 1783.01 is much higher at $f_{TTV}=(35.95_{1.02}^{+0.64})\cdot 10^{-4} d^{-1}$ - which is consistent with all three of: the orbital frequency of 1783.02, the “chopping” frequency between the two candidates, and with the maximal TTV frequency of KOI 1783.01 itself. Also, we constrain the TTV frequency of KOI 1783.02 to be $f_{TTV}=(10.6_{0.77}^{+1.53})\cdot 10^{-4} d^{-1}$, and the second-most significant TTV frequency for KOI 1783.01 (at low confidence) is virtually identical to this frequency - but not close to any expected super-frequency. The two candidates therefore appear to be interacting - but further study is needed. ![Similar to Figure \[KOI\_209\_system\], for the KOI 1783 system.[]{data-label="KOI_1783_system"}](KOI_1783_system.pdf){width="50.00000%"} - **KOI 1831 / Kepler 324:** (Figure \[KOI\_1831\_system\]) Known TTVs on KOIs 1831.01 and 1831.03 are anti-correlated but only polynomial \[H16\], and only the former was statistically validated while the latter is still a candidate. Here we detect the TTV frequency of KOI 1831.01 at $f_{TTV}=(4.24_{-0.47}^{+1.38})\cdot 10^{-4} d^{-1}$ and find it to be in agreement with the predicted 3:2 MMR super-frequency with KOI 1831.03 - but we do not detect the very significant TTVs on KOI 1831.03, as expected, since it is high-amplitude. Additionally, there are a few more significant frequencies in the PA spectrum of KOI 1831.01. This dynamically confirms KOI 1831.03, hitherto just a candidate, and possibly allows probing the absolute masses of the planets. ![Similar to Figure \[KOI\_209\_system\], for the KOI 1831 system.[]{data-label="KOI_1831_system"}](KOI_1831_system.pdf){width="50.00000%"} - **KOI 1955 / Kepler-342:** (Figure \[KOI\_1955\_system\]) We find two significant TTVs in the PA spectrum of 1955.02, also present in 1955.04 (partly also seen by H16). The frequencies are away from the expected 3:2 MMR super-frequency, possibly due to the system being deep in resonance ($\Delta=0.0027$). ![Similar to Figure \[KOI\_209\_system\], for the KOI 1955 system.[]{data-label="KOI_1955_system"}](KOI_1955_system.pdf){width="50.00000%"} - **KOI 2038 / Kepler-85:** (Figure \[KOI\_2038\_system\]) This well studied system (Xie 2013, H16, Hadden & Lithwick 2016) shows significant TTV frequencies on KOIs 2038.01 and 2038.02 that are consistent with the expected 3:2 MMR between them. Here we find the system may be more interconnected, with a possible blended peak on on the 2038.02 PA spectrum related to a 2:1 MMR with KOI 2038.04, and two additional low significance peaks in the PA spectrum of 2038.01 - one occurring at the expected super-frequencies of the 2:1 MMR with 2038.04 (at $f_{\mathrm{Sup}}=410\cdot 10^{-4} d^{-1}$, and another near the unusual 9:4 MMR with 2038.03. ![Similar to Figure \[KOI\_209\_system\], for the KOI 2038 system.[]{data-label="KOI_2038_system"}](KOI_2038_system.pdf){width="50.00000%"} - **KOI 2092 / Kepler-359:** (Figure \[KOI\_2092\_system\]) All three planets in the system were only statistically validated and weak ($<2\sigma$) mass limits were subsequently given by Hadden & Lithwick (2016). Here we find significant low frequency TTVs for all three planets which may arise either from the system residing near or within resonance. ![Similar to Figure \[KOI\_209\_system\], for the KOI 2092 system.[]{data-label="KOI_2092_system"}](KOI_2092_system.pdf){width="50.00000%"} ------------ ---------------------------------- -------------------- ----------------- ------- ---------- --------- ------- ------------------------------ ----------------------------- -------------- ---------------- ---------------------------------- ---------------------------------- ------------ ------------------------------------ KOI $\mathrm{f_{TTV}}$ $\mathrm{P_{TTV}}$ $\Delta \chi^2$ $A$ $T_0$ Conf. STD/Med Frequencies Strob. freq. Previous Comments $\cdot10^{-4} [\mathrm{d}^{-1}]$ $[\mathrm{d}]$ Area Single RMS Corr. \[min\] \[KBJD\] (linear model) $\cdot10^{-4} [\mathrm{d}^{-1}]$ $\cdot10^{-4} [\mathrm{d}^{-1}]$ references 12.01 $7.29^{+0.21}_{-0.26}$ 1371 1932 0.338 -121.947 249.643 0.885 $1.162^{+0.050}_{-0.042}$ $1068.5^{+5.9}_{-6.3}$ 1 4.02 \[16.1, 24.5, 200.7\] $\pm3.1$ 459.2 1 Kepler-448 b 13.01 $1746.12^{+0.25}_{-0.18}$ 6 12659 0.312 -15.347 46.904 0.961 $0.1070^{+0.0075}_{-0.0057}$ $134.143^{+0.129}_{-0.094}$ 1 2.27 \[1718.7, 1771.5\] $\pm2.7$ 1746.5 1 Kepler-13 b 41.01 $115.93^{+0.60}_{-0.34}$ 86 48 0.215 -7.584 5.721 0.940 $5.28^{+0.48}_{-0.69}$ $190.6^{+1.5}_{-1.7}$ 0.994 1.62 – 152.2 Kepler-100 c 42.01 $9.913^{+0.072}_{-0.067}$ 1009 3246 0.165 -83.669 272.222 0.987 $14.95^{+0.25}_{-0.30}$ $523.9^{+3.6}_{-3.6}$ 1 2.46 \[19.4, 33.9, 143.2\] $\pm3.1$ 428.4 1 Kepler-410 A b 46.01 \* $699.69^{+0.75}_{-0.71}$ 14 45 0.258 -2.094 3.791 0.973 $1.35^{+0.19}_{-0.20}$ $134.70^{+0.75}_{-0.72}$ 1 1.15 – 1960.3 Kepler-101 b 49.01 $244.98^{+0.66}_{-0.71}$ 41 42 0.217 -2.765 3.514 0.950 $2.37^{+0.35}_{-0.38}$ $170.8^{+2.3}_{-2.0}$ 0.997 1.33 – 1042.8 63.01 \* $265.79^{+0.31}_{-0.29}$ 38 116 0.244 -16.469 14.878 0.959 $0.441^{+0.032}_{-0.033}$ $160.24^{+0.96}_{-0.91}$ 1 2.27 306.0 $\pm3.1$ 738.1 Kepler-63 b, Active star 64.01 \* $3.2^{+1.2}_{-1.3}$ 3087 240 0.151 -4.605 14.783 0.985 $4.4^{+7.1}_{-1.7}$ $120^{+172}_{-168}$ 1 1.26 – 2478.0 70.01 $335.07^{+0.58}_{-0.53}$ 30 46 0.539 -3.314 6.149 0.749 $1.44^{+0.20}_{-0.24}$ $157.3^{+1.8}_{-1.8}$ 0.995 1.28 – 174.1 Kepler-20 c 70.02 $725.66^{+0.59}_{-0.91}$ 14 14 0.456 -2.296 3.983 0.880 $2.23^{+0.39}_{-0.39}$ $141.9^{+0.8}_{-1.2}$ 0.993 1.22 – 2391.1 Kepler-20 b 72.01 $667.46^{+0.52}_{-0.58}$ 15 23 0.241 -1.439 5.141 0.916 $1.10^{+0.20}_{-0.16}$ $145.12^{+0.88}_{-0.77}$ 0.994 1.32 – 11773.0 Kepler-10 b 75.01 \*\* $3.62^{+0.40}_{-0.29}$ 2759 2308 0.106 -192.179 124.444 0.975 $37.2^{+5.4}_{-5.7}$ $116^{+91}_{-56}$ 1 3.16 \[36.9, 47.1\] $\pm2.5$ 65.7 1 82.01 \* $214.94^{+0.46}_{-0.72}$ 47 47 0.426 -4.175 4.717 0.948 $1.61^{+0.19}_{-0.24}$ $174.9^{+1.4}_{-1.6}$ 1 1.49 – 94.2 Kepler-102 e 84.01 $31.80^{+0.18}_{-0.18}$ 314 317 0.187 -10.790 13.228 0.990 $5.40^{+0.26}_{-0.27}$ $291.3^{+4.2}_{-4.2}$ 1 1.20 – 532.6 1 Kepler-19 b 89.02 $5.39^{+0.67}_{-0.74}$ 1854 182 0.483 -22.188 13.741 0.954 $86^{+40}_{-20}$ $10^{+104}_{-146}$ 1 1.36 24.1 $\pm2.8$ 43.5 1 94.01 \* $80.06^{+0.49}_{-0.52}$ 125 72 0.252 -9.305 8.650 0.937 $0.817^{+0.073}_{-0.086}$ $198.8^{+5.8}_{-5.9}$ 1 1.54 \[65.0, 117.4, 161.4\] $\pm2.8$ 180.1 Kepler-89 d, A mutual event system 94.02 \* $64.13^{+0.28}_{-0.27}$ 156 228 0.145 -12.213 7.622 0.957 $5.86^{+0.51}_{-0.52}$ $149.8^{+3.3}_{-3.4}$ 1 1.67 (&gt;5) 100.3 Kepler-89 c, A mutual event system 94.03 $14.64^{+0.27}_{-0.31}$ 683 299 0.061 -27.076 9.659 0.995 $7.35^{+0.61}_{-0.51}$ $723^{+12}_{-17}$ 1 1.80 42.4 $\pm3.5$ 50.5 1 Kepler-89 e 100.01 \* $339.31^{+0.31}_{-0.35}$ 29 108 0.396 -6.342 17.335 0.656 $2.39^{+0.24}_{-0.22}$ $153.99^{+0.79}_{-0.78}$ 1 1.61 – 744.5 103.01 $38.111^{+0.076}_{-0.079}$ 262 2419 0.194 -37.914 140.433 0.999 $25.56^{+0.61}_{-0.46}$ $329.6^{+1.5}_{-1.4}$ 1 1.54 \[28.3, 48.4, 72.6\] $\pm2.8$ 487.3 1 105.01 $281.23^{+0.61}_{-0.65}$ 36 20 0.376 -2.752 4.218 0.735 $1.20^{+0.24}_{-0.25}$ $150.7^{+1.8}_{-1.8}$ 0.990 1.08 – 578.0 108.02 $10.60^{+0.33}_{-0.34}$ 943 115 0.398 -16.683 18.669 0.863 $20.3^{+1.8}_{-1.9}$ $767.2^{+10.2}_{-9.3}$ 1 1.42 22.2 $\pm2.4$ 39.8 1, 6 Kepler-103 c ------------ ---------------------------------- -------------------- ----------------- ------- ---------- --------- ------- ------------------------------ ----------------------------- -------------- ---------------- ---------------------------------- ---------------------------------- ------------ ------------------------------------ Conclusions {#Conclusions} =========== We introduced the technique of Spectral Approach to TTVs for the detection of transit timing variations. The Spectral Approach is: more sensitive due to the reduced number of free parameters in its model; not limited by short or low-SNR single events because it uses one global fit and not multiple event-by-event fits; unbiased since only the improvement over the linear model matters, and not the properties of linear model itself. New TTV candidates were found, and the overall set has no significant period or depth biases relative to the general *Kepler* candidates population - unlike catalogs resulting from the classical approach to TTV detection. Consequently the Spectral Approach is more sensitive to TTVs of lower amplitude, around smaller- and shorter-period planets, than the classical TTV measurement technique. We also presented the Perturbative Approximation (PA) to the Spectral Approach, a linear approximation which is much faster than the full model, albeit less sensitive to higher-amplitude TTVs, allowing to quickly identify candidates for more computationally-intensive full Spectral Approach fit. PA can also be used to other types of variations, such as impact parameter variations, and this will be further explored in future work. Applying these techniques to *Kepler* data we were able to detect 131 new TTV-bearing stars. The fact that so many new TTVs were detected is interpreted as stemming from the high planetary multiplicity uncovered by *Kepler*: TTVs are not the exception but rather the rule. Of particular importance are: (a) Stars that exhibit multiple sets of transits, which sometimes allow us to link the observed TTVs to a specific planet-planet interaction, and to place constraints on the masses of the planets; (b) Stars that exhibit multiple significant TTV frequencies. (c) Planets that have TTVs that cannot be linked to other planets in the system: these planets are likely affected by other yet-unknown objects in the system. We note that the use of the full PA spectrum, and not just the most significant frequency was found to be useful. The sensitivity and generality of PA for all targets that exhibit three or more transits, paired with its short execution time, make it highly suitable for current and future large-scale surveys ([*e.g.*]{} space-based K2, TESS, and PLATO as well as ground-based SuperWASP and HATNet). Moreover, the execution time of PA is so short that applying it to all  150,000 *Kepler* stars is possible - even those with currently no threshold crossing event at all. The speed will make searching for TTVs on *all* stars feasible. This may be useful since small, low-mass planets and easier to deflect by gravitational interaction with other bodies in the system, and thus planets may (and probably do) exist that have escaped detection because their transit signals have been blurred by TTVs. Such planets may become detectable once TTVs are accounted for using PA. Acknowledgements {#acknowledgements .unnumbered} ================ This project was supported by the Helen Kimmel Center for Planetary Science, the Minerva Center for Life Under Extreme Planetary Conditions and by the I-CORE Program of the PBC and ISF (Center No. 1829/12). AO acknowledges the support of the Koshland Foundation and McDonald-Leapman grant J.-W.X. acknowledges support from the NSFC Grants (11403012, 11333002,11661161014) and a Foundation for the Author of National Excellent Doctoral Dissertation of People’s Republic of China. RS acknowledges the support of the ISF. This paper includes data collected by the Kepler mission. Funding for the Kepler mission is provided by the NASA Science Mission directorate. Some/all of the data presented in this paper were obtained from the Mikulski Archive for Space Telescopes (MAST). STScI is operated by the Association of Universities for Research in Astronomy, Inc., under NASA contract NAS5-26555. Support for MAST for non-HST data is provided by the NASA Office of Space Science via grant NNX09AF08G and by other grants and contracts. This work has made use of services produced by the NASA Exoplanet Science Institute at the California Institute of Technology. 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--- abstract: 'We report on the production of nanodiamonds (NDs) with size via bead assisted sonic disintegration (BASD) of a polycrystalline chemical vapor deposition (CVD) film. The high crystalline quality NDs display intense narrowband () room temperature luminescence at from in situ incorporated silicon vacancy (SiV) centers. We demonstrate bright, narrowband single photon emission with $>100000$ cps. Due to the narrow fluorescence bandwidth as well as the near-infrared emission these NDs are also suitable as fluorescence labels with significantly enhanced performance for in-vivo imaging.' author: - 'E. Neu' - 'C. Arend' - 'E. Gross' - 'F. Guldner' - 'C. Hepp' - 'D.Steinmetz' - 'E. Zscherpel' - 'S. Ghodbane' - 'H. Sternschulte' - 'D. Steinmüller-Nethl' - 'Y. Liang' - 'A. Krueger' - 'C. Becher' title: Narrowband fluorescent nanodiamonds produced from chemical vapor deposition films --- ![\[fig:remnd\]SEM Images of: (a) the polycrystalline starting material, (b) individual NDs after spin coating onto a Si substrate. (c) High resolution transmission electron microscopy (HR-TEM) images of dropcast colloid on a copper grid with lacey carbon film. ](fig1.eps) In recent years fluorescent nanodiamonds (NDs) have attracted increasing attention in a wide field of prospective applications e.g. fluorescence labeling in biological imaging [@Chang2008; @Mohan2010; @Neugart2007] and as solid-state single photon sources (SPSs) [@Beveratos2002]. In these applications they stand out owing to photostable fluorescence [@Chang2008], bio compatibility [@Mohan2010] and feasible surface functionalization [@Neugart2007]. The majority of the experiments performed in these fields rely on the fluorescence of the nitrogen vacancy (NV) center in diamond. NV centers are readily available as single centers in NDs[@Schietinger2009] or created by irradiating diamond powders [@Chang2008; @Beveratos2002]. Recent work also demonstrated the efficient production of NV containing NDs by ball milling of microcrystalline powder [@Boudou2009]. Despite these advantages, NV centers exhibit two major drawbacks for the applications mentioned above: their room temperature fluorescence spectrum spans about [@Schietinger2009] and, due to their zero-phonon-line (ZPL) at , excitation with visible laser light (typ. at ) is necessary. On the other hand, the silicon vacancy (SiV) center delivers superior fluorescence properties. Firstly, it exhibits a ZPL at [@Feng1993; @Wang2006; @Sternschulte1994], lying within the near-infrared window of biological tissue[@Weissleder2003]. Furthermore, due to low electron-phonon-coupling about of the fluorescence is emitted into the ZPL even at room temperature[@Zaitsev2001; @Neu2011] with an ensemble width of only [@Wang2006; @Feng1993]. These properties lead to exceptional advantages in the potential applications of SiV-containing NDs: Firstly, the fluorescence can be excited using red laser light (here: 671 nm) thereby minimizing tissue autofluorescence for in-vivo imaging applications [@Chang2008; @Weissleder2003]. Simultaneously, absorption of the excitation laser is $\approx$20 times lower compared to 532 nm[@Weissleder2003], thus enabling imaging of deep tissue which is crucial for life science applications. Secondly, the significantly reduced fluorescence bandwidth allows for narrow spectral filtering, even further reducing autofluorescence in imaging applications. The narrow-band fluorescence also enables efficient discrimination of single photon emission and background signals for applications in e.g. quantum cryptography. As we show in this work, the SiV centers emit in a spectral region where background emission from the diamond material is low. Combined with the enhanced photon extraction from NDs[@Beveratos2002] SiV centers in NDs thus are promising candidates for practical solid state SPSs. Commonly, SiV centers are observed in all types of chemical vapor deposition (CVD) diamonds including homoepitaxial single-crystalline [@Sternschulte1994] and polycrystalline materials [@Feng1993] due to Si incorporation by etching of Si substrates and CVD reactor walls. These SiV centers produced in situ during CVD growth seem to possess superior fluorescence properties as compared to SiV centers produced by ion implantation: the latter ones were examined as SPSs [@Wang2006] but showed unfavorably low emission rates (1000 counts per second (cps)). On the other hand, single SiV centers produced in situ in CVD grown NDs demonstrated very bright emission with recently [@Neu2011]. These substrate bound CVD NDs, however, are not applicable as fluorescence labels requiring NDs in solution or for enhanced SPSs which require spatial nanomanipulation to achieve coupling to photonic or nanoplasmonic structures[@Schietinger2009]. We here report the production of fluorescent NDs containing in situ produced SiV centers from polycrystalline CVD films via the bead assisted sonic disintegration (BASD) method. These NDs combine the advantageous SiV fluorescence properties and its feasible production during the CVD process with the extended applicability of NDs dispersed in solution. As starting material for ND production we employ a polycrystalline diamond film grown by rho-BeSt Coating GmbH (Innsbruck) using hot filament CVD on ND seeded Si[@supplmat2011]. Scanning electron microscope (SEM) images indicate a film thickness of and high crystalline quality (grain size , Fig. \[fig:remnd\](a)). Raman spectroscopy reveals a distinct diamond Raman line (, inset Fig. \[fig:plrmanens\]); its width of indicates high crystalline quality with moderate stress distribution [@Zaitsev2001]. A weak G-band at around is attributed to residual sp$^2$ carbon. The film thus provides a high quality starting material for ND production. Photoluminescence measurements (Fig. \[fig:plrmanens\]) display a pronounced SiV ZPL at (width in accordance with the literature [@Wang2006; @Feng1993]). Additionally, we observe a broad band fluorescence that strongly diminishes beyond 700 nm[@supplmat2011]. The characterized diamond film is used for ND production after removing the substrate by chemical etching. ![\[fig:plrmanens\]Photoluminescence spectra recorded under 532 nm laser excitation from the polycrystalline starting material and an ensemble of the produced NDs. The inset shows the Raman spectrum of the starting material and the NDs recorded with 488 nm laser light.](fig2.eps) The BASD process, that some of us disclosed recently [@Ozawa2007; @Liang2009], enables the deagglomeration of strongly agglomerated nanoparticles by charging ceramic microbeads to a sonicated suspension. The microjets or shock waves induced by ultrasonic cavitations propel the beads (here: ZrO$_2$, size) leading to the disintegration of large agglomerates and the eventual formation of colloidal solutions of primary nanoparticles. So far, the method has only been applied for the deagglomeration of nanoparticles bound by inter-particle forces. Here, we apply the BASD technique as a top down approach for the production of nanoparticles from a CVD diamond film by separating the grains and crushing the $\mu$m-sized crystals. The BASD process is followed by several purification steps[@supplmat2011], delivering a colloidal suspension of NDs in deionized water. The average particle size, determined by dynamic light scattering directly in the resulting solution, shows a distribution maximum at [@supplmat2011]. As the cumulative size distribution shows that more than of all particles have a size [@supplmat2011], these NDs are highly suitable for in-vivo imaging applications[@Mohan2010]. HRTEM and SEM observation verified the diamond nature of the nanoparticles as well as the nanometric size (Fig. \[fig:remnd\](b)$\backslash$(c)). The faceted shape of the individual NDs indicates the crushing of the film’s crystallites along lattice planes. No contamination from the BASD process was found in the final colloidal solution[@supplmat2011]. ![\[fig:ndhist\](a) Photoluminescence spectra of individual NDs. Inset: of ND (1), fit taking into account background and instrument response revealing g$^{(2)}$(0)$\approx$0.05, (b) Histogram of ZPL positions and widths (mean value 6.8 nm). ](fig3.eps) ![\[fig:ndcryo\]Photoluminescence spectra at cryogenic temperatures. (1) SiV luminescence from 1.65 $\mu$m thick polycrystalline film at , (2) SiV luminescence from high quality 100 nm thick homoepitaxial film at , (3)-(5) spectra of individual NDs at , Inset: of ND (3) ](fig4.eps) To determine the fluorescence properties the ND solution is diluted and spin coated onto Si or Ir substrates. In a first step, we investigate a large ensemble of NDs on a sample with high density. Raman measurements certify conservation of the crystalline quality throughout the production process (Raman line , width , inset Fig. \[fig:plrmanens\]). Thus, the NDs provide high crystalline quality hosts for SiV centers, indispensable for achieving bright SiV luminescence[@Neu2011]. Fig. \[fig:plrmanens\] displays a typical fluorescence spectrum of a ND ensemble. We achieve significant reduction of broadband fluorescence, while the SiV luminescence remains basically unchanged (ZPL , width ). The reduced background is very promising with respect to applications of the NDs as solid state SPSs.\ To perform single ND spectroscopy on a low density sample we use a homebuilt confocal microscope setup [@Neu2011] (NA 0.8, excitation ). The spectra of four individual NDs are shown in Fig. \[fig:ndhist\](a), displaying intense SiV ZPLs. A histogram of the observed line positions is given in Fig. \[fig:ndhist\](b), displaying peak wavelengths between and . The ’ensemble ZPL’ widths of 3.8 to confirm narrowband luminescence for all NDs investigated. The spread of the line positions and widths is caused by varying stress in the NDs[@Zaitsev2001; @Neu2011]. We perform polarization dependent excitation studies, yielding up to visibility, evidencing (partial) alignment of the color centers inside the NDs. We point out that the observed fluorescence was photostable under excitation with up to . The measured ensemble fluorescence rates did not saturate, partially exceeding the maximum count rate of our photon counters (). Therefore, the NDs containing larger ensembles of SiV centers are applicable as bright, narrowband, photostable fluorescence labels. In addition to these ensemble ZPLs we observe lines as narrow as (see Fig. \[fig:ndhist\](a),(1)-(3)). Comparable line widths have been recently reported for single SiV centers [@Neu2011], thus we tentatively attribute these lines to shifted, bright single SiV centers. We find that some of the NDs (Fig. \[fig:ndhist\](a),(1)) show dominant bright emission ($>100000$ cps) from a single SiV center, verified by nonclassical intensity autocorrelation [@Beveratos2002; @Neu2011] (Fig. \[fig:ndhist\](a) inset). We thus demonstrate a SiV based SPS from a colloidal ND solution featuring a ZPL linewidth of $\approx$ at room temperature. To gain further insight into the ND fluorescence we perform low temperature (4.7 - 30 K) spectroscopy. Fig. \[fig:ndcryo\] displays luminescence spectra from individual NDs, a high quality homoepitaxial and a polycrystalline CVD film. The observed lines blue-shift as compared to the room temperature ZPL in agreement with the literature[@Feng1993]. The spectrum of the homoepitaxial film displays a four line fine structure indicating a split excited and ground state [@Sternschulte1994], while this structure is washed out in the polycrystalline film due to inhomogeneous broadening. The spectra of the NDs also show a fine structure (individual line widths ), but are more complex due to sub-ensembles of SiV centers under different stress, corresponding to the broadening in the polycrystalline film. Again some of the NDs show dominant emission from a single SiV center (Fig. \[fig:ndcryo\] inset), evidencing low temperature single photon emission. In summary we have demonstrated that BASD of diamond films produces fluorescent NDs that preserve or even enhance favorable properties of the starting material i.e. high brightness, narrowband fluorescence, low background emission and high photostability. We demonstrate bright ($>100000$ cps) single photon emission from SiV centers at room temperature. The fluorescent NDs also offer great potential as optimized fluorescence labels, where additional Si doping could further enhance the performance. For single photon applications, smaller NDs produced from purer diamond material are desirable. As the method presented here is easily scalable it pioneers the production of large amounts of NDs with engineered defect properties from a variety of diamond films. We acknowledge funding by the DFG, the European Commission (EQUIND, DINAMO, DRIVE) and the BMBF (EPHQUAM 01BL0903). SEM measurements were performed by J. Schmauch (UdS). [15]{}ifxundefined \[1\][ ifx[\#1]{} ]{}ifnum \[1\][ \#1firstoftwo secondoftwo ]{}ifx \[1\][ \#1firstoftwo secondoftwo ]{}““\#1””@noop \[0\][secondoftwo]{}sanitize@url \[0\][‘\ 12‘\$12 ‘&12‘\#12‘12‘\_12‘%12]{}@startlink\[1\]@endlink\[0\]@bib@innerbibempty [****,  ()](\doibase {10.1038/nnano.2008.99}) [****,  ()](\doibase {10.1021/nl1021909}) [****,  ()](\doibase {10.1021/nl0716303}) @noop [****,  ()]{} [****,  ()](\doibase {10.1021/nl900384c}) [****,  ()](\doibase {10.1088/0957-4484/20/23/235602}) @noop [****,  ()]{} @noop [****,  ()]{} [****,  ()](\doibase 10.1103/PhysRevB.50.14554) @noop [****,  ()]{} @noop [**]{} (, ) @noop [****,  ()]{} @noop [****,  ()](\doibase {10.1002/adma.200601452}) [****,  ()](\doibase {10.1021/nn900339s}) @noop [****,  ()]{} **Supplementary material for:\ Narrowband fluorescent nanodiamonds produced from chemical vapor deposition films\ ** *The supplementary material covers details of the BASD process, including the employed purification steps, as well as the data of the nanodiamond (ND) size measurements by dynamic light scattering (DLS). We give additional information on the luminescence spectra discussed in the paper.*\ **Additional discussion on fluorescence spectra and starting material:**\ The polycrystalline diamond film used as a starting material has been grown by rho-BeSt using a hot filament CVD Process. As substrate a 4’ ND seeded {100}-oriented Si wafer has been used. The growth was performed using CH$_4$ in H$_2$. Taking into account the SEM images, the film provided approx. 0.053 g of diamond material.\ In Fig. \[fig:plrmanens\_suppl\] (cf. Fig 2, original manuscript) additional lines between 680 and are observed in the fluorescence spectrum. ![\[fig:plrmanens\_suppl\]Photoluminescence spectra recorded under 532 nm laser excitation from the polycrystalline starting material and an ensemble of the produced NDs. The marked areas show the tantalum related lines.](Plramanens_suppl.eps){width="35.00000%"} As these lines are not relevant for the SiV related topics they are not discussed in the paper. This triplet has been tentatively attributed to the incorporation of tantalum from the hot filament process [@Harris1996]. As these fluorescence lines are rather weak and nearly coincide with the maximum of the broad fluorescence background observed, the application of these centers as fluorescence markers or single photon sources is unfavorable.\ **Additional information on the BASD process:**\ ![\[fig:suppl\_DLS\_sol\](a) size distribution of the ND colloid, (b) cumulative size distribution of the colloidal solution. (c) ND colloid in deionized water obtained from a BASD-treated CVD diamond film](suppl_DLS_sol.eps){width="50.00000%"} As described in the manuscript we apply a BASD process to crush a polycrystalline diamond film. So far, the method has only been applied for the deagglomeration of nanoparticles bound by inter-particle forces (electrostatic, $\pi$-$\pi$ interactions, hydrogen bonding, bonds between surface groups). Here, the BASD technique has been applied for the production of nanoparticles from a CVD diamond film, as these films are not easily processed in a conventional grinding tool such as a mortar or ball mill due to technological restrictions (available amounts of starting material, applicable forces, necessity to crush the crystallites along crystal planes etc.). In preliminary experiments to this study we observed that beadless ultrasonic treatment was not effective for the crushing of diamond films. Therefore, the BASD process and the following purification have been carried out: Sheets of the diamond film and ZrO$_2$ microbeads (50 $\mu$m) were treated in of dimethyl sulfoxide (DMSO) for using a powerful sonicator () equipped with a horn-type sonotrode. of toluene were added to the reaction mixture. The sediments were separated by centrifugation and washed with of acetone. In order to remove impurities like sonotrode abrasion, amorphous carbon and nanozirconia fragments, the residue was first stirred overnight in a 1:1:1 mixture of conc. H$_2$SO$_4$, HNO$_3$ and HClO$_4$ at . Secondly, the zirconia beads have been removed by centrifugations. The resulting supernatant was then stirred in overnight and centrifuged. The sediment was washed in repeated centrifugation-redispersion cycles with deionized water until and stored as a colloidal solution (Fig. \[fig:suppl\_DLS\_sol\](c)) in deionized water. The average particle size, i.e. hydrodynamic diameter determined by dynamic light scattering (DLS) directly in the colloidal solution of the resulting ND particles, is well below with the distribution maximum at (Fig. \[fig:suppl\_DLS\_sol\](a)) . The cumulative size distribution shows that more than of all particles have a size (Fig. \[fig:suppl\_DLS\_sol\](b)). EDX measurements after the purification proved the absence of zirconia contamination.
--- abstract: 'We address an important issue as to whether bulk-sensitive data of Raman scattering, optical conductivity, magnetic penetration depth, directional point-contact tunneling spectra, and nonmagnetic pair-breaking effect in optimally electron-doped Nd$_{1.85}$Ce$_{0.15}$CuO$_{4-y}$ support a nodeless $s$-wave or $d$-wave superconducting gap. We numerically calculate Raman intensities, directional point-contact tunneling spectra, and nonmagnetic pair-breaking effect in terms of both $s$-wave and $d$-wave gap symmetries. We find that all these bulk-sensitive data are in quantitative agreement with a nearly isotropic $s$-wave gap. The fact that $T_{c}$ is nearly independent of the residual resistivity rules out any $d$-wave gap symmetry.' author: - 'Guo-meng Zhao$^{1,2,*}$' title: 'Unambiguous evidence for nearly isotropic $s$-wave gap in the bulk of optimally electron-doped Nd$_{1.85}$Ce$_{0.15}$CuO$_{4-y}$ ' --- The gap symmetry of high-temperature cuprate superconductors has been a topic of intense debate for over twenty years. For hole-doped cuprates, bulk-sensitive experiments probing low-energy excitations in the superconducting state have consistently pointed towards the existence of line nodes in the gap function of hole-doped cuprates [@Hardy; @Jacobs; @Lee; @Chiao], which is consistent with either $d$-wave gap (having four line nodes) or extended $s$-wave gap (having eight line nodes). Other bulk-sensitive experiments on hole-doped cuprates [@Bha; @Li] can be quantitatively explained by extended $s$-wave gap [@Zhao2001]. In contrast, the issue as to whether there exist line nodes in the gap function of electron-doped ($n$-type) cuprates remains controversial. Phase and surface-sensitive experiments [@Tsu] provided evidence for pure $d$-wave order-parameter (OP) symmetry in optimally doped and overdoped $n$-type cuprates. Surface-sensitive angle-resolved photoemission spectroscopy (ARPES) [@Arm01; @Matsui] showed a $d$-wave gap with a maximum gap size of about 2.5 meV. This gap size would imply a $T_{c}$ of about 14 K at the surface, which is significantly lower than the bulk $T_{c}$ of 26 K (Ref. [@Matsui]). Earlier magnetic penetration depth data [@Alff] of optimally electron-doped cuprates with $T_{c}$ = 24 K were shown to be consistent with nodeless $s$-wave gap symmetry. Later on, a $T^{2}$ dependence of the penetration depth at low temperatures was observed in Pr$_{1.85}$Ce$_{0.15}$CuO$_{4-y}$ ($T_{c}$ = 20 K), which appears to support $d$-wave gap symmetry in the dirty limit [@Pr]. But the same data can be quantitatively explained [@Zhao2001] in terms of a nodeless $s$-wave gap if one takes into account an extrinsic effect due to current-induced nucleation of vortex-antivortex pairs at defects. Extensive penetration depth data [@Kim] of Pr$_{2-x}$Ce$_{x}$CuO$_{4-y}$ confirmed the nodeless gap symmetry at all the doping levels except for a deeply underdoped Pr$_{1.885}$Ce$_{0.115}$CuO$_{4-y}$ sample with $T_{c}$ = 12 K. Point-contact tunneling spectra [@Kas; @Bis; @Qaz; @Shan05; @Shan08] also showed no zero-bias conductance peak (ZBCP) at all the doping levels except for a deeply underdoped Pr$_{1.87}$Ce$_{0.13}$CuO$_{4-y}$ with $T_{c}$ = 12 K. Therefore, the penetration depth and point-contact tunneling spectra consistently suggest that the gap symmetry in deeply underdoped samples should be $d$-wave and change to a nodeless $s$-wave when the doping level is above a critical value. This scenario can naturally explain the $d$-wave gap symmetry inferred from surface-sensitive experiments if surfaces or interfaces are deeply underdoped. Experiments on hole-doped cuprates [@Bet; @Mann] indeed show that surfaces and interfaces are significantly underdoped. Here we focus on optimally electron-doped Nd$_{1.85}$Ce$_{0.15}$CuO$_{4-y}$ (NCCO) to unambiguously address this important issue as to whether bulk-sensitive data of Raman scattering, optical conductivity, magnetic penetration depth, directional point-contact tunneling spectra, and nonmagnetic pair-breaking effect support a nodeless $s$-wave gap. We numerically calculate Raman intensities, directional point-contact tunneling spectra, and nonmagnetic pair-breaking effect in terms of both $s$-wave and $d$-wave gap symmetries. We find that all these bulk-sensitive data are in quantitative agreement with a nearly isotropic $s$-wave gap. The fact that $T_{c}$ is nearly independent of the residual resistivity rules out any $d$-wave gap symmetry. ![ Raman intensities at 8 K for Nd$_{1.85}$Ce$_{0.15}$CuO$_{4-y}$ ($T_{c}$ = 22$\pm$1 K) in the $B_{1g}$ (Fig. 1a) and $B_{2g}$ (Fig. 1b) symmetries. The data are digitized from Ref. [@Blu]. The solid lines are numerically calculated curves in terms of an anisotropic s-wave gap function: $\Delta$ = $3.59 (1-0.11\cos 4\theta)$ meV with $\Gamma$ = 1.4 meV for the $B_{1g}$ channel and $\Gamma$ = 1.9 meV for the $B_{2g}$ channel; the dashed lines are the numerically calculated curve in terms of a non-monotonic $d$-wave gap function: $\Delta$ = $3.25 (1.43\cos 2\theta-0.43\cos 6\theta)$ meV with $\Gamma$ = 1.6 meV for both channels. ](Fig1.eps){height="12cm"} Bulk-sensitive Raman scattering has been proved to be a very powerful tool to study the anisotropy of the superconducting energy gap. Experiments carried out with different polarization orientations pick up the contributions to the light scattering on different parts of the Fermi surface. The $B_{1g}$ spectra provide information on the light scattering primarily in the neighborhood of the $k_{x}$ and $k_{y}$ axes while $B_{2g}$ spectra probe mainly along the diagonals, where ($k_{x}$, $k_{y}$) = $\vec{k}$ is the in-plane wave vector of electrons. The electronic Raman intensity is proportional to the imaginary part of Raman susceptibility $\chi_{\gamma\gamma} (\vec{q},\omega)$ in the limit of $\vec{q}$ approaching 0. At zero temperature, the imaginary part of $\chi_{\gamma\gamma} (\omega)$ is given by [@Branch] $$\begin{aligned} \label{Raman} Im\chi_{\gamma\gamma} (\omega) =\sum_{\vec{k}}\frac{\gamma^{2} (\vec{k})\Delta^{2}(\vec{k})}{E^{2}(\vec{k})}[\frac{\Gamma}{(\omega - 2E(\vec{k}))^{2} + \Gamma^{2}}\nonumber \\ -\frac{\Gamma}{(\omega + 2E(\vec{k}))^{2} + \Gamma^{2}}],\end{aligned}$$ where $\Delta (\vec{k})$ is the momentum-dependent superconducting energy gap, $E(\vec{k})$ = $\sqrt{\Delta^{2}(\vec{k}) + \epsilon^{2}(\vec{k})}$, $\Gamma$ is the parameter associated with the life-time broadening of the quasiparticles, $\epsilon (\vec{k})$ is the band-dispersion relation, and $\gamma (\vec{k})$ is the Raman vertex which is proportional to $\cos k_{x}a - \cos k_{y}a$ for $B_{1g}$ symmetry and to $\sin k_{x}a\sin k_{y}a$ for $B_{2g}$ symmetry (where $a$ is the lattice constant) [@Dev]. By fitting ARPES data, the band-dispersion relation $\epsilon (\vec{k})$ (in units of meV) for Nd$_{1.85}$Ce$_{0.15}$CuO$_{4-y}$ was found to be [@Mar] $$\begin{aligned} \epsilon (\vec{k}) = -460(\cos k_{x}a + \cos k_{y}a) + 220\cos k_{x}a\cos k_{y}a \nonumber \\ -70(\cos 2k_{x}a + \cos 2k_{y}a) +60(\cos 2k_{x}a\cos k_{y}a \nonumber \\ +\cos k_{x}a\cos 2k_{y}a)- 27\end{aligned}$$ Figure 1 shows $B_{1g}$ and $B_{2g}$ Raman intensities at 8 K for Nd$_{1.85}$Ce$_{0.15}$CuO$_{4-y}$ with $T_{c}$ = 22$\pm$1 K. The data are digitized from Ref. [@Blu]. The solid lines are numerically calculated curves using Eqs. 1 and 2, $\Gamma$ = 1.4 meV for the $B_{1g}$ channel, $\Gamma$ = 1.9 meV for the $B_{2g}$ channel, and $\Delta$ = $3.59 (1-0.11\cos 4\theta)$ meV, where $\theta$ is measured from the Cu-O bonding direction. It is remarkable that the calculated curves match very well with the experimental data. Furthermore, the minimum value $\Delta_{min}$ of this gap function is 3.2 meV, which is close to that (3 meV) deduced from the magnetic penetration depth [@Alff]. The optical reflectance of a similar Nd$_{1.85}$Ce$_{0.15}$CuO$_{4-y}$ crystal with $T_{c}$ = 23 K also shows a minimum gap of about 3.1 meV at 10 K (Ref. [@Homes97]). For comparison, we also numerically calculate Raman intensities in terms of a non-monotonic $d$-wave gap function: $\Delta$ = $3.25 (1.43\cos 2\theta-0.43\cos 6\theta)$ meV (dashed lines). This gap size, which is a factor of 1.7 larger than that extracted from ARPES [@Matsui], matches the peak position of the $B_{2g}$ Raman spectrum. It is apparent that the $B_{2g}$ Raman spectrum is inconsistent with any $d$-wave gap function with nodes along the Cu-Cu directions. Further evidence for the nodeless $s$-wave gap symmetry comes from the directional point-contact tunneling spectra of optimally doped Nd$_{1.85}$Ce$_{0.15}$CuO$_{4-y}$ ($T_{c}$ = 25 K). It was argued that single-particle tunneling experiments along the CuO$_{2}$ planes can probe the bulk electronic density of states since the mean free path is far larger than the thickness of the possibly degraded surface layer [@Pon]. Figure 2 shows the in-plane point-contact tunneling spectra of Nd$_{1.85}$Ce$_{0.15}$CuO$_{4-y}$ measured along (100) direction (Fig. 2a) and (110) direction (Fig. 2b), respectively. The data are digitized from Ref. [@Shan05]. One can calculate point-contact tunneling spectra using the Blonder-Tinkham-Klapwijk (BTK) theory [@BTK]. In this model, two parameters are introduced to describe the effective potential barrier ($Z$) and the superconducting energy gap $\Delta$. As a supplement, the quasiparticle energy $E$ is replaced by $E-i\Gamma$, where $\Gamma$ is the broadening parameter characterizing the finite lifetime of the quasiparticles. Based on the BTK theory, Shan [*et. al.*]{} calculate the tunneling conductance in terms of the isotropic $s$-wave gap function [@Shan05]. The agreement between the calculated curve and data is excellent for each tunneling spectrum. However, the gap sizes that used to fit the tunneling spectra along the (100) and (110) directions are slightly different. This implies that the gap is not isotropic. ![In-plane point-contact tunneling spectra of Nd$_{1.85}$Ce$_{0.15}$CuO$_{4-y}$ ( $T_{c}$ = 25 K) measured along (100) direction (Fig. 2a) and (110) direction (Fig. 2b), respectively. The data are digitized from Ref. [@Shan05]. The solid line in Fig. 2a is the numerically calculated curve using $Z$ = 2.8, $\Gamma$ = 0.86 meV, and $\alpha$ = 0 and the solid line in Fig. 2b is the calculated curve using $Z$ = 3.0, $\Gamma$ = 0.65 meV, and $\alpha$ = $\pi$/4. The gap function used in the calculations is $\Delta$ = $3.52 (1-0.17\cos 4\theta)$ meV. ](Fig2.eps){height="12cm"} For the extended anisotropic BTK model [@Tan], another parameter $\alpha$ is introduced to distinguish between different tunneling directions. In Fig. 2, we compare the tunneling spectra with the calculated curves based on the extended anisotropic BTK model and an anisotropic $s$-wave gap function: $\Delta$ = $3.52 (1-0.17\cos 4\theta)$ meV. The solid line in Fig. 2a is the numerically calculated curve using $Z$ = 2.8, $\Gamma$ = 0.86 meV, and $\alpha$ = 0 and the solid line in Fig. 2b is the calculated curve using $Z$ = 3.0, $\Gamma$ = 0.65 meV, and $\alpha$ = $\pi$/4. It is apparent that the calculated curves are in excellent agreement with the data. ![Comparison of the tunneling spectra of NCCO with the calculated curves based on a nonmonotonic $d$-wave gap function: $\Delta$ = $3.0 (1.43\cos 2\theta-0.43\cos 6\theta)$ meV. The solid lines are the numerically calculated curve using $Z$ = 3.0, $\Gamma$ = 0.65 meV, and $\alpha$ = 0 for the spectrum along (100) direction and $\alpha$ = $\pi$/4 for the spectrum along (110) direction, respectively. ](Fig3.eps){height="12cm"} In Figure 3, we compare the tunneling spectra with the calculated curves in terms of a nonmonotonic $d$-wave gap function: $\Delta$ = $3.0 (1.43\cos 2\theta-0.43\cos 6\theta)$ meV. The solid lines are the numerically calculated curves using $Z$ = 3.0, $\Gamma$ = 0.65 meV, $\alpha$ = 0 for the spectrum along (100) direction, and $\alpha$ = $\pi$/4 for the spectrum along (110) direction. One can see that, for the tunneling spectrum along (100) direction, the calculated curve coincides with the data at high bias voltages but significant deviations occur at low bias voltages. For tunneling spectrum along (110) direction, ZBCP is clearly seen in the calculated curve, in sharp contrast to the data. Therefore, the tunneling spectra cannot be explained by $d$-wave gap symmetry. Finally, the most powerful way to distinguish between any $d$-wave and anisotropic $s$-wave gap symmetries is to study the response of a superconductor to nonmagnetic impurities or disorder. The nonmagnetic impurity pair-breaking effect is both bulk- and phase-sensitive. This is because the rate of $T_{c}$ suppression by nonmagnetic impurities or defects in a two-dimensional superconductor [@Op] is determined by the value of the Fermi surface (FS) average $<\Delta (\vec{k})>_{FS}$, which depends sensitively on the phase of the gap function. More specifically, the rate is proportional to a parameter $\chi$ = $1-(<\Delta (\vec{k})>_{FS})^{2}$/$<\Delta^{2}(\vec{k})>_{FS}$. It is clear that $\chi$ = 0 for isotropic $s$-wave superconductors while $\chi$ = 1 for $d$-wave and $g$-wave superconductors. For the anisotropic $s$-wave gap: $\Delta$ = $3.59 (1-0.11\cos 4\theta)$ meV, $\chi$ = 0.006. An equation to describe the pair-breaking effect by nonmagnetic impurities (or defects) is given by [@Op] $$\label{pairbreak} \ln \frac{T_{c0}}{T_{c}}= \chi[\Psi (\frac{1}{2} + \frac{0.122 (\hbar\Omega_{p}^*)^{2}\rho_{r}}{T_{c}}) - \Psi (\frac{1}{2})],$$ where $\hbar\Omega_{p}^{*}$ is the renormalized plasma energy [@Op; @Rad] in units of eV, $\rho_{r}$ is the residual resistivity in units of $\mu\Omega$cm, and $\Psi$ is the digamma function. The renormalized plasma energy can be independently determined from optical conductivity. Optical data of Pr$_{1.85}$Ce$_{0.15}$CuO$_{4-y}$ indicate $\hbar\Omega_{p}^{*}$ = 1.64 eV (Ref. [@Homes]). We will use this value of $\hbar\Omega_{p}^{*}$ to calculate $T_{c}$ as a function of the residual resistivity in terms of both $d$-wave and an anisotropic $s$-wave gap function inferred from the Raman spectra above. ![$T_{c}$ as a function of residual resistivity in optimally doped Nd$_{1.85}$Ce$_{0.15}$CuO$_{4-y}$. The data are from Refs. [@Alff; @Shan05; @On]. The solid line is numerically calculated curve in terms of any $d$-wave gap and the dotted line is the calculated curve for an $s$-wave gap function proportional to $(1-0.11\cos 4\theta)$. ](Fig4.eps){height="6.5cm"} Figure 4 shows $T_{c}$ as a function of residual resistivity in optimally doped Nd$_{1.85}$Ce$_{0.15}$CuO$_{4-y}$. The data are from Refs. [@Alff; @Shan05; @On]. The solid line is the numerically calculated curve in terms of any $d$-wave gap and the dotted line is the calculated curve for an $s$-wave gap function proportional to $(1-0.11\cos 4\theta)$. For the $d$-wave gap symmetry, the parameter-free calculation (solid line) shows that $T_{c}$ will be suppressed to zero at a very small residual resistivity of 10.7 $\mu\Omega$cm while the measured $T_{c}$ is nearly independent of the residual resistivity. 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--- abstract: 'The effect of boundary conditions on the vacuum structure of quantum field theories is analysed from a quantum information viewpoint. In particular, we analyse the role of boundary conditions on boundary entropy and entanglement entropy. The analysis of boundary effects on massless free field theories points out the relevance of boundary conditions as a new rich source of information about the vacuum structure. In all cases the entropy does not increase along the flow from the ultraviolet to the infrared.' address: 'Departamento de Física Teórica. Facultad de Ciencias. Universidad de Zaragoza, 50009 Zaragoza. Spain' author: - 'M. Asorey and J. M. Muñoz-Castañeda' title: Vacuum Boundary Effects --- Introduction ============ In quantum field theory the vacuum state encodes all physical properties of the theory. Indeed, any other state can be generated by the action of field operators on the vacuum. In particular, the effects generated by non-trivial topological structures of space or change of boundary conditions can be directly analysed from the changes induced on the vacuum structure. Among the most famous vacuum effects are the phenomenon of spontaneous symmetry breaking and the Casimir effect [@casimir]. In particle physics, the main interest usually focuses on the behaviour of Green’s and other quantum field correlation functions at short distances which provides information about high energy particle scattering processes. These observables are very insensitive to space topology or field boundary conditions [@karpacz]. However, for strongly correlated or confining theories long distance properties become very important, for instance, to point out the existence or not of confinement or mass gap. The existence of deconfining transitions in those theories (e.g. non-abelian gauge theories) can be directly extracted from the analysis of the structure of the vacuum state. Another rich source of information about the theory is encoded in the behaviour of non-local observables like free energy or entropy that can be defined by exploiting analogies with thermodynamics. The interest on observables of this type has been recently boosted by the development of quantum information theory. The entanglement entropy [@sorkin] provides a good measure of the vacuum entanglement structure. It can also be used to point out the existence of phase transitions since it is unbounded for critical systems and bounded for systems with a finite mass gap [@guifre]. It has been also pointed out that the confinement mechanism might be related to vacuum entanglement [@kleb]. Another thermodynamic observable, the boundary entropy [@cardy][@affleck] is related to the number of boundary states. Both new types of entropy do not scale with the volume of the space, unlike the standard bulk entropy and other extensive quantities. The entanglement entropy scales in the critical case with the area of the boundary where the fluctuating modes of the vacuum are traced out [@sorkin][@srednicki]. This behaviour is characteristic of black hole physics and is one of the key features of the AdS/CFT correspondence. By their own nature it is quite possible that both new entropies shall depend on the global properties of the configuration space. In this note we analyse the dependence of those quantities on the space topology and field boundary conditions as well as its physical implications for quantum field theories. Boundary conditions and conformal invariance ============================================ Let us consider a real scalar free field theory defined in a bounded domain $\Omega$ in $\IR^D$ with regular and smooth boundary $\partial \Omega$. The quantum dynamics is governed by the Hamiltonian =-12\^2 +12(, ). \[prima\] Unitarity requires that $\CH$ has to be selfadjoint. In particular, this implies that one must fix the boundary conditions of the fields $\phi$ in a way that the Laplace-Beltrami operator $-\Delta$ is selfadjoint and positive. The boundary conditions which define a selfadjoint operator $-\Delta$ are given by [@aim] \[const\] in terms of an unitary operator $U\in \CU(L^2(\partial\Omega,\C))$ which acts on the boundary values $\varphi$ of the quantum fields $\phi$ and their normal derivatives $\partial_n\varphi=\dot\varphi$. Notice that not all unitary operators give rise to positive Laplace-Beltrami operators, but to have a consistent quantum field theory for all values of $m$ one needs to consider only boundary conditions which satisfy both requirements. The set of boundary conditions which are compatible with unitarity is given by unitary matrices $U$ with eigenvalues $\lambda={\rm e}^{i \alpha}$ in the upper unit semi-circumference $0\leq \alpha\leq \pi $. For a single real scalar field defined on the two-dimensional space-time $\IR\times[0,L]$ the set of compatible boundary conditions is a four-dimensional manifold which can be covered by two charts parametrised by L [(0) (L)]{} = A [(0) (L)]{} \[uno\] where $A=-i(\I-U)/(\I+U)$ is any hermitian matrix with $A\geq 0$ , and [ (L) L (L)]{} = B [(0) L (0) ]{} \[dos\] where $B=\begin{pmatrix} {a & b \cr c & d}\end{pmatrix}$ is any real matrix with $ad+bc=-1$, $ ac \leq 0$ and $bd \leq 0$. In the massless case $m=0$ the theory is conformally invariant. However, most of the compatible boundary conditions (\[uno\]) (\[dos\]) break conformal invariance [@cardy]. Only the boundary conditions corresponding to unitary matrices $U$ with eigenvalues $\pm 1$ preserve conformal invariance [@agm; @agm2]. In the two-dimensional case the set of conformally invariant boundary conditions {,-, U\_= [&&-]{} ; (0,2\] }U(2), \[uni\] is given by Neumann ($U=\I$), Dirichlet ($U=-\I$) and quasiperiodic ($U_\alpha$) boundary conditions [@agm]. All other compatible boundary conditions break conformal invariance and are not invariant under renormalization group transformations. They describe renormalised trajectories of the renormalization group flowing towards one of the conformally invariant boundary conditions [@agm2]. Boundary effects in conformal field theories ============================================ The infrared properties of quantum field theory are very sensitive to quantum field boundary conditions [@karpacz]. In particular, the physical properties of the quantum vacuum, free energy and vacuum energy exhibit a very strong dependence on the type of boundary conditions. The vacuum state of the free field theory is gaussian ()=  [e]{}\^[-12 (, )]{} and the vacuum energy density $\CE_0= \tr \sqrt{-\Delta+m^2}\,$ is ultraviolet divergent. However, for finite cylindric domains of the form $S^{D-1}\times[0,L]$ the finite size corrections $\epsilon_c$ of the asymptotic expansion of the vacuum energy density for large values of cylinder base radius $\Lambda$ and generatrix $L$ with $\Lambda>>L>>1$ \_0=\_B+\_b 1[L]{} + 1[L\^[D+1]{}]{} [\_c(m L)]{} +(1) \[ve\] are not divergent [@casimir]. In the massless limit $m\to 0$ the coefficient $\epsilon_c$ of this term becomes universal (i.e. independent of $L$) but is highly dependent on the boundary conditions. For instance, in two dimensions for quasi-periodic boundary conditions this first finite size correction is \_c= - \^2. \[qp\] The values and signs of this finite size contribution to the energy are very different for periodic ($\alpha=\pi/2,\epsilon_c=-\pi/6$), antiperiodic ($\alpha=3\pi/2,\epsilon_c=\pi/12$) and Zaremba ($\alpha=\pi ,\epsilon_c=\pi/48$) boundary conditions [@bfsv]-[@klp]. In higher dimensions we have for domains of the form $S^{1}\times[0,L]$ the values of $\epsilon_c$: $-\zeta(3)/(2\pi)$) for periodic, $3\,\zeta(3)/(8\pi)$ for antiperiodic and $3\, \zeta(3)/( 6 4\pi)$ for Zaremba boundary conditions, where $\zeta(3)=1.2020569$ is Apéry’s constant [@elizalde]. Similarly, in three-dimensional cylindric domains $S^{2}\times[0,L]$ we have for the same boundary conditions $-\pi^2/90$, $7 \pi^2/720$ and ${7 \pi^2/11520}$, respectively [@elizalde]. In a similar manner the free energy of the system at finite temperature $1/T$ with the boundary conditions (\[const\]) has the following asymptotic expansion for large volumes and low temperature $0<<L<<T<<\Lambda$ [@affleck; @Cardy], f=-=[ f\_B]{} T + f\_b + + (1[T]{}, 1), where $f_B=\epsilon_B$, $f_b=\epsilon_b$ and $f_c=\epsilon_c$. This is in agreement with the asymptotic expansion of vacuum energy density (\[ve\]) and for the same reason does not present any logarithmic dependence in the smaller transverse size scale $L$. In the asymptotic regime of low temperature and large volumes $0<<T<<L<<\Lambda$ we have f=- =[ f\_B]{} T + \_c(mT) + (1[L]{}, 1). There is a similar expansion for the entropy $$\hbox{S}= (1-T\partial_T)\log \hbox{Z}= - (D+1)\frac{\hbox{ }\Lambda^{D-1} L} {T^D}\tilde{f}_c(mT)+\frac{\hbox{ }m \Lambda^{D-1} L} {T^{D-1}}\tilde{f}'_c(mT) +\, { s_b }+ \CO\left(\frac1{L}, \frac1{\Lambda}\right).$$ The third term of this expansion $s_b$, known as boundary entropy [@cardy][@affleck], is finite and depends on the boundary conditions of the fields. In two dimensional conformal theories this entropy $s_b=\log g$ can be formally associated with the number of boundary states $g$ [@cardy], but in many cases $g={\rm e}^{\log s_b}$ is not integer and does not correspond to a simple counting of boundary states [@affleck]. It has been conjectured that the quantities $g$ and $s$ evolve with the renormalization group flow in a non-increasing way [@affleck] $$s_{ _{UV}}\geq s_{ _{IR}},\quad g_{ _{UV}}\geq g_{_{IR}}$$ as it corresponds to any type of thermodynamic entropy [@affleck][@friedan]. This conjecture is known as $g$-theorem and has been verified in many cases [@affleck2][@friedan] although not yet proved for the boundary renormalization group flow. The conjecture can be verified in the case of a two-dimensional free real scalar field defined on $\Rm\times [0,L]$. The partition function for anti-periodic boundary conditions, once properly renormalised, can be exactly calculated and it is given by Z\_a= q\^[1[24]{}]{}\_[n=1]{}\^(1-q\^[n-1[2]{}]{})\^[-2]{}= 12\^[-1[12]{}]{}\_[n=1]{}\^(1-\^[2n-1]{})\^[2]{}, \[antiperiodic\] where ${q}=e^{-2\pi T/L}$ and $\tilde{q}=e^{-2\pi L/T}$. From (\[antiperiodic\]) it follows that Casimir coefficient is in this case $\epsilon_c=\frac{\pi}{12}$. For Zaremba boundary conditions [@klebanov] we have Z\_z= q\^[1[96]{}]{}\_[n=1]{}\^(1-q\^[-1[4]{}]{})\^[-1]{} = \^[-1[12]{}]{}\_[n=1]{}\^(1-\^[4[n]{}-2]{}), \[zaremba\] which leads to the Casimir coefficient $\epsilon_c=\frac{\pi}{48}$. For periodic boundary conditions there are zero modes which generate infrared divergences. The partition function (density) is given by [@polchinski] z\_[p]{}= [q]{}\^[-1[12]{}]{}\_[n=1]{}\^(1-[q]{}\^n)\^[-2]{}= \^[-1[12]{}]{}\_[n=1]{}\^(1-\^n)\^[-2]{}. \[peri\] But, the infrared problem is so severe that affects the consistency of the theory [@Coleman]. In any quantum field theory the Schwinger functions must satisfy the Osterwalder-Schrader reflection positivity property in order to preserve unitarity and causality. However, in a free theory of two-dimensional massless bosons the two point function is neither positive nor reflection positive [@gift]. One way of solving all these problems is to consider a compactification of the scalar field $\Phi= {\rm e}^{i \phi/R}$ to a circle of unit radius. In that case the correlators of the compactified field $\Phi$ satisfy the reflection positivity requirement and theory becomes consistent [@gift]. In that case the partition function acquires some additional contributions dues the compactification of zero-modes. In particular, these contributions give rise to the following partition function Z\_[p]{}\^R &=& [q]{}\^[-1[12]{}]{}\_[n=1]{}\^(1-[q]{}\^n)\^[-2]{} \_[n,m=-]{}\^\^[ R\^2 n\^2+ ]{} \[peri2\]\ &=& \^[-1[12]{}]{}\_[n=1]{}\^(1-\^n)\^[-2]{} \_[n,m=-]{}\^\^[ R\^2 n\^2+ ]{} \[peri22\] for periodic boundary conditions. However, for the rest of quasiperiodic boundary conditions ($\alpha\neq \pi/2 $) there is no contribution of the compactification of zero modes and the partition function is directly given by Z\_[a]{}\^R &=& q\^[1[24]{}-12([-12]{})\^[2]{}]{} \_[n=-]{}\^(1-q\^[|n-|]{})\^[-1]{}\ &=& \^[-1[12]{}]{} ([2 ]{})\^[-1]{} \_[n=1]{}\^|1-e\^[2i]{}\^[n]{}|\^[-2]{} \[peri5\] where $\epsilon=|\frac{\alpha}{2\pi }- \frac14|$. In particular, this means that for antiperiodic and Zaremba boundary conditions there is no modification of (\[antiperiodic\]) and (\[zaremba\]), respectively. For Neumann boundary conditions the partition function is also modified by the presence of compact zero modes Z\_[N]{}\^R &=& [q]{}\^[-1[48]{}]{} \_[n=1]{}\^(1-[q]{}\^[[n/2]{}]{})\^[-1]{} \_[n=1]{}\^\^[ ]{}\ &=& R \^[-1[12]{}]{} \_[n=1]{}\^(1-\^[[2n]{}]{})\^[-1]{} \_[n=1]{}\^\^[ R\^2 n\^2]{} \[neumann2\] in a similar way that for the theory with Dirichlet boundary conditions, where Z\_[D]{}\^R&=& [q]{}\^[-1[48]{}]{}\_[n=1]{}\^(1-[q]{}\^[[n/2]{}]{})\^[-1]{} \_[m=-]{}\^\^[ R\^2 m\^2]{}[ ]{}\ &=& \^[-1[12]{}]{}\_[n=1]{}\^(1-\^[[2n]{}]{})\^[-1]{} \_[m=-]{}\^\^. \[dirichlet2\] The boundary entropy can easily be computed for all those cases and the results are: $$\begin{array}{llll} s^\alpha_{b} &=-\log{(2\sin \pi\epsilon}) \ &g_\alpha=(2 \sin \pi \epsilon)^{-1} & \hbox{ quasiperiodic b.c.} \cr s^D_{b}&=-\log 2R\sqrt{\pi} \ & g_{_D}=({2R\sqrt{\pi}})^{-1} & \hbox{ Dirichlet b.c.}\\ s^Z_{b}&=-\frac12\log 2 \ & g_{_Z}={2}^{-\frac1{2}} & \hbox{ Zaremba b.c.} \\ s^N_{b}&=\phantom{-}\log R\sqrt{\pi} \ & g_{_N}={R\sqrt{\pi}}{} &\hbox{ Neumann b.c.} \end{array}$$ The singularity observed for quasiperiodic boundary conditions at $\epsilon=0$ is due to the existence of zero-modes which once properly incorporated into the compact theory give rise to the correct value for periodic boundary conditions (\[peri2\]) (\[peri22\]) with vanishing boundary entropy. Notice also that $g_{_Z}= \sqrt{g_{_D} g_{_N}}$ as corresponds to the factorisation property of counting boundary states. The g-theorem holds along the renormalised flow of Robin boundary conditions $$U=\begin{pmatrix} {{\rm e}^{i \beta_0} &0 \cr 0&{\rm e}^{i \beta_L} } \end{pmatrix},$$ which interpolate between Dirichlet ($U=-\I$) to Neumann ($U=\I$) boundary conditions through Zaremba ($U=\sigma_3$) boundary conditions [@agm3] $$g_{_D} > g_{_Z}> g_{_N}$$ provided that $R< 1/\sqrt{2\pi}$. The boundary entropy exhibits a monotone behaviour similar to that of the central charge or the bulk entropy. Entanglement Entropy ==================== There is another type of entropy associated to the vacuum state of a field theory. If we ignore some field degrees of freedom of the theory one can consider the effective physical (mixed) states by tracing out those degrees of freedom. In this way mixed states with finite entropies can effectively appear in quantum field theory at zero temperature from pure states. The mechanism of tracing out degrees of freedom is a kind of quantum version of the renormalization group. In particular, the vacuum state generates by this mechanism a family of mixed states whose entropies provide measures of its degree of entanglement. These mixed states are generated by integration of the fluctuating modes of the vacuum state $\Psi_0$ in bounded domains $\Omega_1$ of the physical space $\Rm^D$ [@sorkin], i.e. \_[\_[\_1]{}]{}=\_[\_1]{} \_0\^\_0(x) d\^D x. The entropy of this state $ S_{\Omega_1}= -Tr \,\rho_{_{\Omega_1}} \log \rho_{_{\Omega_1}}$ (vacuum entanglement entropy) is ultraviolet divergent, but once regularised exhibit a very interesting asymptotic behaviour which is similar to that of the boundary entropy analysed in the previous section [@srednicki][@friedan][@callan][@dowker][@kabat]. For massless scalar theories the entropy presents the following asymptotic behaviour S\_[\_1]{}= \_[i=0]{}\^[D-1]{} C\_i ()\^[i]{}+( ), \[al\] in terms of the diameter $L_1$ of $\Omega_1$ and the ultraviolet short distances cut-off $a$ introduced to split apart the domain $\Omega_1$ and its complement $\Rm^D \hbox{\, $\backslash$ \,} \Omega_1$. In the three-dimensional case, this asymptotic behaviour follows an area law similar to the black hole area law [@sorkin; @srednicki]. In general, for $D>1$ the coefficients $C_i$ are not universal because they are regularization dependent. However, for one-dimensional spaces, although the formula (\[al\]) suggests that $C_0$ could be universal, it does not happen. In fact, the asymptotic behaviour of the entanglement entropy is not given in that case by (\[al\]) because that entropy acquires a leading logarithmic correction S\_[\_1]{}=C +C\_0, \[eone\] which obviously implies that the constant term is highly dependent on the regularization method. However, it turns out that the value of the coefficient of this logarithmic term $C$ is universal and equal to $1/3$ of the central charge $c$ of the conformal invariant theory. In the case of a massless scalar boson $c=1$ and $C=1/3$ [@holz]. The question is whether this value is dependent or not on the boundary conditions of the fields when the theory is defined on a large bounded domain $\Omega\supset\Omega_1.$ It is remarkable that coefficient $c_1=1/3$ turns out to be independent of the choice of boundary condition in $\Omega=(0,L)$ when $\Omega_1=(L/2-l/2, L/2+l/2)$ is chosen to have half of the size of the interval. This result can be easily understood as a consequence of the fact that the entanglement entropy is basically due to the behaviour of field correlations at the interface between $\Omega_1$ and its complement $\Omega \hbox{\, $\backslash$ \,} \Omega_1$ which does not involve the boundary values of the fields. On the other hand the finite part $C_0$ is highly dependent on the ultraviolet regularization method. However, when $\Omega_1$ reaches the boundary of the whole space $\Omega$ the entropy has the same asymptotic behaviour [@calcar; @calcar2] S\_l= + g+ 12 C\_0, \[eon\] but with a different coefficient for the asymptotic logarithmic term and a different finite term which is related to the boundary entropy [@affleck] and, thus also dependent on the boundary condition. The behaviour of this quantity along the boundary renormalization group flow has then the same monotone behaviour that the boundary entropy. A similar phenomenon occurs in 2+1 dimensions with the constant term. In general, the entropy is given by S\_[\_1]{}= C\_1 + C + C\_0. The logarithmic term is absent for domains $\Omega$ and $\Omega_1$ with smooth boundaries $\partial\Omega$ and $\partial\Omega_1$, whenever $\Omega\backslash\Omega_1$ is a connected manifold [@Fradkin]. In a regularized theory the smoothness condition requires that the curvature of the boundaries must be always much larger than the ultraviolet cut-off $a$ [@hamma]. In that case, the remaining constant $C_0$ has a special behaviour because not only is regularisation independent but also independent on the size of $\Omega_1$. $C_0$ can be split in two terms $C_0=C_0^\prime+C_0^\ast$, one $C_0^\prime$ which contains all possible dependences on the prescription used for the definition of the $\Omega_1$ perimeter $L_1$, and another one $C_0^\ast$ which is absolutely prescription independent. In a massive theory, if $L_1$ is much larger than the inverse of the mass gap $1/m$, there is a prescription which uniquely fixes the ambiguities involved in such a splitting [@kitaev] [@wen]. If $\Omega_1$ is decomposed as the disjoint union of three similar domains $\Omega_1=\Omega_\alpha\cup\Omega_\beta\cup \Omega_\gamma$, one can define C\_0\^=C\_0\^[\_1]{} -C\_0\^[\_\_]{}-C\_0\^[\_\_]{} -C\_0\^[\_\_]{}+C\_0\^[\_]{}+C\_0\^[\_]{}+C\_0\^[\_]{}, and the result is independent of the $\Omega_1$ decomposition and the perimeter definition prescription. The constant $C_0^\ast$ is also shape independent and only really depends on the topology of the domain $\Omega\, \backslash \, \Omega_1$. It defines a topological invariant entropy $S_{\rm top}=C_0^\ast$ associated to the quantum vacuum [@kitaev] [@wen], which measures its degree of topological entanglement. It can be shown that $S_{\rm top}=- \log \CD$, where $\CD$ is the total quantum dimension of the underlying topological theory. In our case case it is easy to show that $\CD=1$, which means the vanishing of the topological entanglement entropy, and that result is independent of the boundary conditions. In more general theories like the SU(2) WZWN theory with level $k$ the topological entanglement entropy is given by [@kitaev] S\_[top]{}=. The quantum dimension $\CD$ is non-integer in that case but it is a real topological invariant. Conclusions =========== The novel thermodynamic quantities associated to field theories like boundary entropy and vacuum entanglement entropy reveal new interesting properties of vacuum structure. The boundary entropy is associated to the existence of boundary states and, thus, is very sensitive to the boundary conditions of the fields. The role of the vacuum entanglement entropy focuses on the measure of the amount of entanglement of the quantum vacuum and is absolutely independent of the type of boundary condition, whenever the domain where the quantum fluctuations of the fields are integrated out does not reach the boundary of the space. However, when this domain reaches the boundary, the entanglement entropy becomes dependent on the boundary conditions, displaying a monotone behaviour along the boundary renormalization group flow similar to that of the boundary entropy. We have explicitly verified the behaviour of boundary and entanglement entropies under changes of boundary conditions for low dimensional massless free field theories. The boundary entropy varies for quasiperiodic boundary conditions and Robin boundary conditions, whereas the entanglement entropy only changes when the entanglement domain reaches the boundary or changes its topology. The same behaviour appears in three–dimensional field theories where the finite term of the asymptotic behaviour of the entanglement entropy can be related to a new topological invariant (topological entanglement entropy). For free scalar field theories we have shown that this topological invariant is trivial for connected convex domains, but self-interacting field theories and non-connected domains might have non-trivial topological entanglement entropy, which provides a basis for robust codes in quantum computation [@kitaev] In all analysed cases the boundary entropy does not increase along the boundary renormalization group flow from the ultraviolet to the infrared [@affleck][@agm3]. There are two interesting problems which remain open: the effect of interactions on both types of entropies associated to the quantum vacuum and their behaviour for topological field theories. Both problems deserve further analysis. Acknowledgements {#acknowledgements .unnumbered} ================ We thank M. Aguado, D. Garca Alvarez and J.I. Latorre, for interesting discussions. M. A. also thanks the organizers of the [*5th International Symposium on Quantum Theory and Symmetries*]{}, and specially M. del Olmo, for their hospitality in Valladolid during the meeting. This work is partially supported by CICYT (grant FPA2006-2315) and DGIID-DGA (grant2007-E24/2). References {#references .unnumbered} ========== [10]{} H. B. G. Casimir, Proc. K. ned. Akad. Wet. [**51**]{}(1948) 793 M. Asorey, J. Geom. Phys. [**11**]{}(1993)94 L. Bombelli, R.K. Koul, J. Lee and R. Sorkin, Phys. Rev. [**D 34**]{} (1986) 373 G. Vidal, J. I. Latorre, E. Rico and A. Kitaev, Phys. Rev. Lett. [**90**]{} (2003) 227902 I. R. Klebanov, D. Kutasov and A. Murugan, arXiv preprint \[arXiv:0709.2140\] (2007) J. Cardy, Nucl. Phys. [**B 324**]{} (1989) 581 I. Affleck and A.W.W. Ludwig, Phys. Rev. Lett. [**67**]{} (1991) 161 M. Srednicki, Phys. Rev. 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--- abstract: 'Mahjong is a very popular tile-based game commonly played by four players. Each player begins with a hand of 13 tiles and, in turn, players draw and discard (i.e., change) tiles until they complete a legal hand using a 14th tile. In this paper, we initiate a mathematical and AI study of the Mahjong game and try to answer two fundamental questions: how bad is a hand of 14 tiles? and which tile should I discard? We define and characterise the notion of deficiency and present an optimal policy to discard a tile in order to increase the chance of completing a legal hand within $k$ tile changes for each $k\geq 1$.' address: - 'Centre for Quantum Software and Information, University of Technology Sydney, Sydney, Australia' - 'School of Computer Science, Shaanxi Normal University, Xi’an, China' author: - Sanjiang Li - Xueqing Yan bibliography: - 'mahjong.bib' title: 'Let’s Play Mahjong!' --- Mahjong ,14-tile ,complete ,deficiency ,decomposition ,pseudo-decomposition Introduction {#S:1} ============ From the very beginning of AI research, from checker [@Samuel59], chess [@shannon1988programming], Go [@Silver+16] to poker [@bowling2015heads] and StarCraft II[^1], games have played as test-beds of many AI techniques and ideas. In the past decades, we have seen AI programs that can beat best human players in perfect information games including checker, chess and Go, where players know everything occurred in the game before making a decision. Imperfect information games are more challenging. Very recently, important progress has been made in solving the two-player heads-up limit Texas hold’em poker [@bowling2015heads] and its no-limit version [@BrownS17], which are the smallest variants of poker played competitively by humans. In this paper, we initiate a mathematical and AI study of the more popular and more complicated Mahjong game. Mahjong is a very popular tile-based multiplayer game played worldwide. The game is played with a set of 144 tiles based on Chinese characters and symbols (see Figure \[fig:mjtiles\]) and has many variations in tiles used, rules, and scores [@Wiki_mahjong]. Each player begins with a hand of 13 tiles and, in turn, players draw and discard (i.e., change) tiles until they complete a legal hand using a 14th tile. In this paper, for simplicity, we only consider Mahjong-0, the very basic and essential version of Mahjong. The other variations can be dealt with analogously. There are only three types of tiles used in Mahjong-0: - Bamboos: $B1, B2, ..., B9$, each with four identical tiles - Characters: $C1, C2, ..., C9$, each with four identical tiles - Dots: $D1, D2, ..., D9$, each with four identical tiles In this paper, we call *Bamboo* ($B$), Character ($C$), Dot ($D$) *colours* of the tiles, and write ${\mathcal{M}}_{0}$ for the set of tiles in Mahjong-0, which include in total 108 tiles. ![Mahjong tiles, from https://en.wikipedia.org/wiki/Mahjong[]{data-label="fig:mjtiles"}](mjtiles.png){width="95.00000%"} \[fig:my\_label\] An *eye* is a pair of identical tiles, a *pong* (*kong*) is a sequence of three (four) identical tiles. A *chow* is a sequence of three consecutive tiles of the same colour. A *meld* is either a pong or a chow. In the analysis of this paper, we need the following non-standard notion. \[dfn:pmeld\] A *pseudochow* (*pchow* for short) is a pair of tiles of the same colour such that it becomes a chow if we add an appropriate tile with the same colour. A *pseudomeld* (*pmeld* for short) is a pchow or a pair. We say a tile $c$ *completes* a meld $(ab)$ if $(abc)$ (after alphabetic sorting) is a meld. Similarly, we say a pair is completed from a single tile $t$ if it is obtained by adding an identical tile to $t$. For example, $(B3B4B5)$ is a chow, $(C1C1)$ is an eye, $(B7B7B7)$ is a pong, $(D9D9D9D9)$ is a kong, and $(B1B3)$ and $(C2C3)$ are two pchows. The remainder of this paper is organised as follows. In Section 2, we introduce basic notions related to pure and hybrid 14-tiles, and characterise how bad a pure or hybrid 14-tile is in, respectively, Sections 3 and 4. Section 5 then presents an optimal policy for discarding tiles, in order to increase the possibility of completing a 14-tile within at most $k$ tile changes. Concluding remarks and directions for future research are given in Section 6. Pure and hybrid 14-tiles ======================== Recall that each player begins with a set of 13 tiles (often called a *hand*). Then players may change a tile by either drawing a new tile or robbing a tile, say $t$, discarded by another player if the player has two identical tiles $t$ and wants to consolidate a pong $(ttt)$ or if $t$ completes her 13-hand into a legal hand.[^2] In this paper, we take the following defined 14-tiles as our object of study. \[dfn:14-tile\] A 14-tile is a sequence $S$ of 14 tiles from ${\mathcal{M}}_0$. A sequence of tiles is *pure* if they are all of the same colour and is *hybrid* if otherwise. Apparently, no identical tiles can appear more than four times in a 14-tile. To be precise, we often write a tile $t$ as a pair $(c,n)$ such that $c \in \{0,1,2\}$ denotes the colour of $t$ and $n$ denotes the number of $t$. Here we assume that $0, 1,2$ denote, respectively, Bamboo, Character, and Dot. For example, $(0,3)$ and $(1,5)$ denote $B3$ and $C5$ respectively. Moreover, we assume that each 14-tile is represented in the standard form in the following sense. \[dfn:standard14-tile\] The *standard form* of a (hybrid) 14-tile is a sequence $H$ of pairs $(c,n)$ with $0\leq c \leq 2$ and $1\leq n \leq 9$. We denote as $H[i]=(c_i,n_i)$ the $(i+1)$-th tile in $H$ and require, for each $0\leq i \leq 13$, that $c_i \leq c_{i+1}$ and $n_i \leq n_{i+1}$ if $c_i = c_{i+1}$. We assume that there are no five identical tiles. For the purpose of this work, a pure 14-tile of Bamboos is not different from a pure 14-tile of Dots or a pure 14-tile of Characters. Thus we often write a pure 14-tile simply as a 14-tuple of integers from 1 to 9 if the colour of the tiles is clear from the context or not important. Formally, we have A *pure* 14-tile $S$ is a sequence of 14 integers with non-decreasing integer value from 1 to 9 and no value appears more than four times. Suppose $S$ is a sequence of 14 integers. Write $S[i]$ for the $(i+1)$-th value of $S$ for $0 \leq i \leq 13$, i.e., we assume, as in Python and many other programming languages, that $S[0]$ represents the first value of $S$. Then $S$ is a pure 14-tile if and only if: - $S[i] \in \mathbb{N}$ and $1\leq S[i] \leq 9$ for each $0\leq i \leq 13$; and - $S[i] \leq S[i+1]$ for any $0\leq i \leq 12$; and - $S[i] < S[i+4]$ for any $0 \leq i \leq 9$. For each pure 14-tile $S$, we define its *set sequence*, written ${{\sf{Set}}}(S)$, as the subsequence of $S$ obtained by deleting, for each tile $(c,n)$ in $S$, all but the first occurrences of $(c,n)$ in $S$, and define its *residual sequence* as ${{\sf{Res}}}(S) \equiv S\setminus {{\sf{Set}}}(S)$.[^3] For example, consider the following pure 14-tile $S$ and its set and residual sequences: $$\begin{aligned} S \quad &=\ (1,1,1,2,2,3,3,4,7,7,7, 8,8,9,9) \\ {{\sf{Set}}}(S) &=\ (1,2,3,4,7,8,9) \\ {{\sf{Res}}}(S) & =\ (1,1,2,3,7,7,8,9)\end{aligned}$$ For convenience, we also write these sequences compactly as\ $S=(111223347778899)$, ${{\sf{Set}}}(S)=(1234789)$, and ${{\sf{Res}}}(S)=(11237789)$. \[dfn:complete&decomposition\] A 14-tile over ${\mathcal{M}}_0$ is *legal* or *complete* if it can be decomposed into four melds and one eye. Given a complete 14-tile $T$, a *decomposition* $\pi$ of $T$ is a sequence of five subsequences of $V$ such that $\pi(5)$ is a pair (the eye) and, for $1\leq i\leq 4$, each $\pi(i)$ is a meld. \[dfn:14tile\_graph\] Two 14-tiles $S, T$ over ${\mathcal{M}}_0$ are *neighbours* if $S[i] = T[i]$ holds for all but exactly one $i \in [0,13]$. The 14-tile graph is the pair $(\mathcal{T},E)$, where $\mathcal{T}$ is the collection of all 14-tiles and two 14-tiles $S,T$ are connected by an edge if $S,T$ are neighbours. The distance between two 14-tiles $S,T$ is the length of a shortest path from $S$ to $T$ in the 14-tile graph. \[dfn:dfncy\] The *deficiency number* (or simply *deficiency*) of a 14-tile $S$ is defined recursively: - A 14-tile has deficiency 0 if it is complete; - A 14-tile has deficiency 1 if it is not complete but has a neighbour which is complete; - In general, for $\ell\geq 0$, a 14-tile has deficiency $\ell+1$ if it has no deficiency smaller than or equal to $\ell$ and it has a neighbour which has deficiency $\ell$. If the deficiency of $S$ is $\ell$, we write ${{\sf{dfncy}}}(S) = \ell$. For example, the following 14-tile is complete and, thus, has deficiency 0. $$\begin{aligned} \label{mjsol1} H &= (B1B2B2B3B3B4B7B7B7)(C1C1)(D4D5D6)\end{aligned}$$ In this example, we have the following decomposition $$\begin{aligned} \pi &= (B1B2B3)(B2B3B4)(B7B7B7)(D4D5D6)(C1C1).\end{aligned}$$ A 14-tile $T$ is often not complete. The deficiency number of $T$ measures how bad $T$ is by counting the number of necessary tile changes to make it complete. In the following two sections, we consider how to compute the deficiency number of a pure and, respectively, a hybrid 14-tile. Deficiency of pure 14-tiles =========================== In this section we prove the following theorem. \[thm:pure\] Any pure 14-tile has deficiency less than or equal to 3. To prove this theorem, we need to make several simple observations. First, there exists a pure 14-tile that has deficiency 3: $$\begin{aligned} \label{14-tile_dfncy=3} V & = (1 1 2 2 5 5 6 6 8 888 99)\end{aligned}$$ Indeed, this pure 14-tile contains only one meld, viz. $(888)$. To get four melds, we need to replace some tile with another one (with the same colour). Since each tile can be used to construct at most one meld or one eye, we need at least 3 tile changes to obtain four melds. Suppose without loss of generality this pure 14-tile is in colour $B$. Then we may replace both $B2$ with $B4$, obtaining two chows $(B4B5B6)$, and replace one $B8$ with $B9$, obtaining a pong $(B9B9B9)$. This example leads to the following fact. \[fact&lt;=3\] There exists a pure 14-tile with deficiency 3. By Definition \[dfn:pmeld\], a pair of tiles say $(Bi,Bj)$ is a pmeld if $|j-i| \leq 2$. In what follows, we assume that a meld (kong) can also be counted as a pmeld (pong). \[fact2\] Any four tiles of the same colour contain a pmeld. Suppose $V=(abcd)$ is a set of four tiles of the same colour and $1\leq a \leq b \leq c \leq d \leq 9$. If $a=b$ or $b=c$ or $c=d$, we have a pair and thus a pmeld. Now, suppose $1\leq a < b < c < d \leq 9$. It’s easy to see that $b-a > 2$, $c-b > 2$, and $d-c > 2$ cannot happen in the same time. Thus $V$ has at least one pmeld. The following notion of ‘disjoint’ pmelds/melds is important. \[dfn:disjoint\] Suppose $V$ is a sorted tuple of $k$ ($3\leq k\leq 14$) tiles of the same colour. We say two sorted pmelds $(uv),(xy)$ contained in $V$ are *disjoint*, if $(uv)$ and $(xy)$ are not identical pairs and there exist indexes $0\leq i<j<p<q<k$ such that $u=V[i], v=V[j], x=V[p]$ and $y=V[q]$. Similarly, two sorted melds $(abc)$ and $(xyz)$ are *disjoint* if there exist indexes $0\leq i_1<i_2<i_3<i_4<i_5<i_6<k$ such that $a=V[i_1], b=V[i_2], c=V[i_3], x=V[i_4], y=V[i_5]$, and $z=V[i_6]$. In the same sense, we can also define when a meld and a pmeld are disjoint, and when $V$ contains three or more disjoint pmelds. Consider the tuple $W=(1248)$ of four tiles of the same colour. It has two pmelds, viz. (12) and (24), but they are not disjoint in our sense. Furthermore, consider $V=(111158)$. It has two identical pairs $(11)$, which are in a sense competing with each other and cannot both be useful in forming a legal hand. That’s why we don’t consider them as disjoint pmelds. For six tiles of the same colour, we have the following result. \[fact3\] Suppose $V$ is a tuple of six tiles of the same colour. If $V$ contains no kong, then $V$ contains at least two disjoint pmelds. If, in addition, $V$ contains six pairwise different tiles, then $V$ contains a chow and a disjoint pchow or contains three disjoint pchows. Suppose $V=(abcdef)$ has no kong. Here we don’t assume that $V$ is sorted. First, we consider the case when $V$ contains at least one pair, say $a=b$. By Lemma \[fact2\], there is a pmeld in $(cdef)$, which cannot be $(aa)$ as $V$ has no kong. Thus $V$ contains two disjoint pmelds if it has no kong. Second, suppose $V$ contains six different tiles and $a<b<c<d<e<f$. It is easy to see that, among $b-a$, $c-b$, $d-c$, $e-d$, and $f-e$, there exist at most one that is greater than 2. In fact, this is because $8= 9-1 \geq f-a = (f-e) + (e-d) + (d-c) + (c-b) + (b-a)\geq 1 + 1 + 1 + 1 + 1 = 5$ and $8 < 3+3+1+1+1 = 9$. This shows that $V$ contains one chow and a pchow or contains three disjoint pchows in this case. We next show that if $V$ contains two disjoint melds, then it can be completed within two tile changes. \[prop0\] If $V$ contains two disjoint melds, then ${{\sf{dfncy}}}(V)\leq 2$. The proof is not difficult but lengthy, see \[sec:appendix1\]. For any pure 14-tile $V$, recall that ${{\sf{Set}}}(V)$ is the set sequence of $V$, containing all different tiles in $V$, and ${{\sf{Res}}}(V)\equiv V\setminus{{\sf{Set}}}(V)$, containing all the rest tiles. We prove Theorem \[thm:pure\] case by case. Suppose $|{{\sf{Set}}}(V)|=8$ or $|{{\sf{Set}}}(V)|=9$. We have at least two chows in $\{(123),(456),(789)\}$. By Proposition \[prop0\], we know ${{\sf{dfncy}}}(V)\leq 2$. Suppose $|{{\sf{Set}}}(V)| = 4$ or $|{{\sf{Set}}}(V)| = 5$. Then $V$ contains at least two different pongs in both cases. By Proposition \[prop0\], we have ${{\sf{dfncy}}}(V) \leq 2$. Suppose $|{{\sf{Set}}}(V)|=6$ or $|{{\sf{Set}}}(V)|=7$. If $V$ contains two or more disjoint melds, by Proposition \[prop0\], we know ${{\sf{dfncy}}}(V)\leq 2$. Suppose $|{{\sf{Set}}}(V)|=7$ and $V$ does not contain two disjoint melds. Let ${{\sf{Set}}}(V)$ $=(abcdefg)$ with $a<b<c<d<e<f<g$. It is easy to see that ${{\sf{Set}}}(V)$ contains one chow. We note that ${{\sf{Res}}}(V)$ must contain a pair, this is because, otherwise, we have ${{\sf{Res}}}(V)={{\sf{Set}}}(V)$ and $V$ shall contain two chows. Thus $V$ consists of one kong or pong and at least four different pairs. Clearly, $V$ can be completed within three tile changes in this case. That is, ${{\sf{dfncy}}}(V)\leq 3$. Suppose $|{{\sf{Set}}}(V)|=6$ and $V$ does not contain two disjoint melds. In particular, it has no two differnt pongs. Let ${{\sf{Set}}}(V)=(abcdef)$. This implies that $V$ consists of one kong and five different pairs. Clearly, $V$ can be completed within three tile changes, by completing any three pairs into pongs. This shows ${{\sf{dfncy}}}(V)\leq 3$. It is not difficult to design brute force searching algorithms for counting how many pure 14-tiles that are complete and how many pure 14-tiles that have deficiency 1, 2, 3. Our implementation results show that there are - 118800 valid pure 14-tiles - 13259 complete pure 14-tiles - 91065 pure tiles with deficiency 1 - 14386 pure tiles with deficiency 2 - 90 pure tiles with deficiency 3 One natural question is, do we have more direct methods to determine the deficiency number of a 14-tile by, e.g., examining its structure? This is surely possibly. For example, we have the following characterisation for pure 14-tiles with deficiency 3. \[lem:7\] For any pure 14-tile $V$, $V$ has deficiency 3 if and only if either - $V$ consists of a kong and 5 different pairs; or - $V$ consists of five different pairs, a pong $(p,p,p)$ and a single tile $s$, and there exists $x$ in $V$ such that $(psx)$ (after ordering) is a chow in $V$ and $V\setminus(psx)$ contains no chows. Consider $V = (11222344558899)$ as an example. $V$ consists of a pong $(222)$, a single tile $(3)$ and five pairs $(11)$, $(44)$, $(55)$, $(88)$, $(99)$. Thus ${{\sf{dfncy}}}(V)=3$. Deficiency of hybrid 14-tiles ============================= When a hybrid 14-tile $H$ is complete? How many tile changes are necessary to complete it if it is not? These questions can be answered by computing the deficiency number of $H$. There is one problem: the deficiency number of $H$ is defined in a recursive way and, to tell if $H$ is $k+1$, one need to know in principle all 14-tiles with deficiency $k$. It will be more convenient if we can determine, just like Proposition \[lem:7\], the deficiency of $H$ by analysing its structure directly. In this section, we present a method for determining if a 14-tile is complete, and show that the worst 14-tiles have deficiency 6 and give a characterisation for these 14-tiles, and then give a more direct method for measuring the ‘worseness’ of a hybrid 14-tile $H$, and prove that it is exactly the deficiency of $H$. The best and the worst 14-tiles ------------------------------- We first show how to decide if a hybrid 14-tile $H$ is complete. For convenience, we introduce the following notations. \[dfn:H\_b\] Given a hybrid 14-tile $H$, we write $n_b(H)$, $n_c(H)$, and $n_d(H)$ for the numbers of Bamboos, Characters, and Dots in $H$. We denote by $H_b$, $H_c$, and $H_d$ for, respectively, the subsequences of Bamboos, Characters, and Dots in $H$. A hybrid 14-tile $H$ is called a bcd-type 14-tile if $n_b\geq n_c\geq n_d$. For the hybrid 14-tile $H$ in Eq. , we have $H_b=(122334777)$, $H_c=(11)$, and $H_d=(456)$, where we omit the colour symbol $B,C,D$ in $H_b,H_c$ and $H_d$. Since $n_b(H)=9$, $n_c(H)=2$, $n_d(H)=3$, $H$ is not a bcd-type 14-tile. But it is easy to see that every hybrid 14-tile $T$ has a corresponding bcd-type 14-tile $S$ such that $S$ and $T$ are identical in essence. \[prop:complete\_14tile\] A bcd-type 14-tile $H$ is complete if and only if one of the following conditions is satisfied: - $n_b=14$, $n_c=n_d=0$ and $H$ is a complete pure 14-tile; - $n_b=12$, $n_c=2$, $n_d=0$, $H_b$ consists of four disjoint melds and $H_c$ consists of a pair; - $n_b=11$, $n_c=3$, $n_d=0$, $H_b$ consists of three disjoint melds and a pair and $H_c$ consists of a meld; - $n_b = 9$, $n_c = 5$, $n_d = 0$, $H_b$ consists of three disjoint melds and $H_c$ consists of a meld and a pair; - $n_b=9$, $n_c=3$, $n_d=2$, $H_b$ consists of three disjoint melds, $H_c$ consists of a meld, and $H_d$ consists of a pair; - $n_b=8$, $n_c=3$, $n_d=3$, $H_b$ consists of two disjoint melds and a pair and each of $H_c$ and $H_d$ consists of a meld; - $n_b=8$, $n_c=6$, $n_d=0$, $H_b$ consists of two disjoint melds and a pair and $H_c$ consists of two disjoint melds; - $n_b=6$, $n_c=6$, $n_d=2$, each of $H_b$ and $H_c$ consists of two disjoint melds and $H_d$ consists of a pair; - $n_b=6$, $n_c=5$, $n_d=3$, $H_b$ consists of two disjoint melds, $H_c$ consists of a meld and a pair, and $H_d$ consists of a meld. Knowing how to decide if a hybrid 14-tile is complete, we next present one method for deciding the deficiency number of a hybrid 14-tile. \[lem:distance\] A 14-tile $H$ has deficiency $\leq k$ if and only if there exists a sequence of $s\leq k$ 14-tiles $H_1$, $H_2$, ..., $H_s$, such that $H_s$ is complete, and $H_i$ is a neighbour of $H_{i-1}$ for every $1\leq i\leq s$, where $H_0 = H$. We prove this by using induction on $k$. When $k=0$, the result trivially holds, as a 14-tile $H$ has deficiency $\leq 0$ if and only if it is complete. In general, suppose the result holds for any $i\leq k$. We prove that it also holds for $i=k+1$. On one hand, suppose $H$ has deficiency $\leq k+1$. If ${{\sf{dfncy}}}(H) \leq k$, then by inductive hypothesis, there exists a sequence of length $\leq k$ that satisfies the desired condition. Now assume ${{\sf{dfncy}}}(H)=k+1$. By definition, there exists a neighbour $T$ of $H$ such that ${{\sf{dfncy}}}(T)=k$. By inductive hypothesis again, there exists a sequence $T_0=T$, $T_1$, ..., $T_s$ with $s\leq k$ such that $T_s$ is complete and $T_i$ is a neighbour of $T_{i-1}$ for $1\leq i\leq s$. Apparently, $H$, $T_0$, $T_1$, ..., $T_s$ is a desired sequence of 14-tiles with length $\leq k+1$ for $H$. On the other hand, suppose there exists a desired sequence of $s\leq k+1$ 14-tiles $H_1$, $H_2$, ..., $H_{s}$ for $H=H_0$. If $s\leq k$, then by inductive hypothesis, ${{\sf{dfncy}}}(H)\leq k < k+1$. If $s=k+1$, then, by inductive hypothesis and that $H_2$, $H_3$, ..., $H_s$ is a desired sequence of length $k$ for $H_1$, we know ${{\sf{dfncy}}}(H_1)\leq k$. Since $H_1$ is a neighbour of $H$, we know by definition ${{\sf{dfncy}}}(H)\leq k+1$. Since a tile change from a 14-tile $H$ leads to a neighbour of $H$, we have the following corollary. \[coro:1\] A 14-tile $H$ has deficiency $k$ if and only if it can be changed into a complete 14-tile by $k$ tile changes but cannot be changed into a complete 14-tile by $k-1$ or less tile changes. We next show that every 14-tile can be completed within six tile changes. \[prop:dfncy&lt;=6\] For any 14-tile $H$, the deficiency of $H$ is not greater than 6. Without loss of generality, we assume that $H$ is a bcd-type 14-tile, i.e., $n_b(H)\geq n_c(H) \geq n_d(H)$. Clearly, $n_b(H)\geq 5$. To show ${{\sf{dfncy}}}(H)\leq 6$, by Corollary \[coro:1\], we show that $H$ can be completed within six tile changes. To this end, we need only show that $H$ contains two disjoint melds, or a meld and a disjoint pmeld, or three disjoint pmelds such that all involved pmelds are completable. Suppose $H_b$ has at least eight tiles, i.e., $n_b(H)\geq 8$. If $H_b$ contains a pong $(aaa)$, then by Lemma \[fact2\] the remaining five or more Bamboo tiles contain at least one pmeld which does not involve $a$. In this case, we have a pong and a disjoint pmeld. If $H_b$ contains no pongs but has two different pairs, by Lemma \[fact2\] again, the remaining four or more Bamboo tiles contain a pmeld. If $H_b$ contains no pongs and has only one pair, then by Lemma \[fact3\] the remaining six or more tiles contain three disjoint pmelds. In case $H_b$ contains no pairs, it is easy to see that any set of eight different bamboo tiles contains two disjoint chows. If $6\leq n_b(H) \leq 7$, then by $n_b(H) \geq n_c(H) \geq n_d(H)$, we know $n_c(H) \geq 4$ and thus, by Lemma \[fact2\], $H_c$ contains a pmeld. By Lemma \[fact3\], $H_b$ contains either a kong (hence a pong) or two disjoint pmelds. So in this case, $H$ contains a pong and a disjoint pmeld or three disjoint melds. If $n_b(H)=5$, then we have $n_c(H)=5$ and $n_d(H)=4$. By Lemma \[fact2\], $H$ contains three disjoint pmelds. Note that a pair $(xx)$ cannot be completed into a pong if and only if two identical $x$ have been used in the other melds or pmelds; and a pchow $(xy)$ cannot be completed into a chow $(xyz)$ if and only if all four identical $z$ have been used in the other melds or pmelds. Apparently, this is not the case for any pmeld involved here. Therefore, $H$ can be completed within six tile changes. For example, the following 14-tile has deficiency 6 as it only has three disjoint pmelds, e.g., $(B1B2)$, $(C2C2)$, and $(D6D8)$. $$\begin{aligned} \label{eq:14-tile_dfncy6} H &= (B1B1B2B5B8)(C1C2C2C5C8)(D3D6D8D9).\end{aligned}$$ As proved in Theorem \[thm:pure\], if $H$ is pure, then ${{\sf{dfncy}}}(H) \leq 3$. Similarly, if at least nine tiles are of the same colour, then the deficiency of $H$ is smaller than 6. \[prop:dfncy&lt;=5\] Given a 14-tile $H$, if $H_b$ contains 9 tiles, then ${{\sf{dfncy}}}(H)\leq 5$. We show that $H_b$ contains two disjoint melds, or one meld and two disjoint pmelds, or four disjoint pmelds. We discuss this case by case. Note that if $H_b$ contains two pongs, or two disjoint melds, or four disjoint pairs, we are done. Suppose $H_b$ contains only one pong $(aaa)$. By Lemma \[fact3\] the remaining 6 tiles contains two pmelds $(bc)$ and $(de)$. Assume $b<c$ and $d<e$. If $a$ completes both $(bc)$ and $(de)$, then we have two chows. If $a$ completes only one of $(bc)$ and $(de)$, we have a pair $(aa)$, a chow and a pmeld. If $a$ completes neither $(bc)$ nor $(de)$, then we can complete $(bc)$ and $(de)$ in two tile changes without violating the constraint that there are no five identical tiles in ${\mathcal{M}}_0$. Suppose $H_b$ contains no pongs. If $H_b$ contains only three different pairs, then $H_b$ has the form $(aa)(bb)(cc)$ $(d)(e)(f)$ with $a<b<c$ and $d<e<f$. - Suppose $e-d \leq 2$ or $f-e \leq 2$. Then we have three pairs and one pmeld. - Suppose $e-d>2$ and $f-e>2$. If $(abc)$ is a meld, then we already have at least two melds. If $(abc)$ is not a meld and $(xyz)$ is a meld, where $x,y \in \{a,b,c\}$ and $z \in \{d,e,f\}$, then we have one meld $(xyz)$, one pmeld $(xy)$ and one pair. If there is no meld in $H_b$, since $H_b$ has six different tiles, we can show that there exist $x\in \{a,b,c\}$ and $y,z\in \{d,e,f\}$ such that $(yx)$ and $(xz)$ are two pmelds. Together with two other pairs, we also have four disjoint pmelds in $H_b$. Suppose $H_b$ contains no pongs and has only two different pairs. Then ${{\sf{Set}}}(H_b)$ has seven different tiles and thus at least one chow $(def)$ ($d<e<f$). Write $(aa)$ and $(bb)$ for the two pairs in $H_b$, where $a<b$. If $a,b$ are not in $\{d,e,f\}$, then we have one meld and two different pairs. If $a,b$ are both in $\{d,e,f\}$, then $(ab)$ is a pchow. Since there is a pmeld in the other four tiles of $H_b$, we also have one chow and two pmelds in this case. If, for example, $a\in \{d,e,f\}$ but $b\not\in\{d,e,f\}$, then we have one chow $(def)$, one pair $(bb)$, and another pmeld in the remaining four tiles of $H_b$. Suppose $H_b$ has only one pair. Then, since ${{\sf{Set}}}(H_b)$ has 8 different tiles, there are two disjoint chows in $H_b$. This shows that ${{\sf{dfncy}}}(H) \leq 5$ if $H_b$ contains 9 tiles. Next, we give a characterisation of the worst 14-tiles. To this end, we need the following notion of worst pure $k$-tiles for $1\leq k \leq 8$: Let $k$ be an integer between 1 and 8. A *pure $k$-tile* $V$ is a sequence of integers in $[0,9]$ such that no integer appears more than four times and $V[i] \leq V[i+1]$ for any $0\leq i<k-1$. We say a pure $k$-tile $V$ is a worst $k$-tile if - when $1\leq k\leq 3$, $V$ contains no pmelds; - when $k=4$ or $k=5$, $V$ contains no meld and contains only one pmeld; - when $k=6$, $V$ contains at most one meld or up to two disjoint pmelds; - when $k=7$, $V$ contains up to two disjoint pmelds; - when $k=8$, $V$ contains at most one meld and a disjoint pmeld, or up to three disjoint pmelds. \[prop:dfncy=6\] For any 14-tile $H$, ${{\sf{dfncy}}}(H)=6$ if and only if one of the following situation occurs: - $n_b(H) = 8$, $n_d(H) = n_c(H) = 3$, $H_c$ and $H_d$ contain no pmeld and $H_b$ is a worst pure 8-tile. - $n_b(H) = 7$, $n_c(H) =5$, $n_d(H)=2$, $H_b$ is a worst pure 7-tile and $H_c$ a worst pure 5-tile and $H_d$ has no pmeld. - $n_b(H) = 6$, $n_c(H) =5$, $n_d(H)=3$, $H_b$ is a worst pure 6-tile and $H_c$ a worst pure 5-tile, and $H_d$ has no pmeld. - $n_b(H) = n_c(H) =5$, $n_d(H)=4$, $H_b$ and $H_c$ are worst pure 5-tiles and $H_d$ a worst pure 4-tile. A direct method for measuring the worseness of a 14-tile -------------------------------------------------------- While in principle we can determine if the deficiency of a hybrid 14-tile is not large than $k$ by recursively checking if it has a neighbour with deficiency smaller than $k$, this is far from efficient. In this subsection, we provide a direct method for measuring the worseness of a 14-tile. The notion of decomposition may also be extended to incomplete 14-tiles. Given any 14-tile $T$, a *pseudo-decomposition* (*p-decomposition* for short) of $T$ is a sequence $\pi$ of five subsequences, $\pi(1),...,\pi(5)$, of $T$ such that - $\pi(5)$ is a pair, a single tile, or empty; - for $1\leq i\leq 4$, each $\pi(i)$ is a meld, a pmeld, a single tile, or empty. We write $\pi(0)\equiv T \setminus \bigcup_{i=1}^5 \pi(i)$ for the sequence of remaining tiles of $T$. \[dfn:normal\_dcmp\] A p-decomposition $\pi$ of a 14-tile $T$ is *completable* if there exists a decomposition $\pi^*$ of a complete 14-tile $S$ such that each $\pi(i)$ is a subsequence of $\pi^*(i)$. If this is the case, we call $S$ a *completion* of $T$ based upon $\pi$. The *cost* of a completable p-decomposition $\pi$, written ${{\sf{cost}}}_T(\pi)$, is the number of necessary tile changes to complete $\pi(5)$ into a pair and complete each $\pi(i)$ into a meld for $1\leq i\leq 4$. A completable p-decomposition $\pi$ is *saturated* if no tiles in $\pi(0)$ can be added into any $\pi(i)$ to form another completable p-decomposition. A p-decomposition $\pi$ of $T$ is called *normal* if $\pi$ is saturated and has a completion $S$ which is a closest complete 14-tile of $T$. There may exist many different p-decompositions of $T$. If a p-decomposition $\pi$ is incompletable, we write ${{\sf{cost}}}_T(\pi)=\infty$. In the following, we show that ${{\sf{dfncy}}}(T)$ is identical to the minimal cost among all p-decompositions of $T$. The following results will be useful, where we say a p-decomposition $\pi_1$ *refines* another p-decomposition $\pi_2$ if $\pi_1(i)\supseteq \pi_2(i)$ for any $1\leq i\leq 5$. \[lem:saturated\] Suppose $\pi$ is a p-decomposition of a 14-tile $T$. If $\pi$ is completable but not saturated, then there exists a saturated completable p-decomposition $\pi^*$ such that $\pi^*$ refines $\pi$ and ${{\sf{cost}}}_T(\pi^*) < {{\sf{cost}}}_T(\pi)$. Suppose $\pi$ itself is not saturated. By definition, there exist $t\in \pi(0)$ and $1\leq i\leq 5$ such that $\pi(i)$ is not a meld (a pair) when $i\leq 4$ ($i=5$) and adding $t$ to $\pi(i)$ results a new completable p-decomposition $\pi^*$. Clearly, ${{\sf{cost}}}_T(\pi^*)={{\sf{cost}}}_T(\pi)-1$. If $\pi^*$ is not saturated, then we can further refine it till it is saturated. The conclusion then follows immediately. The above lemma shows that we need only consider saturated completable p-decompositions. The following result states that the cost of a p-decomposition of $T$ is identical to the distance between $T$ and any completion of $T$ based upon $\pi$. \[lem:dcmp-dist\] Suppose $\pi$ is a saturated completable p-decomposition of a 14-tile $T$. Then, for any completion $S$ of $T$ based upon $\pi$, we have ${{\sf{dist}}}(S,T)={{\sf{cost}}}_T(\pi)$. Note that any completion $S$ is obtained by filling the ‘holes’ in $\pi$. Since $\pi$ is saturated, these holes cannot be filled by using tiles in $\pi(0)$. Thus, each hole must be filled by a tile change. Therefore, $S$ has distance ${{\sf{cost}}}_T(\pi)$ with $T$. The following result shows that, very often, a saturated p-decomposition has no empty subsequences. \[prop:normal\] Suppose $T$ is a 14-tile and $\pi$ is a completable p-decomposition which has an empty subsequence. If $\pi(0)$ contains 4 or more tiles or $\pi(5)\not=\varnothing$, then there exists another completable p-decomposition $\pi^*$ which has no empty subsequences and ${{\sf{cost}}}_T(\pi^*) < {{\sf{cost}}}_T(\pi)$. See \[sec:appendix2\]. As a corollary, we have \[coro:emptyss\] Suppose $T$ is a 14-tile and $\pi$ a completable p-decomposition with $n\geq 2$ empty subsequences. There exists a completable p-decomposition $\pi^*$ s.t. ${{\sf{cost}}}_T(\pi^*) \leq{{\sf{cost}}}_T(\pi)-n$. If $\pi(5)=\varnothing$, then $\pi(0)$ contains at least $3n-1\geq 5$ tiles. We take the case when $n=2$ as an example. The other cases are analogous. By the proof of Proposition \[prop:normal\], there exists a tile $t\in\pi(0)$ such that $T$ contains at most three identical tiles $t$ and thus we could refine $\pi$ by letting $\pi(5)=(tt)$ if $(tt)$ is contained in $T$ and letting $\pi(5)=(t)$ otherwise. If the first case holds, we already have the desired p-decomposition. In the second case, we have four tiles left in $\pi(0)$ and can apply Proposition \[prop:normal\] to obtain a new completable p-decomposition with a smaller cost. This also gives us the desired p-decomposition. \[ex:T9\] Consider the 14-tile $$\begin{aligned} \label{eq:T9} T &=(B1B1B2B2B2B2B3B3)(C1C2C8)(D2D2D8)\end{aligned}$$ and its p-decompositions $$\begin{aligned} \label{eq:D(T9)0} \pi_0 &= (B1B2B3)(B2B2B2)(B1B3)(D2D2)(D8) \\ \label{eq:D(T9)1} \pi_1 &= (B1B2B3)(B1B2B3)(B3)(D2D2)(D8)\\ \label{eq:D(T9)2} \pi_2 &= (B1B2B3)(B1B2B3)(C1C2)(D2D2)(B2B2).\end{aligned}$$ The pmeld $(B1B3)$ in $\pi_0$ cannot be completed, as there are already four $B2$ in $\pi_0$. Both $\pi_1$ and $\pi_2$ are saturated and completable. For example, $\pi_1$ can be completed into $$(B1B2B3)(B1B2B3)(B3\underline{B4}\underline{B5})(D2D2\underline{D2})(D8\underline{D8})$$ and $\pi_2$ can be completed into $$(B1B2B3)(B1B2B3)(C1C2\underline{C3})(D2D2\underline{D2})(B2B2).$$ We have ${{\sf{cost}}}_T(\pi_1)=4$ and ${{\sf{cost}}}_T(\pi_2)=2$. Since ${{\sf{dfncy}}}(T)=2$, $\pi_2$ is a p-decomposition with the minimal cost. \[thm:m-cost\] For any 14-tile $T$, the minimum cost among all p-decompositions of $T$ is the deficiency of $T$, i.e., $$\begin{aligned} {{\sf{dfncy}}}(T) = \min\{{{\sf{cost}}}_T(\pi) \mid \mbox{$\pi$ is a p-decomposition of $T$}\} .\end{aligned}$$ Let $k={{\sf{dfncy}}}(T)$. We show there exists a p-decomposition $\pi$ of $T$ with ${{\sf{cost}}}_T(\pi)=k$. By Corollary \[coro:1\], the distance of any complete 14-tile to $T$ is not less than $k$ and there exists a complete 14-tile $S$ that has distance $k$ with $T$. Let $\pi^*$ be a decomposition of $S$ and let $\pi$ be the restriction of $\pi^*$ to $T$, that is, all tiles not in $T$ are removed from $\pi^*$. There are exactly $k$ tiles removed from $\pi^*$ and thus, by definition, we have $k={{\sf{cost}}}_T(\pi)$. On the other hand, for any p-decomposition $\pi$, if it is incompletable, then ${{\sf{cost}}}_T(\pi)=\infty$; and if it is completable, then by Lemma \[lem:saturated\] there exists a saturated p-decomposition $\pi'$ of $T$ such that ${{\sf{cost}}}_T(\pi')\leq {{\sf{cost}}}_T(\pi)$. Assume $\pi$ is a saturated p-decomposition $\pi$ of $T$. By Lemma \[lem:dcmp-dist\], there exists a completion $S$ of $T$ based upon $\pi$ such that the distance between $T$ and $S$ is the same as ${{\sf{cost}}}_T(\pi)$. This shows that $k\leq {{\sf{cost}}}_T(\pi)$ for any saturated p-decomposition $\pi$ of $T$. Therefore, $k$ is the minimum cost among all p-decompositions of $T$. In what follows, we say a p-decomposition $\pi$ of a 14-tile $T$ is *minimal* if ${{\sf{cost}}}_T(\pi) = {{\sf{dfncy}}}(T)$. We next present a method for determining the deficiency of a 14-tile by trying to construct a minimal p-decomposition of $T$. #### Our procedure We construct a quadtree, where each node is denoted by a word $\alpha$ in the alphabet $\Sigma=\{1,2,3,4\}$ and we attach to each $\alpha$ a subsequence $S_\alpha$ of $T$, which denotes the set of tiles remaining to be processed, and a p-decomposition $\pi_\alpha$ of $T$, and maintain a queue $Q$ of nodes to be explored and a global current best value ${{\sf{val}}}$. Initially, the root node, denoted as the empty word $\varepsilon$, is put in $Q$. We define $S_\varepsilon=T$, the p-decomposition $\pi_\varepsilon$ with $\pi_\varepsilon(i)=\varnothing$ for $0\leq i\leq 5$, and let ${{\sf{val}}}=6$, the largest possible deficiency number. Suppose a node $\alpha$ is popped out from the queue $Q$ and suppose ${{\sf{val}}}$ is the current best value and $S_\alpha$ and $\pi_\alpha$ are, respectively, the subsequence and the p-decomposition associated with $\alpha$. We now *expand* $\alpha$ as follows. Let $a=S_\alpha[0]$ be the first tile in the subsequence $S_\alpha$. We add up to four child nodes $\alpha1$ to $\alpha4$ under the node $\alpha$. For each $\ell\in\{1,2,3,4\}$, the subsequence $S_{\alpha\ell}$ and the p-decomposition $\pi_{\alpha\ell}$ are obtained by *reducing* $S_\alpha$ and *refining* $\pi_\alpha$ respectively. #### Notation Given a tile $t=(c,n)$, we write $t^+$ ($t^{++}$, resp.) for $(c,n+1)$ ($(c,n+2)$, resp.) as long as $1\leq n\leq 8$ ($1\leq n\leq 7$, resp.); and write $ t^-$ ( $t^{--}$, resp.) for $(c,n-1)$ ($(c,n-2)$, resp.) as long as $2\leq n \leq 9$ ($3\leq n\leq 9$, resp.). - Define $S_{\alpha1}=S_\alpha\setminus (a)$ and $\pi_{\alpha1}=\pi_\alpha$. - If $a^+$ or $a^{++}$ is in $S_\alpha$,[^4] define $S_{\alpha2}=S_\alpha \setminus (aa^+a^{++})$.[^5] Suppose $i$ is the first index in $\{1,2,3,4\}$ such that $\pi_\alpha(i)=\varnothing$. Define $\pi_{\alpha2}(i)=(aa^+a^{++})\cap S_\alpha$ and $\pi_{\alpha2}(j)= \pi_\alpha(j)$ for $j\not=i$.[^6] - If $(aa)\subseteq S_\alpha$ and $\pi_\alpha(5)$ is empty, define $S_{\alpha3} = S_\alpha \setminus (aa)$ and $\pi_{\alpha3}(5)=(aa)$ and $\pi_{\alpha3}(j)=\pi_\alpha(j)$ for $1\leq j\leq 4$. - If $(aa)\subseteq S_\alpha$, define $S_{\alpha4} = S_\alpha \setminus (aaa)$. Suppose $i$ is the first index in $\{1,2,3,4\}$ such that $\pi_\alpha(i)=\varnothing$. Define $\pi_{\alpha4}(i)=(aaa)\cap S_\alpha$ and $\pi_{\alpha4}(j)= \pi_\alpha(j)$ for $j\not=i$. For each child node $\alpha\ell$, if $\pi_{\alpha\ell}(i)\not=\varnothing$ for all $1\leq i\leq 5$, then we compare the value ${{\sf{cost}}}_T(\pi_{\alpha\ell})$ with the current best value ${{\sf{val}}}$, update ${{\sf{val}}}$ as ${{\sf{cost}}}_T(\pi_{\alpha\ell})$ if the latter is smaller, and terminate this branch. Suppose $\pi_{\alpha\ell}(i)$ is empty for some $i$ but there are no more tiles left to process, i.e., $S_{\alpha\ell}=\varnothing$. We may recycle from $\pi_{\alpha\ell}(0)$ some discarded tiles and put it in some subsequence of $\pi_{\alpha\ell}$ to get a refined completable p-decomposition. Suppose $\pi_{\alpha\ell}(5)\not=\varnothing$ or $\pi_{\alpha\ell}(0)$ contains four or more tiles. Let $n$ be the number of empty $\pi_{\alpha\ell}(i)$ for $1\leq i\leq 5$. Then, by Corollary \[coro:emptyss\], there exists another completable p-decomposition $\pi'$ such that ${{\sf{cost}}}_{T}(\pi') \leq {{\sf{cost}}}_T(\pi_{\alpha\ell})-n$. We then directly compare ${{\sf{cost}}}_T(\pi_{\alpha\ell})-n$ with ${{\sf{val}}}$ and update ${{\sf{val}}}$ as ${{\sf{cost}}}_T(\pi_{\alpha\ell})-n$ if the latter is smaller, and terminate this branch. Suppose $S_{\alpha\ell}=\varnothing$, $\pi_{\alpha\ell}(5)=\varnothing$ and $\pi_{\alpha\ell}(0)$ contains two (three) different tiles $u,v$ ($u,v,w$). We set ${{\sf{cost}}}={{\sf{cost}}}_T(\pi_{\alpha\ell})$ if $T$ contains two kongs $(uuuu)$ and $(vvvv)$ (three kongs $(uuuu)$, $(vvvv)$, and $(wwww)$), and set ${{\sf{cost}}}={{\sf{cost}}}_T(\pi_{\alpha\ell})-1$ otherwise. Then we compare ${{\sf{cost}}}$ with ${{\sf{val}}}$ and update ${{\sf{val}}}$ as ${{\sf{cost}}}$ if the latter is smaller, and terminate this branch. If $S_{\alpha\ell}\not=\varnothing$ and $\pi_{\alpha\ell}$ has an empty subsequence, then we put $\alpha\ell$ in $Q$. After $\alpha$ is expanded, if $Q$ is nonempty and ${{\sf{val}}}>0$, then we pop out another node $\beta$ from $Q$ and expand $\beta$ as above. The whole procedure is stopped either when we have ${{\sf{val}}}=0$ or when $Q$ is empty. Decision making =============== Given a 14-tile $H$, we have developed methods for deciding if $H$ is complete, and in case it is not complete, measuring how good or bad it is. The next important task is to decide, in case $H$ is not complete, which tile should the agent discard? We suppose the agent maintains a *knowledge base* $\omega$, which contains all her information about the available tiles. For simplicity, we represent $\omega$ as a 27-tuple, where $\omega[9c+n]$ ($0\leq c\leq 2$ and $1\leq n \leq 9$) denotes the number of identical tiles $t=(c,n)$ the agent *believes* to be available. Initially, we have $\omega[9c+n] = 4$ for each tile $t=(c,n)$. When all players have their hands, the agent also has her hand $H$ and updates her $\omega$ accordingly as $$\begin{aligned} \omega[9c+n] = 4- \mbox{the number of $(c,n)$ in $H$}. $$ Then she continues to modify $\omega$ according to the process of the game. For example, if one player discards a tile $t=(c,n)$ and no pong is formed from the discard of $t$, then the agent updates its $\omega$ by decreasing by one its $\omega[9c+n]$ and leaves the other items unchanged. More advanced techniques will be investigated in future work, where we may record the history of every player, and *infer* for each tile $t=(c,n)$, what is the most likely value of $\omega[9c+n]$. Now suppose the agent has a 14-tile $T$ and her current knowledge base is $\omega$. For simplicity, we write $\omega[9c+n]$ as $\omega(c,n)$ for any tile $t=(c,n)$ and write $$\begin{aligned} {\lVert\omega\rVert} \equiv \sum\{\omega(c,n) \mid 0\leq c\leq 2, 1\leq n\leq 9\}\end{aligned}$$ for the number of available tiles. Our task is to decide which tile she should discard. To this end, for each $0\leq i \leq 13$, we associate $i$ with a value $\delta_{T,\omega}(i)$ which is defined as $$\begin{aligned} \label{eq:delta} \delta_{T,\omega}(i) &= \sum_{0\leq c\leq 2, 1\leq n\leq 9}\Big\{\omega(c,n) \mid {{\sf{dfncy}}}(T[i/(c,n)]) < {{\sf{dfncy}}}(T)\Big\}\end{aligned}$$ Apparently, we change $T[i]$ to an available tile $(c,n)$ only if it is profitable, i.e., if the 14-tile obtained by replacing $T[i]$ with $(c,n)$, written $T[i/(c,n)]$, has a smaller deficiency number. For each index $i$, $\delta_{T,\omega}(i)$ denotes the number of available tiles $(c,n)$ (identical tiles are counted differently) such that $T[i/(c,n)]$ has a smaller deficiency than $T$. After $\delta_{T,\omega}(i)$ is computed for each $0\leq i\leq 13$, we then could pick one index $i$ that has the maximum value and discard $T[i]$, i.e., $$\begin{aligned} \label{eq:discard} {{\sf{discard}}}(T,\omega) = \operatorname*{arg\:max}_{T[i]\;:\; 0\leq i\leq 13} \delta_{T,\omega}(i).\end{aligned}$$ \[ex:decision\] For example, consider $$\begin{aligned} H &= (B1B1B1B8B8B9)(C1C5C5C5)(D1D5D6D7)\\ \omega &= (111111111)(111111111)(111111111)\end{aligned}$$ Here, for convenience, we assume that for each tile $t=(c,n)$ there is only one identical tile available, i.e., $\omega[i] = 1$ for any $0\leq i \leq 26$. Then we have ${{\sf{dfncy}}}(H)=2$ and $\delta_{H,\omega}=[0,0,0,3,3,7,6,0,0,0,6,0,0,0]$. In particular, we have $\delta_H(5)=7$, $\delta_H(6)=6$, and $\delta_H(10)=6$. Thus, according to Eq. , we should discard $H[5]=B9$. In this case, changing $B9$ as any of $B8,C1,C2,C3$, $D1,D2,D3$ will decrease the deficiency number from $2$ to $1$. Eq.  provides a very good, and easy to compute, heuristics for deciding which tile the agent should discard when her goal is to win as early as possible. It is, however, not an optimal policy. \[ex:optimal\] Consider the following example. $$\begin{aligned} H &= (B1B2B3B7B8B9)(C2C2C2)(D1D2D3D5D9)\\ \omega &= (000000000)(000000000)(010110001) $$ Suppose our goal is to decide, given the knowledge base $\omega$, which tile in $H$ to discard if we want to increase our chance to *win within 2 tile changes*. If we adopt the heuristics given in Eq. \[eq:discard\], then by ${{\sf{dfncy}}}(H)=1$ and $\delta_{H,\omega}(12) = \delta_{H,\omega}(13) = 1$ and $\delta_{H,\omega}(i) = 0$ for $0\leq i < 12$, we should discard either $T[12]=D5$ or $T[13]=D9$. But we will see this is not exact, and discarding $D9$ is more profitable. To this end, we need to compute, for each $0\leq i\leq 13$, the chance of completing $H$ if we discard $H[i]$. Note that $D2,D4,D5$ and $D9$ are the only tiles available. In the first tile change, if we discard any Bamboo tile, then, as there are no Bamboo tiles available, the corresponding chow will never be completed again. To make the revised 14-tile complete, we need at least three more tile changes. That is, the success chance is 0. This is also true if we discard a $C2$, a $D2$ or a $D3$ first. But if we first discard any of $D1$, $D5$ and $D9$, then we still have a chance to complete $T$ within two tile changes. Suppose we discard $D5$ in the first tile change and replace it with an available tile $t=(c,n)$. If $t$ happens to be $D9$, then we have a complete 14-tile. The chance of replacing $D5$ with $D9$ in the first tile change is $1/4$ as there are, according to $\omega$, only 4 tiles are available. If $t$ is not $D9$ (say it is $D2$), we cannot make a pair using it, as there is no more $D2$ left. To complete $H[12/t]$ in one tile change, we have to discard the new obtained tile $t$ and change it to $D9$. The chance of this is equal to $$\begin{aligned} &\mbox{({\it the chance of changing $D5$ to $t$}) $\times$ ({\it the chance of changing $t$ to $D9$})}, \end{aligned}$$ which is $1/4 \times 1/3$ (note that, after changing $D5$ to $t$, we have only 3 tiles available). Therefore, the chance of completing $H$ within two tile changes if we first discard $D5$ is $1/4+3\times (1/4\times 1/3)=1/2$. Similarly, if we discard $D9$ first, then the chance of completing $H$ in one tile change is also $1/4$. Suppose $D9$ is replaced with a $D2$ or another $D9$ in the first tile change. Then, in order to complete the revised 14-tile, we have to discard the new given tile and replace it with $D5$. The success chance is also $1/4 \times 1/3$ for each of $D2$ and $D9$. But, if we replace $D9$ with $D4$, then, to complete $H[D9/D4]$, we may either discard $D4$ and replace it with $D5$ or discard $D1$ and replace it with either $D2$ or $D5$. Apparently, in the second tile change, it is more profitable to discard $D1$ instead of $D4$. The success chance is $1/4 \times \max(1/3,2/3)=1/4\times 2/3$. Thus the chance of completing $H$ within two tile changes if we first discard $D9$ is $1/4 + 2\times (1/4\times 1/3) + 1/4\times 2/3=7/12>1/2$. Lastly, suppose we discard $D1$ first. Then if we get a $D2$, then we cannot complete $H$ by one more tile change; if we get a $D4$, then need replace $D5$ with $D9$ or replace $D9$ with either $D2$ or $D5$; if we get a $D5$ ($D9$), then we need replace $D9$ ($D5$) with $D4$. The success chance is $0 + 1/4 \times \max(1/3,2/3) + 1/4\times 1/3 + 1/4\times 1/3 = 1/3$. In summary, discarding $D9$ is more profitable than discarding any other tile. From the above example, we can see that the value of a tile depends on how many tile changes we could have before the game is finished. For any 14-tile $T$, the *step 0 value* of $T$ is defined as $$\begin{aligned} {{\sf{val}}}_0(T) &= \begin{cases} 1, & \mbox{if $T$ is complete}\\ 0, & \text{otherwise} \end{cases}\end{aligned}$$ Suppose $T$ is an incomplete 14-tile and $\omega$ is the current knowledge base of the agent. For any $0\leq i\leq 13$ and any $k>0$, we define the *step $k$ value* of the $i$-th tile of $T$ w.r.t. $\omega$, written ${{\sf{val}}}_k(T,\omega,i)$, as the chance of completing $T$ within $k$ tile changes if $T[i]$ is discarded first. For the special case when $k=1$, ${{\sf{val}}}_1(T,\omega,i)$ is indeed the success chance of obtaining a complete 14-tile by replacing $T[i]$ with an available tile. Using the above notion, we further introduce a notion that measures the chance of completing an incomplete 14-tile $T$ within $k$ tile changes. Suppose $T$ is an incomplete 14-tile and $\omega$ is the current knowledge base of the agent. For any $k>0$, the *step $k$ value* of $T$ w.r.t. $\omega$ is defined as $$\begin{aligned} {{\sf{val}}}_k(T,\omega) \equiv \max_{0\leq i\leq 13} {{\sf{val}}}_k(T,\omega,i).\end{aligned}$$ In case $T$ is complete, we define ${{\sf{val}}}_k(T,\omega)=1$. Clearly, ${{\sf{val}}}_k(T,\omega)=0$ if ${{\sf{dfncy}}}(T)>k$ or $0<{{\sf{dfncy}}}(T)\leq k$ but ${\lVert\omega\rVert}=0$. We next give a recursive method for computing ${{\sf{val}}}_k(T,\omega,i)$. First of all, we note that ${{\sf{val}}}_1(T,\omega,i)$ is the weighted sum over all available tiles $(c,n)$ of the chance of obtaining a complete 14-tile by changing $T[i]$ as $(c,n)$, where the chance of selecting $(c,n)$ to replace $T[i]$ is $\omega(c,n)/{\lVert\omega\rVert}$. Thus we have For any incomplete 14-tile $T$, any knowledge base $\omega$ with ${\lVert\omega\rVert}>0$, and any $0\leq i\leq 13$, we have $$\begin{aligned} \label{eq:val1} {{\sf{val}}}_1(T,\omega,i) &= \sum_{(c,n)\;:\;\omega(c,n)>0} \frac{\omega(c,n)}{{\lVert\omega\rVert}} \times {{\sf{val}}}_0 \big(T[i/(c,n)] \big) .\end{aligned}$$ In case ${\lVert\omega\rVert}=0$, we have ${{\sf{val}}}_1(T,\omega)=0$ for any $i$. Comparing equations and , it is easy to see that $$\begin{aligned} \label{eq:delta=val1} {{\sf{val}}}_1(T,\omega,i) = {\delta_{T,\omega}(i)}/{{\lVert\omega\rVert}} \quad\quad (0\leq i < 14).\end{aligned}$$ If $T$ is not complete, the chance of completing $T$ within $k$ tile changes is the weighted sum of the chances of completing $T[i/(c,n)]$ within $k-1$ tile changes over $(c,n)$ with $\omega(c,n)>0$. Thus we have the following characterisation: For any incomplete 14-tile $T$, any knowledge base $\omega$ with ${\lVert\omega\rVert}>0$, and any $0\leq i\leq 13$, we have $$\begin{aligned} {{\sf{val}}}_k(T, \omega, i) &= \sum_{(c,n)\ :\ \omega(c,n)>0} \frac{\omega(c,n)}{{\lVert\omega\rVert}} \times {{\sf{val}}}_{k-1}\big(T[i/(c,n)],\omega\!-\!(c,n)\big)\end{aligned}$$ where $\omega\!-\!(c,n)$ denotes the knowledge base obtained by decreasing $\omega(c,n)$ by 1, as this tile $(c,n)$ becomes unavailable after replacing $T[i]$ with it. The above two lemmas give a recursive method for selecting the tile to discard: #### The best tile to discard Suppose $T$ is an incomplete 14-tile and $\omega$ is a knowledge base with ${\lVert\omega\rVert}>0$. Then $T[i] = {{\sf{discard}}}_k(T,\omega)$ is the tile that has the best chance for completing $T$ within $k$ tile changes if we discard $T[i]$ first, where $$\begin{aligned} \label{eq:discard-k} {{\sf{discard}}}_k(T,\omega) = \operatorname*{arg\:max}_{T[i]\ :\ 0\leq i\leq 13} {{\sf{val}}}_{k}(T,\omega,i)\end{aligned}$$ By Eq. , ${{\sf{discard}}}_1(T,\omega)$ is the same as the tile ${{\sf{discard}}}(T,\omega)$ defined in Eq. . Conclusions and Future Work =========================== In this paper, we have initiated a mathematical and AI study of the Mahjong game. The definition of the deficiency number as well as the notions of knowledge base and step $k$ value will play important role in devising efficient computer programs for playing Mahjong. This is the topic of an ongoing research. Despite of its extremely popularity, there are very few mathematical or AI research papers which are devoted to the study of the Mahjong game. To our best knowledge, [@Cheng18] is the first serious attempt to study the Mahjong game using mathematical (mainly elementary combinatorial theory) techniques. In their paper, Cheng, Li and Li studied a special combinatorial problem in Mahjong game, viz., the so-called *$k$-gate problem*. A pure 13-tile $T$ is called a *nine-gate* if $T$ becomes complete if we add any tile in the same colour. In general, for $1\leq k\leq 9$, $T$ is called an $k$-gate if there are $k$ tiles with different values such that each of these $k$ tiles can complete $T$ but no any other tile can do the same job. It is easy to see that the $k$-gate problem can also be described and solved in the formalism developed in our work. To find all $k$-gates, one need only decide, for each pure 13-tile $T$, and any tile $t=i$ ($1\leq i\leq 9$) with the same colour, whether there are exactly $k$ tiles such that $T$ plus tile $i$ is complete. There are at least three directions to extend the above work. First, we may include more tiles in ${\mathcal{M}}_0$, e.g., the *winds*, the *dragons*, and the *bonus* (see Figure \[fig:mjtiles\]). Second, we may expand and/or restrict the set of legal 14-tiles. For examples, we may allow any seven pairs as a complete 14-tile, or we may require that any complete 14-tile has at most two colours. Third, different complete 14-tiles may have different scores. For example, a pure complete 14-tile may worth much more than a hybrid one. Future work will address these issues and adapt our methods according to different score systems. Proof of Proposition \[prop0\] {#sec:appendix1} ============================== If $V$ contains four disjoint melds, then $V$ can be completed within two tile changes by constructing a new pair. Suppose $V$ contains three, but no more, disjoint melds. Let $W=(uvwxy)$ be the (not sorted) subsequence containing the rest of $V$. Since $W$ does not contain a meld, it has at most two pairs. Suppose $p,q,r$ are any three pairwise different tiles in $W$. We assert that there is one of $p,q,r$ which can be completed into a meld within two tile changes. This is because, if, say, $p$ cannot be completed into a pong, then $V\setminus W$ must contain at least two identical tiles $p$, i.e., $(pp)\subseteq V\setminus W$. That is, if $(pp)\subseteq W$, then $V$ must contain the kong $(pppp)$; if $(pp)$ is not contained in $W$, then $V$ must contain the pong $(ppp)$. Similarly, suppose $(pab)$ is a chow that is not completable. Then $(ab)$ is not contained in $W$ and, hence, either $a$ or $b$ is not in $W$. If $a\in W$, then $V\setminus W$ must contain $(bbbb)$; if $b\in W$, then $V\setminus W$ must contain $(aaaa)$; if neither $a$ nor $b$ is in $W$, then $V$ must contain either $(aaaa)$ or $(bbbb)$. Thus, if none of $p,q,r$ is completable, then $T$ must contain three pongs and at least one kong $(zzzz)$ other than $(pppp),(qqqq),(rrrr)$. The existence of $(zzzz)$ is due to that $(pqr)$ is not a meld. This, however, contradicts the assumption that $V$ contains no more than three disjoint melds. Suppose $W$ contains two pairs say $u=v$ and $x=y$. If $w$ can be completed into a meld, then either $x$ or $u$ is not involved in the meld. Thus completing $w$ into a meld and using either $(xy)$ or $(uv)$ as the eye will complete $V$ within two tile changes. If $w$ cannot be completed into a meld, then $V$ contains $(www)$ and, because no chow $(wab)$ is completable, there exists a tile $w'\in \{a,b\}$ and $w'\not=u,x$ such that $(w'w'w'w')\subset V$. If $V$ contains both $(xxx)$ and $(uuu)$, then, together with $(www)$ and $(w'w'w')$, we shall have four pongs in $V$, a contradiction. Thus either $(xxx)$ or $(uuu)$ is not contained in $V$. This implies that in this subcase $V$ can be completed by completing $(xy)$ into $(xxx)$ or completing $(uv)$ into $(uuu)$ using one tile change. Suppose $W$ contains exactly one pair, say, $u=v$ and $u,x,y,w$ are pairwise different. If none of $u,x,y,w$ can be completed into a pong, then by previous analysis $V$ should contain $(uuuu)$, $(xxx), (yyy)$ and $(www)$, which contradicts the assumption that $W$ does not contain four disjoint melds. If there is one of $x,y,w$ that is completable into a pong or a chow does not involve $u=v$, then $V$ can be completed within two tile changes. Suppose this is not the case. Then $V$ should contain $(xxx),(yyy),(www)$ but not $(uuu)$, and one of $x,y,w$, say $x$, can be completed into a chow $(xuz)$. Then $V$ can be completed within two tile changes by constructing the chow $(xuz)$ and use the pair $(uv)$ as eye. If $W$ contains no pair, then $u,v,w,x,y$ are pairwise different. Suppose $u<v<w<x<y$. After routine check, we can see that there exist two tiles $z_1,z_2$ such that $z_1<v<w<x<z_2$ and $(uvwz_1)$ and $(wxyz_2)$ contain two disjoint chows.[^7] If neither $z_1$ nor $z_2$ is available, then $V\setminus W$ contains two kongs $(z_1z_1z_1z_1)$ and $(z_2z_2z_2z_2)$. This is impossible as $V\setminus W$ consists of three melds and $z_2>z_1+2$. Suppose without loss of generality $z_1$ is available and $(vwz_1)$ is a chow. If $V$ does not contain all of $(uuuu),(xxxx)$ and $(yyyy)$, then we may construct a pair, thus complete $V$, by one more tile change. In case $V$ does contain all of $(uuuu),(xxxx)$ and $(yyyy)$, then the other two tiles of $V$ are $v$ and $w$. Together with three pongs $(uuu),(xxx),(yyy)$, we may complete $V$ by forming the chow in $(wxyz_2)$ (by replacing $u$ with $z_2$) and form a pair $(vv)$ by replacing with $v$ the tile in $(wxy)$ that is not in the chow involving $z_2$. In the last, we consider the case when $V$ contains two, and no more, disjoint melds. Let $W$ be the subsequence containing the rest eight tiles of $V$. Since it has no meld, $W$ contains two to four pairs. For any two pairs $(xx)$ and $(yy)$ contained in $W$, if neither can be completed into a pong, then $V\setminus W$ contains two identical $x$ and two identical $y$. Since $V\setminus W$ consists of two melds, the two melds must be chows with form $(xya)$ and $(xyb)$, where $a=b$ is possible. This implies, however, that $V$ contains three disjoint melds $(xxx)$, $(yyy)$ and $(xya)$, a contradiction. Therefore, if $W$ contains $2\leq k\leq 4$ different pairs, then $k-1$ of these pairs can be completed into pongs. If $k\geq 3$, then $V$ can be completed within two tile changes. If $k=2$, then let $(xx)$ and $(yy)$ be the two pairs and $u,v,w,z$ be the rest four tiles (not sorted) in $W$. By Lemma \[fact2\], there is a pchow contained in $(uvwz)$. Let $(uv)$ be this pchow. If $(uv)$ is completable, then together with a completable pair $(xx)$ or $(yy)$, we can complete $V$ within two tile changes. Suppose this is not the case and $(uv)$ is not completable. Then, for any $a$ such that $(uva)$ (after sorting) is a chow, $V\setminus W$ contains the kong $(aaaa)$. Because $V\setminus W$ consists of two melds, this is possible only if $a$ is between $u,v$ and $V\setminus W = (aaa)(uva)$. Note that $a$ cannot be either $x$ or $y$. Both pairs are completable as $V\setminus W = (aaa)(uva)$ does not contain $x$ or $y$. Therefore, we can complete $V$ by complete both $(xx)$ and $(yy)$, obtaining four melds $(xxx),(yyy),(uva),(uva)$ and a pair $(aa)$. Proof of Proposition \[prop:normal\] {#sec:appendix2} ==================================== By Lemma \[lem:saturated\], if $\pi$ is not saturated, there exists a saturated p-decomposition with a smaller cost. In the following, we assume $\pi$ is saturated. Suppose $\pi(5)\not=\varnothing$. Without loss of generality, let $\pi(4)=\varnothing$. Clearly, $\pi(0)$ contains at least 3 tiles. Since $\pi$ is saturated, $\pi(0)$ does not contain any meld, nor any pmeld or single tile that is completable. This implies that, for any $x$ in $\pi(0)$, $T$ must contain the pong $(xxx)$. Moreover, for any chow $(xyz)$ ($x,y,z$ may be not ordered) containing $x$, $T$ must contain either $(yyyy)$ or $(zzzz)$. In particular, for any $x\in \pi(0)$, $T$ must - contain $(xxx)$, and - contain either $(x^{--}x^{--}x^{--}x^{--})$ or $(x^{-}x^{-}x^{-}x^{-})$ if $x\geq 3$, and - contain either $(x^{-}x^{-}x^{-}x^{-})$ or $(x^{+}x^{+}x^{+}x^{+})$ if $2\leq x\leq 8$, and - contain either $(x^{++}x^{++}x^{++}x^{++})$ or $(x^{+}x^{+}x^{+}x^{+})$ if $x\leq 7$. Let $u,v,w$ be three tiles contained in $\pi(0)$. By our assumption, $(u,v,w)$ is neither a chow nor a pong. #### Case 1 Suppose $u,v,w$ are in different colours. By the previous analysis, $T$ should contain $(uuu),(vvv),(www)$ and at least three kongs in different colours. This is impossible as $T$ is a 14-tile. #### Case 2 Suppose $u<v$ are in the same colour but $w$ is in a different colour. As above, $T$ should contain $(uuu),(vvv),(www)$ and at least two kongs in different colours. This is also impossible as $T$ is a 14-tile. #### Case 3 Suppose $u<v<w$ are in the same colour. As above, $T$ should contain $(uuu),(vvv)$, $(www)$. Moreover, if $w<8$ ($u>2$), then $T$ should contain $(xxxx)$ for some $x>w$ ($x<u$); if $w>v+1$ ($v>u+1$), then $T$ should contain $(yyyy)$ for some $v<y<w$ ($u<y<v$). Because $(uvw)$ is not a chow, we have $w>u+2$, and thus have either $w>v+1$ or $u>u+1$. To not violate the constraint that $T$ contains 14 tiles, we can have at most one kong $(xxxx)$ with $x\not\in \{u,v,w\}$. This is only possible when $u\leq 2$, $w\geq 8$ and either $v=u+1$ or $v=w-1$, which implies $w-v\geq 5$. But in this case, $T$ should also contain either $(w^-w^-w^-w^-)$ or $(w^{--}w^{--}w^{--}w^{--})$ and either $(v^+v^+v^+v^+)$ or $(v^{++}v^{++}v^{++}v^{++})$. A contradiction. #### Case 4 Suppose $u=v<w$ are in the same colour. In this case, $T$ should contain $(uuuu)$ and $(www)$. When $u=v=1$ and $w=2$, if $T$ does not contain $(2222)$, then it should contain $(3333)$; if $T$ contains $(2222)$, then it should also contain either $(3333)$ or $(4444)$. In the first subcase, we should have that $(11223333)$ is contained in $\bigcup_{i=1}^5 \pi(i)$ and $(11112223333)$ is contained in $T$. Assume $x,y,z$ are the rest tiles contained in $T$. Since $x,y,z$ cannot be $1,2,3$, we have a new completable decomposition $\hat{\pi}=(111)(222)(333)(13)(x)$, which has cost smaller than $\pi$. In the second subcase, without loss of generality, we assume that $T$ contains $(1111),(2222)$ and $(4444)$ and two additional tiles $x,y$. Let $\hat{\pi}=(111)(222)(444)(12)(x)$. Then $\hat{\pi}$ is a completable p-decomposition which has a smaller cost than $\pi$. The case when $u=v=8$ and $w=9$ is dually analogous. Now suppose $1<u=v<8$ and $u<w$. If $w=u+1$, then, as $(uw)$ is not completable, $\bigcup_{i=1}^5 \pi(i)$ must contain $(u^-u^-u^-u^-)$ and $(w^+w^+w^+w^+)$. Recall that $T$ already contains $(uuuu)$ and $(www)$. This contradicts the fact that $T$ is a 14-tile. Similarly, if $w=u+2$, then $\bigcup_{i=1}^5 \pi(i)$ contains $(u^+u^+u^+u^+)$. Let $x,y,z$ be the rest three tiles in $T$ and assume without loss of generality $x,y,z$ are not $w$. This shows that $T$ has a better p-decomposition $$\hat{\pi}=(uu^+w)(uu^+w)(uu^+w)(uu^+)(x).$$ If $w>u+2$, then we shall have at least two tiles $x,y$ not in $\{u,w\}$ s.t. $T$ contains $(xxxx)$ and $(yyyy)$. This, again, violates the assumption that $T$ is a 14-tile. #### Case 5 The case when $u<v=w$ are in the same colour is analogous to the previous one. Suppose $\pi(5)=\varnothing$ and $\pi(0)$ contains 4 or more tiles. Since $\pi$ is saturated, there are no pairs in $\pi(0)$. If there is a tile $t$ in $\pi(0)$ such that $T$ contains 3 or less identical tiles $t$, then we may replace $\pi(5)$ with the single tile $(t)$ and obtain a new completable p-decomposition with a smaller cost. If this is not possible for any $x\in \pi(0)$, then, for each $x\in \pi(0)$, $T$ should contain the pong $(xxx)$. As there are four or more tiles in $\pi(0)$, we have a new completable p-decomposition $\pi^*$ consists of four pongs and, possibly, an empty $\pi^*(5)$, which has a smaller cost. As a direct corollary, we know \[coro1\] Let $T$ be a 14-tile and $\pi$ a completable p-decomposition of $T$. If $\pi(i)=\varnothing$ for some $1\leq i\leq 4$, then there exists another completable p-decomposition $\hat{\pi}$ of $T$ such that $\hat{\pi}(i)\not=\varnothing$ for any $1\leq i\leq 4$ and ${{\sf{cost}}}_T(\hat{\pi}) < {{\sf{cost}}}_T(\pi)$. [^1]: https://deepmind.com/blog/alphastar-mastering-real-time-strategy-game-starcraft-ii/ [^2]: In ${\mathcal{M}}_0$ we assume that kongs play no role in forming legal hands. [^3]: In this paper, we regard a sequence as a set and identical items in the same sequence are considered as different elements in the corresponding set. Thus, suppose $W$ is a subsequence of $V$, $V\setminus W$ denotes the subsequence with items in $W$ removed from $V$. [^4]: If this is not the case, then this node is not introduced. Similar assumption applies to the other two child nodes. [^5]: Note that if, for example, $(aa^+a^{++}) \cap S_\alpha=(aa^{++})$, then $S_\alpha \setminus (aa^+a^{++})=S_\alpha \setminus (aa^{++})$. [^6]: Note that if $(a^-aa^+)$ is contained in $S$, then this chow shall have been put in some $\pi$ in a previous step when $a^-$ is examined. [^7]: This is because $v-u>2$ and $y-x>2$ cannot happen simultaneously. For example, if $W=(12459)$, then we may select $z_1=3$ and $z_2=6$, and we have chows $(123)$ and $(456)$.
--- abstract: 'We analyze the asymptotic behavior of a $2$-dimensional integral current which is almost minimizing in a suitable sense at a singular point. Our analysis is the second half of an argument which shows the discreteness of the singular set for the following three classes of $2$-dimensional currents: area minimizing in Riemannian manifolds, semicalibrated and spherical cross sections of $3$-dimensional area minimizing cones.' author: - 'Camillo De Lellis, Emanuele Spadaro and Luca Spolaor' bibliography: - 'references-Cal.bib' title: 'Regularity theory for $2$-dimensional almost minimal currents III: blowup' --- This paper is the fourth and last in a series of works aimed at establishing an optimal regularity theory for $2$-dimensional integral currents which are almost minimizing in a suitable sense. Building upon the monumental work of Almgren [@Alm], Chang in [@Chang] established that $2$-dimensional area-minimizing currents in Riemannian manifolds are classical minimal surfaces, namely they are regular (in the interior) except for a discrete set of branching singularities. The argument of Chang is however not entirely complete since a key starting point of his analysis, the existence of the so-called “branched center manifold”, is only sketched in the appendix of [@Chang] and requires the understanding (and a suitable modification) of the most involved portion of the monograph [@Alm]. An alternative proof of Chang’s theorem has been found by Rivière and Tian in [@RT1] for the special case of $J$-holomorphic curves. Later on the approach of Rivière and Tian has been generalized by Bellettini and Rivière in [@BeRi] to handle a case which is not covered by [@Chang], namely that of special Legendrian cycles in $\mathbb S^5$. Meanwhile the first and second author revisited Almgren’s theory giving a much shorter version of his program for proving that area-minimizing currents are regular up to a set of Hausdorff codimension $2$, cf. [@DS1; @DS2; @DS3; @DS4; @DS5]. In this note and its companion papers [@DSS2; @DSS3] we build upon the latter works in order to give a complete regularity theory which includes both the theorems of Chang and Bellettini-Rivière as special cases. In order to be more precise, we introduce the following terminology (cf. [@DSS1 Definition 0.3]). \[d:semicalibrated\] Let $\Sigma \subset R^{m+n}$ be a $C^2$ submanifold and $U\subset {{\mathbb R}}^{m+n}$ an open set. - An $m$-dimensional integral current $T$ with finite mass and ${{\rm spt}}(T)\subset \Sigma\cap U$ is area-minimizing in $\Sigma\cap U$ if ${{\mathbf{M}}}(T + \partial S)\geq {{\mathbf{M}}}(T)$ for any $m+1$-dimensional integral current $S$ with ${{\rm spt}}(S) \subset \subset \Sigma\cap U$. - A semicalibration (in $\Sigma$) is a $C^1$ $m$-form $\omega$ on $\Sigma$ such that $\|\omega_x\|_c \leq 1$ at every $x\in \Sigma$, where $\|\cdot \|_c$ denotes the comass norm on $\Lambda^m T_x \Sigma$. An $m$-dimensional integral current $T$ with ${{\rm spt}}(T)\subset \Sigma$ is [*semicalibrated*]{} by $\omega$ if $\omega_x (\vec{T}) = 1$ for $\|T\|$-a.e. $x$. - An $m$-dimensional integral current $T$ supported in $\partial {{\mathbf{B}}}_{\bar R} (p) \subset {{\mathbb R}}^{m+n}$ is a [*spherical cross-section of an area-minimizing cone*]{} if ${p{{\times\hspace{-0.6em}\times\,}}T}$ is area-minimizing. In what follows, given an integer rectifiable current $T$, we denote by ${{\rm Reg}}(T)$ the subset of ${{\rm spt}}(T)\setminus {{\rm spt}}(\partial T)$ consisting of those points $x$ for which there is a neighborhood $U$ such that $T{\mathop{\hbox{\vrule height 7pt width .3pt depth 0pt \vrule height .3pt width 5pt depth 0pt}}\nolimits}U$ is a (costant multiple of) a regular submanifold. Correspondingly, ${{\rm Sing}}(T)$ is the set ${{\rm spt}}(T)\setminus ({{\rm spt}}(\partial T)\cup {{\rm Reg}}(T))$. Observe that ${{\rm Reg}}(T)$ is relatively open in ${{\rm spt}}(T) \setminus {{\rm spt}}(\partial T)$ and thus ${{\rm Sing}}(T)$ is relatively closed. The main result of this and the works [@DSS2; @DSS3] is then the following \[t:finale\] Let $m=2$ and $T$ be as in (a), (b) or (c) of Definition \[d:semicalibrated\]. Assume in addition that $\Sigma$ is of class $C^{3,{{\varepsilon}}_0}$ (in case (a) and (b)) and $\omega$ of class $C^{2,{{\varepsilon}}_0}$ (in case (b)) for some positive ${{\varepsilon}}_0$. Then ${{\rm Sing}}(T)$ is discrete. Clearly Chang’s result is covered by case (a). As for the case of special Lagrangian cycles considered by Bellettini and Rivière in [@BeRi] observe that they form a special subclass of both (b) and (c). Indeed these cycles arise as spherical cross-sections of $3$-dimensional special lagrangian cones: as such they are then spherical cross sections of area-minimizing cones but they are also semicalibrated by a specific smooth form on $\mathbb S^5$. Following the Almgren-Chang program, Theorem \[t:finale\] will be established through a suitable “blow-up argument”: this argument is presented here but requires several tools. The first important tool is the theory of multiple-valued functions, for which we will use the results and terminology of the papers [@DS1; @DS2]. The second tool is a suitable approximation result for area-minimizing currents with graphs of multiple valued functions, which for the case at hand has been established in the preceding note [@DSS2]. The last tool is the so-called “branched center manifold”: this has been constructed in the paper [@DSS3]. We note in passing that all our arguments use heavily the uniqueness of tangent cones for $T$. This result is a, by now classical, theorem of White for area-minimizing $2$-dimensional currents in the euclidean space, cf. [@Wh]. Chang extended it to case (a) in the appendix of [@Chang], whereas Pumberger and Rivière covered case (b) in [@PuRi]. A general derivation of these results for a wide class of almost minimizers has been given in [@DSS1]: the theorems in there cover, in particular, all the cases of Definition \[d:semicalibrated\]. The proof of Theorem \[t:finale\] is based, as in [@Chang], on an induction statement, cf. Theorem \[t:induttivo\] below. Although Theorem \[t:induttivo\] is already stated in [@DSS3], we will recall it in the next section, where we also show how it implies Theorem \[t:finale\]. This and the previous paper [@DSS3] can be hence thought as the two halves in the proof of Theorem \[t:induttivo\]. After introducing some terminology, in Section \[s:2\_pezzi\] we will state the first half in Theorem \[t:cm\] (which is proved in [@DSS3]) and the second half in Theorem \[t:bu\], and we will then show how they fit together to prove Theorem \[t:induttivo\]. The remaining sections (the biggest portion of this note) are then dedicated to prove Theorem \[t:bu\]. [**Acknowledgments**]{} The research of Camillo De Lellis and Luca Spolaor has been supported by the ERC grant RAM (Regularity for Area Minimizing currents), ERC 306247. Preliminaries and the main induction statement ============================================== Basic notation and first main assumptions ----------------------------------------- For the notation concerning submanifolds $\Sigma\subset {{\mathbb R}}^{2+n}$ we refer to [@DS3 Section 1]. With ${{\mathbf{B}}}_r (p)$ and $B_r (x)$ we denote, respectively, the open ball with radius $r$ and center $p$ in $\mathbb R^{2+n}$ and the open ball with radius $r$ and center $x$ in $\mathbb R^2$. ${{\mathbf{C}}}_r (p)$ and ${{\mathbf{C}}}_r (x)$ will always denote the cylinder $B_r (x)\times \mathbb R^n$, where $p=(x,y)\in {{\mathbb R}}^2\times {{\mathbb R}}^n$. We will often need to consider cylinders whose bases are parallel to other $2$-dimensional planes, as well as balls in $m$-dimensional affine planes. We then introduce the notation $B_r (p, \pi)$ for ${{\mathbf{B}}}_r (p) \cap (p+\pi)$ and ${{\mathbf{C}}}_r (p, \pi)$ for $B_r (p, \pi) + \pi^\perp$. $e_i$ will denote the unit vectors in the standard basis, $\pi_0$ the (oriented) plane ${{\mathbb R}}^2\times \{0\}$ and $\vec \pi_0$ the $2$-vector $e_1\wedge e_2$ orienting it. Given an $m$-dimensional plane $\pi$, we denote by ${{\mathbf{p}}}_\pi$ and ${{\mathbf{p}}}_\pi^\perp$ the orthogonal projections onto, respectively, $\pi$ and its orthogonal complement $\pi^\perp$. For what concerns integral currents we use the definitions and the notation of [@Sim]. Since $\pi$ is used recurrently for $2$-dimensional planes, the $2$-dimensional area of the unit circle in ${{\mathbb R}}^2$ will be denoted by $\omega_2$. By [@DSS2 Lemma 1.1] in case (b) we can assume, without loss of generality, that the ambient manifold $\Sigma$ coincides with the euclidean space ${{\mathbb R}}^{2+n}$. In the rest of the paper we will therefore always make the following \[ipotesi\_base\] $T$ is an integral current of dimension $2$ with bounded support and it satisfies one of the three conditions (a), (b) or (c) in Definition \[d:semicalibrated\]. Moreover - In case (a), $\Sigma\subset{{\mathbb R}}^{2+n}$ is a $C^{3, {{\varepsilon}}_0}$ submanifold of dimension $2 + \bar n = 2 + n - l$, which is the graph of an entire function $\Psi: {{\mathbb R}}^{2+\bar n}\to {{\mathbb R}}^l$ and satisfies the bounds $$\label{e:Sigma} \|D \Psi\|_0 \leq c_0 \quad \mbox{and}\quad {\mathbf{A}}:= \|A_\Sigma\|_0 \leq c_0,$$ where $c_0$ is a positive (small) dimensional constant and ${{\varepsilon}}_0\in ]0,1[$. - In case (b) we assume that $\Sigma = {{\mathbb R}}^{2+n}$ and that the semicalibrating form $\omega$ is $C^{2, {{\varepsilon}}_0}$. - In case (c) we assume that $T$ is supported in $\Sigma = \partial {{\mathbf{B}}}_R (p_0)$ for some $p_0$ with $|p_0|=R$, so that $0\in \partial {{\mathbf{B}}}_R (p_0)$. We assume also that $T_0 \partial {{\mathbf{B}}}_R (p_0)$ is ${{\mathbb R}}^{2+n-1}$ (namely $p_0= (0, \ldots, 0, \pm |p_0|)$ and we let $\Psi: {{\mathbb R}}^{2+n-1}\to {{\mathbb R}}$ be a smooth extension to the whole space of the function which describes $\Sigma$ in ${{\mathbf{B}}}_2 (0)$. We assume then that holds, which is equivalent to the requirement that $R^{-1}$ be sufficiently small. In addition, since the conclusion of Theorem \[t:finale\] is local, by [@DSS1 Proposition 0.4] we can also assume to be always in the following situation. \[ipotesi\_base\_2\] In addition to Assumption \[ipotesi\_base\] we assume the following: - $\partial T {\mathop{\hbox{\vrule height 7pt width .3pt depth 0pt \vrule height .3pt width 5pt depth 0pt}}\nolimits}{{\mathbf{C}}}_2 (0, \pi_0)=0$; - $0\in {{\rm spt}}(T)$ and the tangent cone at $0$ is given by $\Theta (T, 0) \a{\pi_0}$ where $\Theta (T, 0)\in {{\mathbb N}}\setminus \{0\}$; - $T$ is irreducible in any neighborhood $U$ of $0$ in the following sense: it is not possible to find $S$, $Z$ non-zero integer rectifiable currents in $U$ with $\partial S = \partial Z =0$ (in $U$), $T= S+Z$ and ${{\rm spt}}(S)\cap {{\rm spt}}(Z)=\{0\}$. In order to justify point (iii), observe that we can argue as in the proof of [@DSS1 Theorem 3.1]: assuming that in a certain neighborhood $U$ there is a decomposition $T= S+Z$ as above, it follows from [@DSS1 Proposition 2.2] that both $S$ and $Z$ fall in one of the classes of Definition \[d:semicalibrated\]. In turn this implies that $\Theta (S, 0), \Theta (Z, 0)\in {{\mathbb N}}\setminus \{0\}$ and thus $\Theta (S, 0) < \Theta (T, 0)$. We can then replace $T$ with either $S$ or $Z$. Let $T_1=S$ and argue similarly if it is not irreducibile: obviously we can apply one more time the argument above and find a $T_2$ which satisfies all the requirements and has $0<\Theta (T_2, 0) < \Theta (T_1, 0)$. This process must stop after at most $N = \Theta (T, 0)$ steps: the final current is then necessarily irreducible. Branching model --------------- We next introduce an object which will play a key role in the rest of our work, because it is the basic local model of the singular behavior of a $2$-dimensional area-minimizing current: for each positive natural number $Q$ we will denote by ${{\mathfrak{B}}}_{Q,\rho}$ the flat Riemann surface which is a disk with a conical singularity, in the origin, of angle $2\pi Q$ and radius $\rho>0$. More precisely we have \[d:Riemann\_surface\] ${{\mathfrak{B}}}_{Q,\rho}$ is topologically an open $2$-dimensional disk, which we identify with the topological space $\{(z,w)\in \mathbb C^2 : w^Q=z, |z|<\rho\}$. For each $(z_0, w_0)\neq 0$ in ${{\mathfrak{B}}}_{Q,\rho}$ we consider the connected component ${{\mathfrak{D}}}(z_0, w_0)$ of ${{\mathfrak{B}}}_{Q,\rho}\cap \{(z,w):|z-z_0| < |z_0|/2\}$ which contains $(z_0, w_0)$. We then consider the smooth manifold given by the atlas $$\{({{\mathfrak{D}}}(z,w)), (x_1, x_2)): (z,w)\in {{\mathfrak{B}}}_{Q,\rho} \setminus \{0\}\}\, ,$$ where $(x_1, x_2)$ is the function which gives the real and imaginary part of the first complex coordinate of a generic point of ${{\mathfrak{B}}}_{Q,\rho}$. On such smooth manifold we consider the following flat Riemannian metric: on each ${{\mathfrak{D}}}(z,w)$ with the chart $(x_1, x_2)$ the metric tensor is the usual euclidean one $dx_1^2+dx_2^2$. Such metric will be called the [*canonical flat metric*]{} and denoted by $e_Q$. The coordinates $(x_1, x_2) = z$ will be called [*standard flat coordinates*]{}. When $Q=1$ we can extend smoothly the metric tensor to the origin and we obtain the usual euclidean $2$-dimensional disk. For $Q>1$ the metric tensor does not extend smoothly to $0$, but we can nonetheless complete the induced geodesic distance on ${{\mathfrak{B}}}_{Q,\rho}$ in a neighborhood of $0$: for $(z,w)\neq 0$ the distance to the origin will then correspond to $|z|$. The resulting metric space is a well-known object in the literature, namely a flat Riemann surface with an isolated conical singularity at the origin (see for instance [@Zorich]). Note that for each $z_0$ and $0 <r\leq \min \{|z_0|, \rho -|z_0|\} $ the set ${{\mathfrak{B}}}_{Q, \rho}\cap \{|z-z_0|<r\}$ consists then of $Q$ nonintersecting $2$-dimensional disks, each of which is a geodesic ball of ${{\mathfrak{B}}}_{Q, \rho}$ with radius $r$ and center $(z_0, w_i)$ for some $w_i\in {{\mathbb C}}$ with $w_i^Q = z_0$. We then denote each of them by $B_r (z_0,w_i)$ and treat it as a standard disk in the euclidean $2$-dimensional plane (which is correct from the metric point of view). We use however the same notation for the distance disk $B_r (0)$, namely for the set $\{(z,w): |z|<r\}$, although the latter is [*not isometric*]{} to the standard euclidean disk. Since this might be create some ambiguity, we will use the specification ${{\mathbb R}}^2 \supset B_r (0)$ when referring to the standard disk in ${{\mathbb R}}^2$. Admissible $Q$-branchings ------------------------- When one of (or both) the parameters $Q$ and $\rho$ are clear from the context, the corresponding subscript (or both) will be omitted. We will consider repeatedly functions $u$ defined on ${{\mathfrak{B}}}$. We will always treat each point of ${{\mathfrak{B}}}$ as an element of ${{\mathbb C}}^2$, mostly using $z$ and $w$ for the horizontal and vertical complex coordinates. Often ${{\mathbb C}}$ will be identified with ${{\mathbb R}}^2$ and thus the coordinate $z$ will be treated as a two-dimensional real vector, avoiding the more cumbersome notation $(x_1, x_2)$. \[d:admissible\] Let $\alpha\in ]0,1[$, $b>1$, $Q\in {{\mathbb N}}\setminus \{0\}$ and $n\in {{\mathbb N}}\setminus \{0\}$. An admissible $\alpha$-smooth and $b$-separated $Q$-branching in ${{\mathbb R}}^{2+n}$ (shortly a $Q$-branching) is the graph $${{\rm Gr}}(u) := \{(z, u(z,w)): (z,w) \in {{\mathfrak{B}}}_{Q, 2\rho}\} \subset {{\mathbb R}}^{2+n}\,$$ of a map $u: {{\mathfrak{B}}}_{Q,2\rho} \to \mathbb R^n$ satisfying the following assumptions. For some constants $C_i>0$ we have - $u$ is continuous, $u\in C^{3,\alpha}$ on ${{\mathfrak{B}}}_{Q, \rho} \setminus \{0\}$ and $u(0)=0$; - $|D^j u (z,w)|\leq C_i |z|^{1-j+\alpha}$ $\forall (z,w)\neq 0$ and $j\in \{0,1,2,3\}$; - $[D^3 u]_{\alpha, B_r (z,w)} \leq C_i |z|^{-2}$ for every $(z,w)\neq 0$ with $|z|=2r$; - If $Q>1$, then there is a positive constant $c_s\in ]0,1[$ such that $$\label{e:separation} \min \{|u (z,w)-u(z,w')|: w\neq w'\} \geq 4 c_s |z|^b \qquad \mbox{for all $(z,w)\neq 0$.}$$ The map ${{\bm{\Phi}}}(z,w):= (z, u (z,w))$ will be called the [*graphical parametrization*]{} of the $Q$-branching. Any $Q$-branching as in the Definition above is an immersed disk in ${{\mathbb R}}^{2+n}$ and can be given a natural structure as integer rectifiable current, which will be denoted by ${{\mathbf{G}}}_u$. For $Q=1$ a map $u$ as in Definition \[d:admissible\] is a (single valued) $C^{1,\alpha}$ map $u: {{\mathbb R}}^2\supset B_{2\rho} (0)\to {{\mathbb R}}^n$. Although the term branching is not appropriate in this case, the advantage of our setup is that $Q=1$ will not be a special case in the induction statement of Theorem \[t:induttivo\] below. Observe that for $Q>1$ the map $u$ can be thought as a $Q$-valued map $u: {{\mathbb R}}^2\supset B_\rho (0) \to {{\mathcal{A}}_Q({{\mathbb R}}^{n})}$, setting $u(z)= \sum_{(z,w_i)\in {{\mathfrak{B}}}} \a{u(z,w_i)}$ for $z\neq 0$ and $u (0) = Q\a{0}$. The notation ${{\rm Gr}}(u)$ and ${{\mathbf{G}}}_u$ is then coherent with the corresponding objects defined in [@DS2] for general $Q$-valued maps. The inductive statement ----------------------- Before coming to the key inductive statement, we need to introduce some more terminology. \[d:horned\] Let ${{\rm Gr}}(u)$ be a $b$-separated $Q$-branching. For every $a>b$ we define the [*horned neighborhood*]{} ${{\mathbf{V}}}_{u,a}$ of ${{\rm Gr}}(u)$ to be $${{\mathbf{V}}}_{u, a} := \{(x,y)\in {{\mathbb R}}^2\times {{\mathbb R}}^n: \exists (x,w) \in {{\mathfrak{B}}}_{Q, 2\rho} \mbox{ with } |y-u(x,w)|< c_s |x|^a\}\, ,$$ where $c_s$ is the constant in . \[d:excess\] Given an $m$-dimensional current $T$ in $\mathbb R^{m+n}$ with finite mass, its [*excess*]{} in the ball ${{\mathbf{B}}}_r (x)$ and in the cylinder ${{\mathbf{C}}}_r (p, \pi')$ with respect to the $m$-plane $\pi$ are $$\begin{aligned} {{\mathbf{E}}}(T,{{\mathbf{B}}}_r (p),\pi) &:= \left(2\omega_m\,r^m\right)^{-1}\int_{{{\mathbf{B}}}_r (p)} |\vec T - \vec \pi|^2 \, d\|T\|\\ {{\mathbf{E}}}(T, {{\mathbf{C}}}_r(p, \pi'), \pi) &:= \left(2\omega_m\,r^m\right)^{-1}\int_{{{\mathbf{C}}}_r (p, \pi')} |\vec T - \vec \pi|^2 \, d\|T\|\, .\end{aligned}$$ For cylinders we omit the third entry when $\pi=\pi'$, i.e. ${{\mathbf{E}}}(T, {{\mathbf{C}}}_r (p, \pi)) := {{\mathbf{E}}}(T, {{\mathbf{C}}}_r (p, \pi), \pi)$. In order to define the spherical excess we consider $T$ as in Assumption \[ipotesi\_base\] and we say that $\pi$ [*optimizes the excess*]{} of $T$ in a ball ${{\mathbf{B}}}_r (x)$ if - In case (b) $$\label{e:optimal_pi} {{\mathbf{E}}}(T,{{\mathbf{B}}}_r (x)):=\min_\tau {{\mathbf{E}}}(T, {{\mathbf{B}}}_r (x), \tau) = {{\mathbf{E}}}(T,{{\mathbf{B}}}_r (x),\pi);$$ - In case (a) and (c) $\pi\subset T_x \Sigma$ and $$\label{e:optimal_pi_2} {{\mathbf{E}}}(T,{{\mathbf{B}}}_r (x)):=\min_{\tau\subset T_x \Sigma} {{\mathbf{E}}}(T, {{\mathbf{B}}}_r (x), \tau) = {{\mathbf{E}}}(T,{{\mathbf{B}}}_r (x),\pi)\, .$$ Note in particular that, in case (a) and (c), ${{\mathbf{E}}}(T, {{\mathbf{B}}}_r (x))$ differs from the quantity defined in [@DS5 Definition 1.1], where, although $\Sigma$ does not coincide with the ambient euclidean space, $\tau$ is allowed to vary among [*all*]{} planes, as in case (b). Thus a notation more consistent with that of [@DS5] would be, in case (a) and (c), ${{\mathbf{E}}}^\Sigma (T, {{\mathbf{B}}}_r (x))$. However, the difference is a minor one and we prefer to keep our notation simpler. Our main induction assumption is then the following \[induttiva\] $T$ is as in Assumption \[ipotesi\_base\] and \[ipotesi\_base\_2\]. For some constants $\bar Q\in \mathbb N\setminus \{0\}$ and $0<\alpha< \frac{1}{2\bar Q}$ there is an $\alpha$-admissible $\bar Q$-branching ${{\rm Gr}}(u)$ with $u: {{\mathfrak{B}}}_{\bar{Q}, 2}\to {{\mathbb R}}^n$ such that - If $\bar Q>1$, $u$ is $b$-separated for some $b>1$; a choice of some $b>1$ is fixed also in the case $\bar Q =1$, although in this case the separation condition is empty. - ${{\rm spt}}(T) \subset {{\mathbf{V}}}_{u,a}\cup\{0\}$ for some $a>b$; - There exist $\gamma >0$ and a $C_i>0$ with the following property. Let $p = (x_0,y_0)\in {{\rm spt}}(T)\cap {{\mathbf{C}}}_{\sqrt{2}} (0)$ and $4 d := |x_0|>0$, let $V$ be the connected component of ${{\mathbf{V}}}_{u,a}\cap \{(x,y): |x-x_0| < d\}$ containing $p$ and let $\pi (p)$ be the plane tangent to ${{\rm Gr}}(u)$ at the only point of the form $(x_0, u(x_0, w_i))$ which is contained in $V$. Then $$\label{e:effetto_energia} {{\mathbf{E}}}(T{\mathop{\hbox{\vrule height 7pt width .3pt depth 0pt \vrule height .3pt width 5pt depth 0pt}}\nolimits}V,{{\mathbf{B}}}_\sigma (p), \pi (p)) \leq C_i^2 d^{2\gamma -2} \sigma^2 \qquad \forall \sigma\in \left[{\textstyle{\frac{1}{2}}}d^{(b+1)/2}, d\right]\, .$$ The main inductive step is then the following theorem, where we denote by $T_{p,r}$ the rescaled current $(\iota_{p,r})_\sharp T$, through the map $\iota_{p,r} (q) := (q-p)/r$. \[t:induttivo\] Let $T$ be as in Assumption \[induttiva\] for some $\bar Q = Q_0$. Then, - either $T$ is, in a neighborhood of $0$, a $Q$ multiple of a $\bar{Q}$-branching ${{\rm Gr}}(v)$; - or there are $r>0$ and $Q_1>Q$ such that $T_{0, r}$ satisfies Assumption \[induttiva\] with $\bar Q = Q_1$. Theorem \[t:finale\] follows then easily from Theorem \[t:induttivo\] and [@DSS1]. Proof of Theorem \[t:finale\] ----------------------------- As already mentioned, without loss of generality we can assume that Assumption \[ipotesi\_base\] holds, cf. [@DSS1 Lemma 1.1] (the bounds on ${\mathbf{A}}$ and $\Psi$ can be achieved by a simple scaling argument). Fix now a point $p$ in ${{\rm spt}}(T)\setminus {{\rm spt}}(\partial T)$. Our aim is to show that $T$ is regular in a punctured neighborhood of $p$. Without loss of generality we can assume that $p$ is the origin. Upon suitably decomposing $T$ in some neighborhood of $0$ we can easily assume that (I) in Assumption \[induttiva\] holds, cf. the argument of Step 4 in the proof of [@DSS1 Theorem 3.1]. Thus, upon suitably rescaling and rotating $T$ we can assume that $\pi_0$ is the unique tangent cone to $T$ at $0$, cf. [@DSS1 Theorem 3.1]. In fact, by [@DSS1 Theorem 3.1] $T$ satisfies Assumption \[induttiva\] with $\bar{Q}=1$: it suffices to chose $u\equiv 0$ as admissible smooth branching. If $T$ were not regular in any punctured neighborhood of $0$, we could then apply Theorem \[t:induttivo\] inductively to find a sequence of rescalings $T_{0, \rho_j}$ with $\rho_j\downarrow 0$ which satisfy Assumption \[induttiva\] with $\bar Q=Q_j$ for some strictly increasing sequence of integers. It is however elementary that the density $\Theta (0, T)$ bounds $Q_j$ from above, which is a contradiction. The branched center manifold and the blow-up theorem {#s:2_pezzi} ==================================================== From now on we fix $T$ satisfying Assumption \[induttiva\]. Observe that, without loss of generality, we are always free to rescale homothetically our current $T$ with a factor larger than $1$ and ignore whatever portion falls outside ${{\mathbf{C}}}_2 (0)$. We will do this several times, with factors which will be assumed to be sufficiently large. Hence, if we can prove that something holds in a sufficiently small neighborhood of $0$, then we can assume, withouth loss of generality, that it holds on ${{\mathbf{C}}}_2$. For this reason we can assume that the constants $C_i$ in Definition \[d:admissible\] and Assumption \[induttiva\] is as small as we want. In turns this implies that there is a well-defined orthogonal projection ${{\mathbf{P}}}: {{\mathbf{V}}}_{u,a}\cap {{\mathbf{C}}}_1 \to {{\rm Gr}}(u)\cap {{\mathbf{C}}}_2$, which is a $C^{2,\alpha}$ map. We next recall [@DSS3 Lemma 2.1]: \[l:density\] Let $T$ and $u$ be as in Assumption \[induttiva\] for some $\bar Q$. Then the nearest point projection ${{\mathbf{P}}}: {{\mathbf{V}}}_{u,a} \cap {{\mathbf{C}}}_1 \to {{\rm Gr}}(u)$ is a well-defined $C^{2, \alpha}$ map. In addition there is $Q\in {{\mathbb N}}\setminus \{0\}$ such that $\Theta (0, T) = Q \bar{Q}$ and the unique tangent cone to $T$ at $0$ is $Q\bar{Q} \a{\pi_0}$. Finally, after possibly rescaling $T$, $\Theta (p, T) \leq Q$ for every $p\in {{\mathbf{C}}}_2\setminus \{0\}$ and, for every $x\in B_2 (0)$, each connected component of $(x\times {{\mathbb R}}^n) \cap {{\mathbf{V}}}_{u,a}$ contains at least one point of ${{\rm spt}}(T)$. Since we will assume during the rest of the paper that the above discussion applies, we summarize the relevant conclusions in the following \[piu\_fogli\] $T$ satisfies Assumption \[induttiva\] for some $\bar{Q}$ and with $C_i$ sufficiently small. $Q\geq 1$ is an integer, $\Theta (0, T) = Q \bar{Q}$ and $\Theta (p, T) \leq Q$ for all $p\in {{\mathbf{C}}}_2\setminus \{0\}$. The overall plan to prove Theorem \[t:induttivo\] is then the following: - We construct first a branched center manifold, i.e. a second admissible smooth branching ${{\bm{\varphi}}}$ on ${{\mathfrak{B}}}_{\bar{Q}}$, and a corresponding $Q$-valued map $N$ defined on the normal bundle of ${{\rm Gr}}({{\bm{\varphi}}})$, which approximates $T$ with a very high degree of accuracy (in particular more accurately than $u$) and whose average ${{\bm{\eta}}}\circ N$ is very small; - Assuming that alternative (a) in Theorem \[t:induttivo\] does not hold, we study the asymptotic behavior of $N$ around $0$ and use it to build a new admissible smooth branching $v$ on some ${{\mathfrak{B}}}_{k \bar{Q}}$ where $k\geq 2$ is a factor of $Q$: this map will then be the one sought in alternative (b) of Theorem \[t:induttivo\] and a suitable rescaling of $T$ will lie in a horned neighborhood of its graph. The first part of the program is the one achieved in [@DSS3], whereas the second part will be completed in this note. Note that, when $Q=1$, from (BU) we will conclude that alternative (a) necessarily holds: this will be a simple corollary of the general case, but we observe that it could also be proved resorting to the classical Allard’s regularity theorem. Smallness condition ------------------- In several occasions we will need that the ambient manifold $\Sigma$ is suitably flat and that the excess of the current $T$ is suitably small. This can, however, be easily achieved after scaling. More precisely we recall [@DSS3 Lemma 2.3]. \[l:piccolezza\] Let $T$ be as in the Assumptions \[induttiva\] and \[piu\_fogli\]. After possibly rescaling, rotating and modifying $\Sigma$ outside ${{\mathbf{C}}}_2 (0)$ we can assume that, in case (a) and (c) of Definition \[d:semicalibrated\], - $\Sigma$ is a complete submanifold of ${{\mathbb R}}^{2+n}$; - $T_0 \Sigma = {{\mathbb R}}^{2+\bar{n}}\times \{0\}$ and, $\forall p\in \Sigma$, $\Sigma$ is the graph of a $C^{3,{{\varepsilon}}_0}$ map $\Psi_p: T_p \Sigma \to (T_p\Sigma)^\perp$. Under these assumptions, we denote by ${{\bm{c}}}$ and ${{\bm m}_0}$ the following quantities $$\begin{aligned} &{{\bm{c}}}:= \sup \{\|D\Psi_p\|_{C^{2,{{\varepsilon}}_0}}:p\in \Sigma\}\qquad \mbox{in the cases (a) and (c) of Definition \ref{d:semicalibrated}}\\ &{{\bm{c}}}:=\|d\omega\|_{C^{1,{{\varepsilon}}_0}} \qquad \qquad\;\;\qquad\qquad\mbox{in case (b) of Definition \ref{d:semicalibrated}}\\ &{{\bm m}_0}:= \max \left\{ {{\bm{c}}}^2, {{\mathbf{E}}}(T, {{\mathbf{C}}}_2, \pi_0), C_i^2, c_s^2\right\}\, ,\end{aligned}$$ where $C_i$ and $c_s$ are the constants appearing in Definition \[d:admissible\] and Assumption \[induttiva\]. Then, for any ${{\varepsilon}}_2>0$, after possibly rescaling the current by a large factor, we can assume $$\label{e:smallness} {{\bm m}_0}\leq {{\varepsilon}}_2\, .$$ In order to carry on the plan outlined in the previous subsection, it is convenient to use a different parametrization of $Q$-branchings. If we remove the origin, any admissible $Q$-branching is a Riemannian submanifold of ${{\mathbb R}}^{2+n}\setminus \{0\}$: this gives a Riemannian tensor $g := {{\bm{\Phi}}}^\sharp e$ (where $e$ denotes the euclidean metric on ${{\mathbb R}}^{2+n}$) on the puntured disk ${{\mathfrak{B}}}_{Q, 2\rho}\setminus \{0\}$. Note that in $(z,w)$ the difference between the metric tensor $g$ and the canonical flat metric $e_Q$ can be estimated by (a constant times) $|z|^{2\alpha}$: thus, as it happens for the canonical flat metric $e_Q$, when $Q>1$ it is not possible to extend the metric $g$ to the origin. However, using well-known arguments in differential geometry, we can find a conformal map from ${{\mathfrak{B}}}_{Q, r}$ onto a neighborhood of $0$ which maps the conical singularity of ${{\mathfrak{B}}}_{Q,r}$ in the conical singularity of the $Q$-branching. In fact, we need the following accurate estimates for such a map, cf. [@DSS3 Proposition 2.4]: \[p:conformal\] Given an admissible $b$-separated $\alpha$-smooth $Q$-branching ${{\rm Gr}}(u)$ with $\alpha <1/(2Q)$ there exist a constant $C_0 (Q, \alpha)>0$, a radius $r>0$ and functions ${{\bm{\Psi}}}\colon {{\mathfrak{B}}}_{Q,r} \to {{\rm Gr}}(u)$ and $\lambda\colon {{\mathfrak{B}}}_{Q,r} \to {{\mathbb R}}_+$ such that - ${{\bm{\Psi}}}$ is a homeomorphism of ${{\mathfrak{B}}}_{Q,r}$ with a neighborhood of $0$ in ${{\rm Gr}}(u)$; - ${{\bm{\Psi}}}\in C^{3,\alpha} ({{\mathfrak{B}}}_{Q,r}\setminus \{0\})$, with the estimates $$\begin{aligned} |D^l \big({{\bm{\Psi}}}(z,w) - (z,0)\big)| \leq & C_0 C_i |z|^{1+\alpha-l} \qquad \mbox{for $l=0,\dots,3$, $z\neq 0$}\, ,\label{e:conformal2}\\ [D^3 {{\bm{\Psi}}}]_{\alpha, B_r (z,w)} \leq & C_0 C_i |z|^{-2} \qquad \mbox{for $z\neq 0$ and $r = |z|/2$}\, ;\label{e:conformal3}\end{aligned}$$ - ${{\bm{\Psi}}}$ is a conformal map with conformal factor $\lambda$, namely, if we denote by $e$ the ambient euclidean metric in ${{\mathbb R}}^{2+n}$ and by $e_Q$ the canonical euclidean metric of ${{\mathfrak{B}}}_{Q,r}$, $$\label{e:conformal} g:= {{\bm{\Psi}}}^\sharp e = \lambda\, e_Q\, \qquad \mbox{on ${{\mathfrak{B}}}_{Q,r}\setminus \{0\}$.}$$ - The conformal factor $\lambda$ satisfies $$\begin{aligned} |D^l (\lambda -1) (z,w)| \leq &C_0 C_i |z|^{2\alpha-l} \qquad \mbox{for $l=0,1,\ldots, 2$}\label{e:conformal4}\\ [D^2 \lambda]_{\alpha, B_r (z,w)} \leq & C_0 C_i |z|^{\alpha-2}\qquad \mbox{for $z\neq 0$ and $r = |z|/2$}\, .\label{e:conformal5}\end{aligned}$$ A map ${{\bm{\Psi}}}$ as in Proposition \[p:conformal\] will be called a [*conformal parametrization*]{} of an admissible $Q$-branching. The center manifold and the approximation ----------------------------------------- We are now ready to state the two “halves” of Theorem \[t:induttivo\]. The first one is [@DSS3 Theorem 2.6], which we recall here for the reader’s convenienve. \[t:cm\] Let $T$ be as in Assumptions \[induttiva\] and \[piu\_fogli\]. Then there exist $\eta_0, \gamma_0, r_0,C>0$, $b>1$, an admissible $b$-separated $\gamma_0$-smooth $\bar Q$-branching ${{\mathcal{M}}}$, a corresponding conformal parametrization ${{\bm{\Psi}}}: {{\mathfrak{B}}}_{\bar Q,2} \to {{\mathcal{M}}}$ and a $Q$-valued map ${{\mathpzc{N}}}: {{\mathfrak{B}}}_{\bar Q,2} \to \I{Q}({{\mathbb R}}^{2+n})$ with the following properties: - $\bar Q Q = \Theta (T, 0)$ and $$\begin{aligned} |D({{\bm{\Psi}}}(z,w) -(z,0))|\leq &\, C {{\bm m}_0}^{1/2} |z|^{\gamma_0}\\ |D^2 {{\bm{\Psi}}}(z,w)|+ |z|^{-1} |D^3 {{\bm{\Psi}}}(z,w)| \leq &\, C {{\bm m}_0}^{\sfrac{1}{2}} |z|^{\gamma_0-1}\, ;\end{aligned}$$ in particular, if we denote by $A_{{\mathcal{M}}}$ the second fundamental form of ${{\mathcal{M}}}\setminus \{0\}$, $$|A_{{\mathcal{M}}}({{\bm{\Psi}}}(z,w))| + |z|^{-1} |D_{{{\mathcal{M}}}} A_{{\mathcal{M}}}({{\bm{\Psi}}}(z,w))| \leq C {{\bm m}_0}^{\sfrac{1}{2}} |z|^{\gamma_0-1}\, .$$ - ${{\mathpzc{N}}}\,_i (z,w)$ is orthogonal to the tangent plane, at ${{\bm{\Psi}}}(z,w)$, to ${{\mathcal{M}}}$. - If we define $S:= T_{0,r_0}$, then ${{\rm spt}}(S)\cap {{\mathbf{C}}}_1 \setminus \{0\}$ is contained in a suitable horned neighborhood of the $\bar Q$-branching, where the orthogonal projection ${{\mathbf{P}}}$ onto it is well-defined. Moreover, for every $r\in]0,1[$ we have $$\begin{aligned} \label{e:Ndecay} \|{{\mathpzc{N}}}\vert_{B_r}\|_0 + \sup_{p\in {{\rm spt}}(S) \cap {{\mathbf{P}}}^{-1} ({{\bm{\Psi}}}(B_r))} |p- {{\mathbf{P}}}(p)|\leq C {{\bm m}_0}^{\sfrac{1}{4}} r^{1+\sfrac{\gamma_0}{2}}\, .\end{aligned}$$ - If we define $${{\mathbf{D}}}(r) := \int_{B_r} |D{{\mathpzc{N}}}|^2 \quad \text{and}\quad {{\mathbf{H}}}(r) := \int_{{\partial}B_r} |{{\mathpzc{N}}}|^2\, ,$$ $${{\mathbf{F}}}(r) := \int_0^r\frac{{{\mathbf{H}}}(t)}{t^{2-\,\gamma_0}}\,dt \quad\text{and}\quad {{\mathbf{\Lambda}}}(r):= {{\mathbf{D}}}(r)+{{\mathbf{F}}}(r)\, ,$$ then the following estimates hold for every $r\in ]0,1[$: $$\begin{aligned} {{\rm {Lip}}}({{\mathpzc{N}}}\vert_{B_r}) \leq & C \min\{{{\mathbf{\Lambda}}}^{\eta_0}(r), {{\bm m}_0}^{\eta_0}r^{\eta_0}\} \label{e:Lip_N}\\ {{\bm m}_0}^{\eta_0} \,\int_{B_r} |z|^{\gamma_0 -1} |{{\bm{\eta}}}\circ {{\mathpzc{N}}}(z,w)| \leq & C\, {{\mathbf{\Lambda}}}^{\eta_0}(r)\,{{\mathbf{D}}}(r)+C\,{{\mathbf{F}}}(r)\, .\label{e:media_pesata}\end{aligned}$$ - Finally, if we set $${{\mathpzc{F}}}(z,w):= \sum_i\a{{{\bm{\Psi}}}(z,w) + {{\mathpzc{N}}}\,_i(z,w)}\, ,$$ then $$\begin{aligned} \|S-{\mathbf{T}}_{{\mathpzc{F}}}\|\big({{\mathbf{P}}}^{-1}({{\bm{\Psi}}}(B_r))\big) \leq & C\,{{\mathbf{\Lambda}}}^{\eta_0}(r)\,{{\mathbf{D}}}(r)+ C\,{{\mathbf{F}}}(r)\, . \label{e:diff masse} \end{aligned}$$ The second main step is the analysis of the asymptotic behaviour of ${{\mathpzc{N}}}$ around the origin, which is the main focus of this paper. \[r:W12\] In order to state it, we agree to define $W^{1,2}$ functions on ${{\mathfrak{B}}}$ in the following fashion: removing the origin $0$ from ${{\mathfrak{B}}}$ we have a $C^3_{loc}$ (flat) Riemannian manifold embedded in ${{\mathbb R}}^{4}$ and we can define $W^{1,2}$ maps on it following [@DS1]. Alternatively we can use the conformal parametrization ${{\mathbf{W}}}: {{\mathbb R}}^2 = {{\mathbb C}}\to {{\mathfrak{B}}}_{\bar Q}$ given by ${{\mathbf{W}}}(z)= (z^{\bar Q}, z)$ and agree that $u\in W^{1,2} ({{\mathfrak{B}}}_{\bar Q})$ if $u\circ {{\mathbf{W}}}$ is in $W^{1,2} ({{\mathbb R}}^2)$. Since discrete sets have zero $2$-capacity, it is immediate to verify that these two definitions are equivalent. In a similar fashion, we will ignore the origin when integrating by parts Lipschitz vector fields, treating ${{\mathfrak{B}}}_{\bar Q}$ as a $C^1$ Riemannian manifold. It is straightforward to show that our assumption is correct, for instance removing a disk of radius $\varepsilon$ centered at the origin, integrating by parts and then letting $\varepsilon\downarrow 0$. \[t:bu\] Under the assumptions of Theorem \[t:cm\], the following dichotomy holds: - either there exists $s>0$ such that ${{\mathpzc{N}}}\vert_{B_s} \equiv Q \a{0}$; - or there exist constants $I_0>1$, $a_0, \bar r, C>0$ and an $I_0$-homogeneous nontrivial Dir-minimizing function $g: {{\mathfrak{B}}}_{\bar{Q}} \to \I{Q}({{\mathbb R}}^{2+n})$ such that - ${{\bm{\eta}}}\circ g \equiv 0$, - $g = \sum_i \a{(0, \bar{g}_i , 0)}$, where $\bar{g}_i (x)\in {{\mathbb R}}^{\bar{n}}$ and $(0, \bar g_i (x), 0)\in {{\mathbb R}}^2\times {{\mathbb R}}^{\bar n} \times {{\mathbb R}}^l$, - and the following estimates hold: $$\begin{aligned} {{\mathcal{G}}}\big({{\mathpzc{N}}}(z,w),g(z,w)\big) \leq &\; C |z|^{I_0+a_0}\quad\qquad\qquad\qquad \forall\;(z,w) \in {{\mathfrak{B}}}_{Q}, \;|z|<\bar r,\label{e:uniform convergence}\\ \int_{B_{r+\rho}\setminus B_{r-\rho}}|D{{\mathpzc{N}}}|^2 \leq &\; C\,r^{2I_0+a_0} + C\,r^{2I_0-1}\,\rho \qquad \forall\; 4\,\rho\leq r < 1,\label{e:energia N}\\ {{\mathbf{H}}}(r) \leq &\; C\,r\,{{\mathbf{D}}}(r)\quad\qquad\qquad\qquad\; \forall\; r < 1.\label{e:Poincare0}\end{aligned}$$ \[r:ultimo\_step\] Note that, when $\bar{Q} = \Theta (T, 0)$, we necessarily have $Q=1$ and the second alternative is excluded. In particular we conclude that $T$ coincides with $\a{{{\mathcal{M}}}}$ in a neighborhood of $0$ and thus it is a regular submanifold in a [*punctured neighborhood*]{} of $0$. \[r:bound\_energia\] By a simple dyadic argument it follows from and that $$\label{e:consequences} \int_{B_r}|D{{\mathpzc{N}}}|^2 \leq C\,r^{2I_0} \quad\text{and}\quad {{\mathbf{F}}}(r) \leq C\,r^{2I_0+\gamma_0} \quad \forall\;r < 1.$$ Below we show how to conclude Theorem \[t:induttivo\] from Theorem \[t:bu\]. The remaining part of the paper is dedicated to the proof of the latter, which will be split in six sections each corresponding to one of the following steps. - In Section \[s:AM\] we will deduce an almost minimizing property for the map ${{\mathpzc{N}}}$ in terms of its Dirichlet energy. - In Section \[s:harmonic\] we will apply the almost minimizing property and compare the Dirichlet energy of ${{\mathpzc{N}}}$ with that of a suitable harmonic extension of its boundary value on any given ball. - In Section \[s:poincare\] we use the comparison above and a first variation argument to derive a suitable Poincaré-type inequality for ${{\mathpzc{N}}}$. - In Section \[s:refined\] we compute again the first variations of the Dirichlet energy of ${{\mathpzc{N}}}$ and use the Poincaré inequality to bound efficiently several error terms. - Using the latter bounds, in Section \[s:decay\] we will prove an almost monotonicity property for the frequency function and the existence and boundedness of its limit, which is indeed the number $I_0$ of Theorem \[t:bu\]. The almost minimality of ${{\mathpzc{N}}}$ will then allow us to conclude an exponential rate of decay to this limit. - From the decay of the previous step we will capture in Section \[s:bu\] the asymptotic behaviour of ${{\mathpzc{N}}}$ and show the existence of the map $g$ of Theorem \[t:bu\]. The overall strategy follows the ideas and some of the computations in [@Chang]. However several adjustments are needed to carry on the proof in the cases (b) and (c) of Definition \[d:semicalibrated\]. In particular in Section \[s:decay\] we need to introduce a suitable modification of the usual frequency function to handle case (b). Proof of the inductive step --------------------------- In the next sections we will prove Theorem \[t:bu\]. We start observing that if case (a) of Theorem \[t:induttivo\] does not hold, then we are necessarily in case (ii) of Theorem \[t:bu\]. Therefore we only need to prove that Theorem \[t:bu\](ii) implies Theorem \[t:induttivo\](b). We divide the proof in different steps. [**Step 1.**]{} For a reason which will become clear later, it is convenient to slightly modify the map $g$ to a multivalued map $n (z,w) = \sum_i \a{n_i (z,w)}$ in such a way that $n_i (z,w)$ is orthogonal to ${{\mathcal{M}}}$ at ${{\bm{\Psi}}}(z,w)$. To achieve this it suffices to project $g_i (z,w) = (0, \bar{g}_i (z,w),0)$ on the normal bundle. Observe that, by the estimates on $|A_{{\mathcal{M}}}|$ and ${{\bm{\Psi}}}$, we have $$\begin{aligned} |g_i (z,w) - n_i (z,w)|\leq & C C_i |z|^{\gamma_0} |g_i (z,w)|\, ,\label{e:proietta_1}\\ |Dn| (z,w) \leq & |Dg| (z,w) + C C_i |z|^{\gamma_0 -1} |g| (z,w)\,\label{e:proietta_2}.\end{aligned}$$ We introduce the function $H: {{\mathfrak{B}}}_{\bar Q} \to \I{Q}({{\mathbb R}}^{2+n})$ given by $$H(z,w) = \sum_{i=1}^{Q}\a{H_i(z,w)} := \sum_{i=1}^{Q} \a{{{\bm{\Psi}}}(z,w) + n_i (z,w)}\, .$$ Note that, since $g$ is $I_0$-homogeneous and ${\textup{Dir}}$-minimizer, by [@DS1 Proposition 5.1] there is a constant $C$ such that $$\label{e:separazione omogenea 1} |g_i(z,w) - g_j(z,w)| \geq 2 C\,|z|^{I_0} \quad\text{whenever }g_i(z,w) \neq g_j(z,w)\, .$$ In fact [@DS1 Proposition 5.1] is stated for maps with domain ${{\mathbf{C}}}= {{\mathbb R}}^2$. However, if we define the map ${{\mathbf{W}}}: {{\mathbb C}}\to {{\mathfrak{B}}}_{\bar Q, \infty}$ as ${{\mathbf{W}}}(z) = (z^{\bar Q}, z)$, by the conformality of ${{\mathbf{W}}}$ it is easy to check that $g\circ {{\mathbf{W}}}$ is ${\textup{Dir}}$-minimizer and $I_0 Q$ homogeneous. By and , provided $z$ is small enough we have $$\label{e:separazione omogenea} |H_i(z,w) - H_j(z,w)| \geq C\,|z|^{I_0} \quad\text{whenever }H_i(z,w) \neq H_j(z,w).$$ Let $\bar a\in ]0, a_0[$ be a constant to be fixed momentarily and $\zeta := I_0 + \sfrac{\bar a}{2}>1$. Set $${{\mathbf{V}}}_{H, \zeta} := \big\{ H_i(z,w) + p \in {{\mathbb R}}^{2+n} : |p|< |z|^\zeta,\; i=1,\ldots, Q\big\}.$$ We claim that there exists $s>0$ such that ${{\rm spt}}(T) \cap {{\mathbf{B}}}_s\setminus \{0\} \subset {{\mathbf{V}}}_{H,\zeta}$. In order to prove this claim, we distinguish two cases. First we consider any point $p \in {{\rm spt}}(T) \cap {{\rm spt}}({\mathbf{T}}_{{\mathpzc{F}}})\setminus \{0\}$. In this case $p = {{\bm{\Psi}}}(z,w) + {{\mathpzc{N}}}\, _i(z,w)$ for some $(z,w) \in {{\mathfrak{B}}}_{\bar Q}\setminus \{0\}$ and for some $i=1,\ldots, Q$. Without loss of generality, by we can assume $|{{\mathpzc{N}}}\,_i(z,w) - g_i(z,w)| \leq C |z|^{I_0+\bar a}$, i.e. $$\begin{aligned} \label{e:uniform graph} |p-H_i(z,w)| & = |{{\mathpzc{N}}}\,_i (z,w) - n_i (z,w)|\leq |{{\mathpzc{N}}}\,_i (z,w)-g_i (z,w)|+ |g_i (z,w)- n_i (z,w)|\nonumber\\ &\leq C |z|^{I_0+\bar a} + C|z|^{\gamma_0+I_0},\end{aligned}$$ which in particular implies ${{\rm spt}}(T) \cap {{\rm spt}}({\mathbf{T}}_{{\mathpzc{F}}}) \cap {{\mathbf{B}}}_s \subset {{\mathbf{V}}}_{H,\zeta}$ if $s$ is sufficiently small and we impose $\frac{\bar a}{2}< \gamma_0$. For the second case we consider a point $p \in {{\rm spt}}(T) \setminus {{\rm spt}}({\mathbf{T}}_{{\mathpzc{F}}})$ and assume by contradiction that $p \not\in {{\mathbf{V}}}_{H,\zeta}$. In particular, in view of we have that $$\label{e:grafico_escluso} B:={{\mathbf{B}}}_{\frac{|z|^{\zeta}}{2}}(p) \cap {{\rm spt}}({\mathbf{T}}_{{\mathpzc{F}}}) = \emptyset$$ if $|z|$ is sufficiently small. By the monotonicity formula we know that $\|T\|(B) \geq C\, |z|^{2\zeta}$; nevertheless since $B \subset {{\mathbf{P}}}^{-1}(B_{2|z|}\setminus B_{\sfrac{|z|}{2}})$, implies $\|T\| (B) \leq \|T - {\mathbf{T}}_{{{\mathpzc{F}}}}\| (B)$ and from and we conclude $\|T\|(B) \leq C\, |z|^{2I_0+2\,\kappa}$ with $\kappa = \min\{2\,\eta_0\,I_0, \gamma_0\}$. This gives a contradiction if $\bar a < 2\kappa$. [**Step 2.**]{} From the previous step we can infer that $g$ is a constant multiple of an irreducible function, namely there exists $Q'>0$ such that ${{\textrm{card}}}(g(z,w)) = Q'$ for every $(z,w) \neq (0,0)$ and there exists a continuous map $h: {{\mathfrak{B}}}_{\bar Q Q'} \to {{\mathbb R}}^{2+n}$ such that $$\label{e:irreducibility} g(z,w) = \frac{Q}{Q'} \sum_{\tilde z = z, \; \tilde w^{Q'} = w} \a{h(\tilde z, \tilde w)}.$$ If this is not the case, by [@DS1 Proposition 5.1] we can decompose $g$ in the superposition of irreducible functions, i.e. there exists a unique decomposition $g=\sum_{j=1}^J k_j g_j$ where $g_j:{{\mathfrak{B}}}_{Q} \to \I{q_j}({{\mathbb R}}^{n})$ are Dir-minimizing $I_0$-homogeneous functions, for some choice of positive integers $J,k_j, q_j$ such that $\sum_{j=1}^Jk_jq_j=Q$. Denoting by $H^j$ the corresponding maps $$H^j(z,w) := \sum_{l=1}^{q_j}\a{{{\bm{\Psi}}}(z,w) + (n^j)_l(z,w)}$$ and by ${{\mathbf{V}}}_{H^j,\zeta}$ the corresponding horned neighborhoods $${{\mathbf{V}}}_{H^j, \zeta} := \big\{ (H^j)_l(z,w) + p \in {{\mathbb R}}^{2+n} : |p|< |z|^\zeta,\; l=1,\ldots, q_j\big\},$$ it follows from that the closures of the $V_{\zeta, H_i}$ intersect only at the origin. Setting $T_i := T{\mathop{\hbox{\vrule height 7pt width .3pt depth 0pt \vrule height .3pt width 5pt depth 0pt}}\nolimits}V_{\zeta, H_i}$, we infer that $T= \sum_i T_i$ with ${{\rm spt}}(T_i) \cap {{\rm spt}}(T_j) = \{0\}$, against the irreducibility of $T$. Note that, since ${{\bm{\eta}}}\circ g=0$ it also follows that $Q'>1$. Having established , let us define ${{\bm{\Theta}}}:{{\mathfrak{B}}}_{\bar QQ'} \to {{\mathbb R}}^{2+n}$ as $${{\bm{\Theta}}}(\tilde z,\tilde w) := {{\bm{\Psi}}}(\tilde z, \tilde w^{Q'}) + h^n (\tilde z, \tilde w) \quad \forall\;(\tilde z, \tilde w) \in {{\mathfrak{B}}}_{\bar QQ'}\, ,$$ where $h^n (\tilde{z}, \tilde w)$ is the projection of $h (\tilde{z}, \tilde{w})$ on the space normal to ${{\mathcal{M}}}$ at the point ${{\bm{\Psi}}}(\tilde{z}, \tilde w^{Q'})$. It follows that ${{\rm Im}}(H) = {{\rm Im}}({{\bm{\Theta}}})$ is an admissible $\bar Q Q'$-branching (the Hölder regularity for the graphical parametrization follows from the fact that $I_0 >1$). Moreover, from the homogeneity of $g$ we easily infer that ${{\rm Im}}({{\bm{\Theta}}})$ is $I_0$-separated (for a suitable constant $c_s$). Note that for $\zeta':=I_0+\bar a/4$ and $s$ sufficiently small ${{\mathbf{V}}}_{H,\zeta}\cap{{\mathbf{B}}}_s\subset {{\mathbf{V}}}_{{{\bm{\Theta}}},\zeta'}\cap {{\mathbf{B}}}_s$. [**Step 3.**]{} Finally we prove the condition (Dec) of Assumption \[induttiva\]. Let $(z,w) \in {{\mathfrak{B}}}_{\bar Q}$ with $0<|z|<\sqrt{2}$, let $V$ be the connected component of ${{\mathbf{V}}}_{{{\bm{\Theta}}}, \zeta'}\cap \{(x,y): |x-z| < d\}$ with $d:=\sfrac{|z|}{4}$ containing ${{\bm{\Theta}}}(z,w)$, and $p\in {{\rm spt}}(T) \cap V$ with coordinates $p=(z,y)$. Denote by $\pi$ the oriented two-vector for ${{\rm Im}}({{\bm{\Theta}}})$ at ${{\bm{\Theta}}}(z,w)$, and consider $\rho \in [\frac12 d^{\sfrac{(I_0+1)}{2}},d]$. Since ${{\mathbf{B}}}_{\rho}(p)\cap {{\rm spt}}(T) \subset {{\mathbf{P}}}^{-1}({{\bm{\Psi}}}(B_{|z|+2\rho}\setminus B_{|z|-2\rho}))$, we start estimating as follows $$\begin{aligned} \int_{{{\mathbf{B}}}_\rho(p)} |\vec T - \vec\pi|^2 \, d\|T{\mathop{\hbox{\vrule height 7pt width .3pt depth 0pt \vrule height .3pt width 5pt depth 0pt}}\nolimits}V\| &\leq \int_{{{\mathbf{B}}}_\rho(p)\cap V} |\vec {\mathbf{T}}_{{\mathpzc{F}}}- \vec\pi|^2 \, d\|{\mathbf{T}}_{{\mathpzc{F}}}\| + \|T-{\mathbf{T}}_{{\mathpzc{F}}}\|({{\mathbf{p}}}^{-1}(B_{|x_0|+2\rho}))\notag\\ & \stackrel{\eqref{e:diff masse}}{\leq} \int_{{{\mathbf{B}}}_\rho(p)\cap V} |\vec {\mathbf{T}}_{{\mathpzc{F}}}- \vec\pi|^2 \, d\|{\mathbf{T}}_{{\mathpzc{F}}}\| + C\,|z|^{2I_0 + 2 \kappa}.\end{aligned}$$ Next, note that for $|z|$ small enough ${{\mathbf{P}}}({{\mathbf{B}}}_\rho(p)\cap {{\mathbf{V}}}_{{{\bm{\Theta}}}, \zeta'}) \subset {{\bm{\Psi}}}(B_{2\rho}(z,w))$. We can consider the set of indices $A \subset \{1,\ldots,Q\}$ such that ${{\mathpzc{F}}}_i(z,w) \in V$ for $i \in A$ and estimate as follows $$\begin{aligned} \int_{{{\mathbf{B}}}_\rho(p)\cap V} |\vec {\mathbf{T}}_{{\mathpzc{F}}}- \vec\pi|^2 \, d\|{\mathbf{T}}_{{\mathpzc{F}}}\| & \leq C\sum_{i\in A}\int_{B_{2\rho}(z,w)} |\vec {\mathbf{T}}_{{{\mathpzc{F}}}_i (\zeta, \omega)} - \vec {\mathbf{T}}_{{{\bm{\Theta}}}(\zeta, \omega)}|^2\, d\zeta +C\,\rho^2\,{{\rm {Lip}}}(D{{\bm{\Theta}}}\vert_{B_{2\rho}(z,w)})^2\notag\\ & \leq C\sum_{i\in A}\int_{B_{2\rho}(z,w)} |\vec {\mathbf{T}}_{{{\mathpzc{F}}}_i (\zeta, \omega)} - \vec {\mathbf{T}}_{{{\bm{\Psi}}}(\zeta, \omega)}|^2\, d\zeta\notag\\ &\quad+ C\,\int_{B_{2\rho}(z,w)} |\vec {\mathbf{T}}_{{{\bm{\Psi}}}(\zeta, \omega)} - \vec {\mathbf{T}}_{{{\bm{\Theta}}}(\zeta, \omega)}|^2\, d\zeta + C\,\rho^4\,|z|^{2\theta-2}, \end{aligned}$$ where $\theta:=\min\{\gamma_0, I_0-1\}$ and we used that $|D^2{{\bm{\Theta}}}|(z,w) \leq C\,|z|^{\theta-1}$. We can finally use the computation of the excess in curvilinear coordinates in [@DS2 Proposition 3.4] to get $$\begin{aligned} \sum_i\int_{B_{2\rho}(z,w)} |\vec {\mathbf{T}}_{{{\mathpzc{F}}}_i (\zeta, \omega)} - \vec {\mathbf{T}}_{{{\bm{\Psi}}}(\zeta, \omega)}|^2 & \leq C\int_{B_{2\rho}(z,w)} \left(|D{{\mathpzc{N}}}|^2 + |\zeta|^{2\gamma_0-2} |{{\mathpzc{N}}}|^2\right) \notag\\ &\stackrel{\eqref{e:consequences}}{\leq} C\int_{B_{|z|+2\rho}\setminus B_{|z|-2\rho}} |D{{\mathpzc{N}}}|^2 + C\,|z|^{2I_0+2\gamma_0}\\ &\stackrel{\eqref{e:energia N}}{\leq} C\,|z|^{2I_0+a_0}+C\,|z|^{2I_0-1}\,\rho\,,\end{aligned}$$ and similarly $$\begin{aligned} \int_{B_{2\rho}(z,w)} |\vec {\mathbf{T}}_{{{\bm{\Theta}}}(\zeta, \omega)} - \vec {\mathbf{T}}_{{{\bm{\Psi}}}(\zeta, \omega)}|^2& \leq C\int_{B_{2\rho}(z,w)} \left(|Dn|^2 + |\zeta|^{2\gamma_0-2} |n|^2\right)\notag\\ & \leq C\int_{B_{2\rho}(z,w)} \left(|Dg|^2 + |\zeta|^{2\gamma_0-2} |g|^2\right)\notag\\ &\leq C\,|z|^{2I_0-2}\rho^2+C\,|z|^{2I_0+2\gamma_0}\, \end{aligned}$$ (observe that, in order to apply [@DS2 Proposition 3.4] we need that $n$ takes value into the normal bundle). Collecting all the estimates together, we have that there exists a suitable constant $\varpi$ such that $$\label{e:dec finale} \int_{{{\mathbf{B}}}_\rho(p)} |\vec T - \vec\pi|^2 \, d\|T{\mathop{\hbox{\vrule height 7pt width .3pt depth 0pt \vrule height .3pt width 5pt depth 0pt}}\nolimits}V\| \leq C\,|z|^{2I_0 + 2 \varpi} + C\,\rho\,|z|^{2I_0-1} + C\,\rho^4\,|z|^{2\varpi-2} \leq |z|^{\gamma-2}\rho^4,$$ where the last inequality is easily verified for $\gamma>0$ and $|z|$ small enough. This shows (Dec) in Assumption \[induttiva\] and completes the proof. Dirichlet almost minimizing property {#s:AM} ==================================== The normal approximation ${{\mathpzc{N}}}$ inherits from $T$ an almost minimizing property for the Dirichlet energy, where the errors involved are in fact expressed in terms of some specific norms of ${{\mathpzc{N}}}$ itself and of its competitors. For techincal reasons we introduce the maps $F := \sum_{i=1}^{Q}\a{p + N_i(p)}$, where $N := {{\mathpzc{N}}}\circ {{\bm{\Psi}}}^{-1}$. In order to state the almost minimizing property of ${{\mathpzc{N}}}$ we introduce an appropriate notion of competitor. \[d:competitors\] A Lipschitz map ${{\mathpzc{L}}}\colon B_r\to \I{Q}({{\mathbb R}}^{n+2})$ is called a competitor for ${{\mathpzc{N}}}$ in the ball $B_r$ if - ${{\mathpzc{L}}}\vert_{\partial B_r}= {{\mathpzc{N}}}\vert_{{\partial}B_r}$; - ${{\rm spt}}({{\mathpzc{G}}}(z, w)) \subset \Sigma$ for all $(z,w)\in B_r$, where ${{\mathpzc{G}}}(z,w) := \sum_{j=1}^{Q}\a{{{\bm{\Psi}}}(z,w) +{{\mathpzc{L}}}_j(z,w)}$. We are now ready to state the almost minimizing property for ${{\mathpzc{N}}}$. We use the notation ${{\mathbf{p}}}_{T_p \Sigma}$ for the orthogonal projection on the tangent space to $\Sigma$ at $p$. We recall that, given our choice of coordinates, ${{\mathbf{p}}}_{T_0 \Sigma}$ is the projection on ${{\mathbb R}}^{2+\bar{n}}\times \{0\}$. Since this projection will be used several times, we will denote it by ${{\mathbf{p}}}_0$. By the $C^{3,{{\varepsilon}}}$ regularity of $\Sigma$, there exists a map $\Psi_0\in C^{3,{{\varepsilon}}}({{\mathbb R}}^{2+\bar{n}},{{\mathbb R}}^l)$ such that $$\Psi_0 (0)= 0\, ,\; D\Psi_0 (0)=0\quad \textup{and}\quad {{\rm Gr}}(\Psi_0) = \Sigma\,.$$ Next, for each function ${{\mathpzc{L}}}$ satisfying Condition (b) in Definition \[d:competitors\] we consider the map $\bar{{\mathpzc{L}}}:= {{\mathbf{p}}}_0 \circ {{\mathpzc{L}}}$, which is a multivalued $\bar{{\mathpzc{L}}}: {{\mathfrak{B}}}\to {{\mathcal{A}}_Q}({{\mathbb R}}^{2+\bar{n}})$. We observe that it is possible to determine ${{\mathpzc{L}}}$ from $\bar{{\mathpzc{L}}}$. In particular, fix coordinates $(\xi, \eta)\in {{\mathbb R}}^{2+\bar{n}}\times {{\mathbb R}}^{l}$ and let ${{\mathpzc{L}}}= \sum \a{{{\mathpzc{L}}}_i}$, $\bar{{\mathpzc{L}}}= \sum \a{\bar {{\mathpzc{L}}}_i}$, where $\bar {{\mathpzc{L}}}_i = {{\mathbf{p}}}_0 \circ {{\mathpzc{L}}}_i$. Then the formula relating ${{\mathpzc{L}}}_i$ and $\bar {{\mathpzc{L}}}_i$ is $$\label{e:relazione} {{\mathpzc{L}}}_i (z,w) = \big(\bar {{\mathpzc{L}}}_i (z,w), \Psi_0 \big({{\mathbf{p}}}_0 ({{\bm{\Psi}}}(z,w)) + \bar{{\mathpzc{L}}}_i (z,w)\big) - \Psi_0 ({{\mathbf{p}}}_0 ({{\bm{\Psi}}}(z,w))\big)\, .$$ \[p:amin\] There exists a constant $C_{\ref{p:amin}}>0$ such that the following holds. If $r\in (0,1)$ and ${{\mathpzc{L}}}\colon B_r\to \I{Q}({{\mathbb R}}^{2+n})$ is a Lipschitz competitor for ${{\mathpzc{N}}}$ with $\|{{\mathpzc{L}}}\|_\infty \leq r$ and ${{\rm {Lip}}}({{\mathpzc{L}}})\leq C_{\ref{p:amin}}^{-1}$, then $$\begin{aligned} \label{e:amin} \int_{B_r}|D{{\mathpzc{N}}}|^2 &\leq (1+C_{\ref{p:amin}}\,r)\,\int_{B_r} |D\bar {{\mathpzc{L}}}|^2 + C_{\ref{p:amin}}\,\textup{Err}_1({{\mathpzc{N}}},B_r) + C_{\ref{p:amin}}\,\textup{Err}_2({{\mathpzc{L}}},B_r) + C_{\ref{p:amin}}\, r^2 {{\mathbf{D}}}' (r)\, ,\end{aligned}$$ where $\bar {{\mathpzc{L}}}:={{\mathbf{p}}}_0 \circ {{\mathpzc{L}}}$ and the the error terms $\textup{Err}_1({{\mathpzc{N}}},B_r)$, $\textup{Err}_2({{\mathpzc{L}}},B_r)$ are given by the following expressions: $$\begin{aligned} \label{e:error1} \textup{Err}_1({{\mathpzc{N}}},B_r) & = {{\mathbf{\Lambda}}}^{\eta_0}(r)\,{{\mathbf{D}}}(r)+{{\mathbf{F}}}(r)+{{\mathbf{H}}}(r) +{{\bm m}_0}^{\sfrac12}\, r^{1+\gamma_0}\int_{{\partial}B_r} |{{\bm{\eta}}}\circ {{\mathpzc{N}}}|\, \end{aligned}$$ and $$\begin{aligned} \label{e:error2} \textup{Err}_2({{\mathpzc{L}}},B_r) & = {{\bm m}_0}^{\sfrac12}\,\int_{B_r} |z|^{\gamma_0 -1} |{{\bm{\eta}}}\circ {{\mathpzc{L}}}|\, \,.\end{aligned}$$ For the proof of Proposition \[p:amin\] we consider separately the three cases: - $T$ is mass minimizing; - $T$ is semicalibrated; - $T$ is the cross-section of a mass minimizing three-dimensional cone. For notational convenience we set $L := {{\mathpzc{L}}}\circ {{\bm{\Psi}}}^{-1}$, $G := {{\mathpzc{G}}}\circ {{\bm{\Psi}}}^{-1}$. Observe also that, by Lemma \[l:poinc\] and \[l:proiezione\], it is enough to prove that $$\label{e:amin2} \int_{B_r}|D{{\mathpzc{N}}}|^2 \leq (1+C_{\ref{p:amin}}\,r)\,\int_{B_r} |D {{\mathpzc{L}}}|^2 + C\,\textup{Err}_1({{\mathpzc{N}}},B_r) + C \,\textup{Err}_2({{\mathpzc{L}}},B_r)+\frac{C}{r}\int_{B_r}|{{\mathpzc{L}}}|^2 + C r^2{{\mathbf{D}}}'(r)\,.$$ Indeed Lemma \[l:proiezione\] implies that $$\begin{aligned} \int_{B_r} |D {{\mathpzc{L}}}|^2 \leq & (1+Cr) \int_{B_r} |D \bar{{\mathpzc{L}}}|^2 + C r \int_{\partial B_r} |\bar {{\mathpzc{L}}}|^2 \leq (1+Cr) \int_{B_r} |D\bar{{\mathpzc{L}}}|^2 + Cr \int _{\partial B_r}|{{\mathpzc{L}}}|^2\\ = & (1+Cr) \int_{B_r} |D\bar{{\mathpzc{L}}}|^2 + Cr \int_{\partial B_r} |{{\mathpzc{N}}}|^2 \leq (1+Cr)\int_{B_r} |D\bar {{\mathpzc{L}}}|^2 + C\, \textup{Err}_1 ({{\mathpzc{N}}}, B_r)\, ,\end{aligned}$$ whereas Lemma \[l:poinc\] implies $$\begin{aligned} \frac{1}{r} \int_{B_r} |{{\mathpzc{L}}}|^2 \leq & Cr \int_{B_r} |D{{\mathpzc{L}}}|^2 + C \int_{\partial B_r} |{{\mathpzc{L}}}|^2 \leq C r \int_{B_r} |D\bar{{\mathpzc{L}}}|^2 + C \, \textup{Err}_1 ({{\mathpzc{N}}}, B_r)\, .\end{aligned}$$ Proof of Proposition \[p:amin\] case (a): $T$ mass minimizing. -------------------------------------------------------------- We fix ${{\mathpzc{L}}}$, $\bar {{\mathpzc{L}}}$, $L$, ${{\mathpzc{G}}}$, $\bar {{\mathpzc{G}}}$ and $G$ as above. Let us set $$\label{e:competitore} Z:= T - {\mathbf{T}}_{{{\mathpzc{F}}}|_{B_r}} + {\mathbf{T}}_{{{\mathpzc{G}}}}\, .$$ Since ${{\mathpzc{F}}}|_{\partial B_r} = {{\mathpzc{G}}}|_{\partial B_r}$, from [@DS2] it follows that $\partial ({\mathbf{T}}_{{{\mathpzc{G}}}} - {\mathbf{T}}_{{{\mathpzc{F}}}|_{B_r}}) =0$. Moreover ${{\rm spt}}(Z) \subset \Sigma$ and therefore we must have ${{\mathbf{M}}}(T) \leq {{\mathbf{M}}}(Z)$. Taking into account , we conclude that $$\begin{aligned} \label{e:prima} {{\mathbf{M}}}({\mathbf{T}}_{{{\mathpzc{F}}}|_{B_r}}) &\leq {{\mathbf{M}}}(T{\mathop{\hbox{\vrule height 7pt width .3pt depth 0pt \vrule height .3pt width 5pt depth 0pt}}\nolimits}{{\mathbf{p}}}^{-1} ({{\bm{\Psi}}}(B_r))) + \|T-{\mathbf{T}}_{{{\mathpzc{F}}}|_{B_r}}\|( {{\mathbf{p}}}^{-1} ({{\bm{\Psi}}}(B_r)))\notag\\ &\leq {{\mathbf{M}}}({\mathbf{T}}_{{{\mathpzc{G}}}}) +2 \,\|T-{\mathbf{T}}_{{{\mathpzc{F}}}|_{B_r}}\|( {{\mathbf{p}}}^{-1} ({{\bm{\Psi}}}(B_r))) \notag\\ &\leq {{\mathbf{M}}}({\mathbf{T}}_{{{\mathpzc{G}}}})+C\, \textup{Err}_1 ({{\mathpzc{N}}}, B_r) \, .\end{aligned}$$ Observe now that ${\mathbf{T}}_{{{\mathpzc{F}}}|_{B_r}} = {\mathbf{T}}_{F|_{{{\bm{\Psi}}}(B_r)}}$ and we can use the Taylor expansion in [@DS2] to compute: $$\begin{aligned} \label{e:mass1} {{\mathbf{M}}}({\mathbf{T}}_{{{\mathpzc{F}}}|_{B_r}}) &\geq Q {{\mathcal{H}}}^2({{\bm{\Psi}}}(B_r))+\frac{1}{2}\int_{{{\bm{\Psi}}}(B_r)}|D N|^2 - Q\int_{{{\bm{\Psi}}}(B_r)} \langle {{\bm{\eta}}}\circ N,H_{{\mathcal{M}}}\rangle\notag\\ &\quad -C\int_{{{\bm{\Psi}}}(B_r)} \Big(|A_{{{\mathcal{M}}}}|^2 | N|^2+|DN|^4\Big)\, ,\end{aligned}$$ where $H_{{{\mathcal{M}}}}$ denotes the mean curvature vector of ${{\mathcal{M}}}$. Note that in order to apply the Taylor expansion in [@DS2] we need the manifold ${{\mathcal{M}}}$ to be $C^2$, with an apriori bound on the $C^2$ norm. However, if we take $T_F {\mathop{\hbox{\vrule height 7pt width .3pt depth 0pt \vrule height .3pt width 5pt depth 0pt}}\nolimits}{{\mathbf{B}}}_r \setminus {{\mathbf{B}}}_{r/2}$ and rescale by a factor $1/r$, the corresponding rescaled current, map and manifold fall under the assumptions of the Taylor expansion in [@DS2]. We can then scale back to find the corresponding inequalities for $T{\mathop{\hbox{\vrule height 7pt width .3pt depth 0pt \vrule height .3pt width 5pt depth 0pt}}\nolimits}{{\mathbf{B}}}_r \setminus {{\mathbf{B}}}_{r/2}$ and sum over dyadic annuli to conclude . Using the conformality of ${{\bm{\Psi}}}$ we conclude $$\int_{{{\bm{\Psi}}}(B_r)}|D N|^2 = \int_{B_r} |D{{\mathpzc{N}}}|^2\, ,$$ As for the other terms, we recall $$\begin{aligned} &\int_{{{\bm{\Psi}}}(B_r)} |\langle {{\bm{\eta}}}\circ N, H_{{{\mathcal{M}}}}\rangle| \leq C {{\bm m}_0}^{\sfrac{1}{2}} \int_{B_r} |{{\bm{\eta}}}\circ {{\mathpzc{N}}}| \stackrel{\eqref{e:media_pesata}}{\leq} C \textup{Err}_1 ({{\mathpzc{N}}}, B_r)\, , \\ &\int_{{{\bm{\Psi}}}(B_r)} |DN|^4 \leq C {{\rm {Lip}}}({{\mathpzc{N}}}|_{B_r})^2 \int_{B_r} |D{{\mathpzc{N}}}|^2 \stackrel{\eqref{e:Lip_N}}{\leq} C \textup{Err}_1 ({{\mathpzc{N}}}, B_r)\, ,\\ &\int_{{{\bm{\Psi}}}(B_r)} |A_{{\mathcal{M}}}|^2 |N|^2 \leq C {{\bm m}_0}\int_{B_r} |z|^{2\gamma_0-2} |{{\mathpzc{N}}}|^2 = C {{\bm m}_0}\int_0^r \frac{{{\mathbf{H}}}(s)}{s^{2-2\gamma_0}}\, ds \leq C \textup{Err}_1 ({{\mathpzc{N}}}, B_r)\, .\end{aligned}$$ Combining the latter estimates with and we achieve $$\label{e:competitore2} \frac{1}{2} \int_{B_r} |D{{\mathpzc{N}}}|^2 \leq C \textup{Err}_1 ({{\mathpzc{N}}}, B_r) + {{\mathbf{M}}}({\mathbf{T}}_G) - Q {{\mathcal{H}}}^2 ({{\bm{\Psi}}}(B_r (x))\, .$$ Next, fix an orthonormal frame $\xi_1, \xi_2$ on $B_r$ and, using the area formula from [@DS2], compute $$\begin{aligned} {{\mathbf{M}}}({\mathbf{T}}_G)= & \int_{{{\bm{\Psi}}}(B_r)} \sum_i |(\xi_1 + DL_i \cdot \xi_1) \wedge (\xi_2 + DL_i \cdot \xi_2)|\\ \leq &\frac{1}{2} \int_{{{\bm{\Psi}}}(B_r)} \sum_i \left(|\xi_1 + DL_i \cdot \xi_1)|^2 + |\xi_2 + DL_i \cdot \xi_2|^2\right)\\ = & Q {{\mathcal{H}}}^2 ({{\bm{\Psi}}}(B_r)) + \frac{1}{2} \int_{{{\bm{\Psi}}}(B_r)} |DL|^2\\ &\quad + Q \int_{{{\bm{\Psi}}}(B_r)} \left( \langle D ({{\bm{\eta}}}\circ L) \cdot \xi_1, \xi_1\rangle + \langle D ({{\bm{\eta}}}\circ L) \cdot \xi_2, \xi_2 \rangle \right)\, .\end{aligned}$$ By conformality the second summand in the last inequality equals $\frac{1}{2}\int_{B_r} |D{{\mathpzc{L}}}|^2$. We integrate by parts the third summand. Recall that ${{\bm{\eta}}}\circ L={{\bm{\eta}}}\circ N$ on ${{\bm{\Psi}}}(\partial B_r) = \partial ({{\bm{\Psi}}}(B_r))$: since ${{\bm{\eta}}}\circ N$ is orthogonal to $\xi_i$ the boundary term vanishes. Moreover, since the origin is a singularity, we must in fact integrate by parts in $B_r\setminus B_{\varepsilon}$ and then let $\varepsilon \to 0$. A specific choice of $\xi_i$ is $\xi_i = \lambda^{-\sfrac{1}{2}} D {{\bm{\Psi}}}\cdot e_i$, where $e_1, e_2$ is the parallel frame on ${{\mathfrak{B}}}_Q$ naturally induced by the standard flat coordinates. It then turns out that $$|D_{\xi_1} \xi_1 + D_{\xi_2} \xi_2|({{\bm{\Psi}}}(z,w)) \leq C {{\bm m}_0}^{\sfrac{1}{2}} |z|^{\gamma_0-1}\, .$$ In particular $|D_{\xi_1} \xi_1 + D_{\xi_2} \xi_2|$ is integrable on $B_r$ and we can therefore conclude $$\begin{aligned} {{\mathbf{M}}}({\mathbf{T}}_G) - Q {{\mathcal{H}}}^2 ({{\bm{\Psi}}}(B_r)) \leq & \frac{1}{2} \int_{{{\bm{\Psi}}}(B_r)} |DL|^2 + Q \int_{{{\bm{\Psi}}}(B_r)} \langle {{\bm{\eta}}}\circ L, D_{\xi_1} \xi_1 + D_{\xi_2} \xi_2 \rangle\nonumber\\ \leq & \frac{1}{2} \int_{B_r} |D{{\mathpzc{L}}}|^2 + C \textup{Err}_2 ({{\mathpzc{L}}}, B_r)\, .\label{e:competitore3}\end{aligned}$$ Combining and we conclude . Proof of Proposition \[p:amin\] case (b): $T$ semicalibrated ------------------------------------------------------------ We proceed as in the previous step and define the current $Z$ as in . If $S$ is any current such that $$\partial S = T-Z = {\mathbf{T}}_{{{\mathpzc{F}}}|_{B_r}} - {\mathbf{T}}_{{\mathpzc{G}}}= {\mathbf{T}}_{F|_{{{\bm{\Psi}}}(B_r)}} - {\mathbf{T}}_G\, ,$$ then the semicalibrated condition gives $${{\mathbf{M}}}(T) \leq {{\mathbf{M}}}(Z) + S (d\omega)\, ,$$ where $\omega$ is the calibrating form. In particular, in order to conclude the proof it suffices to find an $S$ such that $$\label{e:errore_semical} |S (d\omega)| \leq C\,\textup{Err}_1({{\mathpzc{N}}},B_r) + C\,\textup{Err}_2({{\mathpzc{L}}},B_r)+\frac{C}{r}\int_{B_r}|{{\mathpzc{L}}}|^2\, :$$ combining the latter inequality with the estimates of the previous subsection we reach the desired inequality. We first define $H_i: [0,1] \times {{\bm{\Psi}}}(B_r) \to \I{Q}({{\mathbb R}}^{2+n})$ for $i=1,2$ by $$\begin{gathered} [0,1]\times {{\bm{\Psi}}}(B_r) \ni (t,p) \mapsto H_1(t,p):= \sum_{i=1}^{Q}\a{p+t\, N_i(p)}\in \I{Q}({{\mathbb R}}^{2+n})\notag\\ [0,1]\times {{\bm{\Psi}}}(B_r) \ni (t,p) \mapsto H_2(t,p):= \sum_{i=1}^{Q}\a{p+(1-t) \, L_i(p)}\in \I{Q}({{\mathbb R}}^{2+n})\notag\,.\end{gathered}$$ We choose $S:=S_1+S_2$, where $S_i:={\mathbf{T}}_{H_i}$ for $i=1,2$. Thanks to the homotopy formula in [@DS2], we get $$\begin{aligned} {\partial}S_1 & = {\mathbf{T}}_{F|_{{{\bm{\Psi}}}(B_r)}}-Q \a{{{\mathcal{M}}}} -{\mathbf{T}}_{H_1\vert_{[0,1]\times {{\bm{\Psi}}}(\partial B_r)} },\\ {\partial}S_2 & = Q\a{{{\mathcal{M}}}} - {\mathbf{T}}_{G|_{{{\bm{\Psi}}}(B_r)}} + {\mathbf{T}}_{H_2\vert_{[0,1]\times {{\bm{\Psi}}}(\partial B_r)}} .\end{aligned}$$ On the other hand since $N=L$ on ${{\bm{\Psi}}}(\partial B_r)$, we conclude $\partial S = \partial (S_1+S_2) = T-Z$. We next estimate $|S_1 (d\omega)|$ and $|S_2 (d\omega)|$. Since the estimates are analogous, we give the details only for the first. We start from the formula $$S_1(d\omega) =\int_{{{\bm{\Psi}}}(B_r)}\int_0^1 \sum_{i=1}^{Q} \big\langle \vec\zeta_i(t,p), d\omega((H_1)_i(t,p)) \big\rangle\, d{{\mathcal{H}}}^2(p)\,dt,$$ with $$\begin{aligned} \vec{\zeta}_i(t,p) & = \big(\xi_1+t\,\nabla_{\xi_1}N_i(p)\big)\wedge\big(\xi_2+t\,\nabla_{\xi_2} N_i(p)\big) \wedge N_i(p)\\ &=:\xi_1\wedge \xi_2 \wedge N_i(p) + \vec{E}_i(t, p)\,,\end{aligned}$$ and $$\begin{aligned} |\vec E_i(t,p)| \leq C\,(|D N|(p) + |DN|^2 (p))\,|N|(p).\end{aligned}$$ Next we note that $$\begin{aligned} d\omega((H_1)_i(t,p)) = d\omega(p) + I(t,p),\end{aligned}$$ where $I(t,p)$ can be estimated by $$\begin{aligned} |I(t,p)| & = |d\omega((H_1)_i(t,p)) - d\omega(p)| \leq C\, \|D^2\omega\|_{L^\infty}\, | N|(p).\end{aligned}$$ Therefore, we have $$\begin{aligned} &\Big\vert\sum_{i=1}^{Q}\big\langle \vec\zeta_i(t,p), d\omega((H_1)_i(t,p)) \big\rangle \Big\vert \leq \sum_{i=1}^{Q}\langle\xi_1\wedge \xi_2 \wedge N_i(p), d\omega(p) \rangle + \|d\omega\|_{L^\infty} \sum_{i=1}^{Q} |\vec E_i(t,p)|\notag\\ &\quad + C\sum_{i=1}^{Q}\Big((|N_i| +|\vec E_i|)\, |I|\Big)(t,p)\notag\\ & \leq C {{\bm m}_0}^{\sfrac{1}{2}} \,|{{\bm{\eta}}}\circ N| + C |N|^2(p) + C |D N|(p)\,| N|(p) + Cr |DN|^2 (p)\, ,\notag\end{aligned}$$ where we have only used the bound $|N| (p) \leq C r$ on ${{\bm{\Psi}}}(B_r)$. Arguing similarly for $S_2$ (observe that we have the bound $|L| (p) \leq C r$) and estimating $|N||DN| + |L| |DL| \leq r^{-1} (|N|^2 + |L|^2) + Cr (|DN|^2 + |DL|^2)$, we achieve $$\begin{aligned} |S_1(d\omega)| + |S_2(d\omega)| &\leq C\,{{\bm m}_0}^{\sfrac{1}{2}} \int_{{{\bm{\Psi}}}(B_r)}\big(|{{\bm{\eta}}}\circ N| +|{{\bm{\eta}}}\circ L|\big)+ C\, r^{-1}\int_{{{\bm{\Psi}}}(B_r)}\big(| N|^2 +| L|^2\big)\\ &\quad + C r\int_{{{\bm{\Psi}}}(B_r)}\big(|D N|^2+|D L|^2\big),\end{aligned}$$ and we conclude as above by a change of variable and Theorem \[t:cm\]. Proof of Proposition \[p:amin\] in case (c): $T$ is the cross-section of a three dimensional area minimizing cone ----------------------------------------------------------------------------------------------------------------- Recall that in this case ${{\rm spt}}(T) \subset \partial {{\mathbf{B}}}_R (p_0)$, where $p_0 = (0, \ldots , 0, R) = R e_{n+2}$ and $R^{-1} \leq {{\bm m}_0}^{\sfrac{1}{2}}$. For the computations of this subsection it is indeed convenient to change coordinates so that $p_0$ is in fact the origin, whereas $\Psi (0,0)$ is the point $(0, \ldots , 0, -R)$. In these new coordinates we then have ${{\mathcal{M}}}, {{\rm spt}}(T), {{\rm Im}}({{\mathpzc{F}}}) \subset \partial {{\mathbf{B}}}_R (0)$. These coordinates will however be used only in here, whereas in the next sections we will return to the usual ones. We introduce the following notation: ${{\mathcal{C}}}(r)$ is the cone over ${{\bm{\Psi}}}(B_r)$ with vertex $0$, i.e. $${{\mathcal{C}}}(r):=\big\{\rho p \in {{\mathbb R}}^{n+2}\,:\, \rho\in [0,1], \,p\in {{\bm{\Psi}}}(B_r)\big\},$$ with the orientation compatible with that of $0 {{\times\hspace{-0.6em}\times\,}}\a{{{\mathcal{M}}}}$. We extend $F$ to $\tilde{F}: {{\mathcal{C}}}(r)\to {{\mathcal{A}}_Q}({{\mathbb R}}^{n+2})$ by setting $\tilde F (\rho p) := \rho\, F(p)$ for every $p\in {{\bm{\Psi}}}(B_r)$. In order to estimate the Dirichlet energy of $N$ in terms of that of $L$, we construct a suitable function $K: {{\mathcal{C}}}(r) \to \I{Q}({{\mathbb R}}^{n+2})$ (depending on $L$ and $N$) such that $K \vert_{{\partial}{{\mathcal{C}}}(r)} = \tilde{F} \vert_{{\partial}{{\mathcal{C}}}(r)}$: we can then test the minimizing property of $0 {{\times\hspace{-0.6em}\times\,}}T$ comparing its mass with that of the current $$Z:= 0{{\times\hspace{-0.6em}\times\,}}T - {\mathbf{T}}_{\tilde{F}} + {\mathbf{T}}_K = 0 {{\times\hspace{-0.6em}\times\,}}(T- {\mathbf{T}}_{F|_{{{\bm{\Psi}}}(B_r)}}) + {\mathbf{T}}_K$$ which is easily recognized to satisfy ${\partial}Z = {\partial}(0{{\times\hspace{-0.6em}\times\,}}T)$. In particular, using the minimality of $0{{\times\hspace{-0.6em}\times\,}}T$, we conclude $$\label{e:minimalita'_cono} R^{-1} {{\mathbf{M}}}(0{{\times\hspace{-0.6em}\times\,}}{\mathbf{T}}_{F|_{{{\bm{\Psi}}}(B_r)}}) \leq R^{-1} {{\mathbf{M}}}({\mathbf{T}}_K) + C \textup{Err}_1 ({{\mathpzc{N}}}, B_r)\, .$$ We consider the space of parameters $[0,1] \times B_r$ and recall that the points in ${{\mathfrak{B}}}_{Q}$ are identified by two complex coordinates $(z,w) \in {{\mathbb C}}\times {{\mathbb C}}$. For the definition of $K$ we need to introduce the following sets $$\begin{aligned} A_1 := & \left\{(\rho, z, w) \in [0,1] \times B_r : 1-r\leq \rho \leq 1, \; |z| \leq \frac{\rho + 2\,r -1}{2}\right\},\\ A_2 := &\left\{(\rho, z, w) \in [0,1] \times B_r : 1-2\,r\leq \rho \leq 1 -r, \; |z| \leq \frac{1-\rho}{2}\right\},\\ B:= & [1-2\,r, 1] \times B_r \setminus \big( A_1 \cup A_2 \big),\end{aligned}$$ We then define the function ${{\mathpzc{H}}}:[0,1] \times B_r \to \I{Q}({{\mathbb R}}^{n+2})$ given by $$\label{e:competitor} {{\mathpzc{H}}}(\rho,z,w) := \begin{cases} \rho\, {{\mathpzc{L}}}(z,w) & \text{if } \rho \leq 1 - 2\, r,\\ \rho\,l_1(\rho)\, {{\mathpzc{N}}}\left(\frac{2\,r\,z}{\rho+2r -1},\frac{(2\,r)^{\sfrac1{Q}}}{(\rho + 2\,r -1)^{\sfrac1{Q}}}w\right) & \text{if } (\rho,z,w) \in A_1,\\ - \rho\,l_1(\rho)\, {{\mathpzc{L}}}\left(\frac{2\,r\,z}{1-\rho},\frac{(2\,r)^{\sfrac1{Q}}}{(1-\rho)^{\sfrac1{Q}}}w\right) & \text{if } (\rho,z,w) \in A_2,\\ \rho\,l_2(|z|)\, {{\mathpzc{N}}}\left(\frac{r\,z}{|z|},\frac{r^{\sfrac1{Q}}}{|z|^{\sfrac1{Q}}}w\right) & \text{if } (\rho,z,w) \in B, \end{cases}$$ where $l_1, l_2 : {{\mathbb R}}\to {{\mathbb R}}$ are the affine functions $$l_1(t) := \frac{t+r-1}{r}\quad \text{and}\quad l_2(t) := \frac{2\,t-r}{r}.$$ The following are simple properties of ${{\mathpzc{H}}}$ which can be easily verified: - ${{\mathpzc{H}}}(1,z,w) = {{\mathpzc{N}}}(z,w)$ for every $(z,w) \in B_r$, as $(1, z, w) \in A_1$ and $l_1(1)=1$; - ${{\mathpzc{H}}}(\rho, z, w) = \rho\,{{\mathpzc{N}}}(z,w)$ for every $\rho \in [0,1]$ and for every $z$ with $|z| = r$, as ${{\mathpzc{L}}}\vert_{{\partial}B_r} = {{\mathpzc{N}}}\vert_{{\partial}B_r}$ and $l_2(r) =1$; - ${{\mathpzc{H}}}$ is well-defined and continuous, as ${{\mathpzc{H}}}\equiv 0$ in $A_1 \cap A_2$ from $l_1(1-r) =0$, $${{\mathpzc{H}}}(\rho, z,w) = \textstyle{ \rho \,\frac{\rho + r -1}{r}\, {{\mathpzc{N}}}\left(\frac{r\,z}{|z|},\frac{r^{\sfrac1{Q}}}{|z|^{\sfrac1{Q}}}z\right) \quad \text{in } A_1\cap {\partial}B, }$$ and $${{\mathpzc{H}}}(\rho, z,w) = \textstyle{ \rho \,\frac{\rho + r -1}{r}\, {{\mathpzc{N}}}\left(\frac{r\,z}{|z|},\frac{r^{\sfrac1{Q}}}{|z|^{\sfrac1{Q}}}w\right) \quad \text{in } A_2\cap {\partial}B. }$$ The competitor map $K: {{\mathcal{C}}}(r) \to \I{Q}({{\mathbb R}}^{n+2})$ is now given by $$K(\rho\,p) := \sum_{i=1}^Q \a{\rho\,p + H_i(\rho\,p)} \quad \text{with } H(\rho\, p) := {{\mathpzc{H}}}(\rho, {{\bm{\Psi}}}^{-1}(p)).$$ Note that by (1) and (2) above it follows that $K\vert_{{\partial}{{\mathcal{C}}}(r)} = \tilde F\vert_{{\partial}{{\mathcal{C}}}(r)}$. We start now estimating the masses of the various currents introduced above. Since ${{\rm spt}}({\mathbf{T}}_F) \subset \partial {{\mathbf{B}}}_R (0)$, it follows that ${{\mathbf{M}}}(0{{\times\hspace{-0.6em}\times\,}}{\mathbf{T}}_F) = R {{\mathbf{M}}}({\mathbf{T}}_F) /3$ and, by the expansion of the mass of ${\mathbf{T}}_F$, we have that $$\begin{aligned} \label{e:mass F} {{\mathbf{M}}}({\mathbf{T}}_{F|_{{{\bm{\Psi}}}(B_r)}}) & \geq Q\, {{\mathcal{H}}}^2({{\bm{\Psi}}}(B_r)) + \frac{1}{2} \int_{B_r} |D{{\mathpzc{N}}}|^2 - C \textup{Err}_1 ({{\mathpzc{N}}}, B_r)\, .\end{aligned}$$ Combining the latter estimate with we conclude $$\label{e:minimalita'_cono_2} \int_{B_r} |D{{\mathpzc{N}}}|^2 \leq 6R^{-1} {{\mathbf{M}}}({\mathbf{T}}_K) - 2Q {{\mathcal{H}}}^2 ({{\bm{\Psi}}}(B_r)) + C\,\textup{Err}_1 ({{\mathpzc{N}}}, B_r)\, .$$ For what concerns the mass of ${\mathbf{T}}_K$, recalling that $p+{{\rm spt}}(L(p))\in \partial {{\mathbf{B}}}_R (0)$ for every $p \in {{\bm{\Psi}}}(B_r)$, we deduce that $${{\mathbf{M}}}({\mathbf{T}}_K{\mathop{\hbox{\vrule height 7pt width .3pt depth 0pt \vrule height .3pt width 5pt depth 0pt}}\nolimits}{{\mathbf{B}}}_{R (1-2r})) = {{\mathbf{M}}}(0 {{\times\hspace{-0.6em}\times\,}}{\mathbf{T}}_G {\mathop{\hbox{\vrule height 7pt width .3pt depth 0pt \vrule height .3pt width 5pt depth 0pt}}\nolimits}{{\mathbf{B}}}_{R (1-2r)}) = R \frac{(1-2r)^3{{\mathbf{M}}}({\mathbf{T}}_G)}{3}$$ and $${{\mathbf{M}}}({\mathbf{T}}_G) \leq Q\, {{\mathcal{H}}}^2({{\bm{\Psi}}}(B_r)) + \frac{1}{2} \int_{B_r} |D{{\mathpzc{L}}}|^2 + \textup{Err}_2 ({{\mathpzc{L}}}, B_r)\, .$$ In particular we conclude $$\label{e:stima_cono_1} 6 R^{-1} {{\mathbf{M}}}({\mathbf{T}}_K{\mathop{\hbox{\vrule height 7pt width .3pt depth 0pt \vrule height .3pt width 5pt depth 0pt}}\nolimits}{{\mathbf{B}}}_{R(1-2r)}) \leq 2Q (1-2r)^3 {{\mathcal{H}}}^2 ({{\bm{\Psi}}}(B_r)) + \int_{B_r} |D{{\mathpzc{L}}}|^2 + \textup{Err}_2 ({{\mathpzc{L}}}, B_r)\, .$$ Next we pass to estimating ${{\mathbf{M}}}({\mathbf{T}}_K {\mathop{\hbox{\vrule height 7pt width .3pt depth 0pt \vrule height .3pt width 5pt depth 0pt}}\nolimits}{{\mathbf{B}}}_R\setminus {{\mathbf{B}}}_{R(1-2r)})$. In order to carry on our estimates we use the area formula for multifunctions, cf. [@DS2]. In particular we fix an orthonormal frame $\xi_1, \xi_2$ for ${{\mathcal{M}}}$ as in the proof of case (a) and we let $\xi_3 = R^{-1} \partial_t$ be normal to them in $T {{\mathcal{C}}}(r)$, i.e. pointing in the radial direction of the cone. We then have $${{\mathbf{M}}}({\mathbf{T}}_K {\mathop{\hbox{\vrule height 7pt width .3pt depth 0pt \vrule height .3pt width 5pt depth 0pt}}\nolimits}({{\mathbf{B}}}_R\setminus {{\mathbf{B}}}_{R(1-2r)}) = \int_{{{\mathcal{C}}}(r)} \sum_i \underbrace{|(\xi_1 + DH_i \cdot \xi_1)\wedge (\xi_2 + DH_i \cdot \xi_2)\wedge (\xi_3 + DH_i\cdot \xi_3)|}_{(A)}\, .$$ Using the Taylor expansion for $(A)$, cf. [@DS2], we can bound $$\begin{aligned} \label{e:stima K} & R^{-1} {{\mathbf{M}}}({\mathbf{T}}_K {\mathop{\hbox{\vrule height 7pt width .3pt depth 0pt \vrule height .3pt width 5pt depth 0pt}}\nolimits}({{\mathbf{B}}}_R\setminus {{\mathbf{B}}}_{R(1-2r)})) \leq Q R^{-1}\, {{\mathcal{H}}}^3\big({{\mathcal{C}}}(r) \cap {{\mathbf{B}}}_1 \setminus {{\mathbf{B}}}_{1-2r}\big)\nonumber\\ &\qquad+ Q\, R^{-1} \int_{1-2r}^1 \int_{{{\bm{\Psi}}}(B_r)} \frac{d}{dt} [({{\bm{\eta}}}\circ H)(t p)] t^2 dt \notag\\ &\qquad + Q\, R^{-1} \int_{1-2r}^1 \int_{{{\bm{\Psi}}}(B_r)} \sum_{i=1}^2 \langle \nabla_{\xi_i} ({{\bm{\eta}}}\circ H), \xi_i\rangle \, t^2 dt + CR^{-1} \int_{1-2r}^1 \int_{{{\bm{\Psi}}}(B_r)} |D H|^2\,t^2 dt\, .\end{aligned}$$ The linear terms can be integrated by parts: since $\nabla_p ({{\bm{\eta}}}\circ H)(tp) = \frac{d}{dt}({{\bm{\eta}}}\circ H)(tp)$, we have $$\begin{aligned} \label{e:linear 1} \int_{1-2r}^1 \int_{{{\bm{\Psi}}}(B_r)} \frac{d}{dt} [({{\bm{\eta}}}\circ H)(t p)] t^2 dt & = \int_{{{\bm{\Psi}}}(B_r)} \left\langle ({{\bm{\eta}}}\circ H)(p) - (1-2r)^2({{\bm{\eta}}}\circ H)\big((1-2r)p\big), p \right\rangle\notag\\ &\quad - 2\int_{1-2r}^1\int_{{{\bm{\Psi}}}(B_r)} \langle ({{\bm{\eta}}}\circ H)(t p), p\rangle\,t dt\end{aligned}$$ $$\begin{aligned} \label{e:linear 2} \int_{1-2r}^1 \int_{{{\bm{\Psi}}}(B_r)} \sum_{i=1}^2 \langle \nabla_{\xi_i} ({{\bm{\eta}}}\circ H), \xi_i\rangle \,t^2 dt & = - \int_{1-2r}^1 \int_{{{\bm{\Psi}}}(B_r)} \langle ({{\bm{\eta}}}\circ H), H_{{{\mathcal{M}}}}\rangle\, t^2 dt.\end{aligned}$$ Therefore, by a simple change of coordinates we can estimate $$\begin{aligned} \label{e:stima K bis} &R^{-1} {{\mathbf{M}}}({\mathbf{T}}_K {\mathop{\hbox{\vrule height 7pt width .3pt depth 0pt \vrule height .3pt width 5pt depth 0pt}}\nolimits}({{\mathbf{B}}}_R\setminus {{\mathbf{B}}}_{R(1-r)})) \leq \frac{Q\, \big(1 - (1-2r)^{3}\big)}{3}\, {{\mathcal{H}}}^2\big({{\bm{\Psi}}}(B_r)\big)\\ &\qquad + C {{\bm m}_0}^{\sfrac{1}{2}} \int_{B_r} \big(|{{\bm{\eta}}}\circ {{\mathpzc{N}}}| + |{{\bm{\eta}}}\circ {{\mathpzc{L}}}|\big) + C {{\bm m}_0}^{\sfrac{1}{2}} \int_{1-2r}^1 \int_{B_r} |D {{\mathpzc{H}}}|^2(t,z,w) \,dz\, dt\\ &\qquad + C{{\bm m}_0}^{\sfrac{1}{2}}\int_{1-2r}^1 \int_{B_r} |z|^{\gamma_0 -1} |{{\bm{\eta}}}\circ {{\mathpzc{H}}}|(t, z, w)dz\, dt\, .\end{aligned}$$ In order to bound the various integrands of , we start with the following general remark. Assume that $\chi: [1-2r,1] \times B_r \to [0,+\infty)$ has the structure $$\label{e:chi} \chi(\rho,x,y) = \begin{cases} \chi_1\left(\frac{2\,r\,z}{\rho+2r -1},\frac{(2\,r)^{\sfrac1{Q}}}{(\rho + 2\,r -1)^{\sfrac1{Q}}}w\right) & \text{if } (\rho,z,w) \in A_1,\\ \chi_2\left(\frac{2\,r\,z}{1-\rho},\frac{(2\,r)^{\sfrac1{Q}}}{(1-\rho)^{\sfrac1{Q}}}w\right) & \text{if } (\rho,z,w) \in A_2,\\ \chi_3\left(\frac{r\,z}{|z|},\frac{r^{\sfrac1{Q}}}{|z|^{\sfrac1{Q}}}w\right) & \text{if } (\rho,z,w) \in B, \end{cases}$$ for some $\chi_1, \chi_2, \chi_3:B_r \to [0,+\infty)$. Then one can compute the integral of $\chi$ in the following way: $$\int_{1-2r}^1 \int_{B_{r}} \chi(t,z,w)\,dz\,dt = \int_{A_1}\chi(t,z,w)\,dz\,dt + \int_{A_2}\chi(t,z,w)\,dz\,dt + \int_{B}\chi(t,z,w)\,dz\,dt,$$ and one can easily compute that $$\begin{aligned} &\int_{A_1} \chi (t,z,w)\, dz\, dt = \int_{1-r}^1 \int_{B_{\frac{t+2r-1}{2}}} \chi_1(t,z,w)\, dz\, dt\notag\\ =& \int_{1-r}^1 \int_{B_{\frac{t+2r-1}{2}}} \chi_1\left(\frac{2\,r\,z}{t+2r -1},\frac{(2\,r)^{\sfrac1{Q}}}{(t + 2\,r -1)^{\sfrac1{Q}}}w\right)dz\, dt\notag\\ =& \int_{1-r}^1\left(\frac{t+2r-1}{2r}\right)^2 \int_{B_{r}} \chi_1(z,w) dz\, dt \leq r \int_{B_{r}} \chi_1(z,w) dz\, dt\, .\label{e:int A1}\end{aligned}$$ Similarly $$\begin{aligned} \label{e:int A2} \int_{A_2} \chi (t,z,w) dz\, dt \leq r \int_{B_{r}} \chi_2(z,w)\, dt,\end{aligned}$$ and $$\begin{aligned} \label{e:int B} \int_{B} \chi(t,z,w) dz\, dt= & \int_{1-r}^1 dt \int^{r}_{\frac{t+2r-1}{2}} \frac{s}{r}\, ds \int_{{\partial}B_r} \chi_3(z,w)\,dz\nonumber\\ &\quad + \int_{1-2r}^{1-r} \int_{\frac{1-t}{2}}r \frac{s}{r}\, ds\int_{{\partial}B_r} \chi_3(z,w)\,dz \leq r^2\int_{{\partial}B_{r}} \chi_3(z,w)\,dz\, .\end{aligned}$$ By direct computations one verifies that the integrands in are all bounded from above by functions $\chi$ with the structure : in particular, - $|z|^{\gamma_0 -1} |{{\bm{\eta}}}\circ {{\mathpzc{H}}}|(t,z,w) \leq \chi(t,z,w)$ if we choose $$\chi_1(z,w) = \chi_3(z,w) = |z|^{\gamma_0 -1} |{{\bm{\eta}}}\circ {{\mathpzc{N}}}|(z,w) \quad \text{and}\quad \chi_2(z,w) = |z|^{\gamma_0 -1} |{{\bm{\eta}}}\circ {{\mathpzc{L}}}|(x,y);$$ - $|D{{\mathpzc{H}}}|^2(t,z,w)\leq \chi(t,z,w)$ if we choose $$\begin{gathered} \chi_1(z,w) = \chi_3(z,w) = \frac{C}{r^2} |{{\mathpzc{N}}}|^2(z,w) + C\, |D{{\mathpzc{N}}}|^2(z,w)\\ \chi_2(z,w) = \frac{C}{r^2} |{{\mathpzc{L}}}|^2(z,w) + C\,|DL|^2(z,w).\end{gathered}$$ for some dimensional constant $C>0$. It then turns out from , , and (i), (ii), (iii) that $$\begin{aligned} \label{e:stima K tris} 6 R^{-1} {{\mathbf{M}}}({\mathbf{T}}_K {\mathop{\hbox{\vrule height 7pt width .3pt depth 0pt \vrule height .3pt width 5pt depth 0pt}}\nolimits}({{\mathbf{B}}}_R\setminus {{\mathbf{B}}}_{R(1-r)}))& \leq Q\, \big(1 - (1-2r)^{3}\big)\, {{\mathcal{H}}}^2\big({{\bm{\Psi}}}(B_r)\big)\notag\\ &\qquad + C\, \textup{Err}_1({{\mathpzc{N}}}, B_r) + C\, \textup{Err}_2 ({{\mathpzc{L}}}, B_r)\, .\end{aligned}$$ Summing and we conclude $$6 R^{-1} {{\mathbf{M}}}({\mathbf{T}}_K) \leq 2 Q {{\mathcal{H}}}^2 ({{\bm{\Psi}}}(B_r)) + \int_{B_r} |D{{\mathpzc{L}}}|^2 + C\, \textup{Err}_1 ({{\mathpzc{N}}}, B_r) + C\, \textup{Err}_2 ({{\mathpzc{L}}}, B_r)\, .$$ Combining the latter estimate with we conclude the proof. Harmonic competitor {#s:harmonic} =================== The most natural choice for the competitor ${{\mathpzc{L}}}$ is a suitable “harmonic” extension of the boundary value ${{\mathpzc{N}}}|_{\partial B_r}$. Following the ideas of [@Chang] we estimate carefully the energy of such competitor. To this purpose it is useful to introduce “polar” coordinates with center $0$ in ${{\mathfrak{B}}}$ and split accordingly the Dirichlet integrand in radial and angular parts. More precisely, consider $(z_0,w_0) = ((\xi_0, \zeta_0), w_0) \in \partial B_r$ and take, locally, the standard flat coordinates $z = (x_1, x_2)$ of Definition \[d:Riemann\_surface\]. We then denote by $\nu$ the exterior unit vector normal to $\partial B_r$ at $(z_0, w_0)$ and by $\tau$ the corresponding tangent unit vector obtained by rotating $\nu$ of an angle $\pi/2$ in the counterclockwise direction, namely $$\nu := |z_0|^{-1} \left( \xi_0 \frac{\partial}{\partial x_1} + \zeta_0 \frac{\partial}{\partial x_2}\right) \qquad \mbox{and}\qquad \tau := |z_0|^{-1} \left( - \zeta_0 \frac{\partial}{\partial x_1} + \xi_0 \frac{\partial}{\partial x_2}\right)\, .$$ The directional derivatives of any (multi)function $f$ on ${{\mathfrak{B}}}$ gives then two (multi)functions $$D_\nu f = \sum_i \a{Df_i\cdot \nu}\qquad \mbox{and} \qquad D_\tau f = \sum_i \a{Df_i \cdot \tau}\, .$$ The Dirichlet integrand $|Df|^2$ enjoys then the splitting $$|Df|^2 = |D_\nu f|^2 + |D_\tau f|^2\, .$$ For the rigorous justification of these identities see [@DS1]. \[p:harmonic\] There are constants $C>0$, $\sigma >0$ such that, for every $r\in (0,1)$ there exists a competitor ${{\mathpzc{L}}}\colon B_r\to \I{Q} ({{\mathbb R}}^{2+{n}})$ for ${{\mathpzc{N}}}$ with the following additional properties: - ${{\rm {Lip}}}({{\mathpzc{L}}}) \leq C_{\ref{p:amin}}^{-1} $, $\|{{\mathpzc{L}}}\|_0 \leq C r$. - The following estimates hold: $$\begin{gathered} \int_{B_r}|D\bar {{\mathpzc{L}}}|^2\leq C\, r \int_{{\partial}B_r} |D\bar {{\mathpzc{N}}}|^2 \leq C r {{\mathbf{D}}}' (r) \, , \label{e:D<rD'}\\ \int_{B_r}|z|^{\gamma_0-1}|{{\bm{\eta}}}\circ {{\mathpzc{L}}}| \leq C\,r^{\gamma_0} \int_{{\partial}B_r} |{{\bm{\eta}}}\circ {{\mathpzc{N}}}| + C\, {{\mathbf{H}}}(r) \, . \label{e:comp_mean_H}\end{gathered}$$ - For every $a>0$ there exists $b_0>0$ such that, for all $b \in (0, b_0)$, the following estimate holds: $$\begin{aligned} \label{e:harmonic energy} (2\,a+b)\int_{B_r} |D\bar {{\mathpzc{L}}}|^2 & \leq r \int_{{\partial}B_r} |D_\tau {{\mathpzc{N}}}|^2 + \frac{a\,(a+b)}{r}\int_{{\partial}B_r} |{{\mathpzc{N}}}|^2 + C r^{1+\sigma} {{\mathbf{D}}}' (r)\, .\end{aligned}$$ Using this competitor in Proposition \[p:amin\], we then infer the following corollary. \[c:amin\_ineq\] For every $r\in (0,1)$ the following inequality holds $$\begin{aligned} \label{e:AM1 bis} {{\mathbf{D}}}(r) & \leq C\,r\,{{\mathbf{D}}}'(r) + C\, {{\mathbf{H}}}(r)+ C\,{{\mathbf{F}}}(r) +C\,{{\bm m}_0}^{\sfrac12}\, r^{\gamma_0} \int_{{\partial}B_r}|{{\bm{\eta}}}\circ {{\mathpzc{N}}}|\,.\end{aligned}$$ For every $a>0$ there exists $b_0>0$ such that, for all $b \in (0, b_0)$ and all $r\in ]0,1[$ $$\begin{aligned} \label{e:AM2 bis} {{\mathbf{D}}}(r) &\leq (1+Cr) \left[\frac{r}{\,(2\,a+b)}\, \int_{{\partial}B_r} |D_\tau {{\mathpzc{N}}}|^2 + \frac{a\,(a+b)}{r\,(2\,a+b)}{{\mathbf{H}}}(r)\right]+C\, \mathcal{E}_{QM}(r)+ C r^{1+\sigma} {{\mathbf{D}}}' (r)\,,\end{aligned}$$ with $$\mathcal{E}_{QM}(r)\leq {{\mathbf{\Lambda}}}(r)^{\eta_0} {{\mathbf{D}}}(r) + {{\mathbf{F}}}(r) + {{\mathbf{H}}}(r) + {{\bm m}_0}^{\sfrac12}\, r^{\gamma_0} \int_{{\partial}B_r}|{{\bm{\eta}}}\circ {{\mathpzc{N}}}|\, .$$ Recalling that ${{\mathbf{H}}}(r) \leq C r \|{{\mathpzc{N}}}\|^2_{\partial B_r} \leq C r^{3+\gamma_0}$ we easily infer that ${{\mathbf{\Lambda}}}(r) \leq C r^2$ and thus the inequalities follow readily from Proposition \[p:amin\] and Proposition \[p:harmonic\]. Proof of Proposition \[p:harmonic\]: Step 1 ------------------------------------------- First of all we observe that it suffices to exhibit $\bar {{\mathpzc{L}}}$, as ${{\mathpzc{L}}}$ can be recovered from it via the formula . Moreover, it suffices to show the estimates with $\bar {{\mathpzc{N}}}$ in place of ${{\mathpzc{N}}}$ in the right hand side, because we obviously have $|\bar {{\mathpzc{N}}}|\leq |{{\mathpzc{N}}}|$ and $|D\bar {{\mathpzc{N}}}|\leq |D{{\mathpzc{N}}}|$. Next we wish to relate ${{\bm{\eta}}}\circ {{\mathpzc{L}}}$ and ${{\bm{\eta}}}\circ \bar {{\mathpzc{L}}}$ for two maps satisfying the relation . Note that by a simple Taylor expansion we have $$|{{\bm{\eta}}}\circ {{\mathpzc{L}}}|\leq C |{{\bm{\eta}}}\circ \bar {{\mathpzc{L}}}| + C {{\mathcal{G}}}(\bar {{\mathpzc{L}}}, {{\bm{\eta}}}\circ \bar {{\mathpzc{L}}})^2\, ,$$ where the constant $C$ depends on the $C^2$ norm of $\Psi_0$. In particular we record the following conclusion: $$\label{e:dettagliuccio} \int_{B_r} |z|^{\gamma_0-1} |{{\bm{\eta}}}\circ {{\mathpzc{L}}}| \leq C \int_{B_r} |z|^{\gamma_0-1} |{{\bm{\eta}}}\circ \bar {{\mathpzc{L}}}| + C \int_{B_r} |z|^{\gamma_0-1} |\bar {{\mathpzc{L}}}|^2\, .$$ In this step we exhibit an “harmonic”[^1] competitor ${{\mathpzc{H}}}$ which satisfies all the requirements of the proposition except for the Lipschitz estimate. In fact we will show that there is a $W^{1,2}$ map ${{\mathpzc{H}}}: B_r \to {{\mathcal{A}}_Q}({{\mathbb R}}^{2+\bar n})$ such that $$\begin{aligned} &{{\mathpzc{H}}}|_{\partial B_r} = \bar {{\mathpzc{N}}}|_{\partial B_r} \qquad \mbox{and}\qquad \|{{\mathpzc{H}}}\|_{L^\infty (B_r)} \leq Q \|\bar {{\mathpzc{N}}}\|_{L^\infty (\partial B_r)}\label{e:armonica1}\\ &\int_{B_r} |D{{\mathpzc{H}}}|^2 \leq C r \int_{\partial B_r} |D\bar {{\mathpzc{N}}}|^2 \label{e:armonica2}\\ &\int_{B_r} |z|^{\gamma_0-1} |{{\bm{\eta}}}\circ {{\mathpzc{H}}}| \leq C r^{\gamma_0} \int_{\partial B_r} |{{\bm{\eta}}}\circ \bar {{\mathpzc{N}}}|\label{e:armonica3}\\ &\int_{B_r} |z|^{\gamma_0-1} |{{\mathpzc{H}}}|^2 \leq C r^{\gamma_0} \int_{\partial B_r} |\bar {{\mathpzc{N}}}|^2\label{e:armonica5}\\ &(2\,a+b)\int_{B_r} |D\bar {{\mathpzc{H}}}|^2 \leq r \int_{{\partial}B_r} |D_\tau \bar {{\mathpzc{N}}}|^2 + \frac{a\,(a+b)}{r}\int_{{\partial}B_r} |\bar {{\mathpzc{N}}}|^2\, .\label{e:armonica4}\end{aligned}$$ In these estimates we do not use any of the particular properties of $\bar{{\mathpzc{N}}}$ and indeed for any Lipschitz multivalued map $\bar{{\mathpzc{N}}}: B_r \to {{\mathcal{A}}_Q}({{\mathbb R}}^{2+\bar{n}})$ there is such an “harmonic” competitor. Therefore, given the scaling invariance of the estimates, we will assume without loss of generality that $r=1$. Let $D_r :=\{|z|<r\}$ denote the disk of radius $r$ in ${{\mathbb R}}^2$, which we identify with the complex plane. We start by defining the “winding map” ${{\mathbf{W}}}: \bar D_1 \to {{\mathfrak{B}}}$ given (in complex notation) by $${{\mathbf{W}}}(z) := (z^{\bar Q}, z)\, .$$ We then consider the multivalued map ${{\mathpzc{U}}}:=\bar {{\mathpzc{N}}}\circ {{\mathbf{W}}}$. Let $\theta \mapsto u (\theta)$ be its trace on $\partial D_1 (0)$, which we parametrize with the angle $\theta\in [0,2\pi]$. According to [@DS1 Proposition 1.5] we can decompose $u$ in a superposition of simple functions $u (\theta)=\sum_{j=1}^J u_j(\theta)$ such that, for every $j=1,\dots,J$, $$u_j(\theta)=\sum_{i=1}^{Q_j}\a{\gamma_j\left(\frac{\theta+2\pi i}{Q_j}\right)}\, ,$$ where the $\gamma_j:[0, 2\pi]\to {{\mathbb R}}^{2+\bar n}$ are periodic Lipschitz functions. Next consider the Fourier’s expansion of each $\gamma_j$ $$\gamma_j(\theta)=\frac{a_{j,0}}{2}+\sum_{l=1}^\infty \left(a_{j,l}\cos(l\theta)+b_{j,l}\sin (l\theta)\right)\,,$$ and its harmonic extension, which in polar coordinates $(\rho, \theta)$ reads as $$\label{e:zeta_j} \zeta_j(\rho,\theta):=\frac{a_{j,0}}{2}+\sum_{l=1}^\infty \rho^l \big(a_{j,l}\cos(l\theta )+b_{j,l}\sin (l\theta)\big)\, .$$ We then can define the “harmonic” competitor for ${{\mathpzc{U}}}$, which is the $Q$-valued map $${{\mathpzc{V}}}(\rho, \theta) := \sum_{j=1}^J\sum_{i=1}^{Q_j}\a{\zeta_j\left(\rho^{\sfrac{1}{Q_j}}, \frac{\theta+2\pi i}{Q_j}\right)}$$ and the “harmonic” competitor for $\bar{{\mathpzc{N}}}$, which is ${{\mathpzc{H}}}= {{\mathpzc{V}}}\circ {{\mathbf{W}}}^{-1}$. Observe that the first claim in is obvious, whereas the second claim follows from the maximum principle for classical harmonic functions. Simple computations and the conformality of ${{\mathbf{W}}}$, see for instance [@DS1 Proof of Proposition 5.2], yield $$\begin{aligned} \int_{B_1} |D {{\mathpzc{H}}}|^2 = \int_{D_1} |D{{\mathpzc{V}}}|^2 = & \pi \sum_{j=1}^{J} \sum_{l=1}^\infty l \big(|a_{j,l}|^2+|b_{j,l}|^2\big)\,, \label{e:D}\\ \int_{{\partial}B_1} |D_\tau {{\mathpzc{H}}}|^2= &\frac{\pi}{\bar Q}\sum_{j=1}^J\sum_{l=1}^\infty \frac{l^2}{Q_j}\big(|a_{j,l}|^2+|b_{j,l}|^2\big)\,,\label{e:D'}\end{aligned}$$ $$\begin{aligned} \int_{{\partial}B_1} |{{\mathpzc{H}}}|^2=& \pi \bar Q \sum_{j=1}^JQ_j\Big(\frac{|a_{j,0}|^2}{2}+\sum_{l=1}^\infty \big(|a_{j,l}|^2+|b_{j,l}|^2\big)\Big)\label{e:h'}\, .\end{aligned}$$ Clearly, follows from the first and second inequality, with the constant $C = \bar{Q} Q_1 \leq \bar{Q} Q$, assuming that $Q_1 = \max \{Q_1, \ldots, Q_j\}$. follows from the fact that, for any chosen $a>0$, if $b_0$ is sufficiently small and $0<b< b_0$, then $$(2a+b) \ell \leq \frac{\ell^2}{\bar Q Q_j} + \bar{Q} Q_j \ell a (a+b) \qquad \forall \ell \in {{\mathbb N}}\, .$$ The latter claim is elementary and the reader can consult, for instance, Step 2 in the proof of [@DS1 Proposition 5.2]. Observe next that ${{\bm{\eta}}}\circ {{\mathpzc{V}}}$ is the classical harmonic extension of the single-valued function ${{\bm{\eta}}}\circ {{\mathpzc{U}}}|_{\partial D_1}$. We then have the classical estimates $$\|{{\bm{\eta}}}\circ {{\mathpzc{V}}}\|_{L^\infty (D_{2^{ \sfrac{1}{\bar Q} }})}+ \|{{\bm{\eta}}}\circ {{\mathpzc{V}}}\|_{L^1 (D_1)} \leq C \|{{\bm{\eta}}}\circ {{\mathpzc{U}}}\|_{L^1 (\partial D_1)}\, .$$ In particular we conclude easily $$\|{{\bm{\eta}}}\circ {{\mathpzc{H}}}\|_{L^\infty (B_{1/2})} + \|{{\bm{\eta}}}\circ {{\mathpzc{H}}}\|_{L^1 (B_1\setminus B_{1/2})} \leq C \int_{\partial B_1} |{{\bm{\eta}}}\circ \bar{{\mathpzc{N}}}|\, ,$$ because the change of variables ${{\mathbf{W}}}^{-1}$ is smooth on $B_1\setminus B_{1/2}$. The integrability of $|z|^{\gamma_0-1}$ on $B_1$ gives then $$\begin{aligned} \int_{B_1} |z|^{\gamma_0-1} |{{\bm{\eta}}}\circ {{\mathpzc{H}}}(z,w)|\, dz\leq & C \|{{\bm{\eta}}}\circ {{\mathpzc{H}}}\|_{L^\infty (B_{1/2})} + C\|{{\bm{\eta}}}\circ {{\mathpzc{H}}}\|_{L^1 (B_1\setminus B_{1/2})}\, , \end{aligned}$$ which in turn completes the proof of . A similar argument proves . Using the classical theory of single valued harmonic functions we see indeed that $\|\zeta_j\|_{L^2 (B_1)} + \|\zeta_j\|_{L^\infty (B_{1/2})} \leq C \|\gamma_j\|_{L^2 (\partial B_1)}$ and thus, using the fact that ${{\mathbf{W}}}$ is smooth on $B_1\setminus B_{1/2}$, we conclude that $$\|{{\mathpzc{H}}}\|^2_{L^\infty (B_{1/2})} + \|{{\mathpzc{H}}}\|^2_{L^2 (B_1\setminus B_{1/2})} \leq C \int_{\partial B_1} |\bar {{\mathpzc{N}}}|^2\, .$$ From this we easily conclude . Proof of Proposition \[p:harmonic\]: Step 2 ------------------------------------------- We keep the notation of the previous paragraphs and assume that $\bar {{\mathpzc{N}}}$ is defined in $B_1$, after scaling. The specific scaling that we are using is the one which preserves the Lipschitz constant and is given by $$\bar{{\mathpzc{N}}}(z,w) \mapsto r^{-1} \bar {{\mathpzc{N}}}\big(r z, r^{\sfrac{1}{\bar{Q}}} w\big)\,$$ and by abuse of notation we keep the symbols $\bar {{\mathpzc{N}}}$, $\bar {{\mathpzc{L}}}$, etc. for all the rescaled maps. Under this scaling we then have the estimates $\|\bar {{\mathpzc{N}}}\|_{L^\infty}\leq C {{\bm m}_0}^{\sfrac{1}{4}} r^{\gamma_0/2}$ and ${{\rm {Lip}}}(\bar {{\mathpzc{N}}}) \leq {{\mathbf{\Lambda}}}(r)^{\eta_0}$ and we want to show that we can modify ${{\mathpzc{H}}}$ to a competitor $\bar {{\mathpzc{L}}}$ with ${{\rm {Lip}}}(\bar{{\mathpzc{L}}})\leq C_{\ref{p:amin}}^{-1}$, satisfying $$\begin{aligned} &\bar {{\mathpzc{L}}}|_{\partial B_1} = \bar {{\mathpzc{N}}}|_{\partial B_1} \qquad \mbox{and}\qquad \|\bar {{\mathpzc{L}}}\|_{L^\infty (B_1)} \leq C \|\bar {{\mathpzc{N}}}\|_{L^\infty (\partial B_1)}\label{e:app_arm_1}\\ &\int_{B_1} |D\bar {{\mathpzc{L}}}|^2 \leq C(1+r^\sigma) \int_{B_1} |D{{\mathpzc{H}}}|^2 + C {{\mathbf{\Lambda}}}(r)^\sigma \int_{\partial B_1} |D\bar{{\mathpzc{N}}}|^2 \label{e:app_arm_2}\\ &\int_{B_r} |z|^{\gamma_0-1} |\bar{{\mathpzc{L}}}|^2 \leq C \int_{\partial B_1} |\bar {{\mathpzc{N}}}|^2\, \label{e:app_arm_5}\\ &\int_{B_1} |z|^{\gamma_0-1} |{{\bm{\eta}}}\circ \bar{{\mathpzc{L}}}| \leq C \int_{\partial B_1} |{{\bm{\eta}}}\circ \bar {{\mathpzc{N}}}|\, .\label{e:app_arm_3}\end{aligned}$$ Observe that the harmonic functions $\zeta_j$ defined in are Lipschitz in every ball $D_{1-t}$ for $0<t<1$ with an estimate of the form $$\label{e:Lipschitz_sciocca} \|D \zeta_j\|_{L^\infty (D_{1-t})} \leq \frac{C}{t} {{\rm {Lip}}}(\gamma_j) \leq \frac{C}{t} {{\rm {Lip}}}(\bar {{\mathpzc{N}}}) \leq \frac{C {{\mathbf{\Lambda}}}(r)^{\eta_0}}{t}\, .$$ They are not Lipschitz up to the boundary $\partial D_1$ because the Dirichlet to Neumann map $\gamma_j \to \frac{\partial \zeta_j}{\rho} (1, \cdot)$ does not map $L^\infty$ into $L^\infty$. However we have the estimate $$\|D \zeta_j \|_{L^p (D_1)} \leq C_p \|\gamma_j\|_{W^{1,p} (\partial D_1)} \leq C_p{{\mathbf{\Lambda}}}(r)^{\eta_0}\,$$ for every $p<\infty$. In particular, we can bound $$\|\zeta_j (1-t, \cdot)-\gamma_j \|_{W^{1,1} (\partial D_1)} \leq C_2 t^{\sfrac{1}{2}}{{\mathbf{\Lambda}}}(r)^{\eta_0} \, ,$$ which in turn implies $$\label{e:Lipschitz_scema} \max |\zeta_j (1-t, \theta) - \gamma_j (\theta)| \leq C_2 t^{\sfrac{1}{2}}{{\mathbf{\Lambda}}}(r)^{\eta_0}\, .$$ Choose $t:= {{\mathbf{\Lambda}}}(r)^{\sfrac{\eta_0}{2}}$ and define a new map $\xi_j$ as $$\xi_j (\rho, \theta) := \left\{ \begin{array}{ll} \zeta_j (\rho, \theta) \quad & \mbox{for $\rho\leq 1-t$}\\ \\ \textstyle{\frac{1-\rho}{t}} \zeta_j (1-t, \theta) + \textstyle{\frac{\rho - (1-t)}{t}} \gamma_j (\theta) \quad &\mbox{for $1-t \leq \rho \leq 1$.} \end{array} \right.$$ Now, and imply that $\|D\zeta_j\|\leq C {{\mathbf{\Lambda}}}(r)^{\sfrac{\eta_0}{2}}$. Moreover we obviously have $$\begin{aligned} \int_{D_1} |D\xi_j|^2 \leq & \int_{D_1} |D\zeta_j|^2 + C {{\mathbf{\Lambda}}}(r)^{\eta_0} \Big( \int_{\partial D_{1-t}} |D \zeta_j|^2 + \int_{\partial D_1} |D\gamma_j|^2\Big)\nonumber\\ \leq & \int_{D_1} |D\zeta_j|^2 + C r{{\mathbf{\Lambda}}}(r)^{\eta_0} \int_{\partial B_1} |D\gamma_j|^2\, .\label{e:int_en}\end{aligned}$$ We can now define two “intermediate” maps $${{\mathpzc{V}}}^0 (\rho, \theta) := \sum_{j=1}^J\sum_{i=1}^{Q_j}\a{\xi_j\left(\rho^{\sfrac{1}{Q_j}}, \frac{\theta+2\pi i}{Q_j}\right)}$$ and ${{\mathpzc{L}}}^0 := {{\mathpzc{V}}}^0\circ {{\mathbf{W}}}^{-1}$. It is then immediate to see that ${{\mathpzc{L}}}^0$ enjoys the bound ${{\rm {Lip}}}({{\mathpzc{L}}}^0) \leq C {{\mathbf{\Lambda}}}(r)^{\sfrac{\eta_0}{2}}$ on the domain $B_1\setminus B_{1/4}$ and that all the estimates , and . On the other hand the differential $D {{\mathpzc{L}}}^0$ is singular in the origin and in fact it is rather easy to see that we have the bound $$\label{e:higher} |D{{\mathpzc{L}}}^0 (z,w)|^2 \leq C |z|^{2-\sfrac{2}{(Q\bar Q)}} \int_{B_1} |D{{\mathpzc{L}}}^0|^2\, .$$ In order to produce $\bar {{\mathpzc{L}}}$ we need to smooth the singularity of ${{\mathpzc{L}}}^0$ at the origin. There are several ways to do this and we present here one possibility. First of all we fix $2<p< 2 Q \bar Q/(2 Q \bar Q -2)$ and observe that yields the estimate $$\label{e:higher2} \int_{B_{3/4}} |D{{\mathpzc{L}}}^0 (z,w)|^p \leq C \Big( \int_{B_1} |D{{\mathpzc{L}}}^0|^2 \Big)^{\sfrac{p}{2}}\, .$$ Next we define $$M |D{{\mathpzc{L}}}^0 (z,w)|:= \sup_{\rho< 1/4} \frac{1}{\rho^2} \int_{B_\rho (z,w)} |D{{\mathpzc{L}}}^0 (z,w)|$$ and let $$A := \{(z,w) : M |D{{\mathpzc{L}}}^0 (z,w)| \geq c_0\}$$ where $c_0$ is a constant to be chosen later. Observe that, given the Lipschitz bound for ${{\mathpzc{L}}}^0$ outside the origin, for $r$ sufficiently small the set $A$ is contained in $B_{1/2}$. Arguing as in the proof of [@DS1 Proposition 4.4] we have the Lipschitz estimate ${{\rm {Lip}}}({{\mathpzc{L}}}^0) \leq C c_0$ on $B_1\setminus A$, where $C$ is a dimensional constant. We can then use the Lipschitz extension of [@DS1 Theorem 1.7] to extend ${{\mathpzc{L}}}^0$ to $\bar {{\mathpzc{L}}}$ on $A$ so that ${{\rm {Lip}}}({{\mathpzc{L}}}) \leq C c_0$. Choosing $c_0$ accordingly we achieve the desired Lipschitz bound on $B_1$. As for and observe that the extension satisfies $$\|\bar {{\mathpzc{L}}}\|^2_{L^\infty (B_{1/2})} \leq C \|{{\mathpzc{H}}}\|^2_{L^\infty (B_{3/4})}$$ and coincides with ${{\mathpzc{L}}}_0$ on $B_1\setminus B_{1/2}$. As for , it would suffice to show that $|{{\bm{\eta}}}\circ \bar{{\mathpzc{L}}}|\leq C |{{\bm{\eta}}}\circ \bar {{\mathpzc{N}}}|$. This can be easily achieved in the following way: we make a Lipschitz extension of ${{\mathpzc{L}}}^0$, subtract from each sheet the average and then sum back to each sheet a Lipschitz extension of ${{\bm{\eta}}}\circ {{\mathpzc{L}}}^0$. As for we compute $$\begin{aligned} \int |D\bar {{\mathpzc{L}}}|^2 \leq & \int |D{{\mathpzc{L}}}^0|^2 + C c_0^2 |A| \leq \int |D{{\mathpzc{L}}}^0|^2 + C c_0^{2-p} \int_{B_{3/4}} |D{{\mathpzc{L}}}^0|^p\notag\\ \leq & \int |D{{\mathpzc{L}}}^0|^2 \Big(1+ C c_0^{2-p} \Big(\int |D{{\mathpzc{L}}}^0|^2\Big)^{\sfrac{p}{2}-1}\Big)\, .\label{e:int_en_2}\end{aligned}$$ Observe that $p/2 -1 > 0$ and that by and $$\int |D{{\mathpzc{L}}}^0|^2 \leq \int |D{{\mathpzc{H}}}|^2 + C {{\mathbf{\Lambda}}}(r)^{\sfrac{\sigma}{2}} \int_{\partial B_1} |D \bar {{\mathpzc{N}}}|^2 \leq C \int_{\partial B_1} |D \bar {{\mathpzc{N}}}|^2 \leq C r^{\sigma}\, .$$ so that $$\int_{B_1} |D\bar {{\mathpzc{L}}}|^2 \leq (1+C\, r^\sigma) \int_{B_1} |D{{\mathpzc{H}}}|^2 + C r^\sigma \int_{\partial B_1} |D\bar{{\mathpzc{N}}}|^2 \stackrel{\eqref{e:armonica2}}{\leq} \int_{B_1} |D{{\mathpzc{H}}}|^2 + C r^\sigma \int_{\partial B_1} |D\bar{{\mathpzc{N}}}|^2\, .$$ Outer variations and the poincaré inequality {#s:poincare} ============================================ In this section we begin to exploit the first variations of the area functional on $T$ in conjunction with the estimates of the previous section. The main conclusion will be the following Poincaré inequality: \[t:poincare\] There exists a constant $C_{\ref{t:poincare}}>0$ such that if $r$ is sufficiently small, then $$\label{e:poincare} {{\mathbf{H}}}(r) \leq C_{\ref{t:poincare}}\, r\, {{\mathbf{D}}}(r)\,.$$ We record however the two main tools used to prove Theorem \[t:poincare\], since they will be useful in the future. The first one is an elementary computation. In order to state it we introduce the quantity $$\label{e:E} {{\mathbf{E}}}(r) := \int_{{\partial}B_r} \sum_{j=1}^{Q} \langle {{\mathpzc{N}}}\,_j, D_\nu {{\mathpzc{N}}}\,_j\rangle\, .$$ \[l:H’\] ${{\mathbf{H}}}$ is a Lipschitz function and the following identity holds for a.e. $r \in (0,1)$ $$\label{e:H'} {{\mathbf{H}}}' (r) = \frac{{{\mathbf{H}}}(r)}{r} + 2\,{{\mathbf{E}}}(r)\,.$$ The second identity is a consequence of the first variations of $T$ under specific vector fields, which we call “outer variations”: such variations “stretch” the normal bundle of ${{\mathcal{M}}}$ suitably and they are defined using the map ${{\mathpzc{N}}}$. In the case of semicalibrated currents it is convenient to modify the Dirichlet energy suitably to gain a new quantity which enjoys better estimates. Thus, from now on ${{\mathbf{\Omega}}}$ will denote ${{\mathbf{D}}}$ in the cases (a) and (c) of Definition \[d:semicalibrated\], whereas in the case (b) it will be given by $$\begin{aligned} {{\mathbf{\Omega}}}(r) := &{{\mathbf{D}}}(r) + {{\mathbf{L}}}(r) \\ := &{{\mathbf{D}}}(r) + \int_{{{\bm{\Psi}}}(B_r)}\sum_{i=1}^{Q}\, \langle \xi_1 (p) \wedge D_{\xi_2} N_i (p) \wedge N_i (p) + D_{\xi_1} N_i (p) \wedge \xi_2 (p) \wedge N_i (p),d\omega (p)\rangle\, dp\, .\end{aligned}$$ \[p:OV\] There exist constants $C_{\ref{p:OV}}>0$ and $\kappa >0$ such that, if $r>0$ is small enough, then the inequality $$\label{e:OV} \left|{{\mathbf{\Omega}}}(r)- {{\mathbf{E}}}(r)\right| \leq C_{\ref{p:OV}}\,\mathcal{E}_{OV}(r)\,$$ holds with $$\begin{aligned} \label{e:stima_Eov_1} \mathcal{E}_{OV}(r)& = {{\mathbf{\Lambda}}}(r)^{\kappa} \Big({{\mathbf{D}}}(r)+ \frac{{{\mathbf{H}}}(r)}{r} + r {{\mathbf{D}}}' (r)\Big) + {{\mathbf{F}}}(r) + r^{1+\gamma_0} \frac{d}{dr}\|T-{\mathbf{T}}_F\|({{\mathbf{p}}}^{-1}({{\bm{\Psi}}}(B_r))) \,.\end{aligned}$$ Moreover $$\begin{aligned} \label{e:L0} |{{\mathbf{L}}}(r)| \leq & C\,{{\bm m}_0}^{\sfrac12}\,r^{2-\gamma_0} {{\mathbf{D}}}(r) + C\,{{\bm m}_0}^{\sfrac12}\,{{\mathbf{F}}}(r).\end{aligned}$$ Proof of Lemma \[l:H’\] ----------------------- The Lipschitz regularity of ${{\mathbf{H}}}$ follows from the Lipschitz regularity of ${{\mathpzc{N}}}$. Consider next the map $i_r : {{\mathfrak{B}}}\to {{\mathfrak{B}}}$ given by $i_r(z,w)=\left(rz, r^{\sfrac{1}{\bar Q}}w\right)$. By a simple change of variables we compute $${{\mathbf{H}}}(r)=\int_{{\partial}B_1} |{{\mathpzc{N}}}|^2(i_r(z',w'))\, r\, .$$ The formula is then an elementary computation using the chain rule for multifunctions, cf. [@DS1]. Proof of Proposition \[p:OV\] {#ss:OV} ----------------------------- The inequality is a simple consequence of $$|{{\mathbf{L}}}(r)| \leq C{{\bm m}_0}^{\sfrac{1}{2}} \int_{B_r} |D{{\mathpzc{N}}}| |{{\mathpzc{N}}}| \leq C{{\bm m}_0}^{\sfrac{1}{2}} \int_{B_r} |z|^{2-\gamma_0} |D{{\mathpzc{N}}}|^2 + C {{\bm m}_0}^{\sfrac{1}{2}} \int_{B_r} |z|^{\gamma_0-2} |{{\mathpzc{N}}}|^2\, .$$ In order to show we fix a test function $\phi\in C^\infty_c ({{\mathbb R}})$, nonnegative, symmetric, with support in $]-1,1[$ and monotone decreasing on $[0,1]$. We then follow [@DS5 Section 3.3] and, having fixed $r$, we define the vector field $X^o$ on ${{\mathbf{V}}}_{u,a}$ via $$X^o (p) := \varphi ({{\mathbf{p}}}(p)) (p-{{\mathbf{p}}}(p))\qquad \mbox{where}\qquad \varphi ({{\bm{\Psi}}}(z,w)) = \phi \left(\textstyle{\frac{|z|}{r}}\right)\, .$$ For $r$ small enough, by we can argue as in [@DS5 Section 3.3] and deduce via the change of coordinates given by ${{\bm{\Psi}}}$, that $$\label{e:OV_0} \delta {\mathbf{T}}_F(X)= \int_{{{\mathfrak{B}}}}\phi \big( \textstyle{\frac{|z|}{r}}\big)\,|D{{\mathpzc{N}}}|^2 + r^{-1} \int_{{{\mathfrak{B}}}} \phi' \big( \textstyle{\frac{|z|}{r}}\big) \,\sum_{j=1}^{Q} \langle {{\mathpzc{N}}}\,_j, D_\nu {{\mathpzc{N}}}\,_j\rangle + \sum_{i=1}^3 \textup{Err}_i^o,$$ with $$\begin{aligned} \label{e:outer_resto_1} {\rm Err}_1^o & = \Big\vert\int_{{\mathcal{M}}}\varphi\, \langle H_{{\mathcal{M}}}, {{\bm{\eta}}}\circ {N}\rangle\Big\vert \leq C\, {{\bm m}_0}^{\sfrac12}\int_{B_r} \,|z|^{\gamma_0 - 1}\, |{{\bm{\eta}}}\circ {{\mathpzc{N}}}| \stackrel{\eqref{e:media_pesata}}{\leq} C\, {{\mathbf{\Lambda}}}^{\eta_0}(r)\,{{\mathbf{D}}}(r) + C\,{{\mathbf{F}}}(r)\,,\end{aligned}$$ $$\label{e:outer_resto_2} {\rm Err}_2^o \leq C \int_{{\mathcal{M}}}|\varphi| \,|A_{{{\mathcal{M}}}}|^2\,|N|^2 \leq C\, {{\mathbf{F}}}(r)\,,$$ $$\begin{aligned} \label{e:outer_resto_3} {\rm Err}_3^o \leq & C \int_{{\mathcal{M}}}\Big(|\varphi| \big(|D{N}|^2\, |{N}|\, |A_{{{\mathcal{M}}}}| + |D{N}|^4\big) + |D\varphi|\, \big(|DN|^3 \,|{N}| + |D{N}|\, |{N}|^2 \,|A_{{{\mathcal{M}}}}|\big)\Big)\notag\\ \leq & C \int_{B_r} \Big[\Big(\frac{|{{{\mathpzc{N}}}}|^2}{|z|^{2-2\,\gamma_0}}+ |D{{{\mathpzc{N}}}}|^4\Big)- r^{-1} \phi' ( \textstyle{\frac{|z|}{r}})\,r^{1+\gamma_0}\,|D {{\mathpzc{N}}}|^3 - r^{-1} \phi' ( \textstyle{\frac{|z|}{r}})\,|D {{\mathpzc{N}}}|\,\frac{|{{\mathpzc{N}}}|^2}{|z|^{1-\gamma_0}}\Big] \notag\\ \stackrel{\eqref{e:Lip_N}\&\eqref{e:Ndecay}}{\leq} & C\,{{\mathbf{\Lambda}}}^{\eta_0}(r)\,{{\mathbf{D}}}(r)+C{{\mathbf{F}}}(r) - C \,{{\mathbf{\Lambda}}}(r)^{\eta_0}\,\int_{B_r} r^{-1} \phi' ( \textstyle{\frac{|z|}{r}})\,\frac{|{{\mathpzc{N}}}|^2}{|z|^{1-\gamma_0}}\notag\\ &\qquad - C \,r^{1+\gamma_0}\,{{\mathbf{\Lambda}}}^{\eta_0}\,\int_{B_r} r^{-1} \phi' ( \textstyle{\frac{|z|}{r}})\,|D{{\mathpzc{N}}}|^2\,.\end{aligned}$$ (We recall that $\phi' \leq 0$ on $[0,1])$). We next drop the superscript from $X^o$ and we distinguish two situations: - In the cases (a) and (c) of Definition \[d:semicalibrated\], we denote by $X^\perp$ and $X^T$ the projections of $X$ on the normal and the tangential bundle of $\Sigma$, respectively. Then $\delta T(X^T) =0$ and therefore $$|\delta {\mathbf{T}}_F(X)| \leq \underbrace{|\delta {\mathbf{T}}_F(X)-\delta T(X)|}_{\textup{Err}_4^o}+\underbrace{|\delta T(X^\perp)|}_{\textup{Err}^o_5};$$ - In case (b), since $\delta T(X) = T(dw{\mathop{\hbox{\vrule height .3pt width 5pt depth 0pt \vrule height 7pt width .3pt depth 0pt}}\nolimits}X)$, we estimate $$\big|\delta {\mathbf{T}}_F(X) - {\mathbf{T}}_F(d\omega{\mathop{\hbox{\vrule height .3pt width 5pt depth 0pt \vrule height 7pt width .3pt depth 0pt}}\nolimits}X)\big| \leq \underbrace{|\delta {\mathbf{T}}_F(X)-\delta T(X)|+ |T(d\omega {\mathop{\hbox{\vrule height .3pt width 5pt depth 0pt \vrule height 7pt width .3pt depth 0pt}}\nolimits}X)-{\mathbf{T}}_F(d\omega{\mathop{\hbox{\vrule height .3pt width 5pt depth 0pt \vrule height 7pt width .3pt depth 0pt}}\nolimits}X)|}_{\textup{Err}_4^o}\, .$$ In both cases we have $$\begin{aligned} \textup{Err}^o_4 \leq & Q \int_{{{\rm spt}}(T)\setminus {{\rm Im}}(F)}\left|{{\textup {div}}}_{\vec T} X\right|\, d\|T\| + Q\int_{{{\rm Im}}(F)\setminus {{\rm spt}}(T)} \left|{{\textup {div}}}_{\vec {\mathbf{T}}_F} X\right|\, d\|{\mathbf{T}}_F\|\\ &\quad + Q \|d\omega\|_\infty \int |X| d\|T-{\mathbf{T}}_F\|\, ,\end{aligned}$$ where we use the convention that $\omega=0$ in the cases (a) and (c). We then can estimate $$\begin{aligned} \textup{Err}^o_4&\leq C\,\int \left(\textstyle{{\varphi}'({{\mathbf{p}}}(p)) \,|p-{{\mathbf{p}}}(p)| +{\varphi}({{\mathbf{p}}}(p))}\right) \,d\|T-{\mathbf{T}}_F\|\notag\\ & \stackrel{\eqref{e:Ndecay}\&\eqref{e:diff masse}}{\leq} C\,{{\mathbf{\Lambda}}}^{\eta_0}(r)\,{{\mathbf{D}}}(r) +C\, {{\mathbf{F}}}(r) + C\, r^{1+\gamma_0}\underbrace{\int |\nabla {\varphi}({{\mathbf{p}}}(p))| \,|p-{{\mathbf{p}}}(p)|\,d\|T-{\mathbf{T}}_F\|}_{S ({\varphi})}\,.\label{e:Err4_o}\end{aligned}$$ In case (b) we have that $$\begin{aligned} {\mathbf{T}}_F(d\omega{\mathop{\hbox{\vrule height .3pt width 5pt depth 0pt \vrule height 7pt width .3pt depth 0pt}}\nolimits}X) &= \int_{{{\mathcal{M}}}} {\varphi}\,\sum_{i=1}^{Q} \langle (\xi_1+D_{\xi_1} N_i)\wedge(\xi_2+D_{\xi_2} N_i\cdot\xi_2)\wedge N_i\,,\,d\omega(p+ N_i(p))\, .\end{aligned}$$ Clearly $$\begin{aligned} &\Big|{\mathbf{T}}_F (d\omega {\mathop{\hbox{\vrule height .3pt width 5pt depth 0pt \vrule height 7pt width .3pt depth 0pt}}\nolimits}X) - \int_{{{\mathcal{M}}}} {\varphi}\,\sum_{i=1}^{Q} \langle (\xi_1+D_{\xi_1} N_i)\wedge(\xi_2+D_{\xi_2} N_i\cdot\xi_2)\wedge N_i\,,\,d\omega(p)\rangle\Big|\\ \leq &\; C \|d\omega\|_1 \int {\varphi}|N|^2 \end{aligned}$$ and we can therefore conclude $$\begin{aligned} &\Big|{\mathbf{T}}_F (d\omega {\mathop{\hbox{\vrule height .3pt width 5pt depth 0pt \vrule height 7pt width .3pt depth 0pt}}\nolimits}X) - \int_{{{\mathcal{M}}}} {\varphi}\, \sum_{i=1}^{Q}\, \langle \xi_1 (p) \wedge D_{\xi_2} N_i (p) \wedge N_i (p) + D_{\xi_1} N_i (p) \wedge \xi_2 (p) \wedge N_i (p),d\omega(p)\rangle\Big|\\ \leq & C\|d\omega\|_0 \int {\varphi}|N| |DN|^2 + C \|d\omega\|_0 \int {\varphi}|{{\bm{\eta}}}\circ N|+C \|d\omega\|_1 \int {\varphi}|N|^2\, . \end{aligned}$$ Letting $\phi$ converge to the characteristic function of the interval $[-1,1]$, we reach the conclusion . The only term which needs some care is the term $S ({\varphi})$ in . Note that we can approximate the characterstic function of $[-1, 1]$ with an increasing sequence of functions $\phi_j$ with the property that $|\phi'_j|\leq C j$, $0\leq \phi_j \leq 1$ and $\phi_j \equiv 1$ on $[-1+1/j, 1-1/j]$. Then we would have $$\limsup_j S ({\varphi}_j) \leq C \limsup_j \frac{ j}{r} \|T - {\mathbf{T}}_F\| ({{\bm{\Psi}}}(B_r\setminus B_{r (1-1/j)})) \leq C \frac{d}{dr} \|T - {\mathbf{T}}_F\| ({{\bm{\Psi}}}(B_r))\, ,$$ by the monotonicity of the function $r\mapsto \|T - {\mathbf{T}}_F\| ({{\bm{\Psi}}}(B_r))$. In the cases (a) and (c) we follow the same argument, but we need to bound the additional term $\textup{Err}^o_5$. In order to deal with the latter term we argue as in [@DS5 Section 4.1]. In particular we bound $$\begin{aligned} \textup{Err}^o_5 &\leq \left|\int {{\textup {div}}}_{\vec T} X^\perp\, d\|T\| \right|\notag\\ &\leq \underbrace{\int_{{{\rm spt}}(T)\setminus {{\rm Im}}(F)}\left|{{\textup {div}}}_{\vec T} X\right|\, d\|T\| + \int_{{{\rm Im}}(F)\setminus {{\rm spt}}(T)} \left|{{\textup {div}}}_{\vec {\mathbf{T}}_F} X\right|\, d\|{\mathbf{T}}_F\|}_{I_1}\notag\\ &\qquad + \underbrace{\left|\int\langle X^\perp,h(\vec{{\mathbf{T}}}_F(p))\rangle\,d\|{\mathbf{T}}_F\|\right|}_{I_2}\, ,\end{aligned}$$ where $h (v_1\wedge v_2) := \sum_{i=1}^2 A_\Sigma (v_i, v_i)$. Since the projection on the normal to $\Sigma$ is a $C^{2,{{\varepsilon}}_0}$ map, $X^\perp$ enjoys the same $C^1$ bounds as $X$ and $I_1$ can be controlled as $\textup{Err}^o_4$. The term $I_2$ can be estimated using $$|X^{o\perp} (p)| = \varphi\, |{{\mathbf{p}}}_{T_p \Sigma^\perp} (p-{{\mathbf{p}}}(p))| \leq C \mathbf{c} (\Sigma)\,{\varphi}\, |p-{{\mathbf{p}}}(p)|^2 \leq C {{\bm m}_0}^{\sfrac{1}{2}}\,{\varphi}\, |p-{{\mathbf{p}}}(p)|^2\quad\forall\;p\in\Sigma.$$ In particular we achieve $I_2\leq C {{\mathbf{H}}}(r)$, which concludes the proof. Proof of Theorem \[t:poincare\] ------------------------------- In order to prove the theorem we start estimating the error term ${{\mathbf{F}}}$. \[l:F\] There exist a constant $C_{\ref{l:F}}>0$ (depending on $\gamma_0$) such that $$\label{e:F} {{\mathbf{F}}}(r)\leq C_{\ref{l:F}}\,r^{\gamma_0 -1}\,{{\mathbf{H}}}(r)+C_{\ref{l:F}}\,r^{\gamma_0}\, {{\mathbf{D}}}(r) \quad \forall\;r \in (0,1).$$ Using and an integration by parts we infer that $$\begin{aligned} \label{e:F1} \gamma_0\int_0^r\frac{{{\mathbf{H}}}(\rho)}{\rho^{2-\gamma_0}}\,d\rho &= \frac{{{\mathbf{H}}}(\rho)}{\rho^{1-\gamma_0}}\Big\vert_0^r-\int_0^r\frac{d}{d\rho}\left(\frac{{{\mathbf{H}}}(\rho)}{\rho}\right)\,\rho^{ \gamma_0}\,d\rho =\frac{{{\mathbf{H}}}(r)}{r^{1-\gamma_0}}-\int_0^r\frac{2\,{{\mathbf{E}}}(\rho)}{\rho^{1-\gamma_0}}\,d\rho.\end{aligned}$$ The Cauchy–Schwarz inequality yields then the following bound for every ${{\varepsilon}}$: $$\begin{aligned} \label{e:CS per E} |{{\mathbf{E}}}(r)| &\leq \frac{{{\varepsilon}}}{r} \int_{{\partial}B_r} |{{\mathpzc{N}}}|^2 + \frac{r}{4{{\varepsilon}}} \int_{{\partial}B_r} |D{{\mathpzc{N}}}|^2 = {{\varepsilon}}\,\frac{{{\mathbf{H}}}(r)}{r} + \frac{r\,{{\mathbf{D}}}'(r)}{4{{\varepsilon}}}.\end{aligned}$$ Therefore, by choosing ${{\varepsilon}}=\sfrac{\gamma_0}{2}$, we deduce from and . In view of Lemma \[l:F\], for $r$ sufficiently small, the almost minimizing condition reads as $${{\mathbf{D}}}(r)\leq C\,r\,{{\mathbf{D}}}'(r)+C\,\frac{{{\mathbf{H}}}(r)}{r^{1-\gamma_0}} +C\,{{\bm m}_0}^{\sfrac12}\,r^{\gamma_0}\,\int_{{\partial}B_r} |{{\bm{\eta}}}\circ {{\mathpzc{N}}}|\,.$$ Dividing by the radius and integrating we get $$\begin{aligned} \label{e:AM1 tris} \int_0^r \frac{{{\mathbf{D}}}(s)}{s}\,ds &\leq C \, \int_0^r \left( {{\mathbf{D}}}'(\rho) +\,\frac{{{\mathbf{H}}}(\rho)}{\rho^{2-\gamma_0}}+ \, \rho^{\gamma_0-1}\,\int_{{\partial}B_\rho} |{{\bm{\eta}}}\circ {{\mathpzc{N}}}|\right)\,d\rho \notag\\ &\stackrel{\eqref{e:F}}{\leq} C \,{{\mathbf{D}}}(r) +C\, {{\mathbf{F}}}(r) + C\,{{\bm m}_0}^{\sfrac12}\int_{B_r} \frac{|{{\bm{\eta}}}\circ {{\mathpzc{N}}}|}{|z|^{1-\gamma_0}} \notag\\ &\stackrel{\eqref{e:media_pesata}}{\leq} C\, {{\mathbf{D}}}(r)+C\,({{\mathbf{\Lambda}}}^{\eta_0}(r)\, {{\mathbf{D}}}(r) +{{\mathbf{F}}}(r))\leq C\, {{\mathbf{D}}}(r) + C\, r^{\gamma_0-1} {{\mathbf{H}}}(r)\, .\end{aligned}$$ Therefore, using Lemma \[l:H’\] we deduce that $$\begin{aligned} \frac{{{\mathbf{H}}}(r)}{r} & = \int_0^r \frac{2\,{{\mathbf{E}}}(\rho)}{\rho}\, dt \stackrel{\eqref{e:OV}}{\leq} C\,\int_0^r \frac{{{\mathbf{D}}}(\rho)}{\rho}\, d\rho \notag\\ &\qquad +C\,\int_0^r \left( \frac{{{\mathbf{H}}}(\rho)}{\rho^{2-2\gamma_0}}+ \rho^{\gamma_0} \,{{\mathbf{D}}}'(\rho)+ \rho^{\gamma_0} \frac{d}{d\rho}\|T-{\mathbf{T}}_F\|({{\mathbf{p}}}^{-1}({{\bm{\Psi}}}(B_\rho))) \right) \, d\rho\notag\\ & \stackrel{\eqref{e:AM1 tris}}{\leq} C\,{{\mathbf{D}}}(r) + C\,\frac{{{\mathbf{H}}}(r)}{r^{1-\gamma_0}}+C\,r^{\gamma_0}{{\mathbf{D}}}(r)+C\,{{\mathbf{F}}}(r)+C\,r^{\gamma_0}\|T-{\mathbf{T}}_F\|({{\mathbf{p}}}^{-1}({{\bm{\Psi}}}(B_r))) \notag\\ &\stackrel{\eqref{e:diff masse}\&\eqref{e:F}}{\leq} C\,{{\mathbf{D}}}(r) + C\,\frac{{{\mathbf{H}}}(r)}{r^{1-\gamma_0}}\, .\end{aligned}$$ For $r$ sufficiently small this concludes the proof. Inner variations and key estimates {#s:refined} ================================== Using the Poincaré inequality in Theorem \[t:poincare\], we can give very simple estimates of the error terms in the “inner variations” of the current $T$. The latter corresponds to deformations of $T$ along appropriate vector fields which are tangent to ${{\mathcal{M}}}$. In order to state our main conclusion we need to introduce yet another quantity $$\label{e:G} {{\mathbf{G}}}(r) := \int_{{\partial}B_r} \left|D_\nu {{\mathpzc{N}}}\right|^2\, .$$ \[p:IV\] There exist constants $C_{\ref{p:IV}}>0$ and $\eta>0$ such that, if $r>0$ is small enough, than the following holds $$\label{e:inn} \left|{{\mathbf{D}}}' (r) - 2\,{{\mathbf{G}}}(r)\right| \leq C\, \mathcal{E}_{IV}(r)\,,$$ where $$\begin{aligned} \label{e:stima_Eiv_1} \mathcal{E}_{IV}(r) &= r^{2\eta-1} {{\mathbf{D}}}(r)+ {{\mathbf{D}}}(r)^{\eta}\,{{\mathbf{D}}}'(r)+\frac{{{\bm m}_0}^{\sfrac12}}{r^{1-\gamma_0}} \int_{{\partial}B_r}|{{\bm{\eta}}}\circ {{\mathpzc{N}}}(z,w)|\notag\\ &+\frac{d}{dr}\|T-{\mathbf{T}}_F\|({{\mathbf{p}}}^{-1}({{\bm{\Psi}}}(B_r)))\,.\end{aligned}$$ For further use we summarize in the next lemma a set of inequalities which will be used in the next sections and which are direct consequences of all the conclusions derived so far \[l:tutte le stime\] There exist constant $C_{\ref{l:tutte le stime}}>0$ and $\eta>0$ such that for every $r$ sufficiently small the following holds: $$\begin{aligned} {{\mathbf{F}}}(r) + r {{\mathbf{F}}}' (r) \leq & C_{\ref{l:tutte le stime}}\,r^{\gamma_0}{{\mathbf{D}}}(r)\label{e:F finale}\\ |{{\mathbf{L}}}(r)|\leq & C_{\ref{l:tutte le stime}} \,r\,{{\mathbf{D}}}(r)\label{e:L finale}\\ |{{\mathbf{L}}}'(r)| \leq & C_{\ref{l:tutte le stime}}\,\left({{\mathbf{H}}}(r)\,{{\mathbf{D}}}'(r)\right)^{\sfrac12}\label{e:L' finale}\\ \mathcal{E}_{OV} \leq &C_{\ref{l:tutte le stime}}\, {{\mathbf{D}}}^{1+\eta} (r) + C_{\ref{l:tutte le stime}}\,{{\mathbf{F}}}(r)+C_{\ref{l:tutte le stime}} r {{\mathbf{D}}}^\eta (r) {{\mathbf{D}}}' (r) + C_{\ref{l:tutte le stime}}\, r\,\mathcal{E}_{BP}(r),\label{e:OV finale}\\ \mathcal{E}_{IV}(r)\leq & C_{\ref{l:tutte le stime}}\, r^{2\eta-1} {{\mathbf{D}}}(r)+C_{\ref{l:tutte le stime}}\,{{\mathbf{D}}}(r)^{\eta}\,{{\mathbf{D}}}'(r)+C_{\ref{l:tutte le stime}}\,\mathcal{E}_{BP}(r),\label{e:IV finale}\end{aligned}$$ where $$\mathcal{E}_{BP}(r):=\frac{{{\bm m}_0}^{\sfrac12}}{r^{1-\gamma_0}} \int_{{\partial}B_r}|{{\bm{\eta}}}\circ {{\mathpzc{N}}}|+\frac{d}{dr}\|T-{\mathbf{T}}_F\|({{\mathbf{p}}}^{-1}({{\bm{\Psi}}}(B_r)))$$ Moreover, for every $a>0$ there exist constants $b_0(a),C(a)>0$ such that $$\label{e:AM finale 1} {{\mathbf{D}}}(r) \leq \frac{r\,{{\mathbf{D}}}'(r)}{2(2\,a+b)} + \frac{a(a+b)\,{{\mathbf{H}}}(r)}{r(2\,a+b)} +C(a)\,r\, \mathcal{E}_{IV}(r) \quad \forall \; b<b_0(a).$$ An important corollary of the previous lemma is the following \[c:integrability\] There exists a constant $C_{\ref{c:integrability}}>0$ such that, if $\eta$ is the constant of Lemma \[l:tutte le stime\], then for every $0 \leq \gamma<\eta$ and $r$ sufficiently small, the nonnegative functions $\frac{\mathcal{E}_{IV}(r)}{r^\gamma\,{{\mathbf{D}}}(r)}$ $\frac{\mathcal{E}_{OV}(r)}{r^{1+\gamma}{{\mathbf{D}}}(r)}$ are both integrable. Moreover, if we define the functions $$\begin{aligned} {{\mathbf{\Sigma}}}_{IV} (r) := & \int_0^r \frac{\mathcal{E}_{IV}(s)}{s^\gamma\,{{\mathbf{D}}}(s)}\, ds\, ,\label{e:def_SIV}\\ {{\mathbf{\Sigma}}}_{OV} (r) := & \int_0^r \frac{\mathcal{E}_{OV}(s)}{s^\gamma\,{{\mathbf{D}}}(s)}\, ds\, ,\label{e:def_SOV}\\ {{\mathbf{\Sigma}}}(r) := & {{\mathbf{\Sigma}}}_{IV} (r) + {{\mathbf{\Sigma}}}_{OV} (r)\, ,\end{aligned}$$ then $$\label{e:bSigma} {{\mathbf{\Sigma}}}(r) \leq C_{\ref{c:integrability}}\,r^{\eta-\gamma}\,.$$ Proof of Proposition \[p:IV\] ----------------------------- We evaluate the first variation of $T$ along a suitably defined vector field $X$. To this aim we fix a function $\phi\in C^\infty_c (]-1,1[)$, symmetric, nonnegative and identically one on $]-1+1/j, 1-1/j[$ and with the property that $|\phi'|\leq C j$. Then we introduce the vector field $Y\colon {{\mathcal{M}}}\to {{\mathbb R}}^{n+2}$ defined, for every $(z,w)\in {{\mathfrak{B}}}\setminus \{0\}$, by $$Y({{\bm{\Psi}}}(z,w)) := \textstyle{\frac{|z|}{r}}\,\phi(\textstyle{\frac{|z|}{r}}) \, D_\nu {{\bm{\Psi}}}(z,w) \in T_{{{\bm{\Psi}}}(z,w)} {{\mathcal{M}}}\,,$$ and extended to be $0$ at the origin. Next we define the vector field $X_{i}\colon {{\mathbf{V}}}_{a,u} \to {{\mathbb R}}^{n+2}$ by $X_i (p):= Y({{\mathbf{p}}}(p))$. Note that $X_i$ is the infinitesimal generator of a one parameter family of diffeomorphisms $\Phi_{{\varepsilon}}$ defined as $\Phi_{{\varepsilon}}(p):= \Gamma_{{\varepsilon}}({{\mathbf{p}}}(p)) + p - {{\mathbf{p}}}(p)$, where $\Gamma_{{\varepsilon}}$ is the one-parameter family of biLipschitz homeomorphisms of ${{\mathcal{M}}}$ generated by $Y$. In fact, since $\Gamma_\varepsilon$ fixes the origin, we can consider it as a $C^{2, \gamma_0}$ map of ${{\mathcal{M}}}\setminus \{0\}$ onto itself. Note moreover that $X_i$ is Lipschitz on the entire ${{\mathfrak{B}}}$. Observe that, by Lemma \[l:F\] and the Poincaré inequality, ${{\mathbf{F}}}(r)\leq C\, r^{\gamma_0}\,{{\mathbf{D}}}(r)$, so that ${{\mathbf{\Lambda}}}(r)\leq C \,{{\mathbf{D}}}(r)$. Moreover, $$\label{e:DY_bound} |D_{{{\mathcal{M}}}} Y|({{\bm{\Psi}}}(z,w))+|{{\textup {div}}}_{{\mathcal{M}}}\, Y|({{\bm{\Psi}}}(z,w))\leq - C r^{-2} |z|\,\phi'(\textstyle{\frac{|z|}{r}})+ Cr^{-1} \,\phi(\textstyle{\frac{|z|}{r}}) \,,$$ where we recall that $\phi'\leq 0$ on $[0,1]$. If $r$ is small enough, by we can argue as in [@DS5 Section 3.3] and deduce that $$\frac{1}{2}\left|\int_{{{\mathcal{M}}}}\left( |D{N}|^2 \,{\rm div}_{{{\mathcal{M}}}}Y-2\sum_{i=1}^{Q} \langle D{N}_i\colon \left(D{N}_i\cdot D_{{{\mathcal{M}}}}Y\right)\rangle\right)\right| \leq \sum_{k=1}^5 \textup{Err}^i_k\,,$$ where the error terms can be bounded in the following manner. First of all, $$\begin{aligned} {\rm Err}_1^i &= Q \left| \int_{{\mathcal{M}}}\big(\langle H_{{\mathcal{M}}}, {{\bm{\eta}}}\circ {N}\rangle\,{{\textup {div}}}_{{\mathcal{M}}}Y+\langle D_Y H_{{\mathcal{M}}},{{\bm{\eta}}}\circ {N}\rangle \big) \right| \\ &\leq Cr^{-1} \,{{\bm m}_0}^{\sfrac12}\, \int_{{{\mathfrak{B}}}} \left(\phi \big( \textstyle{\frac{|z|}{r}}\big)\,|z|^{\gamma_0-1}\,|{{\bm{\eta}}}\circ {{\mathpzc{N}}}(z,w)| - \phi' \big( \textstyle{\frac{|z|}{r}}\big)\, |z|^{\gamma_0-1}\,|{{\bm{\eta}}}\circ {{\mathpzc{N}}}(z,w)|\right)\\ &\stackrel{\eqref{e:media_pesata}}{\leq} Cr^{-1}\, {{\mathbf{D}}}^{1+\eta}(r) - C\,{{\bm m}_0}^{\sfrac12}\,r^{\gamma_0-1} \int_{B_r} r^{-1} \phi' \big( \textstyle{\frac{|z|}{r}}\big)\,|{{\bm{\eta}}}\circ {{\mathpzc{N}}}(z,w)|\,,\end{aligned}$$ where in the first inequality we used and the fact that $$\langle D_YH_{{\mathcal{M}}},{{\bm{\eta}}}\circ {N}\rangle\leq |Y|\,|D H_{{{\mathcal{M}}}}|\,|{{\bm{\eta}}}\circ N|\leq C\, \textstyle{\frac{|z|}{r}} \phi ( \textstyle{\frac{|z|}{r}})\,|z|^{\gamma_0-2}\,|{{\bm{\eta}}}\circ {{\mathpzc{N}}}|\,.$$ As for $\textup{Err}^i_2$ and $\textup{Err}_i^3$ we have $$\begin{aligned} {\rm Err}_2^i &= C \int_{{\mathcal{M}}}|A_{{{\mathcal{M}}}}|^2\big(|DY|\,|N|^2+|Y|\,|N|\,|DN|\big)\\ &\leq C\, {{\bm m}_0}\, \int_{{{\mathfrak{B}}}} \left[r^{-1}\left(- \textstyle{\frac{|z|}{r}}\phi'(\big( \textstyle{\frac{|z|}{r}}\big))+ \phi \big( \textstyle{\frac{|z|}{r}}\big)\right) \frac{|{{\mathpzc{N}}}|^2}{|z|^{2-2\gamma_0}}+ {\textstyle{\frac{|z|}{r}}}\, \phi\big( {\textstyle{\frac{|z|}{r^2}}}\big) \frac{|{{\mathpzc{N}}}|\,|D{{\mathpzc{N}}}|}{|z|^{2-2\gamma_0}}\right]\\ & \leq C \,{{\bm m}_0}\,r^{\gamma_0-1} \,{{\mathbf{D}}}(r)- Cr^{-1} \, \int_{B_r}r^{-1} \phi' \big( {\textstyle{\frac{|z|}{r}}}\big) \, \frac{|{{\mathpzc{N}}}|^2}{|z|^{1-\gamma_0}}\,,\end{aligned}$$ and $$\begin{aligned} {\rm Err}_3^i &\leq C \int_{{\mathcal{M}}}\Big(|Y|\,|A_{{{\mathcal{M}}}}|\,|DN|^2\big(|N| + |DN|\big) + |DY| \big(|A_{{{\mathcal{M}}}}| \,|DN|\, |N|^2 + |DN|^4\big)\Big)\\ &\leq C r^{\gamma_0-1} {{\mathbf{D}}}(r)- C\,{{\mathbf{D}}}(r)^{\eta}\,\int_{{{\mathfrak{B}}}} r^{-1} \phi'\big( {\textstyle{\frac{|z|}{r}}}\big)\, |D{{\mathpzc{N}}}|^2 +C\,r^{-1}\,{{\mathbf{D}}}(r)^{\eta}\,\int_{{{\mathfrak{B}}}}r^{-1} \phi\big( {\textstyle{\frac{|z|}{r}}}\big)\, \frac{|{{\mathpzc{N}}}|^2}{|z|^{2-\gamma_0}}\, .\end{aligned}$$ The errors ${\rm Err}_4^i$ and ${\rm Err}_5^i$ are the same as ${\rm Err}_4^o$ and ${\rm Err}_5^o$ respectively, in Section \[ss:OV\], evaluated along a different vector field. Proceeding in the same way as in the estimate of ${\rm Err}_4^o$, we deduce $$\begin{aligned} {\rm Err}_4^i &=\int_{{{\rm spt}}(T)\setminus {{\rm Im}}(F)} \left|{{\textup {div}}}_{\vec T} X_i\right|\, d\|T\| + \int_{{{\rm Im}}(F)\setminus {{\rm spt}}(T)} \left|{{\textup {div}}}_{\vec {\mathbf{T}}_F} X_i \right|\, d\|{\mathbf{T}}_F\|\\ &\leq C\, r^{\gamma_0-1}\,{{\mathbf{D}}}(r)+ C \underbrace{\int \alpha\,d\|T-{\mathbf{T}}_F\|}_{S(\phi)}\,.\end{aligned}$$ where $\alpha (p) = \varphi ({{\mathbf{p}}}(p))$ and $\varphi ({{\bm{\Psi}}}(z,w)) =r^{-2} |z| \phi \big(r^{-1} |z|) - r^{-1} \phi' (r^{-1} |z|)$. In particular using and the fact that $- \phi' \leq C j$ on $[0,1]$, we infer $$S (\phi) \leq C r^{\gamma_0-1} {{\mathbf{D}}}(r) + C \frac{j}{r} \|T - {\mathbf{T}}_F\| ({{\mathbf{p}}}^{-1} ({{\bm{\Psi}}}(B_r\setminus B_{r(1-1/j)}))\, .$$ As for $\textup{Err}_i^5$, we observe that it only appears in the cases (a) and (c) and arguing as in Section \[ss:OV\] we can bound it as $$\textup{Err}_i^5 \leq I_1 + \underbrace{\left|\int\langle X_i^\perp,h(\vec{{\mathbf{T}}}_F(p))\rangle\,d\|{\mathbf{T}}_F\|\right|}_{I_2}\, ,$$ where $h (v_1\wedge v_2) := \sum_{i=1}^2 A_\Sigma (v_i, v_i)$ and $I_1$ enjoys the same bounds as $\textup{Err}_i^4$. Following the argument of [@DS5 Section 4.3] we can see that $I_2$ enjoys the same bounds as $\textup{Err}_i^1$ and $\textup{Err}_i^2$. To conclude the proof notice that, with analogous computation as in [@DS1 Proposition 3.1], $$\label{e:dir_var} \frac{d}{d{{\varepsilon}}}\Big\vert_{{{\varepsilon}}=0}\int_{{{\mathcal{M}}}} |D(N\circ \Gamma_{{\varepsilon}})|^2=\int_{{{\mathcal{M}}}}\left(2\sum_{i=1}^{Q} \langle DN_i\colon \left(DN_i\cdot D_{{{\mathcal{M}}}}Y\right)\rangle-|DN|^2 \,{\rm div}_{{{\mathcal{M}}}}Y \right)\,.$$ However, by the conformal invariance of the Dirichlet energy, we have $$\int_{{{\mathcal{M}}}} |D(N\circ \Gamma_{{\varepsilon}})|^2 = \int_{{\mathfrak{B}}}|D ({{\mathpzc{N}}}\circ \hat{\Gamma}_{{\varepsilon}})|^2\, ,$$ where $\hat{\Gamma}_{{\varepsilon}}$ is the one parameter family of diffeomorphisms generated by the vector field $\hat{Y}\colon {{\mathfrak{B}}}\to {{\mathfrak{B}}}$ defined by $$\hat{Y}(z,w):= \frac{|z|}{r} \,\phi\left(\frac{|z|}{r}\right)\nu\,.$$ Hence $$\label{e:dir_var1} \frac{d}{d{{\varepsilon}}}\Big\vert_{{{\varepsilon}}=0}\int_{{{\mathcal{M}}}} |D(N\circ \Gamma_{{\varepsilon}})|^2= \int_{{{\mathfrak{B}}}}\left(2\sum_{i=1}^{Q} \langle D{{\mathpzc{N}}}\,_i\colon \left(D{{\mathpzc{N}}}\,_i\cdot D\hat{Y}\right)\rangle-|D{{\mathpzc{N}}}|^2 \,{\rm div}\, \hat{Y} \right)\, ,$$ where the differentiation is taken with respect to the (local) flat structure of ${{\mathfrak{B}}}$. In particular we conclude $$\label{e:dir_var3} \frac{d}{d{{\varepsilon}}}\Big\vert_{{{\varepsilon}}=0}\int_{{{\mathcal{M}}}} |D(N\circ \Gamma_{{\varepsilon}})|^2 = \int_{B_r} \frac{|z|}{r^2}\,\phi'\left(\frac{|z|}{r}\right) (2 |D_\nu {{\mathpzc{N}}}|^2 - |D{{\mathpzc{N}}}|^2)\,.$$ Collecting together , and the error estimates, and letting $\phi$ converge to the to the indicator function of $[-1,1]$ (namely letting $j\uparrow \infty$) we conclude the proof. Proof of Lemma \[l:tutte le stime\] ----------------------------------- The lemma is a very simple corollary of the estimates proven so far. is a simple consequence of the Poincaré inequality and of . Similarly, by Lemma \[l:F\], we have that ${{\mathbf{\Lambda}}}(r) \leq C\, {{\mathbf{D}}}(r)$, and therefore follows in view of . The same arguments hold for . Next for we can estimate as follows: $$\begin{aligned} \label{e:L} |{{\mathbf{L}}}(r)| & \leq C\,{{\bm m}_0}^{\sfrac12}\,\int_{B_r} |{{\mathpzc{N}}}|\,|D{{\mathpzc{N}}}| \leq C_{}\,{{\bm m}_0}^{\sfrac12}\,\left(\int_0^r {{\mathbf{H}}}(t)\,dt\right)^{\frac12} {{\mathbf{D}}}^{\frac12}(r)\notag \\ &\stackrel{\eqref{t:poincare}}{\leq} C_{}\,{{\bm m}_0}^{\sfrac12}\,\left(C_{\ref{t:poincare}}\int_0^r t\,{{\mathbf{D}}}(t)\,dt\right)^{\frac12} {{\mathbf{D}}}^{\frac12}(r) \leq C\,{{\bm m}_0}^{\sfrac12}\,r\,{{\mathbf{D}}}(r)\, .\end{aligned}$$ Similarly $$\begin{aligned} \label{e:L'} |{{\mathbf{L}}}'(r)| & \leq C\,{{\bm m}_0}^{\sfrac12}\,\int_{{\partial}B_r} |{{\mathpzc{N}}}|\,|D{{\mathpzc{N}}}| \leq C\,{{\bm m}_0}^{\sfrac12}\,\left(\,{{\mathbf{D}}}'(r)\,{{\mathbf{H}}}(r)\right)^{\frac12}\,.\end{aligned}$$ Finally, we notice that Proposition \[p:IV\] implies $$\left|\frac{{{\mathbf{D}}}'(r)}{2}- \int_{{\partial}B_r} |D_\tau {{\mathpzc{N}}}|^2\right| \leq C\, \mathcal{E}_{IV}(r).$$ Therefore, using the almost minimizing property in and the Poincaré inequality we infer that $$\begin{aligned} {{\mathbf{D}}}(r) & \leq (1+C\,r)\left[\frac{r\,{{\mathbf{D}}}'(r)}{2(2\,a+b)} + \frac{a(a+b)\,{{\mathbf{H}}}(r)}{r(2\,a+b)}\right] +C(a)\,r\,\mathcal{E}_{IV}(r)+ \mathcal{E}_{QM}(r)+C\,r^{1+\sigma}\,{{\mathbf{D}}}'(r)\,.\end{aligned}$$ Absorbing the error term $r^{1+\sigma}\,{{\mathbf{D}}}'(r)$ and dividing by $(1+C\,r^\sigma)$ we get $$\begin{aligned} {{\mathbf{D}}}(r) & \leq \frac{r\,{{\mathbf{D}}}'(r)}{2(2\,a+b)} + \frac{a(a+b)\,{{\mathbf{H}}}(r)}{r(2\,a+b)} +C(a)\,r\,\mathcal{E}_{IV}(r)+ \mathcal{E}_{QM}(r)+C\,r^\sigma {{\mathbf{D}}}(r)\, ,\end{aligned}$$ from which follows straightforwardly by noticing that $\mathcal{E}_{QM}(r)+r\,{{\mathbf{D}}}(r)\leq C\, r\,\mathcal{E}_{IV}(r)$. Proof of Corollary \[c:integrability\] -------------------------------------- Recall first thet $\eta<\gamma_0$. We start with $\mathcal{E}_{BP}(r)$. Notice that, using ${{\mathbf{H}}}(t)\leq C \,t\, {{\mathbf{D}}}(t)$ together with the definition of ${{\mathbf{F}}}(r)$, we have $$\int_0^r\left(\frac{1}{t^{\gamma}\,{{\mathbf{D}}}(t)}\right)' {{\mathbf{F}}}(t)\,dt\leq C\,\frac{{{\mathbf{F}}}(r)}{r^\gamma \,{{\mathbf{D}}}(r)}+C\int_0^r \frac{1}{t^\gamma\,{{\mathbf{D}}}(t)}\,\frac{{{\mathbf{H}}}(t)}{t^{2-\gamma_0}}\,dt\leq C r^{\gamma_0-\gamma}$$ Next, by a simple integration by parts and the fact that ${{\mathbf{D}}}(r)\leq C r^2$, we deduce $$\begin{aligned} &\int_0^r\frac{1}{t^{\gamma}{{\mathbf{D}}}(t)} \frac{d}{dt}\|T-{\mathbf{T}}_F\|({{\mathbf{p}}}^{-1}({{\bm{\Psi}}}(B_t)))\,dt=\frac{1}{r^{\gamma}{{\mathbf{D}}}(r)} \|T-{\mathbf{T}}_F\|({{\mathbf{p}}}^{-1}({{\bm{\Psi}}}(B_r))) \notag\\ &\qquad\qquad+\int_0^r \left(\frac{1}{t^\gamma {{\mathbf{D}}}(t)}\right)'\|T-{\mathbf{T}}_F\|({{\mathbf{p}}}^{-1}({{\bm{\Psi}}}(B_t)))\,dt\notag\\ \stackrel{\eqref{e:diff masse}}{\leq} & C\, \frac{{{\mathbf{D}}}^{1+\eta}(r)+{{\mathbf{F}}}(r)}{r^\gamma\,{{\mathbf{D}}}(r)}+\int_0^r \left(\frac{1}{t^\gamma {{\mathbf{D}}}(t)}\right)' \, \left({{\mathbf{D}}}(t)^{1+\eta}+{{\mathbf{F}}}(t)\right)\,dt \leq C \,r^{\eta-\gamma}\,.\end{aligned}$$ In a similar fashion we have $$\begin{aligned} \int_0^r\frac{{{\bm m}_0}^{\sfrac12}}{t^\gamma\,{{\mathbf{D}}}(t)} &\int_{{\partial}B_t}\frac{|{{\bm{\eta}}}\circ {{\mathpzc{N}}}(z,w)|}{t^{1-\gamma_0}}\,dt \leq\frac{{{\bm m}_0}^{\sfrac12}}{r^{\gamma}\,{{\mathbf{D}}}(r)}\int_{B_r}\frac{|{{\bm{\eta}}}\circ {{\mathpzc{N}}}(z,w)|}{|z|^{1-\gamma_0}}\notag\\ &+\int_0^r \left(\frac{1}{t^\gamma {{\mathbf{D}}}(t)}\right)'\,{{\bm m}_0}^{\sfrac12}\int_{B_t}\frac{|{{\bm{\eta}}}\circ {{\mathpzc{N}}}(z,w)|}{|z|^{1-\gamma_0}}\notag\\ &\stackrel{\eqref{e:media_pesata}}{\leq} C\, \frac{{{\mathbf{D}}}^{1+\eta}(r)+{{\mathbf{F}}}(r)}{r^\gamma\,{{\mathbf{D}}}(r)}+\int_0^r \left(\frac{1}{t^\gamma {{\mathbf{D}}}(t)}\right)' \, \left({{\mathbf{D}}}(t)^{1+\eta}+{{\mathbf{F}}}(t)\right)\,dt \leq C \,r^{\eta-\gamma}\,.\end{aligned}$$ so that $$\int_0^r\frac{\mathcal{E}_{BP}(t)}{t^{\gamma}\,{{\mathbf{D}}}(t)}\,dt\leq C\, r^{\eta-\gamma}$$ To conclude, we estimate separately the two nonnegative functions ${{\mathbf{\Sigma}}}_{IV}$ and ${{\mathbf{\Sigma}}}_{OV}$. In particular $$\begin{aligned} {{\mathbf{\Sigma}}}_{IV} (r) = & \int_{0}^r \frac{\mathcal{E}_{IV}(t)}{t^\gamma\,{{\mathbf{D}}}(t)}\,dt \stackrel{\eqref{e:IV finale}}{\leq} 2\,C_{\ref{l:tutte le stime}}\, \int_0^r \left(t^{\gamma_0-\gamma-1}+t^{-\gamma}{{\mathbf{D}}}(t)^{\eta-1}\,{{\mathbf{D}}}'(t)+\frac{\mathcal{E}_{BP}(t)}{t^{\gamma}\,{{\mathbf{D}}}(t)} \right)\,dt\notag\\ \leq &\, C\,r^{\eta-\gamma} \left(1+{{\mathbf{D}}}(t)^{\sfrac{\eta}{2}}\right) \leq C\,r^{\eta-\gamma}\,,\end{aligned}$$ where in the second inequality we used ${{\mathbf{D}}}(t)\leq C\, t^{2}$. Finally $$\begin{aligned} {{\mathbf{\Sigma}}}_{OV} (r) = & \int_{0}^r \frac{\mathcal{E}_{OV}(t)}{t^{1+\gamma}{{\mathbf{D}}}(t)}\,dt \notag\\ \stackrel{\eqref{e:OV finale}}{\leq} C_{\ref{l:tutte le stime}}& \,\int_{0}^r \left(\frac{{{\mathbf{D}}}^{\eta}(t)}{t^{1+\gamma}} + \frac{{{\mathbf{F}}}(t)}{t^{1+\gamma}{{\mathbf{D}}}(t)}+t^{-\gamma} {{\mathbf{D}}}^{\eta-1} (t){{\mathbf{D}}}' (t) +\frac{\mathcal{E}_{BP}(t)}{t^{\gamma}\,{{\mathbf{D}}}(t)}\right)\,dt\notag\\ \leq&\, C\, r^{\eta-\gamma}\,.\end{aligned}$$ Almost monotonicity and decay of the frequency function {#s:decay} ======================================================= In this section we study the asymptotic behaviour of the normal approximation ${{\mathpzc{N}}}$. The first step consists in proving approximate monotonicity and decay estimates for the frequency function. For every $r \in (0,1)$ such that ${{\mathbf{H}}}(r)>0$, we set ${{\mathbf{\bar I}}}(r) := \frac{r\,{{\mathbf{\Omega}}}(r)}{{{\mathbf{H}}}(r)}$ where we recall that $${{\mathbf{\Omega}}}(r) := \left\{ \begin{array}{ll} {{\mathbf{D}}}(r) \quad & \text{in the cases (a) and (b) of Definition \ref{d:semicalibrated};}\\ {{\mathbf{D}}}(r) + {{\mathbf{L}}}(r) \quad &\text{in case (c).} \end{array} \right.$$ Furthermore we define $\bar {{\mathbf{K}}}(r):=\bar {{\mathbf{I}}}(r)^{-1}$ whenever ${{\mathbf{\Omega}}}(r)\neq 0$. By there exists $r_0>0$ such that $$\label{e:r_0} \frac{1}{2} {{\mathbf{D}}}(r) \leq (1-C\,r){{\mathbf{D}}}(r)\leq {{\mathbf{\Omega}}}(r)\leq (1+C\,r) {{\mathbf{D}}}(r)\leq 2\,{{\mathbf{D}}}(r) \qquad \forall r\leq r_0\, .$$ Having fixed $r_0$, $\bar {{\mathbf{K}}}(r)$ is well defined whenever ${{\mathbf{D}}}(r) >0$ and hence, by the Poincaré inequality, whenever ${{\mathbf{\bar I}}}(r)$ is defined. Moreover, if for some $\rho\leq r_0$ $\bar {{\mathbf{K}}}(\rho)$ is not well defined, that is ${{\mathbf{\Omega}}}(\rho) =0$, then obviously ${{\mathbf{\Omega}}}(r) = {{\mathbf{D}}}(r) = 0$ for every $r\leq \rho$. We are now ready to state the first important monotonicity estimate. From now on we assume of having fixed a $\gamma$ \[t:monot\] There exists a constant $C_{\ref{t:monot}}>0$ with the following property: if ${{\mathbf{D}}}(r) > 0$ for some $r\leq r_0$, then (setting $\gamma =0$ in and ) the function $$\label{e:monot} \bar {{\mathbf{K}}}(r)\,\exp ( -4r -4 {{\mathbf{\Sigma}}}_{IV}(r))-4\,{{\mathbf{\Sigma}}}_{OV}(r)$$ is monotone non-increasing on any interval $[a,b]$ where ${{\mathbf{D}}}$ is nowhere $0$. In particular, either there is $\bar{r} >0$ such that ${{\mathbf{D}}}(\bar{r})=0$ or $\bar {{\mathbf{K}}}$ is well-defined on $]0, r_0[$ and the limit $K_0:=\lim_{r\to 0}\bar {{\mathbf{K}}}(r)$ exists. A fundamental consequence of Theorem \[t:monot\] is the following dichotomy. \[c:bdd\_freq\] There exists $\bar{r}>0$ such that - either $\bar {{\mathbf{K}}}(r)$ is well-defined for every $r \in ]0,r_0[$, the limit $$\label{e:K_0} K_0:=\lim_{r\downarrow0} \bar {{\mathbf{K}}}(r)$$ is positive and thus there is a constant $C$ and a radius $\bar{r}$ such that $$\label{e:H=rD} C^{-1}\,r\,{{\mathbf{D}}}(r) \leq {{\mathbf{H}}}(r) \leq C\,r\,{{\mathbf{D}}}(r) \qquad \forall\; r \in ]0, \bar{r}[\, ;$$ - or $T{\mathop{\hbox{\vrule height 7pt width .3pt depth 0pt \vrule height .3pt width 5pt depth 0pt}}\nolimits}{{\mathbf{p}}}^{-1} ({{\bm{\Psi}}}(B_{\bar{r}})) = Q \a{{{\bm{\Psi}}}(B_{\bar r})}$ for some positive $\bar{r}$. In turn, using the above dichotomy we will show \[t:decay\] Assume that condition (i) in Theorem \[t:bu\] fails. Then the frequency ${{\mathbf{\bar I}}}(r)$ is well-defined for every sufficiently small $r$ and its limit $I_0= \lim_{r\to 0} {{\mathbf{\bar I}}}(r) = K_0^{-1}$ exists and it is finite and positive. Moreover there exist constants $\lambda, C_{\ref{t:decay}}, H_0, D_0>0$ such that, for every $r$ sufficiently small the following holds: $$\begin{gathered} \label{e:decay} \big\vert {{\mathbf{I}}}(r) - I_0 \big\vert +\left\vert \frac{{{\mathbf{H}}}(r)}{r^{2I_0+1}} - H_0 \right\vert+ \left\vert \frac{{{\mathbf{D}}}(r)}{r^{2I_0}} - D_0 \right\vert \leq C_{\ref{t:decay}}\,r^{\lambda}\,.\end{gathered}$$ Proof of Theorem \[t:monot\] ---------------------------- In the first step we claim the monotonicity of the function $\bar {{\mathbf{K}}}(r)\,\exp ( -{{\mathbf{\Sigma}}}_{IV}(r))-2\,{{\mathbf{\Sigma}}}_{OV}(r)$ on any interval contained in $[a,b]$ on which ${{\mathbf{D}}}$ is everywhere positive. Recalling that ${{\mathbf{\Omega}}}$ and ${{\mathbf{H}}}$ are absolutely continuous functions, we can compute the following derivative: for every $r \in [a,b]$ $$\begin{aligned} \label{e:I'1} \bar {{\mathbf{K}}}'(r)& = \left(\frac{{{\mathbf{H}}}(r)}{r}\right)'\,\frac{1}{{{\mathbf{\Omega}}}(r)}-\frac{{{\mathbf{H}}}(r)}{r}\,\frac{{{\mathbf{\Omega}}}'(r)}{{{\mathbf{\Omega}}}^2(r)}\notag\\ &\stackrel{\eqref{e:H'}}{\leq}\frac{1}{r{{\mathbf{\Omega}}}^2(r)}\big(2\,{{\mathbf{E}}}(r)\,{{\mathbf{\Omega}}}(r)-{{\mathbf{D}}}'(r)\,{{\mathbf{H}}}(r)+|{{\mathbf{L}}}'(r)|\,{{\mathbf{H}}}(r)\big)\,.\end{aligned}$$ Then, either $\bar {{\mathbf{K}}}'\leq 0$, or the RHS of the inequality above is positive, that is $${{\mathbf{D}}}'(r)\,{{\mathbf{H}}}(r)\leq 2{{\mathbf{E}}}(r)\,{{\mathbf{\Omega}}}(r)+|{{\mathbf{L}}}'(r)|\,{{\mathbf{H}}}(r)\stackrel{\eqref{e:L' finale}}{\leq}2{{\mathbf{E}}}(r)\,{{\mathbf{\Omega}}}(r)+r\,{{\mathbf{D}}}'(r)\,{{\mathbf{H}}}(r)+\frac{{{\mathbf{H}}}^2(r)}{r}\, .$$ In turn, using ${{\mathbf{H}}}(r)\leq C\,r\,{{\mathbf{D}}}(r)\leq C\,r\,{{\mathbf{\Omega}}}(r)$, the latter inequality implies $$\label{e:controllo_D'H} {{\mathbf{D}}}'(r)\,{{\mathbf{H}}}(r)\leq C\,{{\mathbf{E}}}(r)\,{{\mathbf{\Omega}}}(r)+C\,r\,{{\mathbf{\Omega}}}^2(r)\,.$$ From this we deduce $${{\mathbf{E}}}^2(r)\leq {{\mathbf{H}}}(r)\,{{\mathbf{G}}}(r)\leq {{\mathbf{H}}}(r)\,{{\mathbf{D}}}'(r)\leq C {{\mathbf{\Omega}}}^2(r)+\frac{{{\mathbf{E}}}^2(r)}{2}\,$$ which implies that ${{\mathbf{E}}}(r)\leq C{{\mathbf{\Omega}}}(r)$ and so, by , $$\label{e:fanghualin} |{{\mathbf{L}}}'(r)|\leq C \,{{\bm m}_0}^{\sfrac12}\,({{\mathbf{D}}}'(r)\,{{\mathbf{H}}}(r))^{\sfrac12}\leq C\,{{\bm m}_0}^{\sfrac12}\,{{\mathbf{\Omega}}}(r)\,.$$ Next using again the Cauchy-Schwarz inequality and , we have $$\begin{aligned} {{\mathbf{\Omega}}}(r)\,{{\mathbf{E}}}(r) &\leq \,{{\mathbf{\Omega}}}(r)\,{{\mathbf{H}}}(r)^{\sfrac12} \,{{\mathbf{G}}}(r)^{\sfrac12} \leq\, \frac{{{\mathbf{\Omega}}}(r)^2}{2}+\frac{{{\mathbf{H}}}(r)\,{{\mathbf{G}}}(r)}{2}\\ &\leq \frac{{{\mathbf{\Omega}}}(r){{\mathbf{E}}}(r)}{2}+\frac{{{\mathbf{\Omega}}}(r)\,\mathcal{E}_{OV}(r)}{2}+ \frac{{{\mathbf{H}}}(r)\,{{\mathbf{G}}}(r)}{2}\,,\end{aligned}$$ which implies $$\label{e:mon1} {{\mathbf{\Omega}}}(r)\,{{\mathbf{E}}}(r) \leq {{\mathbf{H}}}(r)\,{{\mathbf{G}}}(r)+{{\mathbf{\Omega}}}(r)\,\mathcal{E}_{OV}(r)\,.$$ Collecting all these estimates together and using , we conclude that, if $\bar {{\mathbf{K}}}' (r)\geq 0$, then $$\begin{aligned} &\bar {{\mathbf{K}}}'(r) \stackrel{\eqref{e:I'1}\&\eqref{e:mon1}}{\leq} \frac{1}{r{{\mathbf{\Omega}}}^2(r)}\big(2\,{{\mathbf{H}}}(r)\,{{\mathbf{G}}}(r)-{{\mathbf{D}}}'(r)\,{{\mathbf{H}}}(r)+|{{\mathbf{L}}}'(r)|\,{{\mathbf{H}}}(r)+2\,{{\mathbf{\Omega}}}(r)\,\mathcal{E}_{OV}(r)\big)\notag\\ \stackrel{\eqref{e:inn}\&\eqref{e:fanghualin}}{\leq} &\frac{1}{r{{\mathbf{\Omega}}}^2(r)}\big(2\,{{\mathbf{H}}}(r)\,{{\mathbf{G}}}(r)-2\,{{\mathbf{H}}}(r)\,{{\mathbf{G}}}(r)+{{\mathbf{\Omega}}}(r)\,{{\mathbf{H}}}(r)+{{\mathbf{H}}}(r)\,\mathcal{E}_{IV}(r)+2\,{{\mathbf{\Omega}}}(r)\,\mathcal{E}_{OV}(r)\big)\notag\\ \leq &\, 2\frac{\mathcal{E}_{OV}(r)}{r{{\mathbf{\Omega}}}(r)}+\bar {{\mathbf{K}}}(r)\, \left(1+\frac{\mathcal{E}_{IV}(r)}{{{\mathbf{\Omega}}}(r)}\right) \leq 4\,\frac{\mathcal{E}_{OV}(r)}{r{{\mathbf{D}}}(r)}+4\,\bar {{\mathbf{K}}}(r)\, \left(1+\frac{\mathcal{E}_{IV}(r)}{{{\mathbf{D}}}(r)}\right)\, .\end{aligned}$$ On the other hand the final inequality $$\bar {{\mathbf{K}}}' (r) \leq 4 \frac{\mathcal{E}_{OV}(r)}{r{{\mathbf{D}}}(r)}+4\,\bar {{\mathbf{K}}}(r)\, \left(1+\frac{\mathcal{E}_{IV}(r)}{{{\mathbf{D}}}(r)}\right)$$ is certainly correct when $\bar {{\mathbf{K}}}' (r)\leq 0$, because the right hand side is positive. The monotonicity of the function in is then obvious. Next, as already observed, either ${{\mathbf{D}}}$ is always positive, or it vanishes on some interval $]0, \bar{r}[$. If ${{\mathbf{D}}}$ is always positive, then $\bar {{\mathbf{K}}}$ is well defined on $]0, r_0[$ and the existence of the limit $K_0 := \lim_{r\downarrow 0} \bar {{\mathbf{K}}}(r)$ is a direct consequence of and Corollary \[c:integrability\]. Proof of Corollary \[c:bdd\_freq\] ---------------------------------- First of all observe that, if ${{\mathbf{D}}}(\bar r)$ vanishes, then ${{\mathpzc{N}}}\equiv Q\a{0}$ on $B_{\bar r}$. In particular by we conclude that we are in the alternative (B). We can thus assume, without loss of generality, that ${{\mathbf{D}}}$ is positive on $]0, r_0[$. Assuming that $K_0$ vanishes we will then reach a contradiction. Under the assumption $K_0=0$, consider the monotonicity of $\bar {{\mathbf{K}}}(r)\,\exp ( -4 {{\mathbf{\Sigma}}}_{IV}(r)) - 4\,{{\mathbf{\Sigma}}}_{OV}(r)$ between two radii $0<s<r$ and let $s\to 0$ to get $$\bar {{\mathbf{K}}}(r)\leq 4\,e^{4r + 4 {{\mathbf{\Sigma}}}_{IV}(r)}\,{{\mathbf{\Sigma}}}_{OV}(r)\leq C\,{{\mathbf{\Sigma}}}_{OV}(r) \,,$$ where the last inequality holds for $r$ sufficiently small, since ${{\mathbf{\Sigma}}}_{IV}(r)\leq C r^{\eta}$ (recall that we have set $\gamma=0$). Next observe that, since the function ${{\mathbf{\Sigma}}}_{OV}(r)$ is non-decreasing, $$\begin{aligned} \label{e:err1_K} \frac{{{\mathbf{F}}}(r)}{{{\mathbf{D}}}(r)} &\leq \frac{1}{{{\mathbf{D}}}(r)}\int_0^r\frac{{{\mathbf{H}}}(s)}{s^{2-\gamma_0}}\frac{{{\mathbf{D}}}(s)}{{{\mathbf{D}}}(s)}\,ds\leq C\,\int_0^r\frac{\bar {{\mathbf{K}}}(s)}{s^{1-\gamma_0}}\,ds\leq C \,r^{\gamma_0}\,{{\mathbf{\Sigma}}}_{OV}(r)\, .\end{aligned}$$ Moreover, integrating by parts: $$\begin{aligned} \label{e:err2_K} &\int_0^r \frac{1}{{{\mathbf{D}}}(s)}\frac{d}{ds}\|T-{\mathbf{T}}_F\|({{\mathbf{p}}}^{-1} ({{\bm{\Psi}}}(B_s)))\,ds\notag\\ \stackrel{\eqref{e:diff masse}}{\leq} &C \frac{{{\mathbf{D}}}^{1+\eta}(r)+{{\mathbf{F}}}(r)}{{{\mathbf{D}}}(r)}+C\int_0^r\left(\frac{1}{{{\mathbf{D}}}(s)}\right)'\left({{\mathbf{D}}}^{1+\eta}(s)+{{\mathbf{F}}}(s)\right)\,ds\notag\\ \leq & C\,{{\mathbf{D}}}^\eta(r)+C\, r^{\gamma_0} {{\mathbf{\Sigma}}}_{OV}(r)+C\,\frac{{{\mathbf{F}}}(r)}{{{\mathbf{D}}}(r)}+C\,\int_0^r \frac{{{\mathbf{F}}}'(s)}{{{\mathbf{D}}}(s)}\,ds\notag\\ \leq & C\,{{\mathbf{D}}}^\eta(r)+C\, r^{\gamma_0} {{\mathbf{\Sigma}}}_{OV}(r)+C\,\int_0^r \frac{\bar {{\mathbf{K}}}(s)}{s^{1-{\gamma_0}}}\,ds \leq C\,{{\mathbf{D}}}^\eta(r)+C\, r^{\gamma_0} {{\mathbf{\Sigma}}}_{OV}(r)\, ,\end{aligned}$$ where we have used repeatedly . Using the latter in the formula for $\mathcal{E}_{OV}$ we also conclude $$\begin{aligned} &\;{{\mathbf{\Sigma}}}_{OV}(r)\\ \leq &\; C\int_0^r\frac{1}{s{{\mathbf{D}}}(s)}\left( {{\mathbf{D}}}(s)^{1+\eta}+s {{\mathbf{D}}}^\eta (s) {{\mathbf{D}}}' (s) + {{\mathbf{F}}}(s)+s \frac{d}{ds}\|T-{\mathbf{T}}_F\|({{\mathbf{p}}}^{-1} ({{\bm{\Psi}}}(B_s))) \right)\,ds\\ \leq &\; C\,r^{\eta} {{\mathbf{D}}}(r)^{\sfrac{\eta}{2}}+Cr^{\gamma_0}\,{{\mathbf{\Sigma}}}_{OV}(r)\, .\end{aligned}$$ Hence, for $r$ sufficiently small, $$\label{e:K_decay} \bar {{\mathbf{K}}}(r)\leq C {{\mathbf{\Sigma}}}_{OV}(r)\leq C {{\mathbf{D}}}(r)^{\sfrac{\eta}{2}}\,.$$ In particular this implies that $$\label{e:H_decay} {{\mathbf{H}}}(r)\leq C \,r\,{{\mathbf{D}}}(r)^{1+\sfrac{\eta}{2}}\,.$$ Combining this with and the Cauchy-Schwarz inequality, we deduce $$\begin{aligned} \frac{1}{2} \,{{\mathbf{D}}}(r) &\leq {{\mathbf{\Omega}}}(r) \leq \frac{{{\mathbf{E}}}(r)}{r}+\mathcal{E}_{OV}(r) \leq \left(\frac{{{\mathbf{H}}}(r)}{r\,{{\mathbf{D}}}(r)^{\sfrac{\eta}{4}}}\right)^{\sfrac12}\,\left(r\,{{\mathbf{D}}}'(r)\,{{\mathbf{D}}}(r)^{\sfrac{\eta}{4}}\right)^{\sfrac12}+\mathcal{E}_{OV}(r)\\ &\stackrel{\eqref{e:H_decay}}{\leq} C \, {{\mathbf{D}}}(r)^{1+\sfrac{\eta}{4}}+C\, r\, {{\mathbf{D}}}(r)^{\sfrac{\eta}{4}} {{\mathbf{D}}}'(r)+\mathcal{E}_{OV}(r)\,. \end{aligned}$$ Dividing the expression above by $r{{\mathbf{D}}}(r)$, integrating between two radii $0<s<r$ and using the bound ${{\mathbf{D}}}(r)\leq\,C\, r^2$ we obtain $$\begin{aligned} \log\left(\frac{r}{s}\right) \leq C \int_s^r \left( \frac{{{\mathbf{D}}}(\rho)^{\sfrac{\eta}{4}}}{\rho}+{{\mathbf{D}}}(\rho)^{\sfrac{\eta}{4}-1}\, {{\mathbf{D}}}'(\rho) +\frac{\mathcal{E}_{OV}(\rho)}{\rho\,{{\mathbf{D}}}(\rho)}\right)\,d\rho \leq C\, r^{\sfrac{\eta}{2}}\, .\end{aligned}$$ Sending $s\to 0$ we get a contradiction. Proof of Theorem \[t:decay\] ---------------------------- Clearly, if (i) in Theorem \[t:bu\] does not hold, then ${{\mathbf{D}}}$ is always positive and we are in alternative (A) of Corollary \[c:bdd\_freq\]. Thus $K_0$ is positive and the first statement is obvious. Let ${{\mathbf{K}}}(r):={{\mathbf{I}}}(r)^{-1}$ and observe that by we have $$(1-C\,r){{\mathbf{I}}}(r)\leq {{\mathbf{\bar I}}}(r)\leq (1+C\,r){{\mathbf{I}}}(r)\,,\quad\forall0\leq r\leq r_0\,,$$ which implies $$(1-C\,r)\bar {{\mathbf{K}}}(r)\leq {{\mathbf{K}}}(r)\leq (1+C\,r)\bar {{\mathbf{K}}}(r)\quad \forall \,0\leq r\leq r_0\,,$$ so that in particular ${{\mathbf{K}}}(r)\leq C\, \bar{{\mathbf{K}}}(r)<\infty$ for every $0<r<r_0$ and ${{\mathbf{K}}}(r)\to K_0$ as $r\to 0$. Using the monotonicity formula of Theorem \[t:monot\] together with Corollary \[c:integrability\] we have $$\label{e:K_alto} \bar {{\mathbf{K}}}(r)-K_0\leq K_0 (\exp (4r + 4 {{\mathbf{\Sigma}}}_{OV} (r)) -1) + 4 {{\mathbf{\Sigma}}}_{IV} (r) \exp (4r + 4 {{\mathbf{\Sigma}}}_{OV} (r))\leq C r^\eta\, .$$ Therefore $$\label{e:uniq_ineq_1} {{\mathbf{K}}}(r)-K_0\leq C\,r^{\eta}+C\, {{\mathbf{K}}}(r) \,r\leq C\,r^{\eta}\,.$$ To control ${{\mathbf{K}}}(r) - K_0$ from below we apply with $a=I_0=\frac{1}{K_0}$ and $b=\lambda \leq \min\{\sfrac{\eta}{2}, b_0(I_0)\}$ to infer, after dividing by $r {{\mathbf{D}}}(r)$, that $$-\frac{{{\mathbf{D}}}'(r)}{{{\mathbf{D}}}(r)}\leq \frac{2}{r} \left(I_0(I_0+\lambda) {{\mathbf{K}}}(r) - (2I_0+\lambda)\right)\,.$$ Multiplying this expression by ${{\mathbf{K}}}(r)>0$ and adding $\sfrac2r$, we get $$\begin{aligned} \label{e:epip} \frac{2}{r}-\frac{{{\mathbf{D}}}'(r)}{{{\mathbf{D}}}(r)}{{\mathbf{K}}}(r) &\leq \frac{2}{r}\left[ 1+I_0(I_0+\lambda) {{\mathbf{K}}}^2(r) - (2I_0+\lambda){{\mathbf{K}}}(r)\right]+ \frac{C\,\mathcal{E}_{IV}(r)}{{{\mathbf{D}}}(r)}\notag\\ &\leq \frac{2}{r}\,I_0\left( {{\mathbf{K}}}(r)-\frac{1}{I_0} \right)\,\left((I_0+\lambda) {{\mathbf{K}}}(r)-1\right)+ \frac{C\,\mathcal{E}_{IV}(r)}{{{\mathbf{D}}}(r)}\end{aligned}$$ Since $(I_0+\lambda){{\mathbf{K}}}(r)$ converges to $1+\lambda K_0$, we easily deduce that for $r$ small enough $(I_0+\lambda){{\mathbf{K}}}(r)-1 \geq \frac{\lambda}{2} K_0$. Using this together with , we deduce from that $$\label{e:epip2} \frac{2}{r}-\frac{{{\mathbf{D}}}'(r)}{{{\mathbf{D}}}(r)}\leq \,\frac{\lambda}{r}\, \left({{\mathbf{K}}}(r)-\frac{1}{I_0} \right)+ \frac{C\,\mathcal{E}_{IV}(r)}{{{\mathbf{D}}}(r)}+C\frac{r^{\eta}}{r}.$$ We next compute $$\begin{aligned} \label{e:I' bis} {{\mathbf{K}}}'(r) &=\left(\frac{{{\mathbf{H}}}(r)}{r}\right)'\frac{1}{{{\mathbf{D}}}(r)}-\frac{{{\mathbf{H}}}(r)}{r{{\mathbf{D}}}(r)}\,\frac{{{\mathbf{D}}}'(r)}{{{\mathbf{D}}}(r)} \stackrel{\eqref{e:H'}\&\eqref{e:r_0}}{\leq} \frac{2\,{{\mathbf{E}}}(r)}{r\,{{\mathbf{D}}}(r)}-\frac{{{\mathbf{D}}}'(r)}{{{\mathbf{D}}}(r)} {{\mathbf{K}}}(r)\notag\\ &\stackrel{\eqref{e:OV}\&\eqref{e:r_0}}{\leq} \frac{2}{r} + C -\frac{{{\mathbf{D}}}'(r)}{{{\mathbf{D}}}(r)} {{\mathbf{K}}}(r)+C\,\frac{\mathcal{E}_{OV}(r)}{r{{\mathbf{D}}}(r)}\notag\\ &\stackrel{\eqref{e:epip2}}{\leq} \,\frac{\lambda}{r}\, \left({{\mathbf{K}}}(r)-\frac{1}{I_0} \right)+ \frac{C\,\mathcal{E}_{IV}(r)}{{{\mathbf{D}}}(r)}+C\,\frac{\mathcal{E}_{OV}(r)}{r{{\mathbf{D}}}(r)}+C\frac{r^{\eta}}{r} \,.\end{aligned}$$ Recalling that $ {{\mathbf{K}}}(r)\leq C$, we deduce $$\begin{aligned} \label{e:I' differenziale 3} \frac{d}{dr}\left[\frac{ {{\mathbf{K}}}(r)-K_0}{r^{\lambda}}\right] \leq C\,\frac{\mathcal{E}_{OV}(r)}{r^{1+\lambda}{{\mathbf{D}}}(r)}+C\, \frac{\mathcal{E}_{IV}(r)}{r^{\lambda}\,{{\mathbf{D}}}(r)}+C\frac{1}{r^{1+\lambda-\eta}}\, .\end{aligned}$$ Integrating on the interval $]s,r[$ and using , we get $${{\mathbf{K}}}(r) - K_0 \leq \frac{r^\lambda}{s^\lambda}\left({{\mathbf{K}}}(s)-K_0\right) + C\,r^{\eta-\lambda}$$ that is ${{\mathbf{K}}}(s)-K_0\geq - C s^{\lambda}$. The inequality $|{{\mathbf{K}}}(r)-K_0|\leq C \,r^\lambda$ easily implies $|{{\mathbf{I}}}(r)-I_0|\leq C \,r^\lambda$. For what concerns the other inequalities we compute $$\begin{aligned} \label{e:log H' 1} \left[\log\left(\frac{{{\mathbf{H}}}(r)}{r^{2I_0+1}}\right)\right]' & = \frac{{{\mathbf{H}}}'(r)}{{{\mathbf{H}}}(r)} - \frac{2\,I_0+1}{r}= \frac{2\,{{\mathbf{E}}}(r)}{r\,{{\mathbf{H}}}(r)} - \frac{2\,I_0}{r} \leq \frac{2\,{{\mathbf{D}}}(r)}{{{\mathbf{H}}}(r)} - \frac{2\,I_0}{r} + C\frac{\mathcal{E}_{OV}(r)}{{{\mathbf{H}}}(r)} \notag\\ &= \frac{2}{r}\left({{\mathbf{I}}}(r)-I_0\right) + C\, \frac{\mathcal{E}_{OV}(r)}{{{\mathbf{H}}}(r)} \, ,\end{aligned}$$ and similarly $$\label{e:log H'2} \left[\log\left(\frac{{{\mathbf{H}}}(r)}{r^{2I_0+1}}\right)\right]' \geq \frac{2}{r}\left({{\mathbf{I}}}(r)-I_0\right) - C\, \frac{\mathcal{E}_{OV}(r)}{{{\mathbf{H}}}(r)}\, .$$ Using that $|{{\mathbf{I}}}(r) - I_0|\leq C r^\lambda$, for $r$ small enough we have the bound $r {{\mathbf{D}}}(r)\leq 2 I_0 {{\mathbf{H}}}(r)$. Hence we can use in the integrals of and to deduce the existence of the limit $$H_0:=\lim_{s\downarrow0}\frac{{{\mathbf{H}}}(s)}{s^{2I_0+1}}, \quad\text{with}\quad \left|\frac{{{\mathbf{H}}}(r)}{r^{2I_0+1}}-H_0\right|\leq C\,r^\lambda.$$ Moreover, from we also infer that for $r$ sufficiently small $$H_0 \geq \frac{{{\mathbf{H}}}(r)}{r^{2I_0+1}} e^{-C\,r^{\lambda}} >0.$$ Finally the last assertion follows simply setting $D_0:= I_0 \cdot H_0$ and from $$\begin{aligned} \left\vert\frac{{{\mathbf{D}}}(r)}{r^{2\,I_0}} - D_0\right\vert &= \left\vert{{\mathbf{I}}}(r) \, \frac{{{\mathbf{H}}}(r)}{r^{2\,I_0+1}} - I_0\, H_0\right\vert\notag\\ & \leq \left\vert{{\mathbf{I}}}(r) - I_0\right\vert\, \frac{{{\mathbf{H}}}(r)}{r^{2\,I_0+1}} + I_0\,\left\vert\frac{{{\mathbf{H}}}(r)}{r^{2\,I_0+1}} - H_0\right\vert \leq C\,r^\lambda.\qedhere\end{aligned}$$ Blow-up and proof of Theorem \[t:bu\] {#s:bu} ===================================== As a consequence of the decay estimate in Theorem \[t:decay\] we can show that suitable rescaling of the normal approximation $N$ converge to a unique limiting profile. To this aim we consider for every $r \in (0,1)$ the functions $f_r : {\partial}B_1 \to \I{Q_1}({{\mathbb R}}^{2+n})$ given by $$f_r(z,w) := \frac{{{\mathpzc{N}}}(i_r(z,w))}{r^{I_0}},$$ where we recall that $i_r(z,w)=\left(rz, r^{\sfrac{1}{\bar Q}}w\right)$. We recall also that $T_0{{\mathcal{M}}}= {{\mathbb R}}^2 \times \{0\}$, and $T_0\Sigma = {{\mathbb R}}^{2} \times {{\mathbb R}}^{\bar n} \times \{0\}$. In the following, with a slight abuse of notation, we write ${{\mathbb R}}^{\bar n}$ for the subspace $\{0\} \times {{\mathbb R}}^{\bar n} \times \{0\}$. The final step in the proof of Theorem \[t:bu\] is then the following proposition. \[p:unique limit\] Assume alternative (i) in Theorem \[t:bu\] fails and let $I_0$ and $\lambda$ be the positive numbers of Theorem \[t:decay\]. Then $I_0>1$ and there exists a function $f_0:{\partial}B_1 \to \I{Q}({{\mathbb R}}^{\bar n})$ such that - ${{\bm{\eta}}}\circ f_0 =0$ and $f_0 \not\equiv Q_1\a{0}$; - for every $r$ sufficiently small $$\label{e:unique limit} {{\mathcal{G}}}(f_r(z,w),f_0(z,w))\leq C\, r^{\sfrac{\lambda}{16}}\,\quad \forall\;(z,w)\in {\partial}B_1\,;$$ - the $I_0$-homogeneous extension $g (z,w):=|z|^{I_0}f_0\left(\textstyle{\frac{z}{|z|},\frac{w}{|w|}}\right)$ is nontrivial and ${\textup{Dir}}$-minimizing. In particular, by ${{\rm Im}}(g) \setminus \{0\} \subset{{\mathbb R}}^{2+n}$ is a real analytic submanifold. Theorem \[t:bu\] follows immediately from Proposition \[p:unique limit\] and Theorem \[t:decay\]. Since we have identitified ${{\mathbb R}}^{\bar n}$ with $\{0\}\times {{\mathbb R}}^{\bar n}\times \{0\}$, it is obvious that the map $g$ has all the properties claimed in (ii), namely it is ${\textup{Dir}}$-minimizing, ${{\bm{\eta}}}\circ g \equiv 0$ and it is nontrivial. is a corollary of provided $a_0 \leq \frac{\lambda}{16}$. Next note that has been shown in Theorem \[t:poincare\]. As for observe that, if $4\rho\leq r < 1$, then, by Theorem \[t:decay\], $$D_0 (r- 2\rho)^{2I_0} - C (r-2\rho)^{2I_0 + \lambda} \leq {{\mathbf{D}}}(r - 2\rho) \leq {{\mathbf{D}}}(r+2\rho) \leq D_0 (r+2\rho)^{2 I_0} + C (r+2\rho)^{2I_0 + \lambda}\, .$$ Since $2 I_0 >2$, follows easily from $$\int_{B_{r+2\rho}\setminus B_{r-2\rho}} |D{{\mathpzc{N}}}|^2 = {{\mathbf{D}}}(r+2\rho) - {{\mathbf{D}}}(r-2\rho)\, ,$$ provided $a_0 \leq \lambda$. The rest of this final section of the note is devoted to the proof Proposition \[p:unique limit\], which is split in several steps. Before starting with it, let us however observe that the conclusion $I_0 >1$ is an obvious consequence of the decay estimates of Theorem \[t:decay\] and the fact that ${{\mathbf{D}}}(r) \leq C r^{2+2\gamma_0}$. Step 1: uniqueness of the limit $f_0$ ------------------------------------- For $r$ sufficienly small and $s \in [\frac{r}{2}, r]$, we start estimating the following quantity: $$\begin{aligned} \label{e:stima differenza 1} \int_{{\partial}B_1}{{\mathcal{G}}}(f_r, f_s)^2 \leq (r-s) \int_{{\partial}B_1}\int_s^r\left\vert \frac{d}{dt}f_t(z,w)\right\vert^2\,dt\, .\end{aligned}$$ Using the differentiability properties of Lipschitz multiple valued functions and the $1$-dimensional theory in [@DS1 Section 1.1.2] (note that $t\mapsto {{\mathpzc{N}}}(i_t(z,w))$ is a Lipschitz map), we easily infer that $$\begin{aligned} \left\vert \frac{d}{dt}f_t(z,w)\right\vert^2 & = \sum_{j=1}^{Q} \left\vert \frac{D{{\mathpzc{N}}}\,_j(i_t(z,w))\cdot z}{t^{I_0}} - I_0\,\frac{{{\mathpzc{N}}}\,_j(i_t(z,w))}{t^{I_0+1}} \right\vert^2\notag\\ & = \frac{|z|^2|{\partial}_{\hat r}{{\mathpzc{N}}}|^2(i_t(z,w))}{t^{2I_0}} - 2\,I_0\,\frac{|z|}{t^{2I_0+1}}\,\sum_{j=1}^{Q}\langle{\partial}_{\hat r}{{\mathpzc{N}}}\,_j, {{\mathpzc{N}}}\,_j \rangle(i_t(z,w)) + \frac{|{{\mathpzc{N}}}|^2(i_t(z,w))}{t^{2I_0+2}}.\end{aligned}$$ Therefore, by the change of variable $(z',w') = i_t(z,w)$ in we infer that $$\begin{aligned} \int_{{\partial}B_1}&{{\mathcal{G}}}(f_r, f_s)^2 \leq \frac{r}{2}\int_{\sfrac{r}{2}}^r \left(\frac{{{\mathbf{G}}}(t)}{t^{2I_0+1}} - 2\,I_0 \frac{{{\mathbf{E}}}(t)}{t^{2I_0+2}} + I_0^2\,\frac{{{\mathbf{H}}}(t)}{t^{2I_0+3}}\right)\,dt\notag\\ &\leq \frac{r}{2} \int_{\sfrac{r}{2}}^r \left(\frac{{{\mathbf{D}}}'(t)}{2t^{2I_0+1}} - 2\,I_0 \frac{{{\mathbf{D}}}(t)}{t^{2I_0+2}} + I_0^2\,\frac{{{\mathbf{H}}}(t)}{t^{2I_0+3}} +C\,\frac{\mathcal{E}_{IV}(t)}{t^{2I_0+1}} + C\,\frac{\mathcal{E}_{OV}(t)}{t^{2I_0+2}} \right)\,dt\notag\\ &= \frac{r}{2} \int_{\sfrac{r}{2}}^r \left[\frac{1}{2t}\left(\frac{{{\mathbf{D}}}(t)}{t^{2I_0}}\right)' + I_0 \frac{{{\mathbf{H}}}(t)}{t^{2I_0+3}}\,\left(I_0-{{\mathbf{I}}}(t)\right) +C\,\frac{\mathcal{E}_{IV}(t)}{t^{2I_0+1}} + C\,\frac{\mathcal{E}_{OV}(t)}{t^{2I_0+2}} \right]\,dt.\end{aligned}$$ Using Theorem \[t:decay\], we can then conclude that $$\begin{aligned} \label{e:stima differenza 3} \int_{{\partial}B_1}{{\mathcal{G}}}(f_r, f_s)^2\, &\leq C\,\left\vert\frac{{{\mathbf{D}}}(r)}{r^{2I_0}} - \frac{{{\mathbf{D}}}\left(\frac{r}{2}\right)}{\left(\frac{r}{2}\right)^{2I_0}}\right\vert +C\,\int_{\sfrac{r}{2}}^r\left[\frac{|I_0-{{\mathbf{I}}}(t)|}{t} +C\,\frac{\mathcal{E}_{IV}(t)}{{{\mathbf{D}}}(t)} + C\,\frac{\mathcal{E}_{OV}(t)}{t\,{{\mathbf{D}}}(t)} \right]\,dt\notag\\ &\leq C\, r^{\lambda}.\end{aligned}$$ By an elementary dyadic argument analogous to that of [@DS1 Theorem 5.3], we then infer the existence of $f_0:{\partial}B_1 \to \I{Q}({{\mathbb R}}^{2+n})$ such that, for $r$ sufficiently small, $$\label{e:unique limit 2} \|{{\mathcal{G}}}(f_r, f_0)\|_{L^2({\partial}B_1)}^2\leq C\, r^{\lambda}.$$ Step 2: uniform convergence --------------------------- Set next $h(z,w):={{\mathcal{G}}}\left(\frac{{{\mathpzc{N}}}(z,w)}{|z|^{I_0}},\frac{{{\mathpzc{N}}}(i_{1/2}(z,w))}{|\sfrac z2|^{I_0}}\right)$. It follows from that for $r$ sufficiently small $$\label{e:diff_height} \int_{B_r} h^2\leq \int_0^r\,\int_{{\partial}B_1} {{\mathcal{G}}}(f_t,f_{\sfrac t2})^2\,t\,dt\stackrel{\eqref{e:stima differenza 3}}{\leq} C \,r^{2+\lambda}\,,$$ and from and $$\label{e:lip_diff} {{\rm {Lip}}}(h|_{B_1\setminus B_s})\leq C\, s^{-I_0}.$$ Moreover, for every $\rho < \sfrac{|z|}{4}$ we claim the estimate $$\begin{gathered} \label{e:en_diff} \int_{B_{\rho}(z,w)}|Dh|^2\leq C\, \rho + C\,|z|^{\lambda}\, .\end{gathered}$$ Indeed $|Dh| \leq C\, \left|D\left(\frac{{{\mathpzc{N}}}}{|z|^{I_0}}\right)\right|$ and by Theorem \[t:decay\] $$\begin{aligned} \int_{B_{\rho}(z,w)}\left|D\left(\frac{{{\mathpzc{N}}}}{|z|^{I_0}}\right)\right|^2 & \leq 2\, \int_{|z|-\rho}^{|z|+\rho} \int_{{\partial}B_t} \left(\frac{|D{{\mathpzc{N}}}|^2}{t^{2I_0}}+I_0^2 \, \frac{|{{\mathpzc{N}}}|^2}{t^{2I_0+2}} \right)\,dt\\ &\leq 2\int_{|z|-\rho}^{|z|+\rho} \left(\left(\frac{{{\mathbf{D}}}(t)}{t^{2I_0}}\right)'+2\,I_0\frac{{{\mathbf{D}}}(t)}{t^{2I_0+1}}+I_0^2\, \frac{{{\mathbf{H}}}(t)}{t^{2I_0+2}}\right)\,dt\\ &\leq C\, \left(|z|+\rho \right)^\lambda + C \log\left( \frac{|z|+\rho}{|z|-\rho} \right)\leq C \,|z|^\lambda + C\,\frac{\rho}{|z|}.\end{aligned}$$ In particular, applying , and with $\rho = |z|^{1+\sfrac{\lambda}{4}}$, we infer that for every point $p = (z,w)\in {{\mathfrak{B}}}_{\bar{Q}}$ with $|z|$ sufficiently small $$\begin{aligned} h(p) & \leq \Bigg\vert h(p) - \mint_{B_{\frac{|z|^{1+\sfrac{\lambda}{4}}}{2^k}} (p)} h\Bigg\vert + \sum_{i=0}^{k-1}\Bigg\vert \mint_{B_{\frac{|z|^{1+\sfrac{\lambda}{4}}}{2^i}} (p)} h- \mint_{B_{\frac{|z|^{1+\sfrac{\lambda}{4}}}{2^{i+1}}} (p)} h\Bigg\vert + \mint_{B_{|z|^{1+\sfrac{\lambda}{4}}}(p)} h\notag\\ &\leq {{\rm {Lip}}}(h|_{B_1 (p) \setminus B_{\sfrac{|z|}{2}} (p)}) \frac{|z|^{1+\sfrac{\lambda}{4}}}{2^k} + C\sum_{i=0}^{k-1}\frac{|z|^{1+\sfrac{\lambda}{4}}}{2^i} \mint_{B_{\frac{|z|^{1+\sfrac{\lambda}{4}}}{2^i}} (p)} |Dh| + \mint_{B_{|z|^{1+\sfrac{\lambda}{4}}} (p)} h\notag\\ &\stackrel{\eqref{e:lip_diff}}{\leq} C\, |z|^{1+\sfrac{\lambda}{4}} + C\,\sum_{i=0}^{k-1}\,\left(\int_{B_{|z|^{1+\sfrac{\lambda}{4}}}} |Dh|^2\right)^{\frac12} + \frac{C}{|z|^{1+\sfrac{\lambda}{4}}}\left(\int_{B_{2|z|}} |h|^2\right)^{\frac12}\, ,\end{aligned}$$ where we have used the standard Poincaré inequality $$\left\vert \mint_{B_r} f - \mint_{B_{\frac{r}{2}}} f\right\vert \leq C\, r\,\mint_{B_r}|Df|\,\quad f\in W^{1,2}.$$ Now choose $k \in {{\mathbb N}}$ such that $\frac{|z|^{1+\sfrac{\lambda}{4}}}{2^k} < |z|^{1+\sfrac{\lambda}{4}+I_0} \leq \frac{|z|^{1+\sfrac{\lambda}{4}}}{2^{k-1}}$ (in particular $k\leq |\log|z||$) and use together with to bound $$h (z,w) \leq C\,|z|^{1+\sfrac{\lambda}{4}} + C\,|\log |z||\,|z|^{\sfrac{\lambda}{8}} + C\,|z|^{\sfrac{\lambda}{4}} \leq C\,|z|^{\sfrac{\lambda}{16}}\,,$$ This gives that, for a sufficiently small $r$, $$\max_{\partial B_1} {{\mathcal{G}}}(f_r, f_{r/2}) \leq C r^{\sfrac{\lambda}{16}}\, .$$ Thus $$\max_{\partial B_1} {{\mathcal{G}}}(f_r, f_0) \leq \sum_{k=0}^\infty {{\mathcal{G}}}(f_{r2^{-k}}, f_{r2^{-k-1}}) \leq C r^{\sfrac{\lambda}{16}}\, .$$ Step 3: nontriviality of the limit and other properties ------------------------------------------------------- To show that $f_0 \neq Q\a{0}$ it is enough to observe that, by Theorem \[t:decay\], $$\int_{{\partial}B_1} |f_0|^2=\lim_{r\to 0} \int_{{\partial}B_1} |f_r|^2 =\lim_{r\to 0}\frac{{{\mathbf{H}}}(r)}{r^{2I_0+1}}=H_0> 0.$$ In order to show that ${{\bm{\eta}}}\circ f_0 \equiv 0$, we notice that by a simple slicing argument combined with there exists a sequence of radii $r_k \in [2^{-k-1}, 2^{-k}]$ such that $$\begin{aligned} \int_{{\partial}B_{r_k}} |{{\bm{\eta}}}\circ {{\mathpzc{N}}}| &\leq 2^{k+1} \int_{B_{2^{-k}}\setminus B_{2^{-k-1}}} |{{\bm{\eta}}}\circ {{\mathpzc{N}}}| \leq C\, r_k^{\gamma_0}\int_{B_{2^{-k}}}|z|^{\gamma_0-1}|{{\bm{\eta}}}\circ {{\mathpzc{N}}}|\notag\\ &\leq C\, r_k^{\gamma_0+2\eta}{{\mathbf{D}}}(2r_k) \leq C\, r_k^{\gamma_0+2\eta + 2I_0},\end{aligned}$$ from which $$\begin{aligned} \int_{{\partial}B_1} |{{\bm{\eta}}}\circ f_0| &=\lim_{r_k\to0}\int_{{\partial}B_1} |{{\bm{\eta}}}\circ f_{r_k}|= \lim_{r_k\to 0}r_k^{-I_0-1}\int_{{\partial}B_r} |{{\bm{\eta}}}\circ {{\mathpzc{N}}}|\\ & \leq C\,\lim_{r_k\to 0} r_k^{\gamma_0+2\eta+I_0-1} =0.\end{aligned}$$ Next we show that $f_0$ takes values in ${{\mathbb R}}^{\bar n}$. We start by showing that $f_0$ must take values in $T_0\Sigma={{\mathbb R}}^{2+\bar{n}}\times \{0\}$. Indeed, if we set $f_r (z,w) := \bar{{\mathpzc{N}}}(i_r (z,w))$, using and $|{{\mathpzc{N}}}|(i_r(z,w)) \leq C\, r^{1+\sfrac{\gamma_0}{2}}$ we conclude $$\int_{{\partial}B_1} {{\mathcal{G}}}(f_r,\bar f_r)^2\leq \frac{Cr^2}{r^{2I_0+1}}\,\int_{{\partial}B_r} |{{\mathpzc{N}}}|^2\leq C r^2 \,,$$ which implies that $f_0(z,w)\in \I{Q}(T_0\Sigma)$. Next observe that $f_r (z,w) = \sum_i \a{{{\mathpzc{N}}}\,_i (i_r (z,w))}$ has the property that each ${{\mathpzc{N}}}\,_i (i_r (z,w))$ is orthogonal to $T_{{{\bm{\Psi}}}(i_r (z,w))} {{\mathcal{M}}}$. In particular, if $|z|=1$ and $r\downarrow 0$, the tangent planes $T_{{{\bm{\Psi}}}(i_r (z,w))} {{\mathcal{M}}}$ converge to ${{\mathbb R}}^2\times \{0\}$: it follows, by the uniform convergence of $f_r$ to $f_0$, that $f_0 (z,w) = \sum_i \a{(f_0)_i (z,w)}$ for some $(f_0)_i (z,w)$ which are orthogonal to ${{\mathbb R}}^2\times \{0\}$. We thus conclude that each $(f_0)_i (z,w)$ belongs to $\{0\}\times {{\mathbb R}}^{\bar n}\times \{0\}$. Step 4: Minimality of $g$ ------------------------- In order to complete the proof of Proposition \[p:unique limit\] we need to show that $g$ is ${\textup{Dir}}$-minimizing. Given the homogeneity of $g$ in the radial direction, it suffices to show that there is no $W^{1,2}$ multifunction $h: B_1 \to {{\mathcal{A}}_Q}({{\mathbb R}}^{\bar{n}})$ which has the same trace of $g$ on $\partial B_1$ and less energy on $B_1$. Assume thus by contradiction that there is an $h\in W^{1,2} (B_1, {{\mathcal{A}}_Q}({{\mathbb R}}^{\bar n}))$ such that $h|_{\partial B_1}$ and $$\int |Dh|^2 \leq \int |Dg|^2 - \delta$$ for some positive $\delta>0$. Recall the definition of $W^{1,2}$ according to Remark \[r:W12\]: using the map ${{\mathbf{W}}}$ in there and the functions $h\circ {{\mathbf{W}}}$ and $g\circ {{\mathbf{W}}}$ we can use the theory of [@DS1] and assume that $h\circ {{\mathbf{W}}}$ is a ${\textup{Dir}}$-minimizer on the euclidean disk $D_1 \subset {{\mathbb R}}^2$. Observe also that, since ${{\bm{\eta}}}\circ g \equiv 0$, we must have ${{\bm{\eta}}}\circ h \equiv 0$ as well. Indeed since $h\circ {{\mathbf{W}}}=g\circ {{\mathbf{W}}}$ on ${\partial}D_1$, we have ${{\bm{\eta}}}\circ h\circ {{\mathbf{W}}}={{\bm{\eta}}}\circ g\circ {{\mathbf{W}}}=0$ on the boundary and considering that $$\int_{D_1} \sum_i |D(h_i \circ {{\mathbf{W}}}-{{\bm{\eta}}}\circ h\circ {{\mathbf{W}}})|^2\leq \int_{D_1}|D (h\circ {{\mathbf{W}}})|^2-Q\int_{D_1}|D ({{\bm{\eta}}}\circ h\circ {{\mathbf{W}}})|^2\, ,$$ the minimality of $h\circ {{\mathbf{W}}}$ forces the Dirichlet energy of ${{\bm{\eta}}}\circ h \circ {{\mathbf{W}}}$ to vanish identically. Using , the decay ${{\mathbf{D}}}(r)\leq C\,r^{2I_0}$ and a Fubini-type argument we can find a sequence of radii $s_j\to 0$ such that $$\label{e:bounds} \int_{{\partial}B_1}|Df_{0}|^2\leq \limsup_j \int_{{\partial}B_1}|Df_{s_j}|^2\leq \limsup_j \frac{{{\mathbf{D}}}'(s_j)}{s_j^{2I_0-1}}\leq C\,.$$ We now wish to “smooth” $h$, i.e. to approximate it with a sequence of Lipschitz maps $h_{{\varepsilon}}$ such that ${{\bm{\eta}}}\circ h_{{\varepsilon}}\equiv 0$, $$\begin{gathered} \int_{B_1} |Dh_{{\varepsilon}}|^2-|Dh|^2\leq {{\varepsilon}}^2 \label{e:est_B1}\\ \int_{{\partial}B_1}{{\mathcal{G}}}(f_0,h_{{{\varepsilon}}})^2+\Bigg|\int_{\partial B_1} |Df_0|^2 - |Dh_{{{\varepsilon}}}|^2\Bigg|\leq {{\varepsilon}}^2 \label{e:est_dB1}\,.\end{gathered}$$ We would like to appeal to [@DS3 Lemma 3.5], but there is the slight technical complication that ${{\mathfrak{B}}}$ is not regular. We postpone this technical step and continue with the argument assuming the existence of the approximations $h_{{\varepsilon}}$. Next we would like to apply [@DS3 Lemma 3.6] to $h_{{\varepsilon}}$ and ${{\mathbf{p}}}_{T_0\Sigma }(f_{s_j})=:\bar f_{s_j}$, to get a family of competitor functions $(\hat f_{s_j})\subset W^{1,2}(B_1,\I{Q}({{\mathbb R}}^{2+\bar n}))$, such that $\hat f_{s_j}|_{{\partial}B_1}= \bar f_{s_j} |_{{\partial}B_1})$ and $$\begin{gathered} \int_{B_1} |D\hat f_{s_j}|^2\leq \int_{B_1} |Dh_{{\varepsilon}}|^2+{{\varepsilon}}\int_{{\partial}B_1} \left(|D_\tau h_{{\varepsilon}}|^2 +|D_\tau \bar f_{s_j}|^2\right)+\frac{C}{{{\varepsilon}}} \int_{{\partial}B_1} {{\mathcal{G}}}(h_{{\varepsilon}},\bar f_{s_j})^2\label{e:est_en_B1}\,,\\ {{\rm {Lip}}}(\hat f_{s_j}) \leq C\left({{\rm {Lip}}}(h_{{\varepsilon}})+{{\rm {Lip}}}(\bar f_{s_j})+\frac{1}{{{\varepsilon}}}\sup_{{\partial}B_1}{{\mathcal{G}}}(\bar f_{s_j},h_{{\varepsilon}})\right)\label{e:est_Lip_B1}\\ {{\bm{\eta}}}\circ \hat{f}_{s_j} = {{\bm{\eta}}}\circ \bar{f}_{s_j} \label{e:est_med_B1_2}\,. \end{gathered}$$ Again, this is not straightforward because [@DS3 Lemma 3.6] is stated for euclidean domains. We postpone this second technical problem and continue with our argument assuming the existence of $\hat{f}_{s_j}$. We are now ready to define our competitor function. We set $\bar {{\mathpzc{L}}}_{s_j} (z,w):=s_j^{I_0} \hat f_{s_j} (i_{\sfrac{1}{s_j}}(z,w))$ and, observing that $\bar {{\mathpzc{L}}}_{s_j}$ takes value in ${{\mathcal{A}}_Q}(T_0 \Sigma)$, we use to define a corresponding ${{\mathpzc{L}}}_{s_j}$, which clearly is a competitor ${{\mathpzc{N}}}$ in $B_{s_j}$ according to Definition \[d:competitors\]. Moreover $${{\rm {Lip}}}({{\mathpzc{L}}}_{s_j})\leq C \, s_j^{I_0+1} {{\rm {Lip}}}(\hat f_{s_j}|_{B_1})\stackrel{\eqref{e:lip_diff}}{\leq} C\, {s_j}^{\eta}\,.$$ Therefore we can apply Proposition \[p:amin\] with $\bar {{\mathpzc{L}}}=\bar {{\mathpzc{L}}}_{s_j}$. In particular, taking into account Theorem \[t:decay\] and , we conclude that $$\int_{B_{s_j}} |D\bar {{\mathpzc{N}}}|^2 \leq (1+ Cs_j) \int_{B_{s_j}} |D\bar{{\mathpzc{L}}}_{s_j}|^2 + C {{\bm m}_0}^{\sfrac{1}{2}} \int_{B_{s_j}} |z|^{\gamma_0-1} |{{\bm{\eta}}}\circ {{\mathpzc{L}}}_{s_j}| + C s_j^{2I_0+\eta}\, .$$ Next, recall the inequality : $$\int_{B_{s_j}} |z|^{\gamma_0-1} |{{\bm{\eta}}}\circ {{\mathpzc{L}}}_{s_j}| \leq C \int_{B_{s_j}} |z|^{\gamma_0-1} |{{\bm{\eta}}}\circ \bar {{\mathpzc{L}}}_{s_j}| + C \int_{B_{s_j}} |z|^{\gamma_0-1} |\bar {{\mathpzc{L}}}_{s_j}|^2\, .$$ By the first term in the right hand side equals indeed $$C \int_{B_{s_j}} |z|^{\gamma_0-1} |{{\bm{\eta}}}\circ \bar {{\mathpzc{N}}}| \leq C s_j^\eta {{\mathbf{D}}}(s_j) \leq C s_j^{2I_0+\eta}\, .$$ For the second term we use the Poincaré inequality $$\label{e:ancora_poinc} \int_{B_{s_j}} |z|^{\gamma_0-1} |\bar {{\mathpzc{L}}}_{s_j}|^2 \leq C s_j^{1+\gamma_0} \int_{B_{s_j}} |D\bar {{\mathpzc{L}}}_{s_j}|^2 + C s_j^{\gamma_0} \int_{\partial B_{s_j}} |\bar {{\mathpzc{L}}}_{s_j}|^2\, ,$$ whose proof will be given in Lemma \[l:poinc\]. Using that $$\int_{\partial B_{s_j}} |\bar {{\mathpzc{L}}}_{s_j}|^2 = \int_{\partial B_{s_j}} |\bar {{\mathpzc{N}}}|^2 ={{\mathbf{H}}}(s_j) \leq C s_j^{2I_0+1}\, ,$$ we easily conclude that $$\label{e:ci_siamo_quasi} \int_{B_{s_j}} |D\bar {{\mathpzc{N}}}|^2 \leq (1+ Cs_j) \int_{B_{s_j}} |D\bar{{\mathpzc{L}}}_{s_j}|^2 + C s_j^{2I_0+\eta}\, .$$ Changing variables and dividing by $s_j^{2I_0}$ we infer that $$\label{e:quasi_2} \int_{B_1} |D \bar{f}_{s_j}| \leq \int_{B_1} |D \hat{f}_{s_j}|^2 + C s_j^\eta\, .$$ Using , and , we conclude $$\begin{aligned} \int_{B_1} |D \bar{f}_{s_j}|^2 \leq & \int_{B_1} |Dh|^2 + C s_j^\eta + C \varepsilon + \frac{C}{\varepsilon} \int_{\partial B_1} {{\mathcal{G}}}(f_0, \bar f_{s_j})^2\\ \leq & \int_{B_1} |Dg|^2 -\delta + C s_j^\eta + C \varepsilon + \frac{C}{\varepsilon} \int_{\partial B_1} {{\mathcal{G}}}(f_0, \bar f_{s_j})^2\, ,\end{aligned}$$ where the constant $C$ is independent of $\varepsilon$. In particular, if we fix $\varepsilon$ sufficiently small and we then let $s_j\downarrow 0$, by the uniform convergence of $f_{s_j}$ to $f_0$ on $\partial B_1$ we conclude $$\limsup_{j\to\infty} \int_{B_1} |D\bar f_{s_j}|^2 \leq \int_{B_1} |Dg|^2 - \frac{\delta}{2}\, .$$ Since however $f_{s_j} \to g$ in $B_1$, the latter inequality contradicts the semicontinuity of the Dirichlet energy. Step 5: Technical leftovers --------------------------- First of all we show the existence of the map $h_{{\varepsilon}}$ as in and . We consider $h\circ {{\mathbf{W}}}$, which is defined on the closed unit disk $\bar{D}_1\subset {{\mathbb R}}^2$. We then can apply [@DS3 Lemma 3.5] to the latter map and generate approximations $\hat h_\varepsilon$ which satisfy the bounds and with $D_1$ in place of $B_1$ and $h\circ {{\mathbf{W}}}$ in place of $h$. The maps $h_\varepsilon:= \hat h_{{\varepsilon}}\circ {{\mathbf{W}}}$ would then satisfy the desired estimates because of the conformality of ${{\mathbf{W}}}^{-1}$ (which keeps the Dirichlet energy invariant) and its regularity in $B_1\setminus \{0\}$ (which results into the loss of a constant factor in ). However the resulting map would not be Lipschitz because of the singularity of ${{\mathbf{W}}}^{-1}$ in the origin. To overcome this difficulty it suffices to perturb slightly $\hat h_\varepsilon$ so that it is constant in a small neighborhood of the origin. As for the condition ${{\bm{\eta}}}\circ h_{{\varepsilon}}\equiv 0$, this can easily be achieved subtracting the average to whichever extension satisfies and . Secondly we show the existence of $\hat{f}_{s_j}$. First of all we observe that the condition can be easily achieved after we prove the existence of a map which satisfies the other two conditions: as above it suffices to subtract the average of this map and add back ${{\bm{\eta}}}\circ \bar{f}_{s_j}$. At this point we observe that it suffices, as above, to compose with the map ${{\mathbf{W}}}$, apply [@DS1 Lemma 2.14] and [@DS2 Lemma 3.6] and compose the resulting map with ${{\mathbf{W}}}^{-1}$: indeed the latter would coincide with $h_\varepsilon \circ {{\mathbf{W}}}$ on $D_{1-{{\varepsilon}}}$ and on the complement ${{\mathbf{W}}}^{-1}$ is regular. Some useful lemmas. {#a:proiezione} =================== The first lemma is a simple version of the Poincaré inequality for $W^{1,2}$ functions. \[l:poinc\] There exists a universal constant $C>0$ such that the following two inequalities hold for every $f\in W^{1,2}(B_r, {{\mathcal{A}}_Q})$ with $B_r\subset {{\mathfrak{B}}}_{Q}$: $$\begin{aligned} \int_{B_r} |f|^2\leq C r^2 \,\int_{B_r} |Df|^2+C r\,\int_{{\partial}B_r}|f|^2 \label{e:poinc1}\\ \int_{B_r} |z|^{\gamma_0-1} |f|^2 \leq C\,r^{1+\gamma_0} \int_{B_r} |D f|^2 + C\,r^{\gamma_0} \int_{\partial B_1} |f|^2\, .\label{e:poinc2}\end{aligned}$$ By approximation we can assume, without loss of generality, that $f$ is Lipschitz and, by scaling, it suffices to show the inequalities and on the ball $B_1$. Fixing $|z|=1$ and integrating along rays $$|f (r z,r^{1/Q} w)|^2 \leq 2 |f (z,w)|^2 + 2 \int_r^1 |Df (tz, t^{1/Q} w)|^2\, dt\, .$$ Using radial coordinates we then conclude $$\int_{B_1} |z|^{\gamma_0-1} |f|^2 \leq C \int_{\partial B_1} |f|^2 + \int_{\partial B_1} \int_0^1 r^\gamma_0 \int_r^1 |Df (tz, t^{\sfrac{1}{Q}} w)|^2\, dt\, dr\, dz\, .$$ Using Fubini the latter integral can be rewritten as $$\int_0^1 \int_{\partial B_1} |Df (tz, t^{\sfrac{1}{Q}} w)|^2 \int_0^t r^{\gamma_0}\, dt\, dz\, dr \leq \int_0^1 t \int_{\partial B_1} |Df (tz, t^{\sfrac{1}{Q}} w)|^2\, dz\, dr\, .$$ This completes the proof of . The proof of is a simple variation of this one and is left to the reader. \[l:proiezione\] Let $\bar {{\mathpzc{L}}}\colon {{\mathfrak{B}}}_{\bar Q} \to \I{Q}({{\mathbb R}}^{2+\bar n})$ be Lipschitz and consider the map ${{\mathpzc{L}}}\colon {{\mathfrak{B}}}_{\bar Q} \to \I{Q}({{\mathbb R}}^{2+n})$ defined by . Then there exists a constant $C:=C(\|\Psi_0\|_{C^3})>0$ such that $$\begin{gathered} {{\mathcal{G}}}({{\mathpzc{L}}},\bar {{\mathpzc{L}}})(z,w)\leq C\,r\,|\bar{{{\mathpzc{L}}}}|(z,w)+ C\, |\bar{{{\mathpzc{L}}}}|^2(z,w)\,,\quad \forall (z,w)\in B_r\label{e:P1}\\ \int_{B_r}|D{{\mathpzc{L}}}|^2\leq (1+Cr)\int_{B_r} |D\bar{{{\mathpzc{L}}}}|^2 +C\,r\,\int_{{\partial}B_r}|\bar{{{\mathpzc{L}}}}|^2\label{e:P2}\, .\end{gathered}$$ For what concerns , observe that $D\Psi(0)=0$ implies $\|D\Psi\|_{L^\infty(B_r)}\leq C r$. Therefore, by the $C^{3}$ regularity of $\Psi$, we get $$\begin{aligned} {{\mathcal{G}}}({{\mathpzc{L}}},\bar {{\mathpzc{L}}})(z,w) &= \sum_{j=1}^{Q} |\Psi({{\mathbf{p}}}_0({{\bm{\Psi}}})+\bar{{{\mathpzc{L}}}}_j)-\Psi({{\mathbf{p}}}_0({{\bm{\Psi}}}))| (z,w)\\ &\leq \|D\Psi\|({{\bm{\Psi}}}(z,w))\,|\bar {{\mathpzc{L}}}|(z,w)+ \|A_{\Sigma}\|\,|\bar {{\mathpzc{L}}}|^2(z,w)\\ &\leq C\,r\,|\bar {{\mathpzc{L}}}|(z,w)+C\,|\bar {{\mathpzc{L}}}|^2\,.\end{aligned}$$ An analogous computation gives $$\int_{B_r}|D{{\mathpzc{L}}}|^2\leq (1+Cr)\int_{B_r} |D\bar{{{\mathpzc{L}}}}|^2 +C\,\int_{B_r}|\bar{{{\mathpzc{L}}}}|^2$$ and we conclude using Lemma \[l:poinc\]. [^1]: We remark that the competitor used here does not coincide, in general, with the Dirichlet minimizer with boundary value $\bar {{\mathpzc{N}}}|_{\partial B_r}$.
--- abstract: 'We prove that large cardinals need not generally exhibit their large cardinal nature in . For example, a supercompact cardinal $\kappa$ need not be weakly compact in , and there can be a proper class of supercompact cardinals in $V$, none of them weakly compact in $\HOD$, with no supercompact cardinals in $\HOD$. Similar results hold for many other types of large cardinals, such as measurable and strong cardinals.' address: - 'Department of Philosophy, Wuhan University, BaYi Road 299, Wuchang District, Wuhan, Hubei Province, P.R.China. 430072' - 'Kurt Gödel Research Center, University of Vienna, Währinger Strasse 25 A-1090 Vienna, Austria' - 'Philosophy, New York University & Mathematics, Philosophy, Computer Science, The Graduate Center of The City University of New York & Mathematics, College of Staten Island of CUNY' author: - Yong Cheng - 'Sy-David Friedman' - Joel David Hamkins bibliography: - 'MathBiblio.bib' - 'HamkinsBiblio.bib' title: Large cardinals need not be large in HOD --- Introduction ============ In this article, we shall prove that large cardinals need not generally exhibit their large cardinal nature in , the inner model of hereditarily ordinal-definable sets, and there can be a divergence in strength between the large cardinals of the ambient set-theoretic universe $V$ and those of $\HOD$. Our general theme concerns the questions:  \[Question.HODquestion\] 1. To what extent must a large cardinal in $V$ exhibit its large cardinal properties in ? 2. To what extent does the existence of large cardinals in $V$ imply the existence of large cardinals in $\HOD$? For large cardinal concepts beyond the weakest notions, we prove, the answers are generally negative. In Theorem \[Theorem.SupercompactNotWCinHOD\], for example, we construct a model with a supercompact cardinal that is not weakly compact in $\HOD$, and Theorem \[Theorem.ProperClassSCNotLargeInHOD\] extends this to a proper class of supercompact cardinals, none of which is weakly compact in $\HOD$, thereby providing some strongly negative instances of (1). The same model has a proper class of supercompact cardinals, but no supercompact cardinals in $\HOD$, providing a negative instance of (2). A natural strengthening of these situations would be a model with a proper class of supercompact cardinals, but no weakly compact cardinals in $\HOD$, but such a situation is impossible for reasons we discuss in Section \[Section.Limitations\]. Conjecture \[Conjecture.Woodin\] is an intriguing positive instance of (2) recently proposed by W. Hugh Woodin, and many other natural possibilities remain as open questions. Large cardinals that are not weakly compact in {#Section.IndividualLC} =============================================== Let us begin with an elementary case, showing that a measurable cardinal $\kappa$ need not be measurable in $\HOD$. We shall subsequently strengthen this in Theorem \[Theorem.SupercompactNotWCinHOD\] to show that even a supercompact cardinal need not be weakly compact in . \[Theorem.MeasurableNotMeasurableInHOD\] If $\kappa$ is measurable, then there is a forcing extension in which $\kappa$ is measurable, but not measurable in . Suppose that $\kappa$ is measurable in $V$, and assume without loss of generality, by forcing if necessary [@Jensen1974:MeasurableCardinalsAndTheGCH], that the  holds. Let $\P$ be the Easton-support $\kappa$-iteration, which forces with $\Q_\gamma=\Add(\gamma,1)$ at every inaccessible cardinal $\gamma\lt\kappa$. Suppose that $G\of\P$ is $V$-generic. The forcing is $\kappa$-c.c. and preserves the inaccessibility of $\kappa$. There is no forcing (yet) at stage $\kappa$. We show first that $\kappa$ is not measurable in $V[G]$. If it were measurable there, then let $j:V[G]\to \Mbar$ be a normal ultrapower embedding. By elementarity, we may decompose $\Mbar=M[j(G)]$ as a forcing extension of the ground model $M=\Union_\alpha j(V_\alpha)$. Since $\kappa$ is inaccessible in $M$, there is nontrivial forcing at stage $\kappa$ in $j(\P)$, and so $j(G)=G*g*\Gtail$ for some $M[G]$-generic filter $g$ for $\Add(\kappa,1)^{M[G]}$. But since $P(\kappa)^V\of M$ and $\P$-names for subsets of $\kappa$ are coded in $H_{\kappa^+}$, it follows that $P(\kappa)^{V[G]}\of M[G]$. In particular, it must be that $g\in M[G]$, contrary to the assumption that $g$ is $M[G]$-generic. Next, we perform additional forcing over $V[G]$. Let $\R$ be the set-support product forcing to code $P(\kappa)^{V[G]}$ into the  or $\Diamond^*_\lambda$ patterns (see Appendix) at regular cardinals above $\kappa^{++}$, with $\leqkappa$-closed forcing also preserving $P(P(\kappa))$. Suppose that $H\of\R$ is $V[G]$-generic, and then lastly let $g\of\kappa$ be $V[G][H]$-generic for $\Add(\kappa,1)$, which is the same in $V[G][H]$ as in $V[G]$. Our final model is $V[G][H][g]$. We claim that $\kappa$ is not measurable in $\HOD^{V[G][H][g]}$. To see this, note first that because $\kappa$ is not measurable in $V[G]$ and $H$ preserves $P(P(\kappa))$, it follows that $\kappa$ is not measurable in $V[G][H]$. Next, observe that since $P(\kappa)^{V[G]}$, which is the same as $P(\kappa)^{V[G][H]}$, is explicitly coded in $V[G][H]$ and this coding is preserved by the subsequent forcing to add $g$ (using Lemma \[Lemma.Diamond\*InvariantBySmallForcing\] in the case of $\Diamond^*$ coding), it follows that $P(\kappa)^{V[G][H]}\of\HOD^{V[G][H][g]}$. Meanwhile, since $\Add(\kappa,1)$ is weakly homogeneous (see appendix for definition), it follows by Lemma \[Lemma.WeaklyHomogenousControlsHOD\] that $\HOD^{V[G][H][g]}\of V[G][H]$. So $\HOD^{V[G][H][g]}$ and $V[G][H]$ agree on $P(\kappa)$, but there is no measure on $\kappa$ in $V[G][H]$, and consequently there can be none in $\HOD^{V[G][H][g]}$, because this model is contained in $V[G][H]$. Meanwhile, $\kappa$ is measurable in $V[G][H][g]$. The standard master-condition lifting arguments show that $\kappa$ remains measurable in $V[G][g]$, and this will be preserved to $V[G][g][H]=V[G][H][g]$. Specifically, fix a normal ultrapower embedding $j:V\to M$, and consider the forcing $j(\P)$, which factors as $\P*\Q_\kappa*\Ptail$. Using $2^\kappa=\kappa^+$, one may construct an $M[G][g]$-generic filter $\Gtail\of\Ptail$ and lift the embedding to $j:V[G]\to M[j(G)]$ with $j(G)=G*g*\Gtail$. Using $g$ as a master condition in $j(\Q_\kappa)$, one similarly constructs an $M[j(G)]$-generic filter $j(g)\of j(\Q_\kappa)$ and lifts the embedding fully to $j:V[G][g]\to M[j(G)][j(g)]$, witnessing that $\kappa$ is measurable in $V[G][g]$. Next, since $\R$ is $\leqkappa$-closed and $\Add(\kappa,1)$ is $\kappa^+$-c.c. in $V[G]$, it follows by Lemma \[Lemma.ClosureDistributive\] that $\R$ remains $\leqkappa$-distributive in $V[G][g]$. Thus, the forcing to add $H$ over $V[G][g]$ adds no new subsets of $\kappa$ not already in $V[G][g]$, and so $\kappa$ remains measurable in $V[G][g][H]$, which is the same as $V[G][H][g]$, as desired. We have therefore produced a model $V[G][H][g]$ in which $\kappa$ is measurable, but not measurable in , as desired. In fact, we could easily have coded much more of $V[G]$ than just $P(\kappa)$. We could have coded some large $V_\theta[G]$, for example, or indeed, we could let $\R$ be the proper class $\leqkappa$-closed forcing to code all the sets of $V[G]$ into the  or $\Diamond^*$ patterns, with the result by homogeneity that $\HOD^{V[G][H][g]}=V[G]$, where $\kappa$ is not measurable, though it is measurable in $V[G][H][g]$ as above. Meanwhile, let us now modify the proof in order to provide a stronger result: If $\kappa$ is a measurable cardinal, then there is a forcing extension in which $\kappa$ remains measurable, but is not weakly compact in $\HOD$. Suppose that $\kappa$ is a measurable cardinal in $V$. By preparatory forcing, if necessary, we may assume that the measurability of $\kappa$ is indestructible by $\Add(\kappa,1)$. (This can be accomplished, for example, by first forcing $2^\kappa=\kappa^+$ and then performing the Easton-support iteration that adds a Cohen subset to every inaccessible cardinal up to and including $\kappa$; alternatively, one can use the lottery preparation [@Hamkins2000:LotteryPreparation].) Consider now the forcing $\Add(\kappa,1)$ to add a Cohen subset to $\kappa$, and suppose that $g\of\kappa$ is the resulting $V$-generic Cohen subset. By Lemma \[Lemma.Kunen\], the forcing $\Add(\kappa,1)$ may be factored as a two-step forcing iteration $\SS*\Tdot$, where the first step $\SS$ is the forcing to add a weakly homogeneous $\kappa$-Suslin tree $T$ and the second step $\Tdot$ simply forces with that tree, adding a branch through it. In our case, we may factor the extension $V[g]$ as $V[T][b]$, where $g\iso T*b$, first adding the weakly homogeneous $\kappa$-Suslin tree $T$ and then adding a branch $b$ through $T$. Since $T$ is a $\kappa$-Suslin tree in $V[T]$, it follows that $\kappa$ is not weakly compact there. In $V[T]$, the tree $T$ has size $\kappa$ and so there is some set $E\of\kappa$ that codes $T$ in some canonical way. Let $\R$ be the Easton product forcing in $V[T]$ that codes $E$ into the  or $\Diamond^*_\lambda$ patterns on the next $\kappa$ many regular cardinals above $\kappa$. Suppose that $H\of\R$ is $V[T]$-generic. This is equivalent to assuming $H$ is $V[g]$-generic, since $T$ is still $\kappa$-Suslin in $V[T][H]$, and so $b$ is $V[T][H]$-generic (as any cofinal branch through a Suslin tree is), making $b$ and $H$ mutually $V[T]$-generic. Thus, because $\R\times T\iso T\times\R$ in $V[T]$, we may view our final model as any of the three forcing iterations $V[T][H][b]=V[T][b][H]=V[g][H]$. By our indestructibility assumption on $\kappa$, it follows that $\kappa$ is measurable in $V[g]$, and the forcing $\R$ is $\leqkappa$-closed in $V[T]$ and consequently remains $\leqkappa$-distributive in $V[g]$ by Lemma \[Lemma.ClosureDistributive\]. Since $\R$ does not add new subsets to $\kappa$, it follows that $\kappa$ remains measurable in $V[g][H]$. But consider now $\HOD^{V[g][H]}$, which is the same as $\HOD^{V[T][H][b]}$. Since $\R$ forces to code $T$ explicitly into the  or $\Diamond^*$ pattern, we know that $T\in\HOD^{V[g][H]}$. Meanwhile, since $T$ is weakly homogeneous in $V[T]$, it remains so in $V[T][H]$ because the automorphisms are still there, and consequently $\HOD^{V[g][H]}=\HOD^{V[T][H][b]}\of V[T][H]$ by Lemma \[Lemma.WeaklyHomogenousControlsHOD\]. Since $T$ is a $\kappa$-Suslin tree in $V[T]$ and hence in $V[T][H]$, as this model has the same subsets of $\kappa$ as $V[T]$, it follows that $\HOD^{V[g][H]}$ can have no branch through $T$, and so $\kappa$ does not have the tree property in $\HOD^{V[g][H]}$. Thus, $\kappa$ is not weakly compact in $\HOD^{V[g][H]}$, while it is measurable in $V[g][H]$, as desired. This argument generalizes to other large cardinals that can be made indestructible by $\Add(\kappa,1)$ (or even merely resurrectible after this forcing) and the coding forcing, allowing us to increase the gap in strength between the property exhibited by the large cardinal and the property it exhibits in $\HOD$. Let us illustrate in the case of a supercompact cardinal. \[Theorem.SupercompactNotWCinHOD\] If $\kappa$ is a supercompact cardinal, then there is a forcing extension in which $\kappa$ remains supercompact, but is not weakly compact in $\HOD$. Suppose that $\kappa$ is a supercompact cardinal. By preparatory forcing, if necessary, we may assume without loss of generality that the supercompactness of $\kappa$ is indestructible by $\ltkappa$-directed forcing over $V$. We now force over $V$ to add a $V$-generic Cohen set $g\of\kappa$. As above, we may factor this forcing as $\Add(\kappa,1)\cong\SS*\Tdot$, first adding a weakly homogeneous $\kappa$-Suslin tree and then forcing with the tree. So the extension $V[g]$ can be viewed as $V[T][b]$, where $T$ is the generic $\kappa$-Suslin tree that is added and $b$ is the $V[T]$-generic branch through $T$. Let $\R$ be the forcing in $V[T]$ to code (a code for) $T$ into the  or $\Diamond^*_\lambda$ patterns on the next $\kappa$ many regular cardinals above $\kappa$. Suppose that $H\of\R$ is $V[T]$-generic, which as before is equivalent to assuming that $H$ is $V[g]$-generic. We may view the extension $V[g][H]$ as $V[T][H][b]$. Since $\R$ is $\ltkappa$-directed closed in $V[T]$, it is also $\ltkappa$-directed closed in $V[g]$, since any subset of $\R$ of size less than $\kappa$ in $V[g]$ is in $V[T]$. Altogether, therefore, the forcing $\Add(\kappa,1)*\R$ is $\ltkappa$-directed closed in $V$, and so our indestructibility assumption on $\kappa$ ensures that $\kappa$ is supercompact in $V[g][H]$. Meanwhile, we claim that $\kappa$ is not weakly compact in $\HOD^{V[g][H]}$. The tree $T$ is in $\HOD^{V[g][H]}$, since we explicitly forced to encode it. But since $V[g][H]=V[T][H][b]$ and $T$ is weakly homogeneous in $V[T]$ and hence in $V[T][H]$, it follows by Lemma \[Lemma.WeaklyHomogenousControlsHOD\] that $\HOD^{V[g][H]}\of V[T][H]$. Since the forcing to add $H$ adds no new subsets of $T$, it follows that $T$ has no cofinal branches in $V[T][H]$ and hence none in $\HOD^{V[g][H]}$, and so the tree property fails for $\kappa$ there, which means that $\kappa$ is not weakly compact in $\HOD^{V[g][H]}$, as desired. Let us now provide with Theorem \[Theorem.GeneralVersionIndividualCardinal\] a general template for the method, allowing us to extend the phenomenon to many other large cardinals. For this purpose, define that a property $\phi(\kappa)$ is [coding compatible]{}, if after forcing to add a Cohen subset $g\of\kappa$, then for any particular $E\of\kappa$ in $V[g]$ there is some further forcing $\R_E\in V[E]$, such that (i) forcing with $\R_E$ over $V[E]$ does not add subsets to $\kappa$; (ii) forcing with $\R_E$ over $V[g]$ makes $E$ ordinal definable; and (iii) forcing with $\R_E$ over $V[g]$ forces $\phi(\kappa)$. It will turn out that many large cardinal properties can be made coding compatible. \[Theorem.GeneralVersionIndividualCardinal\] Suppose that the property $\phi(\kappa)$ is coding compatible. Then there is a forcing extension $V[g][H]$ in which $\phi(\kappa)$ holds, but $\kappa$ is not weakly compact in $\HOD^{V[g][H]}$. Force with $\Add(\kappa,1)$ to add a $V$-generic Cohen subset $g\of\kappa$. By Lemma \[Lemma.Kunen\], we may decompose $\Add(\kappa,1)$ as $\SS*\Tdot$ and view $V[g]=V[T][b]$, where $T$ is a weakly homogeneous $\kappa$-Suslin tree in $V[T]$ and $b\of T$ is a branch through it. In $V[T]$, let $E\of\kappa$ code $T$ in some absolute canonical manner. Since $\phi(\kappa)$ is coding compatible, there is a forcing notion $\R_E\in V[T]$, not adding subsets to $\kappa$ over $V[T]$, such that if $H\of\R_{E}$ is $V[g]$-generic, then $\phi(\kappa)$ holds in $V[g][H]$ and $E$ is ordinal definable there. So we have $T\in\HOD^{V[g][H]}$, since we made $E$ ordinal definable. Since $V[g][H]=V[T][H][b]$ and $T$ is weakly homogeneous in $V[T][H]$, it follows that $\HOD^{V[g][H]}\of V[T][H]$, and since $H$ does not add subsets to $\kappa$ over $V[T]$, it follows that $T$ still has no cofinal branches in $V[T][H]$, and hence none in $\HOD^{V[g][H]}$. Thus, $\kappa$ is not weakly compact in $\HOD^{V[g][H]}$. One could relax the template of Theorem \[Theorem.GeneralVersionIndividualCardinal\] to allow the additional forcing $\R_E$ after $\Add(\kappa,1)$ to add subsets to $\kappa$, provided that it was weakly homogeneous in $V[E]$. The point would be in this case that if $H\of\R_{E}$ is $V[g]$-generic, then we view $V[g][H]$ as $V[T][H][b]$, and the last two steps of forcing $H\times b\of\R_E\times T$ is weakly homogeneous in $V[T]$, which means $\HOD^{V[g][H]}\of V[T]$, where $T$ has no cofinal branches. The utility of Theorem \[Theorem.GeneralVersionIndividualCardinal\] is revealed in Observation \[Observation.LocalGlobal\] and Corollary \[Corollary.LCNotLargeInHOD\]. For this purpose, define that a property $\phi(\kappa)$ is [locally verifiable]{}, if $\phi(\kappa)\Iff\exists\theta\, H_\theta\satisfies\psi(\kappa)$, for some assertion $\psi$. We say $\phi$ is [local]{}, if both $\phi$ and its negation $\neg\phi$ are locally verifiable. It is an excellent exercise to check that the locally verifiable properties are precisely the $\Sigma_2$-definable properties in set theory (see [@Hamkins2014:LocalPropertiesInSetTheory]), and consequently the local properties are the $\Delta_2$ properties, with large cardinal examples including measurability, superstrongness, almost hugeness and many others listed in Corollary \[Corollary.LCNotLargeInHOD\]. \[Observation.LocalGlobal\] 1. Any locally verifiable property that holds in a forcing extension $V[g][h]$, where $g\of\kappa$ is a $V$-generic Cohen subset of $\kappa$ and $h$ is additional (possibly trivial) forcing not adding subsets to $\kappa$, is coding compatible. 2. Any property that is indestructible by $\ltkappa$-directed closed forcing is coding compatible. If a locally verifiable property $\phi(\kappa)$ holds in $V[g][h]$, where $g\of\kappa$ is a Cohen subset of $\kappa$ and $h\of\Q$ is further forcing not adding subsets to $\kappa$, then $\phi(\kappa)$ is witnessed in some $H_\theta^{V[g][h]}$. For any $E\of\kappa$ in $V[g]$, therefore, we may simply force over $V[g][h]$ to code $E$ into the  or $\Diamond^*$ pattern above $\theta$, which will make $E$ definable while preserving $H_\theta$ and hence $\phi(\kappa)$, thereby showing that $\phi(\kappa)$ was coding compatible in $V$. And if $\phi(\kappa)$ is indestructible by $\ltkappa$-directed closed forcing, then it will be preserved by the forcing to add $g\of\kappa$ and then code $E$ above $\kappa$. One doesn’t need full indestructibility, of course, but rather only that the property is preserved after adding the Cohen set and the coding forcing (plus, if necessary, the additional forcing not adding subsets to $\kappa$). For example, any measurable cardinal $\kappa$ can be made indestructible by $\Add(\kappa,1)$ followed by any further $\leqkappa$-distributive forcing. This perspective unifies the local case of Observation \[Observation.LocalGlobal\] with the indestructibility case. \[Corollary.LCNotLargeInHOD\] Suppose that $\kappa$ has any of the following large cardinal properties > - weakly compact, indescribable, totally indescribable, Ramsey, strongly Ramsey, measurable, $\theta$-tall, $\theta$-strong, Woodin, $\theta$-supercompact, superstrong, $n$-superstrong, $\omega$-superstrong, $\lambda$-extendible, almost huge, huge, $n$-huge, rank-into-rank ($I_0, I_1$ and $I_3$); > > - unfoldable, strongly unfoldable, tall, strong, supercompact, superhuge, and many others. > Then there is a forcing extension in which $\kappa$ continues to have the property, but is not weakly compact in . The point is that each of these large cardinal properties can be made coding compatible. In the case of the local properties, standard arguments in the literature show how these large cardinals are preserved by suitable Easton-support iterations of Cohen forcing, forcing to add a Cohen subset, for example, at every regular cardinal stage; this is essentially the canonical forcing of the . (For the superstrong, extendible and rank-into-rank cases, the reader may find it helpful to consult [@DimonteFriedman2014:RankIntoRankHypothesesAndTheFailureOfGCH; @Friedman2006:LargeCardinalsAndLLikeUniverses; @Tsaprounis2013:OnExtendibleCardinalsAndGCH; @Hamkins2001:WholenessAxiom].) If $\kappa$ has any of these local properties, and $V[G]$ is the forcing extension up to stage $\kappa$, then after forcing to add $g\of\kappa$ at stage $\kappa$ and possibly additional forcing $g^*\of\Q^*$—in the case of superstrong, almost huge and others one must force with the rest of the iteration above $\kappa$—then $\kappa$ retains the large cardinal property in $V[G][g][g^*]$, and so by Observation \[Observation.LocalGlobal\], the property was made coding compatible in $V[G]$. Consequently, these instances of the corollary follow from Theorem \[Theorem.GeneralVersionIndividualCardinal\]. For the global properties, as mentioned before the statement of the corollary, one needs to have the large cardinal property after forcing with $\Add(\kappa,1)$ and the coding forcing for a specific $E\of\kappa$ (plus, if this helps, additional forcing not adding subsets to $\kappa$). Results in the literature establish the required degree of indestructibility(see variously [@Laver78; @GitikShelah89; @Hamkins2000:LotteryPreparation; @HamkinsJohnstone2010:IndestructibleStrongUnfoldability; @Hamkins2009:TallCardinals; @GitmanJohnstone:IndestructibilityForRamsey; @Cody2013:EastonsTheoremInThePresenceOfWoodinCardinals; @Hamkins:ForcingAndLargeCardinals]). Although our method is flexible, it does not apply to all large cardinals, and there are several cases left open. For example, can there be an extendible cardinal, which is not extendible in ? (Or not measurable in ? Not weakly compact in ?) The method also does not apply to those large cardinals, such as the strongly compact cardinals, which lack a robust forcing-preservation theory (although of course the case of supercompactness in Corollary \[Corollary.LCNotLargeInHOD\] provides a strongly compact cardinal that is not weakly compact in ). See Section \[Section.Limitations\] for these and other open questions. Proper class of large cardinals, not large in {#Section.ProperClassLC} ============================================== We should like now to extend the phenomenon to have a proper class of various kinds of large cardinals, which are not large in $\HOD$, and furthermore such that there are no such large cardinals in $\HOD$. Let us begin with the case of measurable cardinals. \[Theorem.ProperClassMeasurablesNotLargeInHOD\]There is a class forcing notion $\P$ forcing that 1. All measurable cardinals of the ground model are preserved and no new measurable cardinals are created. 2. There are no measurable cardinals in the $\HOD$ of the extension. 3. The measurable cardinals of the extension are not weakly compact in the $\HOD$ of the extension. One may also ensure that the  holds in the extension and its $\HOD$. In the interesting case, there are many measurable cardinals in the ground model $V$. Let $\Vbar=V[F]$ be the extension arising from the forcing mentioned in the proof of Lemma \[Lemma.MeasurableCCAcoding\], so that the  holds at every inaccessible cardinal in $\Vbar$ and every set of ordinals of $\Vbar$ is coded into the  or $\Diamond^*$ patterns at, say, the triple successors $\delta^{+++}$ of the $\beth$-fixed points $\delta=\beth_\delta$, and furthermore that every measurable cardinal $\kappa$ is indestructible by forcing over $\Vbar$ with $\Add(\kappa,1)$. Now, in $\Vbar$, for each measurable cardinal $\kappa$, consider the forcing $\Add(\kappa,1)*\Rdot(\kappa)$, where as previously we factor $\Add(\kappa,1)$ into two steps $\SS_\kappa*\Tdot_\kappa$, which first adds a homogeneous $\kappa$-Suslin tree $T_\kappa$ and then forces with it, and then $\Rdot(\kappa)$ is the forcing in $\overline{V}[T_\kappa]$ that codes the tree $T_\kappa$ into the  or $\Diamond^*$ patterns at the next $\kappa$ many regular cardinals above $\kappa$, starting above $\kappa^{+++}$. Thus, $\Add(\kappa,1)*\Rdot(\kappa)\iso\SS_\kappa*(\Rdot(\kappa)\dot\times\Tdot_\kappa)$. Let $\P=\Pi_\kappa(\Add(\kappa,1)*\Rdot(\kappa))$ be the Easton-support product of these forcing notions, taken over all measurable cardinals $\kappa$, and let $G\of\P$ be $\Vbar$-generic. Our final model is $\Vbar[G]$, which we shall now argue is as desired. First, we claim that every measurable cardinal $\kappa$ is preserved to $\Vbar[G]$. Since the forcing above $\kappa$ is $\leqkappa$-closed, and the forcing $\R(\kappa)$ is $\leqkappa$-distributive after adding the Cohen subset to $\kappa$, it suffices to argue that $\kappa$ is measurable in the extension $\Vbar[G_\kappa][g_\kappa]$, where $G_\kappa$ performs the forcing at measurable cardinals below $\kappa$ and $g_\kappa\of\kappa$ is the Cohen subset of $\kappa$ added by $\Add(\kappa,1)$ on coordinate $\kappa$. By our indestructibility assumption on $\kappa$ in $\Vbar$, we know that $\kappa$ remains measurable in $V_1=\Vbar[g_\kappa]$, and so it suffices to argue merely that the forcing $\P_\kappa$ preserves the measurability of $\kappa$, forcing over $V_1$. And this can be done in the style of the Kunen-Paris theorem. Namely, fix in $V_1$ any normal ultrapower embedding $j:V_1\to M$ for which $\kappa$ is not measurable in $M$, and consider $j(\P_\kappa)=\P_\kappa\times\P_{\kappa,j(\kappa)}$, where $\P_{\kappa,j(\kappa)}$ is the rest of the product forcing from stage $\kappa$ up to $j(\kappa)$ in $M$. Since $\kappa$ is not measurable in $M$, there is no forcing at coordinate $\kappa$, and so $\P_{\kappa,j(\kappa)}$ is $\leqkappa$-closed in $M$. Since $M^\kappa\of M$ in $V_1$ and $\P_{\kappa,j(\kappa)}$ has $|j(2^\kappa)|^{V_1}=\kappa^+$ many dense subsets in $M$, we may construct by diagonalization in $V_1$ an $M$-generic filter $G_{\kappa,j(\kappa)}\of\P_{\kappa,j(\kappa)}$. It follows that $G_\kappa\times G_{\kappa,j(\kappa)}\of j(\P_\kappa)$ is $M$-generic, and so we may lift the embedding to $j:V_1[G_\kappa]\to M[j(G_\kappa)]$, where $j(G_\kappa)=G_\kappa\times G_{\kappa,j(\kappa)}$, thereby witnessing that $\kappa$ is measurable in $V_1[G_\kappa]$ and hence also in $\Vbar[G]$, as desired. Next, since $\Vbar=V[F]$, where we had forced over the original ground model $V$ with $F\of\bar\P$ of Lemma \[Lemma.MeasurableCCAcoding\], then the combined forcing $\bar\P*\P$ admits a closure point in the sense of [@Hamkins2003:ExtensionsWithApproximationAndCoverProperties], and so by the main theorem of [@Hamkins2003:ExtensionsWithApproximationAndCoverProperties] therefore creates no new measurable cardinals. In particular, the three models $V\of \Vbar\of \Vbar[G]$ have the same measurable cardinals. The same reasoning shows that the large cardinals in $\Vbar[\vec T]$, where $\vec T$ is the sequence of Suslin trees $T_\kappa$ added by $\SS_\kappa$ at coordinate $\kappa$, are also large in $\Vbar$. We claim that $\HOD^{\Vbar[G]}$ has no measurable cardinals. To see this, we argue that $\HOD^{\Vbar[G]}=\Vbar[\vec T]$. The forward inclusion is a consequence of the fact that $\Pi_\kappa(\R(\kappa)\times T_\kappa)$ is weakly homogeneous in $\Vbar[\vec T]$, since the coding forcing is weakly homogeneous and the trees themselves are weakly homogeneous, and so $\HOD^{\Vbar[G]}\of \Vbar[\vec T]$ by Lemma \[Lemma.WeaklyHomogenousControlsHOD\]. Conversely, note that the coding performed by $F$ at the triple successors of the $\beth$-fixed points is preserved to $\Vbar[G]$, since the forcing to add $G$ does not interfere with that coding and furthermore all $\beth$-fixed points are preserved from $\Vbar$ to $\Vbar[G]$. It follows that every set of ordinals in $\Vbar$ is coded into the  or $\Diamond^*$ pattern on such cardinals in $\Vbar[G]$, and this implies $\Vbar\of\HOD^{\Vbar[G]}$. Further, the trees $T_\kappa$ themselves are coded into the  or $\Diamond^*$ pattern on the next $\kappa$-many regular cardinals of $\Vbar[G]$, and so $\vec T$ is definable in $\Vbar[G]$. Thus, $\Vbar[\vec T]\of \HOD^{\Vbar[G]}$, and we conclude $\HOD^{\Vbar[G]}=\Vbar[\vec T]$. So we shall show that there are no measurable cardinals in $\Vbar[\vec T]$. Suppose that $\kappa$ is a measurable cardinal in $\Vbar[\vec T]$. By our remark two paragraphs above, it follows that $\kappa$ was measurable in $\Vbar$ and therefore is one of the coordinates at which forcing is performed. In particular, at stage $\kappa$ we added the $\kappa$-Suslin tree $T_\kappa$, which is Suslin in $\Vbar[T_\kappa]$. The forcing above $\kappa$ cannot affect whether $T_\kappa$ is $\kappa$-Suslin, since it adds no new subsets to $\kappa$. The forcing $\Pi_{\delta<\kappa}\SS_\delta$ that adds the trees $T_\delta$ at measurable cardinals $\delta<\kappa$ is productively $\kappa$-c.c. (meaning it remains $\kappa$-c.c. in the forcing extension), and such forcing cannot add a $\kappa$-branch through a $\kappa$-Suslin tree. Thus, the tree property fails for $\kappa$ in $\Vbar[\vec T]$ and in particular, $\kappa$ is not weakly compact there. So there are no measurable cardinals in $\Vbar[\vec T]$, and consequently no measurable cardinals in $\HOD^{\Vbar[G]}$, establishing statement (2). Furthermore, the measurable cardinals of $\Vbar[G]$ are the same as the measurable cardinals of $\Vbar$, which are not weakly compact in $V[\vec T]=\HOD^{\Vbar[G]}$, establishing statement (3). Finally, to achieve the  in the extension $\Vbar[G]$ and its , one should start in a model of  and use exclusively the $\Diamond^*$ coding, rather than the  coding. \[Theorem.ProperClassSCNotLargeInHOD\]There is a class forcing notion $\P$ forcing that 1. All supercompact cardinals of the ground model are preserved and no new supercompact cardinals are created. 2. There are no supercompact cardinals in the $\HOD$ of the extension. 3. The supercompact cardinals of the extension are not weakly compact in the $\HOD$ of the extension. One may also ensure that the  holds in the extension and its $\HOD$. We simply modify the proof of Theorem \[Theorem.ProperClassMeasurablesNotLargeInHOD\] for the supercompact context. In the interesting case, the ground model $V$ has many supercompact cardinals. We may assume without loss that they have all been made Laver indestructible, by forcing with the global Laver preparation (or if the  is desired, we may assume that they are all indestructible by -preserving directed closed forcing, as in [@Hamkins98:AsYouLikeIt]). When there is a proper class of supercompact cardinals, this implies outright by Observation \[Observation.IndestructibleImpliesV\_kappaSubsetHOD\] that every set of ordinals in $V$ is coded explicitly into the  pattern (and the same idea applies also to the $\Diamond^*$ pattern when  is desired), at whichever type of coding cardinals we might prefer. If the supercompact cardinals are bounded, then we may simply perform additional coding forcing above the bound to ensure that every set is coded in this way as desired. Let $\P$ be the Easton-support product $\Pi_\kappa(\Add(\kappa,1)*\Rdot(\kappa))$ as in Theorem \[Theorem.ProperClassMeasurablesNotLargeInHOD\], but now with forcing only at coordinates $\kappa$ that are fully supercompact. The indestructibility of $\kappa$ in $V$ ensures that the forcing $\P^\kappa$ at coordinates $\kappa$ and above preserves the supercompactness of $\kappa$ to $V[G^\kappa]$, and so we must prove only that $\P_\kappa$ preserves the supercompactness of $\kappa$. If $\kappa$ is not a limit of supercompact cardinals, then this product is small relative to $\kappa$ and therefore preserves the supercompactness of $\kappa$. For the remaining case, assume that $\kappa$ is a supercompact limit of supercompact cardinals. Since the  holds above any supercompact cardinal, we may fix a large strong limit cardinal $\theta$ with $2^\theta=\theta^+$ and with cofinality above $\kappa$, so that $\theta^{\ltkappa}=\theta$, and let $j:V[G^\kappa]\to M$ be a $\theta$-supercompactness embedding in $V[G^\kappa]$, for which $\kappa$ is not $\theta$-supercompact in $M$. It follows that there are no supercompact cardinals in $M$ in the interval $[\kappa,\theta]$, and so $j(\P_\kappa)$ has no forcing at coordinates in the interval $[\kappa,\theta]$. So $j(\P_\kappa)$ factors as $\P_\kappa\times\P_{\kappa,j(\kappa)}$, but the second factor is $\leqtheta$-closed in $M$ and has size $j(\kappa)$. Since $M^\theta\of M$ in $V[G^\kappa]$ and furthermore $|j(\kappa)|^{V[G^\kappa]}=\theta^+$, it follows that in $V[G^\kappa]$ we may construct an $M$-generic filter $G^*\of\P_{\kappa,j(\kappa)}$, and so when this filter is combined with $G_\kappa\of\P_\kappa$, we may lift the embedding to $j:V[G^\kappa][G_\kappa]\to M[j(G_\kappa)]$, where $j(G_\kappa)=G_\kappa\times G^*$. This lifted embedding witnesses that $\kappa$ remains $\theta$-supercompact in $V[G^\kappa][G_\kappa]=V[G]$, as desired. Meanwhile, we may argue as in the proof of Theorem \[Theorem.ProperClassMeasurablesNotLargeInHOD\] that $\HOD^{V[G]}=V[\vec T]$, which is a model in which no supercompact cardinal $\kappa$ of $V$ and hence of $V[G]$ is weakly compact. And further, no new supercompact cardinals are created in $V[\vec T]$, and so $\HOD^{V[G]}$ has no supercompact cardinals at all. An essentially similar argument works with strong cardinals; we omit the proof. \[Theorem.ProperClassStrongNotLargeInHOD\]There is a class forcing notion $\P$ forcing that 1. All strong cardinals of the ground model are preserved and no new strong cardinals are created. 2. There are no strong cardinals in the $\HOD$ of the extension. 3. The strong cardinals of the extension are not weakly compact in the $\HOD$ of the extension. One may also ensure that the  holds in the extension and its $\HOD$. Indeed, the method works with many other kinds of large cardinals, including strongly unfoldable cardinals, strongly Ramsey cardinals, tall cardinals and many others, which we invite the reader to check. Let us consider the case of weakly compact cardinals. There is a class forcing notion $\P$, preserving all weakly compact cardinals, creating no new weakly compact cardinals, and forcing that there are no weakly compact cardinals in the  of the extension. One may also ensure that the  holds in the extension and its . Once again, in the interesting case there will be many weakly compact cardinals in $V$. As in Theorem \[Theorem.ProperClassMeasurablesNotLargeInHOD\], we force to $\Vbar=V[F]$ in such a way that preserves all weakly compact cardinals and creates no new weakly compact cardinals, such that every set of ordinals in $\Vbar$ is coded into the  or $\Diamond^*$ patterns at the triple successors of the $\beth$-fixed points, and such that every weakly compact cardinal $\kappa$ of $\Vbar$ is indestructible by further forcing with $\Add(\kappa,1)$. Next, let $\P$ be the Easton-support product $\Pi_{\kappa}(\Add(\kappa,1)*\Rdot(\kappa))$, taken over all weakly compact cardinals $\kappa$, where $\Rdot(\kappa)$ is as before the forcing to code the $\kappa$-Suslin tree added by the first factor into the  or $\Diamond^*$ patterns at the next $\kappa$ many regular cardinals above $\kappa$, starting above $\kappa^{+++}$ (one could alternatively perform forcing at every inaccessible cardinal). Suppose that $G\of\P$ is $\Vbar$-generic and consider the extension $\Vbar[G]$, our final model. One may see that every weakly compact cardinal of $V$ is preserved to $\Vbar[G]$, using the same argument as in the case of measurable cardinals in Theorem \[Theorem.ProperClassMeasurablesNotLargeInHOD\], namely, if $\kappa$ is weakly compact in $\Vbar$, then it suffices to argue that $\kappa$ is weakly compact in $\Vbar[G_\kappa][g_\kappa]$. The forcing to add the Cohen set $g_\kappa$ preserves weak compactness by the indestructibility assumption on $\Vbar$, and one may lift any weak compactness embedding $j:M\to N$ of $\Vbar[g_\kappa]$ to $j:M[G_\kappa]\to N[j(G_\kappa)]$ by diagonalizing to produce $G_{\kappa,j(\kappa)}$ as in the measurable case (this works whether or not there is forcing at stage $\kappa$ in $j(\P_\kappa)$, and one does not need  at $\kappa$ here in the weakly compact case). Thus, all weakly compact cardinals of $V$ are preserved to $\Vbar[G]$. No new weakly compact cardinals are created in $\Vbar[G]$ (or in $\Vbar[\vec T]$) by the main result of [@Hamkins2003:ExtensionsWithApproximationAndCoverProperties]. Finally, $\HOD^{\Vbar[G]}=\Vbar[\vec T]$ for the same reasons as in the previous cases, and this is a model having no weakly compact cardinals, for the reasons explained in the earlier cases. Questions and Limitations {#Section.Limitations} ========================= There are numerous natural questions left open by the results of this paper. Several large cardinal notions, such as the extendible cardinals, are missing from Corollary \[Corollary.LCNotLargeInHOD\]. Furthermore, although in Section \[Section.ProperClassLC\] we arranged models with a proper class of large cardinals of a particular type, with none of that type in , what we did not arrange is that  had no large cardinals of smaller type. For example, those results did not provide a model with a measurable cardinal, but no weakly compact cardinal in . Such a situation, however, is impossible. Specifically, Philip Welch pointed out that if kappa is an $\omega$- cardinal in $V$, then it is $\omega$- in any inner model (by an absoluteness argument), and therefore there are also weakly compact cardinals below $\kappa$ in any inner model. Similarly, if $\alpha$ is countable in $\HOD$ and there is an $\alpha$- cardinal in $V$, then there is one in $\HOD$ by the same absoluteness argument. A similar observation about $\alpha$- cardinals was made in [@Friedman2002:OsharpAndInnerModels p. 3]. In the same vein, Gitik and Hamkins [@GitikHamkins:LargeCardinalsNecessarilyLargeInHOD] proved that if $\kappa$ is a measurable cardinal, then for any transitive inner model $W$, including $W=\HOD$, the set of cardinals below $\kappa$ that are weakly compact, ineffable, and even superstrongly $\ltkappa$-unfoldable has normal measure one in $V$; indeed, if $\kappa$ is merely subtle, then the set of cardinals below $\kappa$ that are superstrongly $\ltkappa$-unfoldable in $W$ is stationary in $V$ and hence also in $W$. Meanwhile, many other instances remain open. \[Question.LCbutNoneInHOD?\] Can there be a strong cardinal, or a proper class of strong cardinals, but no measurable cardinal in ? Can there be an extendible cardinal which is not weakly compact in ? Can there be a proper class of extendible cardinals, but no supercompact cardinal in $\HOD$? Can there be a supercompact cardinal, which is strongly compact but not supercompact in $\HOD$? There are infinitely many variations of these questions. In the supercompact cardinal context, W. Hugh Woodin has announced in email correspondence with the first and third authors the following remarkable implication, for which he has sketched a proposed proof making use of the  dichotomy theorem of [@Woodin2010:SuitableExtenderModelsI]. \[Conjecture.Woodin\] If there is a supercompact cardinal, then there is a measurable cardinal in . This result would provide a general limitation on the phenomenon we have been exploring in this paper, and one may view Question \[Question.LCbutNoneInHOD?\] as asking whether or not there are analogues of Woodin’s conjecture at large cardinals below supercompact. The theme of this paper has been to show that there can be cardinals that are large in $V$, but small in $\HOD$. A dual theme would be to ask for cardinals that are large in $\HOD$, but small in $V$. On this note, Cummings, Friedman and Golshani [@CummingsFriedmanGolshani:CollapsingTheCardinalOfHOD thm 1.1] proved that $\alpha^+$ can be greater than $(\alpha^+)^\HOD$ for all cardinals $\alpha$ below a measurable cardinal, and this can happen for a club of cardinals $\alpha$ below a supercompact cardinal. They have conjectured that $\alpha^+$ can be measurable in $\HOD$ for club-many cardinals $\alpha$ below a supercompact cardinal. In email correspondence, Woodin has conjectured that “club-many" cannot be replaced by “all", even with “measurable" replaced by “inaccessible". It is also natural to broaden the theme of this paper beyond $\HOD$ and definability altogether, by asking whether the results extend to other natural inner models and extensions. Consider, for example, the stable core $\mathbb S$ of [@Friedman2012:TheStableCore]. As $V$ is generic over $\mathbb S$, one might expect results for the stable core similar to those we have obtained for $\HOD$. This would present new challenges, however, as it is far more difficult to code a set into the stable core than it is to make it ordinal-definable. Meanwhile, we also consider the questions for ground models or inner models generally: \[Question.VsubsetW\] Can there be a supercompact cardinal in a forcing extension $V[G]$, if there are no measurable cardinals in $V$? There are again innumerable variations of these questions to other large cardinal concepts, and we take ourselves to have asked an entire scheme of questions here. For the strong-cardinal analogue of Question \[Question.VsubsetW\], we have the sketch of a proof that there can be a model with no measurable cardinals, but with a strong cardinal in a forcing extension. The second author proved in [@Friedman2002:OsharpAndInnerModels] that if there are no inaccessible cardinals in $L[0^\sharp]$, then there is an $M \subsetneq L[0^\sharp]$ in which  holds and there is no cardinal $\kappa$ which is $\kappa$-Mahlo. It is natural to ask whether similar results hold for larger core models. Appendix: background material {#Section.Background} ============================= We have made use in this article of several general facts and some other background material, whose proofs we include here for completeness. In the main argument, we used the following fact, due to Kunen, that the forcing $\Add(\kappa,1)$ to add a Cohen subset to an inaccessible cardinal $\kappa$ can be factored as first adding a certain weakly homogeneous $\kappa$-Suslin tree and then forcing with that tree. \[Lemma.Kunen\] If $\kappa$ is inaccessible, then there is a strategically $\ltkappa$-closed notion of forcing $\SS$ of size $\kappa$ such that forcing with $\SS$ adds a weakly homogeneous $\kappa$-Suslin tree $T$ and the combined forcing $\SS*\Tdot$ is forcing-equivalent to the forcing $Add(\kappa,1)$ to add a Cohen subset of $\kappa$. See also the detailed accounts in [@CodyGitikHamkinsSchanker2015:LeastWeaklyCompact; @GitmanWelch2011:Ramsey-likeCardinalsII]. The key point is that $\SS*\Tdot$ has a $\ltkappa$-closed dense subset of size $\kappa$, and all such (nontrivial) forcing is equivalent to $\Add(\kappa,1)$. A forcing notion $\Q$ is [weakly homogeneous]{}, if for any two conditions $p,q\in\Q$, there is an automorphism $\pi$ of $\Q$ for which $\pi(q)$ and $p$ are compatible. It follows that if $\varphi$ is a statement in the forcing language involving only check names $\check x$, and some condition $p$ forces $\varphi$, then every condition forces $\varphi$, since otherwise some $q$ forces $\neg\varphi$, but in this case $\pi(q)$ will also force $\neg\varphi$, which is impossible if it is compatible with $p$. A weaker variant of this property is that $\Q$ is [locally homogeneous]{}, if for any $p,q\in\Q$, there are extensions $p^*\leq p$ and $q^*\leq q$ and an isomorphism $\pi$ of $\Q\restrict p^*$ with $\Q\restrict q^*$; it follows again in this case that all conditions force the same assertions with only ground model parameters. \[Lemma.WeaklyHomogenousControlsHOD\] If $\Q$ is a locally homogeneous notion of forcing and $G\of\Q$ is $V$-generic, then $\HOD^{V[G]}\of\HOD(\Q)^V$. In particular, if $\Q$ is also ordinal definable in $V$, then $\HOD^{V[G]}\of\HOD^V$. The point is that if $A\of\Ord$ is defined in $V[G]$ by $\alpha\in A\iff \varphi(\alpha,\beta)$, then we can define $A$ in the ground model as $\set{\alpha\st \one\forces\varphi(\check\alpha,\check\beta)}$, since all conditions must force the same assertions. Several of our arguments use Easton’s lemma: \[Lemma.ClosureDistributive\] Suppose that $G\times H$ is $V$-generic for $\P\times\Q$, where $\P$ is $\ltkappa$-closed and $\Q$ is $\kappa$-c.c. Then $\P$ is $\ltkappa$-distributive in $V[H]$. In other words, $\Ord^\ltkappa\intersect V[G][H]\of V[H]$. In our main argument, we often have need to perform forcing that ensures that a given set of ordinals becomes ordinal definable in the forcing extension. The general idea goes back to McAloon [@McAloon1971:ConsistencyResultsAboutOrdinalDefinability], who forced $V=\HOD$ by various types of coding forcing. One quite commonly sees the forcing to code a given set into the  pattern on a definable block of cardinals. This -controlling forcing is highly homogeneous and directed closed, and it is easy to see that the coding is preserved by any forcing below the place where the coding took place, since small forcing cannot affect the continuum function up high. But there are other coding methods, such as coding into the $\Diamond^*_\kappa$ pattern. The second author and his student Andrew Brook-Taylor [@Brooke-Taylor:Thesis; @Brooke-Taylor2009:LargeCardinalsAndDefinableWellOrders] have emphasized the value of coding via the $\Diamond^*_\kappa$ pattern, in part because this can be done while preserving the , as well as various large cardinals. The main fact is the following. Recall that $\<D_\alpha\mid\alpha<\kappa>$ is a [$\Diamond^*_\kappa$-sequence]{}, where $\kappa$ is an uncountable regular cardinal, if $D_\alpha$ consists of at most $|\alpha|$ many subsets of $\alpha$ and for every $A\of\kappa$ there is a club $C\of\kappa$ such that $A\intersect\alpha\in D_\alpha$ for each $\alpha\in C$. \[Lemma.Diamond\*Coding\] One may force so as to control $\Diamond^*_\kappa$ as desired on the successor cardinals. 1. [@Devlin1979:VariationsOnDiamond; @Brooke-Taylor2009:LargeCardinalsAndDefinableWellOrders] If $\kappa$ is any regular cardinal, then $\Add(\kappa,\kappa^+)$ forces $\neg\Diamond^*_\kappa$. 2. [@CummingsForemanMagidor2001:SquaresScalesAndStationaryReflection; @Brooke-Taylor2009:LargeCardinalsAndDefinableWellOrders] For any infinite successor cardinal $\kappa^+$, there is a $\ltkappa^+$-closed forcing notion of size $2^{\kappa^+}$, which forces $\Diamond^*_{\kappa^+}$. Furthermore, the $\Diamond^*$ coding forcing in each case is locally homogeneous. The application of $\Diamond^*$ coding in this article requires that the coding is preserved by small forcing, so let us also note that here. \[Lemma.Diamond\*InvariantBySmallForcing\] For any regular cardinal $\kappa$, the $\Diamond^*_\kappa$ hypothesis is invariant by small forcing. Furthermore, $\Diamond^*_\kappa$ is preserved by any forcing of size at most $\kappa$ preserving the regularity of $\kappa$. More specifically, what we claim is the following: 1. For any small forcing extension $V[G]$, meaning that $G\of\P$ is $V$-generic for some $\P$ of size less than $\kappa$ in $V$, the principle $\Diamond^*_\kappa$ holds in $V$ if and only if it holds in $V[G]$. 2. If $\Diamond^*_\kappa$ holds in $V$ and $G\of\P$ is $V$-generic, where $\P$ has size at most $\kappa$ and $\kappa$ is regular in $V[G]$, then $\Diamond^*_\kappa$ continues to hold in $V[G]$. For (1), suppose that $\<D_\alpha \mid \alpha < \kappa>$ is a $\Diamond^*_\kappa$-sequence in $V[G]$, where $G\of\P$ is $V$-generic and $|\P|<\kappa$. Fix a name $\dot D$ for the sequence, forced by $\one$ via $\P$ to be a $\Diamond_\kappa^*$-sequence, and corresponding names $\dot D_\alpha$ for each $D_\alpha$. In $V$, for any infinite ordinal $\alpha$ above $|\P|$, let $B_\alpha$ consist simply of all those subsets $X\of\alpha$ for which there is some condition $p\in\P$ forcing $\check X\in\dot D_\alpha$. Since $\P$ is small, it follows that $B_\alpha$ has size at most $|\alpha|$, since any given condition can force at most $|\alpha|$ many sets into $\dot D_\alpha$. If $X\of\kappa$ is any subset of $\kappa$ in $V$, then there is a club $C\of\kappa$ in $V[G]$ such that $X\intersect\alpha\in D_\alpha$ for all $\alpha\in C$. Since these facts are forced by some condition, it follows that $X\intersect\alpha\in B_\alpha$ as well, and since the forcing is small, the club $C$ must contain a club in the ground model $V$, and so $\<B_\alpha\mid\alpha<\kappa>$ witnesses $\Diamond^*_\kappa$ in $V$, establishing the converse implication of statement (1). The forward implication of (1) is generalized by statement (2), which we now prove. Suppose that $\<B_\alpha \mid \alpha<\kappa>$ is a $\Diamond^*_\kappa$-sequence in $V$ and $G\of\P$ is $V$-generic for forcing $\P$ of size at most $\kappa$ that preserves the regularity of $\kappa$. We may assume that $\P\of\kappa$. In the forcing extension $V[G]$, let $D_\alpha$ be the set of all subsets $X\of \alpha$ in $V[G]$ for which there is a nice $\P\intersect\alpha$-name $\dot X$ such that $\dot X_G=X$ and $\dot X$ is coded by an element of $B_\alpha$ (we use nice names simply to make the coding more transparent). If $X\of\kappa$ is in $V[G]$, then it has a nice $\P$-name $\dot X$, which is coded by a subset $X_0\of\kappa$ in $V$, and $X_0\intersect\alpha\in D_\alpha$ on a club of $\alpha<\kappa$. Since also $X_0\intersect\alpha$ codes a nice $\P\intersect\alpha$-name $\dot X_\alpha$ for subset of $\alpha$, on a club of $\alpha$, and furthermore $(\dot X_\alpha)_G=X\intersect\alpha$ for a club of $\alpha$, we have a club of $\alpha$ in $V[G]$ for which $X\intersect\alpha\in D_\alpha$, thereby witnessing $\Diamond^*_\kappa$ in $V[G]$. The following lemma is used in the proof of Theorem \[Theorem.ProperClassMeasurablesNotLargeInHOD\]. \[Lemma.MeasurableCCAcoding\] There is a class forcing notion $\P$ such that if $F\of\P$ is $V$-generic, then 1. $V$ and $V[F]$ have exactly the same measurable cardinals. 2. The  holds in $V[F]$ at every inaccessible cardinal. 3. Every set of ordinals in $V[F]$ arises as the  or $\Diamond^*$ pattern (as desired) on a block of cardinals of the form $\delta^{+++}$ for $\delta=\beth_\delta$. 4. Every measurable cardinal $\kappa$ in $V[F]$ is indestructible by the forcing $\Add(\kappa,1)$. By using $\Diamond^*$ coding and no  coding, one can ensure  in $V[F]$. The proof of the lemma is extremely flexible, and one can use different coding points. Let $\P$ be the Easton-support proper-class forcing iteration that forces the  at all infinite cardinals (including adding a Cohen subset at inaccessible cardinal stages), except that at stage $\eta=\delta^{+++}$ where $\delta=\beth_\delta$, the forcing uses the lottery sum of the forcing $\Add(\eta^+,1)$, which forces the  at $\eta$, with the forcing $\Add(\eta,\eta^{++})$, which forces a violation of the  at $\eta$. Suppose that $F\of\P$ is $V$-generic, and consider the extension $V[F]$. The standard arguments show that every measurable cardinal $\kappa$ of $V$ is preserved to $V[F]$, and furthermore becomes indestructible by $\Add(\kappa,1)$ (and this is why we add a Cohen set at inaccessible stages). The main results of [@Hamkins2003:ExtensionsWithApproximationAndCoverProperties] show that the forcing creates no new measurable cardinals, since the iteration admits a closure point below the first inaccessible cardinal. The forcing ensures the  at all cardinals except those coding points, and a simple density argument shows that every set of ordinals in the extension is coded into the  pattern at those coding points. So the extension exhibits all the desired features. By using $\Diamond^*$ coding instead of  coding, an essentially similar argument also achieves  in $V[F]$. A cardinal $\kappa$ is [$(\Sigma_2,0)$-extendible]{} if $V_\kappa\elesub_{\Sigma_2} V_\theta$ for some ordinal $\theta$ (see [@BagariaHamkinsTsaprounisUsuba:SuperstrongAndOtherLargeCardinalsAreNeverLaverIndestructible]). \[Observation.IndestructibleImpliesV\_kappaSubsetHOD\] If $\kappa$ is a Laver indestructible supercompact cardinal, then $V_\kappa\of\HOD$. In particular, in this case there are many measurable and partially supercompact cardinals in . More specifically, if $\kappa$ is Laver indestructibly $\Sigma_2$-reflecting, or even merely indestructibly $(\Sigma_2,0)$-extendible, then $V_\kappa\of\HOD$. Suppose that $\kappa$ is Laver indestructibly $(\Sigma_2,0)$-extendible. Let $\Q$ be $\ltkappa$-closed forcing that forces to code every set of ordinals in $V_\kappa$ into the  pattern at the next $\kappa$ many regular cardinals above $\kappa$. This can be done with $\ltkappa$-directed closed forcing, and so $\kappa$ remains $(\Sigma_2,0)$-extendible in the corresponding forcing extension $V[G]$, and so there is an ordinal $\theta>\kappa$ such that $V_\kappa\elesub_{\Sigma_2} V_\theta^{V[G]}$. Note that $\theta$ must be large enough that all the coding performed by $G$ is available in $V_\theta^{V[G]}$. Thus, any set $x\in V_\kappa$ is coded into the  pattern of $V_\theta^{V[G]}$, and since this is a $\Sigma_2$ expressible property, it follows that $x$ is already coded into the  pattern in $V_\kappa$. Thus, $x$ is ordinal definable in $V$, and so $V_\kappa\of\HOD$, as desired.
--- author: - 'Pinilla P. Birnstiel T. Ricci L. Dullemond C. P. Uribe A. L. Testi L.' - 'Natta A.' bibliography: - 'Pinilla.bib' date: 'Received October 5 2011/ Accepted December 5 2011' title: Trapping dust particles in the outer regions of protoplanetary disks --- [Dust particles are observed at mm sizes in outer regions of the disk, although theoretically, radial drift does not allow dust particles to form pebbles.]{} [In order to explain grain growth to mm sized particles and their retention in outer regions of protoplanetary disks, as it is observed at sub-mm and mm wavelengths, we investigate if strong inhomogeneities in the gas density profiles can slow down excessive radial drift and can help dust particles to grow.]{} [We use coagulation/fragmentation and disk-structure models, to simulate the evolution of dust in a bumpy surface density profile which we mimic with a sinusoidal disturbance. For different values of the amplitude and length scale of the bumps, we investigate the ability of this model to produce and retain large particles on million years time scales. In addition, we introduced a comparison between the pressure inhomogeneities considered in this work and the pressure profiles that come from magnetorotational instability. Using the Common Astronomy Software Applications ALMA simulator, we study if there are observational signatures of these pressure inhomogeneities that can be seen with ALMA.]{} [We present the favorable conditions to trap dust particles and the corresponding calculations predicting the spectral slope in the mm-wavelength range, to compare with current observations. Finally we present simulated images using different antenna configurations of ALMA at different frequencies, to show that the ring structures will be detectable at the distances of the Taurus Auriga or Ophiucus star forming regions.]{} Introduction {#sec1} ============ The study of planet formation is an important field in astronomy with an increasing research since the middle of the twentieth century, however there are still countless unanswered questions. One of these questions is the observed grain growth to mm sized particles in the outer disk regions ([@1991ApJ...381..250B]; [@2000ApJ...534L.101W; @2005ApJ...626L.109W]; ; [@2005ApJ...631.1134A]; ; [@2007prpl.conf..767N]; [@2009ApJ...701..260I]; ; ; ) that suggests a mechanism operating that prevents the rapid inward drift ([@1997Icar..128..213K], , [@2007Natur.448.1022J]). Different efforts are aimed to explain theoretically the growth from small dust particles to planetesimals, which have led to the development of different numerical models, e.g. [@1981Icar...45..517N], , , , [@2009ApJ...698.1122O], . Due to the fact that circumstellar disks exhibit a wide range of temperatures, they radiate from micron wavelengths to millimeter wavelengths, which is why they can be observed with infrared and radio telescopes. With the construction of different kinds of these telescopes, e.g. Spitzer, Herschel, SMA, EVLA or ALMA, astronomers can observe with more details the material inside accretion disks around young stars. The parallel development of theory and observations have allowed astrophysicists to study the different stages of planet formation, making this topic one of the most active fields in astronomy today. In the first stage of planet formation, the growth from sub-micron sized particles to larger objects is a complex process that contains many physical challenges. In the case of smooth disk with a radial gas pressure profile that is monotonically decreasing, the dust particles drift inwards owing to the fact that the gas moves with sub-keplerian velocity due to the gas pressure gradient. Before a large object can be formed, the radial drift causes dust pebbles to move towards the star. Moreover, the high relative velocities due to turbulence and radial drift cause the solid particles to reach velocities that lead to fragmentation collisions which do not allow dust particles to form larger bodies . The combination of these two problems is called “meter-size barrier” because on timescales as short as 100 years, a one meter sized object at 1 AU moves towards the star due to the radial drift, preventing that any larger object could be formed. The observations in the inner regions of the disk, where planets like Earth should be formed, are very difficult because these regions are so small on the sky that few telescopes can spatially resolve them. Also, these regions are optically thick. However, what amounts to the meter-size barrier in the inner few AU is a “millimeter-size barrier” in the outer regions of the disk. These outer regions ($\gtrsim 50$ AU) are much easier to spatially resolve and are optically thin. Moreover, one can use millimeter observations, which probe precisely the grain size range of the millimeter-size. Therefore, the study of dust growth in the outer disk regions may teach us something about the formation of planets in the inner disk regions. Observations of protoplanetary disks at sub-millimeter and mm wavelengths show that the disks remain dust-rich during several million years with large particles in the outer regions . However, it is still unclear how to prevent the inward drift and how to explain theoretically that mm-sized particles are observed in the outer regions of the disk. Different mechanisms of planetesimal formation have been proposed to resist the rapid inwards drift like: gravitational instabilities [@2002ApJ...580..494Y], the presence of zonal flows or dead zones of viscously accumulated gas which form vortices . With the model presented here, we want to imitate mechanisms that allow to have long-lived pressure inhomogeneities in protoplanetary disks, by artificially adding pressure bumps onto a smooth density profile. Parameter Values ----------------------- -------------------------- $A$ $\{0.1; 0.3; 0.5; 0.7\}$ $f$ $\{0.3; 0.7; 1.0; 3.0\}$ $\alpha$ $10^{-3}$ $R_{\star} [R_\odot]$ $2.5$ $T_{\star} [K]$ $4300$ $M_{disk} [M_\odot]$ $0.05$ $\rho_s[g/cm^3]$ $1.2$ $v_f[m/s]$ $10$ : Parameters of the model[]{data-label="table1"} To confront the millimeter-size barrier, it is necessary to stop the radial drift considering a radial gas pressure profile that is not monotonically decreasing with radius. Instead, we take a pressure profile with local maxima adding a sinusoidal perturbation of the density profile. These perturbations influence directly the pressure, following a simple equation of state for the pressure in the disk. Depending on the size of the particle, the dust grains will be nearly perfectly trapped in the pressure peaks, because a positive pressure gradient can cause those dust particles move outwards. On the other hand, turbulence can mix part of the dust particles out of the bumps, so that overall there may still be some net radial inward drift. More importantly, dust fragmentation may convert part of the large particles into micron size dust particles, which are less easily trapped and thus drift more readily inward. In the work of , they compared the observed fluxes and mm spectral indices from Taurus and Ophiucus star-forming regions with predicted fluxes and spectral indices at mm wavelengths. They neglected the radial drift, forcing the dust particles to stay in the outer disk regions. They aimed to keep the spectral index at low values, which implies that the dust particles could acquire millimeter sizes [@1991ApJ...381..250B]. However, they found over-predictions of the fluxes. As an extension of their work, the purpose of this paper is to model the combination of three processes: the radial drift, the radial turbulent mixing and the dust coagulation/fragmentation cycle in a bumpy surface density profile. Our principal aim is to find out how the presence of pressure bumps can help explain the retainment of dust pebbles in the outer regions of protoplanetary disks, while still allowing for moderate drift and thus obtaning a better match with the observed fluxes and mm spectral indeces. In addition, we show simulated images using different antenna configurations of the complete stage of ALMA, to study if it is possible to detect these kind of inhomogeneities with future ALMA observations. This paper is ordered as follows: Sect. \[sec2\] will describe the coagulation/fragmentation model and the sinusoidal perturbation that we take for the initial condition of the gas surface density. Section \[sec3\] will describe the results of these simulations, the comparison between existing mm observations of young forming disks and the results from our model. We discuss if the type of structures generated by our model can be detectable with future ALMA observations. In Sect. \[sec4\], we explore the relation of our model with predictions of current simulations of the magnetorotational instability (MRI) [@1991ApJ...376..214B]. Finally, Sect. \[sec5\] will summarize our results and the conclusions of this work. ![image](Figure1.pdf){width="18cm"} Dust Evolution Model {#sec2} ==================== We make use of the model presented in to calculate the evolution of dust surface density in a gaseous disk, radial drift, and turbulent mixing. The dust size distribution evolves due to grain growth, cratering and fragmentation. Relative velocities due to Brownian motion, turbulence, radial and azimuthal drift as well as vertical settling are taken into account. In this work, we do not consider the viscous evolution of the gas disk because the aim is to investigate how dust evolution is influenced by stationary perturbations of an otherwise smooth gas surface density. The effects of time dependent perturbation and the evolution of the gas disk will be the subject of future work. For a comprehensive description of the numerical code we refer to . In order to simulate radial pressure maxima that allow the trapping of particles, we consider a perturbation of the gas surface density that it is taken for simplicity as a sinusoidal perturbation such that: $$\Sigma'(r)=\Sigma(r)\left(1+A\cos{\left[ 2\pi \frac{r}{L(r)}\right]}\right), \label{eq1}$$ where the unperturbed gas surface density $\Sigma(r)$ is given by the self similar solution of [@1974MNRAS.168..603L]: $$\Sigma(r)=\Sigma_0 \left(\frac{r}{r_c}\right)^{-\gamma} \exp \left[-\left(\frac{r}{r_c}\right)^{2-\gamma}\right], \label{eq2}$$ where $r_c$ is the characteristic radius, taken to be $60$AU, and $\gamma$ is the viscosity power index equal to $1$, which are the median values found from high angular resolution imaging in the sub-mm of disks in the Ophiucus star forming regions [@2010ApJ...723.1241A]. The wavelength $L(r)$ of the sinusoidal perturbation depends on vertical disk scale-height $H(r)$ by a factor $f$ as $$L(r)=f H(r), \label{eq3}$$ with $H(r)= c_s \Omega^{-1}$, where the isothermal sound speed $c_s$ is defined as $$c_s^2=\frac{k_B T}{\mu m_p}, \label{eq3-1}$$ and the Keplerian angular velocity $\Omega$ is $$\Omega=\sqrt{\frac{GM_\star}{r^3}}, \label{eq3-2}$$ with $k_B$ being the Boltzmann constant, $m_p$ the mass of the proton and $\mu$ is the mean molecular mass, which in proton mass units is taken as $\mu=2.3$. For an ideal gas, the pressure is defined as $$P ( r , z )=c_s^2 \rho( r, z ), \label{eq4}$$ where $\rho(r,z)$ is the gas density, such that $\Sigma'(r)=\int_{-\infty}^\infty \rho(r,z) dz$. With the surface density described by Eq. \[eq1\], we can have pressure bumps such that the wavelength is increasing with radius. These bumps are static, which may not be entirely realistic. However, these can be a good approximation of long-lived, azimuthally extended pressure bumps, that can be e.g. the result of MHD effects . The influence of time-dependent pressure perturbations [e.g. @2004ApJ...608..489L; @2007Icar..188..522O] on the dust growth process will be the topic of future work. Left plot of Fig. \[comparacion\] (dashed lines) shows the behavior of the perturbed surface density for two values of the amplitude and fixed value of the width. The right plot of Fig. \[comparacion\] shows the corresponding pressure gradient. The very fine dust particles move as the gas because they are well coupled to the gas since the stopping time is very short. In the presence of a drag force, the stopping time is defined as the time that a particle, with a certain momentum, needs to be aligned to the gas velocity. However, when the particles are large enough and they are not forced to move as the gas, they experience a head wind, because of the sub-Keplerian velocity of the gas and therefore they lose angular momentum and move inwards. In this case, the resulting drift velocity of the particles is given by [@1977MNRAS.180...57W]: $$u_{\mathrm{drift}}=\frac{1}{\textrm{St}^{-1}+\textrm{St}} \frac{\partial_r P}{\rho \Omega}. \label{eq5}$$ Comparing Eq. \[eq5\] with the expression for the drift velocity given by (Eq. 19), we can notice that the drag term from the radial movement of the gas is not taken here since we are assuming a stationary gas surface density. The Stokes number denoted by St, describes the coupling of the particle to the gas. The Stokes number is defined as the ratio between the eddy turn-over time ($1/\Omega$) and the stopping time. For larger bodies, the Stokes number is much greater than one, which implies that the particles are not affected by the gas drag, consequently they move on Keplerian orbits. When the particles are very small, St $\ll$ 1, they are strongly coupled to the gas. Since the gas is orbiting at sub-Keplerian velocity because its pressure support, there is a relative velocity between the dust particles and the gas. The Stokes number equal to unity is a critical value where the particles are still influenced by the gas drag but they are not completely coupled to the gas, instead they are marginally coupled, and move at speeds between Keplerian and the sub-Keplerian gas velocity. The retainment of dust particles due to the presence of pressure bumps depend on the size of the particles. Since very small particles, with St $\ll$ 1, are tightly coupled to the gas, they do not drift inwards. However, radial drift becomes important when the size of the particles increases and it is strongest when St=1 (see Eq. \[eq5\]). In the Epstein regime, where the ratio between the mean free path of the gas molecules $\lambda_{\mathrm{mfp}}$ and the particle size, denoted by $a$, satisfies that $\lambda_{\mathrm{mfp}}/a\geq 4/9$, the Stokes number is given by: $$\textrm{St}=\frac{a\rho_s}{\Sigma_g}\frac{\pi}{2}, \label{eq6}$$ where $\rho_s$ is the solid density of the dust grains, that is taken to be constant (see Table \[table1\]). In this case particles are small enough to be in this regime. Parameterizing the radial variation of the sound speed via $$c_s\propto r^{-q/2}, \label{eq7}$$ where for a typical disk, the temperature is assumed to be a power law such that $T\propto r^{-q}$, which is an approximation of the temperature profile taken for this model. Therefore, the wavelength of the perturbation $L(r)$ scales as: $$L(r)=f H(r)\propto f r^{(-q+3)/2}. \label{eq8}$$ The pressure bumps have the same amplitude $A$ and wavelength $L(r)$ than the density, because the over pressure is induced adding inhomogeneities to the gas surface density and parameterizing the temperature on the midplane by a power law [@1994ApJ...421..640N]. The model taken here can artificially imitate e.g. the case of zonal flows in protoplanetary disks, where over densities create pressure bumps. Zonal flows are formed due to MRI, which depend on the degree of ionization of the disk, i.e. on the temperature of the disk and other factors as the exposure to cosmic and stellar rays. MRI appears to be the most probable source of turbulence, and if the turbulence is not uniform, there can be excitation of long-lived pressure fluctuations in the radial direction. For instance, [@2009ApJ...697.1269J] performed shearing box simulations of MRI turbulent disk that show large scale radial variations in Maxwell stresses of $10\%$. presented 3D global non-ideal MHD simulations including a dead zone that induces pressure maxima of $20-25\%$. [@2011ApJ...736...85U] showed 3D global MHD simulations, leading to pressure bumps of around $25\%$. On the other hand, when the viscosity drops, the gas surface density changes causing a local inversion of the pressure gradient and an accumulation of dust particles. This matter accumulation causes Rossby wave instabilities [@1999ApJ...513..805L] that lead non-axisymmetric distributions on the disk, which we cannot exactly model at this moment since our dust evolution models are axisymmetric. To constrain the values of the amplitude and wavelength of the perturbation, we take into account three different factors: First, it is important to analyze the necessary conditions to have local outwards movement of the dust particles. Second, we compare our assumptions with current studies on zonal flows [@2009ApJ...697.1269J; @2011ApJ...736...85U]. And third, we only work with small-enough amplitude disturbances such that the disk has angular momentum per unit mass increasing outwards, which means it is Rayleigh stable . The Rayleigh criterion establishes that for a rotating fluid system, in the absence of viscosity, the arrangement of angular momentum per unit mass is stable if and only if it increases monotonically outward [@1961hhs..book.....C], which implies: $$\frac{\partial }{\partial r}(r v_{\phi})>0. \label{eq16}$$ ![Vertically integrated dust density distribution at 1 Myr for $A=0.1$ and $f = 1$ (top) and $A=0.1$ and $f=3$ (bottom). The white line shows the particle size corresponding to a Stokes number of unity, which shows the same shape that the gas surface density $\Sigma'$ of Eq. \[eq1\] (see Eq. \[eq6\]). The blue line represents the maximum size of the particles before they reach fragmentation velocities (fragmentation barrier according to Eq.\[eq9\]).[]{data-label="Fig1"}](Figure2a.pdf "fig:"){width="8.8cm"}\ ![Vertically integrated dust density distribution at 1 Myr for $A=0.1$ and $f = 1$ (top) and $A=0.1$ and $f=3$ (bottom). The white line shows the particle size corresponding to a Stokes number of unity, which shows the same shape that the gas surface density $\Sigma'$ of Eq. \[eq1\] (see Eq. \[eq6\]). The blue line represents the maximum size of the particles before they reach fragmentation velocities (fragmentation barrier according to Eq.\[eq9\]).[]{data-label="Fig1"}](Figure2b.pdf "fig:"){width="8.8cm"} Since turbulence is necessary to have angular momentum transport, instabilities may occur if a magnetic field is present, and in that cases the angular velocity decreases with radius (MRI). For a typical $\alpha-$turbulent disk, the MRI time scale is much greater than the time that the disk needs to recover the Rayleigh stability, this implies that the disk should remain quasi stable at all time [@2010MNRAS.402.2436Y]. Any perturbation in the gas surface density, that is Rayleigh unstable will almost instantly be smeared out sufficiently to make it Rayleigh stable again, thereby lowering its amplitude. This happens on a time scale much shorter than what MRI could ever counteract. The angular velocity of Eq. \[eq16\] is given by: $$v_{\phi}^2=v_k^2+\frac{r}{\rho}\frac{\partial P}{\partial r}=v_k^2(1-2\eta) \label{eq17}$$ with $$\eta=-\frac{1}{2r\Omega^2 \rho}\frac{\partial P}{\partial r}. \label{eq17.0}$$ ![image](Figure3.pdf){width="18cm"} The Rayleigh stability of the disk depends on the amplitude and the width of the bumps. In this case, we want to study the influence of the amplitude of the perturbation, so for this analysis we constrain the value of the wavelength of the perturbation, $f$ equals to unity, such that it stays consistent with the values expected from predictions of zonal flows models by [@2011ApJ...736...85U] (see Fig. \[comparacion\]). Taking the perturbed gas density of Eq. \[eq1\] and $f=1.0$, it is possible to find the upper limit of the amplitude to satisfy the Eq. \[eq16\], i.e. the condition to remain Rayleigh stable at all time. This calculation lets the maximum value of the amplitude $A$ to be at most $\sim 35\%$ of the unperturbed density. Various investigations have been focused on the possibility of Rayleigh instabilities when disks have sharp profiles for the radial density (see e.g. [@1984ApJ...285..818P], [@2001ApJ...551..874L] and [@2010MNRAS.402.2436Y]). These kind of profiles can exist when the temperature in the midplane is not sufficient to ionize the gas [@1996ApJ...457..355G] and as a result the turbulence parameter $\alpha$ is reduced. These regions are known as “dead zones” and these are candidates to be places where planet embryos can be formed. In the boundaries of these regions, it was shown by that it is possible to have a huge vortex with a local bump in the gas surface density. [@1999ApJ...513..805L] demonstrated that these perturbations create an accumulation of gas that leads to the disk to be unstable to Rossby wave instability (RWI). As a comparison of the amplitudes generated by RWI vortices and the amplitudes of our perturbations, the left plot of Fig. \[comparacion\] also shows the azimuthally average gas surface density of a large-scale anticyclonic vortex by [@2011arXiv1109.6177R]. In those cases, the equilibrium of the disk is affected such that the disk may become Rayleigh unstable. Since we are focused on perturbations that allow to the disks stay Rayleigh stable at all evolution time, we do not consider for our perturbed density such kind of amplitudes. On the other hand, since the drift velocity is given by Eq. \[eq5\], to prevent the inwards drift, the value of $\eta$ from Eq. \[eq17.0\] must be negative, this implies that the pressure gradient has to be positive for some regions of the disk. Doing this calculation for the condition that $\eta<0$, we get that the values of the amplitude $A$ have to be at least equal to about $10\%$. In right plot of Fig. \[comparacion\] we see that with an amplitude of 10% the pressure gradient barely reaches positive values in the inner regions of the disk ($\lesssim 50$ AU). Summarizing the upper and lower values of the amplitude should be $0.10\lesssim A \lesssim 0.35$, when the width of the perturbation is taken to be $f=1$. Taking into account the growth-fragmentation cycle and the existence of pressure bumps, the radial drift efficiency can be reduced if the bumps have favorable values for the amplitude and the length scale. When the particles grow by coagulation, they reach a certain size with velocities high enough to cause fragmentation (fragmentation barrier). The two main sources of relative velocities are radial drift and turbulence. In the bumps the radial drift can be zero if the pressure gradient is high enough as well as azimuthal relative velocities; but there are still relative velocities due to the turbulence. Therefore, it is necessary to have a condition, such that the particles do not reach the threshold where they fragment. The maximum turbulent relative velocity between particles, with St $\sim1$, is given by , $$\Delta u_{max}^2\simeq \frac{3}{2}\alpha \mathrm{St} c_s^2, \label{eq12}$$ for St $\lesssim0.1$ Eq.\[eq12\] is off by a factor of 2. Therefore, to break through the mm size barrier, we must have that $\Delta u_\mathrm{max}$ has to be smaller that the fragmentation velocities of the particles $v_f$. Recent collision experiments using silicates and numerical simulations show that there is an intermediate regime between fragmentation and sticking, where particles should bounce. In this work, we do not take into account this regime since there are still many open questions in this field. For example, [@2011ApJ...737...36W] suggest that there is no bouncing regime for ice particles, which may be present in the outer regions of the disks . Laboratory experiments and theoretical work suggest that typical values for fragmentation velocities are of the order of few m s$^{-1}$ for silicate dust . Outside the snow-line, the presence of ices affects the material properties, making the fragmentation velocities increase . Since in this work we assume a radial range from 1AU to 300AU, the fragmentation threshold velocity is taken as $v_f=10$m s$^{-1}$. All the parameters used for this model are summarized in Table  \[table1\]. For particles with $ \mathrm{St} \lesssim 1$, taking the size at which the turbulent relative velocities $\Delta u_\mathrm{max}$ are as high as the fragmentation velocity $v_f$, we can find the maximum value of the grain size, which is $$a_{\mathrm{max}}\simeq\frac{4\Sigma_g}{3\pi \alpha\rho_s} \frac{v_f^2}{c_s^2}, \label{eq9}$$ this $a_{\mathrm{max}}$ is valid only for $ \mathrm{St} \lesssim 1$ because for larger bodies the turbulent relative velocities are lower than the given in Eq. \[eq12\] (for a detailed discussion about turbulent relative velocities see ). The dust grain distribution $n(r,z,a)$ is the number of particles per cubic centimeter per gram interval in particle mass, which depends on the grain mass, distance to the star $r$ and height above the mid-plane $z$, such that $$\rho (r,z)=\int_0^\infty n(r,z,a) \cdot m dm, \label{eq10}$$ ![Particle size corresponding to a Stokes number of unity for $A=0.3$ and $f=1.0$ and location of the fragmentation barrier for two different values of the turbulent parameter $\alpha$. The dashed line corresponds to $r=50$AU to distinguish the maximum size particle in the outer regions of the disk for each case.[]{data-label="comparison_alpha"}](Figure4.pdf){width="8.8cm"} is the total dust volume density. The quantity $ n(r,z,a)$ can change due to grain growth and distribution of masses via fragmentation. The vertically integrated dust surface density distribution per logarithmic bin defined as $$\sigma (r,a)=\int_{-\infty}^{\infty} n(r,z,a)\cdot m\cdot a dz \label{eq11}$$ and the total dust surface density is then $$\Sigma_d(r)=\int_0^\infty \sigma (r,a)d\ln a. \label{eq11}$$ Results {#sec3} ======= Density distribution of dust particles {#subsec3.1} -------------------------------------- ![image](Figure5.pdf){width="18.0cm"} The simulations have been done with a disk of mass $0.05 M_\odot$, with a surface density described by Eq. \[eq1\] from $1.0$ AU to $300$ AU, around a star with one solar mass. The turbulence parameter $\alpha$ is taken to be $10^{-3}$, unless other value is specified. Figure \[Fig1\] shows the vertically integrated dust density distribution, taking into account: coagulation, radial mixing, radial drift and fragmentation, after $1$ Myr of the evolution of the protoplanetary disk. The solid white line shows the particle size corresponding to a Stokes number of unity. From Eq. \[eq6\] we can see that when St=1, the particle size $a$ is proportional to the gas surface density $\Sigma '$, then the solid line reflects the shape of the surface density. The blue line of Fig. \[Fig1\] represents the fragmentation barrier, which illustrates the maximum size of the particles before they reach velocities higher than the fragmentation velocity (see Eq. \[eq9\]). Hence, particles above the fragmentation barrier should fragment down to smaller particles, which again contribute to the growth process. Both plots of Fig. \[Fig1\] have the same amplitude of the sinusoidal perturbation $A=0.1$. The factor $f$ which describes the width of the perturbation, is taken to be $f=1$ for the top plot, and $f=3$ for the bottom plot of Fig. \[Fig1\]. This result shows that: First, the amplitude $A=0.1$ of the perturbation is not high enough to have a positive pressure gradient in those regions (see right plot of Fig. \[comparacion\]) such that particles can be retained in the outer regions of the disk after some Myr. Instead the dust particles are still affected by radial drift and turbulence such that the particles do not grow over mm size in the outer regions. Second, taking a greater value of the factor $f$, at the same amplitude, implies that the retention of particles is even weaker. This is because with a wider perturbation is it harder to have positive pressure gradient. It is expected that for a smaller value of $f$, the pressure gradient is higher since the surface profile should be steeper and therefore a dust trapping much more efficient. However, the diffusion timescale $\tau_{\nu}$ depends quadratically on the length scale $\ell$. Therefore, when the wiggles of the perturbation are taken with a smaller wavelength, the diffusion times become much shorter, implying that the turbulence mixes the dust particles out of the bumps faster, even when the pressure gradient is higher for narrow wiggles. More precisely, $\tau_{\nu}\propto \ell^2 \nu^{-1} $ where the viscosity is defined as $\nu = \alpha c_s h$. For this reason, we notice in Fig. \[width\]-left plot that the trapping is more effective taking values of the width less than one, but only by a very small margin. As a result, the ratio between the final and the initial mass of the dust for $r \in [50, 100] $AU remains almost constant when the width is taken larger than $0.3$. We conclude that for an amplitude of $A$=0.1, the trapping after several million years, does not become more effective when the wavelength of the perturbation is taken shorter. Only when the diffusion timescales become larger or equal to the drift timescales for a given pressure profile, a turbulence parameter and a Stokes number, the dust particles can be retained inside the bumps and therefore they can grow. From Eq. \[eq5\] we can deduce that the drift time scales as $\tau_{\mathrm{drift}}\propto \ell (\partial_{\ell} P)^{-1}$, where inside the bumps $\ell$ is the width of the perturbations (which depends directly on $f$). As a consequence, after an equilibrium state is reached, drift and diffusion timescales are both proportional to the square of the width. Accordingly, for a given value of $f$, the effect of turbulent mixing and radial drift offset. We can notice in Fig. \[width\] that for $f$={0.3, 0.2} there is a small effect in the retention of particles for $\alpha=10^{-3}$ (left plot) and an important effect for $\alpha=10^{-4}$ (right plot). This implies that for these values of $f$ and $A$, the pressure gradient becomes positive enough in the outer regions ($r \in [50, 100] $AU) to have trapping of particles. Since $A$ and $f$ are the same for both cases, the pressure gradient is exactly the same. However, for $\alpha=10^{-3}$, the small efficiency that becomes visible reducing $f$, vanishes after two Myr, because turbulent mixing and radial drift cancel each other. Nevertheless, when $\alpha$ is reduced one order of magnitude (right panel of Fig. \[width\]), the diffusion timescales are now longer. In this case, we have that the drift timescales are shorter than the diffusion timescales, hence the ratio between the final and the initial dust mass increase in average for each $f$. When $f$ is small enough to have positive pressure gradient ($f$={0.3, 0.2}), outward drift wins over turbulent mixing, and as a result there is a visible effectiveness in the trapping of particles. However, there is almost no difference between $f$=0.3 and $f$=0.2. Contrary, this counterbalance effect between radial drift and turbulence when $f$ varies does not happen when the amplitude of the perturbation changes. We fixed the value of the width of the perturbation to unity, because this value is consistent with current model predictions of zonal flows. The comparison between our assumption of the density inhomogeneities and the work from @2011ApJ...736...85U is discussed in Sect. \[sec4\]. In addition, for longer values of the width, we should have higher values of the amplitudes in order to have a positive pressure gradient. In that case, however the disk becomes easily Rayleigh unstable when the amplitude in increased. These are the reasons why we fix the value of the width to unity and not higher. Simulations of MRI-active disks suggest that the typical values for the turbulence parameter $\alpha$ are in the range of $10^{-3}-10^{-2}$ . In this work, we focus on the results for $\alpha=10^{-3}$, because with a larger value of the turbulence the viscous time scales become shorter compared with the growth time scales of the dust, making the particles mix out of the bumps and then drift radially inwards before any mm-sizes are reached. In addition, if $\alpha$ is taken one order of magnitude higher, the fragmentation barrier is lower by about one order of magnitude in grain size, implying that particles do not grow over mm-sizes in the outer regions of the disk (see Eq. \[eq12\]). Fig. \[comparison\_alpha\] shows the location of the fragmentation barrier for the case of $A=0.3$ and $f=1.0$ and two different values of $\alpha$. We can notice that the maximum value of the grain size for the case of $\alpha=10^{-2}$ is of the order of few mm in the outer regions of the disk $r>50$AU, while for $\alpha=10^{-3}$ the grains even reach cm-sizes. Figure \[Fig2\] compares the surface density distribution for two different values of the amplitude of the perturbation $A=0.1$ and $A=0.3$, at different times of evolution. Taking $A=0.3$, we can notice that in the pressure bumps there is high density of dust particles, even after $5$ Myr of evolution for a maximum radius around $80$ AU. For $A=0.3$ and $r\gtrsim 100$AU, there is a small amount of particles above the fragmentation barrier for the different times of evolution. It is important to notice that the line of the fragmentation barrier (\[eq9\]) is calculated taking into account only turbulent relative velocities, since radial and azimuthal turbulence relative velocities were assuming zero at the peaks of the bumps. Particles with St$>1$ are not anymore well coupled to the gas, they are not affected by gas turbulence, then relative velocities due to turbulence are lower, which implies that they can grow over the fragmentation barrier. Moreover, we can see in the right plot of Fig. \[comparacion\], that the pressure gradient for $A=0.3$ and $r\gtrsim 100$AU is always negative, hence for those regions the pressure bumps may not reduce the efficacy of radial drift. Therefore, the total relative velocities for $r\gtrsim 100$AU can be go down, leading some particles (with St$>1$) to grow over the fragmentation barrier. Only the particles with St$<1$ and over the fragmentation barrier should eventually fragment down to smaller particles. For regions $r\lesssim 100$AU, dust particles continuously grow to mm-size particles by coagulation because collision velocities due to turbulence are lower that the taken fragmentation velocity $v_f$. ![Ratio between final and the initial dust mass between $50$ AU and $100$ AU, at different times of evolution. Taking a constant value of the width $f=1.0$ and different values of the amplitude of the perturbed density (Eq. \[eq1\]).[]{data-label="amplitude"}](Figure6.pdf){width="8.8cm"} As we mentioned before, the efficiency of the dust trapping will depend on the amplitude of the pressure bumps. It is expected that for higher amplitude there is more trapping of particles, since the pressure gradient is also higher and positive (see right plot of Fig. \[comparacion\]). Taking the perturbed density of Eq. \[eq1\], we can see in Fig. [\[amplitude\]]{} that between 50 AU and 100 AU from the star, the amount of dust grows significantly from $A=0.1$ and $A=0.3$. From $A=0.3$ to $A=0.5$, there is still a considerably growth, but the rate of growth is slower. From $A=0.5$ to $A=0.7$ the rate remains almost constant, reaching a threshold. When the amplitude is increased, the amount of dust particles retained in the bumps rises to a limit when the dust growth is stopped due to the fact that the particles reach the maximum possible value where they start to fragment due to the high relative velocities. ![image](Figure7.pdf){width="18cm"} As we explained in Sect. \[sec2\], taking a fixed value of $f=1.0$, the disk stays Rayleigh stable for $0.10\lesssim A \lesssim 0.35$, which means that for those values of the amplitude, these kind perturbations can be explained via MRI without any Rayleigh instability present at any evolution time. The amplitude of $A=0.3$ is the one that is more consistent with current MHD simulations of zonal flows [@2011ApJ...736...85U], where pressure reaches radial fluctuations of $25\%$ (see left plot of Fig. \[comparacion\]). The amplitude of $A=0.1$ is more consisted with the case of pressure fluctuation of zonal flows of the order of $10\%$ by [@2009ApJ...697.1269J]. Figure \[ratio\] shows the radial dependence of the dust-to-gas ratio for different times of the simulation, for two values for the perturbation amplitude and wavelength without the gas inward motion. For $A=0.1$ and $f=1.0$ (top-left plot of Fig. \[ratio\]), we can see that the dust-to-gas ratio decreased significantly with time in the whole disk. This implies that the dust particles do not grow considerably with time, which is what we expected due to the fact that with this amplitude the trapping of the particles into the pressure bumps is not effective. Consequently, due to turbulence the dust particles collide, fragment and become even smaller, so they mix and the retention of those small particles, with St$<1$, becomes more difficult. Thus the radial drift is not reduced and particles move inwards. Therefore particles with St$\lesssim1$ drift inwards and the particles that survive are those that are very small St$\ll 1$, and are well coupled to the gas. Hence the dust-gas ratio initially decreases quickly and then becomes almost constant with time, which implies that after several Myr only the very small dust particles remain. Top-right plot of Fig. \[ratio\] shows that taking the same amplitude but a smaller wavelength, the dust to gas radio has the same behavior. This confirms that when $\alpha$ turbulence is constant, a decrease of the wavelength implies shorter diffusion time scales. Therefore the trapping is not more effective even if the pressure gradient is higher for narrow bumps. Conversely, due to the strong over-pressures at $A=0.3$ (bottom plots of Fig. \[ratio\]), the dust-to-gas ratio remains almost constant with time for $r< 100$AU, oscillating radially between $\sim10^{-3}$ to $ \sim10^{-1}$. This oscillating behavior, even after 5 Myr of dust evolution, is possible thanks to the fact that the particles are retained in the bumps and grow enough to make the dust-gas ratio higher inside the bumps. Only around $\sim 100$AU from the star, the dust to gas ratio decreases slowly with time. This implies that for $r< 100$AU, the drift is counteracted by the positive local pressure gradient, allowing that the time scales for the growth are comparable with the disk evolution times, i. e. with the viscous time scales. Changing the width of the perturbation, $f=1.0$ for the left-bottom plot and $f=0.7$ for the right-bottom plot of Fig. \[ratio\], it has just a minor effect over the dust-to gas ratio as was explained before. Comparison to current data of young disks in the millimeter range {#subsec3.2} ----------------------------------------------------------------- In this Section we compare the models predictions of the disk fluxes at millimeter wavelenghts with observational data obtained for young disks in Class II Young Stellar Objects (YSOs). To do this, we calculate the time-dependent flux for the disk models described in Sect. \[sec2\]. For the dust emissivity, we adopted the same dust model as in and , i.e. spherical composite porous grains made of silicates, carbonaceous materials and water ice, with relative abundances from . At each stellocentric radius in the disk, the wavelength-dependent dust emissivity is calculated considering the grain size number density $n (r,z,a)$ derived from the dust evolution models at that radius, as described in Sect. \[sec2\]. ![Comparison of the observed fluxes at mm-wavelengths of young disks in Taurus (red dots; from and Ricci (priv comm)), Ophiucus (blue dots; from ), and Orion Nebula Cluster (green dots; from ) star forming regions with the predictions of the disk models at different times of the disk evolution (star symbols). Disk ages are indicated by numbers, in Myr, above the star symbols. The predicted $\sim1$ mm-fluxes (x-axis) and spectral indices between $\sim$1 and 3mm (y-axis) are for the disk models presented in Sect. \[sec2\] with perturbations characterized by $f=1$ and $A=0.1$ (top panel) or $A=0.3$ (bottom panel). The $\sim 1$mm-flux densities for the Orion disks have been scaled by a factor of (420pc/140pc)$^2$ to account for the different distances estimated for the Orion Nebula Cluster ($\sim 420$pc, ) and Taurus and Ophiucus star forming regions ($\sim 140$pc, , [@2008hsf2.book..351W])[]{data-label="Fig3"}](Figure8a.pdf "fig:"){width="8.8cm"}\ ![Comparison of the observed fluxes at mm-wavelengths of young disks in Taurus (red dots; from and Ricci (priv comm)), Ophiucus (blue dots; from ), and Orion Nebula Cluster (green dots; from ) star forming regions with the predictions of the disk models at different times of the disk evolution (star symbols). Disk ages are indicated by numbers, in Myr, above the star symbols. The predicted $\sim1$ mm-fluxes (x-axis) and spectral indices between $\sim$1 and 3mm (y-axis) are for the disk models presented in Sect. \[sec2\] with perturbations characterized by $f=1$ and $A=0.1$ (top panel) or $A=0.3$ (bottom panel). The $\sim 1$mm-flux densities for the Orion disks have been scaled by a factor of (420pc/140pc)$^2$ to account for the different distances estimated for the Orion Nebula Cluster ($\sim 420$pc, ) and Taurus and Ophiucus star forming regions ($\sim 140$pc, , [@2008hsf2.book..351W])[]{data-label="Fig3"}](Figure8b.pdf "fig:"){width="8.8cm"} The opacity, in the millimeter wavelength regime, can be approximated by a power law [@1993Icar..106...20M], which means that the flux can be approximated to $F_\nu\propto\nu^{\alpha_{\mathrm{mm}}}$, where $\alpha_{\mathrm{mm}}$ is known as the spectral index. The spectral index gives us information about the size distribution of the dust in the disk. Figure \[Fig3\] shows the time-dependent predicted fluxes at $\sim 1$mm ($F_{\mathrm{1mm}}$) and spectral index between $\sim 1$ and $3$mm ($\alpha_{\mathrm{1-3mm}}$) for a disk model with $f=1$ and $A=0.1$ (top panel) or $A=0.3$ (bottom panel). In the same plot mm-data for young disks in Taurus and Ricci (priv comm), Ophiucus and Orion Nubula Cluster are also shown. ![Disk image at 2 Myr and observing wavelength of 0.45 mm, the amplitude of the perturbation is $A=0.3$ and the factor $f=1$ for: disk model with parameters of Table \[table1\] (top), simulated image using full configuration of ALMA (bottom) with a maximum value of baseline of around $3$km and with an observing time of 4 hours. The contour plots are at $\{2, 4, 6, 8\}$ the corresponding rms value (see Table \[table:3\]).[]{data-label="Fig4"}](Figure9a.pdf "fig:"){width="8.8cm"}\ ![Disk image at 2 Myr and observing wavelength of 0.45 mm, the amplitude of the perturbation is $A=0.3$ and the factor $f=1$ for: disk model with parameters of Table \[table1\] (top), simulated image using full configuration of ALMA (bottom) with a maximum value of baseline of around $3$km and with an observing time of 4 hours. The contour plots are at $\{2, 4, 6, 8\}$ the corresponding rms value (see Table \[table:3\]).[]{data-label="Fig4"}](Figure9b.pdf "fig:"){width="8.8cm"} As detailed in the $F_{\rm{1mm}}$ vs $\alpha_{\rm{1-3mm}}$ plot reflects some of the main properties of the dust population in the disk outer regions, which dominate the integrated flux at these long wavelengths. In particular the $1$mm-flux density is proportional to the total dust mass contained in the outer disk. The mm-spectral index is instead related to the sizes of grains: values lower than about 3 are due to emission from grains larger than about $\sim1 $mm, whereas values around 3.5 are due to smaller grains [@2007prpl.conf..767N]. In the case of the disk model with $f=1$ and $A=0.1$, the time evolution of the predicted mm-fluxes clearly reflects the main features of the dust evolution depicted in top plot of Fig. \[Fig3\]. Grains as large as a few millimeters are quickly formed in the disk outer regions ($R \gtrsim 50$ AU), and most of them are initially retained in those regions ($\lesssim 0.5$Myr). As a consequence, the mm-spectral index of the disk is slightly lower than 3 at these early stages. However, the radial drift of mm-sized pebbles becomes soon important and, as already described in Sect. \[sec3\], perturbations with a length-scale of $f=1$ and amplitude of $A=0.1$ are not efficient in retaining mm-sized particles in the outer disk. The 1 mm-flux density significantly decreases because of the loss of dust from the outer regions, especially the mm-sized grains which are efficient emitters at these wavelengths. Given that the spectral index is a proxy for the grain size, that is also affected by radial drift: its value increases with time because of the gradual loss of mm-sized pebbles in the outer disk. For this case, the under-predicted fluxes are not consistent with observational data for disk ages $\gtrsim 1$Myr, i.e. with the mean estimated ages of PMS stars in the Taurus, Ophiucus and Orion regions. ![Comparison between the simulated images for an observing wavelength of $1$ mm and $2$ Myr of evolution, using full antenna configuration of ALMA for two different values of the amplitude of the perturbation: $A=0.1$ (top) and $A=0.3$ (bottom). The contour plots are at $\{2, 4, 6, 8\}$ the corresponding rms value (Table \[table:3\] ).[]{data-label="Fig6"}](Figure10.pdf){width="8.8cm"} Interestingly, a disk with perturbations in the gas surface density with a larger amplitude of $A=0.3$ shows different results. In this case the trapping of particles in the pressure bumps is efficient enough to retain most of the large pebbles formed in the outer disk (see bottom plot of Fig.\[Fig3\]). Since radial drift is much less efficient in this case, the predicted $1$mm-flux density is less affected than in the $A=0.1$ case and, more importantly, the spectral index levels off at a value of $\sim$2.5. This model provides a good match with the bulk of the mm-data for disk ages of a few Myr, as seen in bottom panel of Fig. \[Fig3\]. Note also that the match between the model presented here with $A=0.3$ is better than what has been obtained by . Contrary to the present work, they completely switched off radial drift, therefore they restricted particles to stay artificially in the disk outer regions. Specifically, for a disk with the same unperturbed disk structure presented here, they found a larger $ 1$mm-flux density than what we obtained in the $A=0.3$ perturbation case at few Myr. This indicates that in order to interpret the measured mm-fluxes of young disks both radial drift, and a physical mechanism acting in the disk to trap, although not completely, mm-sized particles in the outer disk are needed. ----------- ------------ ----------------------------- --------------------- -------------------- Amplitude Wavelength Atmospheric conditions Total Flux rms (mm) pwv (mm) $\quad\tau_0$ (Jy) (Jy) 0.45 $0.5\qquad \qquad 0.60$ $6.9\times 10^{-1}$ $7.5\times10^{-4}$ A=0.3 0.66 $1.0\qquad \qquad 0.40$ $7.6\times 10^{-1}$ $3.8\times10^{-4}$ 1.00 $1.5\qquad \qquad 0.20$ $2.2\times 10^{-1}$ $1.4\times10^{-4}$ 3.00 $2.3\qquad \qquad 0.03$ $4.0\times 10^{-1}$ $2.0\times10^{-5}$ A=0.1 1.00 $1.5\qquad \qquad 0.20$ $1.6\times 10^{-2}$ $1.3\times10^{-5}$ ----------- ------------ ----------------------------- --------------------- -------------------- ![image](Figure11.pdf){width="18cm"} ![Spectral index $\alpha_{\mathrm{1-3mm}}$ of the model data (top plot) and the spectral index taking two simulated images at $0.45$mm and $3.0$mm, with a time observation of four hours and using the full configuration (max. baseline of 12 km)[]{data-label="Fig5"}](Figure12.pdf){width="8.8cm"} Future observations with ALMA {#subsec3.3} ----------------------------- The Atacama Large Millimeter/sub-millimeter Array (ALMA) will provide an increase in sensitivity and resolution to observe in more detail the structure and evolution of protoplanetary disks. With a minimum beam diameter of $\sim 5$ mas at $900$GHz, ALMA will offer a resolution down to $2$ AU for disks observed in Orion and sub-AU for disk in Taurus-Auriga [@2010MNRAS.407..181C]. Using the Common Astronomy Software Applications (CASA) ALMA simulator (version 3.2.0), we run simulations to produce realistic ALMA observations of our model using ALMA array of $50$ antennas $12$m-each. The selection of observing mode to obtain the images, was chosen to have simultaneously the most favorable values for the resolution and sensitivity that should be available with ALMA. The spatial resolution depends on the observing frequency and the maximum baseline of the array. We do not take the largest array because for very large baselines, the sensitivity could be not enough for the regions that we will need to observe. Therefore, we used different antenna arrays depending on the observing frequency to get the best possible resolution with enough sensitivity. The sensitivity depends on the number of antennas, the bandwidth (which is taken as $\Delta \nu=8$GHz for continuum observations) and the total observing time that is fixed to four hours for each simulation. The sensitivity also depends on the atmospheric conditions. ALMA is located in Llano de Chajnantor Observatory, where the precipitable water vapor (pwv) varies between $0.5$ mm and $2.0$ mm depending on the observable frequency. For the simulations, we assume that the value of the pwv varies with frequency (see Table \[table:3\]). The synthetic images are fully consistent with the opacity dust distribution discussed in Sec. \[subsec3.2\]. Figure \[Fig4\] shows a comparison between the model image and a simulated ALMA image using the full configuration of ALMA with a maximum baseline of around $3$ km and an observation total time of four hours. This image is for observing wavelength of $0.45$mm (Band 9 of ALMA $620-750$ GHz). It is important to note that the simulated images take into account the atmospheric conditions and the expected receiver noise based on technical information of the antennas, but the residual noise after data calibrations and its uncertainties are not considered. We can note (see Fig. \[Fig4\]-bottom plot) that with one of the full configurations (max. baseline $\sim 3$ km), it is possible to distinguish some ring structures due to the fact that the dust has drifted considerably into the rings compared to the gas. In Fig. \[Fig6\] we note again the importance of having a high value of the amplitude. If the gas surface density of the disk is $A=0.3$, then the effects will be observable with ALMA. Both images of the figure have been compute with the complete antenna configuration of ALMA for an observing wavelength of $1$ mm and $2$ Myr of the disk evolution. We can see that for $A=0.1$ it is not possible to detect any structures around the star even considering a perfect data calibration. It is important to note that due to the trapping of dust particles at peaks of the pressure bumps, the contrasts between rings in the simulated images is very clear, around $\sim 20-25\%$, while the contrast for the gas is almost unrecognized. Figure \[compare\] compares the simulated images at different observing wavelength using different antenna configurations of ALMA. The antenna configuration is chosen by CASA depending on the expected resolution. The best image is obtained at $100$ GHz and a maximum baseline of 16 km (most extended ALMA configuration), where it is possible to detect clearly the most external ring structure and some internal ring structures. Nevertheless, with more compact configurations at different frequencies is still possible to detect some structures from the presence of the pressure bumps which allow the formation of mm-sized particles. However, it is important to take into account that the simulated images of Figs. \[Fig4\], \[Fig6\] \[compare\] and \[Fig5\] are considering a perfect data calibration after observations, for long baselines and high frequencies the calibration effects become more important. Taking the ratio of the images at two different wavelengths, we will have the values of the spectral index $\alpha_{1-3mm}$, which indicates the location of mm-sized grains. With the full configuration of ALMA and a maximum value of the baseline of $12$ km for both observing frequencies, some regions with large particles are distinguished which are the regions with low spectral index $\alpha_{\mathrm{1-3mm}}\lesssim 3$, as it was explained in Sec. \[subsec3.2\]. In Fig. \[Fig5\] is the spectral index of the model data (top plot) and the spectral index taking two simulated images at $0.45$mm and $3.0$mm, with a total observing time of four hours. ![Top plot: Ratio between the surface density at two different azimuthal angles of of the disk from zonal flows simulation by [@2011ApJ...736...85U]. The azimuthal angles are chosen such that for a specific radius, the amplitude of the pressure bump has a maximum $\Sigma_{\mathrm{max}}$ and a minimum $\Sigma_{\mathrm{min}}$. Bottom plot: Azimuthal velocity with respect to Keplerian velocity for the azimuthal and time-averaged surface density of the midplane.[]{data-label="ratio_zonal"}](Figure13a.pdf "fig:"){width="8.8cm"}\ ![Top plot: Ratio between the surface density at two different azimuthal angles of of the disk from zonal flows simulation by [@2011ApJ...736...85U]. The azimuthal angles are chosen such that for a specific radius, the amplitude of the pressure bump has a maximum $\Sigma_{\mathrm{max}}$ and a minimum $\Sigma_{\mathrm{min}}$. Bottom plot: Azimuthal velocity with respect to Keplerian velocity for the azimuthal and time-averaged surface density of the midplane.[]{data-label="ratio_zonal"}](Figure13b.pdf "fig:"){width="8.8cm"} Approach to zonal flows predictions {#sec4} =================================== In this paper we have so far assumed ad-hoc models of pressure bumps. But which processes may cause such long lived bumps in protoplanetary disks? In this section we study zonal flows as a possible explanation of the origin of long lived pressure bumps. One possible cause of pressure bumps originates from MRI turbulence. [@1995ApJ...440..742H] and [@1995ApJ...446..741B] were the first attempts to simulate nonlinear evolution of MRI in accretions disk, taking a box as representation of a small part of the disk. More recent simulations have been done taking higher resolution [see e.g. @2009ApJ...697.1269J] and global setup [@2011ApJ...735..122F]. ![image](Figure14.pdf){width="18.0cm"} In magnetorotational instability, “zonal flows” are excited as a result of the energy transportation from the MRI unstable medium scales, to the largest scales, causing an inverse cascade of magnetic energy, and creating a large-scale variation in the Maxwell stress [@2009ApJ...697.1269J]. Different three-dimensional MHD simulations show that with the presence of zonal flows, pressure bumps can appear if there are drops in magnetic pressure through the disk [@2009ApJ...697.1269J], and [@2011ApJ...736...85U]. Nevertheless, in recent simulations with higher resolution by [@2011ApJ...735..122F], pressure bumps are not formed. There is still not a final answer about how and why these pressure bumps can or cannot be created via zonal flows. Another alternative for the origin of pressure bumps due to MRI is the change of the degree of ionization. The disk becomes MRI active if the degree of ionization is sufficient, for the magnetic field to be well couple to the gas. Variable degrees of ionization in the disk could cause local changes in the magnetic stress, which could induce structures in the density and pressure. The question we now wish to answer is: are the pressure bumps caused by zonal flows of MRI-turbulence strong enough to trap the dust like in the models of Sect. \[sec3\]? To find this out, we take the three-dimensional MHD simulations of MRI-turbulent protoplanetary disks by [@2011ApJ...736...85U]. These models have a resolution of $(N_r, N_{\theta}, N_{\phi})=(256,128, 256)$. It is important to notice that the MRI dynamical timescale is around one orbital period (at one AU) while the dust growth timescale is more than 100 orbital periods. Currently, it is not feasible to study the dust growth process self-consistently in the full time-dependent 3-D model. It would require running the MHD model tens to hundreds of times longer than what is currently affordable. Hence, the strategy is to find first a quasi-steady state of the gas surface density from MRI evolution, in which structures in the pressure survive the entire simulation (around 1000 inner disk orbits). Afterwards, to do the coagulation/fragmentation simulation of the dust in $1$D taking the gas surface density for a specific azimuthal angle in the midplane from the results from MHD simulations. Top plot of Fig. \[ratio\_zonal\] shows the ratio between the time averaged surface density at two different azimuthal angles where the amplitude of a pressure bump is maximum and minimum for a specific radius. We can notice, that the variations on the azimuthal angle are very uniform, around $\sim 5 \%$. This is the reason why we work with the azimuthally averaged density. For our simulations, we assume the pressure structure survives and is stable on dust growth and evolution timescales. The lifetime of these structures as determined by global disk simulations is still uncertain, but it has been observed to be on the order of 10-100 local orbits (at the radial position of the bump) [@2009ApJ...697.1269J; @2011ApJ...736...85U]. It is still an open question whether this behavior can be directly re-scaled to apply to the outer parts of the disk and in any case, the structures should be eventually diffused on turbulent diffusion timescales. In the future, the continuous generation and evolution of these structures should be implemented alongside the dust evolution. However, for lack of a better model of this time-dependence at this stage, we assume these structures to be static. Since these MHD simulations use a radial domain where $r \in [1; 10]$, we rescale this grid logarithmically, so the gas surface density is taken from $10$AU to $100$AU, and also the surface density has been scaled such that the total disk mass is $0.05 M_\odot$, see Fig. \[comparacion\]-left plot (solid-line). Comparing the gas surface density obtained from MHD simulations with the assumed perturbed density $\Sigma'$ (Eq. \[eq1\]), we see that the amplitude of the surface density perturbation from zonal flows is around $25\%$ and comparable with the amplitude of $30\%$ of $\Sigma '$. The widths of the bumps from [@2011ApJ...736...85U] are not uniform, but our assumption of $f=1$ fit well with some of those bumps. The bottom plot of Fig. \[ratio\_zonal\] shows the azimuthal velocity with respect to Keplerian velocity for the azimuthal and time-averaged surface density of the midplane. We can notice that the azimuthal velocity exceeds the Keplerian velocity for some regions of the disk. This implies that for those regions, the presence of zonal flows allow to have a positive pressure gradient leaving dust particles to move outwards. Therefore, the peaks of the pressure bumps created by zonal flows may be regions where dust particles can reach millimeter sizes. Figure \[zonal\] shows the vertically integrated dust density distribution after 2 Myr and 4 Myr of dust evolution (making use of the model by ). We can notice that at that time of evolution the pressure bumps caused by zonal flows are able to retain mm and cm sized particles in the outer regions of the disk. This is the reason why a high mass disk is considered in this case, in order to reach large grains. Around $50$-$60$AU there is clearly a region with high vertically integrated dust density distribution for mm and cm-sized particles. It is important to notice that the peak around $~100$AU is a result of the boundary condition. Conclusions {#sec5} =========== Theoretical models of dust evolution in protoplanetary disks show that the growth from sub-micron sized particles to larger objects is prevented basically by two phenomena: radial drift and fragmentation. Nevertheless, infrared and radio observations show that millimeter sized particles can survive under those circumstances in the outer regions of disks. Therefore, various theoretical efforts have been focused on explaining the survival of those bodies. Taking into account strong inhomogeneities expected to be in the gas density profile e.g. zonal flows, and using the coagulation/fragmentation and disk-structure models by , we have investigated how the presence of pressure bumps can cause the reduction of radial drift, allowing the existence of millimeter sized grains in agreement with observations. In this work, we assumed a sinusoidal function for the gas surface density to simulate pressure bumps. The amplitude and wavelength disturbances are chosen considering the necessary conditions to have outward angular momentum transport in an $\alpha$-turbulent type disk, outward radial drift of dust and reasonable values compare to predictions from the recent work of zonal flows [@2011ApJ...736...85U]. The results presented here suggest that the presence of pressure bumps with a width of the order of the disk scale-height and an amplitude of $30\%$ of the gas surface density of the disk, provide the necessary physical conditions for the survival of larger grains in a disk with properties summarized in Table  \[table1\]. Comparisons between the observed fluxes of the Taurus, Ophiucus and Orion Nebula Cluster star forming regions with the results of the models ratify that the effect of the radial drift is reduced allowing particles to grow. Figure \[Fig3\] shows how models with these kind of disturbances reproduce much better mm-observations than models with full or without radial drift. In addition, we presented a comparison between the bumpy density profile assumed in this work and 3D MHD models of zonal flows that can cause long lived bumps in protoplanetary disks. We showed that the pressure bumps cause by zonal flows of [@2011ApJ...736...85U] are in agreement with the amplitudes and wavelengths used in this work. Therefore, taking those bumps, the survival of dust particles is possible in the outer regions after some Myr. The simulated images using CASA ALMA simulator (version 3.2.0) show that, with different antenna configuration of the final ALMA stage, the ring structures, due to the presence of the pressure bumps, should be detectable. Future ALMA observations will have an important impact for understanding the first stages of planet formation and it will be very important to investigate if the grain growth and retetion can be explained with the presence of these kind of inhomogeneities in the gas density profile. We acknowledge Francesco Trotta for his help with the code that we used in this work to derive the mm-fluxes. We would like to thank the referee, Wladimir Lyra, for his useful suggestions. This work was supported in part through the 3rd funding line of German Excellence Initiative. T. Birnstiel acknowledges ESO Office for Science which provided funding for the visits in Garching. L. Ricci acknowledges the PhD fellowship of the International Max-Planck-Research School. A. L. Uribe acknowledges the CPU time for running the simulations in the Bluegene/P supercomputer and the THEO cluster at the Rechenzentrum Garching (RZG) of the Max Planck Society. Finally, L. Testi acknowledges ASI contract to the INAF-Osservatorio Astrofisico Di Arcetri.
--- abstract: 'We investigate the aggregation and phase separation of thin, living [*T. Tubifex*]{} worms that behave as active polymers. Randomly dispersed active worms spontaneously aggregate to form compact, highly entangled blobs, a process similar to polymer phase separation, and for which we observe power-law growth kinetics. We find that the phase separation of active polymer-like worms does not occur through Ostwald ripening, but through active motion and coalescence of the phase domains. Interestingly, the growth mechanism differs from conventional growth by droplet coalescence: the diffusion constant characterizing the random motion of a worm blob is independent of its size, a phenomenon that can be explained from the fact that the active random motion arises only from the worms at the surface of the blob. This leads to a fundamentally different phase-separation mechanism, that may be unique to active polymers.' author: - 'A. Deblais' - 'D. Bonn' - 'S. Woutersen' title: 'Phase Separation of Active-Polymer-like *T. Tubifex* Worms' --- Driven colloidal particles [@palacci13; @Geyer2019; @Linden2019], self-propelled bots [@Briand2016; @Deblais2018], cells [@Schwarz-Lineka2012; @Duclos2018], animals [@Tennenbaum2015] and humans [@Bain2019] belong to the broad field of active matter: a class of non-equilibrium entities composed of many interacting units that individually consume energy and which can at larger scales collectively generate motion or mechanical stresses [@liverpool00; @speck14; @Linden2019]. Their collective behavior is fascinating: whether it is flocks of birds, schools of fish, or ants collaborating to survive, the activity and interactions of the individual components give rise to highly non-trivial macroscopic phenomena [@Bechinger2012]. Here, we investigate the spontaneous aggregation and eventual phase separation of active-polymer-like living worms. At first sight, the phase separation of active polymers may seem similar to that of solutions [@Dobry1947; @Tanaka2005] and crystallizing solids [@Olson1988], for which detailed theories are available: in both cases, the aggregating particles move randomly and tend to stick together when they are in close proximity. As such, one might naively expect the phase separation of active particles to involve a mechanism similar to Ostwald ripening [@Voorhees1992], where the aggregation is driven by the combined effect of diffusion of the aggregating particles and surface tension of the aggregates. However, recent work has shown that the phase separation of active particles can involve unique mechanisms that rely completely on the activity [@Bechinger2012; @Schwarz-Lineka2012; @Thompson2011; @Cates2015; @Geyer2019; @Linden2019], and our results indicate that this also holds for the phase separation of active polymers. We investigate the phase separation of living [*T. Tubifex*]{} worms in water. These worms are active swimmers, and approximately 300 $\mu$m thick and 10–40 mm long (Supplementary Information Discussion I.1). The thermal random motion of the worms (estimated from the Stokes-Einstein equation) is negligible compared to their active motion, so they constitute a good model system for active polymers [@winkler17; @martin18; @Bianco2018; @martin19; @Mokthari2019; @deblais_preprint_2019]. When randomly distributed over a volume of water, the worms aggregate spontaneously (Fig. 1) into highly entangled ‘blobs’. The biological function of this aggregation is to minimize exposure to dissolved oxygen, high levels of which are poisonous to [*Tubifex*]{} [@walker1971]. We find that we can analyze the aggregation with relatively simple methods, which makes these living worms an excellent system to investigate the phase separation of active polymers. ![**Aggregation and phase separation of *T. Tubifex* worms**. (**a–c**)  Snapshots of active-worm aggregation in a 25$\times$25$\times$2.5 cm volume at $t=0$ (a), 9.5 min (b), and 60 min (c). **d** Snapshots from a 1D experiment in a square tube of 51$\times$1$\times$1$\times$1 cm. (**e–h**) Close ups from another experiment at $t=25$ min (e), 28.5 min (f), 29 min (g), and 31.5 min (h), showing the coalescence of two blobs. See Supplementary Information for the videos.[]{data-label="fig:photos"}](fig1.png){width="\linewidth"} ![image](Fig2.pdf){width="\linewidth"} In the experiments, we disperse a specific number of worms in a thermostated water volume and observe their aggregation in real time by recording videos (see Supplementary Information). We investigate two geometries: in the simplest, the worms are dispersed in a 25$\times$25$\times$2.5 cm volume of water. In this geometry worm motion is effectively two-dimensional (2D), since the worms are denser than water and therefore always located at or close to the bottom of the water volume (Supplementary Information Fig. S2). In other experiments the water volume is a 51$\times$1$\times$1 cm channel in which the worms are confined in an effectively one-dimensional space (1D). Figures \[fig:photos\]a–c show snapshots from a typical 2D experiment; Fig. \[fig:photos\]d from a typical 1D experiment (Supplementary Information videos 1 and 2). In both geometries, as time progresses, the worms form ever larger aggregates until after about 1 hour all worms are concentrated into a few large aggregates, which have the shape of a slightly flattened sphere (Supplementary Information Discussion II.1). This shape is a compromise between minimum exposed surface and minimum gravitational energy. In the following, we refer to these worm aggregates as ‘blobs’. The aggregation of the active worms seems similar to that generally observed for polymer phase separation. Hence, one might expect that the aggregation occurs through Ostwald ripening, in which larger aggregates grow at the cost of smaller ones [@jonesbook]. This mechanism is driven by the reduction of the total surface tension with increasing average blob size, and we do in fact measure a finite surface tension for the worm blobs (see Supplementary Information). However, closer inspection (Figures \[fig:photos\] e–h) shows that blobs of all sizes are growing, and that the growth does not occur by Ostwald ripening, but rather by the merging of smaller aggregates into larger ones. Such growth by coalescence of diffusing droplets has been investigated previously in the context of the growth of vapor-deposited thin films and of droplets on a surface [@family88; @steyer91; @meakin92]. To quantify our observations, we characterize the average blob size $\langle R \rangle$ as a function of time by taking the 2D Fourier transform of the images and determining the wave-vector magnitude $q_{\rm max}$ at which the spectrum reaches its maximum intensity (see Figure \[fig:fourier\]a for an example). With increasing time, $q_{\rm max}^{-1}$ shifts to lower values, corresponding to an increasing average blob size $\langle R \rangle$. Figure \[fig:fourier\]b shows the average blob size $q_{\rm max}^{-1}$ determined in this way as a function of time, at several temperatures. We find that the growth exhibits power law behavior with $\langle R_{2D} \rangle \sim t^{1/3}$ in the 2D experiments, and $\langle R_{1D} \rangle \sim t^{1/4}$ in the 1D experiments, independently of the initial worm concentration (Supplementary Information Fig. S3). In addition, we observe in the 2D experiments that the power-law growth behavior is independent of temperature. We also investigate the motion of individual worms by recording image sequences of single, isolated worms at different temperatures (Supplementary Information Discussion I). Figure \[fig:fourier\]c shows an example of the centre-of-mass trajectory of an individual worm at various temperatures. The motion is a random walk with an effective diffusion constant that increases with temperature, as is confirmed by extracting the mean square displacement from which we retrieve the diffusion coefficient (Supplementary Information Discussion I.2), see Fig. \[fig:fourier\]e. We find that by rescaling the time axis $\tau \propto D_{\rm worm} t$ all the growth curves collapse onto a single master curve $\propto\tau^{1/3}$ (Fig. \[fig:fourier\]f), confirming that the aggregation kinetics is determined by the random motion of the the worms. ![**Simulation of phase-separation dynamics.** **a** Snapshots of simulated blob growth by coalescence of randomly moving spherical blobs, with blob diffusion constant $D_{\rm blob}$ inversely proportional to the blob radius (top), and independent of blob radius (bottom). **b** Average blob radius $\langle R \rangle$ as a function of time obtained from the simulations. In both cases the growth follows a power law (indicated by lines), with exponents of 0.15 and 0.3 respectively.[]{data-label="fig:sim"}](Fig3.pdf){width="8.5cm"} To shed more light on the observed power-law growth kinetics, we perform computer simulations of 2D growth by coalescence of diffusing droplets, using an approach similar to that of Ref.  (Supplementary Information Discussion III). We assume that the blobs are spherical and move randomly in a 2D space, and that two blobs with radii $R_1$ and $R_2$ at positions ${\bf r}_1$ and ${\bf r}_2$ coalesce when $|{\bf r}_1-{\bf r}_2|<R_1+R_2$ to form a new blob with radius $R=\sqrt[3]{R_1^3+R_2^3}$ at position ${\bf r}=(R_1^3 {\bf r}_1+R_2^3{\bf r}_2)/(R_1^3+R_2^3)$. The simulation starts with a random distribution of mono-disperse spheres with radius 1 which represent the individual worms, in a 2D space of size 200$\times$200. At every time step, each droplet is moved in a random direction by a distance equal to the diffusion constant $D_{\rm blob}$. Fig. \[fig:sim\]a shows snapshots from two simulations with two different $D_{\rm blob}(R)$ functional dependencies, and Fig. \[fig:sim\]b the time-dependent mass-weighted average size $\langle R \rangle = \sum_i R_i^4 /\sum_i R_i^3$, where both summations run over all the spheres present in the volume (similar results are obtained when using the number-weighted average, see Supplementary Information Fig. S4). If we assume that the effective diffusion constant $D_{\rm blob}$ of a randomly moving blob depends on its radius $R$ as $D_{\rm blob}=R^{-1}$, as is the case for blobs undergoing Brownian random motion (Stokes-Einstein equation) [@jonesbook], we obtain power-law growth of the average blob size with an exponent of $\sim$0.15 (purple circles in Fig. \[fig:sim\]b). This exponent is much smaller the experimentally observed value of $\sim$0.3 (Fig. \[fig:fourier\]f). It may be noted that the exponent obtained from the simulation is different from that in Ref. [@steyer91] because in our case we have conservation of total mass, and so depletion in the space between the blobs. We believe that the discrepancy between the simulated and experimentally observed power-law exponents is due to the active motion of the worms. For a blob of active worms, the effective diffusion constant characterizing the random motion may not be inversely proportional to the blob size, as it is for a particle undergoing conventional, Brownian random motion [@jonesbook]. To investigate this idea in more detail, we perform additional experiments in which we determine the effective diffusion constants of blobs with different sizes, see Figure \[fig:blobdiff\]. ![**Blob diffusion.** Effective diffusion constant as a function of average blob size $\langle R_{\rm blob} \rangle$ at T=20$\degree C$ as determined from the slopes of experimental MSDs (see Supplementary Information). The error bars are mostly due to sample-to-sample variability. The purple line shows the expected scaling for particles undergoing Brownian random motion ($D_{blob} \sim {\langle R_{\rm blob} \rangle}^{-1}$). The experimental data (blue symbols) indicate a diffusion constant independent of blob size (dotted line). Lower insets: photographs of blobs of different sizes from which we measured the diffusion coefficient.[]{data-label="fig:blobdiff"}](Fig4.pdf){width="7.cm"} Interestingly, the effective diffusion constant does not depend on $R_{\rm blob}$ as $D_{\rm blob}\propto R_{\rm blob}^{-1}$, but appears to be independent of the radius of the blob. This means that the random motion of the worm blobs differs fundamentally from conventional, Brownian random motion, In the latter case, $D_{\rm blob}=k_{\rm B}T/6\pi\eta R_{\rm blob}$, with $k_{\rm B}$ Boltzmann’s constant and $\eta$ the viscosity of the liquid surrounding the particle [@einstein1905; @jonesbook]. The difference can be explained by considering the origin of the random motion: Brownian random motion of a particle is caused by the thermal motion of the surrounding molecules, whereas the random motion of a worm blob is due to the active motion of the constituent worms, and this can lead to a different dependence of $D_{\rm blob}$ on $R_{\rm blob}$. To rationalize the observed size-independent blob-diffusion constant for active worms, let us assume that an individual worm exerts a swimming force of fixed magnitude $|F_0|$ in a direction that varies randomly with a correlation time $\tau$. Since the worms inside the entangled blob are effectively immobilized, only the worms at the outer surface of a blob contribute swimming force. The number of worms at the surface of a blob of radius $R_{\rm blob}$ is $N_{\rm surf}\sim R_{\rm blob}^2$, so the total random force exerted by these $N_{\rm surf}$ worms has an average magnitude $\langle |F_{\rm drive}|\rangle\sim|F_0|\sqrt{N_{\rm surf}}\sim |F_0|R$ (and a correlation time $\tau$). The drag force on the blob as a function of speed $v$ is given by Stokes’ law, $F_{\rm drag}\sim vR_{\rm blob}$. To obtain the steady-state speed $v$ we equate the driving and drag forces, and obtain a velocity $v$ that is independent of the blob radius $R_{\rm blob}$. Assuming that the random walk of the blobs occurs by random steps in which it has approximately the steady-state speed, this result implies that a blob-diffusion constant that is independent of blob size. Again simulating the blob growth, but now using a size-independent blob diffusion constant ($D_{\rm blob}=1$), we obtain a growth exponent of $\sim0.3$ (red points and line in Fig. \[fig:sim\]b), in good agreement with the experimentally observed value. Thus a size-independent diffusion constant for the random motion of the blobs explains the observed power-law growth of the worm blobs (Figs. \[fig:fourier\] and \[fig:sim\]), at least in the 2D experiments. In the 1D experiments, the situation is complicated by the fact that blob motion slows down when the blob size becomes comparable to the channel width, possibly due to friction at the glass/worm surface. This makes quantitative analysis of the blob diffusion difficult. However, the fact that in this case the diffusion constant does decrease with $R_{\rm blob}$, and that at the same time the growth exponent is closer to that predicted by the conventional droplet-coalescence model does provide a qualitative confirmation of the above ideas. To conclude, the active motion of [*T. Tubifex*]{} worms leads to a phase-separation mechanism that is fundamentally different from that of normal polymers. It involves growth by blob coalescence rather than Ostwald ripening, and with power-law behavior that is different from that of conventional growth by diffusing-droplet coalescence. The observed phase-separation mechanism and power-law kinetics seem to be due only to the active nature of the living worms and to the immobilization of the worms at the inside of the entangled blobs; it may therefore be a generic phenomenon, also occurring in other types of polymeric active matter. We hope that the results presented here will stimulate further experimental and theoretical work in this direction. *Acknowledgement.* We thank the workshop of the University of Amsterdam for their skilful technical assistance, and aquarium shop Holgen for providing us with fresh batches of [*Tubifex*]{}. We thank Moslem Mbehjati for his contribution to the first experiments. A.D. acknowledges the funding from the European Union’s Horizon 2020 research and innovation program under the Individual Marie Skłodowska-Curie fellowship grant agreement number 798455. Author Contributions: S.W. and D.B. conceived the idea; all authors performed the experiments, discussed the results and wrote the manuscript. 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--- abstract: | There exist different kinds of averaging of the differences of the energy-momentum and angular momentum in normal coordinates [**NC(P)**]{} which give tensorial quantities. The obtained averaged quantities are equivalent mathematically because they differ only by constant scalar dimensional factors. One of these averaging was used in our papers \[1-8\] giving the [*canonical superenergy and angular supermomentum tensors*]{}. In this paper we present another averaging of the differences of the energy-momentum and angular momentum which gives tensorial quantities with proper dimensions of the energy-momentum and angular momentum densities. But these averaged relative energy-momentum and angular momentum tensors, closely related to the canonical superenergy and angular supermomentum tensors, [*depend on some fundamental length $L>0$*]{}. The averaged relative energy-momentum and angular momentum tensors of the gravitational field obtained in the paper can be applied, like the canonical superenergy and angular supermomentum tensors, to [*coordinate independent*]{} analysis (local and in special cases also global) of this field. We have applied the averaged relative energy-momentum tensors to analyze vacuum gravitational energy and momentum and to analyze energy and momentum of the Friedman (and also more general) universes. The obtained results are interesting, e.g., the averaged relative energy density is [*positive definite*]{} for the all Friedman universes. address: 'Institute of Physics, University of Szczecin, Wielkopolska 15; 70-451 Szczecin, POLAND[^1]' author: - Janusz Garecki title: 'The averaged tensors of the relative energy-momentum and angular momentum in general relativity and some of their applications' --- The averaged relative energy-momentum and angular momentum tensors in general relativity ======================================================================================== In the papers \[1-8\] we have defined the canonical superenergy and angular supermomentum tensors, matter and gravitation, in general relativity ([**GR**]{}) and studied their properties and physical applications. In the case of the gravitational field these tensors gave us some substitutes of the non-existing gravitational energy-momentum and gravitational angular momentum tensors. The canonical superenergy and angular supermomentum tensors were obtained pointwise as a result of some special averaging of the differences of the energy-momentum and angular momentum in normal coordinates [**NC(P)**]{}. The role of the normal coordinates [**NC(P)**]{} is, of course, auxilliary, only to extract tensorial quantities even from pseudotensorial ones. The dimensions of the components of the canonical superenergy and angular supermomentum tensors can be written down as:\[the dimensions of the components of an energy-momentum or angular momentum tensor (or pseudotensor)\]$\times m^{-2}$. In this paper we propose a new averaging of the energy-momentum and angular momentum differences in [**NC(P)**]{} which is very like to the averaging used in \[1-8\] and which gives the averaged quantities with proper dimensionality of the energy-momentum and angular momentum densities. Namely, we propose the following general definition of the averaged tensor (or pseudotensor) $T_a^b$ $$<{\it T_a^{~b}}(P)> := \displaystyle\lim_{\varepsilon\to 0}{\int\limits_{\Omega}{\bigl[{\it T_{(a)}^{~~~(b)}}(y) - {\it T_{(a)}^{~~~(b)}}(P)\bigr] d\Omega}\over\varepsilon^2/2\int\limits_{\Omega}d\Omega},$$ where $${\it T_{(a)}^{~~~(b)}}(y) := T_i^{~k}(y){}e^i_{~(a)}(y){}e_k^{~(b)}(y),$$ $${\it T_{(a)}^{~~~(b)}}(P):= T_i^{~k}(P){}e^i_{~(a)}(P){}e_k^{~(b)}(P) = {\it T_a^{~b}}(P)$$ are the tetrad (or physical) components of a tensor or a pseudotensor $T_i^{~k}(y)$ which describes an energy-momentum distribution, $y$ is the collection of normal coordinates [**NC(P)**]{} at a given point [**P**]{}, $e^i_{~(a)}(y),~e_k^{~(b)}(y)$ denote an orthonormal tetrad field and its dual, respectively, $$e^i_{~(a)}(P) = \delta^i_a,~e_k^{~(a)}(P) =\delta^a_k,~e^i_{~(a)}(y)e_i^{~(b)}(y) =\delta_a^b,$$ and they are parallelly propagated along geodesics through [**P**]{}. For a sufficiently small domain $\Omega$ which surrounds [**P**]{} we require $$\int\limits_{\Omega}{y^id\Omega} = 0,~~\int\limits_{\Omega}{y^iy^kd\Omega} = \delta^{ik} M,$$ where $$M = \int\limits_{\Omega}{(y^0)^2 d\Omega} = \int\limits_{\Omega} {(y^1)^2 d\Omega} = \int\limits_{\Omega}{(y^2)^2 d\Omega} = \int\limits_{\Omega}{(y^3)^2 d\Omega},$$ is a common value of the moments of inertia of the domain $\Omega$ with respect to the subspaces $y^i = 0, ~~(i = 0,1,2,3)$. The procedure of averaging of an energy-momentum tensor or an energy-momentum pseudotensor given in (1) is a four-dimensional modification of the proposition by Mashhoon \[9-12\]. Let us choose $\Omega$ as a small analytic ball defined by $$(y^0)^2 + (y^1)^2 + (y^2)^2 + (y^3)^2 \leq R^2 = \varepsilon^2 L^2,$$ which can be described in a covariant way in terms of the auxiliary positive-definite metric $h^{ik} := 2v^iv^k - g^{ik}$, where $v^i$ are the components of the four-velocity of an observer [**O**]{} at rest at [**P**]{} (see, e.g., \[1-8\]). $\varepsilon$ means a small parameter: $\varepsilon\in (0;1)$ and $L >0$ is a fundamental length. Since at [**P**]{} the tetrad and normal components are equal, from now on we will write the components of any quantity at [**P**]{} without (tetrad) brackets, e.g., $T_a^{~b}(P)$ instead of $T_{(a)}^{~~~(b)}(P)$ and so on. Let us now make the following expansions for the energy-momentum tensor of matter $T_i^{~k}(y)$ and for $e^i_{~(a)}(y), e_k^{~(b)}(y)$ \[13\] $$\begin{aligned} T_i^{~k}(y) &=& {\hat T}_i^{~k} + \nabla_l {\hat T}_i^{~k}y^l +1/2{{\hat T}_i^{~k}}{}_{,lm} y^ly^m + R_3\nonumber\\ &=& {\hat T}_i^{~k} +\nabla_l{\hat T}_i^{~k} y^l +1/2\biggl[\nabla_{(l}\nabla_{m)}{\hat T}_i^{~k}\nonumber\\ &-&1/3{\hat R}^c_{~(l\vert i\vert m)}{}{\hat T}_c^{~k} +1/3 {\hat R}^k_{~(l\vert c\vert m)}{} {\hat T}_i^{~c}\biggr] y^ly^m + R_3,\end{aligned}$$ $$e^i_{~(a)}(y) = {\hat e}^i_{~(a)} + 1/6{\hat R}^i_{~lkm}{\hat e}^k_{~(a)} y^ly^m + R_3,$$ $$e_k^{~(b)}(y) = {\hat e}_k^{~(b)} -1/6{\hat R}^p_{~lkm}{\hat e}_p^{~(b)} y^ly^m + R_3,$$ which give (1) in the form $$<_m T_a^{~b}(P)> =\displaystyle\lim_{\varepsilon\to 0}{\int\limits_{\Omega}\bigl[\nabla_l{\hat T}_a^{~b} y^l + 1/2\nabla_{(l}\nabla_{m)}{\hat T}_a^{~b}y^ly^m + THO\bigr]d\Omega\over \varepsilon^2/2\int\limits_{\Omega}d\Omega},$$ where $THO$ means the terms of higher order in the expansion of the differences $T_{(a)}^{~~~(b)}(y) - T_{(a)}^{~~~(b)}(P) = T_{(a)}^{~~~(b)}(y) - T_a^{~b}(P)$; $R_3$ is the remainder of the third order and $\nabla$ denotes covariant differentiation. Hat denotes the value of an object at [**P**]{} and the round brackets denote symmetrization from which the indices inside vertical lines, e.g., $(a\vert c\vert b)$ are excluded. The first and $THO$ terms in the numerator of (11) do not contribute to $<_m T_a^{~b}(P)>$. Hence, we finally get from (11) $$<_m T_a^{~b}(P)> = _m S_a^{~b}(P) {L^2\over 6},$$ where $$_m S_a^{~b}(P) := \delta^{lm}\nabla_{(l}\nabla_{m)}{\hat T}_a^{~b}$$ is the [*canonical superenergy tensor of matter*]{} \[1-8\]. By introducing the four velocity ${\hat v}^l \dot = \delta^l_0,~v^lv_l =1$ of an observer [**O**]{} at rest at [**P**]{} and the local metric ${\hat g}^{ab}\dot = \eta^{ab}$, where $\eta^{ab}$ is the inverse Minkowski metric, one can write (13) in a covariant way as $$_m S_a^{~b} (P;v^l) = \bigl(2{\hat v}^l{\hat v}^m - {\hat g}^{lm}\bigr)\nabla_{(l}\nabla_{m)} {\hat T}_a^{~b}.$$ The sign $\dot =$ means that an equality is valid only in some special coordinates. The matter superenergy tensor $_m S_a^{~b}(P;v^l)$ [*is symmetric*]{}. As a result of an averaging the tensor $_m S_a^{~b}(P;v^l)$, and in consequence the averaged tensor $<_m T_a^{~b}(P;v^l)>$, do not satisfy any local conservation laws in general relativity. However, these tensors satisfy trivial local conservation laws[^2] in special relativity (see, e.g., \[1-8\]). Now let us take the gravitational field and make the expansion $$\begin{aligned} _E t_i^{~k}(y) &=& {\alpha\over 9}\biggl[{\hat B}^k_{~ilm} + {\hat P}^k_{~ilm}\nonumber\\ &-& {\delta_i^k\over 2}{\hat R}^{abc}_{~~~l}\bigl({\hat R}_{abcm} + {\hat R}_{acbm}\bigr) + 2\beta^2\delta_i^k{\hat E}_{(l\vert g} {\hat E}^g_{~\vert m)}\nonumber\\ &-& 3\beta^2{\hat E}_{i(l\vert}{\hat E}^k_{~\vert m)} + 2\beta{\hat R}^k_{~(gi)(l\vert}{\hat E}^g_{~\vert m)}\biggr] y^ly^m + R_3.\end{aligned}$$ Here $_E t_i^{~k}$ mean the components of the canonical Einstein energy-momentum pseudotensor of the gravitational field. In a holonomic frame we have $$\begin{aligned} _E t_i^{~k}&=&\alpha\Bigl\{\delta^k_i g^{ms}\bigl(\Gamma^l_{mr}\Gamma^r_{sl}-\Gamma^r_{ms}\Gamma^l_{rl}\bigr)\nonumber\\ &+& g^{ms}_{~~,i}\bigl[\Gamma^k_{ms}-1/2\bigl(\Gamma^k_{tp} g^{tp}- \Gamma^l_{tl} g^{kt}\bigr)g_{ms}\nonumber\\ & -&1/2\bigl(\delta^k_s \Gamma^l_{ml} + \delta^k_m \Gamma^l_{sl}\bigr)\bigr]\Bigr\}.\end{aligned}$$ $$\alpha = {c^4\over 16\pi G} = {1\over 2\beta},~E_i^{~k} := T_i^{~k} - 1/2\delta_i^k T,$$ and, in any frame, $$B^b_{~alm} := 2R^{bik}_{~~~(l\vert}{}R_{aik\vert m)} -1/2\delta^b_a{}R^{ijk}_{~~~l} R_{ijkm},$$ is the [*Bel-Robinson tensor*]{}, while the tensor $$P^b_{~alm} := 2 R^{bik}_{~~~(l\vert}{}R_{aki\vert m)} -1/2\delta_a^b{} R^{ijk}_{~~~l} R_{ikjm}$$ is very closely related to the former[^3]. The expansion (15) with the help of (9) and (10) gives the following averaged gravitational relative energy-momentum tensor $$<_g t_a^{~b}(P;v^l)> = _g S_a^{~b}(P;v^l){L^2\over 6},$$ where the tensor $_g S_a^{~b}(P;v^l)$ is the [*canonical superenergy tensor*]{} for the gravitational field \[1-8\]. We have \[1-8\] $$\begin{aligned} _g S_a^{~b}(P;v^l) &=& {2\alpha\over 9}\bigl(2{\hat v}^l{\hat v}^m - {\hat g}^{lm}\bigr)\biggl[{\hat B}^b_{~alm} + {\hat P}^b_{~alm}\nonumber\\ &-& 1/2\delta_a^b{\hat R}^{ijk}_{~~~m}\bigl({\hat R}_{ijkl} + {\hat R}_{ikjl}\bigr) + 2\beta^2\delta_a^b{\hat E}_{(l\vert g}{\hat E}^g_{~\vert m)}\nonumber\\ &-& 3\beta^2{\hat E}_{a(l\vert}{\hat E}^b_{~\vert m)} + 2\beta{\hat R}^b_{~(ag)(l\vert} {\hat E}^g_{~\vert m)}\biggr].\end{aligned}$$ In vacuum the tensor $_g S_a^{~b}(P;v^l)$ reduces to the simpler form $$_g S_a^{~b} (P;v^l) = {8\alpha\over 9}\bigl(2{\hat v}^l{\hat v}^m - {\hat g}^{lm}\bigr)\biggl[{\hat R}^{b(ik)}_{~~~~~(l\vert}{\hat R}_{aik\vert m)} -1/2\delta_a^b{\hat R}^{i(kp)}_{~~~~~(l\vert}{\hat R}_{ikp\vert m)}\biggr],$$ which is symmetric and the quadratic form $_g S_{ab}(P;v^l){\hat v}^a{\hat v}^b$ is [*positive-definite*]{}. In vacuum we also have the [*local conservation laws*]{} $$\nabla_b~{_g {\hat S}_a^{~b}} = 0.$$ and the analogous laws satisfied by the averaged tensor $<_g t_a^{~b}(P;v^l)>$. The averaged energy-momentum tensors $<_m T_a^{~b}(P;v^l)>$ and $<_g t_a^{~b}(P;v^l)>$ can be considered as the [*averaged tensors of the relative energy-momentum*]{}. They can also be interpreted as the [*fluxes*]{} of the appropriate canonical superenergy. It is easily seen from the formulas (12) and (20). Now let us consider the [*averaged angular momentum tensors*]{} in [**GR**]{}. The constructive definition of these tensors, in analogy to the definition of the averaged energy-momentum tensors, is as follows. In normal coordinates [**NC(P)**]{} we define $$<M^{(a)(b)(c)}(P)> = <M^{abc}(P)> :=\displaystyle\lim_{\varepsilon\to 0}{ \int\limits_{\Omega}{\bigl[M^{(a)(b)(c)}(y) - M^{(a)(b)(c)}(P)\bigr]d\Omega}\over\varepsilon^2/2\int\limits_{\Omega}d\Omega},$$ where $$M^{(a)(b)(c)}(y) := M^{ikl}(y){}e_i^{~(a)}(y){}e_k^{~(b)}(y){}e_l^{~(c)}(y),$$ $$M^{(a)(b)(c)}(P) := M^{ikl}(P) e_i^{~(a)}(P){}e_k^{~(b)}(P){}e_l^{~(c)}(P) = M^{ikl}(P)\delta_i^a\delta_k^b\delta_l^c = M^{abc}(P),$$ are the [*physical*]{} (or tetrad) components of the field $M^{ikl}(y)= (-)M^{kil}(y)$ which describes the angular momentum densities [^4]. As in (2) and (3) , $e^i_{~(a)}(y),~~e_k^{~(b)}(y)$ denote mutually dual orthonormal tetrads parallelly propagated along geodesics through [**P**]{} such that $e^i_{~(a)}(P) = \delta^i_a,~~e_k^{~(b)}(P) = \delta_k^b$. The compact four-dimensional domain $\Omega$ is defined in the same way as in the formula (1) and we will again take $\Omega$ as a sufficiently small four-dimensional ball with centre at [**P**]{} and with radius $R = \varepsilon L$. At [**P**]{} the tetrad and normal components of an object are equal. We apply this once more and omit tetrad brackets for the indices of any quantity attached to the point [**P**]{}; for example, we write $M^{abc}(P)$ instead of $M^{(a)(b)(c)}(P)$ and so on. For matter as $M^{ikl}(y)$ we take $$_m M^{ikl}(y) =\sqrt{\vert g\vert}\bigl[y^i T^{kl}(y) - y^k T^{il}(y)\bigr],$$ where $T^{ik}(y) = T^{ki}(y)$ are the components of a symmetric energy-momentum tensor of matter and $y^i$ denote the normal coordinates [**NC(P)**]{}. The formula (27) gives us the total angular momentum densities, orbital and spinorial, because the symmetric energy-momentum tensor of matter $T^{ik} = T^{ki}$ comes from the canonical one by using the [*Belinfante-Rosenfeld*]{} symmetrization procedure and, therefore, includes the canonical spin of matter \[14\]. For the gravitational field we take the gravitational angular momentum pseudotensor proposed by Bergmann and Thomson \[14,18\] which in a [**NC(P)**]{} (and in any other holonomic frame) reads $$_g M^{ikl}(y) = _F U^{i[kl]}(y) - _F U^{k[il]}(y)+\sqrt{\vert g\vert}\bigl(y^i _{BT} t^{kl} - y^k _{BT} t^{il}\bigr),$$ where, in a holonomic frame, $$_F U^{i[kl]}:= g^{im}{}_F U_m^{~[kl]} =\alpha g^{im}{g_{ma}\over\sqrt{\vert g\vert}}\biggl[(-g)\bigl(g^{ka} g^{lb} - g^{la} g^{kb}\bigr)\biggr]_{,b}$$ are [*Freud’s superpotentials*]{} with the first index raised and $$_{BT} t^{kl}:= g^{ki}{} _Et_i^{~l} + {g^{mk}_{~~,p}\over\sqrt{\vert g\vert}} {} _F U_m^{~[lp]}$$ are the components of the [*Bergmann-Thomson*]{} gravitational energy-momentum pseudotensor \[14,18\]. $_E t_i^{~k}$ mean the components of the [*Einstein canonical gravitational energy-momentum pseudotensor*]{} of the gravitational field. The Bergmann-Thomson gravitational angular pseudotensor is most closely related to the Einstein canonical energy-momentum complex $_E K_i^{~k}:= \sqrt{\vert g\vert}\bigl(T_i^{~k} + _E t_i^{~k}\bigr)$, matter and gravitation, and it has better physical and transformational properties than the famous gravitational angular momentum pseudotensor proposed by Landau and Lifschitz \[15-17\]. This is why we apply it here. One can interpret the Bergmann-Thomson gravitational angular momentum pseudotensor as the sum of the [*spinorial part*]{} $$S^{ikl} := _F U^{i[kl]} - _F U^{k[il]}$$ and the [*orbital part*]{} $$O^{ikl} := \sqrt{\vert g\vert}\bigl(y^i{}_{BT} t^{kl} - y^k{}_{BT} t^{il}\bigr)$$ of the gravitational angular momentum “densities”. Substitution of (27) and (28) (expanded up to third order), (9),(10) and the expansion $$\sqrt{\vert g\vert} = 1 -1/6{\hat R}_{ab}y^ay^b + R_3 = 1-1/6\beta{\hat E}_{ab}y^ay^b + R_3,$$ into (24) gives us the following [*averaged angular momentum tensors*]{} for matter and gravitation respectively $$<_m M^{abc}(P;v^l)> = _m S^{abc}(P;v^l){L^2\over 6},$$ $$<_g M^{abc}(P;v^l)> = _g S^{abc}(P;v^l){L^2\over 6}.$$ Here $$_m S^{abc}(P;v^l) = 2\bigl[\bigl(2{\hat v}^a{\hat v}^p - {\hat g}^{ap}\bigr)\nabla_p{\hat T}^{bc} - \bigl(2{\hat v}^b{\hat v}^p - {\hat g}^{bp}\bigr) \nabla_p {\hat T}^{ac}\bigr],$$ and $$\begin{aligned} _g S^{abc}(P;v^l) &=& \alpha\bigl(2{\hat v}^p{\hat v}^t - {\hat g}^{pt}\bigr)\biggl[\beta\bigl({\hat g}^{ac}{\hat g}^{br} -{\hat g}^{bc}{\hat g}^{ar}\bigr)\nabla_{(t}{\hat E}_{pr)}\nonumber\\ &+& 2{\hat g}^{ar}\nabla_{(t}{\hat R}^{(b}_{~~p}{}^{c)}_{~~r)} - 2{\hat g}^{br} \nabla_{(t}{\hat R}^{(a}_{~~p}{}^{c)}_{~~r)}\nonumber\\ &+& 2/3{\hat g}^{bc}\bigl(\nabla_r{\hat R}^r_{~(t}{}^{a}_{~p)} -\beta\nabla_{(p} {\hat E}^a_{~t)}\bigr) -2/3{\hat g}^{ac}\bigl(\nabla_r {\hat R}^r_{~(t}{}^b_{~p)} - \beta\nabla_{(p} {\hat E}^b_{~t)}\bigr)\biggr]\end{aligned}$$ are the components of the [*canonical angular supermomentum tensors*]{} for matter and gravitation, respectively \[4,6,8\]. In special relativity the averaged tensor $<_m M^{abc}(P;v^l)>$, and the canonical angular supermomentum tensors for matter $_m S^{abc}(P;v^l)$ satisfy trivial conservation laws \[1-8\]. In the framework of the [**GR**]{} only the tensors $_g S^{abc}(P;v^l)$ and $<_g M^{abc}(P;v^l)>$ satisfy local conservation laws in vacuum. In vacuum, when $T_{ik} = 0 \Longleftrightarrow E_{ik} := T_{ik} -1/2 g_{ik}T = 0$, the canonical gravitational angular supermomentum tensor $_g S^{abc}(P;v^l) = (-) _g S^{bac}(P;v^l)$ given by (37) simplifies to $$_g S^{abc}(P;v^l) =2\alpha\bigl(2{\hat v}^p{\hat v}^t - {\hat g}^{pt}\bigr)\biggl[{\hat g}^{ar}\nabla_{(p}{\hat R}^{(b}_{~~t}{}^{c)}_{~~r)} -{\hat g}^{br}\nabla_{(p} {\hat R}^{(a}_{~~t}{}^{c)}_{~~r)}\biggr].$$ Some remarks are in order: 1. The orbital part $O^{ikl} =\sqrt{\vert g\vert}\bigl(y^i_{BT} t^{kl} - y^k _{BT} t^{il}\bigr)$ of the $_g M^{ikl}$ [*does not contribute*]{} to the tensor $_g S^{abc}(P;v^l)$ and, therefore, also to the tensor $<_g M^{abc}(P;v^l)>$. Only the spinorial part $S^{ikl} = _F U^{i[kl]} - _F U^{k[il]}$ gives nonzero contribution to these tensors. 2. The averaged angular nomentum tensors $<_gM^{abc}(P;v^l)>, ~~<_m M^{abc}(P;v^l)>$, like as the canonical angular supermomentum tensors, [*do not need*]{} any radius-vector for existing. The averaged tensors $< _m M^{abc}(P;v^l)>,~~<_g M^{abc}(P;v^l)>$, likely as the averaged relative energy-momentum tensors, can be interpreted as the [*averaged tensors of the relative angular momentum*]{}[^5] and also as the [*fluxes*]{} of the appropriate angular supermomentum. The formulas (12),(20),(34) and (35) give the direct link beteween the canonical superenergy and angular supermomentum tensors $$_gS_a^{~b}(P;v^l),~_m S_a^{~b}(P;v^l), ~_g S^{abc}(P;v^l), ~~_m S^{abc}(P;v^l)$$ and the averaged relative energy-momentum and angular momentum tensors $$<_g t_a^{~b}(P;v^l)>, <_m T_a^{~b}(P;v^l)>, <_g M^{abc}(P;v^l)>, <_m S^{abc}(P;v^l)>.$$ Namely, it is easily seen from these formulas that the averaged relative energy-momentum and angular momentum tensors [*differ*]{} from the canonical superenergy and angular supermomentum tensors [*only*]{} by the constant scalar multiplicator ${L^2\over 6}$, where $L>0$ means some fundamental length. Thus, from the mathematical point of view, these two kind of tensors are equivalent. Physically they [*are not*]{} because their components have different dimension. Moreover the averaged energy-momentum and angular momentum tensors depend on a fundamental length $L>0$, i.e., they need introduction a supplementary element into [**GR**]{}[^6]. Owing to the last fact and the formulas (12),(20), (34), (35) it seems that the canonical superenergy and angular supermomentum tensors are [*more fundamental*]{} than the averaged energy-momentum and angular momentum tensors. But the averaged energy-momentum and angular momentum tensors have one important superiority over the canonical superenergy and angular supermomentum tensors: their components [*possesse proper dimensions*]{} of the energy-momentum and angular momentum densities. The averaged tensors $$<_g t_a^{~b}(P;v^l)>, ~<_m T_a^{~b}(P;v^l)>,~~ <_g M^{abc}(P;v^l)>, ~<_m M^{abc}(P;v^l)>$$ depend on the four-velocity ${\vec v}$ of a fiducial observer [**O**]{} which is at rest at the beginning [**P**]{} of the normal coordinates [**NC(P)**]{} used for averaging and on some fundamental length $L>0$. After fixing the fundamental length $L$ one can determine univocally these tensors along the world line of an observer [**O**]{}. In general one can [*unambiguously determine*]{} these tensors (after fixing $L$) in the whole spacetime or in some domain $\Omega$ if in the spacetime or in the domain $\Omega$ a geometrically distinguised timelike unit vector field ${\vec v}$ exists. An example of such a kind of the spacetime is given by Friedman universes. One can try to fix[^7] the fundamental length $L$, e.g., by using loop quantum gravity. Namely, one can take as $L$ the smallest length $l$ over which the classical model of the spacetime is admissible. Following loop quantum gravity \[19-29\] one can say about continuous classical differential geometry already just a few orders of magnitude above the Planck scale, e.g., for distances $l\geq 100L_P = 100\sqrt{{G\hbar\over c^3}} \approx 10^{-33}$ m. So, one can take as the fundamental length $L$ the value $L = 100 L_P \approx 10^{-33}$ m.[^8] After fixing the fundamental length $L$ one has the averaged relative energy-momentum and angular momentum tensors as precisely defined as the canonical superenergy and angular supermomentum tensors are. The averaged tensors (with $L$ fixed or no) $$<_mT_a^{~b}(P;v^l)>, <_g t_a^{~b}(P;v^l)>, <_m M^{abc}(P;v^l)>, <_g M^{abc}(P;v^l)>$$ give us as good tool to a local analysis ( and also to global analysis iff in spacetime a privileged global unit timelike vector field exists)) of the gravitational and matter fields as the canonical superenergy and angular supermomentum tensors $$_m S_a^{~b}(P;v^l), ~_g S_a^{~b}(P;v^l), ~_m M^{abc}(P;v^l), ~_g M^{abc}(P;v^l)$$ give. For example, one can apply the averaged energy-momentum and angular momentum tensors to the all problems which have been analyzed in the papers \[1-8\] by using the canonical superenergy and angular supermomentum tensors. Some applications of the averaged relative energy-momentum tensors ================================================================== In this paper we apply the averaged gravitational relative energy-momentum tensor$<_g t_a^{~b}(P;v^l)>$ only to decide if free vacuum gravitational field has energy-momentum; especially, if gravitational waves carry any energy-momentum, and the averaged gravitational and matter relative energy-momentum tensors to analyze the energy and momentum of the Friedman universes. Albrow and Tryon were the first who assumed that the net energy of the closed Friedman universes may be equal to zero \[30-31\]. We will show in this paper that this assumption is, most probably, [*incorrect*]{}. Let us begin from the vacuum gravitational energy and momentum. The problem was revived recently because some authors conjectured \[32-36\], by using coordinate dependent[^9] pseudotensors and double index complexes, that the energy and momentum in general relativity are confined only to the regions of non-vanishing energy-momentum tensor of matter and that the gravitational waves carry no energy and momentum. The argumentation is the following. For some solutions to the Einstein equations and in some special coordinates, e.g., in Bonnor’s spacetime \[37\] in Bonnor’s or in Kerr-Schild coordinates, the Einstein canonical gravitational energy-momentum pseudotensor (and other most frequently used gravitational energy-momentum pseudotensors also) [*globally vanishes*]{} outside of the domain in which $T^{ik}\not= 0$. The analogous global vanishing of the canonical pseudotensor $_E t_a^{~b}$ we have for the plane and for the plane-fronted gravitational waves in, e.g., null coframe \[3,38\]. But one should emphasize that all these results are [*coordinate dependent*]{} \[3,7,38\], i.e., in [*other coordinates*]{} the used gravitational energy-momentum pseudotensors [*do not vanish*]{} in vacuum. Moreover, one should interpret physically the global vanishing of the canonical pseudotensor (and other pseudotensors also) in some coordinates in vacuum as a [*global cancellation*]{} of the energy-momentum of the real gravitational field which has $R_{iklm}\not= 0$ with energy-momentum of the inertial forces field which has $R_{iklm}=0$; [*not as a proof of vanishing of the energy-momentum of the real gravitational field*]{}. It is because the all used pseudotensors were entirely constructed from the Levi-Civita’s connection $\Gamma^i_{~kl} = \Gamma^i_{~lk}$ and from the metric $g_{ik}$ which describe a mixture of the real gravitational field ($R_{iklm}\not= 0$) and an inertial forces field ($R_{iklm}= 0$). In order to get the coordinate independent results about energy-momentum of the [*the real gravitational field*]{} one must use tensorial expressions which depend on curvature tensor, like the averaged gravitational relative energy-momentum tensor $<_g t_a^{~b}(P;v^l)>$. This tensor vanishes iff $R_{iklm}=0$, i.e., iff the spacetime is flat and we have no real gravitational field. When calculated, the averaged gravitational relative energy-momentum tensor$<_g t_a^{~b}(P;v^l)>$ always gives the [*positive-definite*]{} averaged free relative gravitational energy density and, in the case of a gravitational wave, its non-zero flux. It is easily seen from the our papers \[1-8,38\] in which we have used the canonical gravitational superenergy tensor and from the formula (20) of this paper which gives the direct connection between the averaged relative gravitational energy-momentum tensor and the canonical gravitational superenergy tensor. Thus, the conjecture about localization of the gravitational energy only to the regions of the non-vanishing energy-momentum tensor of matter [*is incorrect*]{} for the real gravitational field which has $R_{iklm}\not= 0$. It is interesting that the gravitational angular momentum pseudotensor (28) [*does not vanish*]{} in Bonnor’s spacetime and in Bonnor’s coordinates [*outside*]{} of the domain in which $T^{ik}\not= 0$. This important fact which, as I think, is unknown for the authors of the conjecture, gives other [*direct proof*]{} that this conjecture [*is incorrect*]{}. If the conjecture were correct, then we would have an absurd situation: the energy-momentum density–free vacuum gravitational field has non-vanishing “densities” of the angular momentum. In a similar way as above one can use the averaged gravitational relative angular momentum tensor $<_g M^{abc}(P;v^l)>$ to coordinate independent analysis of the angular momentum of the real gravitational field. Now, let us pass to the problem of the energy and momentum of the Friedman universes. Of course, the problem of the global energy and global linear (or angular) momentum for Friedman universes (and also for more general universes) is not [*well-posed*]{} from the physical point of view because these universes are not asymptotically flat spacetimes \[39\]. Despite this important fact recently many authors concluded \[40-50\] that the energy and momentum of the Friedman universes, flat and closed, are equal to zero locally and globally (flat universes) or only globally (closed universes). Such conclusion, which has a mathematical sense, originated from calculations performed in special comoving coordinates called “Cartesian coordinates” by using [*coordinate dependent*]{} double index energy-momentum complexes, matter and gravitation. One can introduce in [**GR**]{} many different energy-momentum complexes. The six of them are most frequently used: Einstein’s canonical complex, Landau-Lifshitz complex, Bergmann-Thomson complex, Møller complex, Papapetrou complex and Weinberg energy-momentum complex. These all energy-momentum complexes [*are neither geometrical objects nor coordinate independent objects*]{}, e.g., they can vanish in some coordinates locally or globally and in other coordinates they can be different from zero. It results that the double index energy–momentum complexes and the gravitational energy-momentum pseodotensors [*have no physical meaning*]{} to a local analysis of the gravitational field, e.g., to study gravitational energy-density distribution. They can be reasonably used [*only to calculate the global quantities*]{} for the very precisely defined asymptotically flat spacetimes (in spatial or in null direction). The general opinion is that the best one of the all possible double index energy-momentum complexes from physical and geometrical points of view is the canonical Einstein’s double index energy-momentum complex $_E K_i^{~k} = \sqrt{\vert g\vert}\bigl(T_i^{~k} + _E t_i^{~k}\bigr)$. The global results obtained by use of this canonical energy-momentum complex are usually treated as correct and giving some pattern. In fact, the other double index energy-momentum complexes were constructed following the instruction: they should give the same global results as the Einstein energy-momentum complex gives at least in the simplest cases, e.g., in the case of a closed system. That is why we have confined in the paper (and also in the all our previous papers) only to this double index energy-momentum complex. So, let us consider the results of the formal calculations of the global energy and momentum for Friedman universes in the standard comoving coordinates by using canonical Einstein’s double index energy-momentum complex. Any other sensible double index energy-momentum complex gives equivalent results. 1. In the “Cartesian coordinates” $(t,x,y,z)$ in which the line element has the form[^10] $$ds^2 = dt^2 - R^2(t){(dx^2+dy^2+dz^2)\over[1+k/4(x^2+y^2+z^2)]^2},~~~k = 0,^+_-1,$$ we obtain after simple calculations \[1,5\] that for flat universes the global quantities $P_i ~(i = 0,1,2,3)$, where $P_i$ mean the components of the energy-momentum contained inside of a slice $t = const$, are equal to zero. In this case the all integrands (energy and momentum “densities”) in the integrals on $P_i~~(i = 0,1,2,3)$ identically vanish because they are multiplied by the curvature index $k$. So, one can say that for flat Friedman universes the integral quantities $P_i ~~(i=0,1,2,3)$ vanish locally and globally in the “Cartesian” coordinates[^11]. For closed Friedman universes we also get $P_i = 0,~~(i=0,1,2,3)$, but this time the integrands do not vanish. Only after integration one gets that the integrals representing $P_i, (i =0,1,2,3)$ are equal to zero. In the case of the open Friedman universes one gets $E = P_0 = (-)\infty, ~P_1 = P_2 = P_3 = 0$. The integrands also do not vanish in this case. 2. In the coordinates $(t,\chi,\vartheta,\varphi)$ in which the line element reads $$ds^2 = dt^2 -R^2(t)[d\chi^2 + S^2(\chi)(d\vartheta^2 + \sin^2\vartheta d\varphi^2)],$$ where $$S(\chi) =\bigl\{sin\chi~~if~~ k = 1, ~~\chi~~ if~~ k=0,~~ sh\chi~~ if~~ k =-1\bigr\},$$ one gets drastically different results: $E = P_0 =(-)\infty, ~~ P_1 = (-)\infty,~~P_2 = P_3 = 0$ for flat universes; $ E = P_0 = {\pi\over 2} R(t), ~~P_1 = P_2 = P_3 =0$ for closed univeres and $E = P_0 =(-)\infty, ~~ P_1 =(-)\infty, ~~P_2 = P_3 =0$ for open universes. 3. Finally, in the coordinates $(t,r,\vartheta,\varphi)$ in which the line element has the form $$ds^2 = dt^2 - R^2(t)[{dr^2\over(1-kr^2)} + r^2(d\vartheta^2+\sin^2\vartheta d\varphi^2)],~~k=0,^+_-1,$$ we obtain the following results: $E = P_0 =(-)\infty,~~ P_1 = (-)\infty, ~~P_2 = P_3 = 0$ for flat universes; $E = P_0 = {\pi\over 4}R(t),~~P_1 = (-)\infty,~~P_2 = P_3 = 0$ for closed universes and $E= P_0 =(-)\infty,~~ P_1 =(-)\infty,~~P_2 = P_3 =0$ for open Friedman universes. In the all cases in which the integrands (=“densities” of the calculated four-momentum) do not vanish, these integrands go to zero if $R(t)\longrightarrow 0$. So, these integrands (“densities”of the energy-momentum) are not suitable for analysis of the Big-Bang singularity. The authors which assert that the energy and momentum of the Friedman universes, flat and closed, are equal to zero have performed their calculations only in the “Cartesian” comoving coordinates $(t,x,y,z)$ by using coordinate dependent double index energy-momentum complexes and have got zero results. But in the case of the Friedman universes the “Cartesian”coordinates [*are by no means better*]{} than the comoving coordinates $(t,\chi,\vartheta,\varphi)$ or $(t,r,\vartheta,\varphi)$ in which we have obtained non-zero results. Only in a flat and in an asymptotically flat spacetimes one can distinguish in some reasonable way the Cartesian coordinates; but [*not in the case of the Friedman universes*]{}. So, the conclusion of these authors about vanishing of the energy and linear momentum of the Friedman universes, flat and closed, [*cannot be correct*]{}. By using double index energy-momentum complexes one rather should conclude that the energy and momentum of the Friedman universes explicite depend on the used comoving coordinates and, therefore, that [*they are undetermined*]{} locally and globally. This last conclusion is very sensible because [*one cannot measure the global energy and global momentum of the Friedman (and more general) universes*]{}. One can do this only in the case of an isolated system \[39\]. On the other hand the former conclusion directly follows from the coordinate dependence of the energy-momentum complexes. May be one would try to support the [*mathematically sensible*]{} hypothesis which states that energy and momentum of the Friedman universes, flat and closed, disappear by using coordinate independent expressions, like Pirani’s expression on global energy, matter and gravitation, or like single index Komar’s expression (Komar’s single index complex) on global energy-momentum and global angular momentum, matter and gravitation [^12]. The Pirani’s expression (for the energy only, see, eg., \[51\]) is unique and can be applied in a spacetime having a privileged set of observers whose world-lines form a normal congruence. In such spacetime there exists a family of spatial hypersurfaces which are orthogonal to the four-velocities of this set of observers. The Pirani’s expression is coordinate independent but it has two defects: calculated total energy density, matter and gravitation, [*is not positive-definite*]{}, and, if the congruence is geodesic, then the total energy-density [*is identically zero*]{}, and, in consequence, the global energy [*trivially vanishes in the case*]{}. However, this zero values [*are not a property of the gravitational and matter fields*]{}. They are only a property of the geodesics congruence. In Friedman universes does exist privileged set of observers called [*fundamental or isotropic*]{} observers. For these observers the four-velocity ${\vec v}$ has components $v^k = \delta^k_0$ in a comoving coordinates and the family of the spatial hypersurfaces orthogonal to ${\vec v}$ is given by $t = const.$ But, unfortunately, the congruence of the isotropic observers in Friedman universes is geodesic and, therefore, the Pirani’s expression [*fails*]{} in the case [*giving trivially zero*]{}. On the other hand, coordinate independent Komar’s expression (see, e.g., \[51-53\]) [*needs Killing vector fields*]{}: translational timelike Killing vector field as [*energy descriptor*]{}, translational spatial Killing vector fields as [*descriptors of the linear momentum*]{} and rotational spatial Killing vector fields as [*descriptors of the angular momentum*]{}. Friedman universes admit only six linearly independent spatial Killing vector fields, three translational Killing vector fields and three rotational Killing vector fields (see, e.g., \[54\]). So, one can consider in Friedman universes six coordinate independent integrals (scalars) which correctly represent (from mathematical point of view) the components of the global linear momentum and the components of the global angular momentum (see, eg., \[54\]). These integrals [*trivially*]{} vanish for Friedman universes, i.e., integrands in these integrals [*identically*]{} vanish, independently of the curvature index $k=0,^+_-1$. This is very sensible result and it can be interpreted as a mathematically correct proof that the linear and angular momentum for Friedman universes disappear in a comoving coordinates. But we still have a problem with energy of the Fiedman universes [*because we have no energy descriptor*]{}, i.e., translational timelike Killing vector field, in these universes. Therefore, one cannot use the coordinate independent Komar’s expression in order to calculate correctly from the mathematical point of view the energy of the Friedman universes. If one formally uses in Komar’s expression the four-velocity of the privileged set of the isotropic observers as the energy descriptor, then [*one will get identically zero*]{} because for a geodesic timelike congruence the integrand in this expression, like integrand in Pirani’s expression, identically vanishes. But this vanishing is also only a property of the geodesics congruence. It is not a property of the gravitational and matter fields. Resuming, one cannot use the coordinate independent Pirani’s and Komar’s expressions in order to correctly prove[^13] the statement that the energy of the Friedman universes disappears, i.e., that these universes are complete energetic nonentity. For this purpose one cannot also use the coordinate independent KBL bimetric approach \[55\] because the results obtained in this approach depend not only on the used background but also on mapping of the real spacetime onto this background. Therefore, the [*mathematically sensible*]{} statement that the closed and flat Friedman universes have no energetic content [*is still not satisfactory proved*]{}. It is interesting that the using of the coordinate independent averaged relative energy-momentum tensors to analyze the energetic content of the Friedman universes lead us to [*positive-definite results*]{} for the all Friedman universes. Namely, let us apply the averaged relative energy-momentum tensors for gravitation $<_g t_i^{~k}(P;v^l)>$ and for matter $<_m T_i^{~k}(P;v^l)>$ to calculate the averaged relative energy density for Friedman (and more general) universes. With this aim let us define $$_g\epsilon := <_g t_a^{~b}(P;v^l)>v^av_b$$ —– the averaged relative gravitational energy density, $$_m\epsilon:= <_m T_a^{~b}(P;v^l)>v^av_b$$ —– the averaged relative matter energy density, and $$\epsilon:= _g\epsilon + _m \epsilon$$ —– the averaged relative total energy density. Here $v^a$ are the components of the four-velocity of an observer [**O**]{} which is studying gravitational and matter fields. In Friedman universes, if we take as the observers [**O**]{} the globally defined set of the fundamental observers, then we can also define the global averaged total relative energy $E$ of a Friedman universe $$E := \int\limits_{t = const}\epsilon\sqrt{\vert g\vert} d^3v ~{\dot =}\int\limits_{t = const} \bigl[<_g t_i^{~0}> + <_m T_i^{~0}>\bigr] v^i\sqrt{\vert g\vert} d^3v,$$ and, in analogous way, the global averaged relative energy for matter and for gravitation. Here $d^3v$ means the product of the diferentials of the coordinates which parametrize slices $t = const$ of the Friedman universes, e.g., $d^3v = dxdydz$ in the Cartesian comoving coordinates $(t,x,y,z)$. After something tedious but very simple calculations we will obtain for Friedman universes \[1,2,5\][^14]: 1. $_g\epsilon$, $_m\epsilon$ and, in consequence $\epsilon$, are [*positive definite*]{} for the all Friedman universes. 2. $\displaystyle\lim_{R\to 0}{}_g\epsilon = \displaystyle\lim_{R\to 0} {}_m\epsilon =\displaystyle\lim_{R\to 0}{}\epsilon = +\infty, ~~(k = 0,^+_- 1)$. It follows from this that one can use the averaged relative energy densities to study the Big-Bang singularity. 3. $\displaystyle\lim_{R\to\infty}{} _g\epsilon = \displaystyle\lim_{R\to\infty}{} _m \epsilon = \displaystyle\lim_{R\to\infty}{}\epsilon = 0, ~~(k = 0,-1)$. 4. The global averaged relative energies, gravitation, matter and total, are infinite ($+\infty$) for flat and for open Friedman universes and they are finite and positive for closed Friedman universes. Also the other three invariant integrals which formally represent the components $P_{(\alpha)} ~~(\alpha = 1,2,3)$ of the global averaged relative linear momentum for Friedman universes $$P_{(\alpha)}:= \int\limits_{t = const}\bigl\{<_g t_i ^{~0}> + < _m T_i ^{~0}>\bigr\}e^i_{~(\alpha)}\sqrt{\vert g\vert}d^3v,~~(\alpha = 1,2,3),$$ [*vanish trivially*]{} in a comoving coordinates \[1,2,5\] because the integrands in these integrals (densities of the averaged relative linear momentum components) [*identically vanish*]{} \[1,2,5\]. Here $e^i_{~(\alpha)}, ~~(\alpha = 1,2,3)$ mean the components of the three translational spatial Killing vector fields (descriptors of the linear momentum) which exist in the Friedman universes (see, e.g., \[54\]). We would like to emphasize that the integrals (51) and (52) do not depend on the used coordinates. They depend only on a slice $t = const$. The all above results are very sensible and satisfactory from the physical point of view. We will finish this Section with remark that the analogous situation as for flat Friedman universes one has also for the more general, only homogeneous, Kasner vacuum universes \[15\] and Bianchi–type I universes filled with stiff matter (see, e.g., \[44-50, 56-59\]). Namely, the most frequently used double-index energy-momentum complexes, when used in Cartesian comoving coordinates to analyze of these universes, give zero results locally and globally. Of course, in other comoving coordinates, e.g., in the $t,r,\vartheta,\varphi$ comoving coordinates, we have non-zero and globally divergent results. If one applies the averaged relative energy-momentum tensors $<_g t_a^{~b}(P;v^l)>, \\<_m T_a^{~b}(P;v^l)>$ to analyze of a vacuum Kasner universe and a Bianchi–type I universe filled with stiff matter, then one gets the following, coordinate independent results: 1. The averaged relative gravitational energy of a vacuum Kasner universe has [*positive-definite*]{} density and the same limits when $t\longrightarrow 0$ or when $t\longrightarrow +\infty$ as it was in the case of a flat Friedman universe. Also the suitable integral global quantity defined in analogous way as in the case of the Friedman universes is divergent to $+\infty$. 2. For an expanding Bianchi–type I universe filled with stiff matter the averaged relative gravitational energy density and the averaged relative energy-density for matter are still [*positive-definite*]{} and lead to divergent to $+\infty$ global energies. Thus, one can conclude that these two more general, only homogeneous universes, like Friedman flat universes, also [*are not energetic nonentity*]{}. Concerning of the components of the linear momentum for Kasner vacuum universes and for Bianchi–type I universes filled with stiff matter one can easily check that these components, defined in analogous way as in the case of the Friedman universes, [*identically vanish locally and globally*]{} in a comoving coordinates. Conclusion ========== We have introduced in the paper the averaged tensors of the relative energy-momentum and the averaged tensors of the relative angular momentum, for matter and for gravitation. These tensors are very closely related to the canonical superenergy and angular supermomentum tensors and they can be used to analyze the same problems which we have analyzed in the our papers \[1-8\] with the help of the canonical superenergy and angular supermomentum tensors. The superiority of the averaged relative energy-momentum and angular momentum tensors in comparison with the canonical superenergy and angular supermomentum tensors is the following: the averaged tensors have proper dimensionality of the energy-momentum and angular momentum densities. The averaged relative energy-momentum and relative angular momentum tensors of the gravitational field [*refer to the energy-momentum and angular momentum of the real gravitational field*]{} for which we have $R_{iklm}\not= 0$. These tensors vanish iff $R_{iklm} =0$, i.e., iff [*we have no real gravitational field*]{}. In our opinion the all existing (and projected in near future) detectors of the gravitational waves will measure the averaged relative gravitational energy density and its flux; not the gravitational energy defined by pseudotensors. It is easily seen from the fact that the acting of these detectors relies on the equations of the geodesics deviation which explicitly depend on the curvature tensor. In this paper we have applied the averaged relative gravitational energy-momentum tensor to decide if free vacuum gravitational field has energy and momentum and the averaged gravitational and matter relative energy-momentum tensors to analyze energy and momentum of the Friedman universes and also to analyze the Kasner and Bianchi–type I universes. The latter problem is recently very popular despite the fact that the problem of the global quantities for Friedman universes (and for more general cosmological models also) [*is not well-posed from the physical point of view*]{}. The global energy and momentum [*have physical meaning*]{} only when spacetime is asymptotically flat either in spatial or null direction. Of course, this is not a case of the Friedman and Kasner or Bianchi–type I cosmological models. We have obtained the following results: 1. The real vacuum gravitational field for which we have $R_{iklm}\not= 0$ [*always*]{} possesses his own positive-definite averaged relative energy density and in the cases in which the gravitating system is not at rest, the gravitational field possesses also the non-zero averaged relative linear momentum. 2. The coordinate independent averaged relative energy-momentum tensors, gravitation and matter, give positive-definite densities of the averaged relative energy, matter and gravitation, for the all Friedman universes. Therefore, these tensors indicate that the Friedman universes [*are not energetic nonentity*]{}. They [*are not energetic nonentity*]{} in the following sense: one can construct from the canonical energy-momentum complex, matter and gravitation, non-local tensorial, i.e., coordinate-independent expressions with correct dimensions which give positive-definite energy densities for the all Friedman universes. The averaged relative energy-momentum tensors tensors give also zero values of the averaged relative linear momenta for these universes in a comoving coordinates. The above results directly follow from the results obtained in the our previous papers \[1-5\] in which we have used the canonical superenergy (and angular supermomentum) tensors, gravitation and matter, and from the formulas (12) and (20) of this paper which connect the averaged relative energy-momentum tensors with the canonical superenergy tensors. The coordinate independent results presented in this paper for the Friedman universes are very satisfactory from the physical point of view. Much more satisfactory than the strange, coordinate dependent results which one obtains by using gravitational energy-momentum pseudotensors and double index energy-momentum complexes, matter and gravitation. 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[\[58\]]{} Oktay Aydogdu et al., “Energy Density Associated with the Bianchi Type–II Spacetime”, arXiv:gr-qc/0601133. [\[59\]]{} P. Halpern, “Energy of the Taub cosmological solution”, arXiv:gr-qc/0606095. [\[60\]]{} M.S. Berman, “On the energy of the universe”, arXiv:gr-qc/0605063. [^1]: e-mail: garecki@sus.univ.szczecin.pl [^2]: Trivial local conservation laws because the integral superenergetic quantities or, equivalently, integral averaged relative energy-momentum calculated from them for a closed system in special relativity vanish. [^3]: Very closely related because this tensor has almost the same analytic form as the Bel-Robinson tensor and the same symmetry properties. [^4]: Of course, $M^{abc}(P) = 0$, but we leave $M^{abc}(P)$ in our formulas. [^5]: Of course, the angular momentum is always relative quantity, in principle. Despite that we will keep the term [*relative angular momentum tensors*]{}. [^6]: The fundamental length $L>0$ must be infinitesimally small because its existence violates local Lorentz invariance. It is generally belived that a fundamental length exists in Nature. [^7]: But this is [*not necessary*]{}. One can effectively use the averaged energy-momentum and angular momentum tensors [*without fixing L*]{} explicitly. [^8]: Concerning other propositions fixing of $L$ see, e.g., \[9–12\]. [^9]: By “coordinate dependent” quantity we mean a quantity which is not a tensor (in general–which is not a tensor valued p-form). By “coordinate independent” quantity we mean a tensor quantity (in general – a tensor valued p-form). [^10]: From now on we will use [*geometrized units*]{} in which $G = c =1$. [^11]: It is interesting that the angular momentum “densities” when calculated, e.g., by using Bergmann–Thomson angular momentum complex (28) do not vanish in the case even for flat Friedman universes. [^12]: We would like to remark that the Pirani’s and Komar’s expressions, though coordinate independent, depend (like double index energy-momentum complexes) not only on real gravitational field ($R_{iklm}\not= 0$) but also on inertial forces field ($R_{iklm} = 0$). [^13]: Correctly from the mathematical point of view. [^14]: The results given below are easily seen from the our previous papers \[1,2,5\] and from the formulas (12) and (20) which connect the canonical superenergy tensors used in the papers \[1,2,5\] with the averaged relative energy and momentum tensors which we are using in this paper.
--- author: - 'N. Gupta' - 'R. Srianand' - 'P. Noterdaeme' - 'P. Petitjean' - 'S. Muzahid' bibliography: - '/Users/neeraj/Desktop/mybib.bib' date: 'Received 30 Mar 2013; Accepted 16 Aug 2013' title: '21-cm absorption from galaxies at $z\sim0.3$' --- Introduction ============ It is well known that physical conditions in the diffuse interstellar medium (ISM) of galaxies are influenced by various radiative and mechanical feedback processes associated with in-situ star formation [e.g., @Wolfire03]. Therefore, volume-filling factors of different phases of gas in a galaxy are expected to depend on its star formation history. Of particular interest is the evolution of the volume-filling factor of cold neutral medium (CNM) phase that also serves as a gaseous reservoir for star formation in galaxies. It is expected to contain an imprint of the collective outcome of all the processes that shape the star formation history of the Universe [e.g., @Hopkins06]. Systematic searches of high-$z$ [*intervening*]{} 21-cm absorbers in samples of Mg [ii]{} systems and damped  systems (DLAs) towards QSOs to measure CNM filling factor of galaxies have resulted in detections of 21-cm absorption towards $\sim$10-20% of Mg [ii]{} systems [e.g., @Briggs83; @Lane_phd; @Gupta09; @Kanekar09mg2; @Gupta12] and DLAs [e.g., @Kanekar03; @Curran10; @Srianand12dla]. However, establishing a connection between the redshift evolution of 21-cm absorbers and global star formation rate density is not straight forward due to (i) small number statistics of 21-cm absorbers, (ii) ambiguities regarding the origin of absorbing gas (i.e., gaseous disk, halo or outflowing/infalling gas etc.) and (iii) issues related to the small scale structure in absorbing gas and the extent of radio source. Blind searches of 21-cm absorption with Square Kilometre Array (SKA) pathfinders in the near future are expected to significantly increase the number of intervening 21-cm absorbers over a wide redshift range. To understand the nature of absorbers detected from these, it is important to establish a link between the properties of galaxies and intervening 21-cm absorbers, and address issues related to points (ii) and (iii) mentioned above. At present, both these points are best addressed at $z{\mbox{\raisebox{-0.3em}{$\stackrel{\textstyle <}{\sim}$}}}0.2$ where it is possible to determine properties of absorbing gas on pc and kpc scales via arcsecond and milliarcsecond scale spectroscopy, and detect the galaxy responsible for absorption in 21-cm and various optical emission lines. However, almost all the intervening 21-cm absorption line searches until now have happened at $z{\mbox{\raisebox{-0.3em}{$\stackrel{\textstyle >}{\sim}$}}}$0.2. The notable exceptions in the recent past are @Gupta10, @Borthakur11, and @Darling11 [see also @Carilli92]. The first two are based on samples of QSO sight lines passing through disks or halos of foreground galaxies, which we henceforth refer to as quasar-galaxy pairs (QGPs), whereas the last is a blind search for 21-cm absorption at $z{\mbox{\raisebox{-0.3em}{$\stackrel{\textstyle <}{\sim}$}}}0.06$ using the Arecibo telescope. All three have detected one 21-cm absorber each and showcase difficulties in detecting 21-cm absorbers at low-$z$ in the absence of large catalogs of Mg [ii]{} systems and DLAs. To move forward, we are systematically searching the Sloan Digital Sky Survey (SDSS) for QGPs with angular separation $<$10$^{\prime\prime}$ and background quasar flux density ${\mbox{\raisebox{-0.3em}{$\stackrel{\textstyle >}{\sim}$}}}$100mJy in the Faint Images of the Radio Sky at Twenty-Centimeters (FIRST) catalog. The objective is to search for 21-cm absorption in QGPs with impact parameter $<$30kpc and systematically increase the number of low-$z$ intervening 21-cm absorbers/DLAs. This sample of low-$z$ absorbers can then be used as a comparison sample to understand the nature of high-$z$ 21-cm absorbers and DLAs [see @Krogager12 and references therein for the current status of $z>2$ DLA host galaxies]. In this paper, we report the detection of 21-cm absorption from two foreground galaxies at [$z_{\rm {gal}}$]{}$\sim$0.3 towards quasars SDSSJ084957.97+510829.0 ([$z_{\rm {qso}}$]{}=0.584, hereafter J0849+5108) and SDSSJ144304.53+021419.3 ([$z_{\rm {qso}}$]{}=1.82, hereafter J1443+0214). These two pairs with foreground galaxies at ${\ensuremath{z_{\rm {gal}}}}\sim0.3$ were identified while scanning the SDSS images and spectra of bright radio sources and are interesting in their own right. In both these cases, the measured 21-cm optical depth is consistent with the absorbing gas being a DLA. The nature of background radio source in the case of QGP J0849+5108 has been the subject of considerable debates and is suspected of being affected by foreground galaxy lensing. The quasar sight line is positioned close to a pair of low-$z$ (${\ensuremath{z_{\rm {gal}}}}$$\sim$0.07) interacting spiral galaxies, and the field contains many bright galaxies within 60$^{\prime\prime}$ of the quasar [@Stickel89]. The sight line has been previously searched for 21-cm and molecular absorption lines [@Boisse88; @Wiklind95; @Gupta10]. The QGP J1443+0214 differs in the way it has been identified. Unlike most other QGPs studied to date, this QGP is identified by the novel technique of detecting emission lines from a foreground galaxy in the spectra of QSOs [see e.g., @Noterdaeme10o3; @York12]. The foreground galaxy is otherwise invisible in this case. In Section \[sec:obs\], we present details of our observations. We then detail the properties of individual QGPs in Section \[sec:prop\]. In Section \[sec:disc\], we discuss implications of our results. Throughout this paper we use the $\Lambda$CDM cosmology with $\Omega_m$=0.27, $\Omega_\Lambda$=0.73, and H$_{\rm o}$=71[km s$^{-1}$]{}Mpc$^{-1}$. Radio observations {#sec:obs} ================== Both the pairs were observed with Giant Metrewave Radio Telescope (GMRT) using a baseband bandwidth of 4.17MHz split into 512 spectral channels (resolution$\sim$2.3[km s$^{-1}$]{}) centered on the redshifted 21-cm frequency of foreground galaxies. The QGP J0849+5108 was observed on 2011 July 18 (2.7hrs on source) and J1443+0214 on 2012 June 2 (5.7hrs on source). Standard calibrators were regularly observed during the observations for flux density, bandpass, and phase calibrations. The data were reduced using the Astronomical Image Processing System (AIPS) following standard procedures as in @Gupta10. Both the quasars are represented well by a single Gaussian component, i.e. unresolved in our GMRT images that have spatial resolution of $\sim$4$^{\prime\prime}\times$3$^{\prime\prime}$. The flux densities of J0849+5108 and J1443+0214 were measured to be 245mJy and 163mJy, and spectral rms to be $\sim$2.0mJybeam$^{-1}$channel$^{-1}$ and $\sim$0.9mJybeam$^{-1}$channel$^{-1}$, respectively. In both cases, we detected 21-cm absorption consistent with the redshift of the foreground galaxy. In the case of J0849+5108, a few frequency channels on either side of the detected absorption feature were found to be affected by RFI (see shaded regions in Fig. \[j0849fit1\]). We therefore reobserved this QGP on 2012 July 3 (3.5hrs on source) to confirm the absorption. The flux density of the quasar was measured to be 221mJy, i.e. $\sim$10% lower than the value measured in the first observing run. The velocity shift due to heliocentric motion of the Earth between the two observing runs on this QGP is 5.6[km s$^{-1}$]{}. The repeat observation reproduced the previously detected absorption feature at the expected frequency. This confirms the reality of the absorption. The spectrum presented in Fig. \[j0849fit1\] is obtained by the inverse-rms-square weighted average of the [*normalized*]{} spectra obtained from two observing runs. The 21-cm absorption spectrum towards J1443+0214 is presented in Fig. \[j1443fit\]. In the case of absorber towards J0849+5108, the total integrated 21-cm optical depth, $\int\tau$$dv$= 0.95$\pm$0.06[km s$^{-1}$]{}. Overall, 90% of the total optical depth is contained within 22.5[km s$^{-1}$]{}. In the case of J1443+0214, the total integrated 21-cm optical depth is 3.4$\pm$0.1[km s$^{-1}$]{}, and 90% of the total optical depth is contained within 16.5[km s$^{-1}$]{}. We parametrize both the absorption profiles using multiple Gaussian components. The fits are shown in Figs. \[j0849fit1\] and \[j1443fit\], and details of Gaussian components are provided in Table \[21cmfit\]. [ccccc]{} Quasar & ID & [$z_{\rm abs}$]{}& FWHM & $\tau_{peak}$\ & & & (kms$^{-1}$)\ J0849+5108 & A & 0.312074 & 4$\pm$1 & 0.059$\pm$0.010\ & B & 0.312058 & 13$\pm$2 & 0.035$\pm$0.008\ & C & 0.311992 & 8$\pm$2 & 0.031$\pm$0.004\ J1443+0214 & A & 0.371544 & 8$\pm$1 & 0.297$\pm$0.017\ & B & 0.371535 & 19$\pm$4 & 0.043$\pm$0.015\ \[21cmfit\] Physical properties in individual cases {#sec:prop} ======================================= [$z_{\rm {gal}}$]{}= 0.3120 galaxy towards J0849+5108 ----------------------------------------------------- The background radio source in this case is a well-known object that has a BL Lacertae-like spectrum during outbursts [@Arp79]. The nature of this object has been the subject of considerable debate. There are speculations about its optical spectrum being affected by foreground galaxy lensing and/or reddening [@Stickel89; @Ostman06]. Interestingly, there are two foreground galaxies at angular separation less than 15$^{\prime\prime}$ to the radio source [see Fig. 7 of @Gupta10], and the sight line has been previously searched for 21-cm and molecular absorption [@Boisse88; @Wiklind95]. The galaxy J084957.48+510842.3 ([$z_{\rm {gal}}$]{}=0.073) situated at $\sim$14$^{\prime\prime}$ ($b$$\sim$19kpc) north of the radio source is a member of an interacting pair of galaxies [@Stickel89]. No 21-cm absorption was detected from this galaxy by @Boisse88. The 3$\sigma$ 21-cm optical depth limit, $\int\tau_{3\sigma}dv$, was 0.32[km s$^{-1}$]{}. A deeper 21-cm absorption optical depth limit, $\int\tau_{3\sigma}dv$=0.08[km s$^{-1}$]{}, was obtained for this galaxy using GMRT by @Gupta10. They also reported the detection of Na [i]{} and Ca [ii]{} absorption at [$z_{\rm {gal}}$]{}= 0.3120 in the spectrum of the QSO and associated it to a luminous red galaxy (LRG), identified as ‘G1’ in their Fig. 7, at an angular separation of 2.3$^{\prime\prime}$ ($b$$\sim$14kpc) southeast of the QSO. While the spectrum of this galaxy is not available in the SDSS catalog, Fig. \[j0849fit2\] shows that the SDSS photometry is consistent with an LRG at the absorber’s redshift. Here, we report the detection of 21-cm absorption at the redshift of Na [i]{} and Ca [ii]{} absorption lines (see Fig. \[j0849fit1\]). The 21-cm absorption profile is well fitted with three Gaussian components presented in Table \[21cmfit\]. In principle, the width of 21-cm absorption line can be used to constrain the kinetic temperature, $T_K\le$21.86$\times$FWHM$^2$, of absorbing gas [e.g., @Heiles03]. For J0849+5108, the FWHM of the strongest and narrowest Gaussian component ‘A’ implies kinetic temperature, $T_K<$350K. The observed equivalent widths of Na [i]{} absorption lines correspond to $N$(Na [i]{})$\ge$ (6$\pm$2)$\times 10^{12}$ cm$^{-2}$, assuming the optically thin case. Consequently, we derive log$N$()$\ge21.2\pm0.5$ following the relation $$log\,\frac{N(\ion{Na}{i})}{N(\ion{H}{i})} = -(0.16\pm0.06)\,[log\,N(\ion{H}{i}) - 19.5] - (8.12\pm0.79)$$ from the known correlation between $N$(H [i]{}) and $N$(Na [i]{}) found in our Galaxy [@Ferlet85; @Wakker00]. By assuming an optically thin cloud with spin temperature $T_s$ and covering factor $f_c$, the total integrated 21-cm optical depth estimated from the GMRT spectrum corresponds to the  column density, $N$(H [i]{})=1.7$\times$10$^{20}$($T_s$/100)(1.0/$f_c$)cm$^{-2}$. Therefore, if the absorber was to follow the $N$()-$N$(Na [i]{}) relation seen in our galaxy, the harmonic mean spin temperature $T_s>$$890^{+1560}_{-580}$K. Since such high $T_s$ values are typically seen in high-$z$ DLAs [@Kanekar03; @Srianand12dla], it will be important to have a direct Ly$\alpha$ measurement of $N$(H [i]{}) for this system . The extent of gas detected in absorption is determined by the size of the background source. At milliarcsecond (mas) scales, J0849+5108 exhibits a core-jet morphology with an overall separation of $\sim$4mas, i.e. $\sim$20pc at the [$z_{\rm {gal}}$]{} [@D'ammando12]. The 5GHz Very Long Baseline Array (VLBA) observations simultaneous with the Very Large Array (VLA) observations recover 75% to 85% of the total arcsecond scale flux [see Tables 6 and 7 of @D'ammando12]. The remaining flux originates in structures on scales $>$40mas (i.e., 200pc at [$z_{\rm {gal}}$]{}). This suggests that the region probed in the GMRT spectrum probably corresponds to scales $<$20pc. If the extent of the absorbing gas is $>$20pc then this would imply $f_c>$0.7. Actually, the ‘core’ component is unresolved with size$<$0.3mas (i.e., 1.4pc at [$z_{\rm {gal}}$]{}=0.3120) and contains about 95% of the total mas-scale flux at 5-10GHz. Thus, the derived constraints on $f_c$ and extent of the region probed in 21-cm absorption are conservative. As previously mentioned, while J0849+5108 has a spectral energy distribution (SED) similar to that of a blazar, @Yuan08 identified it as a radio-bright narrow-line Seyfert 1 AGN with an additional contribution to the optical flux coming from a mildly relativistic jet. The intrinsic spectrum is uncertain and most likely very different from the median QSO spectrum of @VandenBerk01. Therefore, unlike the case of QGP J1443+0214 discussed next, it is not possible to obtain an independent estimate of $N$() by SED fitting in this case. [$z_{\rm {gal}}$]{}=0.3714 galaxy towards J1443+0214 ---------------------------------------------------- -- -- -- -- Unlike most QGPs studied until now, where one clearly identifies a foreground galaxy in optical images, the galaxy in the case of QGP J1443+0214 is identified by the detection of emission lines in the SDSS spectrum of QSO. The foreground galaxy is invisible otherwise. The SDSS spectrum \[both in SDSS-II (DR7) and SDSS-III (DR9)\] of QSO J1443+0214 shows broad emission lines at [$z_{\rm em}$]{}= 1.82 and a set of narrow emission lines (i.e., \[O[ii]{}\], \[O[iii]{}\], H$\beta$, and H$\alpha$) at [$z_{\rm {gal}}$]{}=0.3714 (see Fig. \[j1443\_em\]) like the foreground galaxies detected by @Noterdaeme10o3 in the SDSS spectra of high-$z$ QSOs. We do not detect any significant difference in the line strengths between the SDSS-II (DR7) and SDSS-III (DR9) spectra, while the fiber diameters are of 3(15kpc at [$z_{\rm {gal}}$]{}) and 2(10kpc at [$z_{\rm {gal}}$]{}), respectively. Therefore the line emission probably originates in a low surface brightness region within 5kpc to the QSO sight line. This is also confirmed by the non-detection of the galaxy in SDSS images after subtracting the QSO contribution and by the fact that the SED fitting as in Fig. \[j1443red\] (see below) did not require any additional contribution from the galaxy continuum. We estimated the color excess E(B-V) by fitting the SDSS-II SED of QSO J1443+0214 by the SDSS-II QSO composite from @VandenBerk01. The $<$2000Å rest-wavelength coverage for the composite spectrum was taken from @Telfer02. We reddened the composite spectrum using the Galaxy, the LMC, and the SMC extinction curves [see @Srianand08bump; @Noterdaeme09co; @Noterdaeme10co for the detailed procedure]. These extinction curves differ significantly only at $\lambda$$<$2500Å. For J1443+0214, the SMC extinction curve is favored by the ultraviolet photometric points from the GALEX images. The J1443+0214 SED, along with the composite QSO SED reddened using the SMC extinction curve at ${\ensuremath{z_{\rm {gal}}}}= 0.3714$ with the best fit A$_V=0.61$, is shown in Fig. \[j1443red\]. We went on to apply the same procedure to a control sample of 223 SDSS-II QSOs from @Schneider10 with emission redshifts within $\Delta z = \pm 0.005$ from that of J1443+0214, where we removed QSOs with strong BAL activity using @Allen11. We estimated the uncertainty due to intrinsic shape variations from the width of A$_V$ distribution and corrected for the “zero-point” (the median A$_V$ value). The extinction we thus derive for J1443 +0214 is A$_V=0.6\pm0.2$, i.e., $3\sigma$ evidence of reddening along the sight line when compared to the control sample. Since we did not attempt to remove intervening absorbers from the control sample, the estimated A$_V$ and its statistical significance are conservative. Next we use the Balmer decrement to obtain an independent estimate of the extinction. Various emission line fluxes estimated through the Gaussian profile fitting using a higher signal-to-noise ratio DR9 spectrum [@Paris12] are given in Table \[j1443\_emt\]. As noted above, different extinction curves are identical over the wavelength range of emission lines considered here and therefore yield similar values. From the observed Balmer decrement, we estimate the color excess $E({\ensuremath{B\!-\!V}}) = E(\Hb\!-\!{H$\alpha$})/(k(\Hb)-k({H$\alpha$})) \approx $ 0.19 and 0.20 assuming intrinsic ${H$\alpha$}/\Hb=2.86$ and $k(\lambda)$ for the SMC- and the Galaxy-type dust respectively [@Gordon03]. This corresponds to $A_V = 0.6$ for $R_V$ = 2.74 seen in the SMC. The extinction detected towards the QSO is significantly higher than the value (E(B-V)$<$0.03) generally seen towards intervening DLAs and  systems [e.g., @York06; @Khare12]. It is also interesting to note that the extinction derived in the star-forming region using emission lines is remarkably consistent with the value derived along the QSO line of sight using SED fitting. This suggests that the abundance of dust is homogeneous over the length scale corresponding to the impact parameter ($\sim$5kpc), over which the line fluxes are measured. Using the mean $A_V$ vs $N$(H [i]{}) relationship found for the SMC [@Gordon03], we infer $N$(H [i]{}) = $(8\pm3)\times10^{21}$cm$^{-2}$. [c c c c]{} Line & $F_{\rm obs}$ & $L$ & Derived\ & ($10^{-17}$ ergs$^{-1}$cm$^{-2}$) & ($10^{40}$ ergs$^{-1}$) & quantities\ $[$O[ii]{}$]$ & 11.7 & 12.5 & SFR = 0.8 M$_{\odot}$yr$^{-1}$\ $[$O[iii]{}$]\lambda$5007 & 24.8 & 21.4 & R$_{23}$ = 13.8\ H$\beta$ & 3.4 & 3.0 & $E({\ensuremath{B\!-\!V}})$ = 0.20\ H$\alpha$ & 12.1 & 8.5 & SFR = 0.7 M$_{\odot}$yr$^{-1}$\ The profile of the 21-cm absorption line detected towards J1443+0214 is relatively simple (Fig. \[j1443fit\]). In addition to a narrow component that can be modeled with a Gaussian of FWHM$\sim$8[km s$^{-1}$]{} ($T_K<$1400K), a broad component is also required to reasonably fit absorption in wings (Table \[21cmfit\]). The total integrated 21-cm optical depth corresponds to $N$(H [i]{})=6.2$\times$10$^{20}$($T_s$/100)(1.0/$f_c$)cm$^{-2}$, which is less than the $N$(H [i]{}) we inferred from the SED fitting. This could either be due to the covering factor of the gas being much less than 1 or the harmonic mean $T_s$ being as high as $\sim$1400K. No mas-scale images or $N$() measurements using are available to distinguish between these scenarios. Another possibility is that the extinction per hydrogen atom in this absorber is much higher than what is measured in the SMC. It has been suggested that this may be the case in some high-$z$ dusty Mg [ii]{} absorbers with 21-cm absorption [@Gupta12] and DLAs with CO detections [@Noterdaeme10co]. Now we use nebular emission line properties to obtain insight into the nature of the absorbing galaxy. We correct emission line fluxes for extinction and use these to estimate the global metallicity and star formation rate (SFR) of the galaxy (Table \[j1443\_emt\]). For J1443+0214 we estimate the R$_{23}$ ratio, defined as (\[O[ii]{}\] + \[O[iii]{}\])/H$\beta$, to be 13.8. In general, the R$_{23}$ calibration is double-valued with respect to metallicity, and additional line ratios are required to break this degeneracy. For J1443+0214, the measured ratio is a high value at the turnover of the high- and low-metallicity branches of R$_{23}$-metallicity calibration [see e.g., @Kobulnicky99; @Pilyugin00], indicating 12 + log(O/H)$\sim 8.4$ i.e., 0.5Z$_\odot$ using 12 + log(O/H)$_\odot$ = 8.69 [@Asplund09]. From extinction-corrected [H$\alpha$]{} luminosity, $L$(H$\alpha$), and using the following relation from @Kennicutt98, $${\rm SFR(M_\odot\,yr^{-1})} = 7.9\times10^{-42} L({\rm H}\alpha)\,({\rm ergs\,s^{-1}}),$$ we derive SFR$\sim$0.7M$_{\odot}$yr$^{-1}$. Using the following relation from @Kewley04 $${\rm SFR(M_\odot\,yr^{-1})} = (6.6\pm1.7)\times10^{-42} L([{\rm O~{II}}])\,({\rm ergs\,s^{-1}})$$ a similar value of 0.8$\pm$0.2 M$_{\odot}$yr$^{-1}$ is derived using L(O [ii]{}). Thus, the SFR is slightly below the median value, and the metallicity is well within the range of values found for such \[O [iii]{}\]-emitting galaxies by @Noterdaeme10o3 (see Fig. \[sfr\]). In Fig. \[sfr\], the metallicity measurements for galaxies detected by @Noterdaeme10o3 correspond to the upper branch of R$_{23}$-metallicity calibration. The corresponding metallicities on the lower R$_{23}$ branch would be typically 0.4dex lower. The SFRs for these have been estimated using calibrations (Eqs. 23 and 24) from @Argence09 that use uncorrected \[O [ii]{}\] and H$\beta$ luminosities. The error bars in Fig. \[sfr\] indicate the range of SFRs allowed by variations in dust attenuation and metallicity but do not account for fiber losses. [c c c c c c c c c c c]{} Quasar & [$z_{\rm {qso}}$]{}& Galaxy & [$z_{\rm {gal}}$]{}& $b$ & $\int\tau$dv & 12+log(O/H) &  \[O/H\] &SFR & $\Sigma_{\rm SFR}$ & Reference\ & & & & (kpc) & ([km s$^{-1}$]{}) & & & (M$_{\odot}$yr$^{-1}$) &\ J104257+074850 & 2.7 & J104257+074751 & 0.03 & 1.7 & 0.2 &8.4 &$-$0.29 &0.01 &   4.1 & @Borthakur10\ J124157+633241 & 2.6 & J124157+633237 & 0.14 & 11 & 2.9 &8.7 &   0.01 &0.1 &   2.9 & @Gupta10\ J144304+021419 & 1.8 & $-$ & 0.37 &$<$5 & 3.4 &8.4 &   0.29 &0.7 & $>$3.8 & This work\ J163956+112758 & 1.0 & J163956+112802 & 0.08 & 4.0 & 15.7 &8.5 &$-$0.19 &0.3 &   18 & @Srianand13dib\ In addition to QGP J1443+0214, galactic emission lines superimposed on top of QSO continuum in SDSS fiber (diameter 3$^{\prime\prime}$) spectra have been detected for three other 21-cm absorbers. For these four absorbers, we are able to obtain SFR, SFR per unit area ($\Sigma_{\rm SFR}$) and metallicity (see Table \[21cmlit\] and references given in the last column for details). The SFR and metallicity for these 21-cm absorbers are also plotted in Fig. \[sfr\] for comparison with the sample of \[O [iii]{}\]-selected normal galaxies of @Noterdaeme10o3. Figure \[sfr\] suggests that the SFR of 21-cm absorbers is less than the median SFR of \[O [iii]{}\]-selected Mg [ii]{} absorbers. While in the case of J1443+0214 we are sure that all the galactic emission is included in the SDSS fiber, in other cases a large fraction of foreground galaxy emission falls outside the SDSS fiber. Therefore, SFRs for these galaxies are at best lower limits, and are in general underestimated with respect to the higher redshift (0.4$<{\ensuremath{z_{\rm {gal}}}}<$0.7) sample of @Noterdaeme10o3. Therefore, within uncertainties SFRs for these QGPs with 21-cm absorption seem to be consistent with \[O [iii]{}\]-selected Mg [ii]{} absorbers. Discussion {#sec:disc} ========== We have detected 21-cm absorption from two foreground galaxies at $z$$\sim$0.3 towards quasars J0849+5108 (b$\sim$14kpc) and J1443+0214 (b$<$5kpc). In both cases, we infer $N$()${\mbox{\raisebox{-0.3em}{$\stackrel{\textstyle >}{\sim}$}}}$2$\times$10$^{20}$cm$^{-2}$, which qualifies these systems as DLAs. The integrated 21-cm optical depths are 0.95[km s$^{-1}$]{} for J0849+5108 and 3.4[km s$^{-1}$]{} for J1443+0214. As can be noted from Fig. 19 of @Gupta10, such high values are observed among $z<1$ DLAs with host galaxy located at $b<$15kpc. In the case of J0849+5108, the SDSS photometry suggests that the Na [i]{}, Ca [ii]{}, and 21-cm absorption lines detected by us are associated with an LRG having a smooth optical morphology that is typical of a passively evolving early-type galaxy. In the case of J1443+0214, the absorption is probably associated with a low surface brightness (LSB) galaxy identified via optical emission lines within $\sim$5kpc of the QSO sight line. Late-type or LSB galaxies have been identified with a few low-$z$ DLAs and 21-cm absorbers [see Table 4 of @Rao03; @Lebrun97]. However, the case of LRG associated with the absorber towards J0849+5108 is the first example of an early-type galaxy associated with an intervening 21-cm absorber. The gas detected in 21-cm and metal absorption lines in the outskirts (b$\sim$14kpc) of this LRG could be associated with the extended  disks ($N$()$\sim$10$^{20}$cm$^{-2}$) detected in deep  21-cm emission observations of local early-type galaxies [e.g., @Serra12]. In particular, the cold gas detected in 21-cm absorption could be serving as fuel to the renewed star formation activity in the outer regions of the galaxy. This idea is supported by detections of ultraviolet rings that correspond to extended low-level recent or ongoing star formation in outskirts of $z\sim0.1$ early-type galaxies [@Salim12; @Fang12]. DLAs, as well as 21-cm absorption line surveys at $z<2$, are generally based on QSO sight lines selected via  absorption [@Rao06], and it is well known that LRGs offer very little cross-section to  absorption [@Bowen11]. Indeed, in an Mg [ii]{} selected sample of DLAs, sub-DLAs and Lyman limit systems at 0.1$<z<$1.0 @Rao11 find that absorber galaxies comprise a mix of spectral types that are predominantly late type. Of eight DLA galaxies in the part of their sample with sufficient photometry to carry out galaxy template fits, only one is found to be early type. Clearly, much larger samples are required to quantify the prevalence of early-type galaxies in samples of DLA and 21-cm absorber hosts. The techniques that rely solely on the detection of nebular emission lines to identify absorber hosts may miss such cases. Now we turn to low-$z$ 21-cm absorbers with detections of nebular emission lines in QSO spectra. The SFR of absorbing galaxy in the case of J1443+0214 presented here is well below the median SFR found for \[O [iii]{}\]-selected Mg [ii]{} absorbers at 0.4$<z<$0.7 [@Noterdaeme10o3]. A large fraction (87%) of these \[O [iii]{}\]-selected absorbers exhibits strong Mg [ii]{} absorption ($W_{\rm r}>$1Å). This, along with the detection of strong Fe [ii]{} absorption in several cases, suggests that most of these systems could be DLAs [see @Rao06]. @Wolfe03b have used C [ii]{}$^*$ to infer $\Sigma_{\rm SFR}$ in $z>2$ DLAs. For the CNM-model i.e. where QSO sight lines pass through CNM, they measure $\Sigma_{\rm SFR}$=10$^{-2.2}$M$_\odot$yr$^{-1}$kpc$^{-2}$ and measure $\Sigma_{\rm SFR}$=10$^{-1.3}$M$_\odot$yr$^{-1}$kpc$^{-2}$ for the warm neutral medium (WNM)-model. The $\Sigma_{\rm SFR}$ that we measure for low-$z$ QGPs in Table \[21cmlit\] are more in line with the CNM-model. This is not surprising since the presence of CNM in these cases is confirmed by the detection of 21-cm absorption. The abundances we infer for these QGPs using emission lines are in the range, \[O/H\]$\sim$ -0.3 to 0.3 (Table \[21cmlit\]), and are significantly higher than the values generally inferred for $z>2$ DLAs using absorption lines. It will be interesting to obtain ultraviolet spectra of these QGPs to measure $N$() and metallicity. This will also provide a direct measurement of spin temperature and CNM fraction along the sight lines, and will be crucial for understanding low CNM fractions in $z>2$ DLAs inferred from the observations of 21-cm and H$_2$ absorption [e.g., @Kanekar03; @Srianand12dla]. At $z>2$, where it is possible to build large samples of DLAs via ground-based optical surveys [@Noterdaeme12dla], searches to detect DLA host galaxies have mostly resulted in non-detections [e.g., @Moller93; @Rahmani10]. Recently, thanks to powerful instrumentation on the VLT, detections of a few DLA host galaxies have been possible by specifically targeting high-metallicity/$N$() DLAs [see e.g., @Fynbo11; @Noterdaeme12dlagal]. Using a sample of ten DLA hosts at $z>2$, @Krogager12 report anti-correlation (Spearman rank, $r_s$=$-$0.6) between log$N$() and the galaxy impact parameter [see also @Peroux11]. A similar trend is expected between 21-cm optical depth and impact parameter albeit with a large scatter probably due to further dependence on $T_s$ and $f_c$. The strength of this correlation can potentially shed light on processes driving the CNM filling factor of DLAs. In Fig. \[imp\], we plot 21-cm optical depth versus impact parameter for all the $z{\mbox{\raisebox{-0.3em}{$\stackrel{\textstyle <}{\sim}$}}}$0.4 QGPs with 21-cm absorption measurements [@Gupta10; @Borthakur11]. The Spearman rank correlation coefficient test suggests weak ($r_s$=-0.3) anti-correlation between $\int\tau$dv and $b$ for QGPs with 21-cm detection. Milliarcsecond scale images and 21-cm absorption spectra along with a much larger sample, especially with $b<$10kpc, are required to confirm and understand the implication of this trend. Because the number of DLAs and consequently DLA hosts at $z>2$ is expected to steadily increase with the availability of large samples of QSO spectra from the SDSS Baryon Oscillation Spectroscopic Survey [@Noterdaeme12dla], all this clearly motivates the need for large surveys of 21-cm absorption line especially at low-$z$ where it is relatively easy to identify and determine the properties of absorbing galaxy. Such surveys will become possible in the near future with SKA pathfinders and will uncover a new population of high-$N$() absorbers i.e., DLA/sub-DLAs that have escaped optical/ultraviolet absorption line surveys due to biases caused by either dust extinction or pre-selection of QSO sight lines based on various metal absorption lines or on the proximity to a certain galaxy type. Acknowledgments {#acknowledgments .unnumbered} =============== We gratefully acknowledge V. Mohan’s help with the subtraction of QSO contribution from SDSS images and useful discussions with P. Serra. We thank the GMRT staff for their help during the observations. GMRT is run by the National Centre for Radio Astrophysics of the Tata Institute of Fundamental Research. We acknowledge the use of SDSS spectra from the archive (http://www.sdss.org/). RS and PPJ gratefully acknowledge support from the Indo-French Centre for the Promotion of Advanced Research (Project N.4304-2). ====
--- abstract: '[In this paper, we study the complicated dynamics of infinite dimensional random dynamical systems which include deterministic dynamical systems as their special cases in a Polish space. Without assuming any hyperbolicity, we proved if a continuous random map has a positive topological entropy, then it contains a topological horseshoe. We also show that the positive topological entropy implies the chaos in the sense of Li-Yorke. The complicated behavior exhibiting here is induced by the positive entropy but not the randomness of the system. ]{}' address: - | School of Mathematics, Sichuan University, Chengdu 610064, Sichuan, P. R. China\ and School of Mathematical Sciences\ University of Science and Technology of China\ Hefei 230026, Anhui, P. R. China - | Department of Mathematics\ Brigham Young University\ Provo, Utah 84602, USA author: - Wen Huang - Kening Lu title: 'Entropy, Chaos and weak Horseshoe for Infinite Dimensional Random Dynamical Systems' --- [^1] Intoduction =========== Entropy plays an important role in the study of behavior of dynamical systems. It measures the rate of increase in dynamical complexity as the system evolves with time. The measure-theoretic entropy was introduced in 1950’s by Kolmogorov [@Kol] and Sinai [@Sinai2] for studying measurable dynamical systems. Sinai [@Sinai] studied an ergodic measure preserving automorphism $f$ of a Lebesgue space $(X, \mu)$ and proved that if the measure-theoretic entropy of $f$ is positive, then $f$ contains factor automorphisms which are isomorphic to Bernoulli shifts. The topological entropy was first introduced in 1965 by Adler, Konheim and McAndrew for studying dynamical systems in topological spaces. In metric spaces a different definition of topological entropy was introduced by Bowen in 1971 and independently Dinaburg in 1970. A fundamental problem is to characterize the chaotic behavior of orbits of a $C^k$ ($k\geq 0$) dynamical system $f$ topologically or geometrically (in terms of horseshoe) in the presence of positive topological entropy. In his remarkable paper [@K], A. Katok proved that for a measure preserving hyperbolic $C^2$ diffeomorphism on a compact Riemannian manifold, the positive entropy implies the existence of a Smale horseshoe. Without assuming any hyperbolicity, Blanchard, Glasner, Kolyada, and Maass [@BGKM] showed that for a homeomorphism on a compact metric space $X$, the positive topological entropy implies the chaos in the sense of Li-Yorke [@LY], i.e., there is a subset $S$ of $X$, which is a union of countably many Cantor sets, and $\kappa>0$ such that for every pair $x_1,x_2$ of distinct points in $S$, the following holds. $$\begin{aligned} &\liminf_{n\rightarrow +\infty} d(f^n(x_1),f^n(x_2))=0, \\ &\limsup_{n\rightarrow +\infty} d(f^n(x_1),f^n(x_2))\ge \kappa.\end{aligned}$$ However, this result does not yield the existence of a horseshoe. Recently, Lian and Young obtained remarkable results on the implication of positive entropy for infinite dimensional deterministic dynamical systems. In [@LianY1], they extended Katok’s results to $C^2$ differentiable maps with a nonuniformly hyperbolic compact invariant set supported by an invariant measure in a separable Hilbert space. In their second paper [@LianY2], Lian and Young went much further and studied a $C^2$ semiflow in a Hilbert space and proved that if it has a nonuniformly hyperbolic compact invariant set supported by an invariant measure, then the positive entropy implies the existence of horseshoes. In this case, the semiflow may have one simple zero Lyapunov exponent, which implies that the associated time-one map restricted to this invariant set is partially hyperbolic with one-dimensional center direction. This result is new even for flow generated by ordinary differential equations. The proofs of the results obtained by Katok and Lian-Young rely on not only the positive of entropy but also the hyperbolic geometric structures of systems. The horseshoes are constructed by using stable and unstable manifolds. In present paper, we study $C^0$ infinite dimensional random dynamical systems which include deterministic dynamical systems as their special cases in a Polish space. Without assuming any hyperbolicity, we proved if a continuous random map has a positive topological entropy, then it contains a topological horseshoe. We also show that the positive topological entropy implies the chaos in the sense of Li-Yorke. The complicated behavior exhibiting here is induced by the positive entropy but not the randomness of the system. Since there is no any hyperbolic geometric structure available, we take a different approach. We use Rohlin’s theory of Lebesques systems as a basic tool and utilize the disintegration of measures, Pinsker algebra, entropy, and ergodic theory. To overcome the obstacle due to lack of hyperbolicity, we construct an “independent” partition with an equal conditional probability measure $\bar \mu_y$ for almost all $y$ via the disintegration of a measure relative to a factor. This partition is the key for constructing the horseshoe. Other challenges are: (i) the infinite dimensional dynamical systems generated, for example, by parabolic PDEs are not invertible and (ii) the phase space is not locally compact. Let $(\Omega, \mathcal{F}, P)$ be a probability space and $(\theta^n)_{n\in \mathbb{Z}}$ be a measurable ${P}$-measure preserving dynamical system on $\Omega$. A discrete time random dynamical system (or a cocycle) on a metric space $X$ over the dynamical system ${\theta}^n$ is a measurable map $$\phi(n,\cdot,\cdot):\Omega \times X \to X, \quad (\omega, x) \mapsto\phi(n, \omega, x), \quad\text{for } n\in\mathbb{Z}^+$$ such that the map $\phi(n,\omega):=\phi(n,\omega,\cdot)$ forms a cocycle over $\theta^n$: $$\phi(0, \omega)=Id, \quad \hbox{ for all }\; \omega \in \Omega,$$ $$\phi(n+m,\omega)=\phi(n,\theta^{m}\omega)\phi(m,\omega), \quad \hbox{ for all }\; m, n \in \mathbb{Z}^+, \quad\omega \in \Omega.$$ When $\phi(n, \omega, \cdot): X \to X$ is continuous, $\phi(n, \omega, x)$ is called a continuous random dynamical system. Replacing $\mathbb{Z}$ and $\mathbb{Z}^+$ by $\mathbb{R}$ and $\mathbb{R}^+$ repectively gives a continuous time random dynamical system, see Section 2 for details. A typical example of random dynamical systems is the solution operator for a stochastic differential equation: $$d x_t= f_0(x_t) dt + \sum_{k=1}^d f_k(x_t) \circ dB^k_t$$ where $x\in \mathbb{R}^d$, $f_k, 0\leq k\leq d$, are smooth vector fields, and $B_t=(B_t^1, \cdots, B_t^d)$ is the standard d-dimensional Brownian motion defined on the probability space $(\Omega, \mathcal{F}, {P})$ and $d B^k_t$ is the Stratonovich differential. Here, $(\Omega, \mathcal{F}, \mathbb{P})$ is the classic Wiener space, i.e., $\Omega =\{ \omega\,:\, \omega(\cdot) \in C(\mathbb{R}, \mathbb{R}^d), \omega(0)=0\}$ endowed with the open compact topology so that $\Omega$ is a Polish space and $\mathbb{P}$ is the Wiener measure. Define a measurable dynamical system $\theta^t$ on the probability space $(\Omega, \mathcal{F}, {P})$ by the Wiener shift $(\theta^t\omega)(\cdot)=\omega(t+\cdot)-\omega(t)$ for $t>0$. It is well-known that ${P}$ is invariant and ergodic under $\theta^t$. This measurable dynamical system $\theta^t$ is also called a metric dynamical system. It models the noise of the system. We consider a random set ${K}\in {\mathcal F}\otimes\mathcal B(X)$ and use $K(\o)$ to denote its $\o$-section $\{x\in X\;|\;( \o,x)\in K\}$. $K$ is said to be invariant under $\phi(n,\omega)(x)$ if for all $n\in {\mathbb}Z^+$ $$\p(n, \o)K(\o)\subset K( \t^n \o)\quad P-a.s..$$ Examples of such random invariant sets in applications are the global random attractors of dissipative stochastic partial differential equations. We study the complicated dynamics of infinite dimensional random dynamical systems restricted to random invariant sets. We assume that $(\Omega, \mathcal{F},P, \theta)$ is a Polish system (see Section 3) and the phase space $X$ is a Polish space with the distance function $d$. We consider a continuous random dynamical system $\phi(n, \omega, x)$ and write the time-one map $\phi(1, \omega, x)$ as $\phi(\omega)(x):=\phi(1, \omega, x)$. Then $\phi(\omega)$ is the so-called random map. This random map generates the random dynamical system: $$\phi(n, \omega, x)= \begin{cases} \phi(\theta^{n-1}\omega)\cdots \phi(\omega)(x), & n >0, \\ I, & n=0. \\ \end{cases}$$ Our main result can be stated as follows. [**Main Theorem.**]{} *Let $\phi$ be an injective continuous random dynamical system on a Polish space $X$ over an ergodic Polish system $(\Omega, \mathcal{F},P,\theta)$. Let ${K}$ be a random $\phi$-invariant set with compact $\omega$-section ${K}(\omega)$ such that $\phi(\omega)(K(\omega))=K(\theta \omega)$. If the topological entropy is positive, i.e., $$h_{\text{top}}(\phi,\mathcal{K})>0,$$ then* - the dynamics of $\phi$ restricted to ${K}$ is chaotic; - $\phi$ restricted to ${K}$ has a weak horseshoe of two symbols; - in addition, if $(\Omega, \mathcal{F}, {P},\theta)$ is a compact metric system and $K(\omega),\,\omega\in\Omega$ is a strongly compact random set, then $\phi$ restricted to ${K}$ has a full horseshoe of two symbols. $\phi$ has a weak horseshoe of two symbols if there exist subsets $U_1,U_2$ of $X$ such that the following properties hold 1. $U_1$ and $U_2$ are non-empty, bounded, and closed and $d(U_1,U_2)>0$. 2. there is a constant $b>0$ satisfying for $P$-a.e. $\omega\in \Omega$, there exists $M_{b,\omega}\in \mathbb{N}$ such that for any natural number $m\ge M_{b,\omega}$, there is a subset $J_m\subset \{0,1,2,\cdots,m-1\}$ with $|J_m|\ge bm$ (positive density), and for any $s\in \{1,2\}^{J_m}$, there exists $x_s\in K(\omega)$ with $\phi(j,\omega,x_s)\in U_{s(j)}\cap K(\theta^j \omega)$ for any $j\in J_m$. By a [*full horseshoe of two symbols*]{} we mean that there exist subsets $U_1,U_2$ of $X$ such that the following properties hold 1. $U_1$ and $U_2$ are non-empty, bounded, and closed subsets of $X$ and $d(U_1,U_2)>0$. 2. there is a constant $b>0$ satisfying for $P$-a.e. $\omega\in \Omega$, there exists $J(\omega)\subset \mathbb{N}_0$ such that the limit $\lim_{m\rightarrow +\infty}\frac{1}{m}|J(\omega)\cap \{0,1,2,\cdots, m-1\}|$ exists and is larger than or equal to $b$ (positive density), and for any $s\in \{1,2\}^{J(\omega)}$, there exists $x_s\in K(\omega)$ with $\phi(j,\omega,x_s)\in U_{s(j)}\cap K(\theta^j \omega)$ for any $j\in J(\omega)$. The horseshoe here is an extension of Smale’s horseshoe. The main difference is that the time spent by the orbit $\phi(j,\omega,x_s)$ bouncing between $U_1$ and $U_2$ is nonuniform. We point out that the random dynamical systems $\phi$ generated by both random parabolic PDEs and random wave equations are continuous and injective. For deterministic dynamical systems, we have the following result ( see [@HY] for topological dynamical systems in compact metric spaces and [@KL-2006] for $C^*$-dynamics). [**Corollary.**]{} [*Let $f$ be an injective continuous map on a Polish space $X$. Let ${K}$ be a compact invariant set of $f$. If the topological entropy is positive, i.e., $$h_{\text{top}}(f,\mathcal{K})>0,$$ then $f|_K$ has a full horseshoe of two symbols.*]{} Random dynamical systems arise in the modelling of many phenomena in physics, biology, climatology, economics, etc. when uncertainties or random influences, called noises, are taken into account. These random effects are not only introduced to compensate for the defects in some deterministic models, but also are often rather intrinsic phenomena. The need for studying random dynamical systems was pointed out by Ulam and von Neumann [@UvN] in 1945. It has flourished since the 1980’s due to the discovery that stochastic ordinary differential equations generate finite dimensional random dynamical systems through the efforts of Harris, Elworthy, Baxendale, Bismut, Ikeda, Kunita, Watanabe, and others. Random dynamical systems are nonuniform in nature in terms of hyperbolicity. There is an extensive literature on the nonuniformly hyperbolic theory and the ergodic theory for both independent and identically distributed random transformations and stationary random dynamical systems, which we refer to Arnold [@A], Kifer [@K86; @Kifer88; @K01], Ledrappier and Young [@LeY1; @LeY2], Liu and Qian [@LQ], Liu [@L-p1], Kifer and Liu [@KL], and the references therein. The study of infinite dimensional random dynamical systems was initiated by Ruelle in [@Ruelle1; @Ruelle2] where the Oseledets’ multiplicative ergodic theorem and Pesin’s stable manifold theorem were established in a Hilbert space, and the notion of random attractor was introduced. Infinite dimensional random dynamical systems are usually generated by stochastic partial differential equations (SPDEs) and contain randomness in many ways, such as stochastic forcing, uncertain parameters, random sources or inputs, and random initial and boundary conditions. There is a vast amount of works on the ergodic theory, the existence of random attractors and the theory of invariant manifold. We do not attempt to give an exhaustive list of references. Results on ergodic theory can be found in Mane [@Mane], Thieullen [@Thieu], Schaumlöffel and Flandoli [@SchFla], Da Prato and Zabczyk [@DaPZab96], E, Khanin, Mazel, and Sinai [@EKMS], Hairer and Mattingly [@HM], Lian and Lu [@LLu]. For the existence of random attractors we refer to Crauel, Debussche, and Flandoli [@CDF], Crauel and Flandoli [@cr-fl], Schmalfuss [@Schm97c], and Bates, Lu, and Wang [@BLW]. Theory of random invariant manifolds can be found in Da Prato and Debussche [@DaPDeb96], Mohammed and Scheutzow [@MoS], Duan, Lu and Schmalfuss [@DLS], and Mohammed, Zhang, and Zhao [@MZZ]. The problem we study here is about the complicated dynamical behavior on random invariant sets such random attractors. We organize the paper as follows. In Section 2, we introduce basic concepts concerning random dynamical systems and random invariant sets and state our main results. In Section 3, we review some of basic concepts and results from measurable dynamical systems and introduce some basic lemmas. The proof of the main result is given in Section 4 and Section 5. [Statement of Results]{} ======================== In this section, we first review some of the basic concepts on RDS, which are taken from Arnold [@A]. Then we state our main results. Random Dynamical Systems ------------------------ Let $(\Omega, \mathcal{F}, {P})$ be a probability space and $\mathbb{T}$ denote one of the sets: $\mathbb{R}$, $\mathbb{R}^+$, $\mathbb{Z}$, and $\mathbb{Z}^+$. $\mathbb{T}$ is endowed with its Borel $\sigma$-algebra $\mathcal{B}(\mathbb{T})$. Let $\theta=(\theta^t)_{t\in \mathbb{R}}$ be a measurable ${P}$-measure preserving flow on $\Omega$, see Arnold [@A]. $(\Omega, \mathcal{F}, {P}, \theta^t)$ is called a metric dynamical system over the probability space $(\Omega, \mathcal{F}, {P})$. This metric dynamical system models the evolution of noise of the system. For the discrete time metric dynamical system, we replace $\mathbb{R}$ by $\mathbb{Z}$. As an example, we take $(\Omega, \mathcal{F}, {P})$ to be the Wiener space, i.e., $\Omega =\{ \omega\,:\, \omega(\cdot) \in C(\mathbb{R}, U), \omega(0)=0\}$ for some separable Hilbert space $U$ endowed with the compact open topology, $\mathcal{F}=\mathcal{B}(\Omega)$ and ${P}$ is the Wiener measure for a trace class covariance operator $Q$ on $U$. In fact, $\Omega$ is a Polish space. We define a measurable flow $\theta^t$ on the probability space $(\Omega, \mathcal{F}, \mathbb{P})$ by the Wiener shift $(\theta^t\omega)(\cdot)=\omega(t+\cdot)-\omega(t)$ for $t\in\mathbb{R}$. It is well-known that ${P}$ is invariant and ergodic under $\theta^t$. A random dynamical system in a metric space $X$ over the metric dynamical system $(\Omega, \mathcal{F}, {P}, \theta^t)$ is a measurable map $$\phi: \mathbb{T} \times \Omega \times X \to X, \quad (t, \omega, x) \mapsto \phi(t, \omega, x),$$ and $\phi(t, \omega)(x):=\phi(t, \omega, x)$ forms a cocycle: $$\phi(0, \omega)=Id, \quad \hbox{ for all }\; \omega \in \Omega,$$ $$\phi(t+s,\omega)=\phi(t,\theta^{s}\omega)\circ \phi(s,\omega), \quad \hbox{ for all }\; s, t \in \mathbb{T}, \quad\omega \in \Omega.$$ When $\phi(t, \omega, \cdot):X \to X$ is continuous, $\phi$ is called a continuous random dynamical system. As we mentioned in the introduction, a typical example in finite dimensional space is the solution operator of stochastic differential equations. An infinite dimensional random dynamical system can be generated by the solutions of partial differential equations driven by a stochastic process of the form $$u_t = \Delta u+ f(u, x, \theta_t \omega), \quad x\in U$$ with the Dirichlet boundary condition or the Neumann boundary, where $U \subset \mathbb{R}^n$ is a bounded region with a smooth boundary. It can also be generated by the solutions of stochastic partial differential equations of the form $$du =(Au+F(u)) dt + d{W},$$ where $A$ is an elliptic operator, $F$ is a smooth nonlinear functional, and $d{W}$ is a white noise given as the generalized temporal differential of a Wiener process with continuous paths in the phase space. Other examples of random dynamical systems are generated by the solutions of stochastic differential equations driven by a fractional Brownian motion [@GLS]. Random Invariant Sets and Random Attractors ------------------------------------------- We first recall that a multifunction $M=(M(\omega))_{\omega\in\Omega}$ of nonempty closed sets $M(\omega),\,\omega\in\Omega$, contained in $X$ is called [*a random set*]{} if $$\omega\mapsto\inf_{y\in M(\omega)}d(x,y)$$ is a random variable for any $x\in X$. A random set $M$ is invariant for random dynamical system $\phi$ if $$\phi(t,\omega,M(\omega))\subset M(\theta_t\omega)\quad \text{for} \quad t\geq 0.$$ A random set $\mathcal{A}=\{\mathcal{A}(\omega)\}_{\omega \in \Omega}$ of $X$ is called [*a global random attractor*]{} for $\phi$ if the following conditions are satisfied, for $\mathbb{P}$-a.e. $\omega \in \Omega$, - $\mathcal{A}(\omega)$ is compact; - $\{\mathcal{A}(\omega)\}_{\omega \in \Omega}$ satisfies for $t\geq 0$: $$\phi(t, \omega, \mathcal{A}(\omega))=\mathcal{A}(\theta_t\omega);$$ - $\{\mathcal{A}(\omega)\}_{\omega \in \Omega}$ attracts every tempered random set $B = \{B(\omega)\}_{\omega \in \Omega}$, that is, $$\lim_{t \to \infty} d (\phi(t, \theta_{-t}\omega, B(\theta_{-t}\omega)), \mathcal{A}(\omega))=0,$$ where $d$ is the Hausdorff semi-metric given by $d(Y,Z) = \sup_{y \in Y } \inf_{z\in Z} \| y-z\|_{X} $ for any $Y\subseteq X$ and $Z \subseteq X$. The study of global random attractors was initiated by Ruelle [@Ruelle1]. The fundamental theory of global random attractors for stochastic partial differential equations was developed by Crauel, Debussche, and Flandoli [@CDF], Crauel and Flandoli [@cr-fl], Flandoli and Schmalfuss [@fl-schm], Imkeller and Schmalfuss [@imk-schm], and others. It has recently attracted more attention due to new equations and models arising in applications such as stochastic infinite dimensional lattice dynamical systems [@BLL]. Due to the unbounded fluctuations in the systems caused by the white noise, the concept of pull-back global random attractor was introduced to capture the essential dynamics with possibly extremely wide fluctuations. This is significantly different from the deterministic case. Main Result ----------- In this paper, we study the complicated dynamics of infinite dimensional random dynamical systems restricted to random invariant sets such as global attractors. We assume that $(\Omega, \mathcal{F},P, \theta)$ is a Polish system (see Section 3) and $X$ is a Polish space with the distance function $d$. We consider a continuous random dynamical system $\phi(n, \omega, x)$ generated by a random map $\phi(\omega)(x)$ defined on $X$ over the metric dynamical system $(\Omega, \mathcal{F},P, \theta)$, i.e., $$\phi(n, \omega, x)= \begin{cases} \phi(\theta^{n-1}\omega)\cdots \phi(\omega)(x), & n >0, \\ I, & n=0. \\ \end{cases}$$ Here, $\phi(\omega):X\rightarrow X$ is continuous almost surely. We consider a $\phi$-invariant random set ${K}\in {\mathcal F}\otimes\mathcal B(X)$ with compact $\omega$-section ${K}(\omega)$. For $\omega \in \Omega$, $\epsilon>0$ and $n\ge 1$, we define $$d^\omega_n(x,y)=\max \limits_{0\le k\le n-1} d(\phi(k, \omega, x),\phi(k, \omega, y)), \, \text{ for }x,y\in X.$$ A subset $E$ of $K(\omega)$ is called [*$(\omega,n,\epsilon,\phi)$-separated subset of $K(\omega)$*]{} if for all $x,y\in E,x\neq y$, one has $d^\omega_n(x,y)>\epsilon$. We denote by $r_n(K,\omega,\epsilon,\phi)$ the maximal cardinality of all $(\omega,n,\epsilon,\phi)$-separated subset of $K_\omega$. The [*topological entropy*]{} of $(\phi,K)$ based on Bowen and Dinaburg’s definition is given by $$h_{\text{top}}(\phi,K):=\lim_{\epsilon\rightarrow 0} \limsup_{n\rightarrow +\infty} \frac{1}{n} \int_\Omega \log r_n(K,\omega,\epsilon,\phi) d P(\omega).$$ See, Bogenschutz [@B92] and Kifer [@K01] for related notions in the case of $X$ being compact. Let $\phi$ be an injective continuous random dynamical system on Polish space $X$ over an ergodic Polish system $(\Omega, \mathcal{F}, {P},\theta)$. Let $K$ be a $\phi$-invariant random set with compact $\omega$-section $K(\omega)$ such that $\phi(\omega)(K(\omega))=K(\theta \omega)$. Subsets $U_1,U_2$ of $X$ is called [*weak Horseshoe*]{} of $(\phi,K)$, if the following properties hold 1. $U_1$ and $U_2$ are non-empty, closed, and bounded subsets of $X$ and $d(U_1,U_2)>0$. 2. there is a $b>0$ satisfying for $P$-a.e. $\omega\in \Omega$, there exists $M_{b,\omega}\in \mathbb{N}$ such that for any natural number $m\ge M_{b,\omega}$, there is a subset $J_m\subset \{0,1,2,\cdots,m-1\}$ with $|J_m|\ge bm$(positive density), and for any $s\in \{1,2\}^{J_m}$, there exists $x_s\in K(\omega)$ with $\phi(j,\omega,x_s)\in U_{s(j)}\cap K(\theta^j \omega)$ for any $j\in J_m$. The first result is on the existence of a weak horseshoe. \[MTH1\] Let $\phi$ be an injective continuous random dynamical system on Polish space $X$ over an ergodic Polish system $(\Omega, \mathcal{F}, {P},\theta)$. Let $K$ be a $\phi$-invariant random set with compact $\omega$-section $K(\omega)$ such that $\phi(\omega)(K(\omega))=K(\theta \omega)$. If $h_{\text{top}}(\phi,K)>0$, then there exists a weak Horseshoe $\{U_1,U_2\}$ for $(\phi,K)$. Let $(\Omega, \mathcal{F}, {P},\theta)$ be a [*compact metric system*]{}. Namely, $\Omega$ is a compact metric space, $\mathcal{F}$ is the Borel $\sigma$-algebra $\mathcal{B}_{\Omega}$ of $\Omega$, $P$ is a Borel probability measure on $\Omega$ and $\theta:\Omega\rightarrow \Omega$ is a continuous map preserving the measure $P$. A multifunction $M=(M(\omega))_{\omega\in\Omega}$ of nonempty closed sets $M(\omega),\,\omega\in\Omega$, contained in $X$ is called [*a strongly compact random set*]{} if the following conditions are satisfied, for each $\omega \in \Omega$, - $M(\omega)$ is compact. - the function $\omega\mapsto\inf_{y\in M(\omega)}d(x,y)$ is lower semi-continuous for any $x\in X$. It is not hard to see that if a multifunction $M=(M(\omega))_{\omega\in\Omega}$ of nonempty closed sets $M(\omega),\,\omega\in\Omega$, contained in $X$ such that the set $\bigcup_{\omega\in \Omega}\{\omega\}\times M(\omega)$ is a compact subset of $\Omega\times X$, then $M$ is strongly compact random set. By a [*full horseshoe of two symbols*]{} we mean that there exist subsets $U_1,U_2$ of $X$ such that the following properties hold 1. $U_1$ and $U_2$ are non-empty, closed, and bounded subsets of $X$ and $d(U_1,U_2)>0$. 2. there is a constant $b>0$ satisfying for $P$-a.e. $\omega\in \Omega$, there exists $J(\omega)\subset \mathbb{N}_0$ such that the limit $\lim_{m\rightarrow +\infty}\frac{1}{m}|J(\omega)\cap \{0,1,2,\cdots, m-1\}|$ exists and is larger than or equal to $b$ (positive density), and for any $s\in \{1,2\}^{J(\omega)}$, there exists $x_s\in K(\omega)$ with $\phi(j,\omega,x_s)\in U_{s(j)}\cap K(\theta^j \omega)$ for any $j\in J(\omega)$. The second result is the existence of a full horseshoe. \[MTH3\] Let $\phi$ be an injective continuous random dynamical system on Polish space $X$ over an ergodic compact metric system $(\Omega, \mathcal{F}, {P},\theta)$ satisfying the map $(\omega,x)\mapsto \phi(\omega)x$ is a continuous map from $\Omega\times X$ to $X$. Let $K$ be a $\phi$-invariant strongly compact random set such that $\phi(\omega)(K(\omega))=K(\theta \omega)$. If $h_{\text{top}}(\phi,K)>0$, then there exists a full Horseshoe $\{U_1,U_2\}$ for $(\phi,K)$. As a consequence, we have \[Cor1\] Let $\phi$ be an injective continuous on Polish space $X$ and $K$ be a $\phi$-invariant compact set. If $h_{\text{top}}(\phi,K)>0$, then there exists a full Horseshoe $\{U_1,U_2\}$ for $(\phi,K)$. The notion of Li-Yorke chaos was introduced in [@LY] for interval maps. With a small modification this notion can be extended to a metric space. Following the idea of Li and Yorke, a subset $D$ of $K({\omega})$, is called [*$\kappa$-chaotic set*]{} for $(\omega,\phi)$, where $\omega\in \Omega$ and $\kappa>0$, if for every pair $(x_1,x_2)$ of distinct points in $D$, one has $$\liminf_{n\rightarrow +\infty} d(\phi(n, \omega, x_1),\phi(n, \omega, x_2))=0 \text{ and } \limsup_{n\rightarrow +\infty} d(\phi(n, \omega, x_1),\phi(n, \omega, x_2))\ge \kappa.$$ The final result is about the positive entropy implying the existence of Li-Yorke chaos. \[MTH\] Let $\phi$ be an injective continuous random dynamical system on Polish space $X$ over an ergodic Polish system $(\Omega, \mathcal{F}, {P},\theta)$. Let $K$ be a $\phi$-invariant random set with compact $\omega$-section $K(\omega)$ such that $\phi(\omega)(K(\omega))=K(\theta \omega)$. If $h_{\text{top}}(\phi,K)>0$, then there exists $\kappa>0$ such that for ${P}$-a.e. $\omega\in \Omega$ there is a $\kappa$-chaotic set $S(\omega)\subset K(\omega)$ of a union of countably many Cantor sets for $(\omega,\phi)$. Basic Concepts and Lemmas on Measurable Dynamical Systems ========================================================= In this section, we review some of basic concepts and results from the theory of measurable dynamical systems and introduce several lemmas that we need for the proofs of the main theorems. Various Dynamical Systems ------------------------- In this paper for a probability space $(X,\mathcal{B},\mu)$ we always require that $\mathcal{B}$ is countably generated ($\mu$-mod $0$), that is, there exists $\{ A_i\}_{i=1}^\infty \subset \mathcal{B}$ such that for any $A\in \mathcal{B}$ and $\epsilon>0$ there is $i:=i(A,\epsilon)\in \mathbb{N}$ satisfying $\mu(A\Delta A_i)<\epsilon$. [*A measure-theoretic dynamical system*]{} (MDS) $(X,\mathcal{B},\mu,T)$ is a measure-preserving map $T$ on a probability space $(X,\mathcal{B},\mu)$. [*A Polish probability space*]{} $(X,\mathcal{B}_X,\mu)$ means that $X$ is a separable topological space whose topology is metrizable by a complete metric, $\mathcal{B}_X$ is the Borel $\sigma$-algebra, and $\mu$ is a Borel probability measure on $X$. [*A Polish system*]{} $(X,\mathcal{B}_X,\mu,T)$ is a measure-preserving map $T$ on a Polish space $(X,\mathcal{B}_X,\mu)$. [*A Lebesgue system*]{} $(X,\mathcal{B},\mu,T)$ is a measure-preserving map $T$ on a Lebesgue space $(X,\mathcal{B},\mu)$ (see [@R]). For a Polish system $(X,\mathcal{B}_X,\mu,T)$, the MDS $(X,\mathcal{B}_\mu,\mu,T)$ constitutes a Lebesgue system, where $\mathcal{B}_\mu$ is the completion of the Borel $\sigma$-algebra $\mathcal{B}_X$ with respect to $\mu$. A MDS $(Y,\mathcal{D},\nu,S)$ is said to be [*a factor*]{} of $(X,\mathcal{B},\mu,T)$ if there is a measure-preserving map $\pi:(X,\mathcal{B},\mu)\rightarrow (Y,\mathcal{D},\nu)$ such that $\pi T=S\pi$. Equivalently, we say that $(X,\mathcal{B},\mu,T)$ is an extension of $(Y,\mathcal{D},\nu,T)$. In this case, we also say $\pi:(X,\mathcal{B},\mu,T)\rightarrow (Y,\mathcal{D},\nu,S)$ is [*a factor map*]{}. Conditional entropy ------------------- In this subsection, we first recall the notation of the conditional entropy of a MDS. Then we state some results about the conditional entropy. Consider an MDS $(X,\mathcal{B},\mu,T)$. [*A partition*]{} of $X$ is a family of pairwise disjoint sets in $\mathcal{B}$ whose union is $X$. For a given partition $\alpha$ of $X$ and $x\in X$, we denote by $\alpha(x)$ the atom of $\alpha$ containing $x$. We denote the set of finite partitions of $X$ by $\mathcal{P}_X(\mathcal{B})$ or for simplicity $\mathcal{P}_X$. Given two partitions $\alpha,\beta$ of $X$, $\alpha$ is said to be finer than $\beta$ (denote by $\alpha\succeq \beta$) if each element of $\alpha$ is contained in some element of $\beta$. Let $\alpha\vee\beta=\{A\cap B: A\in \alpha,B\in \beta \}$. For any given $\alpha \in \mathcal{P}_X$ and any sub-$\sigma$-algebra $\mathcal{C}$ of $\mathcal{B}$, let $$H_{\mu}(\alpha)=\sum_{A\in \alpha} -\mu(A) \log \mu(A)\ \text{and}\ \ H_{\mu}(\alpha|\mathcal{C})=\sum_{A\in \alpha} \int_X -\mathbb{E}_\mu(1_A|\mathcal{C}) \log \mathbb{E}_\mu(1_A|\mathcal{C}) d \mu,$$ where $\mathbb{E}_\mu(1_A|\mathcal{C})$ is the conditional expectation of the characterization function $1_A$ of $A$ with respect to $\mathcal{C}$. One standard fact states that $H_\mu(\alpha| \mathcal{C})$ increases with respect to $\alpha$ and decreases with respect to $\mathcal{C}$. When $T^{-1}\mathcal{C}\subseteq \mathcal{C}$, it is not hard to see that $H_\mu(\bigvee_{i=0}^{n-1}T^{-i}\alpha|\mathcal{C})$ is non-negative and sub-additive sequence for a given $\alpha\in \mathcal{P}_X$, so we can define $$h_\mu(T,\alpha|\mathcal{C})=\lim_{n\rightarrow +\infty} \frac{1}{n} H_\mu(\bigvee_0^{n-1}T^{-i}\alpha|\mathcal{C})=\inf_{n\ge 1} \frac{1}{n} H_\mu(\bigvee_0^{n-1}T^{-i}\alpha|\mathcal{C}).$$ The [*measure-theoretic entropy*]{} of $\mu$ with respect to $\mathcal{C}$ is defined as $$h_\mu(T|\mathcal{C})=\sup_{\alpha \in \mathcal{P}_X} h_\mu(T,\alpha|\mathcal{C}).$$ Let $\pi:(X,\mathcal{B},\mu,T)\rightarrow (Y,\mathcal{D},\nu,S)$ be a factor map between two MDSes. We define [*the conditional entropy*]{} of $\mu$ with respect to $\pi$ as $$h_\mu(T|\pi)=h_\mu(T|\pi^{-1}(\mathcal{D})).$$ The following result is a generalization of Abramov-Rohlin formula borrowed from [@BC]. \[GAR\] (Generalized Abramov-Rohlin formula) Let $\pi:(X,\mathcal{B},\mu,T)\rightarrow (Y,\mathcal{D},\nu,S)$ and $\psi:(Y,\mathcal{B},\nu,S)\rightarrow (Z,\mathcal{A},\lambda,R)$ be two factor maps between two MDSes. Then $\psi\circ \pi:(X,\mathcal{B},\mu,T)\rightarrow (Z,\mathcal{A},\lambda,R)$ is also a factor map between MDSes and $$h_\mu(T|\psi\circ \pi)=h_\mu(T|\pi)+h_\nu(S|\psi).$$ For a given Lebesgue system $(X,\mathcal{B},\mu,T)$, let $$\bar{X}=\{ \bar{x}=(x_i)_{i\in \mathbb{Z}}\in X^{\mathbb{Z}}: \, Tx_{i}=x_{i+1}, i\in \mathbb{Z}\}$$ and $(\bar{X},\bar{\mathcal{B}},\bar{\mu},\bar{T})$ be the natural extension of $(X,\mathcal{B},\mu,T)$ (see [@R Section 3.7]). More precisely, for $n\in \mathbb{Z}$, let $\Pi_{n,X}:\bar{X}\rightarrow X$ with $\Pi_{n,X}((x_i)_{i\in \mathbb{Z}})=x_n$ for $\bar{x}=(x_i)_{i\in \mathbb{Z}}\in X^{\mathbb{Z}}$. Set $$\bar{\mathcal{B}}_n=\Pi_{n,X}^{-1}(\mathcal{B}).$$ Clearly, $\bar{\mathcal{B}}_i\supseteq \bar{\mathcal{B}}_{i+1}$ for each $i\in \mathbb{Z}$. Let $\mathcal{D}_{\bar{X}}=\bigcup_{i\in \mathbb{Z}}\bar{\mathcal{B}}_i$. Then $\mathcal{D}_{\bar{X}}$ is a algebra of subsets of $\bar X$. The Lebesgue measure $\bar{\mu}$ on $\mathcal{D}_{\bar{X}}$ satisfies $\bar{\mu}(\Pi_{n,X}^{-1}(A))=\mu(A)$ for $A\in \mathcal{B}$ and $n\in \mathbb{Z}$. $\bar{\mathcal{B}}$ is the completion of the $\sigma$-algebra generated by $\mathcal{D}_{\bar{X}}$ with respect to $\bar{\mu}$. The self-map $\bar{T}$ defined on $\bar{X}$ by $$\bar{T}( (x_i)_{i\in \mathbb{Z}})=(Tx_{i})_{i\in \mathbb{Z}}=(x_{i+1})_{i\in \mathbb{Z}}$$ is an invertible measure-preserving map on Lebesgue space $(\bar{X},\bar{\mathcal{B}},\bar{\mu})$. Thus, $(\bar{X},\bar{\mathcal{B}},\bar{\mu},\bar{T})$ is an invertible Lebesgue system. Let $\Pi_X:=\Pi_{0,X}$. Then $\Pi_X:(\bar{X},\bar{\mathcal{B}},\bar{\mu},\bar{T})\rightarrow (X,\mathcal{B},\mu,T)$ is a factor map, which is called [*the natural extension*]{} of $(X,\mathcal{B},\mu,T)$. In [@R61] it is proved that $(\bar{X},\bar{\mathcal{B}},\bar{\mu},\bar{T})$ is ergodic if and only if $(X,\mathcal{B},\mu,T)$ is ergodic. The next lemma is on the entropy of the extended map conditional to the natural extension. Its proof is given in Appendix A. \[na=zero\] Let $\Pi_X:(\bar{X},\bar{\mathcal{B}},\bar{\mu},\bar{T})\rightarrow (X,\mathcal{B},\mu,T)$ be the natural extension of Lebesgue system $(X,\mathcal{B},\mu,T)$. Then $h_{\bar{\mu}}(\bar{T}|\Pi_X)=0$. Relative Pinsker $\sigma$-algebra --------------------------------- In this subsection, we recall some notations and results on the relative Pinsker $\sigma$-algebra. Let $(X,\mathcal{B},\mu,T)$ be an invertible Lebesgue system. A $T$-invariant sub-$\sigma$-algebra $\mathcal{F}$ (i.e., $T^{-1}\mathcal{F}=\mathcal{F}$) of $\mathcal{B}$ determines an invertible Lebesgue factor $(Y,\mathcal{D},\nu,S)$ of $(X,\mathcal{B},\mu,T)$, that is, there exists a factor map $\pi:(X,\mathcal{B},\mu,T)\rightarrow (Y,\mathcal{D},\nu,S)$ between two invertible Lebesgue systems such that $\pi^{-1}(\mathcal{D})=\mathcal{F}$. This factor is unique, up to isomorphism (see for example [@P69 Section 4.1]). For a factor map $\pi:(X,\mathcal{B},\mu,T)\rightarrow (Y,\mathcal{D},\nu,S)$ between two invertible Lebesgue systems, there is a set of conditional probability measures $\{\mu_y\}_{y\in Y}$ with the following properties: - $\mu_y$ is a Lebesgue measure on $X$ with $\mu_y(\pi^{-1}(y))=1$ for all $y\in Y$. - for each $f \in L^1(X,\mu)$, one has $f \in L^1(X,\mu_y)$ for $\nu$-a.e. $y\in Y$, the map $y \mapsto \int_X f\,d\mu_y$ is in $L^1(Y,\nu)$ and $\int_Y \left(\int_X f\,d\mu_y \right)\, d\nu(y)=\int_X f \,d\mu$. In this case, we say that $\mu=\int_Y \mu_y d \nu(y)$ is [*disintegration of $\mu$ relative to the factor $(Y,\mathcal{D},\nu,S)$*]{}. The measures $\{ \mu_y\}$ are essentially unique; that is, $\{ \mu_y\}$ and $\{ \mu_y'\}$ have the above properties, then $\mu_y=\mu_y'$ for $\nu$-a.e. $y\in Y$. In particular, it follows that $T\mu_y=\mu_{S y}$ for $\nu$-a.e. $y\in Y$. The conditional expectations and the conditional measures are related by $$\label{meas3} \mathbb{E}_{\mu}(f|\pi^{-1}\mathcal{D})(x)=\int_X f\,d\mu_{\pi(x)} \ \ \text{for $\mu$-a.e. } x\in X$$ for every $f\in L^1(X,\mathcal{B},\mu)$. The product of $(X,\mathcal{B},\mu,T)$ with itself relative to factor $(Y,\mathcal{D},\nu,S)$ is the MDS $$(X\times X,\mathcal{B}\times \mathcal{B},\mu \times_Y\mu, T\times T),$$ where the measure $$(\mu\times_Y \mu)(B)=\int_{Y}(\mu_y\times \mu_y)(B) \, d \nu(y), \ \forall B\in \mathcal{B}\times \mathcal{B}.$$ The measure $\mu\times_Y \mu$ is $T\times T$-invariant and is supported on $$R_\pi:=\{(x_1,x_2)\in X\times X:\pi(x_1)=\pi(x_2)\}$$ since each measure $\mu_y\times \mu_y$ is supported on $\pi^{-1}(y)\times \pi^{-1}(y)$. In the following, we are to introduce the relative Pinsker $\sigma$-algebra for a factor map between two invertible Lebesgue systems. First, we recall the relative Pinsker formula from [@P81 p.66]. \[RPF\] (Relative Pinsker formula) Let $(X,\mathcal{B},\mu,T)$ be an invertible Lebesgue systems and $\mathcal{A}$ be a sub-$\sigma$ algebra of $\mathcal{B}$ with $T^{-1}\mathcal{A}=\mathcal{A}$. Then for any $\alpha,\beta\in \mathcal{P}_X$, $$h_\mu(T,\alpha\vee \beta|\mathcal{A})=h_\mu(T,\beta|\mathcal{A})+h_\mu(T,\alpha|\beta^T\vee \mathcal{A})$$ where $\beta^T=\bigvee_{i\in \mathbb{Z}}T^{-i}\beta$. Let $\pi:(X,\mathcal{B},\mu,T)\rightarrow (Z,\mathcal{A},\lambda,R)$ be a factor map between two invertible Lebesgue systems. Put $$P_\mu(\pi)=\{ A\in \mathcal{B}: h_\mu(T,\{ A,X\setminus A\}|\pi^{-1}(\mathcal{A}))=0\}.$$ It is well known that from Lemma \[RPF\] it follows that $P_\mu(\pi)$ is the smallest sub-$\sigma$-algebra of $\mathcal{B}$ containing $\{ \alpha\in \mathcal{P}_X: h_\mu(T,\alpha|\pi^{-1}(\mathcal{A}))=0\}$ and $T^{-1}(P_\mu(\pi))=P_\mu(\pi)$. $P_\mu(\pi)$ is called the [*relative Pinsker $\sigma$-algebra*]{} of $(X,\mathcal{B},\mu,T)$ with respect to $\pi$. Note that $$\pi^{-1}(\mathcal{A})\subseteq P_\mu(\pi)\subseteq \mathcal{B}.$$ There exist an invertible Lebesgue system $(Y,\mathcal{D},\nu,S)$ and two factor maps $$\pi_1:(X,\mathcal{B},\mu,T)\rightarrow (Y,\mathcal{D},\nu ,S),\ \ \pi_2:(Y,\mathcal{D},\nu,S)\rightarrow (Z,\mathcal{A},\lambda,R)$$ such that $\pi_2\circ \pi_1=\pi$ and $\pi_1^{-1}(\mathcal{D})=P_\mu(\pi)$ (see for example [@P69]). The factor map $\pi_1:(X,\mathcal{B},\mu,T)\rightarrow (Y,\mathcal{D},\nu ,S)$ is called [*the Pinsker factor of $(X,\mathcal{B},\mu,T)$ with respect to $\pi$*]{}. The following result is well known (see for example [@BGKM Theorem 2.1 and Theorem 2.3] or [@ZGH Lemma 4.1]). \[key-lem\] Let $\pi:(X,\mathcal{B},\mu,T)\rightarrow (Z,\mathcal{A},\lambda,R)$ be a factor map between two invertible Lebesgue systems. Let $\pi_1:(X,\mathcal{B},\mu,T)\rightarrow (Y,\mathcal{D},\nu ,S)$ be the Pinsker factor of $(X,\mathcal{B},\mu,T)$ with respect to $(Z,\mathcal{A},\lambda,R)$ and $\mu=\int_Y \mu_y d \nu$ be the disintegration of $\mu$ relative to the factor $(Y,\mathcal{D},\nu,S)$. If $(X,\mathcal{B},\mu,T)$ is ergodic and $h_\mu(T|\pi)>0$, then 1. $\mu_y$ is non-atomic (that is $\mu_y(\{x\})=0$ for each $x\in X$) for $\nu$-a.e. $y\in Y$. 2. $(X\times X,\mathcal{B}\times \mathcal{B},\mu \times_Y\mu, T\times T)$ is ergodic. The following result is also well known (see for example [@HYZ Lemma 3.3]). \[key-lem-1\] Let $\pi:(X,\mathcal{B},\mu,T)\rightarrow (Z,\mathcal{A},\lambda,R)$ be a factor map between two invertible Lebesgue systems. Let $P_{\mu}(\pi)$ be the relative Pinsker $\sigma$-algebra of $(X,\mathcal{B},\mu,T)$ relative to $\pi$. If $\alpha \in \mathcal{P}_X$, then $$\lim_{m \rightarrow +\infty} h_\mu(T^m,\alpha|\pi^{-1}(\mathcal{A}))=H_\mu(\alpha|P_\mu(\pi)).$$ Entropy of compact Random set ----------------------------- In this subsection, we first introduce the entropy of compact random set for the random dynamical system. We then present the variational principle. Let $\phi(n, \omega, x)$ a random dynamical system (RDS) in a Polish space $X$ over the metric dynamical system $(\Omega, \mathcal{F},P,\theta)$. We assume that $\Omega$ is a Polish system. Note that $\phi(n, \omega, x)$ is generated by a random map $\phi(\omega)(x):=\phi(1, \omega, x)$. We assume $\phi(\omega):X\rightarrow X$ is continuous for $P$-a.e. $\omega\in \Omega$. Suppose that $\phi$ is a RDS on $X$. The map $$\Phi:\Omega\times X\rightarrow \Omega\times X,\ \ (\omega,x)\rightarrow (\theta \omega,\phi(\omega)x)$$ is said to be [*skew product*]{} induced by $\phi$. Let $\mathcal{F}\times \mathcal{B}_X$ be the smallest $\sigma$-algebra on $\Omega\times X$ with respect to which both the canonical projections $\Pi_X:\Omega\times X\rightarrow X$ and $\pi_\Omega:\Omega\times X\rightarrow \Omega$ are measurable. A probability measure $\mu$ on the measurable space $(\Omega\times X,\mathcal{F}\times \mathcal{B}_X)$ is said to have [*marginal $P$*]{} on $\Omega$ if $\mu\circ \pi_\Omega^{-1}=P$. Denote by $\mathcal{P}_P(\Omega\times X)$ the collection of such measures. Let $\mathcal{M}_P(\Omega\times X,f)$ denote the set of $\Phi$-invariant elements of $\mathcal{P}_P(\Omega\times X)$. If $(\Omega, \mathcal{F},P,\theta)$ is an ergodic MDS, then we may consider the set $\mathcal{E}_P(\Omega\times X)$ of ergodic elements in $\mathcal{M}_P(\Omega\times X,f)$. Let $K\in \mathcal{F}\times \mathcal{B}_X$ be a forward $\phi$-invariant random set with compact $\omega$-section $K(\omega)$. Then, there exists $\mu\in \mathcal{M}_P(\Omega\times X)$ that is supported on $K$, i.e., $\mu(K)=1$ (see [@C] and [@A Theorem 1.6.13]). For simplicity, we denote $$\mathcal{M}^K_P(\Omega\times X)=\{ \mu\in \mathcal{M}_P(\Omega\times X):\mu(K)=1\}.$$ Let $\mu\in \mathcal{M}^K_P(\Omega\times X)$. Then, there is a decomposition $d\mu(\omega,x)=d\mu_\omega(x) dP(\omega)$ of $\mu$ into its sample measures $\mu_\omega,\omega\in \Omega$ and $P$ (see [@A] and [@C Section 3]). $\mu_\omega(K_\omega)=1$ for $P$-a.e. $\omega\in \Omega$. Since $K$ is a Borel subset of $\Omega\times X$, $K$ is also Polish space and the Borel $\sigma$-algebra $\mathcal{B}_K$ of $K$ is just $\{ A\cap K: A\in \mathcal{F}\times \mathcal{B}_X\}$. Thus $(K,\mathcal{B}_K,\mu, \Phi)$ is a Polish system and $\pi_\Omega: (K,\mathcal{B}_K,\mu, \Phi)\rightarrow (\Omega, \mathcal{F},P,\theta)$ is a factor map bewteen two Polish systems. [*The entropy of RDS $\phi$*]{} with respect to $\mu$ is defined by $$h_\mu(\phi):=h_\mu(\Phi|\pi_\Omega)=h_\mu(\Phi|\pi_\Omega^{-1}(\mathcal{F})).$$ That is, the entropy of RDS $\phi$ with respect to $\mu$ is the entropy of $(K,\mathcal{B}_K,\mu, \Phi)$ with respect to $\pi_\Omega^{-1}(\mathcal{F})$. By Bogenschutz [@B], $$h_\mu(\phi)=\sup\{ h_\mu(\Phi,\pi^{-1}_X\alpha|\pi_\Omega^{-1}(\mathcal{F})):\alpha \text{ is a finite Borel partition of }X\}.$$ If in addition that $P$ is ergodic, then $$\Gamma=\{ \mu\in \mathcal{P}_P(\Omega\times X):\mu(K)=1,\ \mu \text{ invariant for }\Phi\}$$ is a compact (in the narrow topology of $\mathcal{P}_P(\Omega\times X)$, which is a metrizable topology) and convex, and its extremal points are ergodic by Lemma 6.19 in [@C], particularly there exists $\mu\in \mathcal{E}_P(\Omega\times X)$ that is supported on $K$. We use $\mathcal{E}^K_P(\Omega\times X)$ to denote the set of the ergodic elements of $ \mathcal{M}^K_P(\Omega\times X)$. Then $\mathcal{E}^K_P(\Omega\times X)$ is the set of extremal points of $ \mathcal{M}^K_P(\Omega\times X)$, which is a Borel subset of $ \mathcal{M}^K_P(\Omega\times X)$. Using the approach of Kifer [@K01] and Lemma \[ergodic\], one has the following result. \[VP\] (Variational principle) Let $\phi$ be a continuous RDS on Polish space $X$ with Borel $\sigma$-algebra $\mathcal{B}_X$ over Polish system $(\Omega, \mathcal{F},P,\theta)$. Let $K$ be a forward $\phi$-invariant random set with compact $\omega$-section $K(\omega)$. Then $$h_{\text{top}}(\phi,K)=\sup \{h_\mu(\phi):\mu\in \mathcal{M}_P^K(\Omega\times X,\phi)\}.$$ If in addition $(\Omega, \mathcal{F},P,\theta)$ is ergodic, then $$h_{\text{top}}(\phi,K)=\sup \{h_\mu(\phi):\mu\in \mathcal{E}_P^K(\Omega\times X,\phi)\}.$$ Proof of Theorem \[MTH1\] ========================= In this section, we first introduce a combinatorial lemma and some results on relative Pinsker $\sigma$-algebra and conditional entropy of a finite measurable partition. We then prove Theorem \[MTH1\]. Condition Entropy and a combinatorial lemma ------------------------------------------- Let $\pi:(X,\mathcal{B},\mu,T)\rightarrow (Z,\mathcal{A},\lambda,R)$ be a factor map between two invertible Lebesgue systems. Let $\pi_1:(X,\mathcal{B},\mu,T)\rightarrow (Y,\mathcal{D},\nu ,S)$ be the Pinsker factor of $(X,\mathcal{B},\mu,T)$ with respect to $(Z,\mathcal{A},\lambda,R)$ and $\mu=\int_Y \mu_y d \nu$ be the disintegration of $\mu$ relative to the factor $(Y,\mathcal{D},\nu,S)$. It is well known that for $\nu$-a.e. $y\in Y$, $\mu_y(\pi_1^{-1}(y))=1$. Given $\ell\in \mathbb{N}$, let $\alpha$ be a finite measurable partition of $X$. Define a function $$h_\mu(T^\ell,\alpha,y):=\lim_{n\rightarrow +\infty}H_{\mu_y}(\alpha|\bigvee_{i=1}^n T^{-i\ell}\alpha).$$ Then $h_\mu(T^\ell,\alpha,y)$ is a measurable function on $Y$ and $h_\mu(T^\ell,\alpha,y)\le \log \#(\alpha)$. We have \[lem-5.1\] $\int_Y h_\mu(T^\ell, \alpha,y)d \nu(y)=h_\mu(T^\ell,\alpha|\pi^{-1}(\mathcal{A}))$. First, we show that $$\begin{aligned} \label{peq-en} h_\mu(T^\ell,\alpha|\pi^{-1}(\mathcal{A}))=h_\mu(T^\ell,\alpha|P_\mu(\pi)).\end{aligned}$$ Since $\pi^{-1}(\mathcal{A})\subseteq P_\mu(\pi)$, $h_\mu(T^\ell,\alpha|\pi^{-1}(\mathcal{A}))\ge h_\mu(T^\ell,\alpha|P_\mu(\pi))$. Conversely, let $\beta_k\nearrow P_\mu(\pi)$ be an increasing sequence of finite measurable partitions of $X$. Since $\beta_k\subseteq P_\mu(\pi)$, $h_\mu(T,\beta_k|\pi^{-1}(\mathcal{A}))=0$ for $k\in \mathbb{N}$. Moreover, $$\begin{aligned} h_\mu(T^\ell,\beta_k|\pi^{-1}(\mathcal{A}))&\le h_\mu(T^\ell,\bigvee_{j=0}^{\ell-1}T^{-j}\beta_k|\pi^{-1}(\mathcal{A})) =\ell h_\mu(T,\beta_k|\pi^{-1}(\mathcal{A}))=0\end{aligned}$$ for $k\in \mathbb{N}$. Hence, $h_\mu(T^\ell,\beta_k|\pi^{-1}(\mathcal{A}))=0$ for $k\in \mathbb{N}$. Now by Lemma \[RPF\], for $m,k\in \mathbb{N}$ we have $$\begin{aligned} h_\mu(T^\ell,\alpha|\pi^{-1}(\mathcal{A}))&\le h_\mu(T^{\ell},\alpha\vee \beta_k|\pi^{-1}(\mathcal{A}))\\ &=h_\mu(T^\ell,\beta_k|\pi^{-1}(\mathcal{A}))+h_\mu(T^\ell,\alpha|\bigvee_{t\in \mathbb{Z}}T^{t\ell}\beta_k\vee \pi^{-1}(\mathcal{A}))\\ &=h_\mu(T^\ell,\alpha|\bigvee_{t\in \mathbb{Z}}T^{t\ell}\beta_k\vee \pi^{-1}(\mathcal{A}))\le h_\mu(T^\ell,\alpha|\bigvee_{t\in \mathbb{Z}}T^{t\ell}\beta_k)\\ &=\inf_{n\ge 1} \frac{1}{n}H_\mu(\bigvee_{i=0}^{n-1}T^{-i\ell}\alpha|\bigvee_{t\in \mathbb{Z}}T^{t\ell}\beta_k)\le \frac{1}{m}H_\mu(\bigvee_{i=0}^{m-1}T^{-i\ell}\alpha|\bigvee_{t\in \mathbb{Z}}T^{t\ell}\beta_k)\\ &\le \frac{1}{m}H_\mu(\bigvee_{i=0}^{m-1}T^{-i\ell}\alpha|\beta_k).\end{aligned}$$ Fixing $m\in \mathbb{N}$ and letting $k\nearrow+\infty$ in the above inequality we have $$h_\mu(T^\ell,\alpha|\pi^{-1}(\mathcal{A}))\le \frac{1}{m}H_\mu(\bigvee_{i=0}^{m-1}T^{-i\ell}\alpha|P_\mu(\pi))$$ since $H_\mu(\bigvee_{i=0}^{m-1}T^{-i\ell}\alpha|\beta_k)\searrow H_\mu(\bigvee_{i=0}^{m-1}T^{-i\ell}\alpha|P_\mu(\pi))$ when $k\nearrow+\infty$. Then, letting $m\rightarrow +\infty$, we ontain $h_\mu(T^\ell,\alpha|\pi^{-1}(\mathcal{A})) \le h_\mu(T^{\ell},\alpha|P_\mu(\pi))$. Thus, we have the equality . Next, let $a_n=\int_Y H_{\mu_y}(\bigvee_{i=0}^{n-1}T^{-i\ell}\alpha)d \nu(y)$. Since $T\mu_y=\mu_{Sy}$ for $\nu$-a.e. $y\in Y$ and $\nu$ is $S$-invariant, we have $$\begin{aligned} a_n&=\int_Y H_{\mu_{Sy}}(\bigvee_{i=0}^{n-1}T^{-i\ell}\alpha)d \nu(y)=\int_Y H_{T\mu_{y}}(\bigvee_{i=0}^{n-1}T^{-i\ell}\alpha)d \nu(y) \\ &=\int_Y H_{\mu_{y}}(T^{-1}(\bigvee_{i=0}^{n-1}T^{-i\ell}\alpha))d \nu(y)=\int_Y H_{\mu_{y}}(\bigvee_{i=1}^{n}T^{-i\ell}\alpha)d \nu(y).\end{aligned}$$ Moreover by the monotone convergence Theorem, we have $$\begin{aligned} \lim_{n\rightarrow +\infty} (a_{n+1}-a_n)&=\lim_{n\rightarrow +\infty}\int_Y H_{\mu_y}(\alpha|\bigvee_{i=1}^n T^{-i\ell}\alpha)d\nu(y)\\ &=\int_Y \lim_{n\rightarrow +\infty}H_{\mu_y}(\alpha|\bigvee_{i=1}^n T^{-i\ell}\alpha)d \nu(y)\\ &=\int_Y h_\mu(T^\ell, \alpha,y)d \nu(y).\end{aligned}$$ Moreover, $$\lim_{n\rightarrow +\infty}\frac{a_n}{n}=\lim_{n\rightarrow +\infty} (a_{n+1}-a_n)=\int_Y h_\mu(T^\ell, \alpha,y)d \nu(y).$$ By , we have $a_n=H_\mu(\bigvee_{i=0}^{n-1}T^{-i\ell}\alpha|P_\mu(\pi))$. Combing this with , $$\begin{aligned} h_\mu(T^\ell,\alpha|\pi^{-1}(\mathcal{A}))&=h_\mu(T^\ell,\alpha|P_\mu(\pi))=\lim_{n\rightarrow +\infty}\frac{a_n}{n}\\ &=\int_Y h_\mu(T^\ell, \alpha,y)d \nu(y).\end{aligned}$$ This completes the proof of the lemma. The following lemma is taken from [@R] (Lemma 3’ in $\S$4 No.2). \[rohlin\] Suppose $\mu_y$ is non-atomic for $\nu$-a.e. $y\in Y$. If $B$ is a measurable set $X$ with $\mu_y(B)\ge r>0$ for $\nu$-a.e. $y\in Y$, then for any $0\le s\le r$ there exists a measurable set $B_s$ such that $B_s\subseteq B$ and $\mu_y(B_s)=s$ for $\nu$-a.e. $y\in Y$. \[lem-5.3\] Suppose $\mu_y$ is non-atomic for $\nu$-a.e. $y\in Y$. If $U_1,U_2\in \mathcal{B}$ with $\mu\times_Y \mu(U_1\times U_2)>0$, then there exist a measurable set $A\subset Y$ with $\nu(A)>0$, a positive integer $r>2$ and a measurable partition $\alpha=\{ B_1,B_2,\cdots,B_r\}$ of $X$ such that $\pi^{-1}_1(A)\cap B_i\subseteq U_i$, $i=1,2$ and $\mu_y(B_j)=\frac{1}{r}$, $j=1,2,\cdots,r$, for $\nu$-a.e. $y\in Y$. Set $C_i=\{ y\in Y: \mu_y(U_i)>0\}$, $i=1,2$. Since $$0<\mu\times_Y \mu(U_1\times U_2)=\int_Y \mu_y(U_1)\times \mu_y(U_2) d \nu(y),$$ we have $\nu(C_1\cap C_2)>0$. Thus there exist a positive integer $r>2$ and a measurable set $A\subseteq C_1\cap C_2$ such that $\nu(A)>0$ and $\mu_y(U_i)\ge \frac{2}{r}$ for any $y\in A$, $i=1,2$. Next we construct $B_j$, $j=1,2,\cdots,r$ by induction. Taking $$D_1=\pi_1^{-1}(Y\setminus A)\cup (\pi^{-1}_1(A)\cap U_1),$$ then $\mu_y(D_1)\ge \frac{2}{r}$ for $\nu$-a.e. $y\in Y$. By Lemma \[rohlin\], there exists a measurable set $B_1\subseteq D_1$ such that $\mu_y(B_1)=\frac{1}{r}$ for $\nu$-a.e. $y\in Y$, and $B_1\cap \pi^{-1}_1(A) \subseteq D_1\cap \pi^{-1}_1(A)\subseteq U_1$. Next, by the induction hypothesis, i.e., there are measurable sets $B_k$, $(1\le k\le r)$ satisfying - $B_k\cap B_i=\emptyset$ for $1\le i\le k-1$; - $\mu_y(B_k)=\frac{1}{r}$ for $\nu$-a.e. $y\in Y$; - $B_k\cap \pi_1^{-1}(A)\subseteq U_k$ (when $k>2$, we set $U_k=X$). If $k=r$, we are done. If $k<r$, we set $$D_{k+1}=\begin{cases} \pi_1^{-1}(X\setminus A)\cup (\pi^{-1}_1(A)\cap (U_{2})\setminus B_1) \ \ &\text{ if }k=1 \\ X\setminus \bigcup_{i=1}^k B_i \ \ &\text{ if }k>1. \end{cases}$$ It is clear that $\mu_y(D_{k+1})\ge \frac{1}{r}$ for $\nu$-a.e. $y\in Y$. Using Lemma \[rohlin\] again, there exists a measurable set $B_{k+1}\subseteq D_{k+1}$ such that $\mu_y(B_{k+1})=\frac{1}{r}$ for $\nu$-a.e. $y\in Y$. Then $B_{k+1}$ satisfies $(1)_{k+1}$, $(2)_{k+1}$ and $(3)_{k+1}$. Therefore, by induction, we obtain a measurable partition $\alpha=\{ B_1,B_2,\cdots,B_r\}$ of $X$ such that $\mu_y(B_k)=\frac{1}{r}$, $k=1,2,\cdots,r$, for $\nu$-a.e. $y\in Y$ and $B_i\cap \pi_1^{-1}(A)\subseteq U_i$, $i=1,2$. This completes the proof of the lemma. To prove our main theorem \[MTH1\], we also need the following consequence of Karpovsky-Milman-Alon’s generalization of the Sauer-Perles-Shelah lemma [@Al; @KM; @Sau; @Sh]. ([@KM]) \[com-lem\] Given $r\ge 2$ and $\lambda > 1$ there is a constant $e > 0$ such that, for all $n \in \mathbb{N}$, if $S \subseteq \{1, 2, \cdots , r\}^{\{1,2,\cdots,n\}}$ satisfies $|S|\ge ((r-1)\lambda)^n$ then there is an $I\subset \{1, 2,\cdots , n\}$ with $|I|\ge en$ and $S|_I = \{1, 2, \cdots, r\}^I$, i.e., for any $u\in \{1,2,\cdots,r\}^I$ there is $s\in S$ with $s(j)=u(j)$ for any $j\in I$. Proof of Theorem \[MTH1\] ------------------------- Assume that $ h_{top}(\phi,K)>0$. By Proposition \[VP\] there exists $\mu\in \mathcal{E}_P^K(\Omega\times X)$ such that $h_\mu(\phi)>0$. Since $K$ is a Borel subset of $\Omega\times X$, $K$ is also Polish space and the Borel $\sigma$-algebra $\mathcal{B}_K$ of $K$ is just $\{ A\cap K: A\in \mathcal{F}\times \mathcal{B}_X\}$. Thus $(K,\mathcal{B}_K,\mu, \Phi)$ is an ergodic Polish system and $\pi_\Omega: (K,\mathcal{B}_K,\mu, \Phi)\rightarrow (\Omega, \mathcal{F},P,\theta)$ is a factor map between two Polish systems. Recall that the entropy of RDS $\phi$ with respect to $\mu$ is given by $$h_\mu(\phi):=h_\mu(\Phi|\pi_\Omega)=h_\mu(\Phi|\pi_\Omega^{-1}(\mathcal{F})).$$ Let $\mathcal{B}_P$ be the completion of $\mathcal{F}$ with respect to $P$. Let $\mathcal{B}_\mu$ be the completion of $\mathcal{B}_K$ with respect to $\mu$. Then $\pi_\Omega:(K,\mathcal{B}_\mu,\mu, \Phi)\rightarrow (\Omega, \mathcal{B}_P,P,\theta)$ is a factor map between two Lebesgue systems. Since $\mathcal{B}_P=\mathcal{F}$ (mod $P$) and $\mathcal{B}_\mu=\mathcal{B}_K$ (mod $\mu$), it is clear that $$h_\mu(\Phi|\pi_\Omega^{-1}(\mathcal{B}_P))=h_\mu(\Phi|\pi_\Omega^{-1}(\mathcal{F}))=h_\mu(\phi).$$ Let $\Pi_K:(\bar{K},\bar{\mathcal{B}_{\mu}},\bar{\mu}, \bar{\Phi})\rightarrow (K,\mathcal{B}_\mu,\mu, \Phi)$ be the natural extension of $(K,\mathcal{B}_\mu,\mu, \Phi)$. Let $$\Pi_{\Omega}: (\bar{\Omega}, \bar{\mathcal{B}_{P}},\bar{P},\bar{\theta})\rightarrow (\Omega, \mathcal{B}_P,P,\theta)$$ be the natural extension of $(\Omega, \mathcal{B}_P,P,\theta)$. Define $\bar{\pi}:\bar{K}\rightarrow \bar{\Omega}$ by $\bar{\pi}(((\omega_i,x_i))_{i\in \mathbb{Z}})=(\omega_i)_{i\in \mathbb{Z}}$ for $((\omega_i,x_i))_{i\in \mathbb{Z}}\in \bar{K}$. Then $\bar{\pi}:(\bar{K},\bar{\mathcal{B}_{\mu}},\bar{\mu},\bar{\Phi})\rightarrow (\bar{\Omega},\bar{\mathcal{B}_{P}},\bar{P},\bar{\theta})$ is a factor map between two invertible Lebesgue systems and the following diagram is commutative: $$\begin{CD} (K,\mathcal{B}_\mu,\mu,\Phi) @< \Pi_K<< (\bar{K},\bar{\mathcal{B}_{\mu}},\bar{\mu},\bar{\Phi})\\ @VV\pi_\Omega V @VV\bar{\pi} V \\ (\Omega,\mathcal{B}_P,P,\theta) @< \Pi_\Omega<< (\bar{\Omega},\bar{\mathcal{B}_{P}},\bar{P},\bar{\theta})\\ \end{CD}$$ Now, we show $$\begin{aligned} \label{na=en-1} h_{\bar{\mu}}(\bar{\Phi}|\bar{\pi})=h_\mu(\phi)>0.\end{aligned}$$ In fact, since $\pi_{\Omega}\circ \Pi_K=\Pi_\Omega \circ \bar{\pi}$, we have $$\begin{aligned} \label{main-eq-1} h_{\bar{\mu}}(\bar{\Phi}|\pi_{\Omega}\circ \Pi_K)=h_{\bar{\mu}}(\bar{\Phi}|\Pi_\Omega\circ \bar{\pi}).\end{aligned}$$ By Lemma \[GAR\] and Lemma \[na=zero\], we have $$\begin{aligned} \label{main-eq-2} h_{\bar{\mu}}(\bar{\Phi}|\pi_{\Omega}\circ \Pi_K)=h_{\bar{\mu}}(\bar{\Phi}|\Pi_K)+h_\mu(\Phi|\pi_{\Omega})=h_\mu(\Phi|\pi_{\Omega})=h_\mu(\phi) \end{aligned}$$ and $$\begin{aligned} \label{main-eq-3} h_{\bar{\mu}}(\bar{\Phi}|\Pi_\Omega\circ \bar{\pi})=h_{\bar{\mu}}(\bar{\Phi}|\bar{\pi})+h_{\bar{P}}(\bar{\theta}|\Pi_\Omega)=h_{\bar{\mu}}(\bar{\Phi}|\bar{\pi}).\end{aligned}$$ Using , and , we get . Next, let $P_{\bar{\mu}}(\bar{\pi})$ be the relative Pinsker $\sigma$-algebra of $(\bar{K},\bar{\mathcal{B}_{\mu}},\bar{\mu},\bar{\Phi})$ relative to $\bar{\pi}$. Since ${\bar{\Phi}}^{-1}(P_{\bar{\mu}}(\bar{\pi}))=P_{\bar{\mu}}(\bar{\pi})$ and $$\bar{\pi}^{-1}(\mathcal{\bar B}_{{P}})\subseteq P_{\bar{\mu}}(\bar{\pi})\subseteq \bar{\mathcal{B}_{\mu}},$$ there exist an invertible Lebesgue system $(Y,\mathcal{D},\nu,S)$ and two factor maps $$\pi_1:(\bar{K},\bar{\mathcal{B}_{\mu}},\bar{\mu},\bar{\Phi})\rightarrow (Y,\mathcal{D},\nu,S),\ \pi_2:(Y,\mathcal{D},\nu,S)\rightarrow (\bar{\Omega},\bar{\mathcal{B}_{P}},\bar{P},\bar{\theta})$$ between invertible Lebesgue systems such that $\pi_2\circ \pi_1=\bar{\pi}$ and $\pi_1^{-1}(\mathcal{D})=P_{\bar{\mu}}(\bar{\pi})$ (mod $\bar{\mu}$). That is, $\pi_1$ is the Pinsker factor of $(\bar{K},\bar{\mathcal{B}_{\mu}},\bar{\mu},\bar{\Phi})$ relative to $\bar{\pi}$. Let $\bar{\mu}=\int_Y \bar{\mu}_y d \nu(y)$ be the disintegration of $\bar{\mu}$ relative to the factor $(Y,\mathcal{D},\nu,S)$. Let $$(\bar{K}\times \bar{K},\bar{\mathcal{B}_{\mu}}\times \bar{\mathcal{B}_{\mu}},\bar{\mu} \times_Y\bar{\mu}, \bar{\Phi}\times \bar{\Phi})$$ be the product of $(\bar{K},\bar{\mathcal{B}_{\mu}},\bar{\mu},\bar{\Phi})$ with itself relative to a factor $(Y,\mathcal{D},\nu,S)$. Recall that $$(\bar{\mu}\times_Y \bar{\mu})(B)=\int_{Y}(\bar{\mu}_y\times \bar{\mu}_y)(B) \, d \nu(y), \ B\in \bar{\mathcal{B}_{\mu}}\times \bar{\mathcal{B}_{\mu}}.$$ Since $(K,\mathcal{B}_{\mu},\mu,\Phi)$ is ergodic, we have that $(\bar{K},\bar{\mathcal{B}_{\mu}},\bar{\mu},\bar{\Phi})$ is ergodic. By , $h_{\bar{\mu}}(\bar{\Phi}|\bar{\pi})>0$. By Lemma \[key-lem\], we have 1. $\bar{\mu}_y$ is non-atomic for $\nu$-a.e. $y\in Y$. 2. $(\bar{K}\times \bar{K},\bar{\mathcal{B}_{\mu}}\times \bar{\mathcal{B}_{\mu}},\bar{\mu} \times_Y\bar{\mu}, \bar{\Phi}\times \bar{\Phi})$ is ergodic. Since $(X,d)$ is a separable metric space, there exists a countable dense set $X'$ of $X$. Let $\Gamma=\{ \overline{B}(x,t):x\in X' \text{ and } t\in \mathbb{Q}, t>0\}$, where $\overline{B}(x,t)=\{z\in X: d(z,x)\le t\}$. Clearly, $\Gamma$ is a countable set and each element of $\Gamma$ is a non-empty closed set of $X$. Let $\Theta=\{ \{U_1,U_2\}: U_1,U_2\in \Gamma, d(U_1,U_2)>0\}$. Then $\Theta$ is also a countable set. For $U\subseteq X$, we define $$\begin{aligned} \label{blde} \widetilde{U}=\{ ((\omega_i,x_i))_{i\in \mathbb{Z}}\in \overline{K}: x_0\in U\}.\end{aligned}$$ Recall that $\Delta_{\bar{K}}=\{ (\bar{k},\bar{k}):\bar{k}\in \bar{K}\}$. It is clear that when $\bar{\mu}_y$ is non-atomic, i.e., $\bar{\mu}_y(\{ \bar{k}\})=0$ for $\bar{k}\in \bar{K}$, we have $$\bar{\mu}_y\times \bar{\mu}_y(\Delta_{\bar{K}})=\int_{\bar{K}}\left( \int_{\bar{K}}1_{\Delta_{\bar{K}}}(\bar{k}_1,\bar{k}_2) d\bar{\mu}_y(\bar{k}_2) \right) d\bar{\mu}_y(\bar{k}_1)=\int_{\bar{K}}\bar{\mu}_y(\{\bar{k}_1\}) d \bar{\mu}_y(\bar{k}_1)=0$$ where $1_{\Delta_{\bar{K}}}$ is the characterization function of $\Delta_{\bar{K}}$. By (a1), for $\nu$-a.e. $y\in Y$, $\bar{\mu}_y$ is non-atomic. Thus for $\nu$-a.e. $y\in Y$, $\bar{\mu}_y\times \bar{\mu}_y(\Delta_{\bar{K}})=0$. Moreover, $$\begin{aligned} \bar{\mu}\times_Y \bar{\mu}(\Delta_{\bar{K}})=\int_Y \bar{\mu}_y\times \bar{\mu}_y(\Delta_{\bar{K}}) d\nu(y)=0.\end{aligned}$$ Thus $\bar{\mu}\times_Y \bar{\mu}(\bar{K}\times \bar{K}\setminus\Delta_{\bar{K}})=1$. Moreover, since $\bar{K}\times \bar{K}\setminus\Delta_{\bar{K}}=\bigcup_{i\in \mathbb{Z}}\bigcup_{\{U_1,U_2\}\in \Theta} (\bar{\Phi}\times \bar{\Phi})^i \left( \widetilde{U_1}\times \widetilde{U_2} \right)$ and $\bar{\mu}\times_Y \bar{\mu}$ is $\bar{\Phi}\times \bar{\Phi}$-invariant, there exists $\{U_1,U_2\}\in \Theta$ such that $\bar{\mu}\times_Y \bar{\mu}(\widetilde{U_1}\times \widetilde{U_2})>0$. In the following we want to show that $\{U_1,U_2\}$ is a weak Horseshoe of $(\phi,K)$. First, it is clear that $U_1$ and $U_2$ are non-empty, closed and bounded subsets of $X$ and $d(U_1,U_2)>0$ by the definition of $\Theta$. We divide the remainder of the proof into three steps. [*Step 1.*]{} Since $\bar{\mu}_y$ is non-atomic for $\nu$-a.e. $y\in Y$ and $\bar{\mu}\times_Y \bar{\mu}(\widetilde{U_1}\times \widetilde{U_2})>0$, by Lemma \[lem-5.3\], there exist a measurable set $A\subset Y$ with $\nu(A)>0$, a positive integer $r>2$, and a measurable partition $\alpha=\{ B_1,B_2,\cdots,B_r\}$ of $\bar{K}$ such that $\pi^{-1}_1(A)\cap B_i\subseteq \widetilde{U_i}$, $i=1,2$ and $\bar{\mu}_y(B_j)=\frac{1}{r}$, $j=1,2,\cdots,r$, for $\nu$-a.e. $y\in Y$. By Lemma \[key-lem-1\] $$\lim_{m\rightarrow +\infty} h_{\bar{\mu}}(\bar{\Phi}^m,\alpha|\bar{\pi}^{-1}(\bar{\mathcal{B}_P}))=H_{\bar{\mu}}(\alpha|P_{\bar{\mu}}(\bar{\pi})) =\sum_{j=1}^r \int_Y -\bar{\mu}_y(B_j)\log \bar{\mu}_y(B_j) d \nu(y)=\log r.$$ Thus, there is an $\ell>0$ such that $h_{\bar{\mu}}(\bar{\Phi}^\ell,\alpha|\bar{\pi}^{-1}(\bar{\mathcal{B}_P}))>\nu(Y\setminus A) \cdot \log r+\nu(A)\cdot \log (r-1)$. Let $c:=\frac{1}{3}\left(h_{\bar{\mu}}(\bar{\Phi}^\ell,\alpha|\bar{\pi}^{-1}(\bar{\mathcal{B}_P}))-\left( \nu(Y\setminus A) \cdot \log r+\nu(A)\cdot \log (r-1)\right)\right)$. Then $c>0$. Recall that we defined $\nu$-a.e. on $Y$ a function $$h_{\bar{\mu}}(\bar{\Phi}^\ell,\alpha,y):=\lim_{n\rightarrow +\infty}H_{\bar{\mu}_y}(\alpha|\bigvee_{i=1}^n \bar{\Phi}^{-i\ell}\alpha).$$ By Lemma \[lem-5.1\], we have $\int_Y h_{\bar{\mu}}(\bar{\Phi}^\ell,\alpha,y)d\nu(y)=h_{\bar{\mu}}(\bar{\Phi}^\ell,\alpha|\bar{\pi}^{-1}(\bar{\mathcal{B}_P}))$. For $y\in Y$, we set $$\delta(y):=(h_{\bar{\mu}}(\bar{\Phi}^\ell,\alpha,y)-\log (r-1))\cdot 1_A(y).$$ Since $0\le h_{\bar{\mu}}(\bar{\Phi}^\ell,\alpha,y)\le \log r$, $\delta(y)$ is a bounded measurable function on $Y$ and $$\begin{aligned} \label{key-yle} 1_A(y)\ge \frac{\delta (y)}{\log (\frac{r}{r-1})}.\end{aligned}$$ Now, $$\begin{aligned} \int_Y \delta(y) d\nu(y)&=\int_A h_{\bar{\mu}}(\bar{\Phi}^\ell,\alpha,y)d\nu(y)-\nu(A)\log (r-1)\\ &=\int_Y h_{\bar{\mu}}(\bar{\Phi}^\ell,\alpha,y)d\nu(y)- \int_{Y\setminus A} h_{\bar{\mu}}(\bar{\Phi}^\ell,\alpha,y)d\nu(y)-\nu(A)\log (r-1)\\ &\ge h_{\bar{\mu}}(\bar{\Phi}^\ell,\alpha|\bar{\pi}^{-1}(\bar{\mathcal{B}_P}))- \nu(Y\setminus A) \log r-\nu(A)\log (r-1)\\ &=3c.\end{aligned}$$ By the Birkhoff ergodic Theorem, $\frac{1}{m}\sum_{i=0}^{m-1}1_A(S^{i\ell}y)$ converges $\nu$-a.e. to a function $1_A^*\in L^1(\nu)$; $\frac{1}{m}\sum_{i=0}^{m-1}\delta(S^{i\ell}y)$ converges $\nu$-a.e. to a function $\delta^*\in L^1(\nu)$ and $\int_Y \delta^*(y) d \nu(y)=\int_Y \delta(y) d \nu(y)\ge 3c$. Note that for any $i\in \mathbb{Z}$, $\bar{\Phi}^i \bar{\mu}_y=\bar{\mu}_{S^iy}$ and $\bar{\Phi}^i \alpha \subseteq \bar{\mathcal{B}}_{\bar{\mu}_y}$ for $\nu$-a.e. $y\in Y$, where $\bar{\mathcal{B}}_{\bar{\mu}_y}$ is the completion of $\bar{\mathcal{B}_{\mu}}$ under $\bar{\mu}_y$. Next we define the measurable subset $D$ of $Y$ such that $y\in D$ if and only if the following holds - $\lim_{m\rightarrow +\infty}\frac{1}{m}\sum_{i=0}^{m-1}1_A(S^{i\ell}y)=1_A^*(y)$; - $\lim_{m\rightarrow +\infty}\frac{1}{m}\sum_{i=0}^{m-1}\delta(S^{i\ell}y)=\delta^*(y)\ge 2c$; - for any $i\in \mathbb{Z}$, $\bar{\Phi}^i \bar{\mu}_y=\bar{\mu}_{S^iy}$ and $\bar{\Phi}^i \alpha \subseteq \bar{\mathcal{B}}_{\bar{\mu}_y}$. Since $\int_Y \delta^*(y)d\nu(y)\ge 3c$, one has $\nu(D)>0$. For $y\in D$, let $$\mathcal{S}(y)=\{ \ell k: k\in \mathbb{Z}_+, S^{\ell k}y\in A\}:=\{ a_1(y)<a_2(y)<\cdots\}.$$ Then $S^{a_i(y)}y\in A$ for $i=1,2,\cdots$. Put $c_1=\frac{c}{\ell \log (\frac{r}{r-1})}$. Using , $$\begin{aligned} \label{density} \begin{aligned} &\hskip0.5cm \lim_{m\rightarrow +\infty} \frac{|\mathcal{S}(y)\cap \{0,1,\cdots,m-1\}|}{m}\\ &=\lim_{m\rightarrow +\infty} \frac{1}{\ell m} \sum_{i=0}^{m-1}1_A(S^{i\ell}y)\ge \lim_{m\rightarrow +\infty} \frac{1}{\ell m} \sum_{i=0}^{m-1}\frac{\delta(S^{i\ell}y)}{\log (\frac{r}{r-1})}\\ &=\frac{\delta^*(y)}{\ell \log (\frac{r}{r-1})}\ge \frac{3c}{\ell \log (\frac{r}{r-1})}\\ &=3c_1>0. \end{aligned}\end{aligned}$$ Let $E_0=D$ and $E_i=S^{-i}D\setminus \bigcup_{j=0}^{i-1}S^{-j}D$ for $i=1,2,\cdots$. Since $(Y,\mathcal{D},\nu,S)$ is ergodic and $\nu(D)>0$, $E:=\bigcup_{i=0}^{+\infty}S^{-i}D=\bigcup_{i=0}^{+\infty}E_i$ has full measure, i.e., $\nu(E)=1$. [*Step 2.*]{} Given $z\in E$. There exists unique $i\in \{0,1,2,\cdots\}$ and unique $y\in D$ such that $z\in E_i$ and $z=S^{-i}(y)$. Put $a_j(z)=i+a_j(y)$ for $j=1,2,\cdots$ and $\mathcal{S}(z)=\{a_1(z)<a_2(z)<\cdots\}$. Clearly, $$\begin{aligned} \label{s-density} \lim_{m\rightarrow +\infty} \frac{|\mathcal{S}(z)\cap \{0,1,\cdots,m-1\}|}{m}=\lim_{m\rightarrow +\infty} \frac{|\mathcal{S}(y)\cap \{0,1,\cdots,m-1\}|}{m}\ge 3c_1>0\end{aligned}$$ by . For $k\in \mathbb{N}$, let $$M_z(k):=\{ (j_1,j_2,\cdots,j_k)\in \{1,2,\cdots,r\}^{\{1,2,\cdots,k\}}:\bar{\mu}_z(\pi_1^{-1}(z)\cap (\bigcap_{t=1}^k \bar{\Phi}^{-a_t(z)}B_{j_t}))>0\}.$$ Since $\bar{\mu}_z(\pi_1^{-1}(z))=1$, $\bar{\Phi}^u \bar{\mu}_y=\bar{\mu}_{S^uy}$ for any $u\in \mathbb{Z}$ and $\ell |a_t(y)$ for any $t\in \mathbb{N}$, we have $$\begin{aligned} \log |M_z(k)|&\ge H_{\bar{\mu}_z}(\bigvee_{t=1}^k \bar{\Phi}^{-a_t(z)}\alpha)=H_{\bar{\mu}_{S^{-i}y}}(\bigvee \limits_{t=1}^k \bar{\Phi}^{-(i+a_t(y))}\alpha) \\ &=H_{\bar{\mu}_{y}}(\bigvee_{t=1}^k \bar{\Phi}^{-a_t(y)}\alpha )=H_{\bar{\mu}_y}(\bar{\Phi}^{-a_1(y)}\alpha|\bigvee_{t=2}^k \bar{\Phi}^{-a_t(y)}\alpha)+H_{\bar{\mu}_y}(\bigvee_{t=2}^k \bar{\Phi}^{-a_t(y)}\alpha)\\&=H_{\bar{\mu}_{S^{a_1(y)}y}}(\alpha|\bigvee_{t=2}^k \bar{\Phi}^{-(a_t(y)-a_1(y))}\alpha)+H_{\bar{\mu}_y}(\bigvee_{t=2}^k \bar{\Phi}^{-a_t(y)}\alpha)\\ &\ge h_{\bar{\mu}}(\bar{\Phi}^\ell,\alpha,S^{a_1(y)}y)+H_{\bar{\mu}_y}(\bigvee_{t=2}^k \bar{\Phi}^{-a_t(y)}\alpha)\\ &\cdots \cdots \cdots \cdots \cdots \cdots\\ &\ge \sum_{t=1}^k h_{\bar{\mu}}(\bar{\Phi}^\ell,\alpha,S^{a_t(y)}y).\end{aligned}$$ Since $\frac{a_k(y)}{\ell}\ge k-1$ and $$\lim_{k\rightarrow +\infty} \frac{1}{\frac{a_k(y)}{\ell}+1}\sum_{j=0}^{\frac{a_k}{\ell}}\delta(S^{\ell j }y)=\delta^*(y)\ge 2c,$$ there exists $N_y\in \mathbb{N}$ such that when $k\ge N_y$, $\frac{1}{\frac{a_k(y)}{\ell}+1}\sum \limits_{j=0}^{\frac{a_k}{\ell}}\delta(S^{\ell j }y)\ge c$. Thus for $k\ge N_y$, $$\begin{aligned} \log |M_z(k)|&\ge \sum_{t=1}^k h_{\bar{\mu}}(\bar{\Phi}^\ell,\alpha,S^{a_t(y)}y)\\ &=k\cdot \log (r-1)+\sum_{j=0}^{\frac{a_k}{\ell}}(h_{\bar{\mu}}(\bar{\Phi}^\ell,\alpha,S^{\ell j}y)-\log(r-1))\cdot 1_A(S^{\ell j}y)\\ &=k\cdot \log (r-1)+\sum_{j=0}^{\frac{a_k}{\ell}}\delta(S^{\ell j }y)\\ &\ge k\cdot \log (r-1)+(\frac{a_k(y)}{\ell}+1)c\\ &\ge k\cdot \log (r-1)+k c.\end{aligned}$$ Let $\lambda=2^c$. Then $\lambda>1$ and $$|M_z(k)|\ge ((r-1)\lambda)^k$$ for $k\ge N_y$. By Lemma \[com-lem\], there exists constant $e>0$ ($e$ is just dependent on $r$ and $\lambda$) such that for $k\ge N_y$ there exists $I_{k,z}\subseteq \{1,2,\cdots,k \}$ such that $|I_{k,z}|\ge ek$ and $M_z(k)|_{I_{k,z}}=\{1,2,\cdots,r\}^{I_{k,z}}$. Put $L_{k,z}=\{ a_t(z):t\in I_{k,z}\}$. Then for $k\ge \max\{N_y,\frac{1}{e}\}$, we have $|L_{k,z}|=|I_{k,z}|\ge ek\ge 1$, $J_{k,z}\subseteq \{ 0,1,\cdots,a_k(z)\}$ and for any $s\in \{1,2,\cdots,r\}^{L_{k,z}}$, one has $$\begin{aligned} \bar{\mu}_z(\pi_1^{-1}(z)\cap (\bigcap_{j\in L_{k,z}} \bar{\Phi}^{-j}B_{s(j)}))>0.\end{aligned}$$ By , there exists $W_z\in \mathbb{N}$ such that when $m\ge W_z$, one has $$|\mathcal{S}(z)\cap \{0,1,\cdots,m-1\}|\ge c_1m \ge \max\{N_y,\frac{1}{e}\}$$ For $m\ge W_z$, let $k_m(z)=|\mathcal{S}(z)\cap \{0,1,\cdots,m-1\}|$. Then $k_m(z)\ge c_1m \ge \max\{N_y,\frac{1}{e}\}$ and $$\mathcal{S}(z)\cap \{0,1,\cdots,m-1\}=\{a_1(z),a_2(z),\cdots,a_{k_m(z)}(z)\}.$$ Let $J_{m,z}=L_{k_m(z),z}$. Then $J_{m,z}\subseteq \{a_1(z),a_2(z),\cdots,a_{k_m(z)}(z)\}\subseteq \{0,1,\cdots,m-1\}$, $$|J_{m,z}|=|L_{k_m(z),z}|\ge ek_m(z)\ge ec_1m$$ and for any $s\in \{1,2,\cdots,r\}^{J_{m,z}}$, one has $$\begin{aligned} \bar{\mu}_z(\pi_1^{-1}(z)\cap (\bigcap_{j\in J_{m,z}} \bar{\Phi}^{-j}B_{s(j)}))>0.\end{aligned}$$ [*Step 3.*]{} Let $b=ec_1$. Then $b>0$ and for $P$-a.e. $\omega\in \Omega$, there exists $M_{b,\omega}\in \mathbb{N}$ such that for any natural number $m\ge M_{b,\omega}$ we can find $J_m\subset \{0,1,\cdots,m-1\}$ with $|J_m|\ge bm$, and for any $s\in \{1,2\}^{J_m}$, there exists $x_s\in K(\omega)$ with $\phi(j,\omega,x_s)\in U_{s(j)}\cap K(\theta^j \omega)$ for any $j\in J_m$. Note that $$\Pi_\Omega\circ \pi_2:(Y,\mathcal{D},\nu,S)\rightarrow (\Omega,\mathcal{B}_P, P, \theta)$$ is a factor map between two Lebesgue systems. Since $\nu(E)=1$, there exists $\Omega_1\in \mathcal{B}_P$ satisfying $P(\Omega_1)=1$ and $(\Pi_\Omega\circ \pi_2)^{-1}(\omega)\cap E\neq \emptyset$ for each $\omega\in \Omega_1$. Given $\omega\in \Omega_1$. By the definition of $\Omega_1$, there exists $z\in E$ such that $\Pi_\Omega\circ \pi_2(z)=\omega$. Let $\pi_2(z)=(\omega_i(z))_{i\in \mathbb{Z}}\in \bar{\Omega}$. Then $\omega=\Pi_\Omega ((\omega_i(z))_{i\in \mathbb{Z}})=\omega_0(z)$. By Step 2, we can find $W_z\in \mathbb{N}$ such that for any $m\ge W_z$, there exists $$J_{m,z}\subseteq \{a_1(z),a_2(z),\cdots,a_{k_m(z)}(z)\}\subseteq \{0,1,\cdots,m-1\}$$ with $|J_{m,z}|\ge bm$, and for any $s\in \{1,2,\cdots,r\}^{J_{m,z}}$, one has $$\begin{aligned} \label{key-non-empty-2} \bar{\mu}_z(\pi_1^{-1}(z)\cap (\bigcap_{j\in J_{m,z}} \bar{\Phi}^{-j}B_{s(j)}))>0.\end{aligned}$$ We take $M_{b,\omega}=W_z$. For $m\ge M_{b,\omega}$, take $J_m=J_{m,z}$. Then $J_m\subseteq \{0,1,\cdots,m-1\}$ with $|J_m|\ge bm$. Next we are ready to show that for any $s\in \{1,2\}^{J_m}$, there exists $x_s\in K(\omega)$ with $\phi(j,\omega,x_s)\in U_{s(j)}\cap K(\theta^j \omega)$ for any $j\in J_m$. Given $s\in \{1,2\}^{J_{m}}$. By we know $\bar{\mu}_z(\pi_1^{-1}(z)\cap (\bigcap_{j\in J_{m}} \bar{\Phi}^{-j}B_{s(j)}))>0$. Thus $\pi_1^{-1}(z)\cap (\bigcap_{j\in J_{m}} \bar{\Phi}^{-j}B_{s(j)})\neq \emptyset$ and we can find $((\omega_i^s,x_i^s))_{i\in \mathbb{Z}}\in \pi_1^{-1}(z)\cap (\bigcap_{j\in J_{m}} \bar{\Phi}^{-j}B_{s(j)})\subseteq \bar{K}$. On the hand $$\bar{\pi}(((\omega_i^s,x_i^s))_{i\in \mathbb{Z}})=(\omega_i^s)_{i\in \mathbb{Z}}\in \bar{\Omega}.$$ On the other hand $$\bar{\pi}(((\omega_i^s,x_i^s))_{i\in \mathbb{Z}})=\pi_2\circ \pi_1(((\omega_i^s,x_i^s))_{i\in \mathbb{Z}}) =\pi_2(z)=(\omega_i(z))_{i\in \mathbb{Z}}\in \bar{\Omega}.$$ Hence $(\omega_i^s)_{i\in \mathbb{Z}}=(\omega_i(z))_{i\in \mathbb{Z}}$. Particularly, $\omega_0^s=\omega_0(z)=\omega$. Thus $x_0^s\in K(\omega)$ since $(\omega,x_0^s)=(\omega_0,x_0^s)\in K$. Take $x_s=x_0^s$. Then $x_s\in K(\omega)$. Recall that $$\bar{K}=\{ ((\omega_i,x_i))_{i\in \mathbb{Z}}\in K^{\mathbb{Z}}: \Phi(\omega_i,x_i)=(\omega_{i+1},x_{i+1}) \text{ for }i\in \mathbb{Z}\}.$$ Since $((\omega_i^s,x_i^s))_{i\in \mathbb{Z}}\in \bar{K}$ and $(\omega_0^s,x_0^s)=(\omega,x_s)$, one has $\omega_i^s=\theta^i\omega$ and $x_i^s=\phi(i,\omega,x_s)\in K(\theta^i\omega)$ for any $i\in \mathbb{Z}$. Finally, we show that for any $j\in J_m$, $\phi(j,\omega,x_s)\in U_{s(j)}\cap K(\theta^j \omega)$. Given $j\in J_m$. Since $z\in E$, there exists unique $i_*\ge 0$ and unique $y\in D$ such that $z\in E_{i_*}$ and $z=S^{-i_*}(y)$. Note that $$J_m=J_{m,z}\subseteq \{a_1(z),a_2(z),\cdots,a_{k_m(z)}(z)\}=\{ i_*+a_1(y),i_*+a_2(y),\cdots,i_*+a_{k_m(z)}(y)\},$$ we have $j-i_*\in \{a_1(y),a_2(y),\cdots,a_{k_m(z)}(y)\}$ and so $S^{j-i_*}(y)\in A$, that is, $S^j(z)\in A$. Since $((\omega_i^s,x_i^s))_{i\in \mathbb{Z}}\in \pi_1^{-1}(z)\cap \bar{\Phi}^{-j}B_{s(j)})$, one has $$\begin{aligned} \bar{\Phi}^j(((\omega_i^s,x_i^s))_{i\in \mathbb{Z}})&\in \bar{\Phi}^j(\pi_1^{-1}(z))\cap B_{s(j)}=\pi_1^{-1}(S^jz)\cap B_{s(j)}\\ &\subseteq \pi_1^{-1}(A)\cap B_{s(j)}\subseteq \widetilde{U_{s(j)}}.\end{aligned}$$ Moreover as $\bar{\Phi}^j(((\omega_i^s,x_i^s))_{i\in \mathbb{Z}})=((\omega_{i+j}^s,x_{i+j}^s))_{i\in \mathbb{Z}}$, using we know $x_j^s\in U_{s(j)}$. Combing this with $x_j^s=\phi(j,\omega,x_s)\in K(\theta^j\omega)$, this shows $\phi(j,\omega,x_s)\in U_{s(j)}$ and completes the proof of the theorem. $\square$ Proof of Theorem \[MTH3\] ========================= In this section, we prove a stronger result than Theorem \[MTH1\] when $(\Omega, \mathcal{F}, {P},\theta)$ is a [compact metric system]{} and $K(\omega),\,\omega\in\Omega$ is [ a strongly compact random set]{}. We show that there is a full horseshoe for $(\phi, K)$ instead of a weak horseshoe. As a consequence, we have a full horseshoe for a continuous deterministic dynamical systems $\phi$ on a compact invariant set $K$ with positive topological entropy. 0.1in [**Proof of Theorem \[MTH3\].** ]{} By Theorem \[MTH1\] there exist subsets $U_1,U_2$ of $X$, a constant $b>0$ and $\Omega_0\in \mathcal{F}$ with $P(\Omega_0)=1$ such that 1. $U_1$ and $U_2$ are non-empty closed bounded subsets of $X$ and $d(U_1,U_2)>0$. 2. for each $\omega\in \Omega_0$, there exists $M_{b,\omega}\in \mathbb{N}$ such that for any natural number $m\ge M_{b,\omega}$, there is a subset $J_m(w_*)\subset \{0,1,2,\cdots,m-1\}$ with $|J_m(\omega)|\ge bm$(positive density), and for any $s\in \{1,2\}^{J_m(\omega}$, there exists $x_s\in K(\omega)$ with $\phi(j,\omega,x_s)\in U_{s(j)}\cap K(\theta^j \omega)$ for any $j\in J_m(\omega)$. Let $\mathbb{N}_0=\mathbb{N}\cup \{0\}$ and $T:\Omega\times \{0,1\}^{\mathbb{N}_0}\rightarrow \Omega\times \{0,1\}^{\mathbb{N}_0}$ be the map such that $T(\omega,u)=(\theta\omega, \sigma u)$ for $\omega\in\Omega$ and $u=(u(n))_{n\in \mathbb{N}_0}\in \{0,1\}^{\mathbb{N}}$, where $\sigma u=(u_{n+1})_{n\in \mathbb{N}_0}$ is the left shift map on $\{0,1\}^{\mathbb{N}_0}$. Clearly $\Omega\times \{0,1\}^{\mathbb{N}_0}$ is a compact metric space and $T$ is a continuous self-map on $\Omega\times \{0,1\}^{\mathbb{N}_0}$. Consider the subset $Y$ of $\Omega\times \{0,1\}^{\mathbb{N}_0}$ such that $(\omega,u)\in Y$ if and only if for any $s\in \{1,2\}^J$,put $J=\{n\in \mathbb{N}_0:u(n)=1\}$, there exists $x_s\in K(\omega)$ with $\phi(j,\omega,x_s)\in U_{s(j)}\cap K(\theta^j \omega)$ for any $j\in J$. We have the following claim. [*Claim.*]{} $T(Y)\subseteq Y$ and $Y$ is a closed subset of $\Omega\times \{0,1\}^{\mathbb{N}_0}$. If $(\omega,u)\in Y$, let $J=\{n\in \mathbb{N}_0:u(n)=1\}$ and $J_\sigma=\{n\in \mathbb{N}_0:(\sigma u)(n)=1\}$. Then $J_\sigma+1\subset J$. Now for any $t\in \{1,2\}^{J_\sigma}$, we can find $s\in \{1,2\}^J$ such that $t(n)=s(n+1)$ for $n\in J_\sigma$. As $(\omega,u)\in Y$, there exists $x_s\in K(\omega)$ with $\phi(j,\omega,x_s)\in U_{s(j)}\cap K(\theta^j \omega)$ for any $j\in J$. Thus let $x_t=\phi(\omega)(x_s)$. Then $x_t\in K(\theta \omega)$ and $\phi(j,\theta\omega,x_t)=\phi(j+1,\omega,x_s)\in U_{s(j+1)}\cap K(\theta^j \theta\omega)$ for any $j\in J_\sigma$. Hence $(\theta\omega,\sigma u)\in Y$. This implies $T(Y)\subseteq Y$. Next let $(\omega^i,u^i)\in Y$ and $\lim_{i\rightarrow +\infty}(\omega^i,u^i)=(\omega,u)$ in $\Omega\times \{0,1\}^{\mathbb{N}_0}$. Let $J_i=\{n\in \mathbb{N}_0:u^i(n)=1\}$ for $i\in \mathbb{N}$ and $J=\{n\in \mathbb{N}_0: u(n)=1\}$. If $J=\emptyset$, then it is clear that $(\omega,u)\in Y$ by the definition of $Y$. Now suppose $J\neq\emptyset$, and let $N=\min\{n\in J\}$. Given $s\in \{1,2\}^J$. Note that $\lim_{i\rightarrow +\infty}u^i=u$. There exist $1\le i_1<i_2<\cdots$ such that $$J_{i_k}\cap \{0,1,\cdots,N+r\}=J\cap \{0,1,\cdots,N+r\}$$ for each $1\le r\le k$. Now for each $k$, we take $s^k\in \{1,2\}^{J_{i_k}}$ such that $s^k(j)=s(j)$ for $j\in J\cap \{0,1,\cdots,N+k\}$. As $(\omega^{i_k},u^{i_k})\in Y$, there exists $x_{s^k}\in K(\omega^{i_k})$ with $\phi(j,\omega^{i_k},x_{s^k})\in U_{s^k(j)}\cap K(\theta^j \omega^{i_k})$ for any $j\in J_{i_k}$. Without of loss generality, we assume that the limit $\lim_{k\rightarrow +\infty}x_{s_k}$ exists (if necessarily we take subsequence) and let $x_s:=\lim_{k\rightarrow +\infty}x_{s^k}$. Firstly, as $x_{s^k}\in K(\omega^{i_k})$ and the function $\omega\mapsto\inf_{y\in M(\omega)}d(x,y)$ is lower semi-continuous for any $x\in X$, one has $$\begin{aligned} \inf_{y\in K(\omega)} d(x_s,y)&\le \liminf_{k\rightarrow +\infty} \inf_{y\in K(\omega^{i_k})} d(x_s,y)\\ &\le \liminf_{k\rightarrow +\infty} \inf_{y\in K(\omega^{i_k})} \big( d(x_s,x_{s^k})+d(x_{s^k},y)\big)\\ &=0.\end{aligned}$$ Thus $x_s\in K(\omega)$. Given $n\in J$. As the map $(\omega,x)\mapsto \phi(\omega)x$ is a continuous map, one has $$\phi(n,\omega,x_s)=\lim_{k\rightarrow +\infty}\phi(n,\omega^{i_k},x_{s^k}).$$ Combing this with the fact that $$\phi(n,\omega^{i_k},x_{s^k})\in U_{s^k(n)}\cap K(\theta^n \omega^{i_k})=U_{s(n)}\cap K(\theta^n \omega^{i_k})$$ for $k\ge n$ and the function $\omega\mapsto\inf_{y\in M(\omega)}d(x,y)$ is lower semi-continuous for any $x\in X$, one has $\phi(n,\omega,x_s)\in U_{s(n)}$ and $$\begin{aligned} \inf_{y\in K(\theta^n\omega)} d(\phi(n,\omega,x_s),y)&\le \liminf_{k\rightarrow +\infty} \inf_{y\in K(\theta^n\omega^{i_k})} d(\phi(n,\omega,x_{s}),y)\\ &\le \liminf_{k\rightarrow +\infty} \inf_{y\in K(\theta^n\omega^{i_k})} \big( d(\phi(n,\omega,x_s),\phi(n,\omega^{i_k},x_{s^k}))+d(\phi(n,\omega^{i_k},x_{s^k}),y)\big)\\ &=0.\end{aligned}$$ Thus $\phi(n,\omega,x_s)\in U_{s(n)}\cap K(\theta^n \omega)$ for $n\in J$. This implies $(\omega,u)\in Y$. Hence $Y$ is a closed subset of $\Omega\times \{0,1\}^{\mathbb{N}}$. This completes the proof of claim. Let $G_P$ be the set of all generic points of $(\Omega, \mathcal{F}, {P},\theta)$, that is, $\omega\in G_P$ if and only if $P=\lim_{m\rightarrow +\infty} \frac{1}{m}\sum_{j=0}^{m-1}\delta_{\theta^j \omega}$ in the weak\*-topology. By Birkhoff pointwise ergodic Theorem, $P(G_P)=1$ as $(\Omega, \mathcal{F}, {P},\theta)$ is ergodic. As $P(G_P\cap \Omega_0)=1$, we take $\omega_*\in G_P\cap \Omega_0$. By the above (2), for any natural number $m\ge M_{b,\omega_*}$, there is a subset $J_m(w_*)\subset \{0,1,2,\cdots,m-1\}$ with $|J_m(\omega_*)|\ge bm$(positive density), and for any $s\in \{1,2\}^{J_m(\omega_*)}$, there exists $x_s\in K(\omega_*)$ with $\phi(j,\omega_*,x_s)\in U_{s(j)}\cap K(\theta^j \omega_*)$ for any $j\in J_m(\omega_*)$. For $m\ge M_{b,\omega_*}$. Let $v_m\in \{0,1\}^{\mathbb{N}_0}$ with $v_m(n)=1$ if and only if $n\in J_m(\omega_*)$, that is, $\{n\in \mathbb{N}_0:v_m(n)=1\}=J_m(\omega_*)$. Clearly $(\omega_*,v_m)\in Y$. Assume $\mu_m =\frac{1}{m}\sum_{j=0}^{m-1}\delta_{T^j(\omega_*,v_m)}$ for $m\ge M_{b,\omega_*}$. Then each $\mu_m$ is a Borel probability on $Y$. Let $\mu= \lim_{i\rightarrow +\infty}\mu_{m_i}$ be a limit point of $\{\mu_m\}$ in the weak\*-topology. Clearly, $\mu$ is a $T$-invariant Borel probability on $Y$. Let $\pi:Y\rightarrow \Omega$ be the projection of coordinate. Then $\mu\circ \pi^{-1}=P$. In fact, for any continuous function $f\in C(\Omega)$, $$\begin{aligned} \int_\Omega f d\mu\circ \pi^{-1}&=\int_Y f\circ \pi d\mu=\lim_{i\rightarrow +\infty} \int_Y f\circ \pi d\mu_{m_i}\\ &=\lim_{i\rightarrow +\infty}\frac{1}{m_i}\sum_{j=0}^{m_i-1}f\circ \pi(T^j(\omega_*,v_m))\\ &=\lim_{i\rightarrow +\infty}\frac{1}{m_i}\sum_{j=0}^{m_i-1}f(\theta^j\omega_*)\\ &=\int_\Omega f d P\end{aligned}$$ the last equality comes from the fact $\omega_*$ is a generic point of $(\Omega, \mathcal{F}, {P},\theta)$. Thus $\mu\circ \pi^{-1}=P$. Let $[1]=\{(\omega,u)\in Y: u(0)=1 \}$. Then $[1]$ is a closed and open subset of $Y$. Note that $T^j(\omega,u)=(\theta^j\omega,\sigma^ju)$, one has $$\begin{aligned} \mu([1])&=\lim_{i\rightarrow +\infty}\mu_{m_i}([1])=\lim_{i\rightarrow +\infty}\frac{1}{m_i}|\{j\in \{0,1,\cdots,m_i-1\}:(\sigma^jv_{m_i})(j)=1\}|\\ &=\lim_{i\rightarrow +\infty}\frac{1}{m_i}|\{j\in \{0,1,\cdots,m_i-1\}:v_{m_i}(j)=1\}|\\ &=\lim_{i\rightarrow +\infty}\frac{1}{m_i}|J_{m_i}(\omega_*)|\\ &\ge b.\end{aligned}$$ By the ergodic decomposition, $\mu\circ \pi^{-1}=P$ and the fact that $(\Omega, \mathcal{F}, {P},\theta)$ is ergodic, we know that there exists a $T$-invariant ergodic Borel probablity measure $\nu$ on $Y$ with $\nu([1])\ge \mu([1])\ge b$ and $\nu\circ \pi^{-1}=P$. Let $G_\nu$ be the set of all generic points of $(Y,\mathcal{B}_Y,\nu,T)$, that is, $(\omega,u)\in G_\nu$ if and only if $\nu=\lim_{m\rightarrow +\infty} \frac{1}{m}\sum_{j=0}^{m-1}\delta_{T^j(\omega,u)}$ in the weak\*-topology. Then $G_\nu$ is a Borel subset of $Y$, and $\nu(G_\nu)=1$ by Birkhoff pointwise ergodic Theorem. Let $\Omega_1=\pi(G_\nu)$. Then $\Omega_1$ is a $P$-measurable set since it is the continuous image of a Borel set. Thus $P(\Omega_1)=\nu\circ \pi^{-1}(\Omega_1)=\nu(\pi^{-1}(\Omega_1))=1$. Now for $\omega \in \Omega_1$, there exists $u_\omega\in \{0,1\}^{\mathbb{N}_0}$ such that $(\omega,u_\omega)\in G_\nu$. Let $J(\omega)=\{n\in \mathbb{N}_0:u_\omega(n)=1\}$. As $(\omega,u_\omega)\in G_\nu$ and $[1]$ is closed and open, one has $$\begin{aligned} \nu([1])&=\lim_{m\rightarrow +\infty} \frac{1}{m}\sum_{j=0}^{m-1}\delta_{T^j(\omega,u_\omega)}([1])\\ &=\lim_{m\rightarrow +\infty} \frac{1}{m}|\{j\in \{0,1,\cdots,m-1\}:(\sigma^ju_{\omega})(0)=1\}|\\ &=\lim_{m\rightarrow +\infty}\frac{1}{m}|\{j\in \{0,1,\cdots,m_i-1\}:u_\omega(j)=1\}|\\ &=\lim_{m\rightarrow +\infty}\frac{1}{m}|J(\omega)\cap \{0,1,\cdots,m-1\}|\end{aligned}$$ Thus the limit $\lim_{m\rightarrow +\infty}\frac{1}{m}|J(\omega)\cap \{0,1,2,\cdots m\}|$ exists and is larger than or equal to $b$, as $\nu([1])\ge b$. Note that $(\omega,u_\omega)\in Y$. By the definition of $Y$, one has for any $s\in \{1,2\}^{J(\omega)}$, there exists $x_s\in K(\omega)$ with $\phi(j,\omega,x_s)\in U_{s(j)}\cap K(\theta^j \omega)$ for any $j\in J(\omega)$. Summing up the above discussion, $\{U_1,U_2\}$ is a full Horseshoe for $(\phi,K)$ and completes the proof of the Theorem. $\square$. Proof of Theorem \[MTH\] {#Li-Yorke-chaos} ======================== To prove this theorem, we also need the following result due to Mycielski (see [@Ak Theorem 5.10] and [@AAG Theorem 6.32]). (Mycielski) \[Myc\] Let $Y$ be a perfect compact metric space and $C$ a symmetric dense $G_\delta$ subset of $Y\times Y$. Then there exists a dense Mycielski subset $K\subseteq Y$ such that $K\times K\subseteq C\cup \Delta_Y$, where $\Delta_Y=\{ (y,y):y\in Y\}$. [**Proof of Theorem \[MTH\].**]{} Assume that $ h_{top}(\phi,K)>0$. By Proposition \[VP\] there exists $\mu\in \mathcal{E}_P^K(\Omega\times X)$ such that $h_\mu(\phi)>0$. Since $K$ is a Borel subset of $\Omega\times X$, $K$ is also Polish space and the Borel $\sigma$-algebra $\mathcal{B}_K$ of $K$ is just $\{ A\cap K: A\in \mathcal{F}\times \mathcal{B}_X\}$. Thus $(K,\mathcal{B}_K,\mu, \Phi)$ is an ergodic Polish system and $\pi_\Omega: (K,\mathcal{B}_K,\mu, \Phi)\rightarrow (\Omega, \mathcal{F},P,\theta)$ is a factor map between two Polish systems. Recall that the entropy of RDS $\phi$ with respect to $\mu$ is given by $$h_\mu(\phi):=h_\mu(\Phi|\pi_\Omega)=h_\mu(\Phi|\pi_\Omega^{-1}(\mathcal{F})).$$ Let $\mathcal{B}_P$ be the completion of $\mathcal{F}$ with respect to $P$. Let $\mathcal{B}_\mu$ be the completion of $\mathcal{B}_K$ with respect to $\mu$. Then $\pi_\Omega:(K,\mathcal{B}_\mu,\mu, \Phi)\rightarrow (\Omega, \mathcal{B}_P,P,\theta)$ is a factor map between two Lebesgue systems. Since $\mathcal{B}_P=\mathcal{F}$ (mod $P$) and $\mathcal{B}_\mu=\mathcal{B}_K$ (mod $\mu$), it is clear that $$h_\mu(\Phi|\pi_\Omega^{-1}(\mathcal{B}_P))=h_\mu(\Phi|\pi_\Omega^{-1}(\mathcal{F}))=h_\mu(\phi).$$ Let $\Pi_K:(\bar{K},\bar{\mathcal{B}_{\mu}},\bar{\mu}, \bar{\Phi})\rightarrow (K,\mathcal{B}_\mu,\mu, \Phi)$ be the natural extension of $(K,\mathcal{B}_\mu,\mu, \Phi)$. Let $$\Pi_{\Omega}: (\bar{\Omega}, \bar{\mathcal{B}_{P}},\bar{P},\bar{\theta})\rightarrow (\Omega, \mathcal{B}_P,P,\theta)$$ be the natural extension of $(\Omega, \mathcal{B}_P,P,\theta)$. Define $\bar{\pi}:\bar{K}\rightarrow \bar{\Omega}$ by $\bar{\pi}(((\omega_i,x_i))_{i\in \mathbb{Z}})=(\omega_i)_{i\in \mathbb{Z}}$ for $((\omega_i,x_i))_{i\in \mathbb{Z}}\in \bar{K}$. Then $\bar{\pi}:(\bar{K},\bar{\mathcal{B}_{\mu}},\bar{\mu},\bar{\Phi})\rightarrow (\bar{\Omega},\bar{\mathcal{B}_{P}},\bar{P},\bar{\theta})$ be a factor map between two invertible Lebesgue systems and the following diagram is commutative: $$\begin{CD} (K,\mathcal{B}_\mu,\mu,\Phi) @< \Pi_K<< (\bar{K},\bar{\mathcal{B}_{\mu}},\bar{\mu},\bar{\Phi})\\ @VV\pi V @VV\bar{\pi} V \\ (\Omega,\mathcal{B}_P,P,\theta) @< \Pi_\Omega<< (\bar{\Omega},\bar{\mathcal{B}_{P}},\bar{P},\bar{\theta})\\ \end{CD}$$ We divide the remainder of the proof into four steps. [*Step 1.*]{} Let $P_{\bar{\mu}}(\bar{\pi})$ be the relative Pinsker $\sigma$-algebra of $(\bar{K},\bar{\mathcal{B}_{\mu}},\bar{\mu},\bar{\Phi})$ relative to $\bar{\pi}$. Since ${\bar{\Phi}}^{-1}(P_{\bar{\mu}}(\bar{\pi}))=P_{\bar{\mu}}(\bar{\pi})$ and $$\bar{\pi}^{-1}(\mathcal{\bar B}_{{P}})\subseteq P_{\bar{\mu}}(\bar{\pi})\subseteq \bar{\mathcal{B}_{\mu}},$$ there exist an invertible Lebesgue system $(Y,\mathcal{D},\nu,S)$ and two factor maps $$\pi_1:(\bar{K},\bar{\mathcal{B}_{\mu}},\bar{\mu},\bar{\Phi})\rightarrow (Y,\mathcal{D},\nu,S),\ \pi_2:(Y,\mathcal{D},\nu,S)\rightarrow (\bar{\Omega},\bar{\mathcal{B}_{P}},\bar{P},\bar{\theta})$$ between invertible Lebesgue systems such that $\pi_2\circ \pi_1=\bar{\pi}$ and $\pi_1^{-1}(\mathcal{D})=P_{\bar{\mu}}(\bar{\pi})$ (mod $\bar{\mu}$). That is, $\pi_1$ is the Pinsker factor of $(\bar{K},\bar{\mathcal{B}_{\mu}},\bar{\mu},\bar{\Phi})$ relative to $\bar{\pi}$. Let $\bar{\mu}=\int_Y \bar{\mu}_y d \nu(y)$ be the disintegration of $\bar{\mu}$ relative to the factor $(Y,\mathcal{D},\nu,S)$. Let $$(\bar{K}\times \bar{K},\bar{\mathcal{B}_{\mu}}\times \bar{\mathcal{B}_{\mu}},\bar{\mu} \times_Y\bar{\mu}, \bar{\Phi}\times \bar{\Phi})$$ be the product of $(\bar{K},\bar{\mathcal{B}_{\mu}},\bar{\mu},\bar{\Phi})$ with itself relative to a factor $(Y,\mathcal{D},\nu,S)$. Recall that $$(\bar{\mu}\times_Y \bar{\mu})(B)=\int_{Y}(\bar{\mu}_y\times \bar{\mu}_y)(B) \, d \nu(y), \ B\in \bar{\mathcal{B}_{\mu}}\times \bar{\mathcal{B}_{\mu}}.$$ Since $(K,\mathcal{B}_{\mu},\mu,\Phi)$ is ergodic, we have that $(\bar{K},\bar{\mathcal{B}_{\mu}},\bar{\mu},\bar{\Phi})$ is ergodic. By in the proof of Theorem \[MTH1\], $h_{\bar{\mu}}(\bar{\Phi}|\bar{\pi})>0$. Moreover by Lemma \[key-lem\], we have 1. $\bar{\mu}_y$ is non-atomic for $\nu$-a.e. $y\in Y$. 2. $(\bar{K}\times \bar{K},\bar{\mathcal{B}_{\mu}}\times \bar{\mathcal{B}_{\mu}},\bar{\mu} \times_Y\bar{\mu}, \bar{\Phi}\times \bar{\Phi})$ is ergodic. [*Step 2.*]{} We define a metric $\rho$ on $X^{\mathbb{Z}}$ as follows: $$\rho((x_i^1)_{i\in \mathbb{Z}}, (x_i^2)_{i\in \mathbb{Z}})=\sum_{i\in \mathbb{Z}} \frac{1}{2^{|i|}} \frac{d(x_i^1,x_i^2)}{1+d(x_i^1,x_i^2)} \ \text{ for } (x^1_i)_{i\in \mathbb{Z}},(x_i^2)_{i\in \mathbb{Z}}\in X^{\mathbb{Z}}.$$ For $n\in \mathbb{N}$, let $$A_n=\{ ((\omega^1_i,x^1_i))_{i\in \mathbb{Z}}, ((\omega^2_i,x^2_i))_{i\in \mathbb{Z}})\in \bar{K}\times \bar{K} :\rho((x^1_i)_{i\in \mathbb{Z}},(x_i^2)_{i\in \mathbb{Z}})<\frac{1}{n}\}$$ and $$B_n=\bar{K}\times \bar{K}\setminus A_n= \{ ((\omega^1_i,x^1_i))_{i\in \mathbb{Z}}, ((\omega^2_i,x^2_i))_{i\in \mathbb{Z}})\in \bar{K}\times \bar{K} :\rho((x^1_i)_{i\in \mathbb{Z}},(x_i^2)_{i\in \mathbb{Z}})\ge \frac{1}{n}\} .$$ Then $A_n,B_n$ are measurable subsets of $\bar{K}\times \bar{K}$. Now we claim: [**Claim 1:**]{} $\bar{\mu}\times_Y \bar{\mu}(\bigcup \limits_{n=1}^\infty B_n)=1$ and $\bar{\mu}\times_Y \bar{\mu}(A_n)>0$ for each $n\in \mathbb{N}$. Let $\Delta=\{(((\omega_i^1,x^1_i))_{i\in \mathbb{Z}}, ((\omega_i^2,x^2_i))_{i\in \mathbb{Z}})\in \bar{K}\times \bar{K}:x^1_i=x_i^2 \text{ for each }i\in \mathbb{Z}\}$. Then $$\bar{K}\times \bar{K}\setminus \Delta=\bigcup_{n=1}^\infty B_n.$$ Hence to show $\bar{\mu}\times_Y \bar{\mu}(\bigcup \limits_{n=1}^\infty B_n)=1$, it is sufficient to show that $\bar{\mu}\times \bar{\mu}(\Delta)=0$. For $y\in Y$, $\pi_2(y)\in \Omega$ and $\bar\mu_y(\bar{\pi}^{-1}(\pi_2(y)))=1$. Hence $$\bar{\mu}_y\times \bar{\mu}_y(\Delta)=\bar{\mu}_y\times \bar{\mu}_y(\bar{\pi}^{-1}(\pi_2(y))\times \bar{\pi}^{-1}(\pi_2(y))\cap \Delta)=\bar{\mu}_y\times \bar{\mu}_y(\Delta_{\bar{K}})$$ where $\Delta_{\bar{K}}=\{ (\bar{k},\bar{k}):\bar{k}\in \bar{K}\}$. It is clear that when $\bar{\mu}_y$ is non-atomic, i.e., $\bar{\mu}_y(\{ \bar{k}\})=0$ for $\bar{k}\in \bar{K}$, we have $$\bar{\mu}_y\times \bar{\mu}_y(\Delta_{\bar{K}})=\int_{\bar{K}}\left( \int_{\bar{K}}1_{\Delta_{\bar{K}}}(\bar{k}_1,\bar{k}_2) d\bar{\mu}_y(\bar{k}_2) \right) d\bar{\mu}_y(\bar{k}_1)=\int_{\bar{K}}\bar{\mu}_y(\{\bar{k}_1\}) d \bar{\mu}_y(\bar{k}_1)=0$$ where $1_{\Delta_{\bar{K}}}$ is the characterization function of $\Delta_{\bar{K}}$. By (a1) in step 1, for $\nu$-a.e. $y\in Y$, $\bar{\mu}_y$ is non-atomic. Thus for $\nu$-a.e. $y\in Y$, $\bar{\mu}_y\times \bar{\mu}_y(\Delta) = \bar{\mu}_y\times \bar{\mu}_y(\Delta_{\bar{K}})=0$. Moreover, $$\begin{aligned} \bar{\mu}\times_Y \bar{\mu}(\Delta)=\int_Y \bar{\mu}_y\times \bar{\mu}_y(\Delta) d\nu(y)=0.\end{aligned}$$ This implies $\bar{\mu}\times_Y \bar{\mu}(\bigcup \limits_{n=1}^\infty B_n)=1$. Next, for $n\in \mathbb{N}$, we are to show $\bar{\mu}\times_Y \bar{\mu}(A_n)>0$. For $\bar{z}\in X^{\mathbb{Z}}$, we define $$B_\rho(\bar{z},n)=\{ \bar{x}\in X^{\mathbb{Z}}: \rho(\bar{x},\bar{z})<\frac{1}{2n}\}.$$ Since $(X,d)$ is a separable metric space, $(X^{\mathbb{Z}},\rho)$ is also separable. Thus there exist $\{ \bar{z}_i\}_{i=1}^\infty \subseteq X^{\mathbb{Z}}$ such that $\bigcup_{i=1}^\infty B_\rho(\bar{z}_i,n)=X^{\mathbb{Z}}$. Put $$D_i=\{ ((\omega_i,x_i))_{i\in \mathbb{Z}}\in \bar{K}: (x_i)_{i\in \mathbb{Z}}\in B_\rho(\bar{z}_i,n)\}.$$ Then $\bigcup_{i=1}^\infty D_i=\bar{K}$ and there exists $r\in \mathbb{N}$ such that $\bar{\mu}(D_r)>0$. It is clear that $D_r\times D_r\subseteq A_n$ and $$\begin{aligned} \bar{\mu}\times_Y \bar{\mu}(A_n)&\ge \bar{\mu}\times_Y \bar{\mu}(D_r\times D_r)=\int_Y \bar{\mu}_y\times \bar{\mu}_y(D_r\times D_r)d \nu(y)\\ &=\int_Y (\bar{\mu}_y(D_r))^2 d \nu(y)\ge (\int_Y \bar{\mu}_y(D_r) d \nu(y))^2=\bar{\mu}(D_r)^2>0. \end{aligned}$$ This completes the proof of Claim 1. [*Step 3.*]{} Let $$\begin{aligned} \bar{\Omega}_0=\{& (\omega_i)_{i\in \mathbb{Z}}\in \bar{\Omega}: \phi(\omega_i):X\rightarrow X \text{ is continuous and }K_{\omega_i} \text{ is compact subset of }\\ &\hskip1cm X \text{ for each } i\in \mathbb{Z}\}.\end{aligned}$$ Since $\phi(\omega):X\rightarrow X$ is continuous and $K_\omega$ is compact subset of $X$ for $P$-a.e. $\omega\in \Omega$, the set $$\Omega_0:=\{ \omega\in \Omega: \phi(\omega):X\rightarrow X \text{ is continuous and }K_{\omega} \text{ is compact subset of }X\}$$ is full measure, that is $P(\Omega_0)=1$. Note that $\bar{\Omega}_0=\bigcap_{n\in \mathbb{Z}}\Pi_{n,\Omega}^{-1}(\Omega_0)$, we have $\bar{P}(\bar{\Omega}_0)=1$ since for each $n\in \mathbb{Z}$, $\bar{P}(\Pi_{n,\Omega}^{-1}\Omega_0)= P(\Omega_0)=1$. Moreover $$\label{mes=1} \nu(\pi_2^{-1}(\bar{\Omega}_0))=\bar{P}(\bar{\Omega}_0)=1.$$ By the step 2, there exists $r\in \mathbb{N}$ such that $\bar{\mu}\times_Y\bar{\mu}(B_r)>0$. Let $$\begin{aligned} W=\{ &(\bar{k}_1,\bar{k}_2)\in \bar{K}\times \bar{K}: \lim_{\ell\rightarrow \infty} \frac{1}{\ell}\sum_{i=0}^{\ell-1} 1_{A_n}((\bar{\Phi}\times \bar{\Phi})^i(\bar{k}_1,\bar{k}_2))=\bar{\mu}\times_Y \bar{\mu}(A_n)>0 \\ &\text{ for each }n\in \mathbb{N}\text{ and } \lim_{\ell\rightarrow \infty} \frac{1}{\ell}\sum_{i=0}^{\ell-1} 1_{B_r}((\bar{\Phi}\times \bar{\Phi})^i(\bar{k}_1,\bar{k}_2))=\bar{\mu}\times_Y \bar{\mu}(B_r)>0 \}\end{aligned}$$ where $1_E$ is the characterization function of $E\subseteq \bar{K}\times \bar{K}$. Then, $W$ is a Borel subset of $\bar{K}\times \bar{K}$ and by the Birkhoff ergodic theorem $\bar{\mu}\times_Y \bar{\mu}(W)=1$ since $\bar{\mu}\times_Y \bar{\mu}$ is ergodic (see (a2) in Step 1). Let $$Y_0=\{ y\in Y: \bar{\mu}_y\times \bar{\mu}_y(W)=1, \mu_y \text{ is non-atomic}\}\cap \pi_2^{-1}(\bar{\Omega}_0).$$ Note that $1=\bar{\mu}\times_Y \bar{\mu}(W)=\int_Y \bar{\mu}_y\times \bar{\mu}_y(W) d \nu(y)$ since $\bar{\mu}_y\times \bar{\mu}_y(W)=1$ for $\nu$-a.e. $y\in Y$. Combing this with and (a1) in Step 1, we have $\nu(Y_0)=1$. For $y\in Y$, let $(\omega_i(y))_{i\in\mathbb{Z}}:=\pi_2(y)\in \bar{\Omega}$ and $$\bar{K}_y=\{(x_i)_{i\in \mathbb{Z}}\in \prod_{i\in \mathbb{Z}}K_{\omega_i(y)}: \phi(\omega_i(y))(x_i)=x_{i+1} \text{ for each }i\in\mathbb{Z}\}.$$ For simplicity, we identify $(\Omega\times X)^{\mathbb{Z}}$ with $\Omega^{\mathbb{Z}}\times X^{\mathbb{Z}}$ as follow: $$((\omega_i,x_i))_{i\in \mathbb{Z}}\in (\Omega\times X)^{\mathbb{Z}}\simeq ((\omega_i)_{i\in \mathbb{Z}}, (x_i)_{i\in \mathbb{Z}})\in \Omega^{\mathbb{Z}}\times X^{\mathbb{Z}}.$$ Clearly $$\pi_1^{-1}(y)\subseteq \bar{\pi}^{-1}(\pi_2(y))\subseteq \{\pi_2(y)\}\times \bar{K}_y.$$ Put $\kappa=\frac{1}{8r}$, we have the following claim. [*Claim 2.*]{} For $y\in Y_0$, $\bar{K}_{y}$ is a compact subset of $(X^{\mathbb{Z}},\rho)$ and there exists a Mycielski subset $D_y$ of $\bar{K}_y$ such that for each pair $((x^1_i)_{i\in \mathbb{Z}},(x_i^2)_{i\in \mathbb{Z}})$ of distinct points in $D_y$, we have $$\begin{aligned} \label{sat-LY} \begin{aligned} &\limsup \limits_{n\rightarrow +\infty} \rho((\phi(n, {\omega_i(y)}, x_i^1))_{i\in \mathbb{Z}}, (\phi(n, {\omega_i(y)}, x_i^2))_{i\in \mathbb{Z}})\ge 4\kappa \text{ and }\\ &\liminf \limits_{n\rightarrow +\infty} \rho((\phi(n, {\omega_i(y)}, x_i^1))_{i\in \mathbb{Z}}, (\phi(n, {\omega_i(y)}, x_i^2))_{i\in \mathbb{Z}})=0 \end{aligned} \end{aligned}$$ where $\omega\in \Omega$. Let $y\in Y_0$. First by the choice of $Y_0$, $\phi(\omega_i(y)):(X,d)\rightarrow (X,d)$ is continuous and $K_{\omega_i(y)}$ is compact subset of $(X,d)$ for each $i\in\mathbb{Z}$, where $(\omega_i(y))_{i\in \mathbb{Z}}:=\pi_2(y)$. Thus $\prod_{i\in \mathbb{Z}}K_{\omega_i(y)}$ is a compact subset of $(X^{\mathbb{Z}},\rho)$. Since $\bar{K}_y$ is a closed subset of $\prod_{i\in \mathbb{Z}}K_{\omega_i(y)}$, $\bar{K}_y$ is also a compact subset of $(X^{\mathbb{Z}},\rho)$. We define $$\begin{aligned} E_y&=\{ (x_i)_{i\in \mathbb{Z}}\in \bar K_y: \bar{\mu}_y(\{ \pi_2(y)\} \times (U\cap {\bar K_y}))>0 \text{ for any open neighborhood $U$ of } \\ &\hskip1cm (x_i)_{i\in \mathbb{Z}} \text{ in } (X^{\mathbb{Z}},\rho)\}\end{aligned}$$ Since $\bar{\mu}_y(\pi_1^{-1}(y))=1$ and $\pi_1^{-1}(y)\subseteq \{\pi_2(y)\}\times \bar{K}_y$, we have $\bar{\mu}_y(\{\pi_2(y)\}\times \bar{K}_y)=1$. Moreover, $E_y$ is a closed subset of $\bar{K}_y$ since $\bar{K}_y$ is compact. Thus $E_y$ is compact and $\bar{\mu}_y(\{\pi_2(y)\}\times E_y)=1$. Note that $\bar {\mu}_y$ is non-atomic, $E_y$ is a perfect set. Hence, $(E_y,\rho)$ is a perfect compact metric space. For $n,m\in\mathbb{N}$, put $$\begin{aligned} D_n(y)=\{&((x_i^1)_{i\in \mathbb{Z}},(x_i^2)_{i\in \mathbb{Z}})\in E_y\times E_y: \text{ there exists $k\ge n$ }\\ &\hskip0.5cm \text{ such that }\rho\Big((\phi(k, {\omega_i(y)}, x_i^1))_{i\in \mathbb{Z}}, (\phi(k, {\omega_i(y)}, x_i^2))_{i\in \mathbb{Z}}\Big)>4\kappa\}\end{aligned}$$ and $$\begin{aligned} P_{n,m}(y)=\{&((x_i^1)_{i\in \mathbb{Z}},(x_i^2)_{i\in \mathbb{Z}})\in E_y\times E_y: \text{ there exists $p\ge n$ }\\ &\hskip0.5cm \text{ such that }\rho\left((\phi(p, {\omega_i(y)}, x_i^1))_{i\in \mathbb{Z}}, (\phi(p, {\omega_i(y)}, (x_i^2))_{i\in \mathbb{Z}}\right) <\frac{1}{m} \}. \end{aligned}$$ Since $(x_i)_{i\in \mathbb{Z}}\mapsto (\phi(n, {\omega_i(y)}, x_i))_{i\in \mathbb{Z}}$ is a continuous map from $(X^{\mathbb{Z}},\rho)$ to itself, $D_n(y)$ and $P_{n,m}(y)$ are both open subsets of $(E_y\times E_y,\rho\times \rho)$. Let $$C(y)=\left( \bigcap_{n=1}^\infty D_n(y)\right)\cap \left(\bigcap_{m=1}^\infty (\bigcap_{n=1}^\infty P_{n,m}(y))\right).$$ Then $C(y)$ is a $G_\delta$ subset of $E_{y}\times E_y$ and for any $((x^1_i)_{i\in \mathbb{Z}},(x_i^2)_{i\in \mathbb{Z}})\in C(y)$, $((x^1_i)_{i\in \mathbb{Z}},(x_i^2)_{i\in \mathbb{Z}})$ satisfies . We claim $C(y)$ is dense in $E_{y}\times E_y$. If this is not true, then there exist open subsets $U_1$, $U_2$ of $X^{\mathbb{Z}}$ such that $(U_1\times U_2)\cap (E_y\times E_y)\neq \emptyset$ and $(U_1\times U_2)\cap C(y)=\emptyset$. Since $\bar{\mu}_y(\{\pi_2(y)\}\times E_y)=1$, one has $$\begin{aligned} &\bar{\mu}_y\times \bar{\mu}_y((\pi_2(y)\times (U_1\cap E_y))\times (\pi_2(y)\times (U_2\cap E_y)))\\ =&\bar{\mu_y}(\pi_2(y)\times (U_1\cap E_y))\bar{\mu_y}(\pi_2(y)\times (U_2\cap E_y))\\ =&\bar{\mu_y}(\pi_2(y)\times U_1)\bar{\mu_y}(\pi_2(y)\times U_2)>0.\end{aligned}$$ Combing this with the fact $\bar{\mu}_y\times\bar{\mu}_y(W)=1$, we have $$\begin{aligned} &\bar{\mu}_y\times \bar{\mu}_y(\left((\pi_2(y)\times (U_1\cap E_y))\times (\pi_2(y)\times (U_2\cap E_y))\right)\cap W)\\ =&\bar{\mu}_y\times \bar{\mu}_y((\pi_2(y)\times U_1)\times (\pi_2(y)\times U_2))>0. \end{aligned}$$ Particularly, $((\pi_2(y)\times (U_1\cap E_y))\times (\pi_2(y)\times (U_2\cap E_y))\cap W\neq \emptyset$. Hence there exist $(z^1_i)_{i\in \mathbb{Z}}\in U_1\cap E_y$ and $(z^2_i)_{i\in \mathbb{Z}}\in U_2\cap E_y$ such that $(((\omega_i(y),z_i^1))_{i\in\mathbb{Z}}, ((\omega_i(y),z_i^2))_{i\in \mathbb{Z}})\in W$. Let $n,m\in \mathbb{N}$. Since $((\omega_i(y),z_i^1))_{i\in\mathbb{Z}}, ((\omega_i(y),z_i^2))_{i\in \mathbb{Z}})\in W$, we have $$\lim_{\ell\rightarrow \infty} \frac{1}{\ell}\sum_{i=0}^{\ell-1} 1_{B_r}((\bar{\Phi}\times \bar{\Psi})^i(((\omega_i(y),z_i^1))_{i\in\mathbb{Z}}, ((\omega_i(y),z_i^2))_{i\in \mathbb{Z}})=\bar{\mu}\times_Y \bar{\mu}(B_r)>0$$ Hence there exists $k\ge n$ such that $(\bar{\Phi}\times \bar{\Phi})^k(((\omega_i(y),z_i^1))_{i\in\mathbb{Z}}, ((\omega_i(y),z_i^2))_{i\in \mathbb{Z}})\in B_r$. Note that $$\bar{\Phi}^k(((\omega_i(y),z_i^j))_{i\in\mathbb{Z}}= ((\theta^k\omega_i(y), \phi(k, {\omega_i(y)}, z_i^j)))_{i\in \mathbb{Z}}$$ for $j=1,2$. Thus, we have $$\left( ((\theta^k\omega_i(y), \phi(k, {\omega_i(y)}, z_i^j)))_{i\in \mathbb{Z}},((\theta^k\omega_i(y), \phi(k, {\omega_i(y)}, z_i^j)))_{i\in \mathbb{Z}}\right)\in B_r.$$ Thus $\rho((\phi(k, {\omega_i(y)}, z_i^j))_{i\in \mathbb{Z}}, (\phi(k, {\omega_i(y)}, z_i^j))_{i\in \mathbb{Z}})\ge \frac{1}{r}> 4\kappa$. Hence $$\label{dny} ((z^1_i)_{i\in \mathbb{Z}},(z^2_i)_{i\in \mathbb{Z}})\in D_n(y).$$ Since $((\omega_i(y),z_i^1))_{i\in\mathbb{Z}}, ((\omega_i(y),z_i^2))_{i\in \mathbb{Z}})\in W$, we have $$\lim_{\ell\rightarrow \infty} \frac{1}{\ell}\sum_{i=0}^{\ell-1} 1_{A_m}((\bar{\Phi}\times \bar{\Phi})^i(\bar{k}_1,\bar{k}_2))=\bar{\mu}\times_Y \bar{\mu}(A_m)>0.$$ Thus there exits $p\ge n$ such that $(\bar{\Phi}\times \bar{\Phi})^{p}(((\omega_i(y),z_i^1))_{i\in\mathbb{Z}}, ((\omega_i(y),z_i^2))_{i\in \mathbb{Z}})\in A_m$. Note that $$\bar{\Phi}^p(((\omega_i(y),z_i^j))_{i\in\mathbb{Z}}= ((\theta^p\omega_i(y), \phi(p, {\omega_i(y)}, z_i^j)))_{i\in \mathbb{Z}}$$ for $j=1,2$. We have $$\left( ((\theta^p\omega_i(y), \phi(p, {\omega_i(y)}, z_i^j)))_{i\in \mathbb{Z}},((\theta^p\omega_i(y), \phi(p, {\omega_i(y)}, z_i^j)))_{i\in \mathbb{Z}}\right)\in A_m.$$ Thus $\rho((\phi(p, {\omega_i(y)}, z_i^j))_{i\in \mathbb{Z}}, (\phi(p,{\omega_i(y)}, z_i^j))_{i\in \mathbb{Z}})<\frac{1}{m}$. Hence $$\label{pnmy} ((z^1_i)_{i\in \mathbb{Z}},(z^2_i)_{i\in \mathbb{Z}})\in P_{n,m}(y).$$ Since and are true for any $n,m\in \mathbb{N}$, we have $((z^1_i)_{i\in \mathbb{Z}},(z^2_i)_{i\in \mathbb{Z}})\in C(y)$. Therefore $((z^1_i)_{i\in \mathbb{Z}},(z^2_i)_{i\in \mathbb{Z}})\in (U_1\times U_2)\cap C(y)$, a contradiction with $(U_1\times U_2)\cap C(y)=\emptyset$. This shows that $C(y)$ is a dense $G_\delta$ subset of $E_{y}\times E_y$. Using Lemma \[Myc\] for $(C(y), E_y)$, there exists a dense Mycielski subset $D_y\subseteq E_y$ such that $D_y\times D_y\subseteq C\cup \Delta_{E_y}$, where $\Delta_{E_y}=\{ (\bar{x},\bar{x}):\bar{x}\in E_y\}$. Clearly, $D_y\subseteq E_y\subseteq \bar{K}_y$ and if $((x^1_i)_{i\in \mathbb{Z}},(x_i^2)_{i\in \mathbb{Z}})$ is a pair of distinct points in $D_y$, then $((x^1_i)_{i\in \mathbb{Z}},(x_i^2)_{i\in \mathbb{Z}})\in C(y)$ and hence $((x^1_i)_{i\in \mathbb{Z}},(x_i^2)_{i\in \mathbb{Z}})$ satisfies . This completes the proof of Claim 2. [*Step 4.*]{} For $P$-a.e. $\omega\in \Omega$ there exists a Mycieski chaotic set $S_\omega\subset K_\omega$ for $(\omega,\phi)$. Note that $$\Pi_\Omega\circ \pi_2:(Y,\mathcal{D},\nu,S)\rightarrow (\Omega,\mathcal{F},\mathcal{B}_P,\theta)$$ is a factor map between two Lebesgue systems. Since $\nu(Y_0)=1$, there exists $\Omega_1\in \mathcal{B}_P$ satisfying $P(\Omega_1)=1$ and $(\Pi_\Omega\circ \pi_2)^{-1}(\omega)\cap Y_0\neq \emptyset$ for each $\omega\in \Omega_1$. Let $\omega\in \Omega_1$. Then there exists $y\in Y_0$ such that $\Pi_\Omega\circ \pi_2(y)=\omega$. Since $\pi_2(y)=(\omega_i(y))_{i\in \mathbb{Z}}$, we have $\omega=\omega_0(y)$. By Claim 2, we can find a Mycielski subset $D_y$ of $\bar{K}_y$ such that for each pair $((x^1_i)_{i\in \mathbb{Z}},(x_i^2)_{i\in \mathbb{Z}})$ of distinct points in $D_y$, we have $$\begin{aligned} \label{cha-eq} \begin{aligned} &\limsup \limits_{n\rightarrow +\infty} \rho((\phi(n,{\omega_i(y)}, x_i^1))_{i\in \mathbb{Z}}, (\phi(n,{\omega_i(y)}, x_i^2))_{i\in \mathbb{Z}})\ge 4\kappa \text{ and }\\ &\liminf \limits_{n\rightarrow +\infty} \rho((\phi(n,{\omega_i(y)}, x_i^1))_{i\in \mathbb{Z}}, (\phi(n,{\omega_i(y)}, x_i^2))_{i\in \mathbb{Z}})=0. \end{aligned} \end{aligned}$$ Let $\eta:\bar{K}_y\rightarrow K_{\omega}$ be the natural projection of coordinate with $\eta((x_i)_{i\in \mathbb{Z}})=x_0$ for $(x_i)_{i\in \mathbb{Z}}\in X^{\mathbb{Z}}$. Put $S_\omega=\eta(D_y)$. Then $S_\omega\subseteq K_{\omega}$. In the following we show that $S_\omega$ is a Mycieski chaotic set for $(\omega,f)$. Firstly we claim the map $\eta: D_y\rightarrow S_\omega$ is injective. If this is not true, then there exist two distinct points $(x_i^1)_{i\in \mathbb{Z}},(x_i^2)_{i\in \mathbb{Z}}$ in $D_y$ such that $\eta((x_i^1)_{i\in \mathbb{Z}})=\eta((x_i^2)_{i\in \mathbb{Z}})$, i.e., $x_0^1=x_0^2$. Since $(x_i^1)_{i\in \mathbb{Z}},(x_i^2)_{i\in \mathbb{Z}}\in \bar{K}_y$, we have $$\begin{aligned} \label{eq-n} x_i^1=\phi(i, {\omega_0(y)}, x_0^1)=\phi(i, {\omega_0(y)}, x_0^2)=x_i^2 \end{aligned}$$ for $i\in \mathbb{N}$. Now for $n\in \mathbb{N}$, we have $$(\phi(n,{\omega_i(y)}, x_i^1))_{i\in \mathbb{Z}}=(x^1_{i+n})_{i\in \mathbb{Z}} \text{ and } (\phi(n,{\omega_i(y)}, x_i^2))_{i\in \mathbb{Z}}=(x^2_{i+n})_{i\in \mathbb{Z}}.$$ Thus $$\lim_{n\rightarrow +\infty} \rho((\phi(n,{\omega_i(y)}, x_i^1))_{i\in \mathbb{Z}}, (\phi(n,{\omega_i(y)}, x_i^2))_{i\in \mathbb{Z}})=\lim_{n\rightarrow +\infty}\rho((x^1_{i+n})_{i\in \mathbb{Z}},(x^2_{i+n})_{i\in \mathbb{Z}} )=0$$ where the last equality comes from , a contradiction with $$\lim_{n\rightarrow +\infty} \rho((\phi(n,{\omega_i(y)}, x_i^1))_{i\in \mathbb{Z}}, (\phi(n,{\omega_i(y)}, x_i^2))_{i\in \mathbb{Z}})\ge \kappa$$ since $(x_i^1)_{i\in \mathbb{Z}},(x_i^2)_{i\in \mathbb{Z}}$ are distinct points in $D_y$. This shows that $\eta: D_y\rightarrow S_\omega$ is injective. Since $D_y$ is a Mycielski set, $D_y=\bigcup_{j\in \mathbb{N}}C_j$ where each $C_j$ is cantor set. Since $\eta:(C_j,\rho)\rightarrow (\eta(C_j),d)$ is one to one surjective continuous map and $C_j$ is compact subset of $(X^{\mathbb{Z}},\rho)$, $\eta:C_j\rightarrow \eta(C_j)$ is homeomorphism. Thus $\eta(C_j)$ is a cantor set. Hence $S_\omega=\bigcup_{j\in \mathbb{N}} \eta(C_j)$ is also a mycielski set of $K_\omega$. Finally, we prove that $S_\omega$ is a chaotic set for $(\omega,\phi)$. Let $(x^1,x^2)$ is a pair of distinct points in $S_\omega$. Then there exist $(x_i^1)_{i\in \mathbb{Z}}, (x_i^2)_{i\in \mathbb{Z}}\in D_y$ such that $x_0^1=x^1$ and $x_0^2=x^2$. On the one hand, by $$\begin{aligned} &\hskip0.5cm \liminf \limits_{n\rightarrow +\infty} d(\phi(n, {\omega}, x^1), \phi(n, {\omega}, x^2))=\liminf \limits_{n\rightarrow +\infty} d((\phi(n, {\omega_0(y)}, x_0^1), \phi(n, {\omega_0(y)}, x_0^2))\\ &\le \liminf \limits_{n\rightarrow +\infty} \rho((\phi(n, {\omega_i(y)}, x_i^1))_{i\in \mathbb{Z}}, (\phi(n, {\omega_i(y)}, x_i^2))_{i\in \mathbb{Z}})=0. \end{aligned}$$ Thus $\liminf \limits_{n\rightarrow +\infty} d(f_{\omega}^n(x^1), f_{\omega}^n(x^2))=0$. On the other hand, we take $L\in \mathbb{N}$ such that $\sum_{i\in \mathbb{Z},|i|\ge L} \frac{1}{2^i}<\frac{1}{2}\kappa$. By $$\limsup \limits_{n\rightarrow +\infty} \rho((\phi(n, {\omega_i(y)}, x_i^1))_{i\in \mathbb{Z}}, (\phi(n, {\omega_i(y)}, x_i^2))_{i\in \mathbb{Z}})\ge 4\kappa.$$ Hence there exist natural numbers $n_j,j\in \mathbb{N}$ such that $L<n_1<n_2<\cdots$ and $$\rho((\phi(n_j, {\omega_i(y)}, x_i^1))_{i\in \mathbb{Z}}, (\phi(n_j, {\omega_i(y)}, x_i^2))_{i\in \mathbb{Z}})>\frac{7}{2}\kappa.$$ Since $(x_i^1)_{i\in \mathbb{Z}}, (x_i^2)_{i\in \mathbb{Z}}\in \bar{K}_y$, we have that $\phi(\omega_i(y))x_i^1=x_{i+1}^1$ and $\phi(\omega_i(y))x_i^2=x_{i+1}^2$ for $i\in \mathbb{Z}$. Moreover for each $j\in \mathbb{N}$, $(\phi(n_j,{\omega_i(y)}, x_i^1))_{i\in \mathbb{Z}}=(x_{i+n_j}^1)_{i\in \mathbb{Z}}$ and $(\phi(n_j, {\omega_i(y)}, x_i^2))_{i\in \mathbb{Z}}=(x_{i+n_j}^2)_{i\in \mathbb{Z}}$. Thus $\rho((x_{i+n_j})_{i\in \mathbb{Z}}, (x_{i+n_j}^2)_{i\in \mathbb{Z}})>\frac{7}{2}\kappa$ for each $j\in \mathbb{N}$. Then $$\sum_{i,|i|\le L} \frac{1}{2^|i|}\frac{d(x_{i+n_j}^1,x_{i+n_j}^2)}{1+d(x_{i+n_j}^1,x_{i+n_j}^2)}>\frac{7}{2}\kappa-\sum_{i\in \mathbb{Z},|i|\ge L} \frac{1}{2^i} \ge 3\kappa$$ for $j\in \mathbb{N}$. Thus for each $j\in \mathbb{N}$, there exists $i_j\in \mathbb{Z}$ with $|i_j|\le L$ such that $$\frac{d(x_{i_j+n_j}^1,x_{i_j+n_j}^2)}{1+d(x_{i+n_j}^1,x_{i_j+n_j}^2)}\ge \kappa,$$ which implies $d(x_{i_j+n_j}^1,x_{i_j+n_j}^2)\ge \kappa$. Let $m_j=i_j+n_j>0$ for $j\in \mathbb{N}$. Then $d(x_{m_j}^1,x_{m_j}^2)\ge \kappa$ and $\lim \limits_{j\rightarrow +\infty} m_j=+\infty$. Since $$x_{m_j}^1=\phi(m_j, {\omega_0(y)}, x_0^1)=\phi(m_j, {\omega}, x^1) \text{ and } x_{m_j}^2=\phi(m_j, {\omega_0(y)}, x_0^2)=\phi(m_j, {\omega}, x^2) \text{ }\forall j\in \mathbb{N}$$ we have that $d(\phi(, m_j, \omega, x^1),\phi(m_j, \omega, x^2))=d(x_{m_j}^1,x_{m_j}^2)\ge \kappa$, $j\in \mathbb{N}$ and hence $$\limsup \limits_{n\rightarrow +\infty}d(\phi(n, \omega, x^1),\phi(n, \omega, x^2))\ge \limsup \limits_{j\rightarrow +\infty} d(\phi(m_j, \omega, x^1),\phi(m_j, \omega, x^2))\ge \kappa.$$ Summerizing the above , we have that $$\liminf \limits_{n\rightarrow +\infty} d(\phi(n, {\omega}, x^1), \phi(n, {\omega}, x^2))=0 \text{ and }\limsup \limits_{n\rightarrow +\infty}d(\phi(n, \omega, x^1),\phi(n, \omega, x^2))\ge \kappa$$ for a pair $(x^1,x^2)$ of distinct points in $S_\omega$. This shows that $S_\omega$ is a chaotic set for $(\omega,f)$ and completes the proof of the theorem. $\square$ Proof of Basic Lemmas and Propositions ====================================== In this appendix, we give the proof of Lemma \[na=zero\], and several basic facts and their proofs about the ergodic decomposition, which we used in previous sections. [**Proof of Lemma \[na=zero\]**]{} For $n\in \mathbb{Z}$, let $\Pi_{n,X}:\bar{X}\rightarrow X$ with $\Pi_{n,X}((x_i)_{i\in \mathbb{Z}})=x_n$ for $\bar{x}=(x_i)_{i\in \mathbb{Z}}\in X^{\mathbb{Z}}$. Put $$\bar{\mathcal{B}}_n=\Pi_{n,X}^{-1}(\mathcal{B}_X)$$ and $\mathcal{D}_{\bar{X}}=\bigcup_{i\in \mathbb{Z}}\bar{\mathcal{B}}_i$. Clearly, $$\begin{aligned} \label{increase} \bar{\mathcal{B}}_i\supseteq \bar{\mathcal{B}}_{i+1} \end{aligned}$$ for each $i\in \mathbb{Z}$. Then $\mathcal{D}_{\bar{X}}$ is a sub-algebra of $\bar{\mathcal{B}}$ and the $\sigma$-algebra generated by $\mathcal{D}_{\bar{X}}$ is equal to $\bar{\mathcal{B}}$ (mod $\bar{\mu}$). Thus for any $A\in \bar{\mathcal{B}}$ and $\epsilon>0$, there exists $A_\epsilon\in \mathcal{D}_{\bar{X}}$ such that $\bar{\mu}(A\Delta A_\epsilon)<\epsilon$, where $A\Delta A_\epsilon=(A\setminus A_\epsilon)\cup (A_\epsilon\setminus A)$ (see [@W Theorem 0.7]). Let $\alpha=\{A_1,A_2,\cdots, A_k\}\in \mathcal{P}_{\bar{X}}$ with $k\ge 2$. For $m\in \mathbb{N}$, there exists $\delta=\delta(k,m)>0$ such that for any $\beta=\{B_1,B_2,\cdots,B_k\}\in \mathcal{P}_{\bar{X}}$, if $$\bar{\mu}(\alpha\Delta \beta):=\sum_{i=1}^k \bar{\mu}(A_i\Delta B_k)<\delta$$ then $H_\mu(\alpha|\beta)<\frac{1}{m}$ (see for example [@W Lemma 4.15]). For $i=1,2,\cdots,k-1$, we take $A_i'\in \mathcal{D}_{\bar{X}}$ with $\bar{\mu}(A_i\Delta A_i')<\frac{\delta}{k^3}$. Let $A_k'=X\setminus \bigcup_{j=1}^{k-1}A_j'$. Then $A_k'\in \mathcal{D}_{\bar{X}}$ and $$\bar{\mu}(A_k\Delta A_k')\le \mu(\bigcup_{j=1}^{k-1}A_j\Delta A_j')\le \sum_{j=1}^{k-1} \mu(A_j\Delta A_j')<\frac{\delta}{k^2}$$ where the first inequality comes from the fact $$A_k\Delta A_k'=(X\setminus \bigcup_{j=1}^{k-1}A_j)\Delta (X\setminus \bigcup_{j=1}^{k-1}A_j')\subseteq \bigcup_{j=1}^{k-1} (A_j\Delta A_j').$$ Then let $C_1=A_1'$ and $$C_i=A_i'\setminus \bigcup_{j=1}^{i-1}C_j=A_i'\setminus \bigcup_{j=1}^{i-1}A_j'$$ for $i=2,\cdots,k$. Clearly, $C_i\in \mathcal{D}_{\bar{X}}$ for each $i\in \{1,2,\cdots,k\}$ and $\gamma:=\{C_1,C_2,\cdots,C_k\}\in \mathcal{P}_{\bar{X}}$. Since $\mathcal{D}_{\bar{X}}=\bigcup_{i\in \mathbb{Z}}\bar{\mathcal{B}}_i$, by we have $i_*\in \mathbb{N}$ such that $C_i\in \bar{\mathcal{B}}_{-i_*}$ for each $i\in \{1,2,\cdots,k\}$. Thus, there exists $D_i\in \mathcal{B}$ such that $\Pi_{-i_*,X}^{-1}(D_i)=C_i$ for $i=1,2,\cdots,k$. Let $\tau=\{ D_1,D_2,\cdots,D_k\}$. Then $\tau\in \mathcal{P}_X$ and $$\gamma=\Pi_{-i_*,X}^{-1}\tau=\bar{T}^{-i_*}\Pi_{0,X}^{-1}\tau=\bar{T}^{-i_*}\Pi_X^{-1}\tau.$$ For $i\in \{1,2,\cdots,k\}$, we have $$\begin{aligned} A_i\Delta C_i&=(A_i\setminus C_i)\cup (C_i\setminus A_i)\subseteq (A_i\setminus (A_i'\setminus \bigcup_{j=1}^{i-1}A_j'))\cup (A_i'\setminus A_i)\\ &=(A_i\setminus A_i')\cup (A_i\cap \bigcup_{j=1}^{i-1}A_j') \cup (A_i'\setminus A_i)=(A_i\Delta A_i') \cup (\bigcup_{j=1}^{i-1} A_i\cap A_j')\\ &\subseteq (A_i\Delta A_i') \cup (\bigcup_{j=1}^{i-1} (A_i\cap A_j) \cup (A_i\cap (A_j'\setminus A_j)) )=(A_i\Delta A_i') \cup (\bigcup_{j=1}^{i-1} A_i\cap (A_j'\setminus A_j) \\ &\subseteq \bigcup_{j=1}^i (A_j\Delta A_j').\end{aligned}$$ Moreover, $\bar{\mu}(A_i\Delta C_i)\le \sum_{j=1}^i\mu(A_j\Delta A_j')<\frac{\delta}{k}$. Thus $\bar{\mu}(\alpha \Delta \gamma)=\sum_{i=1}^k \bar{\mu}(A_i\Delta C_i)<\delta$. By the choice of $\delta$, we know $H_{\bar{\mu}}(\alpha|\gamma)<\frac{1}{m}$. Now for $n\in \mathbb{N}$, we have $$\begin{aligned} &\hskip0.5cm H_{\bar{\mu}}(\bigvee_{i=0}^{n-1}{\bar{T}}^{-i}\alpha|\Pi_X^{-1}(\mathcal{B}))= H_{\bar{\mu}}(\bigvee_{i=0}^{n-1}{\bar{T}}^{-i}\alpha|\bar{\mathcal{B}}_0)= H_{\bar{\mu}}(\bigvee_{i=0}^{n-1}{\bar{T}}^{-(i+i_*)}\alpha|{\bar{T}}^{-i_*}\bar{\mathcal{B}}_0)\\ &\le H_{\bar{\mu}}(\bigvee_{i=0}^{n-1}{\bar{T}}^{-(i+i_*)}\alpha|{\bar{T}}^{-i_*}\Pi_X^{-1}(\bigvee_{i=0}^{n-1}T^{-i}\tau))= H_{\bar{\mu}}(\bigvee_{i=0}^{n-1}{\bar{T}}^{-(i+i_*)} \alpha|{\bar{T}}^{-i_*}(\bigvee_{i=0}^{n-1}\bar{T}^{-i}\Pi_X^{-1}\tau))\\ &=H_{\bar{\mu}}(\bigvee_{i=0}^{n-1}{\bar{T}}^{-(i+i_*)} \alpha|\bigvee_{j=0}^{n-1}\bar{T}^{-j}\gamma)\le H_{\bar{\mu}}(\bigvee_{i=0}^{n+i_*-1}{\bar{T}}^{-i} \alpha|\bigvee_{j=0}^{n-1}\bar{T}^{-j}\gamma)\\ &\le \sum_{i=0}^{n+i_*-1}H_{\bar{\mu}}(\bar{T}^{-i} \alpha|\bigvee_{j=0}^{n-1}\bar{T}^{-i}\gamma)\le \left( \sum_{i=0}^{n-1}H_{\bar{\mu}}(\bar{T}^{-i}\alpha|\bar{T}^{-i}\gamma) +\sum_{i=n}^{n+i_*-1}H_{\bar{\mu}}(\bar{T}^{-i}\alpha)\right) \\ &=\left( nH_{\bar{\mu}}(\alpha|\gamma) +i_*H_{\bar{\mu}}(\alpha)\right). \end{aligned}$$ Using the above inequality, we have $$\begin{aligned} h_{\bar{\mu}}(\bar{T},\alpha|\Pi_X^{-1}(\mathcal{B}))&=\lim_{n\rightarrow +\infty} \frac{1}{n}H_{\bar{\mu}}(\bigvee_{i=0}^{n-1}{\bar{T}}^{-i}\alpha|\Pi_X^{-1}(\mathcal{B})) \le \lim_{n\rightarrow +\infty} \frac{1}{n} \left( nH_{\bar{\mu}}(\alpha|\gamma) +i_*H_{\bar{\mu}}(\alpha)\right)\\ &=H_{\bar{\mu}}(\alpha|\gamma)<\frac{1}{m}. \end{aligned}$$ Since $m$ is arbitrary, $h_{\bar{\mu}}(\bar{T},\alpha|\Pi_X^{-1}(\mathcal{B}))=0$. This implies $h_{\bar{\mu}}(\bar{T}|\Pi_X)=0$ since $\alpha$ is arbitrary. The proof is complete. $\square$ Next, we investigate the ergodic decomposition of conditional entropy. Let $(X,\mathcal{B},\mu,T)$ be an invertible Lebesgue system. We consider the sub-$\sigma$-algebra $$I_\mu(T)=\{ A\in \mathcal{B}: \mu(T^{-1}B\Delta B)=0\}.$$ It is well known that there exists a factor map $\phi:(X,\mathcal{B},\mu,T)\rightarrow (E,\mathcal{E},\eta,id_E)$ between two invertible Lebesgue systems such that $\phi^{-1}(\mathcal{E})=I_\mu(T)$ (mod $\mu$) and $(X,\mathcal{B}_e,\mu_e,T)$ is an ergodic invertible Lebesgue system for $\eta$-a.e $e\in E$, where $id_E$ is the identity map from $E$ to itself, $\mu=\int_E \mu_e d \eta(e)$ is the disintegration of $\mu$ relative to the factor $(E,\mathcal{E},\eta,id_E)$, and $(X,\mathcal{B}_e,\mu_e)$ is the corresponding Lebesgue space for $e\in E$ in this disintegration (see [@R], [@G Theorem 3.42]). More precisely, conditional probability measures $\{\mu_e\}_{e\in E}$ with the following properties: - $\mu_e$ is a Lebesgue measure on $X$ with $\mu_e(\phi^{-1}(e))=1$ for all $y\in Y$. - for each $f \in L^1(X,\mathcal{B},\mu)$, one has $f \in L^1(X,\mathcal{B}_e,\mu_e)$ for $\eta$-a.e. $e\in E$, the map $e \mapsto \int_X f\,d\mu_e$ is in $L^1(E,\eta)$ and $\int_E \left(\int_X f\,d\mu_e \right)\, d\eta(e)=\int_X f \,d\mu$. Particularly, for each $A\in \mathcal{B}$, one has $A\in \mathcal{B}_e$ for $\eta$-a.e. $e\in E$. The disintegration $\mu=\int_E \mu_e d \eta(e)$ is called the ergodic decomposition of $\mu$. The following result is well known (see [@R Theorem 8.11], [@J] or [@G Theorem 15.12]). \[edep\] (Ergodic decomposition of entropy for partition) Let $(X,\mathcal{B},\mu,T)$ be an invertible Lebesgue system and $\mu=\int_E \mu_e d \eta(e)$ be the ergodic decomposition of $\mu$. Then for any $\alpha\in \mathcal{P}_X(\mathcal{B})$, $$h_\mu(T,\alpha)=\int_E h_{\mu_e}(T,\alpha) d \eta(e).$$ \[relative-entropy\] Let $(X,\mathcal{B},\mu, T)$ be a MDS. Then for any $\alpha,\beta\in \mathcal{P}_X$, $$\inf_{n\ge 1}\frac{1}{n} H_\mu(\bigvee_{i=0}^{n-1}T^{-i}\alpha|\bigvee_{i=0}^{n-1}T^{-i}\beta)=h_\mu(T,\alpha\vee \beta)-h_\mu(T,\alpha).$$ For $\alpha,\beta\in \mathcal{P}_X$, let $a_n=H_\mu(\bigvee_{i=0}^{n-1}T^{-i}\alpha|\bigvee_{i=0}^{n-1}T^{-i}\beta)$ for $n\in \mathbb{N}$. Then, for $n,m\in \mathbb{N}$, we have $$\begin{aligned} a_{n+m}&=H_\mu(\bigvee_{i=0}^{n+m-1}T^{-i}\alpha|\bigvee_{i=0}^{n+m-1}T^{-i}\beta)\\ &\le H_\mu(\bigvee_{i=0}^{n-1}T^{-i}\alpha|\bigvee_{i=0}^{n+m-1}T^{-i}\beta)+H_\mu(T^{-n}(\bigvee_{i=0}^{m-1}T^{-i}\alpha)|\bigvee_{i=0}^{n+m-1}T^{-i}\beta)\\ &\le a_n+H_\mu(T^{-n}(\bigvee_{i=0}^{m-1}T^{-i}\alpha)|T^{-n}(\bigvee_{i=0}^{m-1}T^{-i}\beta))\\ &=a_n+a_m.\end{aligned}$$ By Theorem 4.9 in [@W], $\inf_{n\ge 1}\frac{1}{n}a_n=\lim_{n\rightarrow +\infty}\frac{1}{n}a_n$. Thus, $$\begin{aligned} &\hskip0.5cm \inf_{n\ge 1}\frac{1}{n} H_\mu(\bigvee_{i=0}^{n-1}T^{-i}\alpha|\bigvee_{i=0}^{n-1}T^{-i}\beta)\\ &=\lim_{n\rightarrow +\infty}\frac{1}{n} H_\mu(\bigvee_{i=0}^{n-1}T^{-i}\alpha|\bigvee_{i=0}^{n-1}T^{-i}\beta)\\ &=\lim_{n\rightarrow +\infty}\left( \frac{1}{n} H_\mu(\bigvee_{i=0}^{n-1}T^{-i}(\alpha\vee \beta))-H_\mu(\bigvee_{i=0}^{n-1}T^{-i}\beta)\right)\\ &=\lim_{n\rightarrow +\infty}\frac{1}{n} H_\mu(\bigvee_{i=0}^{n-1}T^{-i}(\alpha\vee \beta))-\lim_{n\rightarrow +\infty}H_\mu(\bigvee_{i=0}^{n-1}T^{-i}\beta)\\ &=h_\mu(T,\alpha\vee \beta)-h_\mu(T,\beta). \end{aligned}$$ This completes the proof of Lemma. Let $(X,\mathcal{B},\mu,T)$ be an invertible Lebesgue system. A sub-$\sigma$-algebra $\mathcal{C}$ of $\mathcal{B}$ is called [*countably generated*]{} if there exists $\{ A_i\}_{i=1}^\infty \subset \mathcal{C}$ such that $\mathcal{C}$ is the $\sigma$-algebra generated by $\{ A_i\}_{i=1}^\infty$, i.e., $\mathcal{C}$ is the smallest $\sigma$-algebra containing all $A_i, i\in \mathbb{N}$. \[cond-ed\] (Ergodic decomposition of conditional entropy for partition) Let $(X,\mathcal{B},\mu,T)$ be an invertible Lebesgue system, $\mu=\int_E \mu_e d \eta(e)$ be the ergodic decomposition of $\mu$, and $\mathcal{C}$ be a countably generated sub-$\sigma$-algebra of $\mathcal{B}$ with $T^{-1}\mathcal{C}\subseteq \mathcal{C}$. Then, for each $\alpha\in \mathcal{P}_X(\mathcal{B})$, $$h_\mu(T,\alpha|\mathcal{C})=\int_E h_{\mu_e}(T,\alpha|\mathcal{C}) d \eta(e).$$ Let $(X,\mathcal{B}_e,\mu_e,T)$ be the corresponding Lebesgue systems for $e\in E$ in the ergodic decomposition of $\mu$. Since $\mathcal{C}$ is a countably generated sub-$\sigma$-algebra of $\mathcal{B}$, there exists $\{ A_i\}_{i=1}^\infty \subset \mathcal{C}$ such that $\mathcal{C}$ is the $\sigma$-algebra generated by $\{ A_i\}_{i=1}^\infty$. Let $\alpha=\{B_1,B_2,\cdots,B_k\}\in \mathcal{P}_X(\mathcal{B})$. Since $A\in \mathcal{B}_e$ for $\eta$-a.e. $e\in E$ when $A\in \mathcal{B}$, one has that for $\eta$-a.e. $e\in E$, $\{B_1,B_2,\cdots,B_k\} \cup \{A_i:i\in \mathbb{N}\}\subseteq\mathcal{B}_e$. Thus for $\eta$-a.e. $e\in E$, $\mathcal{B}_e$ contains the $\sigma$-algebra generated by $\mathcal{C}\cup (\bigcup_{i\in \mathbb{Z}} T^{i}\{B_1,B_2,\cdots,B_k\} )$ since $\mathcal{B}_e$ is $\sigma$-algebra and $T^{-1}\mathcal{B}_e=\mathcal{B}_e$. Let $\beta_j=\bigvee_{i=1}^j \{ A_i,X\setminus A_i\}$ for $j\in \mathbb{N}$. Then for each $n\in \mathbb{N}$, when $j\rightarrow +\infty$, we have $\bigvee_{i=0}^{n-1}T^{-i}\beta_j\nearrow \mathcal{C}$ (mod $\mu$) and $\bigvee_{i=0}^{n-1}T^{-i}\beta_j\nearrow \mathcal{C}$ (mod $\mu_e$) for $\eta$-a.e. $e\in E$. Thus, $$\begin{aligned} H_\mu(\bigvee_{i=1}^{n-1}T^{-i}\alpha|\mathcal{C})=\lim_{j\rightarrow +\infty}H_\mu(\bigvee_{i=1}^{n-1}T^{-i}\alpha|\bigvee_{i=0}^{n-1}T^{-i}\beta_j)=\inf_{j\ge 1}H_\mu(\bigvee_{i=1}^{n-1}T^{-i}\alpha|\bigvee_{i=0}^{n-1}T^{-i}\beta_j).\end{aligned}$$ Moveover, $$\begin{aligned} h_\mu(T,\alpha|\mathcal{C})&=\inf_{n\ge 1} \frac{1}{n}H_\mu(\bigvee_{i=1}^{n-1}T^{-i}\alpha|\mathcal{C})=\inf_{n\ge 1}\left(\inf_{j\ge 1}\frac{1}{n}H_\mu(\bigvee_{i=0}^{n-1}T^{-i}\alpha|\bigvee_{i=0}^{n-1}T^{-i}\beta_j)\right)\\ &=\inf_{j\ge 1}\left(\inf_{n\ge 1}\frac{1}{n}H_\mu(\bigvee_{i=1}^{n-1}T^{-i}\alpha|\bigvee_{i=0}^{n-1}T^{-i}\beta_j)\right)\\ &=\inf_{j\ge 1}\left(h_\mu(T,\alpha\vee \beta_j)-h_\mu(T,\beta_j)\right). \ \ \text{(by lemma \ref{relative-entropy})}\end{aligned}$$ That is $$\begin{aligned} \label{limit-1} \begin{aligned} h_\mu(T,\alpha|\mathcal{C})&=\inf_{j\ge 1}\left(\inf_{n\ge 1}\frac{1}{n}H_\mu(\bigvee_{i=1}^{n-1}T^{-i}\alpha|\bigvee_{i=0}^{n-1}T^{-i}\beta_j)\right)\\ &=\inf_{j\ge 1}\left(h_\mu(T,\alpha\vee \beta_j)-h_\mu(T,\beta_j)\right). \end{aligned}\end{aligned}$$ Similarly, for $\eta$-a.e. $e\in E$, we have $$\begin{aligned} \label{limit-2} \begin{aligned} h_{\mu_e}(T,\alpha|\mathcal{C})&=\inf_{j\ge 1}\left(\inf_{n\ge 1}\frac{1}{n}H_\mu(\bigvee_{i=1}^{n-1}T^{-i}\alpha|\bigvee_{i=0}^{n-1}T^{-i}\beta_j)\right)\\ &=\inf_{j\ge 1}\left(h_{\mu_e}(T,\alpha\vee \beta_j)-h_{\mu_e}(T,\beta_j)\right). \end{aligned}\end{aligned}$$ Using , Lemma \[edep\], Lemma \[relative-entropy\], and , we obtain $$\begin{aligned} h_\mu(T,\alpha|\mathcal{C})&=\inf_{j\ge 1}\left(h_\mu(T,\alpha\vee \beta_j)-h_\mu(T,\beta_j)\right)\\ &=\inf_{j\ge 1}\int_E \left(h_{\mu_e}(T,\alpha\vee \beta_j)-h_{\mu_e}(T,\beta_j)\right)d \eta(e)\\ &=\inf_{j\ge 1}\int_E \inf_{n\ge 1}\frac{1}{n}H_{\mu_e}(\bigvee_{i=1}^{n-1}T^{-i}\alpha|\bigvee_{i=0}^{n-1}T^{-i}\beta_j) d \eta(e)\\ &=\lim_{j\rightarrow +\infty}\int_E \inf_{n\ge 1}\frac{1}{n}H_{\mu_e}(\bigvee_{i=1}^{n-1}T^{-i}\alpha|\bigvee_{i=0}^{n-1}T^{-i}\beta_j) d \eta(e)\\ &=\int_E \lim_{j\rightarrow +\infty}\left(\inf_{n\ge 1}\frac{1}{n}H_{\mu_e}(\bigvee_{i=1}^{n-1}T^{-i}\alpha|\bigvee_{i=0}^{n-1}T^{-i}\beta_j)\right) d \eta(e) \\ &\hskip1cm \ \text{(by Monotone convergence Theorem)}\\ &=\int_E \inf_{j\ge 1}\left(\inf_{n\ge 1}\frac{1}{n}H_{\mu_e}(\bigvee_{i=1}^{n-1}T^{-i}\alpha|\bigvee_{i=0}^{n-1}T^{-i}\beta_j) \right)d \eta(e)\\ &=\int_E h_{\mu_e}(T,\alpha|\mathcal{C}) \, d \eta(e).\end{aligned}$$ This completes the proof of Lemma. Let $(X,\mathcal{B}_X,\mu,T)$ be a Polish system. Let $\mathcal{B}_\mu$ be the completion of the Borel $\sigma$-algebra $\mathcal{B}_X$ with respect to $\mu$. Then $(X,\mathcal{B}_\mu,\mu,T)$ is a Lebesgue system. Put $$\bar{X}=\{ \bar{x}=(x_i)_{i\in \mathbb{Z}}\in X^{\mathbb{Z}}: \, Tx_{i}=x_{i+1}, i\in \mathbb{Z}\}$$ and let $\Pi_X:(\bar{X},\bar{\mathcal{B}_\mu},\bar{\mu},\bar{T})\rightarrow (X,\mathcal{B}_X,\mu,T)$ be the natural extension of $(X,\mathcal{B}_X,\mu,T)$. Let $\bar{\mu}=\int_E \bar{\mu}_e d \eta(e)$ be the ergodic decomposition of $\bar{\mu}$ and $(\bar{X},(\bar{\mathcal{B}_\mu})_e,\bar{\mu}_e,\bar{T})$ be the corresponding Lebesgue systems for $e\in E$ in the ergodic decomposition of $\bar{\mu}$. Since $\Pi_X^{-1}(\mathcal{B}_X)$ is a countably generated sub-$\sigma$-algebra of $\mathcal{B}_{\bar{\mu}}$, one knows that for $\eta$-a.e. $e\in E$, $\Pi_X^{-1}(\mathcal{B}_X)\subset (\bar{\mathcal{B}_{\mu}})_e$. Therefore, for $\eta$-a.e. $e\in E$, letting $\mu_e(A)=\bar{\mu}_e(\Pi_X^{-1}A)$ for $A\in \mathcal{B}_X$, we have that $(X,\mathcal{B}_X,\mu_e,T)$ is an ergodic Polish system. It is clear that for any $f\in L^1(X,\mathcal{B}_X,\mu)$, one has that $f\circ \Pi_X\in L^1(\bar{X},\Pi_X^{-1}(\mathcal{B}_X),\bar{\mu})$. Hence the map $e\in E \mapsto\int_X f(x) d \mu_e(x)(:=\int_{\bar{X}}f\circ \Pi_X d \bar{\mu}_e)$ is $\eta$-measurable and $$\int_E (\int_X f d \mu_e) d \eta(e)=\int_E (\int_{\bar{X}}f\circ \Pi_X d \bar{\mu}_e) d\eta(e)=\int_{\bar{X}} f\circ \Pi_X d \bar{\mu}=\int_X f d \mu.$$ That is, for any $f\in L^1(X,\mathcal{B}_X,\mu)$, one has $$\label{eq-ed-key} \text{the map }e\in E\mapsto \int_X f(x) d \mu_e(x)\text{ is $\eta$-measurable and } \int_E (\int_X f d \mu_e) d \eta(e)=\int_X f d \mu.$$ In this case, we say that $\mu=\int_E \mu_e d \eta(e)$ is [*the ergodic decomposition of $\mu$*]{}. \[ergodic\] Let $\pi:(X,\mathcal{B}_X,\mu,T)\rightarrow (Y,\mathcal{B}_Y,\nu,S)$ be a factor map between two Polish systems and $(Y,\mathcal{B}_Y,\nu,S)$ be ergodic. If $\mu=\int_E \mu_e d \eta(e)$ is the ergodic decomposition of $\mu$, then 1. for $\eta$-a.e. $e\in E$, $\pi:(X,\mathcal{B}_X,\mu_e,T)\rightarrow (Y,\mathcal{B}_Y,\nu,S)$ is a factor map between two Polish systems. 2. $h_\mu(T|\pi^{-1}(\mathcal{B}_Y))=\int_E h_{\mu_e}(T|\pi^{-1}(\mathcal{B}_Y)) d \eta(e)$. That is, $h_\mu(T|\pi)=\int_E h_{\mu_e}(T|\pi) d \eta(e)$. Let $\mathcal{B}_\nu$ be the completion of the Borel $\sigma$-algebra $\mathcal{B}_Y$ with respect to $\nu$. Let $\mathcal{B}_\mu$ be the completion of the Borel $\sigma$-algebra $\mathcal{B}_X$ with respect to $\mu$. Then $\pi:(X,\mathcal{B}_\mu,\mu, T)\rightarrow (Y, \mathcal{B}_\nu,\nu,S)$ is a factor map between two Lebesgue systems. Put $$\bar{X}=\{ \bar{x}=(x_i)_{i\in \mathbb{Z}}\in X^{\mathbb{Z}}: \, Tx_{i}=x_{i+1}, i\in \mathbb{Z}\}$$ and let $\Pi_X:(\bar{X},\bar{\mathcal{B}_\mu},\bar{\mu},\bar{T})\rightarrow (X,\mathcal{B}_X,\mu,T)$ be the natural extension of $(X,\mathcal{B}_X,\mu,T)$. Let $\bar{\mu}=\int_E \bar{\mu}_e d \eta(e)$ the ergodic decomposition of $\bar{\mu}$ and $(\bar{X},(\bar{\mathcal{B}_\mu})_e,\bar{\mu}_e,\bar{T})$ be the corresponding Lebesgue systems for $e\in E$ in the ergodic decomposition of $\bar{\mu}$. Since $\Pi_X^{-1}(\mathcal{B}_X)$ is a countably generated sub-$\sigma$-algebra of $\mathcal{B}_{\bar{\mu}}$, one knows that for $\eta$-a.e. $e\in E$, $\Pi_X^{-1}(\mathcal{B}_X)\subset (\bar{\mathcal{B}_{\mu}})_e$. For $\eta$-a.e. $e\in E$, letting $\mu_e(A)=\bar{\mu}_e(\Pi_X^{-1}A)$ for $A\in \mathcal{B}_X$, then $(X,\mathcal{B}_X,\mu_e,T)$ is an ergodic Polish system and $\mu=\int_E \mu_e d \eta(e)$ is the ergodic decomposition of $\mu$. For $\eta$-a.e. $e\in E$, we define $\nu_e(B)=\mu_e(\pi^{-1}(B))$ for $B\in \mathcal{B}_Y$. Then $(Y,\mathcal{B}_Y,\nu_e,S)$ is an ergodic Polish system and $\pi:(X,\mathcal{B}_X,\mu_e,T)\rightarrow (Y,\mathcal{B}_Y,\nu_e,S)$ is a factor map between two Polish systems. Thus, the property (1) in the lemma follows from the following claim. [**Claim:**]{} For $\eta$-a.e. $e\in E$, $\nu_e=\nu$. Since $Y$ is a Polish space, there are finite Borel-measurable partitions $\beta_i$, $i\in \mathbb{N}$ of $Y$ such that $\beta_1\preceq \beta_2\preceq\cdots$ and $\mathcal{B}_Y$ is the smallest $\sigma$-algebra containing all $\beta_i$, $i\in \mathbb{N}$. Let $\mathcal{D}$ be the algebra generated by $\{ A:A\in \beta_i \text{ for some }i\in \mathbb{N}\}$. Then $\mathcal{D}$ is a countable set and $\mathcal{D}$ generates the $\sigma$-algebra $\mathcal{B}_Y$. Define $$Y(\nu)=\{ y\in Y: \lim_{n\rightarrow +\infty} \frac{1}{n}\sum_{i=0}^{n-1}1_A(S^iy)=\nu(A) \text{ for any }A\in \mathcal{D}\}$$ where $1_A$ is the characterization function of $A$. Since $\mathcal{D}$ is a countable set, $Y(\nu)\in \mathcal{B}_Y$ and $\nu(Y(\nu))=1$ by the Birkhoff ergodic theorem. Since $\int_E \mu_e(\pi^{-1}(Y(\nu))d \eta(e)=\mu(\pi^{-1}(Y(\nu))=\nu(Y(\nu))=1$, one has $\mu_e(\pi^{-1}(Y(\nu)))=1$ for $\eta$-a.e. $e\in E$. That is, $\nu_e(Y(\nu))=\mu_e(\pi^{-1}(Y(\nu)))=1$ for $\eta$-a.e. $e\in E$. Thus, to show $\nu_e=\nu$ for $\eta$-a.e. $e\in E$, it is sufficient to show that if $e\in E$ such that $(Y,\mathcal{B}_Y,\nu_e,S)$ is an ergodic Polish system and $\nu_e(Y(\nu))=1$, then $\nu_e=\nu$. Let $e\in E$ such that $(Y,\mathcal{B}_Y,\nu_e,S)$ is an ergodic Polish system and $\nu_e(Y(\nu))=1$. Set $$\mathcal{F}_e=\{ B\in \mathcal{B}_Y: \nu_e(B)=\nu(B)\}.$$ We want to show $\mathcal{F}_e=\mathcal{B}_Y$. By the Birkhoff ergodic theorem, there exists $y\in Y(\nu)$ such that $$\lim_{n\rightarrow +\infty} \frac{1}{n}\sum_{i=0}^{n-1}1_A(S^iy)=\nu_e(A) \text{ for any }A\in \mathcal{D}.$$ By the definition of $Y(\nu)$, $\lim_{n\rightarrow +\infty} \frac{1}{n}\sum_{i=0}^{n-1}1_A(S^iy)=\nu(A)$ for any $A\in \mathcal{D}$. Hence, $\nu_e(A)=\nu(A)$ for any $A\in \mathcal{D}$, which implies that $\mathcal{D}\subseteq \mathcal{F}_e$. Note that $\mathcal{F}_e$ is a monotone class. Thus, $\mathcal{F}_e=\mathcal{B}_Y$ follows from that the $\sigma$-algebra generated by $\mathcal{D}$ is the monotone class generated by $\mathcal{D}$. This completes the proof of Claim. Next, we prove the property (2). Since $X$ is Polish space, there are finite Borel-measurable partitions $\alpha_i$ of $X$ such that $\alpha_1\preceq\alpha_2\preceq\cdots$ and the small $\sigma$-algebra containing all $\alpha_i$, $i\in \mathbb{N}$ is $\mathcal{B}_X$. Thus for any a $T$-invariant Borel probability measure $\lambda$ on $(X,\mathcal{B}_X)$ and any sub-$\sigma$-algebra $\mathcal{C}$ of $\mathcal{B}_X$ with $T^{-1}\mathcal{C}\subseteq \mathcal{C}$, one has $$\begin{aligned} \label{p-limit} \lim_{i\rightarrow +\infty}h_\lambda(T,\alpha_i|\mathcal{C})=h_\theta(T|\mathcal{C}). \end{aligned}$$ Since $\Pi_X^{-1}(\pi^{-1}\mathcal{B}_Y)$ is a countably generated sub-$\sigma$-algebra of $\mathcal{B}_X$ and $$\bar{T}^{-1}(\Pi_X^{-1}(\pi^{-1}\mathcal{B}_Y))=\Pi_X^{-1}(T^{-1}(\pi^{-1}\mathcal{B}_Y)=\Pi_X^{-1}(\pi^{-1}(S^{-1}\mathcal{B}_Y))\subseteq \Pi_X^{-1}(\pi^{-1}\mathcal{B}_Y),$$ one has for $i\in \mathbb{N}$ $$\label{eq-kkky} h_{\bar{\mu}}(\bar{T},\Pi_X^{-1}(\alpha_i)|\Pi_X^{-1}(\pi^{-1}\mathcal{B}_Y))=\int_E h_{\bar{\mu}_e}(\bar{T},\Pi_X^{-1}(\alpha_i)|\Pi_X^{-1}(\pi^{-1}\mathcal{B}_Y)d \eta(e)$$ by lemma \[cond-ed\]. 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[ **Dipolar condensed atomic mixtures and miscibility under rotation\ **]{} Lauro Tomio^1^, R. K. Kumar^1,2^ and A. Gammal^3^ $^{\bf 1}$Instituto de Física Teórica, Universidade Estadual Paulista, 01140-070 São Paulo, SP, Brazil.\ $^{\bf 2}$Department of Physics, Centre for Quantum Science, and Dodd-Walls Centre for Photonic and Quantum Technologies, University of Otago, Dunedin 9054, New Zealand.\ $^{\bf 3}$ Instituto de Física, Universidade de São Paulo, 05508-090 São Paulo, Brazil. Abstract {#abstract .unnumbered} ======== [By considering symmetric and asymmetric dipolar coupled mixtures (with dysprosium and erbium isotopes), we report a study on relevant anisotropic effects, related to spatial separation and miscibility, due to dipole-dipole interactions (DDIs) in rotating binary dipolar Bose-Einstein condensates. The binary mixtures are kept in strong pancake-like traps, with repulsive two-body interactions modeled by an effective two-dimensional (2D) coupled Gross-Pitaevskii equation. The DDI are tuned from repulsive to attractive by varying the dipole polarization angle. A clear spatial separation is verified in the densities for attractive DDIs, being angular for symmetric mixtures and radial for asymmetric ones. Also relevant is the mass-imbalance sensibility observed by the vortex-patterns in symmetric and asymmetric-dipolar mixtures. In an extension of this study, here we show how the rotational properties and spatial separation of these dipolar mixture are affected by a quartic term added to the harmonic trap of one of the components.]{} Dipolar Bose-Einstein condensate - Introduction {#sec:intro} =============================================== The experimental realization of Bose-Einstein condensation in chromium ($^{52}$Cr) atoms has opened the new research direction called dipolar quantum gases [@2005-Pfau]. Following the condensation in $^{52}$Cr, many subsequent studies have been carried out by different experimental groups on fermionic and bosonic properties of strongly dipolar ultracold gases, such as with dysprosium (Dy) and erbium (Er) (see [@2018-natphys-Ferlaino] and references therein). More recently, the elementary excitations spectrum of $^{164}$Dy and $^{166}$Er dipolar Bose gases were analyzed in Ref. [@Natale2019], by considering three-dimensional (3D) anisotropic traps across the superfluid-supersolid phase transition. The recent investigations in ultracold laboratories with two-component dipolar Bose-Einstein Condensates (BEC), on stability and miscibility properties, became quite interesting due to the number of control parameters that can be explored in new experimental setups. The parameters which can be controlled are given by the strengths of dipoles, the number of atoms in each component, the inter- and intra-species scattering lengths, as well as confining trap geometries. The stability and pattern formation have been studied in Ref. [@Saito2018], by considering dipolar-dipolar interactions (DDIs) in two-component dipolar BEC systems. Rotational properties of two-component dipolar BEC in concentrically coupled annular traps were also studied in Ref. [@2015Zhang], by assuming a mixture with only one dipolar component. Following previous studies with rotating binary dipolar mixtures and their miscibility properties [@Wilson2009; @Wilson2012; @2016Xiao], the miscible-immiscible transition (MIT) of the dipolar mixtures with $^{162,164}$Dy and $^{168}$Er were also recently studied by us in Ref. [@2017jpco]. For these coupled dipolar systems, the miscible-immiscible stable conditions were analyzed within a full 3D formalism, by considering repulsive contact interactions, within pancake- and cigar-type trap configurations. The rotational properties and vortex-lattice pattern structures of these dipolar mixtures were further investigated by us in Refs. [@2017-Kumar-PRA; @2018-Kumar-JPB; @2019-Kumar-PRA], by changing the inter- to intra-species scattering lengths, as well as the polarization angles of the dipoles. Among the observed characteristics of these strong dipolar binary systems, relevant for further investigations are the possibilities to alter the effective time-averaged DDI from repulsive to attractive, by tuning the polarization angle $\varphi$ of both interacting dipoles from zero to $90^\circ$, respectively. In Ref. [@2018-Kumar-JPB], vortex pattern structures were studied by considering rotating binary mixtures confined by squared optical lattices, whereas in Ref. [@2019-Kumar-PRA], by tuning $\varphi$, our investigation was mainly concerned with rotational properties together with spatial separations of the binary mixtures. Motivated by the above mentioned studies, considering that the interplay between DDIs and contact interactions can bring us different interesting effects in the MIT, showing richer vortex-lattice structures in rotating binary dipolar systems, in the present contribution we are also reporting some new results obtained for the properties of dipolar mixtures confined by a strongly pancake-like two-dimensional (2D) rotating harmonic trap. The effect of a weak quartic perturbation in the $x$-$y$ plane, applied to the first dipolar component, is studied by tuning the polarization angles of the dipoles together with the contact inter-species interactions. Next section, the model formalism is presented with our parametrization and numerical procedure. In section 3, after an analysis of the main results for symmetric and asymmetric binary dipolar mixtures confined by strong pancake-like harmonic traps, we present new results obtained when considering the effect of a weak quartic perturbation added to the harmonic trap of one of the components. Finally, a summary with our conclusions is given in section 4. Model formalism, parametrization and numerical approach {#secII} ======================================================= Model formalism --------------- The coupled dipolar system with condensed two atomic species $i=1,2$, with the respective masses $m_i$ (with $m_1\ge m_2$) are assumed to be confined in strongly pancake-shaped harmonic traps, with fixed aspect ratios, such that $\lambda=\omega_{i,z}/\omega_{i,\perp}=20$ for both species $i=1,2$, where $\omega_{i,z}$ and $\omega_{i,\perp}$ are, respectively, the longitudinal and transverse trap frequencies. The coupled Gross-Pitaevskii (GP) equation is cast in a dimensionless format, with energy and length units given, respectively, by $\hbar \omega_{1,\perp}$ and $l_\perp\equiv \sqrt{\hbar/(m_1\omega_{1,\perp})}$. Correspondingly, the space and time variables are given in units of $l_\perp$ and $1/\omega_{1}$, respectively, such that ${\bf r}\to l_\perp {\bf r}$ and $t\to \tau/\omega_{1}$. Within these units, and by adjusting both trap frequencies such that ${m_2\omega_{2,\perp}^2}={m_1\omega_{1,\perp}^2}$, the dimensionless external 3D trap potential for each one of the species $i$ can be written as $V_{3D,i}({\bf r}) \equiv V_i(x,y) +\frac{1}{2}\lambda^2z^2.$ On the miscibility conditions for binary trapped dipolar systems, more details and discussion can be found in Refs. [@2017-Kumar-PRA; @2018-Kumar-JPB; @2019-Kumar-PRA]. Large values for $\lambda$ allow us to reduce to 2D the original 3D formalism by considering the usual factorization of the 3D wave function as $\psi_{i}(x,y,\tau)\chi_i(z)$, where $\chi_i(z)\equiv \left(\lambda/{\pi}\right)^{1/4}e^{-{\lambda z^{2}}/{2}}$. The two-body contact interactions related to the scattering lengths $a_{ij}$, and DDI parameters are defined as [@2019-Kumar-PRA] [$$g_{ij}\equiv \sqrt{2\pi\lambda} \frac{m_1 a_{ij} N_j}{m_{ij}l_\perp},\;\; \mathrm{d}_{ij}=\frac{N_{j}}{4\pi }\frac{\mu_{0}\mu_{i}\mu _{j}}{\hbar \omega_{1}\,l_\perp^{3}}, \;\; a_{ii}^{(d)} \equiv\frac{1}{12\pi }\frac{m_{i}}{m_{1}}\frac{\mu_{0}\mu _{i}^2}{\hbar \omega_{1}l_\perp^{2}}, \;\; a_{12}^{(d)}=a_{21}^{(d)}=\frac{1}{12\pi }\frac{\mu_{0}\mu_{1}\mu _{2}}{\hbar \omega_{1}l_\perp^{2}}, \label{par}$$]{} where $i,j=1,2$, with $N_j$ being the number of atoms and $m_{ij} = m_im_j/(m_i+m_j)$ the reduced mass of the species $i$ and $j$. In our numerical analysis, the length unit will be assumed being $l_\perp = 1\mu$m $ \approx 1.89\times 10^4 a_0$, with $a_0$ being the Bohr radius. The corresponding 2D coupled GP equation for the two components $\psi_{i}\equiv \psi_{i}(x,y,\tau)$ of the total wave function can be written as [$$\begin{aligned} \mathrm{i}\frac{\partial \psi_{i} }{\partial \tau } ={\bigg [}\frac{-m_{1}}{2m_{i}}\nabla_{2D}^2 + V_i(x,y) - \Omega L_{z}+\hspace{-0.2cm}\sum_{j=1,2}g_{ij}|\psi_{j} |^{2}+\hspace{-0.2cm} \sum_{j=1,2}\mathrm{d}_{ij} \int dx^\prime dy^\prime V^{(d)}(x-x^\prime,y-y^\prime) |\psi_{j}^\prime |^{2} {\bigg ]} \psi_{i} \label{2d-2c}, \end{aligned}$$]{} where $\nabla_{2D}^2\equiv \frac{\partial^2}{\partial x^2}+\frac{\partial^2}{\partial y^2}$, $\psi_{i}^\prime\equiv \psi_{i}(x^\prime,y^\prime,\tau)$, with $V^{(d)}(x,y)$ being the reduced 2D expression for the DDI. The 2D confining potential $V_i(x,y)$ is assumed to be harmonic for both components, as in Ref. [@2019-Kumar-PRA]. However, in the present contribution we are providing an extension to our study reported in Ref. [@2019-Kumar-PRA], by examining the effect, on the pattern distribution and spatial separation of the dipolar mixture, of a quartic term applied to one of the components, which we define as the more-massive one. So, the trap is given by $$V_i(x,y)\equiv V_i(\rho)\equiv \frac{\rho^2}{2} + \kappa_i\rho^4,\;\;{\rm where}\;\; \rho\equiv \sqrt{x^2+y^2},\;\; \kappa_2=0, \label{trap}$$ with $\kappa_{1}\equiv \kappa$ being a dimensionless positive parameter (in principle, assumed to be small), which increases the trap confinement of the more massive component. Experimentally, the quartic potential together with harmonic trap can be created by using far-detuned laser beam propagating along the axis of the trap, perpendicular to the $(x,y)$ plane. So, the width and strength of the quartic trap can be controlled, respectively, by the width and amplitude of the blue-detuned Gaussian laser beam. More details can be found in the reference [@quartic-exp], where experiments with quartic trap in BEC are discussed. Each component of the wave function is assumed normalized to one, $\int_{-\infty}^{\infty}dxdy|\psi _{i}|^{2}=1.$ In Eq.(\[2d-2c\]), $L_z$ is the angular momentum operator (in units of $\hbar$), with $\Omega$ being the corresponding rotation parameter (in units of $\omega_{1}$), which is assumed to be common for both components. The 2D DDI presented in the integrand of the second term shown in Eq. (\[2d-2c\]) can be expressed in the 2D momentum space as the combination of two terms, by considering the orientations of the dipoles $\varphi$ and the projection of the corresponding Fourier transformed $V^{(d)}(x,y)$. One term is perpendicular, with the other parallel to the direction of the dipole inclinations, as described in Refs. [@Wilson2012; @2016Xiao]. By generalizing the description to a polarization field rotating in the $(x,y)$ plane, the two terms can be combined according to the dipole orientations $\varphi$, with the total 2D momentum-space DDI given by [@2019-Kumar-PRA] [$$\begin{aligned} \widetilde{V}^{(d)}( k_x, k_y) &=& \frac{3\cos^2\varphi-1}{2} \left[2 - 3 \sqrt{\frac{\pi}{2\lambda}}k_{\rho} \exp \left( \frac{k_{\rho }^{2}}{2\lambda}\right) {\rm erfc}\left( \frac{k_{\rho }}{\sqrt{2\lambda}}\right) \right] \equiv V_\varphi(k_\rho) \label{DDI-2D},\end{aligned}$$ ]{} where $ k_\rho^2\equiv k_x^2+k_y^2$, with $\rm erfc(x)$ being the complementary error function of $x$. The 2D configuration-space effective DDI is obtained by applying the convolution theorem in Eq. (\[2d-2c\]), performing the inverse 2D Fourier-transform for the product of the DDI and density, such that $\int dx^\prime dy^\prime V^{(d)}(x-x^{\prime },y-y^{\prime })|\psi _{j}^{\prime}|^{2} =\mathcal{F}_{2D}^{-1}\left[ \widetilde{{V}}^{(d)}(k_x,k_y){\widetilde{n}_{j}}(k_x,k_y)\right]. $ From Eq. (\[DDI-2D\]), one should notice that such momentum-space Fourier transform of the dipole-dipole potential changes the sign at some particular large momentum $k_\rho$. However, after applying the convolution theorem with the inverse Fourier transform (by integrating the momentum variables), the corresponding coordinate-space interaction has a definite value, as in the 3D case, which is positive for $\varphi\le\varphi_M$, and negative for $90^\circ\ge\varphi>\varphi_M$, where $\varphi_M\approx 54.7^\circ$ is the so-called “magic angle", in which the DDI is canceled out. Parametrization and numerical approach -------------------------------------- The two binary mixtures ($^{164}$Dy-$^{162}$Dy and $^{168}$Er-$^{164}$Dy) that we are investigating are called, respectively, “symmetric" and “asymmetric" ones; where these terms are related to the dipolar symmetry of the condensed atoms. The corresponding magnetic dipole moments of the three species are the following: $\mu=10\mu_B$ for $^{162,164}$Dy, and $\mu=7\mu_B$ for $^{168}$Er. So, by considering the definitions given in (\[par\]), the strengths of the DDI are $a_{ij}^{(d)}=131\,a_{0}$ ($i,j=1,2$), for the symmetric-dipolar mixture $^{164}$Dy-$^{162}$Dy; and $a_{11}^{(d)}=66\,a_{0} $, $a_{22}^{(d)}=131\,a_{0}$ and $a_{12}^{(d)}=a_{21}^{(d)}=94\,a_{0} $, for the $^{168}$Er-$^{164}$Dy mixture. In all the cases, we assume the number of atoms for both species are identical and fixed at $N_{1}=N_{2}=5000$. The number of atoms are reduced in relation to the ones used in Ref. [@2019-Kumar-PRA], in view of our present aim and numerical convenience. For symmetric-dipolar mixture ($\mu_1=\mu_2$) we have $d_{12}=d_{11}=d_{22}$. In the case of contact interactions, we should consider enough large repulsive scattering lengths in view of our stability requirements. We fix both intra-species contact interactions at $a_{11}=a_{22}=50a_0$, remaining the inter-species one to be explored by varying the ratio parameter $\delta\equiv a_{12}/a_{11}$. Once selected the polarization angle and $\delta$ as the appropriate parameters to alter the miscibility properties of a mixture, we fix other parameters guided by possible realistic settings and stability requirements. For the present approach, we choose $\Omega = 0.75$ for the rotation frequency parameter, larger than the one used in Ref. [@2019-Kumar-PRA] ($\Omega=0.6$), in order to improve the observation of vortex-pattern structures and spatial separation. For the numerical approach to solve the GP formalism (\[2d-2c\]), the split-step Crank-Nicolson method [@CPC0; @CPC] is applied, combined with a standard method for evaluating DDI integrals in momentum space, as described in Ref. [@2019-Kumar-PRA]. In the search for stable solutions, the numerical simulations were carried out in imaginary time on a grid with a maximum of 464 points in both $x-y$ directions, with spatial and time steps $\Delta x=\Delta y=0.05$ and $\Delta t=0.0005$, respectively. In this approach, both wave-function components are renormalized to one at each time step. Symmetric and asymmetric dipolar mixtures - Results {#sec:results} =================================================== Dipolar mixtures confined by identical harmonic pancake-like traps ------------------------------------------------------------------ We focus our study in the two coupled mixtures given by $^{168}$Er-$^{164}$Dy and $^{164}$Dy-$^{162}$Dy, motivated by recent experimental studies with dipolar BEC systems. In our investigation, we have considered harmonic strongly pancake-like trap, as detailed in Ref. [@2019-Kumar-PRA]. First, a detailed analysis of ground state and stability properties was performed in the absence of rotation. In this respect, we understand that our theoretical predictions can be helpful in verifying miscibility properties in on-going experiments under different anisotropic trap configurations. The stability regime was verified for $^{168}$Er-$^{164}$Dy and $^{164}$Dy-$^{162}$Dy mixtures considering the fraction number of atoms for each species as functions of the trap-aspect ratio $\lambda$. From the MIT conditions for homogeneous coupled systems confined in hard-wall barriers, one can observe that the miscibility remains unaffected by the dipolar interactions. In order to estimate the miscibility for non-homogeneous confined binary mixtures, a relevant parameter $\eta$ was defined in Ref. [@2017jpco], by integrating the square-root of the product of the two-component densities, given by $\eta = \int \sqrt{\vert \phi_1 \vert^2 \vert \phi_2 \vert^2} \ d{\mathbf x},$ which varies from $\eta=0$ (complete immiscible mixtures) to $\eta=1$ (for complete miscible mixtures). This parameter is found appropriate for a quantitative estimate of the overlap between the two densities of the coupled system. By considering the natural properties of the mixed elements, the two mixtures, we notice that $^{168}$Er-$^{164}$Dy and $^{164}$Dy-$^{162}$Dy have quite different miscibility behaviors, with $^{164}$Dy-$^{162}$Dy being almost completely miscible ($\eta=0.99$) and $^{168}$Er-$^{164}$Dy partially miscible ($\eta=0.77$), when the other parameters (trap-aspect ratio and number of atoms) are fixed to the same values. Such behavior is clearly due to a mass-imbalance effect, as discussed in Ref. [@2019-Kumar-PRA], which plays a relevant role in the inter-species dipolar strength when compared with the intra-species one. The two binary mixtures ($^{164}$Dy-$^{162}$Dy and $^{168}$Er-$^{164}$Dy) are considered in a rotating frame within quasi-2D settings. Owing to the different miscibility properties, quite distinct vortex patterns are observed between the symmetric and asymmetric mixtures. For the dipolar symmetric mixture, $^{164}$Dy-$^{162}$Dy, we observe the following lattice patterns: triangular, square-shaped, rectangular-shaped, double core, striped, and with domain walls. For the dipolar asymmetric mixture, $^{168}$Er-$^{164}$Dy, we notice triangular, square-shaped, and circular pattern lattices. Further, to analyze the anisotropic properties of dipolar interactions, the polarization angle $\varphi$ of the dipoles was modified with the dipolar interactions being tuned from repulsive to attractive. With the dipoles of the two species polarized in the same direction, perpendicular to the direction of the dipole alignment ($\varphi=0$), the DDI is repulsive. By tuning the polarization angle $\varphi$ from zero to 90$^\circ$ the DDI changes from repulsive to fully attractive, with the DDI being canceled for $\varphi=\varphi_M\approx 54.7^\circ$. The miscibility of the condensed mixture is mainly affected by the inter-species interactions; with the vortex-pattern structures being related to combined effects due to inter- and intra-species interactions, with the vortex-pattern formations obtained with $\varphi=0$ surviving approximately up to $\varphi\approx\varphi_M$. Complete spatial separation between the two-component densities under rotation is verified for large $\varphi$, when the DDI is attractive. As verified, half-space angular separations occur in dipolar-symmetric cases, represented by $^{164}$Dy-$^{162}$Dy; whereas radial-space separations occur for dipolar-asymmetric cases, represented by $^{168}$Er-$^{164}$Dy. Another quite relevant result obtained in Ref. [@2019-Kumar-PRA] is the observed effect of the mass-asymmetry in the miscibility and vortex-pattern structures. The particular mass-imbalance sensitivity can better be appreciated in the dipolar-symmetric mixture $^{164}$Dy-$^{162}$Dy for $\delta=1$, when all the differences between the density patterns should be attributed to the small mass-asymmetry. Next, we report new results with the trap interaction as given by Eq. (\[trap\]), with a quartic term added to the harmonic interaction of the more-massive component of both two mixtures. Dipolar symmetric $^{164}$Dy-$^{162}$Dy mixture, with a quartic trap applied to $^{164}$Dy ------------------------------------------------------------------------------------------ As discussed above, being dipolar symmetric, with $a_{11}=a_{22}=131a_0$, this $^{164}$Dy-$^{162}$Dy BEC mixture exposes more miscible properties. As verified in Ref. [@2019-Kumar-PRA], this mixture in the rotating harmonic trap \[with $\kappa_i=0$ in Eq. (\[trap\])\] shows triangular, squared, rectangular-shaped, double core, striped, and with domain wall vortex lattices regarding the ratio between inter- and intra-species contact interaction. Also, this mixture shows complete spatial separation at large polarization angles, where the DDI is purely attractive. By modifying the external confinement of one of the components, we can introduce some external asymmetry to the mixture. So, in this contribution, for this binary system we start by adding a very weak quartic term in the first component of the mixture, in order to analyze the miscibility and complete spatial separation of the coupled system. We consider two different miscible cases, with $\delta$ = 1 and 1.45. For these particular cases, striped and domain wall vortex structures are observed [@2019-Kumar-PRA], respectively, when both species are under identical rotating harmonic pancake-like traps, with $\lambda=20$ and $\Omega=0.6$. By adding a quartic term to the trap, as explained in section \[secII\], we have also reduced the number of atoms to $N_i=5000$ and increased the frequency to $\Omega=0.75$ in order to improve our observation on the corresponding rotational structure and spatial separation. In this case, ring lattice structures can be verified, also verified even for single component BECs. From our results, in this communication, we select three different orientation angles of the dipoles, given by $\varphi\,=\,0{^\circ}$, $45^{\circ}\,$ and $\,90^{\circ}$, in which the first ($\varphi\,=\,0{^\circ}$) provides complete repulsive DDI, the second ($\varphi\,=\,45{^\circ}$) is weakly repulsive, with the DDI of the third $\varphi\,=\,90{^\circ}$ being complete attractive. As shown by our results presented in Fig. \[fig1\], the small weak quartic perturbation in the trap, given by $\kappa=0.05$, induces radial spatial separations between the condensate densities, displaying ring lattice structure in the second component as shown in the panels (a) to (e) of Fig. \[fig1\]. The quartic trap term added to the first component makes the first component more confined than the second one. So the second component becomes radially phase-separated, changing the previous patterns observed in Ref. [@2019-Kumar-PRA] for $\kappa=0$. Such similar behavior for non-dipolar mixtures was also analyzed theoretically recently in Ref. [@quartic-adhikari]. ![ 2D Dipolar density patterns, $|\protect\psi_{j}|^{2}$, where $j=$1 is for $^{164}$Dy and $j=2$ for $^{162}$Dy, are shown for $\delta=1$ \[(a$_j$) to (c$_j$)\] and $\delta=1.45$ \[(d$_j$) to (f$_j$)\]. The dipole polarization angles ($\varphi=0^\circ,\;45^\circ,\;90^\circ$) are indicated at the top of each column, with the $^{164}$Dy component having the addition of a quartic trap with $\kappa=0.05$. The other parameters are: $N_{j=1,2}=5000$, $\lambda =20$, $a_{11}=a_{22}=50a_{0}$ and $\Omega =0.75$. []{data-label="fig1"}](f01){width="0.5\columnwidth"} ![ 2D Dipolar density patterns, $|\protect\psi_{j}|^{2}$, where $j=$1 is for $^{164}$Dy and $j=2$ for $^{162}$Dy, are shown for $\varphi=90^\circ$ and $\delta=1.45$. The quartic trap added to component 1 is such that $\kappa=0.05$ in the left panel and 0.08 in the right panel. As in Fig. \[fig1\], the other parameters are: $N_{j=1,2}=5000$, $\lambda =20$, $a_{11}=a_{22}=50a_{0}$ and $\Omega =0.75$. []{data-label="fig2"}](f02){width="0.4\columnwidth"} ![ 2D Dipolar density patterns, $|\protect\psi_{j}|^{2}$, where $j=$1 is for $^{168}$Er and $j=2$ for $^{164}$Dy, are shown for $\delta=1$ \[(a$_j$) to (c$_j$)\] and $\delta=1.45$ \[(d$_j$) to (f$_j$)\]. The dipole polarization angles ($\varphi=0^\circ,\;45^\circ,\;90^\circ$) are indicated at the top of each column, with the $^{168}$Er component having the addition of a quartic trap with $\kappa=0.05$. The other parameters are: $N_{j=1,2}=5000$, $\lambda =20$, $a_{11}=a_{22}=50a_{0}$ and $\Omega =0.75$. []{data-label="fig3"}](f03){width="0.5\columnwidth"} ![ 2D Dipolar density patterns, $|\protect\psi_{j}|^{2}$, where $j=$1 is for $^{168}$Er and $j=2$ for $^{164}$Dy, are shown for $\varphi=90^\circ$ and $\delta=1.45$. The quartic trap added to component 1 is such that $\kappa=0.25$ in the left panel and 1 in the right panel. As in Fig. \[fig3\], the other parameters are: $N_{j=1,2}=5000$, $\lambda =20$, $a_{11}=a_{22}=50a_{0}$ and $\Omega =0.75$. []{data-label="fig4"}](f04.png){width="0.4\columnwidth"} To improve our understanding of the phase separation and the effect of the added quartic trap, we studied the dipolar binary system by increasing the strength $\kappa$. We observed that, for $\kappa\ge 0.1$, with large repulsive inter-species interaction $\delta=1.45$, the spatial phase separations of the densities change completely from the previous angular to radial ones. This behavior is indicated in Fig. \[fig2\], where the phase-separated case, displayed for $\varphi=90^\circ$ with $\kappa=0.05$, is being compared with the $\kappa=0.08$ case. So, when $\kappa\ge 0.1$, only radial spatial separation can be observed in the binary mixture. Dipolar asymmetric $^{168}$Er-$^{164}$Dy mixture, with a quartic trap applied to $^{168}$Er ------------------------------------------------------------------------------------------- In this subsection we consider the dipolar asymmetric $^{168}$Er-$^{164}$Dy BEC system, to study the effect of a quartic dipolar trap applied to the first component ($^{168}$Er) of the mixture. As reported in Ref. [@2019-Kumar-PRA] for this dipolar asymmetric case, when $\kappa_i=0$ in the rotating confining harmonic trap given by Eq. (\[trap\]), one should observe triangular, square-shaped, and circular lattices, by varying the inter-species interaction. The interplay between the inter-species repulsive character, shown by increasing $\delta$, together with the attractive role of the DDI as the polarization angle is increased, have shown radial density distributions for the binary mixture such that for $\varphi=0^\circ$ and $\delta\ge 1$ the $^{168}$Er element is at the center of the mixture (surrounded by $^{164}$Dy), moving to the external part when the dipolar interaction becomes more attractive, with $\varphi=90^\circ$, with an exchange of the densities. Now, with the present study, by increasing the external trap interaction with the quartic term, as given by Eq. (\[trap\]) with $\kappa=0.05$, one can already observe some differences in the pattern distribution of the vortices of both mixtures, as shown in Fig. \[fig3\]. However, one should notice that, for the complete spatial separation that occurs for $\varphi=90^\circ$, the position of both elements remains as in the case that $\kappa=0$, implying that the added quartic term is not enough to change the position of the density distributions. More interesting behavior can be observed by increasing the strength of the quartic term, as verified in the Fig. \[fig4\], by considering $\varphi=90^\circ$ with $\delta=1.45$. In the left panels of this figure, we consider $\kappa=0.25$, where we can verify that the previous radial distribution of the densities is being modified with the radius of the first component being reduced. With $\kappa=1$, we finally obtain a radial spatial separation in which the $^{168}$Er condensate is occupying the center, surrounded by the $^{164}$Dy condensate. The densities of the two components interchange their positions in relation to the case that $\kappa=0$, due to the quartic trap term, which is dominating the confinement of the $^{168}$Er condensate. Summary ======= By considering the symmetric and asymmetric dipolar coupled mixtures, respectively given by $^{164}$Dy-$^{162}$Dy and $^{168}$Er-$^{164}$Dy, in this communication we have first discussed rotational properties, miscibility aspects, and spatial separation of these two coupled binary BEC systems, by analyzing an investigation previously reported in Ref. [@2019-Kumar-PRA]. In addition, new results are presented by considering one of the elements of the coupled mixture being confined by a quartic interaction, which is added to the previous harmonic trap potential. The relevance of this study relies on current experimental possibilities in cold-atom laboratories to investigate such dipolar binary systems. The stability regime and miscibility properties due to the DDI of the coupled system are obtained numerically, by solving the corresponding GP equation within a model where the mixture is first confined by strong pancake-like harmonic-trap potential with aspect-ratio $\lambda=20$ and considering repulsive two-body interactions. The DDI are tuned from repulsive to attractive by varying the dipole polarization angle, with a clear spatial separation verified in the densities for attractive DDI, being angular for symmetric mixtures and radial for asymmetric ones in the case that no quartic term is present. In an extension of our previous reported work, by adding the quartic term to the trap interaction, here we show how the density distribution of both binary system, symmetric and asymmetric ones, are affected. As shown, the quartic trap supports radial phase separations with ring lattice for both $^{164}$Dy-$^{162}$Dy and $^{168}$Er-$^{164}$Dy BEC mixtures, modifying the previous vortex-pattern structures and spatial separations obtained without the quartic term interaction. Even a weak quartic trap is enough to modify the angular spatial separation to radial ones in the dipolar $^{164}$Dy-$^{162}$Dy mixture, for attractive dipolar interactions. In the asymmetric $^{168}$Er-$^{164}$Dy dipolar BEC mixture, where we have already radial spatial separation for attractive dipolar interactions even without the quartic term, with the $^{168}$Er element surrounding the other element, we have observed that, for the addition of enough large quartic term to the $^{168}$Er element, there is an exchange of the two coupled densities, with this element moving to the center. So, for asymmetric mixture with repulsive inter-species interaction and attractive DDI, strong quartic trap ($\kappa\ge1$) will prevent exchanges between both densities, which will remain completely radial-separated spatially. Acknowledgements {#acknowledgements .unnumbered} ================ This work refers to a contribution to the proceedings for the 24th European Few Body Conference, Surrey, UK. The authors acknowledge partial support received from Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq) \[LT, RKK and AG\], Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP) \[Contracts 2016/14120-6 (LT), 2016/17612-7 (AG)\]. RKK also acknowledge support from Marsden Fund (Contract UOO1726). [99]{} T. Lahaye, C. Menotti, L. Santos, M. Lewenstein, T. Pfau, [*The physics of dipolar bosonic quantum gases,*]{} [Rep. Prog. Phys.]{} [**72**]{}, 126401 (2009). Doi [ 10.1088/0034-4885/72/12/126401]{}. A. Trautmann, P. Ilzhöfer, G. Durastante, C. Politi, M. Sohmen, M. 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--- abstract: 'Spontaneous wavefunction collapse models, like the Continuous Spontaneous Localization, are designed to suppress macroscopic superpositions, while preserving microscopic quantum phenomena. An observable consequence of collapse models is spontaneous heating of massive objects. Here we calculate the collapse-induced heating rate of astrophysical objects, and the corresponding equilibrium temperature. We apply these results to neutron stars, the densest phase of baryonic matter in the universe. Stronger collapse model parameters imply greater heating, allowing us to derive competitive bounds on model parameters using neutron star observational data, and to propose speculative bounds based on the capabilities of current and future astronomical surveys.' author: - Antoine Tilloy - 'Thomas M. Stace' bibliography: - 'main.bib' title: 'Neutron star heating constraints on wave-function collapse models' --- Collapse models, like the Continuous Spontaneous Localization (CSL) model [@pearle1989; @ghirardi1990], aim at solving the measurement problem of quantum mechanics through a stochastic non-linear modification of the Schrödinger equation [@bassi2003; @bassi2013review]. Such modifications have sometimes been conjectured to be caused by gravity, the most famous example being the Diósi-Penrose (DP) model [@diosi1987; @penrose1996]. In general, collapse models posit an intrinsic (possibly gravitational) noise, which endogenously collapses superpositions of sufficiently macroscopic systems (in a particular basis), while preserving the predictions of quantum mechanics at small scales. One notable consequence of these models is spontaneous heating of massive objects. Neutron stars, which are extremely dense, macroscopic quantum-limited objects, offer a unique system on which to test this prediction. Here, we estimate the equilibrium temperature of a neutron star radiating heat generated from spontaneous collapse models. We find that neutron stars are competitive to constrain the parameter diagram of collapse models. Theoretically or observationally improving upper bounds for neutron star equilibrium temperatures could in principle eliminate historically proposed CSL parameter values. #### Collapse models – Continuous Markovian collapse models modify the Schrödinger equation with a non-linear noise term: $$\label{eq:sto} \partial_t {| \psi_t \rangle } = - \frac{i}{\hbar} H {| \psi_t \rangle } + F(\eta_t,{| \psi_t \rangle })$$ where $\eta_t$ is a white noise process and $F$ some function which is partially constrained by consistency conditions [@gisin1989; @wiseman2001], and is chosen to yield a spontaneous collapse in the position basis. Although this stochastic description of the state vector is required to understand why collapse models actually achieve their purpose and solve the measurement problem, their empirical content is fully contained in the master equation obeyed by $\rho_t = \mathds{E}\big[{| \psi_t \rangle }{\langle \psi_t |}\big]$. For *most* Markovian non-dissipative collapse models proposed so far [@bassi2013review], it takes the form $\partial_t \rho_t = -\frac{i}{\hbar} [H,\rho_t] + \mathcal{D}[\hat{M}]\rho_t$ with $$\label{eq:master} \mathcal{D}[\hat{M}]\rho = - \!\int \! {\mathrm{d}}{\mathbf{x}}{\mathrm{d}}{\mathbf{y}}\, f({\mathbf{x}}-{\mathbf{y}})\! \left[\hat{M}_{r_c}({\mathbf{x}}),\left[\hat{M}_{r_c}({\mathbf{y}}),\rho\right]\right]$$ where $f$ is a positive definite function and $\hat{M}_{r_c}({\mathbf{x}})$ is a regularized mass density operator: $$\label{eq:M} \hat{M}_{r_c}({\mathbf{x}})= g_{r_c} * \hat{M}({\mathbf{x}}) = g_{r_c} * m\, a^\dagger({\mathbf{x}}) a({\mathbf{x}}).$$ In this expression, $m$ in the mass of the particle considered (we will consider neutrons), $a_k^\dagger({\mathbf{x}}), a_k({\mathbf{x}})$ denote the usual (here fermionic) creation and annihilation operators, $g_{r_c}$ is a regulator which smooths the mass density over a length scale $r_c$ and “$*$” denotes the convolution product. Typically, the regulator function is taken to be Gaussian: $$g_{r_c}({\mathbf{x}}) = \ e^{-{\mathbf{x}}^2/(2r_c^2)}/(\sqrt{2\pi r_c^2})^3.$$ The regulator length scale has to be much larger than the Planck length and even the nucleon Compton wave-length, the usual choice being $r_c \simeq 10^{-7} \mathrm{m}$ [@ghirardi1986]. The two most common continuous collapse models are the Continuous Spontaneous Localization (CSL) model and the Diósi-Penrose model (the latter having a heuristic link with gravity): 1. The CSL model is obtained for: $$f^\text{CSL}({\mathbf{x}}-{\mathbf{y}}) = \frac{\gamma}{2 m_N^2} \times \delta({\mathbf{x}}-{\mathbf{y}})$$ where $m_N$ is the mass of a nucleon and $\gamma$ is the collapse “strength”. It is a rate $\times$ distance$^3$, the corresponding rate is $\lambda_{\text{CSL}}\equiv \gamma/(4\pi r_c^2)^{3/2} $ historically fixed at $\lambda_{\text{CSL}}\simeq 10^{-16}$s$^{-1}$ (the so called “GRW” value). 2. The DP model is obtained for: $$\label{eq:DPchoice} f^\text{DP}({\mathbf{x}}-{\mathbf{y}})=\frac{G}{4\hbar} \times \frac{1}{|{\mathbf{x}}-{\mathbf{y}}|}.$$ Because the collapse strength is fixed by the gravitational constant, there is one parameter less [^1]. A modern motivation for \[eq:DPchoice\] is given by attempts at constructing models of fundamentally semiclassical gravity [@tilloy2016; @tilloy2017principle]. We note that, at least at the master equation level, the regulator applied on the mass density operator can equivalently be applied on the kernel $f$: $$\label{eq:masterbis} \mathcal{D}[\hat{M}]\rho = - \!\int \! {\mathrm{d}}{\mathbf{x}}{\mathrm{d}}{\mathbf{y}}\, f_{r_c}({\mathbf{x}}-{\mathbf{y}})\! \left[\hat{M}({\mathbf{x}}),\left[\hat{M}({\mathbf{y}}),\rho\right]\right],$$ with $f_{r_c}=g_{r_c}*f*g_{r_c}$. We also note in passing that the two models we consider here are non-relativistic. Efforts towards developing relativistic collapse models for quantum fields have shown that their construction is possible (albeit challenging, see [*e.g.* ]{}[@diosi1990relat; @pearle1999; @tumulka2006; @bedingham2011; @pearle2015; @tilloy2017qft; @juarez-aubry2018]). Here, we simply assume that such relativistic extensions can be constructed, and that, in the limit where relativistic effects are not dominant, their predictions would be similar to those of the non-relativistic CSL or DP models. #### Spontaneous heating – The additional decoherence term \[eq:master\] in the master equation does not commute with the kinetic part of the Hamiltonian, hence the expectation of the energy $\langle H \rangle_t \equiv {\mathrm{tr}}[H \rho_t]$ is no longer conserved. This spontaneous heating provides a natural test of collapse models [@pearle1994; @bahrami2018; @adler2018]. Recent proposals to test these models have [*e.g.* ]{}been built around ultra cold atoms [@laloe2014], which may provide good platforms to obtain bounds on the parameters in the theory as the heating effect should be significant in relative terms. An alternative, which has been overlooked so far, is to consider instead maximally dense systems, exploiting the mass density dependence of the heating for all collapse models. In this respect, neutron stars are ideal candidates. Neutron star cooling has been studied theoretically and observationally. At early stages when $T_\text{star}\sim 10^9$ K, they cool by various baryonic emission processes, but at later stages, when $T_\text{star}\sim 10^6$ K or colder, the cooling is radiation dominated [@lat94; @lat01; @yak04]. Thus, the equilibrium temperature is attained by the balance of the spontaneous collapse induced heating with Stefan-Boltzmann radiation, so is determined by the heat balance condition , where $$\label{eq:pcollapse} P_{\text{heat}}=\partial_t \langle H \rangle_t= {\mathrm{tr}}[ H\, \mathcal{D}[\hat M] \rho_t]$$ and $$\label{eq:stefan} P_{\text{rad}} = S \sigma T^4$$ where $S$ is the neutron star surface area and is Stefan’s constant. It follows that at equilibrium $T_{\rm star}=\big(P_{\rm heat}/(S\sigma)\big)^{1/4}$. For a system of $N$ fermions with non-relativistic Hamiltonian, one can show that the spontaneous collapse induced heating $P_\text{heat}$ is independent of the potential (which commutes with the mass density) and more surprisingly does not even depend on the quantum state. For the CSL model it reads: $$\label{eq:cslheating} P_\text{heat}^{\rm CSL} = {\mathrm{tr}}[ H\, \mathcal{D}[\hat M] \rho_t] = \frac{3\lambda \hbar^2}{4 r_c^2 m} N,$$ where $N$ is the number of neutrons in the star. Similarly, for the DP model it reads: $$\label{eq:dpheating} P_\text{heat}^\text{DP} = \frac{G\hbar m}{8 \sqrt{\pi} r_c^3} N.$$ #### The CSL model – We take the typical neutron star radius and mass , hence neutrons. For the values historically proposed for the CSL model, $\lambda=10^{-16}\textrm{s}$ and $r_c=10^{-7}\textrm{m}$, one finds $P_\text{heat}\sim 10^{14}\textrm{W}$. On the other hand the lowest observed temperature of an astronomical neutron star is $T^\mathrm{(obs)}=0.28$ MK for the object `PSR J 840-1419` [@keane2013]. This observed temperature corresponds to a radiative dissipation rate of $P^{\rm (obs)}_{\rm rad}\sim10^{26}$ W, well above the power that would be radiated by the CSL model. Hence, the neutron stars we can currently observe are not cold enough to straightforwardly falsify the CSL model. Naturally, neutron stars are expected to cool down to much lower temperatures than the ones we currently manage to see directly [@yak04] and the bound from `PSR J 840-1419` is thus an excessively conservative one. ![CSL parameter diagram – Top: Zones formerly excluded by gravitational wave detectors [@carlesso2016; @helou2017] (red), spontaneous X-ray emission [@piscicchia2017] (blue), and insufficient macroscopic localization [@toros2017]. The value historically proposed by GRW [@ghirardi1986] and the range put forward by Adler [@adler2007] are shown with black dots. The green line delineate the upper left regions that are excluded by currently observed neutron stars (continuous line). More speculative bounds, obtained assuming various equilibrium temperatures for neutron stars, are showed in hashed green.[]{data-label="fig:parameters"}](snip_plot.PNG){width="0.98\columnwidth"} #### The DP model – Following the same reasoning as for the CSL model, we can constrain the only free parameter, the regularization length $r_c$, of the DP model using eq. . The most conservative bound, given by `PSR J 840-1419`, yields $r_c \gtrsim 10^{-13}$m, which excludes a regulator of the order of the neutron radius which was historically conjectured to be a possible cutoff. This lower bound is of the same order of magnitude as the current best one of $4\times10^{-14}$m yielded by constraints from gravitational wave detector data [@helou2017]. The bound improves with decreasing temperatures $r_c \propto T^{-4/3}$. #### Discussion – The analysis presented in this letter makes ‘lumped-element’ approximations that provide robust bounds on the radiated power. For example, we have assumed that the emissivity of a neutron star is unity and that the thermal conductivity throughout the core is large enough that the star temperature is approximately uniform. If these assumptions are relaxed, then the core temperature may be substantially higher than the observed surface temperature. Neutron superfluidity [@lat01] has been hypothesised in the core of neutron stars. This phase will have a corresponding critical temperature $T_c$, which may provide a sensitive thermometric bound on tolerable heat generation rates in the star core: superfluidity will be suppressed if the internal temperature is too high. More generally, heat transfer models that include realistic constitutive models for the neutron star body may thus be able to provide even more stringent bounds on collapse model parameters than the lumped-element approximations we have adopted here. On the other hand, refinements and extensions of the CSL model with colored noise (cCSL) [@bassi2002; @adler2007colored], dissipation (dCSL) [@smirne2014], or both [@ferialdi2012], containing additional parameters (such as a high frequency cutoff or a temperature) are known to yield weaker heating effects. Consequently, the constraints we put forward here would be weaker for these models. In summary, with a conservative estimate of neutron star cooling based on the currently observed coldest neutron stars, one obtains constraints on the CSL model (albeit weaker than from spontaneous X-ray emission studies) and on the DP model ($r_c\gtrsim10^{-13}$m, competitive with state of the art gravitational wave interferometer data). Improving the observational upper bound on neutron star equilibrium temperatures would yield substantial improvements. If we could measure an old, cold neutron star one could test more of the CSL parameter diagram. This motivates a systematic survey of nearby, cold neutron stars. We thank Philip Pearle and Tamara Davis for helpful discussions during development of this work. AT was supported by the Alexander von Humboldt foundation. TMS was supported by the Australian Research Council Centre of Excellence for Engineered Quantum Systems CE170100039. Supplementary material {#supplementary-material .unnumbered} ====================== We compute the power $P_\text{heat}={\mathrm{tr}}[ H\, \mathcal{D}[\hat M] \rho_t]$ generated by spontaneous collapse for a generic continuous Markovian non-dissipative collapse model and then evaluate the result for the CSL and DP models. We consider a rather generic non-relativistic Hamiltonian $\hat{H}=\hat{H}_0 + \hat{V}$ for fermions with: $$\begin{aligned} \hat{H}_0 &= \frac{-\hbar^2}{2m}\int {\mathrm{d}}{\mathbf{x}}a^\dagger({\mathbf{x}}) \nabla^2_{\mathbf{x}}a({\mathbf{x}})\\ \hat{V}&= \int {\mathrm{d}}{\mathbf{x}}{\mathrm{d}}{\mathbf{y}}\, a^\dagger({\mathbf{x}})a({\mathbf{x}}) V({\mathbf{x}}-{\mathbf{y}}) a^\dagger({\mathbf{y}}) a({\mathbf{y}}),\end{aligned}$$ where $a^\dagger({\mathbf{x}})$ and $a({\mathbf{x}})$ are the local anti-commuting creation / annihilation operators $\{a^\dagger({\mathbf{x}}),a({\mathbf{y}})\}=\delta({\mathbf{x}}-{\mathbf{y}})$ and we neglect spin. We note that $\hat{V}$ commutes with the local mass density and thus does not contribute to $P_\text{heat}$. Introducing $\hat{P}_\text{heat}$ such that $P_\text{heat} = {\mathrm{tr}}\left[\hat{P}_\text{heat}\,\rho_t\right]$ we have: $$\begin{split} \hat{P}_\text{heat}=&\frac{\hbar^2}{2} m\int{\mathrm{d}}{\mathbf{x}}{\mathrm{d}}{\mathbf{y}}{\mathrm{d}}{\mathbf{z}}\, f_{r_c}({\mathbf{x}},{\mathbf{y}}) \times\\ &\bigg[a^\dagger({\mathbf{x}}) a({\mathbf{x}}),\Big[a^\dagger({\mathbf{y}}) a({\mathbf{y}}), a^\dagger({\mathbf{z}}) \nabla^2_{\mathbf{z}}a({\mathbf{z}}) \Big]\bigg] \end{split}$$ We shall prove that: $$\hat{P}_\text{heat}= \frac{\hbar^2}{2 } m \left(-2\nabla^2_{\mathbf{x}}f_{r_c}({\mathbf{x}})|_{{\mathbf{x}}=0} \right) \; \hat{N}$$ where $\hat{N} = \int {\mathrm{d}}{\mathbf{x}}a^\dagger({\mathbf{x}}) a({\mathbf{x}})$ is the total number of neutrons. Let us compute the first commutator using the canonical anti-commutation relations and evaluate the ${\mathbf{y}}$ integral: $$\begin{aligned} &\int {\mathrm{d}}{\mathbf{y}}f_{r_c} ({\mathbf{x}}-{\mathbf{y}}) \Big[a^\dagger({\mathbf{y}}) a({\mathbf{y}}), a^\dagger({\mathbf{z}}) \nabla^2_{\mathbf{z}}a({\mathbf{z}}) \Big] \nonumber \\ &=\int {\mathrm{d}}{\mathbf{y}}f_{r_c} ({\mathbf{x}}-{\mathbf{y}}) \big(\delta({\mathbf{y}}-{\mathbf{z}}) a^\dagger({\mathbf{y}})\nabla^2_{\mathbf{z}}a({\mathbf{z}}) \\ &\hskip3.4cm- \nabla^2_{\mathbf{z}}\delta({\mathbf{y}}-{\mathbf{z}})a^\dagger({\mathbf{z}}) a({\mathbf{y}})\big)\nonumber\\ &=a^\dagger({\mathbf{z}}) \left(f_{r_c}({\mathbf{x}}-{\mathbf{z}})\nabla^2_{\mathbf{z}}a({\mathbf{z}})- \nabla^2_{\mathbf{z}}\left[f_{r_c}({\mathbf{x}}-{\mathbf{z}})a({\mathbf{z}})\right]\right) \\ &=- a^\dagger({\mathbf{z}})a({\mathbf{z}}) \nabla_{\mathbf{z}}^2 f_{r_c}({\mathbf{x}}-{\mathbf{z}}) - 2 a^\dagger ({\mathbf{z}}) \nabla_{\mathbf{z}}f_{r_c}({\mathbf{x}}-{\mathbf{z}}) \nabla_{\mathbf{z}}a({\mathbf{z}})\label{eq:twoterms}\end{aligned}$$ Upon insertion in the second commutator, the first term in will vanish. Using again the canonical anti-commutation relations we get: $$\begin{split} [\spadesuit]\equiv& [a^\dagger({\mathbf{x}}) a({\mathbf{x}}), -2 a^\dagger({\mathbf{z}}) \nabla_{\mathbf{z}}f_{r_c}({\mathbf{x}}-{\mathbf{z}}) \nabla_{\mathbf{z}}a({\mathbf{z}})] \\ =&-2 \delta({\mathbf{x}}-{\mathbf{z}}) a^\dagger({\mathbf{x}}) \nabla_{\mathbf{z}}f_{r_c}({\mathbf{x}}-{\mathbf{z}}) \cdot \nabla_{\mathbf{z}}a({\mathbf{z}})\\ &+2 a^\dagger({\mathbf{z}}) \nabla_{\mathbf{z}}f_{r_c}({\mathbf{x}}-{\mathbf{z}}) \cdot \nabla_{\mathbf{z}}\delta({\mathbf{x}}-{\mathbf{z}}) a({\mathbf{x}}) \end{split}$$ Once integrated over ${\mathbf{x}}$, the first term will be proportional to $\nabla_{\mathbf{z}}f_{r_c}({\mathbf{x}}-{\mathbf{z}})|_{{\mathbf{x}}={\mathbf{z}}} = 0$ by symmetry, and thus will not contribute. Using $\nabla_{\mathbf{z}}\delta({\mathbf{x}}-{\mathbf{z}}) = -\nabla_{\mathbf{x}}\delta({\mathbf{x}}-{\mathbf{z}})$ on the second term we get: $$\begin{split} \int \!\! {\mathrm{d}}{\mathbf{x}}[\spadesuit]&=-2 \int\!\! {\mathrm{d}}{\mathbf{x}}a^\dagger({\mathbf{z}}) \nabla_{\mathbf{z}}f_{r_c}({\mathbf{x}}-{\mathbf{z}}) \cdot \nabla_{\mathbf{x}}\delta({\mathbf{x}}-{\mathbf{z}}) a({\mathbf{x}})\\ &= 2 \int\!\! {\mathrm{d}}{\mathbf{x}}\delta({\mathbf{x}}-{\mathbf{z}}) a^\dagger({\mathbf{z}}) \left[\nabla_{\mathbf{x}}\cdot \nabla_{\mathbf{z}}f_{r_c}({\mathbf{x}}-{\mathbf{z}}) a({\mathbf{x}})\right] \\ &=2 \Big[a^\dagger({\mathbf{z}})\nabla_{\mathbf{x}}\cdot \nabla_{\mathbf{z}}f_{r_c}({\mathbf{x}}-{\mathbf{z}}) a({\mathbf{x}})\Big]_{{\mathbf{z}}={\mathbf{x}}} \end{split}$$ In this integration by part, we have neglected boundary terms which would vanish once applied to a density matrix $\rho_t$ sufficiently well behaved at infinity (which is reasonable for a compact object). As before, the gradient of $f_{r_c}$ evaluated in $0$ vanishes and we are left with: $$\int \!\! {\mathrm{d}}{\mathbf{x}}[\spadesuit] = - 2\, \nabla_{\mathbf{x}}^2f_{r_c}({\mathbf{x}})|_{{\mathbf{x}}=0}\, a^\dagger({\mathbf{z}}) a({\mathbf{z}}).$$ Carrying the final integration over ${\mathbf{z}}$ yields as advertised: $$\hat{P}_\text{heat} = \frac{\hbar^2}{2} m \left(-2\,\nabla^2_{\mathbf{x}}f_{r_c}({\mathbf{x}})|_{{\mathbf{x}}=0} \right)\int {\mathrm{d}}{\mathbf{z}}\,a^\dagger({\mathbf{z}}) a({\mathbf{z}}).$$ For the CSL model, we have: $$\begin{aligned} f^{\text{CSL}}_{r_c}({\mathbf{x}}) &=\frac{\gamma}{2 m^2} \, g_{r_c} * g_{r_c}({\mathbf{x}})\\ &=\frac{\gamma}{2 m^2 (\sqrt{4\pi r_c^2})^3}{\mathrm{e}}^{-{\mathbf{x}}^2/(4 r_c^2)}\end{aligned}$$ Hence: $$\begin{aligned} -2\,\nabla^2_{\mathbf{x}}f^\text{CSL}_{r_c}({\mathbf{x}})|_{{\mathbf{x}}=0} = \frac{\gamma}{m^2 (\sqrt{4\pi r_c^2})^3} \frac{3}{2r_c^2} = \frac{3 \lambda}{2 m^2r_c^2 }.\end{aligned}$$ Finally, for the CSL model, this gives: $$P_\text{heat}^{\rm CSL}=\frac{3\lambda \hbar^2}{4 r_c^2 m} N,$$ which depends on the quantum state only through the total number of particles. For the DP model, the regularized kernel $f_{r_c}^\text{DP}$ can easily be evaluated in Fourier space: $$\begin{aligned} f_{r_c}^\text{DP}({\mathbf{x}})&=g_{r_c}*f^\text{DP} * g_{r_c} ({\mathbf{x}})\\ &= 4 \pi \frac{G}{4\hbar} \int \frac{{\mathrm{d}}{\mathbf{k}}}{(2\pi)^3} \frac{{\mathrm{e}}^{-{\mathbf{k}}^2 r_c^2}}{{\mathbf{k}}^2} {\mathrm{e}}^{i{\mathbf{k}}\cdot {\mathbf{x}}}\end{aligned}$$ hence: $$\begin{aligned} - \nabla^2_{\mathbf{x}}f_{r_c}^\text{DP}({\mathbf{x}})|_{{\mathbf{x}}=0} &=4 \pi \frac{G}{4\hbar} \int \frac{{\mathrm{d}}{\mathbf{k}}}{(2\pi)^3} {\mathrm{e}}^{-{\mathbf{k}}^2 r_c^2}\\ &= \frac{G}{8 \sqrt{\pi} r_c^3},\end{aligned}$$ and $$P_\text{heat}^\text{DP} = \frac{G\hbar m}{8 \sqrt{\pi} r_c^3} N.$$ [^1]: Note that the factor $1/4$ in \[eq:DPchoice\] has also been fixed to $1/8$ in the literature.